mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
x : 𝕜
hx : x ∈ Ioc a (a + p)
⊢ ↑x = ↑↑⟨x, hx⟩

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Category.Grp.LargeColimits

Goal state before:
pp: J : Type u
inst✝¹ : Category.{v, u} J
F : J ⥤ AddCommGrpCat
c : Cocone F
inst✝ : DecidableEq J
x✝¹ : Nonempty (IsColimit c)
hc : IsColimit c
χ : CharacterModule (Colimits.Quot F)
j j' : J
u : j ⟶ j'
x✝ : ↑(F.obj j)
⊢ AddEquiv.ulift.symm (χ ((Quot.ι F j) x✝)) = AddEquiv.ulift.symm (χ ((Quot.ι F j) x✝))

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
x y : 𝕜
hx : x ∈ Ico a (a + p)
hy : y ∈ Ico a (a + p)
h : (equivIco p a) ↑x = (equivIco p a) ↑y
⊢ (equivIco p a).toFun ((equivIco p a).invFun ⟨x, hx⟩) = (equivIco p a).toFun ((equivIco p a).invFun ⟨y, hy⟩)

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
x y : 𝕜
hx : x ∈ Ico a (a + p)
hy : y ∈ Ico a (a + p)
h : (equivIco p a) ↑x = (equivIco p a) ↑y
⊢ (equivIco p a).toFun ((equivIco p a).invFun ⟨x, hx⟩) = (equivIco p a).toFun ((equivIco p a).invFun ⟨y, hy⟩)
Module: Mathlib.Algebra.Category.Grp.LargeColimits

Goal state before:
pp: case h
J : Type u
inst✝¹ : Category.{v, u} J
F : J ⥤ AddCommGrpCat
c : Cocone F
inst✝ : DecidableEq J
x✝ : Nonempty (IsColimit c)
hc : IsColimit c
χ : CharacterModule (Colimits.Quot F)
c' : Cocone F :=
  { pt := of (ULift.{w, 0} (AddCircle 1)),
    ι := { app := fun j ↦ ofHom ((AddEquiv.ulift.symm.toAddMonoidHom.comp χ).comp (Quot.ι F j)), naturality := ⋯ } }
j : J
x : ↑(F.obj j)
⊢ AddEquiv.ulift ((Hom.hom (hc.desc c')) ((ConcreteCategory.hom (c.ι.app j)) x)) = χ ((Quot.ι F j) x)

Tactic:
]

Goal state after:
case h
J : Type u
inst✝¹ : Category.{v, u} J
F : J ⥤ AddCommGrpCat
c : Cocone F
inst✝ : DecidableEq J
x✝ : Nonempty (IsColimit c)
hc : IsColimit c
χ : CharacterModule (Colimits.Quot F)
c' : Cocone F :=
  { pt := of (ULift.{w, 0} (AddCircle 1)),
    ι := { app := fun j ↦ ofHom ((AddEquiv.ulift.symm.toAddMonoidHom.comp χ).comp (Quot.ι F j)), naturality := ⋯ } }
j : J
x : ↑(F.obj j)
⊢ AddEquiv.ulift ((Hom.hom (hc.desc c')) ((ConcreteCategory.hom (c.ι.app j)) x)) = χ ((Quot.ι F j) x)
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
x y : 𝕜
hx : x ∈ Ioc a (a + p)
hy : y ∈ Ioc a (a + p)
h : (equivIoc p a) ↑x = (equivIoc p a) ↑y
⊢ (equivIoc p a).toFun ((equivIoc p a).invFun ⟨x, hx⟩) = (equivIoc p a).toFun ((equivIoc p a).invFun ⟨y, hy⟩)

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
x y : 𝕜
hx : x ∈ Ioc a (a + p)
hy : y ∈ Ioc a (a + p)
h : (equivIoc p a) ↑x = (equivIoc p a) ↑y
⊢ (equivIoc p a).toFun ((equivIoc p a).invFun ⟨x, hx⟩) = (equivIoc p a).toFun ((equivIoc p a).invFun ⟨y, hy⟩)
Module: Mathlib.Algebra.Category.Grp.LargeColimits

