awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
ι : Type u_2
inst_3 : LinearOrder ι
s_1 : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih_1 :
  StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑s : Set ι) →
    Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hmono : StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons i s hi) : Set ι)
ih : Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hminmem : s.min' hs ∈ Finset.cons i s hi
hne : Ne (α := MulArchimedeanClass M) (mk (a i)) (mk (∏ i ∈ s, a i))
hlt : LT.lt (α := MulArchimedeanClass M) (mk (a i)) (mk (∏ i ∈ s, a i))
y : ι
hy : y ∈ Finset.cons i s hi
⊢ i ≤ y
have inl : ∀ (hi : y ∉ s) (hmono : StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons y s hi) : Set ι))
  (hminmem : s.min' hs ∈ Finset.cons y s hi) (hne : Ne (α := MulArchimedeanClass M) (mk (a y)) (mk (∏ i ∈ s, a i)))
  (hlt : LT.lt (α := MulArchimedeanClass M) (mk (a y)) (mk (∏ i ∈ s, a i))) (hy : y ∈ Finset.cons y s hi), y ≤ y
Tactic: obtain rfl | hmem := Finset.mem_cons.mp hy
Cleared hypotheses: i, hi, hmono, hminmem, hne, hlt, hy
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
ι : Type u_2
inst_3 : LinearOrder ι
s_1 : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih_1 :
  StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑s : Set ι) →
    Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hmono : StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons i s hi) : Set ι)
ih : Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hminmem : s.min' hs ∈ Finset.cons i s hi
hne : Ne (α := MulArchimedeanClass M) (mk (a i)) (mk (∏ i ∈ s, a i))
hlt : LT.lt (α := MulArchimedeanClass M) (mk (a i)) (mk (∏ i ∈ s, a i))
y : ι
hy : y ∈ Finset.cons i s hi
inl :
  ∀ (hi : y ∉ s) (hmono : StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons y s hi) : Set ι))
    (hminmem : s.min' hs ∈ Finset.cons y s hi) (hne : Ne (α := MulArchimedeanClass M) (mk (a y)) (mk (∏ i ∈ s, a i)))
    (hlt : LT.lt (α := MulArchimedeanClass M) (mk (a y)) (mk (∏ i ∈ s, a i))) (hy : y ∈ Finset.cons y s hi), y ≤ y
⊢ i ≤ y
have inr : ∀ (hmem : y ∈ s), i ≤ y
Tactic: obtain rfl | hmem := Finset.mem_cons.mp hy
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
ι : Type u_2
inst_3 : LinearOrder ι
s_1 : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih_1 :
  StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑s : Set ι) →
    Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hmono : StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons i s hi) : Set ι)
ih : Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hminmem : s.min' hs ∈ Finset.cons i s hi
hne : Ne (α := MulArchimedeanClass M) (mk (a i)) (mk (∏ i ∈ s, a i))
hlt : LT.lt (α := MulArchimedeanClass M) (mk (a i)) (mk (∏ i ∈ s, a i))
y : ι
hy : y ∈ Finset.cons i s hi
hmem : y ∈ s
⊢ i ≤ y
have inr : i < s.min' sorry
Tactic: refine (lt_of_lt_of_le ?_ (Finset.min'_le _ _ hmem)).le
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
ι : Type u_2
inst_3 : LinearOrder ι
s_1 : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih_1 :
  StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑s : Set ι) →
    Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hmono : StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons i s hi) : Set ι)
ih : Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hminmem : s.min' hs ∈ Finset.cons i s hi
hne : Ne (α := MulArchimedeanClass M) (mk (a i)) (mk (∏ i ∈ s, a i))
