l3lab/ntp-mathlib huggingface dataset:
theorem h_apply_fromCoset_nin_range (x : B) (hx : x ∈ f.range) (b : B) (hb : b ∉ f.range) :
    h x (fromCoset ⟨b • f.range, b, rfl⟩) = fromCoset ⟨(x * b) • ↑f.range, x * b, rfl⟩ := by
  change ((τ).symm.trans (g x)).trans τ _ = _
  simp only [tau, MonoidHom.coe_mk, Equiv.toFun_as_coe, Equiv.coe_trans, Function.comp_apply]
  rw [Equiv.symm_swap,
    @Equiv.swap_apply_of_ne_of_ne X' _ (fromCoset ⟨f.range, 1, one_leftCoset _⟩) ∞
      (fromCoset ⟨b • ↑f.range, b, rfl⟩) (fromCoset_ne_of_nin_range _ hb) (by simp)]
  

Proof state:
A B : GroupCat
f : A ⟶ B
x : ↑B
hx : x ∈ MonoidHom.range f
b : ↑B
hb : b ∉ MonoidHom.range f
⊢ (Equiv.swap
        (fromCoset
          { val := ↑(MonoidHom.range f),
            property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = ↑(MonoidHom.range f)) })
        ∞)
      ((g x)
        (fromCoset
          { val := b • ↑(MonoidHom.range f),
            property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = b • ↑(MonoidHom.range f)) })) =
    fromCoset
      { val := (x * b) • ↑(MonoidHom.range f),
        property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = (x * b) • ↑(MonoidHom.range f)) }

Next tactic:
simp only [g_apply_fromCoset, leftCoset_assoc]
theorem h_apply_fromCoset_nin_range (x : B) (hx : x ∈ f.range) (b : B) (hb : b ∉ f.range) :
    h x (fromCoset ⟨b • f.range, b, rfl⟩) = fromCoset ⟨(x * b) • ↑f.range, x * b, rfl⟩ := by
  change ((τ).symm.trans (g x)).trans τ _ = _
  simp only [tau, MonoidHom.coe_mk, Equiv.toFun_as_coe, Equiv.coe_trans, Function.comp_apply]
  rw [Equiv.symm_swap,
    @Equiv.swap_apply_of_ne_of_ne X' _ (fromCoset ⟨f.range, 1, one_leftCoset _⟩) ∞
      (fromCoset ⟨b • ↑f.range, b, rfl⟩) (fromCoset_ne_of_nin_range _ hb) (by simp)]
  simp only [g_apply_fromCoset, leftCoset_assoc]
  

Proof state:
A B : GroupCat
f : A ⟶ B
x : ↑B
hx : x ∈ MonoidHom.range f
b : ↑B
hb : b ∉ MonoidHom.range f
⊢ (Equiv.swap
        (fromCoset
          { val := ↑(MonoidHom.range f),
            property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = ↑(MonoidHom.range f)) })
        ∞)
      (fromCoset
        { val := (x * b) • ↑(MonoidHom.range f),
          property :=
            (_ : (fun x => x ∈ Set.range fun x => x • ↑(MonoidHom.range f)) ((x * b) • ↑(MonoidHom.range f))) }) =
    fromCoset
      { val := (x * b) • ↑(MonoidHom.range f),
        property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = (x * b) • ↑(MonoidHom.range f)) }

Next tactic:
refine' Equiv.swap_apply_of_ne_of_ne (fromCoset_ne_of_nin_range _ fun r => hb _) (by simp)
theorem h_apply_fromCoset_nin_range (x : B) (hx : x ∈ f.range) (b : B) (hb : b ∉ f.range) :
    h x (fromCoset ⟨b • f.range, b, rfl⟩) = fromCoset ⟨(x * b) • ↑f.range, x * b, rfl⟩ := by
  change ((τ).symm.trans (g x)).trans τ _ = _
  simp only [tau, MonoidHom.coe_mk, Equiv.toFun_as_coe, Equiv.coe_trans, Function.comp_apply]
  rw [Equiv.symm_swap,
    @Equiv.swap_apply_of_ne_of_ne X' _ (fromCoset ⟨f.range, 1, one_leftCoset _⟩) ∞
      (fromCoset ⟨b • ↑f.range, b, rfl⟩) (fromCoset_ne_of_nin_range _ hb) (by simp)]
  simp only [g_apply_fromCoset, leftCoset_assoc]
  refine' Equiv.swap_apply_of_ne_of_ne (fromCoset_ne_of_nin_range _ fun r => hb _) (by 

