l3lab/ntp-mathlib huggingface dataset:
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      

Proof state:
case mp.succ.f.f.intro.a
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n✝ : ℕ
n_ih : ∀ (x y : M F), Agree (MIntl.approx x n✝) (MIntl.approx y (n✝ + 1)) → Agree' n✝ x y
a✝ : F.A
f✝¹ f✝ : B F a✝ → M F
hagree : ∀ (i : B F a✝), Agree ((fun j => MIntl.approx (f✝¹ j) n✝) i) ((fun j => MIntl.approx (f✝ j) (n✝ + 1)) i)
⊢ ∀ (i : B F a✝), Agree' n✝ (f✝¹ i) (f✝ i)

Next tactic:
intro i
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      intro i
      

Proof state:
case mp.succ.f.f.intro.a
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n✝ : ℕ
n_ih : ∀ (x y : M F), Agree (MIntl.approx x n✝) (MIntl.approx y (n✝ + 1)) → Agree' n✝ x y
a✝ : F.A
f✝¹ f✝ : B F a✝ → M F
hagree : ∀ (i : B F a✝), Agree ((fun j => MIntl.approx (f✝¹ j) n✝) i) ((fun j => MIntl.approx (f✝ j) (n✝ + 1)) i)
i : B F a✝
⊢ Agree' n✝ (f✝¹ i) (f✝ i)

Next tactic:
apply n_ih
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      intro i
      apply n_ih
      

Proof state:
case mp.succ.f.f.intro.a.h
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n✝ : ℕ
n_ih : ∀ (x y : M F), Agree (MIntl.approx x n✝) (MIntl.approx y (n✝ + 1)) → Agree' n✝ x y
a✝ : F.A
f✝¹ f✝ : B F a✝ → M F
hagree : ∀ (i : B F a✝), Agree ((fun j => MIntl.approx (f✝¹ j) n✝) i) ((fun j => MIntl.approx (f✝ j) (n✝ + 1)) i)
i : B F a✝
⊢ Agree (MIntl.approx (f✝¹ i) n✝) (MIntl.approx (f✝ i) (n✝ + 1))

Next tactic:
apply hagree
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      intro i
      apply n_ih
      apply hagree
  · 

Proof state:
case mpr
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n : ℕ
x y : M F
h : Agree' n x y
⊢ Agree (MIntl.approx x n) (MIntl.approx y (n + 1))

Next tactic:
induction' n with _ n_ih generalizing x y
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      intro i
      apply n_ih
      apply hagree
  · induction' n with _ n_ih generalizing x y
    

Proof state:
case mpr.zero
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
x y : M F
h : Agree' zero x y
⊢ Agree (MIntl.approx x zero) (MIntl.approx y (zero + 1))
case mpr.succ
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n✝ : ℕ
n_ih : ∀ (x y : M F), Agree' n✝ x y → Agree (MIntl.approx x n✝) (MIntl.approx y (n✝ + 1))
x y : M F
h : Agree' (succ n✝) x y
⊢ Agree (MIntl.approx x (succ n✝)) (MIntl.approx y (succ n✝ + 1))

Next tactic:
constructor
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      intro i
      apply n_ih
      apply hagree
  · induction' n with _ n_ih generalizing x y
    constructor
    · 

Proof state:
case mpr.succ
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n✝ : ℕ
n_ih : ∀ (x y : M F), Agree' n✝ x y → Agree (MIntl.approx x n✝) (MIntl.approx y (n✝ + 1))
x y : M F
h : Agree' (succ n✝) x y
⊢ Agree (MIntl.approx x (succ n✝)) (MIntl.approx y (succ n✝ + 1))

Next tactic:
cases' h with _ _ _ a x' y'
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      intro i
      apply n_ih
      apply hagree
  · induction' n with _ n_ih generalizing x y
    constructor
    · cases' h with _ _ _ a x' y'
      

