awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m : Associates M
n : Associates N
hn : n ≠ 0
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ (i : Fin (s + 1)), r = c₁ i
c₂ : Fin (s + 1) → Associates N :=
  fun (t : Fin (s + 1)) ↦
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ t) ⋯)) :
      Associates N)
c₂_def :
  ∀ (t : Fin (s + 1)),
    c₂ t =
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ t) ⋯)) :
        Associates N)
r : Associates N
hr :
  r ≤
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ 1 ^ s) hs')) :
      Associates N)
this_1 : r ≤ n
this :
  LE.le.{u_1} (α := (↑(Set.Iic m) : Type u_1))
    ((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
        (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) r this_1) :
      (↑(Set.Iic m) : Type u_1))
    (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ 1 ^ s) hs' : { x : Associates M // x ∈ Set.Iic m })
i : Fin (s + 1)
hi :
  (↑((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
          (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) r this_1)) :
      Associates M) =
    c₁ i
⊢ ∃ (i : Fin (s + 1)), r = c₂ i
have h : r = c₂ i
Tactic: use i
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m : Associates M
n : Associates N
hn : n ≠ 0
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ (i : Fin (s + 1)), r = c₁ i
c₂ : Fin (s + 1) → Associates N :=
  fun (t : Fin (s + 1)) ↦
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ t) ⋯)) :
      Associates N)
c₂_def :
  ∀ (t : Fin (s + 1)),
    c₂ t =
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ t) ⋯)) :
        Associates N)
r : Associates N
hr :
  r ≤
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ 1 ^ s) hs')) :
      Associates N)
this : r ≤ n
⊢ ∃ (i : Fin (s + 1)), r = c₂ i
have neg : LE.le.{u_1} (α := (↑(Set.Iic m) : Type u_1))
  ((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
      (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) r this) :
    (↑(Set.Iic m) : Type u_1))
  (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ 1 ^ s) hs' : { x : Associates M // x ∈ Set.Iic m })
Tactic: suffices d.symm ⟨r, this⟩ ≤ ⟨c₁ 1 ^ s, hs'⟩ by
      obtain ⟨i, hi⟩ := hc₁''.1 this
      use i
      simp only [c₂_def, ← hi, d.apply_symm_apply, Subtype.coe_eta, Subtype.coe_mk]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m : Associates M
n : Associates N
hn : n ≠ 0
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ (i : Fin (s + 1)), r = c₁ i
c₂ : Fin (s + 1) → Associates N :=
  fun (t : Fin (s + 1)) ↦
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ t) ⋯)) :
      Associates N)
c₂_def :
  ∀ (t : Fin (s + 1)),
    c₂ t =
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ t) ⋯)) :
        Associates N)
r : Associates N
hr :
  r ≤
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ 1 ^ s) hs')) :
      Associates N)
this : r ≤ n
| Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ 1 ^ s) hs'
have rw : ((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
    ((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
      (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ 1 ^ s) hs')) :
  (↑(Set.Iic m) : Type u_1))
Tactic: rw [← d.symm_apply_apply ⟨c₁ 1 ^ s, hs'⟩]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m : Associates M
n : Associates N
hn : n ≠ 0
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ (i : Fin (s + 1)), r = c₁ i
c₂ : Fin (s + 1) → Associates N :=
  fun (t : Fin (s + 1)) ↦
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ t) ⋯)) :
      Associates N)
c₂_def :
  ∀ (t : Fin (s + 1)),
    c₂ t =
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ t) ⋯)) :
        Associates N)
r : Associates N
hr :
  r ≤
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ 1 ^ s) hs')) :
      Associates N)
this : r ≤ n
⊢ LE.le.{u_1} (α := (↑(Set.Iic m) : Type u_1))
    ((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
        (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) r this) :
      (↑(Set.Iic m) : Type u_1))
    (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ 1 ^ s) hs' : { x : Associates M // x ∈ Set.Iic m })
have neg : LE.le.{u_1} (α := (↑(Set.Iic m) : Type u_1))
  ((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
      (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) r this) :
    (↑(Set.Iic m) : Type u_1))
  ((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
      ((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