Proof state:
A B : GroupCat
f : A ⟶ B
x : ↑B
hx : x ∈ MonoidHom.range f
b : ↑B
hb : b ∉ MonoidHom.range f
⊢ fromCoset
      { val := (x * b) • ↑(MonoidHom.range f),
        property :=
          (_ : (fun x => x ∈ Set.range fun x => x • ↑(MonoidHom.range f)) ((x * b) • ↑(MonoidHom.range f))) } ≠
    ∞

Next tactic:
simp
theorem h_apply_fromCoset_nin_range (x : B) (hx : x ∈ f.range) (b : B) (hb : b ∉ f.range) :
    h x (fromCoset ⟨b • f.range, b, rfl⟩) = fromCoset ⟨(x * b) • ↑f.range, x * b, rfl⟩ := by
  change ((τ).symm.trans (g x)).trans τ _ = _
  simp only [tau, MonoidHom.coe_mk, Equiv.toFun_as_coe, Equiv.coe_trans, Function.comp_apply]
  rw [Equiv.symm_swap,
    @Equiv.swap_apply_of_ne_of_ne X' _ (fromCoset ⟨f.range, 1, one_leftCoset _⟩) ∞
      (fromCoset ⟨b • ↑f.range, b, rfl⟩) (fromCoset_ne_of_nin_range _ hb) (by simp)]
  simp only [g_apply_fromCoset, leftCoset_assoc]
  refine' Equiv.swap_apply_of_ne_of_ne (fromCoset_ne_of_nin_range _ fun r => hb _) (by simp)
  

Proof state:
A B : GroupCat
f : A ⟶ B
x : ↑B
hx : x ∈ MonoidHom.range f
b : ↑B
hb : b ∉ MonoidHom.range f
r : x * b ∈ MonoidHom.range f
⊢ b ∈ MonoidHom.range f

Next tactic:
convert Subgroup.mul_mem _ (Subgroup.inv_mem _ hx) r
theorem h_apply_fromCoset_nin_range (x : B) (hx : x ∈ f.range) (b : B) (hb : b ∉ f.range) :
    h x (fromCoset ⟨b • f.range, b, rfl⟩) = fromCoset ⟨(x * b) • ↑f.range, x * b, rfl⟩ := by
  change ((τ).symm.trans (g x)).trans τ _ = _
  simp only [tau, MonoidHom.coe_mk, Equiv.toFun_as_coe, Equiv.coe_trans, Function.comp_apply]
  rw [Equiv.symm_swap,
    @Equiv.swap_apply_of_ne_of_ne X' _ (fromCoset ⟨f.range, 1, one_leftCoset _⟩) ∞
      (fromCoset ⟨b • ↑f.range, b, rfl⟩) (fromCoset_ne_of_nin_range _ hb) (by simp)]
  simp only [g_apply_fromCoset, leftCoset_assoc]
  refine' Equiv.swap_apply_of_ne_of_ne (fromCoset_ne_of_nin_range _ fun r => hb _) (by simp)
  convert Subgroup.mul_mem _ (Subgroup.inv_mem _ hx) r
  

Proof state:
case h.e'_4
A B : GroupCat
f : A ⟶ B
x : ↑B
hx : x ∈ MonoidHom.range f
b : ↑B
hb : b ∉ MonoidHom.range f
r : x * b ∈ MonoidHom.range f
⊢ b = x⁻¹ * (x * b)

Next tactic:
rw [← mul_assoc, mul_left_inv, one_mul]
theorem agree : f.range = { x | h x = g x } := by
  

Proof state:
A B : GroupCat
f : A ⟶ B
⊢ ↑(MonoidHom.range f) = {x | h x = g x}

Next tactic:
refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · 

Proof state:
case refine'_1
A B : GroupCat
f : A ⟶ B
b : ↑B
⊢ b ∈ ↑(MonoidHom.range f) → b ∈ {x | h x = g x}