Proof state:
case mpr.succ.step
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n✝ : ℕ
n_ih : ∀ (x y : M F), Agree' n✝ x y → Agree (MIntl.approx x n✝) (MIntl.approx y (n✝ + 1))
x y : M F
a : F.A
x' y' : B F a → M F
a✝² : ∀ (i : B F a), Agree' n✝ (x' i) (y' i)
a✝¹ : x = M.mk { fst := a, snd := x' }
a✝ : y = M.mk { fst := a, snd := y' }
⊢ Agree (MIntl.approx x (succ n✝)) (MIntl.approx y (succ n✝ + 1))

Next tactic:
induction' x using PFunctor.M.casesOn' with x_a x_f
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      intro i
      apply n_ih
      apply hagree
  · induction' n with _ n_ih generalizing x y
    constructor
    · cases' h with _ _ _ a x' y'
      induction' x using PFunctor.M.casesOn' with x_a x_f
      

Proof state:
case mpr.succ.step.f
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n✝ : ℕ
n_ih : ∀ (x y : M F), Agree' n✝ x y → Agree (MIntl.approx x n✝) (MIntl.approx y (n✝ + 1))
y : M F
a : F.A
x' y' : B F a → M F
a✝² : ∀ (i : B F a), Agree' n✝ (x' i) (y' i)
a✝¹ : y = M.mk { fst := a, snd := y' }
x_a : F.A
x_f : B F x_a → M F
a✝ : M.mk { fst := x_a, snd := x_f } = M.mk { fst := a, snd := x' }
⊢ Agree (MIntl.approx (M.mk { fst := x_a, snd := x_f }) (succ n✝)) (MIntl.approx y (succ n✝ + 1))

Next tactic:
induction' y using PFunctor.M.casesOn' with y_a y_f
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      intro i
      apply n_ih
      apply hagree
  · induction' n with _ n_ih generalizing x y
    constructor
    · cases' h with _ _ _ a x' y'
      induction' x using PFunctor.M.casesOn' with x_a x_f
      induction' y using PFunctor.M.casesOn' with y_a y_f
      

Proof state:
case mpr.succ.step.f.f
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n✝ : ℕ
n_ih : ∀ (x y : M F), Agree' n✝ x y → Agree (MIntl.approx x n✝) (MIntl.approx y (n✝ + 1))
a : F.A
x' y' : B F a → M F
a✝² : ∀ (i : B F a), Agree' n✝ (x' i) (y' i)
x_a : F.A
x_f : B F x_a → M F
a✝¹ : M.mk { fst := x_a, snd := x_f } = M.mk { fst := a, snd := x' }
y_a : F.A
y_f : B F y_a → M F
a✝ : M.mk { fst := y_a, snd := y_f } = M.mk { fst := a, snd := y' }
⊢ Agree (MIntl.approx (M.mk { fst := x_a, snd := x_f }) (succ n✝))
    (MIntl.approx (M.mk { fst := y_a, snd := y_f }) (succ n✝ + 1))

Next tactic:
simp only [approx_mk]
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      intro i
      apply n_ih
      apply hagree
  · induction' n with _ n_ih generalizing x y
    constructor
    · cases' h with _ _ _ a x' y'
      induction' x using PFunctor.M.casesOn' with x_a x_f
      induction' y using PFunctor.M.casesOn' with y_a y_f
      simp only [approx_mk]
      

Proof state:
case mpr.succ.step.f.f
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n✝ : ℕ
n_ih : ∀ (x y : M F), Agree' n✝ x y → Agree (MIntl.approx x n✝) (MIntl.approx y (n✝ + 1))
a : F.A
x' y' : B F a → M F
a✝² : ∀ (i : B F a), Agree' n✝ (x' i) (y' i)
x_a : F.A
x_f : B F x_a → M F
a✝¹ : M.mk { fst := x_a, snd := x_f } = M.mk { fst := a, snd := x' }
y_a : F.A
y_f : B F y_a → M F
a✝ : M.mk { fst := y_a, snd := y_f } = M.mk { fst := a, snd := y' }
⊢ Agree (CofixA.intro x_a fun j => MIntl.approx (x_f j) n✝) (CofixA.intro y_a fun j => MIntl.approx (y_f j) (n✝ + 1))