hlt : LT.lt (α := MulArchimedeanClass M) (mk (a i)) (mk (∏ i ∈ s, a i))
y : ι
hy : y ∈ Finset.cons i s hi
hmem : y ∈ s
⊢ i < s.min' ⋯
have inr : LT.lt (α := MulArchimedeanClass M) ((mk ∘ a) i) ((mk ∘ a) (s.min' hs))
Tactic: apply (hmono.lt_iff_lt (by simp) hminmem).mp
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
ι : Type u_2
inst_3 : LinearOrder ι
s_1 : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih_1 :
  StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑s : Set ι) →
    Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hmono : StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons i s hi) : Set ι)
ih : Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hminmem : s.min' hs ∈ Finset.cons i s hi
hne : Ne (α := MulArchimedeanClass M) (mk (a i)) (mk (∏ i ∈ s, a i))
hlt : LT.lt (α := MulArchimedeanClass M) (mk (a i)) (mk (∏ i ∈ s, a i))
y : ι
hy : y ∈ Finset.cons i s hi
hmem : y ∈ s
⊢ LT.lt (α := MulArchimedeanClass M) ((mk ∘ a) i) ((mk ∘ a) (s.min' hs))
have inr : ∀ (hlt : LT.lt (α := MulArchimedeanClass M) (mk (a i)) (mk (a (s.min' hs)))) (y : ι) (hy : y ∈ Finset.cons i s hi)
  (hmem : y ∈ s), LT.lt (α := MulArchimedeanClass M) ((mk ∘ a) i) ((mk ∘ a) (s.min' hs))
Tactic: rw [ih] at hlt
Cleared hypotheses: hlt, y, hy, hmem
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
ι : Type u_2
inst_3 : LinearOrder ι
s_1 : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih_1 :
  StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑s : Set ι) →
    Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hmono : StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons i s hi) : Set ι)
ih : Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hminmem : s.min' hs ∈ Finset.cons i s hi
hne : Ne (α := MulArchimedeanClass M) (mk (a i)) (mk (∏ i ∈ s, a i))
hlt : LT.lt (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a i))
⊢ Eq (α := MulArchimedeanClass M) (mk (a i * ∏ x ∈ s, a x)) (mk (a ((Finset.cons i s hi).min' ⋯)))
have cons : Eq (α := MulArchimedeanClass M) (mk (a (s.min' hs))) (mk (a ((Finset.cons i s hi).min' sorry)))
Tactic: rw [mul_comm, mk_mul_eq_mk_left hlt, ih]
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
ι : Type u_2
inst_3 : LinearOrder ι
s_1 : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih_1 :
  StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑s : Set ι) →
    Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hmono : StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons i s hi) : Set ι)
ih : Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hminmem : s.min' hs ∈ Finset.cons i s hi
hne : Ne (α := MulArchimedeanClass M) (mk (a i)) (mk (∏ i ∈ s, a i))
hlt : LT.lt (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a i))
⊢ Eq (α := MulArchimedeanClass M) (mk (a (s.min' hs))) (mk (a ((Finset.cons i s hi).min' ⋯)))
have cons : s.min' hs = (Finset.cons i s hi).min' sorry
Tactic: congr 2
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
ι : Type u_2
inst_3 : LinearOrder ι
s_1 : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih_1 :
  StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑s : Set ι) →
    Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hmono : StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons i s hi) : Set ι)
ih : Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hminmem : s.min' hs ∈ Finset.cons i s hi
hne : Ne (α := MulArchimedeanClass M) (mk (a i)) (mk (∏ i ∈ s, a i))
