l3lab/ntp-mathlib huggingface dataset:
/-- The ideal `I` equals the finprod `∏_v v^(val_v(I))`, when both sides are regarded as fractional
ideals of `R`. -/
theorem finprod_heightOneSpectrum_factorization_coe (I : Ideal R) (hI : I ≠ 0) :
    (∏ᶠ v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R⁰ K) ^
      ((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ)) = I := by
  

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^ ↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))) =
    ↑I

Next tactic:
conv_rhs => rw [← Ideal.finprod_heightOneSpectrum_factorization I hI]
/-- The ideal `I` equals the finprod `∏_v v^(val_v(I))`, when both sides are regarded as fractional
ideals of `R`. -/
theorem finprod_heightOneSpectrum_factorization_coe (I : Ideal R) (hI : I ≠ 0) :
    (∏ᶠ v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R⁰ K) ^
      ((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ)) = I := by
  conv_rhs => 

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
| ↑I

Next tactic:
rw [← Ideal.finprod_heightOneSpectrum_factorization I hI]
/-- The ideal `I` equals the finprod `∏_v v^(val_v(I))`, when both sides are regarded as fractional
ideals of `R`. -/
theorem finprod_heightOneSpectrum_factorization_coe (I : Ideal R) (hI : I ≠ 0) :
    (∏ᶠ v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R⁰ K) ^
      ((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ)) = I := by
  conv_rhs => rw [← Ideal.finprod_heightOneSpectrum_factorization I hI]
  

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^ ↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))) =
    ↑(∏ᶠ (v : HeightOneSpectrum R), maxPowDividing v I)

Next tactic:
rw [FractionalIdeal.coeIdeal_finprod R⁰ K (le_refl _)]
/-- The ideal `I` equals the finprod `∏_v v^(val_v(I))`, when both sides are regarded as fractional
ideals of `R`. -/
theorem finprod_heightOneSpectrum_factorization_coe (I : Ideal R) (hI : I ≠ 0) :
    (∏ᶠ v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R⁰ K) ^
      ((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ)) = I := by
  conv_rhs => rw [← Ideal.finprod_heightOneSpectrum_factorization I hI]
  rw [FractionalIdeal.coeIdeal_finprod R⁰ K (le_refl _)]
  

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^ ↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))) =
    ∏ᶠ (a : HeightOneSpectrum R), ↑(maxPowDividing a I)

Next tactic:
simp_rw [IsDedekindDomain.HeightOneSpectrum.maxPowDividing, FractionalIdeal.coeIdeal_pow,
    zpow_ofNat]
private def wf_lbp : WellFounded (lbp p) :=
  ⟨by
    

Proof state:
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
⊢ ∀ (a : ℕ), Acc (Nat.lbp p) a

Next tactic:
let ⟨n, pn⟩ := H
private def wf_lbp : WellFounded (lbp p) :=
  ⟨by
    let ⟨n, pn⟩ := H
    

Proof state:
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
n : ℕ
pn : true ∈ p n ∧ ∀ k < n, (p k).Dom
⊢ ∀ (a : ℕ), Acc (Nat.lbp p) a

Next tactic:
suffices ∀ m k, n ≤ k + m → Acc (lbp p) k by exact fun a => this _ _ (Nat.le_add_left _ _)
private def wf_lbp : WellFounded (lbp p) :=
  ⟨by
    let ⟨n, pn⟩ := H
    suffices ∀ m k, n ≤ k + m → Acc (lbp p) k by 

Proof state:
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
n : ℕ
pn : true ∈ p n ∧ ∀ k < n, (p k).Dom
this : ∀ (m k : ℕ), n ≤ k + m → Acc (Nat.lbp p) k
⊢ ∀ (a : ℕ), Acc (Nat.lbp p) a

Next tactic:
exact fun a => this _ _ (Nat.le_add_left _ _)
private def wf_lbp : WellFounded (lbp p) :=
  ⟨by
    let ⟨n, pn⟩ := H
    suffices ∀ m k, n ≤ k + m → Acc (lbp p) k by exact fun a => this _ _ (Nat.le_add_left _ _)
    

Proof state:
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
n : ℕ
pn : true ∈ p n ∧ ∀ k < n, (p k).Dom
⊢ ∀ (m k : ℕ), n ≤ k + m → Acc (Nat.lbp p) k