Next tactic:
have h_a_1 := mk_inj ‹M.mk ⟨x_a, x_f⟩ = M.mk ⟨a, x'⟩›
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      intro i
      apply n_ih
      apply hagree
  · induction' n with _ n_ih generalizing x y
    constructor
    · cases' h with _ _ _ a x' y'
      induction' x using PFunctor.M.casesOn' with x_a x_f
      induction' y using PFunctor.M.casesOn' with y_a y_f
      simp only [approx_mk]
      have h_a_1 := mk_inj ‹M.mk ⟨x_a, x_f⟩ = M.mk ⟨a, x'⟩›
      

Proof state:
case mpr.succ.step.f.f
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n✝ : ℕ
n_ih : ∀ (x y : M F), Agree' n✝ x y → Agree (MIntl.approx x n✝) (MIntl.approx y (n✝ + 1))
a : F.A
x' y' : B F a → M F
a✝² : ∀ (i : B F a), Agree' n✝ (x' i) (y' i)
x_a : F.A
x_f : B F x_a → M F
a✝¹ : M.mk { fst := x_a, snd := x_f } = M.mk { fst := a, snd := x' }
y_a : F.A
y_f : B F y_a → M F
a✝ : M.mk { fst := y_a, snd := y_f } = M.mk { fst := a, snd := y' }
h_a_1 : { fst := x_a, snd := x_f } = { fst := a, snd := x' }
⊢ Agree (CofixA.intro x_a fun j => MIntl.approx (x_f j) n✝) (CofixA.intro y_a fun j => MIntl.approx (y_f j) (n✝ + 1))

Next tactic:
cases h_a_1
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      intro i
      apply n_ih
      apply hagree
  · induction' n with _ n_ih generalizing x y
    constructor
    · cases' h with _ _ _ a x' y'
      induction' x using PFunctor.M.casesOn' with x_a x_f
      induction' y using PFunctor.M.casesOn' with y_a y_f
      simp only [approx_mk]
      have h_a_1 := mk_inj ‹M.mk ⟨x_a, x_f⟩ = M.mk ⟨a, x'⟩›
      cases h_a_1
      

Proof state:
case mpr.succ.step.f.f.refl
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n✝ : ℕ
n_ih : ∀ (x y : M F), Agree' n✝ x y → Agree (MIntl.approx x n✝) (MIntl.approx y (n✝ + 1))
a : F.A
x' y' : B F a → M F
a✝² : ∀ (i : B F a), Agree' n✝ (x' i) (y' i)
y_a : F.A
y_f : B F y_a → M F
a✝¹ : M.mk { fst := y_a, snd := y_f } = M.mk { fst := a, snd := y' }
a✝ : M.mk { fst := a, snd := x' } = M.mk { fst := a, snd := x' }
⊢ Agree (CofixA.intro a fun j => MIntl.approx (x' j) n✝) (CofixA.intro y_a fun j => MIntl.approx (y_f j) (n✝ + 1))

Next tactic:
replace h_a_2 := mk_inj ‹M.mk ⟨y_a, y_f⟩ = M.mk ⟨a, y'⟩›
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      intro i
      apply n_ih
      apply hagree
  · induction' n with _ n_ih generalizing x y
    constructor
    · cases' h with _ _ _ a x' y'
      induction' x using PFunctor.M.casesOn' with x_a x_f
      induction' y using PFunctor.M.casesOn' with y_a y_f
      simp only [approx_mk]
      have h_a_1 := mk_inj ‹M.mk ⟨x_a, x_f⟩ = M.mk ⟨a, x'⟩›
      cases h_a_1
      replace h_a_2 := mk_inj ‹M.mk ⟨y_a, y_f⟩ = M.mk ⟨a, y'⟩›
      

Proof state:
case mpr.succ.step.f.f.refl
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n✝ : ℕ
n_ih : ∀ (x y : M F), Agree' n✝ x y → Agree (MIntl.approx x n✝) (MIntl.approx y (n✝ + 1))
a : F.A
x' y' : B F a → M F
a✝² : ∀ (i : B F a), Agree' n✝ (x' i) (y' i)
y_a : F.A
y_f : B F y_a → M F
a✝¹ : M.mk { fst := y_a, snd := y_f } = M.mk { fst := a, snd := y' }
a✝ : M.mk { fst := a, snd := x' } = M.mk { fst := a, snd := x' }
h_a_2 : { fst := y_a, snd := y_f } = { fst := a, snd := y' }
⊢ Agree (CofixA.intro a fun j => MIntl.approx (x' j) n✝) (CofixA.intro y_a fun j => MIntl.approx (y_f j) (n✝ + 1))