Goal state before:
pp: case h
J : Type u
inst✝¹ : Category.{v, u} J
F : J ⥤ AddCommGrpCat
c : Cocone F
inst✝ : DecidableEq J
x✝ : Nonempty (IsColimit c)
hc : IsColimit c
χ : CharacterModule (Colimits.Quot F)
c' : Cocone F :=
  { pt := of (ULift.{w, 0} (AddCircle 1)),
    ι := { app := fun j ↦ ofHom ((AddEquiv.ulift.symm.toAddMonoidHom.comp χ).comp (Quot.ι F j)), naturality := ⋯ } }
j : J
x : ↑(F.obj j)
⊢ AddEquiv.ulift ((ConcreteCategory.hom (c'.ι.app j)) x) = χ ((Quot.ι F j) x)

Tactic:
]

Goal state after:
case h
J : Type u
inst✝¹ : Category.{v, u} J
F : J ⥤ AddCommGrpCat
c : Cocone F
inst✝ : DecidableEq J
x✝ : Nonempty (IsColimit c)
hc : IsColimit c
χ : CharacterModule (Colimits.Quot F)
c' : Cocone F :=
  { pt := of (ULift.{w, 0} (AddCircle 1)),
    ι := { app := fun j ↦ ofHom ((AddEquiv.ulift.symm.toAddMonoidHom.comp χ).comp (Quot.ι F j)), naturality := ⋯ } }
j : J
x : ↑(F.obj j)
⊢ AddEquiv.ulift ((ConcreteCategory.hom (c'.ι.app j)) x) = χ ((Quot.ι F j) x)
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
f : 𝕜 → B
x : AddCircle p
x_ne_a : x ≠ ↑a
x_eq_b : x = ↑↑((equivIco p a) x)
⊢ liftIoc p a f ↑↑((equivIco p a) x) = f ↑((equivIco p a) x)

Tactic:
]

Goal state after:
𝕜 : Type u_1
B : Type u_2
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
f : 𝕜 → B
x : AddCircle p
x_ne_a : x ≠ ↑a
x_eq_b : x = ↑↑((equivIco p a) x)
⊢ liftIoc p a f ↑↑((equivIco p a) x) = f ↑((equivIco p a) x)
Module: Mathlib.Algebra.Category.Grp.LargeColimits

Goal state before:
pp: case h
J : Type u
inst✝¹ : Category.{v, u} J
F : J ⥤ AddCommGrpCat
c : Cocone F
inst✝ : DecidableEq J
x✝ : Nonempty (IsColimit c)
hc : IsColimit c
χ : CharacterModule (Colimits.Quot F)
c' : Cocone F :=
  { pt := of (ULift.{w, 0} (AddCircle 1)),
    ι := { app := fun j ↦ ofHom ((AddEquiv.ulift.symm.toAddMonoidHom.comp χ).comp (Quot.ι F j)), naturality := ⋯ } }
j : J
x : ↑(F.obj j)
⊢ χ ((Quot.ι F j) x) = χ ((Quot.ι F j) x)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
ha : a ∈ Ico 0 p
h0 : 0 ∈ Ico 0 (0 + p)
⊢ a ∈ Ico 0 p

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
ha : a ∈ Ico 0 p
h0 : 0 ∈ Ico 0 (0 + p)
⊢ a ∈ Ico 0 p
Module: Mathlib.CategoryTheory.Monoidal.Internal.Types.Basic

Goal state before:
pp: X✝ Y✝ : MonCat
f : X✝ ⟶ Y✝
⊢ η ≫ ↾⇑(hom f) = η

Tactic:
#adaptation_note /-- Prior to https://github.com/leanprover/lean4/pull/12244
      this argument was provided by the auto_param. -/
      simp +instances only
      cat_disch