Next tactic:
intro m k kn
private def wf_lbp : WellFounded (lbp p) :=
  ⟨by
    let ⟨n, pn⟩ := H
    suffices ∀ m k, n ≤ k + m → Acc (lbp p) k by exact fun a => this _ _ (Nat.le_add_left _ _)
    intro m k kn
    

Proof state:
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
n : ℕ
pn : true ∈ p n ∧ ∀ k < n, (p k).Dom
m k : ℕ
kn : n ≤ k + m
⊢ Acc (Nat.lbp p) k

Next tactic:
induction' m with m IH generalizing k
private def wf_lbp : WellFounded (lbp p) :=
  ⟨by
    let ⟨n, pn⟩ := H
    suffices ∀ m k, n ≤ k + m → Acc (lbp p) k by exact fun a => this _ _ (Nat.le_add_left _ _)
    intro m k kn
    induction' m with m IH generalizing k <;> 

Proof state:
case zero
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
n : ℕ
pn : true ∈ p n ∧ ∀ k < n, (p k).Dom
k : ℕ
kn : n ≤ k + zero
⊢ Acc (Nat.lbp p) k

Next tactic:
refine' ⟨_, fun y r => _⟩
private def wf_lbp : WellFounded (lbp p) :=
  ⟨by
    let ⟨n, pn⟩ := H
    suffices ∀ m k, n ≤ k + m → Acc (lbp p) k by exact fun a => this _ _ (Nat.le_add_left _ _)
    intro m k kn
    induction' m with m IH generalizing k <;> 

Proof state:
case succ
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
n : ℕ
pn : true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : ∀ (k : ℕ), n ≤ k + m → Acc (Nat.lbp p) k
k : ℕ
kn : n ≤ k + succ m
⊢ Acc (Nat.lbp p) k

Next tactic:
refine' ⟨_, fun y r => _⟩
private def wf_lbp : WellFounded (lbp p) :=
  ⟨by
    let ⟨n, pn⟩ := H
    suffices ∀ m k, n ≤ k + m → Acc (lbp p) k by exact fun a => this _ _ (Nat.le_add_left _ _)
    intro m k kn
    induction' m with m IH generalizing k <;> refine' ⟨_, fun y r => _⟩ <;> 

Proof state:
case zero
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
n : ℕ
pn : true ∈ p n ∧ ∀ k < n, (p k).Dom
k : ℕ
kn : n ≤ k + zero
y : ℕ
r : Nat.lbp p y k
⊢ Acc (Nat.lbp p) y

Next tactic:
rcases r with ⟨rfl, a⟩
private def wf_lbp : WellFounded (lbp p) :=
  ⟨by
    let ⟨n, pn⟩ := H
    suffices ∀ m k, n ≤ k + m → Acc (lbp p) k by exact fun a => this _ _ (Nat.le_add_left _ _)
    intro m k kn
    induction' m with m IH generalizing k <;> refine' ⟨_, fun y r => _⟩ <;> 

Proof state:
case succ
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
n : ℕ
pn : true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : ∀ (k : ℕ), n ≤ k + m → Acc (Nat.lbp p) k
k : ℕ
kn : n ≤ k + succ m
y : ℕ
r : Nat.lbp p y k
⊢ Acc (Nat.lbp p) y

Next tactic:
rcases r with ⟨rfl, a⟩
private def wf_lbp : WellFounded (lbp p) :=
  ⟨by
    let ⟨n, pn⟩ := H
    suffices ∀ m k, n ≤ k + m → Acc (lbp p) k by exact fun a => this _ _ (Nat.le_add_left _ _)
    intro m k kn
    induction' m with m IH generalizing k <;> refine' ⟨_, fun y r => _⟩ <;> rcases r with ⟨rfl, a⟩
    · 

Proof state:
case zero.intro
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
n : ℕ
pn : true ∈ p n ∧ ∀ k < n, (p k).Dom
k : ℕ
kn : n ≤ k + zero
a : ∀ k_1 ≤ k, false ∈ p k_1
⊢ Acc (Nat.lbp p) (k + 1)

Next tactic:
injection mem_unique pn.1 (a _ kn)
private def wf_lbp : WellFounded (lbp p) :=
  ⟨by
    let ⟨n, pn⟩ := H
    suffices ∀ m k, n ≤ k + m → Acc (lbp p) k by exact fun a => this _ _ (Nat.le_add_left _ _)
    intro m k kn
    induction' m with m IH generalizing k <;> refine' ⟨_, fun y r => _⟩ <;> rcases r with ⟨rfl, a⟩
    · injection mem_unique pn.1 (a _ kn)
    · 

