awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.ChainOfDivisors
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)

Tactic:
(Associates.isAtom_iff <|
    ne_zero_of_dvd_ne_zero hn (d ⟨p, _⟩).prop).mp ⟨?_, fun b hb => ?_⟩

Premises used: Associates.isAtom_iff, Subtype.mk, Subtype.prop, And.intro, Iff.mp, ne_zero_of_dvd_ne_zero, DFunLike.coe
-- Module: Mathlib.RingTheory.ChainOfDivisors
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)
    ⊥
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 ⋯)) :
          Associates N)),
    b = ⊥
⊢ 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)

Tactic:
refine (Associates.isAtom_iff <|
    ne_zero_of_dvd_ne_zero hn (d ⟨p, _⟩).prop).mp ⟨?_, fun b hb => ?_⟩

Premises used: Associates.isAtom_iff, Subtype.mk, Subtype.prop, And.intro, Iff.mp, ne_zero_of_dvd_ne_zero, DFunLike.coe
-- Module: Mathlib.RingTheory.ChainOfDivisors
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 : ¬p = 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)
    ⊥

Tactic:
rw [Ne, ← Associates.isUnit_iff_eq_bot, Associates.isUnit_iff_eq_one,
      coe_factor_orderIso_map_eq_one_iff _ d]

Premises used: coe_factor_orderIso_map_eq_one_iff, Ne, Associates.isUnit_iff_eq_bot, Associates.isUnit_iff_eq_one
-- Module: Mathlib.RingTheory.ChainOfDivisors
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)
hp : 1 ∈ normalizedFactors m
⊢ False

Tactic:
(prime_of_normalized_factor 1 hp).not_unit isUnit_one

Premises used: Prime.not_unit, UniqueFactorizationMonoid.prime_of_normalized_factor, isUnit_one, OfNat.ofNat
-- Module: Mathlib.RingTheory.ChainOfDivisors
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)
hp : 1 ∈ normalizedFactors m
⊢ False

Tactic:
exact (prime_of_normalized_factor 1 hp).not_unit isUnit_one

Premises used: Prime.not_unit, UniqueFactorizationMonoid.prime_of_normalized_factor, isUnit_one, OfNat.ofNat
-- Module: Mathlib.RingTheory.ChainOfDivisors
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 ≤ n

Tactic:
have : b ≤ n := le_trans (le_of_lt hb) (d ⟨p, dvd_of_mem_normalizedFactors hp⟩).prop
-- Module: Mathlib.RingTheory.ChainOfDivisors
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
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 = ⊥
⊢ b = ⊥

Tactic:
obtain ⟨x, hx⟩ := d.surjective ⟨b, this⟩

Premises used: OrderIso.surjective, Subtype.mk
-- Module: Mathlib.RingTheory.ChainOfDivisors
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 })
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 ⋯)) :
          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 = ⊥
⊢ b = ⊥

Tactic:
rw [← Subtype.coe_mk (p := (· ≤ n)) b this, ← hx] at hb

Premises used: Subtype.coe_mk
-- Module: Mathlib.RingTheory.ChainOfDivisors
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 = ⊥
⊢ 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

Premises used: OrderIso.map_bot
-- Module: Mathlib.RingTheory.ChainOfDivisors
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 = ⊥
hx₁ : Eq (α := (↑(Set.Iic n) : Type u_2)) ⊥ (Subtype.mk (p := fun (x : Associates N) ↦ x ∈ Set.Iic n) b this_1)
⊢ x = ⊥

Tactic:
rw [this, OrderIso.map_bot d] at hx

Premises used: OrderIso.map_bot
-- Module: Mathlib.RingTheory.ChainOfDivisors
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 = ⊥

Tactic:
(Subtype.mk_eq_bot_iff ?_ _).mp hx.symm

Premises used: Iff.mp, Eq.symm, Subtype.mk_eq_bot_iff
-- Module: Mathlib.RingTheory.ChainOfDivisors
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 = ⊥
refine : ⊥ ∈ Set.Iic n
⊢ b = ⊥