Next tactic:
cases h_a_2
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      intro i
      apply n_ih
      apply hagree
  · induction' n with _ n_ih generalizing x y
    constructor
    · cases' h with _ _ _ a x' y'
      induction' x using PFunctor.M.casesOn' with x_a x_f
      induction' y using PFunctor.M.casesOn' with y_a y_f
      simp only [approx_mk]
      have h_a_1 := mk_inj ‹M.mk ⟨x_a, x_f⟩ = M.mk ⟨a, x'⟩›
      cases h_a_1
      replace h_a_2 := mk_inj ‹M.mk ⟨y_a, y_f⟩ = M.mk ⟨a, y'⟩›
      cases h_a_2
      

Proof state:
case mpr.succ.step.f.f.refl.refl
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n✝ : ℕ
n_ih : ∀ (x y : M F), Agree' n✝ x y → Agree (MIntl.approx x n✝) (MIntl.approx y (n✝ + 1))
a : F.A
x' y' : B F a → M F
a✝² : ∀ (i : B F a), Agree' n✝ (x' i) (y' i)
a✝¹ : M.mk { fst := a, snd := x' } = M.mk { fst := a, snd := x' }
a✝ : M.mk { fst := a, snd := y' } = M.mk { fst := a, snd := y' }
⊢ Agree (CofixA.intro a fun j => MIntl.approx (x' j) n✝) (CofixA.intro a fun j => MIntl.approx (y' j) (n✝ + 1))

Next tactic:
constructor
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      intro i
      apply n_ih
      apply hagree
  · induction' n with _ n_ih generalizing x y
    constructor
    · cases' h with _ _ _ a x' y'
      induction' x using PFunctor.M.casesOn' with x_a x_f
      induction' y using PFunctor.M.casesOn' with y_a y_f
      simp only [approx_mk]
      have h_a_1 := mk_inj ‹M.mk ⟨x_a, x_f⟩ = M.mk ⟨a, x'⟩›
      cases h_a_1
      replace h_a_2 := mk_inj ‹M.mk ⟨y_a, y_f⟩ = M.mk ⟨a, y'⟩›
      cases h_a_2
      constructor
      

Proof state:
case mpr.succ.step.f.f.refl.refl.a
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n✝ : ℕ
n_ih : ∀ (x y : M F), Agree' n✝ x y → Agree (MIntl.approx x n✝) (MIntl.approx y (n✝ + 1))
a : F.A
x' y' : B F a → M F
a✝² : ∀ (i : B F a), Agree' n✝ (x' i) (y' i)
a✝¹ : M.mk { fst := a, snd := x' } = M.mk { fst := a, snd := x' }
a✝ : M.mk { fst := a, snd := y' } = M.mk { fst := a, snd := y' }
⊢ ∀ (i : B F a), Agree (MIntl.approx (x' i) n✝) (MIntl.approx (y' i) (n✝ + 1))

Next tactic:
intro i
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      intro i
      apply n_ih
      apply hagree
  · induction' n with _ n_ih generalizing x y
    constructor
    · cases' h with _ _ _ a x' y'
      induction' x using PFunctor.M.casesOn' with x_a x_f
      induction' y using PFunctor.M.casesOn' with y_a y_f
      simp only [approx_mk]
      have h_a_1 := mk_inj ‹M.mk ⟨x_a, x_f⟩ = M.mk ⟨a, x'⟩›
      cases h_a_1
      replace h_a_2 := mk_inj ‹M.mk ⟨y_a, y_f⟩ = M.mk ⟨a, y'⟩›
      cases h_a_2
      constructor
      intro i
      