hlt : LT.lt (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a i))
⊢ s.min' hs = (Finset.cons i s hi).min' ⋯
have cons : ∀ y ∈ Finset.cons i s hi, s.min' hs ≤ y
Tactic: refine le_antisymm (Finset.le_min' _ _ _ ?_) (Finset.min'_le _ _ hminmem)
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
ι : Type u_2
inst_3 : LinearOrder ι
s_1 : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih_1 :
  StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑s : Set ι) →
    Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hmono : StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons i s hi) : Set ι)
ih : Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hminmem : s.min' hs ∈ Finset.cons i s hi
hne : Ne (α := MulArchimedeanClass M) (mk (a i)) (mk (∏ i ∈ s, a i))
hlt : LT.lt (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a i))
y : ι
hy : y ∈ Finset.cons i s hi
⊢ s.min' hs ≤ y
have cons : ∀ (hi : y ∉ s) (hmono : StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons y s hi) : Set ι))
  (hminmem : s.min' hs ∈ Finset.cons y s hi) (hne : Ne (α := MulArchimedeanClass M) (mk (a y)) (mk (∏ i ∈ s, a i)))
  (hlt : LT.lt (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a y))) (hy : y ∈ Finset.cons y s hi), s.min' hs ≤ y
Tactic: obtain rfl | hmem := Finset.mem_cons.mp hy
Cleared hypotheses: i, hi, hmono, hminmem, hne, hlt, hy
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
ι : Type u_2
inst_3 : LinearOrder ι
s_1 : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih_1 :
  StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑s : Set ι) →
    Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hmono : StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons i s hi) : Set ι)
ih : Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
hminmem : s.min' hs ∈ Finset.cons i s hi
hne : Ne (α := MulArchimedeanClass M) (mk (a i)) (mk (∏ i ∈ s, a i))
hlt : LT.lt (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a i))
y : ι
hy : y ∈ Finset.cons i s hi
cons :
  ∀ (hi : y ∉ s) (hmono : StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons y s hi) : Set ι))
    (hminmem : s.min' hs ∈ Finset.cons y s hi) (hne : Ne (α := MulArchimedeanClass M) (mk (a y)) (mk (∏ i ∈ s, a i)))
    (hlt : LT.lt (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a y))) (hy : y ∈ Finset.cons y s hi),
    s.min' hs ≤ y
⊢ s.min' hs ≤ y
have cons₁ : ∀ (hmem : y ∈ s), s.min' hs ≤ y
Tactic: obtain rfl | hmem := Finset.mem_cons.mp hy
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
ι : Type u_2
inst_3 : LinearOrder ι
s_1 : Finset ι
a : ι → M
s : Finset ι
hs : s.Nonempty
ih_1 :
  StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑s : Set ι) →
    Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
ih : Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
y : ι
hi : y ∉ s
hmono : StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons y s hi) : Set ι)
hminmem : s.min' hs ∈ Finset.cons y s hi
hne : Ne (α := MulArchimedeanClass M) (mk (a y)) (mk (∏ i ∈ s, a i))
hlt : LT.lt (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a y))
hy : y ∈ Finset.cons y s hi
⊢ s.min' hs ≤ y
have apply : LT.lt (α := MulArchimedeanClass M) ((mk ∘ a) (s.min' hs)) ((mk ∘ a) y)
Tactic: apply ((hmono.lt_iff_lt hminmem (by simp)).mp ?_).le
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (hypothesis)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
ι : Type u_2