Proof state:
case succ.intro
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
n : ℕ
pn : true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : ∀ (k : ℕ), n ≤ k + m → Acc (Nat.lbp p) k
k : ℕ
kn : n ≤ k + succ m
a : ∀ k_1 ≤ k, false ∈ p k_1
⊢ Acc (Nat.lbp p) (k + 1)

Next tactic:
exact IH _ (by rw [Nat.add_right_comm]; exact kn)
private def wf_lbp : WellFounded (lbp p) :=
  ⟨by
    let ⟨n, pn⟩ := H
    suffices ∀ m k, n ≤ k + m → Acc (lbp p) k by exact fun a => this _ _ (Nat.le_add_left _ _)
    intro m k kn
    induction' m with m IH generalizing k <;> refine' ⟨_, fun y r => _⟩ <;> rcases r with ⟨rfl, a⟩
    · injection mem_unique pn.1 (a _ kn)
    · exact IH _ (by 

Proof state:
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
n : ℕ
pn : true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : ∀ (k : ℕ), n ≤ k + m → Acc (Nat.lbp p) k
k : ℕ
kn : n ≤ k + succ m
a : ∀ k_1 ≤ k, false ∈ p k_1
⊢ n ≤ k + 1 + m

Next tactic:
rw [Nat.add_right_comm]
private def wf_lbp : WellFounded (lbp p) :=
  ⟨by
    let ⟨n, pn⟩ := H
    suffices ∀ m k, n ≤ k + m → Acc (lbp p) k by exact fun a => this _ _ (Nat.le_add_left _ _)
    intro m k kn
    induction' m with m IH generalizing k <;> refine' ⟨_, fun y r => _⟩ <;> rcases r with ⟨rfl, a⟩
    · injection mem_unique pn.1 (a _ kn)
    · exact IH _ (by rw [Nat.add_right_comm]; 

Proof state:
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
n : ℕ
pn : true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : ∀ (k : ℕ), n ≤ k + m → Acc (Nat.lbp p) k
k : ℕ
kn : n ≤ k + succ m
a : ∀ k_1 ≤ k, false ∈ p k_1
⊢ n ≤ k + m + 1

Next tactic:
exact kn
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
  suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun n => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      

Proof state:
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
⊢ (x : ℕ) →
    ((y : ℕ) → Nat.lbp p y x → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }) →
      (∀ n < x, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }

Next tactic:
intro m IH al
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
  suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun n => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      

Proof state:
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : (y : ℕ) → Nat.lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }
al : ∀ n < m, false ∈ p n
⊢ { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }

Next tactic:
have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · exact h₂ _ h₃
        · rw [h₃]
          exact h₁.fst
        · injection mem_unique h₁ (al _ h₃)
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
  suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun n => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        

Proof state:
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : (y : ℕ) → Nat.lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }
al : ∀ n < m, false ∈ p n
⊢ (p m).Dom

Next tactic:
rcases H with ⟨n, h₁, h₂⟩
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
  suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun n => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        

Proof state:
case intro.intro
p : ℕ →. Bool
m : ℕ
IH : (y : ℕ) → Nat.lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }
al : ∀ n < m, false ∈ p n
n : ℕ
h₁ : true ∈ p n
h₂ : ∀ k < n, (p k).Dom
⊢ (p m).Dom

Next tactic:
rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
  suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun n => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · 

Proof state:
case intro.intro.inl
p : ℕ →. Bool
m : ℕ
IH : (y : ℕ) → Nat.lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }
al : ∀ n < m, false ∈ p n
n : ℕ
h₁ : true ∈ p n
h₂ : ∀ k < n, (p k).Dom
h₃ : m < n
⊢ (p m).Dom

Next tactic:
exact h₂ _ h₃
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
  suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun n => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · exact h₂ _ h₃
        · 

Proof state:
case intro.intro.inr.inl
p : ℕ →. Bool
m : ℕ
IH : (y : ℕ) → Nat.lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }
al : ∀ n < m, false ∈ p n
n : ℕ
h₁ : true ∈ p n
h₂ : ∀ k < n, (p k).Dom
h₃ : m = n
⊢ (p m).Dom

Next tactic:
rw [h₃]
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
  suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun n => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · exact h₂ _ h₃
        · rw [h₃]
          