        (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ 1 ^ s) hs')) :
    (↑(Set.Iic m) : Type u_1))
Tactic: conv_rhs => rw [← d.symm_apply_apply ⟨c₁ 1 ^ s, hs'⟩]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m : Associates M
n : Associates N
hn : n ≠ 0
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ (i : Fin (s + 1)), r = c₁ i
c₂ : Fin (s + 1) → Associates N :=
  fun (t : Fin (s + 1)) ↦
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ t) ⋯)) :
      Associates N)
c₂_def :
  ∀ (t : Fin (s + 1)),
    c₂ t =
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ t) ⋯)) :
        Associates N)
r : Associates N
hr :
  r ≤
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ 1 ^ s) hs')) :
      Associates N)
this : r ≤ n
⊢ LE.le.{u_1} (α := (↑(Set.Iic m) : Type u_1))
    ((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
        (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) r this) :
      (↑(Set.Iic m) : Type u_1))
    ((d.symm : (a : (↑(Set.Iic n) : Type u_2)) → (↑(Set.Iic m) : Type u_1))
        ((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ 1 ^ s) hs')) :
      (↑(Set.Iic m) : Type u_1))
have neg : LE.le (α := (↑(Set.Iic n) : Type u_2)) (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) r this)
  ((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
    (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ 1 ^ s) hs'))
Tactic: rw [d.symm.le_iff_le]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m : Associates M
n : Associates N
hn : n ≠ 0
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ (i : Fin (s + 1)), r = c₁ i
c₂ : Fin (s + 1) → Associates N :=
  fun (t : Fin (s + 1)) ↦
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ t) ⋯)) :
      Associates N)
c₂_def :
  ∀ (t : Fin (s + 1)),
    c₂ t =
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ t) ⋯)) :
        Associates N)
r : Associates N
i : Fin (s + 1)
hr : r = c₂ i
⊢ r ≤
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ 1 ^ s) hs')) :
      Associates N)
have neg : LE.le (α := (↑(Set.Iic m) : Type u_1)) (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ i) sorry)
  (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) (c₁ 1 ^ s) hs')
Tactic: rw [hr, c₂_def, Subtype.coe_le_coe, d.le_iff_le]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
⊢ Prime.{u_2} (M := Associates N)
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
      Associates N)
have rw : Irreducible.{u_2} (M := Associates N)
  (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
        (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p sorry)) :
    Associates N)
Tactic: rw [← irreducible_iff_prime]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
⊢ Irreducible.{u_2} (M := Associates N)
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
      Associates N)
have refine_1 : Ne.{u_2 + 1} (α := Associates N)
  (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
        (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p sorry)) :
    Associates N)
  ⊥
Tactic: refine (Associates.isAtom_iff <|
    ne_zero_of_dvd_ne_zero hn (d ⟨p, _⟩).prop).mp ⟨?_, fun b hb => ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
refine_1 :
  Ne.{u_2 + 1} (α := Associates N)
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
      Associates N)
    ⊥
⊢ Irreducible.{u_2} (M := Associates N)
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
      Associates N)
have refine_2 : ∀ (b : Associates N)
  (hb :
    b <
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p sorry)) :
        Associates N)),
  b = ⊥
Tactic: refine (Associates.isAtom_iff <|
    ne_zero_of_dvd_ne_zero hn (d ⟨p, _⟩).prop).mp ⟨?_, fun b hb => ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
⊢ Ne.{u_2 + 1} (α := Associates N)
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
      Associates N)
    ⊥
have refine_1 : ¬p = 1
Tactic: rw [Ne, ← Associates.isUnit_iff_eq_bot, Associates.isUnit_iff_eq_one,
      coe_factor_orderIso_map_eq_one_iff _ d]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (hypothesis)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
b : Associates N
hb :
  b <
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
      Associates N)
⊢ b = ⊥
have this : b ≤ n
Tactic: have : b ≤ n := le_trans (le_of_lt hb) (d ⟨p, dvd_of_mem_normalizedFactors hp⟩).prop
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
b : Associates N
hb :
  b <
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
      Associates N)
this : b ≤ n
⊢ b = ⊥