inst_3 : LinearOrder ι
s_1 : Finset ι
a : ι → M
s : Finset ι
hs : s.Nonempty
ih_1 :
  StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑s : Set ι) →
    Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
ih : Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))
y : ι
hi : y ∉ s
hmono : StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons y s hi) : Set ι)
hminmem : s.min' hs ∈ Finset.cons y s hi
hne : Ne (α := MulArchimedeanClass M) (mk (a y)) (mk (∏ i ∈ s, a i))
hlt : LT.lt (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a y))
hy : y ∈ Finset.cons y s hi
⊢ LT.lt (α := MulArchimedeanClass M) ((mk ∘ a) (s.min' hs)) ((mk ∘ a) y)
have hlt₁ : LT.lt (α := MulArchimedeanClass M) (mk (a (s.min' hs))) (mk (a y))
Tactic: rw [ih] at hlt
Cleared hypotheses: hlt, hy
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
ι : Type u_2
inst_3 : LinearOrder ι
s : Finset ι
hnonempty : s.Nonempty
a : ι → M
⊢ StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑s : Set ι) →
    Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hnonempty)))
have singleton : ∀ (i : ι),
  StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑{i} : Set ι) →
    Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ {i}, a i)) (mk (a (Finset.min' {i} sorry)))
Tactic: induction hnonempty using Finset.Nonempty.cons_induction with
  | singleton i => simp
  | cons i s hi hs ih =>
    intro hmono
    obtain ih := ih (hmono.mono (by simp))
    rw [Finset.prod_cons]
    have hminmem : s.min' hs ∈ (Finset.cons i s hi) :=
      Finset.mem_cons_of_mem (Finset.min'_mem _ _)
    have hne : mk (a i) ≠ mk (a (s.min' hs)) := by
      by_contra h
      obtain eq := hmono.injOn (by simp) hminmem h
      rw [eq] at hi
      exact hi (Finset.min'_mem _ hs)
    rw [← ih] at hne
    obtain hlt | hlt := lt_or_gt_of_ne hne
    · rw [mk_mul_eq_mk_left hlt]
      congr
      apply le_antisymm (Finset.le_min' _ _ _ ?_) (Finset.min'_le _ _ (by simp))
      intro y hy
      obtain rfl | hmem := Finset.mem_cons.mp hy
      · rfl
      · refine (lt_of_lt_of_le ?_ (Finset.min'_le _ _ hmem)).le
        apply (hmono.lt_iff_lt (by simp) hminmem).mp
        rw [ih] at hlt
        exact hlt
    · rw [mul_comm, mk_mul_eq_mk_left hlt, ih]
      congr 2
      refine le_antisymm (Finset.le_min' _ _ _ ?_) (Finset.min'_le _ _ hminmem)
      intro y hy
      obtain rfl | hmem := Finset.mem_cons.mp hy
      · apply ((hmono.lt_iff_lt hminmem (by simp)).mp ?_).le
        rw [ih] at hlt
        exact hlt
      · exact Finset.min'_le _ _ hmem
Cleared hypotheses: hnonempty
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
ι : Type u_2
inst_3 : LinearOrder ι
s : Finset ι
hnonempty : s.Nonempty
a : ι → M
singleton :
  ∀ (i : ι),
    StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑{i} : Set ι) →
      Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ {i}, a i)) (mk (a (Finset.min' {i} ⋯)))
⊢ StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑s : Set ι) →
    Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hnonempty)))
have cons : ∀ (i : ι) (s : Finset ι) (hi : i ∉ s) (hs : s.Nonempty)
  (ih :
    StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑s : Set ι) →
      Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ s, a i)) (mk (a (s.min' hs)))),
  StrictMonoOn (β := MulArchimedeanClass M) (mk ∘ a) (↑(Finset.cons i s hi) : Set ι) →
    Eq (α := MulArchimedeanClass M) (mk (∏ i ∈ Finset.cons i s hi, a i)) (mk (a ((Finset.cons i s hi).min' sorry)))