Next tactic:
rintro ⟨a, rfl⟩
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · rintro ⟨a, rfl⟩
    

Proof state:
case refine'_1.intro
A B : GroupCat
f : A ⟶ B
a : ↑A
⊢ f a ∈ {x | h x = g x}

Next tactic:
change h (f a) = g (f a)
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · rintro ⟨a, rfl⟩
    change h (f a) = g (f a)
    

Proof state:
case refine'_1.intro
A B : GroupCat
f : A ⟶ B
a : ↑A
⊢ h (f a) = g (f a)

Next tactic:
ext ⟨⟨_, ⟨y, rfl⟩⟩⟩
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · rintro ⟨a, rfl⟩
    change h (f a) = g (f a)
    ext ⟨⟨_, ⟨y, rfl⟩⟩⟩
    · 

Proof state:
case refine'_1.intro.H.fromCoset.mk.intro
A B : GroupCat
f : A ⟶ B
a : ↑A
y : ↑B
⊢ (h (f a))
      (fromCoset
        { val := (fun x => x • ↑(MonoidHom.range f)) y,
          property := (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (fun x => x • ↑(MonoidHom.range f)) y) }) =
    (g (f a))
      (fromCoset
        { val := (fun x => x • ↑(MonoidHom.range f)) y,
          property := (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (fun x => x • ↑(MonoidHom.range f)) y) })

Next tactic:
rw [g_apply_fromCoset]
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · rintro ⟨a, rfl⟩
    change h (f a) = g (f a)
    ext ⟨⟨_, ⟨y, rfl⟩⟩⟩
    · rw [g_apply_fromCoset]
      

Proof state:
case refine'_1.intro.H.fromCoset.mk.intro
A B : GroupCat
f : A ⟶ B
a : ↑A
y : ↑B
⊢ (h (f a))
      (fromCoset
        { val := (fun x => x • ↑(MonoidHom.range f)) y,
          property := (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (fun x => x • ↑(MonoidHom.range f)) y) }) =
    fromCoset
      {
        val :=
          f a •
            ↑{ val := (fun x => x • ↑(MonoidHom.range f)) y,
                property :=
                  (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (fun x => x • ↑(MonoidHom.range f)) y) },
        property :=
          (_ :
            f a •
                ↑{ val := (fun x => x • ↑(MonoidHom.range f)) y,
                    property :=
                      (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (fun x => x • ↑(MonoidHom.range f)) y) } ∈
              Set.range fun x => x • ↑(MonoidHom.range f)) }

Next tactic:
by_cases m : y ∈ f.range
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · rintro ⟨a, rfl⟩
    change h (f a) = g (f a)
    ext ⟨⟨_, ⟨y, rfl⟩⟩⟩
    · rw [g_apply_fromCoset]
      by_cases m : y ∈ f.range
      · 

Proof state:
case pos
A B : GroupCat
f : A ⟶ B
a : ↑A
y : ↑B
m : y ∈ MonoidHom.range f
⊢ (h (f a))
      (fromCoset
        { val := (fun x => x • ↑(MonoidHom.range f)) y,
          property := (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (fun x => x • ↑(MonoidHom.range f)) y) }) =
    fromCoset
      {
        val :=
          f a •
            ↑{ val := (fun x => x • ↑(MonoidHom.range f)) y,
                property :=
                  (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (fun x => x • ↑(MonoidHom.range f)) y) },
        property :=
          (_ :
            f a •
                ↑{ val := (fun x => x • ↑(MonoidHom.range f)) y,
                    property :=
                      (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (fun x => x • ↑(MonoidHom.range f)) y) } ∈
              Set.range fun x => x • ↑(MonoidHom.range f)) }

Next tactic:
rw [h_apply_fromCoset' _ _ _ m, fromCoset_eq_of_mem_range _ m]
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · rintro ⟨a, rfl⟩
    change h (f a) = g (f a)
    ext ⟨⟨_, ⟨y, rfl⟩⟩⟩
    · rw [g_apply_fromCoset]
      by_cases m : y ∈ f.range
      · rw [h_apply_fromCoset' _ _ _ m, fromCoset_eq_of_mem_range _ m]
        