Proof state:
case intro.intro.inr.inl
p : ℕ →. Bool
m : ℕ
IH : (y : ℕ) → Nat.lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }
al : ∀ n < m, false ∈ p n
n : ℕ
h₁ : true ∈ p n
h₂ : ∀ k < n, (p k).Dom
h₃ : m = n
⊢ (p n).Dom

Next tactic:
exact h₁.fst
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
  suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun n => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · exact h₂ _ h₃
        · rw [h₃]
          exact h₁.fst
        · 

Proof state:
case intro.intro.inr.inr
p : ℕ →. Bool
m : ℕ
IH : (y : ℕ) → Nat.lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }
al : ∀ n < m, false ∈ p n
n : ℕ
h₁ : true ∈ p n
h₂ : ∀ k < n, (p k).Dom
h₃ : n < m
⊢ (p m).Dom

Next tactic:
injection mem_unique h₁ (al _ h₃)
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
  suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun n => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · exact h₂ _ h₃
        · rw [h₃]
          exact h₁.fst
        · injection mem_unique h₁ (al _ h₃)
      

Proof state:
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : (y : ℕ) → Nat.lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }
al : ∀ n < m, false ∈ p n
pm : (p m).Dom
⊢ { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }

Next tactic:
cases e : (p m).get pm
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
  suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun n => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · exact h₂ _ h₃
        · rw [h₃]
          exact h₁.fst
        · injection mem_unique h₁ (al _ h₃)
      cases e : (p m).get pm
      · 

Proof state:
case false
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : (y : ℕ) → Nat.lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }
al : ∀ n < m, false ∈ p n
pm : (p m).Dom
e : Part.get (p m) pm = false
⊢ { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }

Next tactic:
suffices ∀ᵉ k ≤ m, false ∈ p k from IH _ ⟨rfl, this⟩ fun n h => this _ (le_of_lt_succ h)
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
  suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun n => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · exact h₂ _ h₃
        · rw [h₃]
          exact h₁.fst
        · injection mem_unique h₁ (al _ h₃)
      cases e : (p m).get pm
      · suffices ∀ᵉ k ≤ m, false ∈ p k from IH _ ⟨rfl, this⟩ fun n h => this _ (le_of_lt_succ h)
        

Proof state:
case false
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : (y : ℕ) → Nat.lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }
al : ∀ n < m, false ∈ p n
pm : (p m).Dom
e : Part.get (p m) pm = false
⊢ ∀ k ≤ m, false ∈ p k

Next tactic:
intro n h
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
  suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun n => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · exact h₂ _ h₃
        · rw [h₃]
          exact h₁.fst
        · injection mem_unique h₁ (al _ h₃)
      cases e : (p m).get pm
      · suffices ∀ᵉ k ≤ m, false ∈ p k from IH _ ⟨rfl, this⟩ fun n h => this _ (le_of_lt_succ h)
        intro n h
        

Proof state:
case false
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : (y : ℕ) → Nat.lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }
al : ∀ n < m, false ∈ p n
pm : (p m).Dom
e : Part.get (p m) pm = false
n : ℕ
h : n ≤ m
⊢ false ∈ p n

Next tactic:
cases' h.lt_or_eq_dec with h h
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
  suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun n => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · exact h₂ _ h₃
        · rw [h₃]
          exact h₁.fst
        · injection mem_unique h₁ (al _ h₃)
      cases e : (p m).get pm
      · suffices ∀ᵉ k ≤ m, false ∈ p k from IH _ ⟨rfl, this⟩ fun n h => this _ (le_of_lt_succ h)
        intro n h
        cases' h.lt_or_eq_dec with h h
        · 

Proof state:
case false.inl
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : (y : ℕ) → Nat.lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }
al : ∀ n < m, false ∈ p n
pm : (p m).Dom
e : Part.get (p m) pm = false
n : ℕ
h✝ : n ≤ m
h : n < m
⊢ false ∈ p n

Next tactic:
exact al _ h
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
  suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun n => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · exact h₂ _ h₃
        · rw [h₃]
          exact h₁.fst
        · injection mem_unique h₁ (al _ h₃)
      cases e : (p m).get pm
      · suffices ∀ᵉ k ≤ m, false ∈ p k from IH _ ⟨rfl, this⟩ fun n h => this _ (le_of_lt_succ h)
        intro n h
        cases' h.lt_or_eq_dec with h h
        · exact al _ h
        · 