Tactic: induction hnonempty using Finset.Nonempty.cons_induction with
  | singleton i => simp
  | cons i s hi hs ih =>
    intro hmono
    obtain ih := ih (hmono.mono (by simp))
    rw [Finset.prod_cons]
    have hminmem : s.min' hs ∈ (Finset.cons i s hi) :=
      Finset.mem_cons_of_mem (Finset.min'_mem _ _)
    have hne : mk (a i) ≠ mk (a (s.min' hs)) := by
      by_contra h
      obtain eq := hmono.injOn (by simp) hminmem h
      rw [eq] at hi
      exact hi (Finset.min'_mem _ hs)
    rw [← ih] at hne
    obtain hlt | hlt := lt_or_gt_of_ne hne
    · rw [mk_mul_eq_mk_left hlt]
      congr
      apply le_antisymm (Finset.le_min' _ _ _ ?_) (Finset.min'_le _ _ (by simp))
      intro y hy
      obtain rfl | hmem := Finset.mem_cons.mp hy
      · rfl
      · refine (lt_of_lt_of_le ?_ (Finset.min'_le _ _ hmem)).le
        apply (hmono.lt_iff_lt (by simp) hminmem).mp
        rw [ih] at hlt
        exact hlt
    · rw [mul_comm, mk_mul_eq_mk_left hlt, ih]
      congr 2
      refine le_antisymm (Finset.le_min' _ _ _ ?_) (Finset.min'_le _ _ hminmem)
      intro y hy
      obtain rfl | hmem := Finset.mem_cons.mp hy
      · apply ((hmono.lt_iff_lt hminmem (by simp)).mp ?_).le
        rw [ih] at hlt
        exact hlt
      · exact Finset.min'_le _ _ hmem
Cleared hypotheses: hnonempty
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (hypothesis)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
a b : M
h : LT.lt (α := MulArchimedeanClass M) (mk a) (mk b)
hpos : 1 ≤ a
⊢ b < a
have h₁ : HPow.hPow (β := ℕ) |b|ₘ 1 < |a|ₘ
Tactic: obtain h := mk_lt_mk.mp h 1
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (hypothesis)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
a b : M
h_1 : LT.lt (α := MulArchimedeanClass M) (mk a) (mk b)
hpos : 1 ≤ a
h : HPow.hPow (β := ℕ) |b|ₘ 1 < |a|ₘ
⊢ b < a
have h₁ : a⁻¹ < b ∧ b < a
Tactic: rw [pow_one, mabs_lt, mabs_eq_self.mpr hpos] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (hypothesis)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
a b : M
h : LT.lt (α := MulArchimedeanClass M) (mk a) (mk b)
hneg : a ≤ 1
⊢ a < b
have h₁ : HPow.hPow (β := ℕ) |b|ₘ 1 < |a|ₘ
Tactic: obtain h := mk_lt_mk.mp h 1
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (hypothesis)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
a b : M
h_1 : LT.lt (α := MulArchimedeanClass M) (mk a) (mk b)
hneg : a ≤ 1
h : HPow.hPow (β := ℕ) |b|ₘ 1 < |a|ₘ
⊢ a < b
have h₁ : a < b ∧ b < a⁻¹
Tactic: rw [pow_one, mabs_lt, mabs_eq_inv_self.mpr hneg, inv_inv] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
a b : M
ha : 1 < a
hab : LT.lt (α := MulArchimedeanClass M) (mk a) (mk (b / a))
⊢ 1 < b
have suffices : a⁻¹ < b / a
Tactic: suffices a⁻¹ < b / a by
    simpa using this
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
a b : M
ha : 1 < a
hab : LT.lt (α := MulArchimedeanClass M) (mk a) (mk (b / a))
⊢ a⁻¹ < b / a
have h : LT.lt (α := MulArchimedeanClass M) (mk a⁻¹) (mk (b / a))
Tactic: apply lt_of_mk_lt_mk_of_le_one
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
a b : M
ha : 1 < a
hab : LT.lt (α := MulArchimedeanClass M) (mk a) (mk (b / a))
h : LT.lt (α := MulArchimedeanClass M) (mk a⁻¹) (mk (b / a))
⊢ a⁻¹ < b / a
have hneg : a⁻¹ ≤ 1
Tactic: apply lt_of_mk_lt_mk_of_le_one
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
h : ∀ (a : M), a ≠ 1 → ∀ (b : M), b ≠ 1 → Eq (α := MulArchimedeanClass M) (mk a) (mk b)
x y : M