Proof state:
case pos
A B : GroupCat
f : A ⟶ B
a : ↑A
y : ↑B
m : y ∈ MonoidHom.range f
⊢ fromCoset
      { val := ↑(MonoidHom.range f),
        property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = ↑(MonoidHom.range f)) } =
    fromCoset
      {
        val :=
          f a •
            ↑{ val := (fun x => x • ↑(MonoidHom.range f)) y,
                property :=
                  (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (fun x => x • ↑(MonoidHom.range f)) y) },
        property :=
          (_ :
            f a •
                ↑{ val := (fun x => x • ↑(MonoidHom.range f)) y,
                    property :=
                      (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (fun x => x • ↑(MonoidHom.range f)) y) } ∈
              Set.range fun x => x • ↑(MonoidHom.range f)) }

Next tactic:
change fromCoset _ = fromCoset ⟨f a • (y • _), _⟩
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · rintro ⟨a, rfl⟩
    change h (f a) = g (f a)
    ext ⟨⟨_, ⟨y, rfl⟩⟩⟩
    · rw [g_apply_fromCoset]
      by_cases m : y ∈ f.range
      · rw [h_apply_fromCoset' _ _ _ m, fromCoset_eq_of_mem_range _ m]
        change fromCoset _ = fromCoset ⟨f a • (y • _), _⟩
        

Proof state:
case pos
A B : GroupCat
f : A ⟶ B
a : ↑A
y : ↑B
m : y ∈ MonoidHom.range f
⊢ fromCoset
      { val := ↑(MonoidHom.range f),
        property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = ↑(MonoidHom.range f)) } =
    fromCoset
      { val := f a • y • ↑(MonoidHom.range f),
        property :=
          (_ :
            f a •
                ↑{ val := (fun x => x • ↑(MonoidHom.range f)) y,
                    property :=
                      (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (fun x => x • ↑(MonoidHom.range f)) y) } ∈
              Set.range fun x => x • ↑(MonoidHom.range f)) }

Next tactic:
simp only [← fromCoset_eq_of_mem_range _ (Subgroup.mul_mem _ ⟨a, rfl⟩ m), smul_smul]
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · rintro ⟨a, rfl⟩
    change h (f a) = g (f a)
    ext ⟨⟨_, ⟨y, rfl⟩⟩⟩
    · rw [g_apply_fromCoset]
      by_cases m : y ∈ f.range
      · rw [h_apply_fromCoset' _ _ _ m, fromCoset_eq_of_mem_range _ m]
        change fromCoset _ = fromCoset ⟨f a • (y • _), _⟩
        simp only [← fromCoset_eq_of_mem_range _ (Subgroup.mul_mem _ ⟨a, rfl⟩ m), smul_smul]
      · 

Proof state:
case neg
A B : GroupCat
f : A ⟶ B
a : ↑A
y : ↑B
m : y ∉ MonoidHom.range f
⊢ (h (f a))
      (fromCoset
        { val := (fun x => x • ↑(MonoidHom.range f)) y,
          property := (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (fun x => x • ↑(MonoidHom.range f)) y) }) =
    fromCoset
      {
        val :=
          f a •
            ↑{ val := (fun x => x • ↑(MonoidHom.range f)) y,
                property :=
                  (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (fun x => x • ↑(MonoidHom.range f)) y) },
        property :=
          (_ :
            f a •
                ↑{ val := (fun x => x • ↑(MonoidHom.range f)) y,
                    property :=
                      (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (fun x => x • ↑(MonoidHom.range f)) y) } ∈
              Set.range fun x => x • ↑(MonoidHom.range f)) }

Next tactic:
rw [h_apply_fromCoset_nin_range f (f a) ⟨_, rfl⟩ _ m]
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · rintro ⟨a, rfl⟩
    change h (f a) = g (f a)
    ext ⟨⟨_, ⟨y, rfl⟩⟩⟩
    · rw [g_apply_fromCoset]
      by_cases m : y ∈ f.range
      · rw [h_apply_fromCoset' _ _ _ m, fromCoset_eq_of_mem_range _ m]
        change fromCoset _ = fromCoset ⟨f a • (y • _), _⟩
        simp only [← fromCoset_eq_of_mem_range _ (Subgroup.mul_mem _ ⟨a, rfl⟩ m), smul_smul]
      · rw [h_apply_fromCoset_nin_range f (f a) ⟨_, rfl⟩ _ m]
        