have refine_2 : ∀ (x : (↑(Set.Iic m) : Type u_1))
  (hx :
    Eq.{u_2 + 1} (α := (↑(Set.Iic n) : Type u_2))
      ((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2)) x : (↑(Set.Iic n) : Type u_2))
      (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) b this : { x : Associates N // x ∈ Set.Iic n })),
  b = ⊥
Tactic: obtain ⟨x, hx⟩ := d.surjective ⟨b, this⟩
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
b : Associates N
hb :
  b <
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
      Associates N)
this : b ≤ n
x : (↑(Set.Iic m) : Type u_1)
hx :
  Eq.{u_2 + 1} (α := (↑(Set.Iic n) : Type u_2))
    ((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2)) x : (↑(Set.Iic n) : Type u_2))
    (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) b this : { x : Associates N // x ∈ Set.Iic n })
⊢ b = ⊥
have refine_2 : ∀ (x : (↑(Set.Iic m) : Type u_1))
  (hb :
    LT.lt.{u_2} (α := Associates N)
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2)) x) : Associates N)
      (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p sorry)) :
        Associates N))
  (hx :
    Eq.{u_2 + 1} (α := (↑(Set.Iic n) : Type u_2))
      ((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2)) x : (↑(Set.Iic n) : Type u_2))
      (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) b this : { x : Associates N // x ∈ Set.Iic n })),
  b = ⊥
Tactic: rw [← Subtype.coe_mk (p := (· ≤ n)) b this, ← hx] at hb
Cleared hypotheses: hb, x, hx
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (hypothesis)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
b : Associates N
this_1 : b ≤ n
x : (↑(Set.Iic m) : Type u_1)
hb :
  LT.lt.{u_2} (α := Associates N)
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2)) x) : Associates N)
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
      Associates N)
hx :
  Eq.{u_2 + 1} (α := (↑(Set.Iic n) : Type u_2))
    ((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2)) x : (↑(Set.Iic n) : Type u_2))
    (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) b this_1 : { x : Associates N // x ∈ Set.Iic n })
this_2 : OrderBot { l : Associates M // l ≤ m } := Subtype.orderBot (p := fun (l : Associates M) ↦ l ≤ m) ⋯
this_3 : OrderBot { l : Associates N // l ≤ n } := Subtype.orderBot (p := fun (l : Associates N) ↦ l ≤ n) ⋯
this : x = ⊥
⊢ b = ⊥
have hx₁ : Eq (α := (↑(Set.Iic n) : Type u_2)) ⊥ (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) b this_1)
Tactic: rw [this, OrderIso.map_bot d] at hx
Cleared hypotheses: hx, this_2, this_3, this
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
b : Associates N
this_1 : b ≤ n
x : (↑(Set.Iic m) : Type u_1)
hb :
  LT.lt.{u_2} (α := Associates N)
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2)) x) : Associates N)
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
      Associates N)
hx : Eq (α := (↑(Set.Iic n) : Type u_2)) ⊥ (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) b this_1)
this_2 : OrderBot { l : Associates M // l ≤ m } := Subtype.orderBot (p := fun (l : Associates M) ↦ l ≤ m) ⋯
this_3 : OrderBot { l : Associates N // l ≤ n } := Subtype.orderBot (p := fun (l : Associates N) ↦ l ≤ n) ⋯
this : x = ⊥
⊢ b = ⊥
have refine : ⊥ ∈ Set.Iic n
Tactic: refine (Subtype.mk_eq_bot_iff ?_ _).mp hx.symm
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : IsCancelMulZero M
N : Type u_2
inst_2 : CommMonoidWithZero N
inst_3 : UniqueFactorizationMonoid N
inst_4 : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : OrderIso.{u_1, u_2} (↑(Set.Iic m) : Type u_1) (↑(Set.Iic n) : Type u_2)
b : Associates N
this_1 : b ≤ n
x : (↑(Set.Iic m) : Type u_1)
hb :
  LT.lt.{u_2} (α := Associates N)
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2)) x) : Associates N)
    (↑((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2))
          (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) p ⋯)) :
      Associates N)
hx :
  Eq.{u_2 + 1} (α := (↑(Set.Iic n) : Type u_2))
    ((d : (a : (↑(Set.Iic m) : Type u_1)) → (↑(Set.Iic n) : Type u_2)) x : (↑(Set.Iic n) : Type u_2))
    (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) b this_1 : { x : Associates N // x ∈ Set.Iic n })
this_2 : OrderBot { l : Associates M // l ≤ m } := Subtype.orderBot (p := fun (l : Associates M) ↦ l ≤ m) ⋯
this : OrderBot { l : Associates N // l ≤ n } := Subtype.orderBot (p := fun (l : Associates N) ↦ l ≤ n) ⋯
⊢ b = ⊥
have refine_2 : x = ⊥
Tactic: suffices x = ⊥ by
      rw [this, OrderIso.map_bot d] at hx
      refine (Subtype.mk_eq_bot_iff ?_ _).mp hx.symm
      simp