Proof state:
case mpr.succ.step.f.f.refl.refl.a
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n✝ : ℕ
n_ih : ∀ (x y : M F), Agree' n✝ x y → Agree (MIntl.approx x n✝) (MIntl.approx y (n✝ + 1))
a : F.A
x' y' : B F a → M F
a✝² : ∀ (i : B F a), Agree' n✝ (x' i) (y' i)
a✝¹ : M.mk { fst := a, snd := x' } = M.mk { fst := a, snd := x' }
a✝ : M.mk { fst := a, snd := y' } = M.mk { fst := a, snd := y' }
i : B F a
⊢ Agree (MIntl.approx (x' i) n✝) (MIntl.approx (y' i) (n✝ + 1))

Next tactic:
apply n_ih
theorem agree_iff_agree' {n : ℕ} (x y : M F) :
    Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y := by
  constructor <;> intro h
  · induction' n with _ n_ih generalizing x y
    constructor
    · induction x using PFunctor.M.casesOn'
      induction y using PFunctor.M.casesOn'
      simp only [approx_mk] at h
      cases' h with _ _ _ _ _ _ hagree
      constructor <;> try rfl
      intro i
      apply n_ih
      apply hagree
  · induction' n with _ n_ih generalizing x y
    constructor
    · cases' h with _ _ _ a x' y'
      induction' x using PFunctor.M.casesOn' with x_a x_f
      induction' y using PFunctor.M.casesOn' with y_a y_f
      simp only [approx_mk]
      have h_a_1 := mk_inj ‹M.mk ⟨x_a, x_f⟩ = M.mk ⟨a, x'⟩›
      cases h_a_1
      replace h_a_2 := mk_inj ‹M.mk ⟨y_a, y_f⟩ = M.mk ⟨a, y'⟩›
      cases h_a_2
      constructor
      intro i
      apply n_ih
      

Proof state:
case mpr.succ.step.f.f.refl.refl.a.h
F : PFunctor.{u}
X : Type u_1
f : X → ↑F X
n✝ : ℕ
n_ih : ∀ (x y : M F), Agree' n✝ x y → Agree (MIntl.approx x n✝) (MIntl.approx y (n✝ + 1))
a : F.A
x' y' : B F a → M F
a✝² : ∀ (i : B F a), Agree' n✝ (x' i) (y' i)
a✝¹ : M.mk { fst := a, snd := x' } = M.mk { fst := a, snd := x' }
a✝ : M.mk { fst := a, snd := y' } = M.mk { fst := a, snd := y' }
i : B F a
⊢ Agree' n✝ (x' i) (y' i)

Next tactic:
simp [*]
@[simp]
theorem cases_mk {r : M F → Sort*} (x : F (M F)) (f : ∀ x : F (M F), r (M.mk x)) :
    PFunctor.M.cases f (M.mk x) = f x := by
  

Proof state:
F : PFunctor.{u}
X : Type u_1
f✝ : X → ↑F X
r : M F → Sort u_2
x : ↑F (M F)
f : (x : ↑F (M F)) → r (M.mk x)
⊢ M.cases f (M.mk x) = f x

Next tactic:
dsimp only [M.mk, PFunctor.M.cases, dest, head, Approx.sMk, head']
@[simp]
theorem cases_mk {r : M F → Sort*} (x : F (M F)) (f : ∀ x : F (M F), r (M.mk x)) :
    PFunctor.M.cases f (M.mk x) = f x := by
  dsimp only [M.mk, PFunctor.M.cases, dest, head, Approx.sMk, head']
  

Proof state:
F : PFunctor.{u}
X : Type u_1
f✝ : X → ↑F X
r : M F → Sort u_2
x : ↑F (M F)
f : (x : ↑F (M F)) → r (M.mk x)
⊢ Eq.mpr
      (_ :
        r { approx := Approx.sMk x, consistent := (_ : AllAgree (Approx.sMk x)) } =
          r
            {
              approx :=
                Approx.sMk
                  { fst := x.fst,
                    snd := fun i =>
                      children { approx := Approx.sMk x, consistent := (_ : AllAgree (Approx.sMk x)) } i },
              consistent :=
                (_ :
                  AllAgree
                    (Approx.sMk
                      { fst := x.fst,
                        snd := fun i =>
                          children { approx := Approx.sMk x, consistent := (_ : AllAgree (Approx.sMk x)) } i })) })
      (f
        { fst := x.fst,
          snd := fun i => children { approx := Approx.sMk x, consistent := (_ : AllAgree (Approx.sMk x)) } i }) =
    f x