Proof state:
case neg
A B : GroupCat
f : A ⟶ B
a : ↑A
y : ↑B
m : y ∉ MonoidHom.range f
⊢ fromCoset
      { val := (f a * y) • ↑(MonoidHom.range f),
        property := (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (f a * y) • ↑(MonoidHom.range f)) } =
    fromCoset
      {
        val :=
          f a •
            ↑{ val := (fun x => x • ↑(MonoidHom.range f)) y,
                property :=
                  (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (fun x => x • ↑(MonoidHom.range f)) y) },
        property :=
          (_ :
            f a •
                ↑{ val := (fun x => x • ↑(MonoidHom.range f)) y,
                    property :=
                      (_ : ∃ y_1, (fun x => x • ↑(MonoidHom.range f)) y_1 = (fun x => x • ↑(MonoidHom.range f)) y) } ∈
              Set.range fun x => x • ↑(MonoidHom.range f)) }

Next tactic:
simp only [leftCoset_assoc]
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · rintro ⟨a, rfl⟩
    change h (f a) = g (f a)
    ext ⟨⟨_, ⟨y, rfl⟩⟩⟩
    · rw [g_apply_fromCoset]
      by_cases m : y ∈ f.range
      · rw [h_apply_fromCoset' _ _ _ m, fromCoset_eq_of_mem_range _ m]
        change fromCoset _ = fromCoset ⟨f a • (y • _), _⟩
        simp only [← fromCoset_eq_of_mem_range _ (Subgroup.mul_mem _ ⟨a, rfl⟩ m), smul_smul]
      · rw [h_apply_fromCoset_nin_range f (f a) ⟨_, rfl⟩ _ m]
        simp only [leftCoset_assoc]
    · 

Proof state:
case refine'_1.intro.H.infinity
A B : GroupCat
f : A ⟶ B
a : ↑A
⊢ (h (f a)) ∞ = (g (f a)) ∞

Next tactic:
rw [g_apply_infinity, h_apply_infinity f (f a) ⟨_, rfl⟩]
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · rintro ⟨a, rfl⟩
    change h (f a) = g (f a)
    ext ⟨⟨_, ⟨y, rfl⟩⟩⟩
    · rw [g_apply_fromCoset]
      by_cases m : y ∈ f.range
      · rw [h_apply_fromCoset' _ _ _ m, fromCoset_eq_of_mem_range _ m]
        change fromCoset _ = fromCoset ⟨f a • (y • _), _⟩
        simp only [← fromCoset_eq_of_mem_range _ (Subgroup.mul_mem _ ⟨a, rfl⟩ m), smul_smul]
      · rw [h_apply_fromCoset_nin_range f (f a) ⟨_, rfl⟩ _ m]
        simp only [leftCoset_assoc]
    · rw [g_apply_infinity, h_apply_infinity f (f a) ⟨_, rfl⟩]
  · 

Proof state:
case refine'_2
A B : GroupCat
f : A ⟶ B
b : ↑B
hb : h b = g b
r : b ∉ ↑(MonoidHom.range f)
⊢ False

Next tactic:
have eq1 : (h b) (fromCoset ⟨f.range, 1, one_leftCoset _⟩) =
        fromCoset ⟨f.range, 1, one_leftCoset _⟩ := by
      change ((τ).symm.trans (g b)).trans τ _ = _
      dsimp [tau]
      simp [g_apply_infinity f]
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · rintro ⟨a, rfl⟩
    change h (f a) = g (f a)
    ext ⟨⟨_, ⟨y, rfl⟩⟩⟩
    · rw [g_apply_fromCoset]
      by_cases m : y ∈ f.range
      · rw [h_apply_fromCoset' _ _ _ m, fromCoset_eq_of_mem_range _ m]
        change fromCoset _ = fromCoset ⟨f a • (y • _), _⟩
        simp only [← fromCoset_eq_of_mem_range _ (Subgroup.mul_mem _ ⟨a, rfl⟩ m), smul_smul]
      · rw [h_apply_fromCoset_nin_range f (f a) ⟨_, rfl⟩ _ m]
        simp only [leftCoset_assoc]
    · rw [g_apply_infinity, h_apply_infinity f (f a) ⟨_, rfl⟩]
  · have eq1 : (h b) (fromCoset ⟨f.range, 1, one_leftCoset _⟩) =
        fromCoset ⟨f.range, 1, one_leftCoset _⟩ := by
      