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
ha : a ∈ Ico 0 p
h0 : 0 ∈ Ico 0 (0 + p)
ha' : a ∈ Ico 0 (0 + p)
⊢ ↑a = 0 ↔ ↑a = 0

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Internal.Types.Basic

Goal state before:
pp: X✝ Y✝ : MonCat
f : X✝ ⟶ Y✝
⊢ μ ≫ ↾⇑(hom f) = MonoidalCategoryStruct.tensorHom (↾⇑(hom f)) (↾⇑(hom f)) ≫ μ

Tactic:
#adaptation_note /-- Prior to https://github.com/leanprover/lean4/pull/12244
      this argument was provided by the auto_param. -/
      simp +instances only
      cat_disch

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case H
𝕜 : Type u_1
inst✝⁵ : AddCommGroup 𝕜
p : 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
x : AddCircle p
z✝ : 𝕜
hx : ↑z✝ ≠ ↑a
⊢ Filter.map (⇑(equivIco p a) ∘ QuotientAddGroup.mk) (𝓝 z✝) ≤ 𝓝 ((equivIco p a) ↑z✝)

Tactic:
]

Goal state after:
case H
𝕜 : Type u_1
inst✝⁵ : AddCommGroup 𝕜
p : 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
x : AddCircle p
z✝ : 𝕜
hx : ↑z✝ ≠ ↑a
⊢ Filter.map (⇑(equivIco p a) ∘ QuotientAddGroup.mk) (𝓝 z✝) ≤ 𝓝 ((equivIco p a) ↑z✝)
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case H
𝕜 : Type u_1
inst✝⁵ : AddCommGroup 𝕜
p : 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
x : AddCircle p
z✝ : 𝕜
hx : ↑z✝ ≠ ↑a
⊢ Filter.map (⇑(equivIoc p a) ∘ QuotientAddGroup.mk) (𝓝 z✝) ≤ 𝓝 ((equivIoc p a) ↑z✝)

Tactic:
]

Goal state after:
case H
𝕜 : Type u_1
inst✝⁵ : AddCommGroup 𝕜
p : 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
x : AddCircle p
z✝ : 𝕜
hx : ↑z✝ ≠ ↑a
⊢ Filter.map (⇑(equivIoc p a) ∘ QuotientAddGroup.mk) (𝓝 z✝) ≤ 𝓝 ((equivIoc p a) ↑z✝)
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
⊢ (range fun x ↦ ↑↑x) = univ

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
⊢ (range fun x ↦ ↑↑x) = univ
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: case inl
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SeparableSpace α
h : {x | IsTop x} = ∅
s : Set α
s_count : s.Countable
hs : Dense s
this : atTop = generate (Ici '' s)
⊢ (generate (Ici '' s)).IsCountablyGenerated

Tactic:
]

Goal state after:
case inl
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SeparableSpace α
h : {x | IsTop x} = ∅
s : Set α
s_count : s.Countable
hs : Dense s
this : atTop = generate (Ici '' s)
⊢ (generate (Ici '' s)).IsCountablyGenerated
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: case inr
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SeparableSpace α
x : α
hx : x ∈ {x | IsTop x}
⊢ (pure x).IsCountablyGenerated

Tactic:
]

Goal state after:
case inr
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SeparableSpace α
x : α
hx : x ∈ {x | IsTop x}
⊢ (pure x).IsCountablyGenerated
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
inst✝ : Archimedean 𝕜
α : Type u_3
f g : AddCircle p → α
a : 𝕜
h : ∀ x ∈ Ico a (a + p), f ↑x = g ↑x
⊢ EqOn f g (QuotientAddGroup.mk '' Ico a (a + p))