hy : 1 < y
⊢ ∃ (n : ℕ), x ≤ y ^ n
have pos : ∀ (hx : x ≤ 1), ∃ (n : ℕ), x ≤ y ^ n
Tactic: by_cases! hx : x ≤ 1
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
h : ∀ (a : M), a ≠ 1 → ∀ (b : M), b ≠ 1 → Eq (α := MulArchimedeanClass M) (mk a) (mk b)
x y : M
hy : 1 < y
pos : ∀ (hx : x ≤ 1), ∃ (n : ℕ), x ≤ y ^ n
⊢ ∃ (n : ℕ), x ≤ y ^ n
have neg : ∀ (hx : 1 < x), ∃ (n : ℕ), x ≤ y ^ n
Tactic: by_cases! hx : x ≤ 1
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
h : ∀ (a : M), a ≠ 1 → ∀ (b : M), b ≠ 1 → Eq (α := MulArchimedeanClass M) (mk a) (mk b)
x y : M
hy : 1 < y
hx : x ≤ 1
⊢ ∃ (n : ℕ), x ≤ y ^ n
have h₁ : x ≤ HPow.hPow (β := ℕ) y 0
Tactic: use 0
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (hypothesis)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
h : ∀ (a : M), a ≠ 1 → ∀ (b : M), b ≠ 1 → Eq (α := MulArchimedeanClass M) (mk a) (mk b)
x y : M
hy : 1 < y
hx : 1 < x
⊢ ∃ (n : ℕ), x ≤ y ^ n
have hxy : Eq (α := MulArchimedeanClass M) (mk x) (mk y)
Tactic: have hxy : mk x = mk y := h x hx.ne.symm y hy.ne.symm
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
h : ∀ (a : M), a ≠ 1 → ∀ (b : M), b ≠ 1 → Eq (α := MulArchimedeanClass M) (mk a) (mk b)
x y : M
hy : 1 < y
hx : 1 < x
hxy : Eq (α := MulArchimedeanClass M) (mk x) (mk y)
⊢ ∃ (n : ℕ), x ≤ y ^ n
have neg : (∃ (m : ℕ), |y|ₘ ≤ |x|ₘ ^ m) → ∀ (m : ℕ) (hm : |x|ₘ ≤ |y|ₘ ^ m), ∃ (n : ℕ), x ≤ y ^ n
Tactic: obtain ⟨_, ⟨m, hm⟩⟩ := mk_eq_mk.mp hxy
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (hypothesis)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
h : ∀ (a : M), a ≠ 1 → ∀ (b : M), b ≠ 1 → Eq (α := MulArchimedeanClass M) (mk a) (mk b)
x y : M
hy : 1 < y
hx : 1 < x
hxy : Eq (α := MulArchimedeanClass M) (mk x) (mk y)
left : ∃ (m : ℕ), |y|ₘ ≤ |x|ₘ ^ m
m : ℕ
hm : |x|ₘ ≤ |y|ₘ ^ m
⊢ ∃ (n : ℕ), x ≤ y ^ n
have hm₁ : x ≤ y ^ m
Tactic: rw [mabs_eq_self.mpr hx.le, mabs_eq_self.mpr hy.le] at hm
Cleared hypotheses: hm
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (hypothesis)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
a b : M
inst_3 : MulArchimedean M
ha : a ≠ 1
hb : b ≠ 1
⊢ Eq (α := MulArchimedeanClass M) (mk a) (mk b)
have hm : ∃ (n : ℕ), |b|ₘ ≤ |a|ₘ ^ n
Tactic: obtain hm := MulArchimedean.arch |b|ₘ (show 1 < |a|ₘ by simpa using ha)
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (hypothesis)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
a b : M
inst_3 : MulArchimedean M
ha : a ≠ 1
hb : b ≠ 1
hm : ∃ (n : ℕ), |b|ₘ ≤ |a|ₘ ^ n
⊢ Eq (α := MulArchimedeanClass M) (mk a) (mk b)
have hn : ∃ (n : ℕ), |a|ₘ ≤ |b|ₘ ^ n
Tactic: obtain hn := MulArchimedean.arch |a|ₘ (show 1 < |b|ₘ by simpa using hb)
-- Lean 4 proof step from Mathlib.Algebra.Order.Archimedean.Class (goal)
Goal:
M : Type u_1
inst : CommGroup M
inst_1 : LinearOrder M
inst_2 : IsOrderedMonoid M
a b : M
N : Type u_2
inst_3 : CommGroup N
inst_4 : LinearOrder N
inst_5 : IsOrderedMonoid N
f : M →*o N
h : Eq (α := MulArchimedeanClass M) (mk a) (mk b)
⊢ Eq.{u_2 + 1} (α := MulArchimedeanClass N) (mk ((f : (a : M) → N) a) : MulArchimedeanClass N)
    (mk ((f : (a : M) → N) b) : MulArchimedeanClass N)
have rw : Eq.{u_2 + 1} (α := MulArchimedeanClass N)
  (((orderHom f : OrderHom.{u_1, u_2} (MulArchimedeanClass M) (MulArchimedeanClass N)) :
        (a : MulArchimedeanClass M) → MulArchimedeanClass N)
      (mk a) :
    MulArchimedeanClass N)
  (mk ((f : (a : M) → N) b) : MulArchimedeanClass N)
Tactic: rw [← orderHom_mk, ← orderHom_mk, h]