Proof state:
A B : GroupCat
f : A ⟶ B
b : ↑B
hb : h b = g b
r : b ∉ ↑(MonoidHom.range f)
⊢ (h b)
      (fromCoset
        { val := ↑(MonoidHom.range f),
          property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = ↑(MonoidHom.range f)) }) =
    fromCoset
      { val := ↑(MonoidHom.range f),
        property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = ↑(MonoidHom.range f)) }

Next tactic:
change ((τ).symm.trans (g b)).trans τ _ = _
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · rintro ⟨a, rfl⟩
    change h (f a) = g (f a)
    ext ⟨⟨_, ⟨y, rfl⟩⟩⟩
    · rw [g_apply_fromCoset]
      by_cases m : y ∈ f.range
      · rw [h_apply_fromCoset' _ _ _ m, fromCoset_eq_of_mem_range _ m]
        change fromCoset _ = fromCoset ⟨f a • (y • _), _⟩
        simp only [← fromCoset_eq_of_mem_range _ (Subgroup.mul_mem _ ⟨a, rfl⟩ m), smul_smul]
      · rw [h_apply_fromCoset_nin_range f (f a) ⟨_, rfl⟩ _ m]
        simp only [leftCoset_assoc]
    · rw [g_apply_infinity, h_apply_infinity f (f a) ⟨_, rfl⟩]
  · have eq1 : (h b) (fromCoset ⟨f.range, 1, one_leftCoset _⟩) =
        fromCoset ⟨f.range, 1, one_leftCoset _⟩ := by
      change ((τ).symm.trans (g b)).trans τ _ = _
      

Proof state:
A B : GroupCat
f : A ⟶ B
b : ↑B
hb : h b = g b
r : b ∉ ↑(MonoidHom.range f)
⊢ ((τ.symm.trans (g b)).trans τ)
      (fromCoset
        { val := ↑(MonoidHom.range f),
          property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = ↑(MonoidHom.range f)) }) =
    fromCoset
      { val := ↑(MonoidHom.range f),
        property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = ↑(MonoidHom.range f)) }

Next tactic:
dsimp [tau]
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · rintro ⟨a, rfl⟩
    change h (f a) = g (f a)
    ext ⟨⟨_, ⟨y, rfl⟩⟩⟩
    · rw [g_apply_fromCoset]
      by_cases m : y ∈ f.range
      · rw [h_apply_fromCoset' _ _ _ m, fromCoset_eq_of_mem_range _ m]
        change fromCoset _ = fromCoset ⟨f a • (y • _), _⟩
        simp only [← fromCoset_eq_of_mem_range _ (Subgroup.mul_mem _ ⟨a, rfl⟩ m), smul_smul]
      · rw [h_apply_fromCoset_nin_range f (f a) ⟨_, rfl⟩ _ m]
        simp only [leftCoset_assoc]
    · rw [g_apply_infinity, h_apply_infinity f (f a) ⟨_, rfl⟩]
  · have eq1 : (h b) (fromCoset ⟨f.range, 1, one_leftCoset _⟩) =
        fromCoset ⟨f.range, 1, one_leftCoset _⟩ := by
      change ((τ).symm.trans (g b)).trans τ _ = _
      dsimp [tau]
      

Proof state:
A B : GroupCat
f : A ⟶ B
b : ↑B
hb : h b = g b
r : b ∉ ↑(MonoidHom.range f)
⊢ (Equiv.swap
        (fromCoset
          { val := Set.range ⇑f, property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = ↑(MonoidHom.range f)) })
        ∞)
      ((g b)
        ((Equiv.swap
            (fromCoset
              { val := Set.range ⇑f,
                property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = ↑(MonoidHom.range f)) })
            ∞)
          (fromCoset { val := Set.range ⇑f, property := (_ : ∃ y, y • Set.range ⇑f = Set.range ⇑f) }))) =
    fromCoset { val := Set.range ⇑f, property := (_ : ∃ y, y • Set.range ⇑f = Set.range ⇑f) }