Proof state:
case false.inr
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : (y : ℕ) → Nat.lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }
al : ∀ n < m, false ∈ p n
pm : (p m).Dom
e : Part.get (p m) pm = false
n : ℕ
h✝ : n ≤ m
h : n = m
⊢ false ∈ p n

Next tactic:
rw [h]
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
  suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun n => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · exact h₂ _ h₃
        · rw [h₃]
          exact h₁.fst
        · injection mem_unique h₁ (al _ h₃)
      cases e : (p m).get pm
      · suffices ∀ᵉ k ≤ m, false ∈ p k from IH _ ⟨rfl, this⟩ fun n h => this _ (le_of_lt_succ h)
        intro n h
        cases' h.lt_or_eq_dec with h h
        · exact al _ h
        · rw [h]
          

Proof state:
case false.inr
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : (y : ℕ) → Nat.lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }
al : ∀ n < m, false ∈ p n
pm : (p m).Dom
e : Part.get (p m) pm = false
n : ℕ
h✝ : n ≤ m
h : n = m
⊢ false ∈ p m

Next tactic:
exact ⟨_, e⟩
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
  suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun n => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · exact h₂ _ h₃
        · rw [h₃]
          exact h₁.fst
        · injection mem_unique h₁ (al _ h₃)
      cases e : (p m).get pm
      · suffices ∀ᵉ k ≤ m, false ∈ p k from IH _ ⟨rfl, this⟩ fun n h => this _ (le_of_lt_succ h)
        intro n h
        cases' h.lt_or_eq_dec with h h
        · exact al _ h
        · rw [h]
          exact ⟨_, e⟩
      · 

Proof state:
case true
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : (y : ℕ) → Nat.lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }
al : ∀ n < m, false ∈ p n
pm : (p m).Dom
e : Part.get (p m) pm = true
⊢ { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }

Next tactic:
exact ⟨m, ⟨_, e⟩, al⟩
@[simp]
theorem mem_rfind {p : ℕ →. Bool} {n : ℕ} :
    n ∈ rfind p ↔ true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m :=
  ⟨fun h => ⟨rfind_spec h, @rfind_min _ _ h⟩, fun ⟨h₁, h₂⟩ => by
    

Proof state:
p : ℕ →. Bool
n : ℕ
x✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m
h₁ : true ∈ p n
h₂ : ∀ {m : ℕ}, m < n → false ∈ p m
⊢ n ∈ rfind p

Next tactic:
let ⟨m, hm⟩ := dom_iff_mem.1 <| (@rfind_dom p).2 ⟨_, h₁, fun {m} mn => (h₂ mn).fst⟩
@[simp]
theorem mem_rfind {p : ℕ →. Bool} {n : ℕ} :
    n ∈ rfind p ↔ true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m :=
  ⟨fun h => ⟨rfind_spec h, @rfind_min _ _ h⟩, fun ⟨h₁, h₂⟩ => by
    let ⟨m, hm⟩ := dom_iff_mem.1 <| (@rfind_dom p).2 ⟨_, h₁, fun {m} mn => (h₂ mn).fst⟩
    

Proof state:
p : ℕ →. Bool
n : ℕ
x✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m
h₁ : true ∈ p n
h₂ : ∀ {m : ℕ}, m < n → false ∈ p m
m : ℕ
hm : m ∈ rfind p
⊢ n ∈ rfind p

Next tactic:
rcases lt_trichotomy m n with (h | h | h)
@[simp]
theorem mem_rfind {p : ℕ →. Bool} {n : ℕ} :
    n ∈ rfind p ↔ true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m :=
  ⟨fun h => ⟨rfind_spec h, @rfind_min _ _ h⟩, fun ⟨h₁, h₂⟩ => by
    let ⟨m, hm⟩ := dom_iff_mem.1 <| (@rfind_dom p).2 ⟨_, h₁, fun {m} mn => (h₂ mn).fst⟩
    rcases lt_trichotomy m n with (h | h | h)
    · 

Proof state:
case inl
p : ℕ →. Bool
n : ℕ
x✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m
h₁ : true ∈ p n
h₂ : ∀ {m : ℕ}, m < n → false ∈ p m
m : ℕ
hm : m ∈ rfind p
h : m < n
⊢ n ∈ rfind p

Next tactic:
injection mem_unique (h₂ h) (rfind_spec hm)