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
inst✝ : Archimedean 𝕜
α : Type u_3
f g : AddCircle p → α
a : 𝕜
h : ∀ x ∈ Ico a (a + p), f ↑x = g ↑x
⊢ EqOn f g (QuotientAddGroup.mk '' Ico a (a + p))
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
inst✝ : Archimedean 𝕜
α : Type u_3
f g : AddCircle p → α
a : 𝕜
h : ∀ x ∈ Ioc a (a + p), f ↑x = g ↑x
⊢ EqOn f g (QuotientAddGroup.mk '' Ioc a (a + p))

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝³ : AddCommGroup 𝕜
p : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsOrderedAddMonoid 𝕜
hp : Fact (0 < p)
inst✝ : Archimedean 𝕜
α : Type u_3
f g : AddCircle p → α
a : 𝕜
h : ∀ x ∈ Ioc a (a + p), f ↑x = g ↑x
⊢ EqOn f g (QuotientAddGroup.mk '' Ioc a (a + p))
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: X : Type u_1
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
Y : Type u_2
inst✝² : ConditionallyCompleteLinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
x : X
hf : LeftOrdContinuous f
⊢ Tendsto f (𝓝[≤] x) (𝓝 (f x))

Tactic:
]

Goal state after:
X : Type u_1
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
Y : Type u_2
inst✝² : ConditionallyCompleteLinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
x : X
hf : LeftOrdContinuous f
⊢ Tendsto f (𝓝[≤] x) (𝓝 (f x))
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
inst✝ : Field 𝕜
p q : 𝕜
hp : p ≠ 0
hq : q ≠ 0
⊢ zmultiples q = zmultiples q

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : Field 𝕜
p q r : 𝕜
m : ℕ
⊢ m • ↑(r / q) = m • ↑(r / q)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : Field 𝕜
p q r : 𝕜
m : ℤ
⊢ m • ↑(r / q) = m • ↑(r / q)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
inst✝³ : Field 𝕜
p q : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
inst✝ : FloorRing 𝕜
n : ℤ
x : AddCircle p
hn : ↑n ≠ 0
⊢ (QuotientAddGroup.mk' (zmultiples p)) ↑((equivIco p 0) x) = x

Tactic:
]

Goal state after:
𝕜 : Type u_1
B : Type u_2
inst✝³ : Field 𝕜
p q : 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
inst✝ : FloorRing 𝕜
n : ℤ
x : AddCircle p
hn : ↑n ≠ 0
⊢ (QuotientAddGroup.mk' (zmultiples p)) ↑((equivIco p 0) x) = x
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < n
⊢ n • ↑(p / ↑n) = 0 ∧ ∀ m < n, 0 < m → m • ↑(p / ↑n) ≠ 0

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < n
⊢ n • ↑(p / ↑n) = 0 ∧ ∀ m < n, 0 < m → m • ↑(p / ↑n) ≠ 0
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < ↑n
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < ↑n
m : ℕ
hn : m < n
h0 : 0 < m
⊢ ¬∃ n_1, n_1 • p = ↑m * (p / ↑n)

Tactic:
]

Goal state after:
case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < ↑n
m : ℕ
hn : m < n
h0 : 0 < m
⊢ ¬∃ n_1, n_1 • p = ↑m * (p / ↑n)
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < ↑n
m : ℕ
hn : m < n
h0 : 0 < m
k : ℕ
hk : n * k = m
⊢ False

Tactic:
]

Goal state after:
case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < ↑n
m : ℕ
hn : m < n
h0 : 0 < m
k : ℕ
hk : n * k = m
⊢ False
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n m : ℕ
hn : 0 < n
⊢ m.gcd n * (addOrderOf ↑(p / ↑n) / (addOrderOf ↑(p / ↑n)).gcd m) = n

pp: case h
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n m : ℕ
hn : 0 < n
⊢ IsOfFinAddOrder ↑(p / ↑n)