Next tactic:
simp [g_apply_infinity f]
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · rintro ⟨a, rfl⟩
    change h (f a) = g (f a)
    ext ⟨⟨_, ⟨y, rfl⟩⟩⟩
    · rw [g_apply_fromCoset]
      by_cases m : y ∈ f.range
      · rw [h_apply_fromCoset' _ _ _ m, fromCoset_eq_of_mem_range _ m]
        change fromCoset _ = fromCoset ⟨f a • (y • _), _⟩
        simp only [← fromCoset_eq_of_mem_range _ (Subgroup.mul_mem _ ⟨a, rfl⟩ m), smul_smul]
      · rw [h_apply_fromCoset_nin_range f (f a) ⟨_, rfl⟩ _ m]
        simp only [leftCoset_assoc]
    · rw [g_apply_infinity, h_apply_infinity f (f a) ⟨_, rfl⟩]
  · have eq1 : (h b) (fromCoset ⟨f.range, 1, one_leftCoset _⟩) =
        fromCoset ⟨f.range, 1, one_leftCoset _⟩ := by
      change ((τ).symm.trans (g b)).trans τ _ = _
      dsimp [tau]
      simp [g_apply_infinity f]
    

Proof state:
case refine'_2
A B : GroupCat
f : A ⟶ B
b : ↑B
hb : h b = g b
r : b ∉ ↑(MonoidHom.range f)
eq1 :
  (h b)
      (fromCoset
        { val := ↑(MonoidHom.range f),
          property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = ↑(MonoidHom.range f)) }) =
    fromCoset
      { val := ↑(MonoidHom.range f),
        property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = ↑(MonoidHom.range f)) }
⊢ False

Next tactic:
have eq2 :
      g b (fromCoset ⟨f.range, 1, one_leftCoset _⟩) = fromCoset ⟨b • ↑f.range, b, rfl⟩ := rfl
theorem agree : f.range = { x | h x = g x } := by
  refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩
  · rintro ⟨a, rfl⟩
    change h (f a) = g (f a)
    ext ⟨⟨_, ⟨y, rfl⟩⟩⟩
    · rw [g_apply_fromCoset]
      by_cases m : y ∈ f.range
      · rw [h_apply_fromCoset' _ _ _ m, fromCoset_eq_of_mem_range _ m]
        change fromCoset _ = fromCoset ⟨f a • (y • _), _⟩
        simp only [← fromCoset_eq_of_mem_range _ (Subgroup.mul_mem _ ⟨a, rfl⟩ m), smul_smul]
      · rw [h_apply_fromCoset_nin_range f (f a) ⟨_, rfl⟩ _ m]
        simp only [leftCoset_assoc]
    · rw [g_apply_infinity, h_apply_infinity f (f a) ⟨_, rfl⟩]
  · have eq1 : (h b) (fromCoset ⟨f.range, 1, one_leftCoset _⟩) =
        fromCoset ⟨f.range, 1, one_leftCoset _⟩ := by
      change ((τ).symm.trans (g b)).trans τ _ = _
      dsimp [tau]
      simp [g_apply_infinity f]
    have eq2 :
      g b (fromCoset ⟨f.range, 1, one_leftCoset _⟩) = fromCoset ⟨b • ↑f.range, b, rfl⟩ := rfl
    

Proof state:
case refine'_2
A B : GroupCat
f : A ⟶ B
b : ↑B
hb : h b = g b
r : b ∉ ↑(MonoidHom.range f)
eq1 :
  (h b)
      (fromCoset
        { val := ↑(MonoidHom.range f),
          property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = ↑(MonoidHom.range f)) }) =
    fromCoset
      { val := ↑(MonoidHom.range f),
        property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = ↑(MonoidHom.range f)) }
eq2 :
  (g b)
      (fromCoset
        { val := ↑(MonoidHom.range f),
          property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = ↑(MonoidHom.range f)) }) =
    fromCoset
      { val := b • ↑(MonoidHom.range f),
        property := (_ : ∃ y, (fun x => x • ↑(MonoidHom.range f)) y = b • ↑(MonoidHom.range f)) }
⊢ False

Next tactic:
exact (fromCoset_ne_of_nin_range _ r).symm (by rw [← eq1, ← eq2, FunLike.congr_fun hb])