Next tactic:
cases x
@[simp]
theorem cases_mk {r : M F → Sort*} (x : F (M F)) (f : ∀ x : F (M F), r (M.mk x)) :
    PFunctor.M.cases f (M.mk x) = f x := by
  dsimp only [M.mk, PFunctor.M.cases, dest, head, Approx.sMk, head']
  cases x; 

Proof state:
case mk
F : PFunctor.{u}
X : Type u_1
f✝ : X → ↑F X
r : M F → Sort u_2
f : (x : ↑F (M F)) → r (M.mk x)
fst✝ : F.A
snd✝ : B F fst✝ → M F
⊢ Eq.mpr
      (_ :
        r
            { approx := Approx.sMk { fst := fst✝, snd := snd✝ },
              consistent := (_ : AllAgree (Approx.sMk { fst := fst✝, snd := snd✝ })) } =
          r
            {
              approx :=
                Approx.sMk
                  { fst := { fst := fst✝, snd := snd✝ }.fst,
                    snd := fun i =>
                      children
                        { approx := Approx.sMk { fst := fst✝, snd := snd✝ },
                          consistent := (_ : AllAgree (Approx.sMk { fst := fst✝, snd := snd✝ })) }
                        i },
              consistent :=
                (_ :
                  AllAgree
                    (Approx.sMk
                      { fst := { fst := fst✝, snd := snd✝ }.fst,
                        snd := fun i =>
                          children
                            { approx := Approx.sMk { fst := fst✝, snd := snd✝ },
                              consistent := (_ : AllAgree (Approx.sMk { fst := fst✝, snd := snd✝ })) }
                            i })) })
      (f
        { fst := { fst := fst✝, snd := snd✝ }.fst,
          snd := fun i =>
            children
              { approx := Approx.sMk { fst := fst✝, snd := snd✝ },
                consistent := (_ : AllAgree (Approx.sMk { fst := fst✝, snd := snd✝ })) }
              i }) =
    f { fst := fst✝, snd := snd✝ }

Next tactic:
dsimp only [Approx.sMk]
@[simp]
theorem cases_mk {r : M F → Sort*} (x : F (M F)) (f : ∀ x : F (M F), r (M.mk x)) :
    PFunctor.M.cases f (M.mk x) = f x := by
  dsimp only [M.mk, PFunctor.M.cases, dest, head, Approx.sMk, head']
  cases x; dsimp only [Approx.sMk]
  

Proof state:
case mk
F : PFunctor.{u}
X : Type u_1
f✝ : X → ↑F X
r : M F → Sort u_2
f : (x : ↑F (M F)) → r (M.mk x)
fst✝ : F.A
snd✝ : B F fst✝ → M F
⊢ Eq.mpr
      (_ :
        r
            { approx := Approx.sMk { fst := fst✝, snd := snd✝ },
              consistent := (_ : AllAgree (Approx.sMk { fst := fst✝, snd := snd✝ })) } =
          r
            {
              approx :=
                Approx.sMk
                  { fst := fst✝,
                    snd := fun i =>
                      children
                        { approx := Approx.sMk { fst := fst✝, snd := snd✝ },
                          consistent := (_ : AllAgree (Approx.sMk { fst := fst✝, snd := snd✝ })) }
                        i },
              consistent :=
                (_ :
                  AllAgree
                    (Approx.sMk
                      { fst := fst✝,
                        snd := fun i =>
                          children
                            { approx := Approx.sMk { fst := fst✝, snd := snd✝ },
                              consistent := (_ : AllAgree (Approx.sMk { fst := fst✝, snd := snd✝ })) }
                            i })) })
      (f
        { fst := fst✝,
          snd := fun i =>
            children
              { approx := Approx.sMk { fst := fst✝, snd := snd✝ },
                consistent := (_ : AllAgree (Approx.sMk { fst := fst✝, snd := snd✝ })) }
              i }) =
    f { fst := fst✝, snd := snd✝ }

Next tactic:
simp only [Eq.mpr]