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n m : ℕ
hn : 0 < n
⊢ m.gcd n * (addOrderOf ↑(p / ↑n) / (addOrderOf ↑(p / ↑n)).gcd m) = n

case h
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n m : ℕ
hn : 0 < n
⊢ IsOfFinAddOrder ↑(p / ↑n)
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n m : ℕ
hn : 0 < n
⊢ n = n

pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n m : ℕ
hn : 0 < n
⊢ n.gcd m ∣ n

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n m : ℕ
hn : 0 < n
⊢ n.gcd m ∣ n
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case h
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n m : ℕ
hn : 0 < n
⊢ 0 < n

Tactic:
]

Goal state after:
case h
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n m : ℕ
hn : 0 < n
⊢ 0 < n
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
m n : ℕ
hn : 0 < n
h : m.gcd n = 1
⊢ addOrderOf ↑(↑m / ↑n * p) = addOrderOf ↑(↑m / ↑n * p)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
q : ℚ
this : ↑↑q.den ≠ 0
⊢ q.den = q.den

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
u : AddCircle p
n : ℕ
h : 0 < n
⊢ (∃ m < n, ↑(↑m / ↑n * p) = u) → addOrderOf u ∣ n

Tactic:
]

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
u : AddCircle p
n : ℕ
h : 0 < n
⊢ (∃ m < n, ↑(↑m / ↑n * p) = u) → addOrderOf u ∣ n
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case refine_1.h
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < n
m : ℕ
⊢ ?refine_1.h.b * addOrderOf ↑(↑m / ↑n * p) = n

pp: case refine_1.h.b
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < n
m : ℕ
⊢ ℕ

Tactic:
]

Goal state after:
case refine_1.h
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < n
m : ℕ
⊢ ?refine_1.h.b * addOrderOf ↑(↑m / ↑n * p) = n

case refine_1.h.b
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
h : 0 < n
m : ℕ
⊢ ℕ
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
u : AddCircle p
n : ℕ
h : 0 < n
k : 𝕜
hk : ∃ n_1, n_1 • p = n • k
⊢ ∃ m < n, ↑(↑m / ↑n * p) = ↑k

Tactic:
]

Goal state after:
case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
u : AddCircle p
n : ℕ
h : 0 < n
k : 𝕜
hk : ∃ n_1, n_1 • p = n • k
⊢ ∃ m < n, ↑(↑m / ↑n * p) = ↑k
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
u : AddCircle p
n : ℕ
h : 0 < n
k : 𝕜
a : ℤ
ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k
h0 : ↑n ≠ 0
⊢ ↑(↑(a.natMod ↑n) / ↑n * p) = ↑k

Tactic:
]

Goal state after:
case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
u : AddCircle p
n : ℕ
h : 0 < n
k : 𝕜
a : ℤ
ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k
h0 : ↑n ≠ 0
⊢ ↑(↑(a.natMod ↑n) / ↑n * p) = ↑k
Module: Mathlib.CategoryTheory.Monoidal.CommMon_

Goal state before:
pp: C : Type u₁
inst✝¹³ : Category.{v₁, u₁} C
inst✝¹² : MonoidalCategory C
inst✝¹¹ : BraidedCategory C
D : Type u₂
inst✝¹⁰ : Category.{v₂, u₂} D
inst✝⁹ : MonoidalCategory D
inst✝⁸ : BraidedCategory D
E : Type u₃
inst✝⁷ : Category.{v₃, u₃} E
inst✝⁶ : MonoidalCategory E
inst✝⁵ : BraidedCategory E
F F' : C ⥤ D
G : D ⥤ E
inst✝⁴ : F.LaxBraided
inst✝³ : F'.LaxBraided
inst✝² : G.LaxBraided
M : C
inst✝¹ : MonObj M
inst✝ : IsCommMonObj M
⊢ LaxMonoidal.μ F M M ≫ F.map μ = LaxMonoidal.μ F M M ≫ F.map μ