Tactic:
refine (Subtype.mk_eq_bot_iff ?_ _).mp hx.symm

Premises used: Iff.mp, Eq.symm, Subtype.mk_eq_bot_iff
-- Module: Mathlib.RingTheory.ChainOfDivisors
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 = ⊥
⊢ ⊥ ∈ Set.Iic n

Tactic:
simp
-- Module: Mathlib.RingTheory.ChainOfDivisors
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) ⋯
refine_2 : x = ⊥
⊢ b = ⊥

Tactic:
suffices x = ⊥ by
      rw [this, OrderIso.map_bot d] at hx
      refine (Subtype.mk_eq_bot_iff ?_ _).mp hx.symm
      simp
-- Module: Mathlib.RingTheory.ChainOfDivisors
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) ⋯
refine_2 :
  ∀ (a : Associates M) (ha : a ∈ Set.Iic m)
    (hb :
      LT.lt.{u_2} (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) a ha)) :
          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))
            (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) a ha) :
          (↑(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 })),
    Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) a ha = ⊥
⊢ x = ⊥

Tactic:
obtain ⟨a, ha⟩ := x
-- Module: Mathlib.RingTheory.ChainOfDivisors
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
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) ⋯
a : Associates M
ha : a ∈ Set.Iic m
hb :
  LT.lt.{u_2} (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) a ha)) :
      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))
        (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) a ha) :
      (↑(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 })
refine_2 : a = ⊥
refine_2₁ : ⊥ ∈ Set.Iic m
⊢ Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) a ha = ⊥

Tactic:
rw [Subtype.mk_eq_bot_iff]

Premises used: Subtype.mk_eq_bot_iff
-- Module: Mathlib.RingTheory.ChainOfDivisors
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
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) ⋯
a : Associates M
ha : a ∈ Set.Iic m
hb :
  LT.lt.{u_2} (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) a ha)) :
      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))
        (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) a ha) :
      (↑(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 })
⊢ a = ⊥

Tactic:
((Associates.isAtom_iff <| Prime.ne_zero <| prime_of_normalized_factor p hp).mpr <|
              irreducible_of_normalized_factor p hp).right
          a (Subtype.mk_lt_mk.mp <| d.lt_iff_lt.mp hb)

Premises used: Iff.mpr, Associates.isAtom_iff, OrderIso.lt_iff_lt, Subtype.mk_lt_mk, UniqueFactorizationMonoid.prime_of_normalized_factor, And.right, Prime.ne_zero, Iff.mp, UniqueFactorizationMonoid.irreducible_of_normalized_factor
-- Module: Mathlib.RingTheory.ChainOfDivisors
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
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) ⋯
a : Associates M
ha : a ∈ Set.Iic m
hb :
  LT.lt.{u_2} (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) a ha)) :
      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))
        (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) a ha) :
      (↑(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 })
⊢ a = ⊥

Tactic:
exact
        ((Associates.isAtom_iff <| Prime.ne_zero <| prime_of_normalized_factor p hp).mpr <|
              irreducible_of_normalized_factor p hp).right
          a (Subtype.mk_lt_mk.mp <| d.lt_iff_lt.mp hb)

Premises used: Iff.mpr, Associates.isAtom_iff, OrderIso.lt_iff_lt, Subtype.mk_lt_mk, UniqueFactorizationMonoid.prime_of_normalized_factor, And.right, Prime.ne_zero, Iff.mp, UniqueFactorizationMonoid.irreducible_of_normalized_factor
-- Module: Mathlib.RingTheory.ChainOfDivisors
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
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) ⋯
a : Associates M
ha : a ∈ Set.Iic m
hb :
  LT.lt.{u_2} (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) a ha)) :
      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))
        (Subtype.mk (p := fun (x : Associates M) ↦ x ∈ Set.Iic m) a ha) :
      (↑(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 })
⊢ ⊥ ∈ Set.Iic m

Tactic:
simp