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
u : AddCircle p
n : ℕ
h : 0 < n
k : 𝕜
a : ℤ
ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k
h0 : ↑n ≠ 0
⊢ 0 = 0

pp: case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
u : AddCircle p
n : ℕ
h : 0 < n
k : 𝕜
a : ℤ
ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k
h0 : ↑n ≠ 0
⊢ 0 ≤ a % ↑n

Tactic:
]

Goal state after:
case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
u : AddCircle p
n : ℕ
h : 0 < n
k : 𝕜
a : ℤ
ha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k
h0 : ↑n ≠ 0
⊢ 0 ≤ a % ↑n
Module: Mathlib.CategoryTheory.Monoidal.CommMon_

Goal state before:
pp: C : Type u₁
inst✝¹¹ : Category.{v₁, u₁} C
inst✝¹⁰ : MonoidalCategory C
inst✝⁹ : BraidedCategory C
D : Type u₂
inst✝⁸ : Category.{v₂, u₂} D
inst✝⁷ : MonoidalCategory D
inst✝⁶ : BraidedCategory D
E : Type u₃
inst✝⁵ : Category.{v₃, u₃} E
inst✝⁴ : MonoidalCategory E
inst✝³ : BraidedCategory E
F F' : C ⥤ D
G : D ⥤ E
inst✝² : F.LaxBraided
inst✝¹ : F'.LaxBraided
inst✝ : G.LaxBraided
A : CommMon C
⊢ LaxMonoidal.μ F A.X A.X ≫ F.map μ = LaxMonoidal.μ F A.X A.X ≫ F.map μ

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case e'_2
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
u : AddCircle p
k : 𝕜
h : 0 < addOrderOf ↑k
m : ℕ
hm : m < addOrderOf ↑k
hk : ↑(↑m / ↑(addOrderOf ↑k) * p) = ↑k
⊢ m.gcd (addOrderOf ↑k) * addOrderOf ↑k = m.gcd (addOrderOf ↑k) * addOrderOf ↑k

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case refine_1.refine_1
𝕜 : Type u_1
B : Type u_2
inst✝² : Field 𝕜
p q : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
hn : 0 < n
m₁ m₂ : ↑{m | m < n ∧ m.gcd n = 1}
h : ↑m₁ • ↑(p / ↑n) = ↑m₂ • ↑(p / ↑n)
⊢ ↑m₁ ∈ Iio n

Tactic:
]

Goal state after:
case refine_1.refine_1
𝕜 : Type u_1
B : Type u_2
inst✝² : Field 𝕜
p q : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
hn : 0 < n
m₁ m₂ : ↑{m | m < n ∧ m.gcd n = 1}
h : ↑m₁ • ↑(p / ↑n) = ↑m₂ • ↑(p / ↑n)
⊢ ↑m₁ ∈ Iio n
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: case refine_1.refine_2
𝕜 : Type u_1
B : Type u_2
inst✝² : Field 𝕜
p q : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
hn : 0 < n
m₁ m₂ : ↑{m | m < n ∧ m.gcd n = 1}
h : ↑m₁ • ↑(p / ↑n) = ↑m₂ • ↑(p / ↑n)
⊢ ↑m₂ ∈ Iio n

Tactic:
]

Goal state after:
case refine_1.refine_2
𝕜 : Type u_1
B : Type u_2
inst✝² : Field 𝕜
p q : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
n : ℕ
hn : 0 < n
m₁ m₂ : ↑{m | m < n ∧ m.gcd n = 1}
h : ↑m₁ • ↑(p / ↑n) = ↑m₂ • ↑(p / ↑n)
⊢ ↑m₂ ∈ Iio n
Module: Mathlib.Topology.Instances.AddCircle.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
u : AddCircle p
hu : ¬IsOfFinAddOrder u
⊢ {x | ¬IsOfFinAddOrder x}.Infinite

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝² : Field 𝕜
p : 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
hp : Fact (0 < p)
u : AddCircle p
hu : ¬IsOfFinAddOrder u
⊢ {x | ¬IsOfFinAddOrder x}.Infinite