mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.BigOperators.Group.Finset.Piecewise

Goal state before:
pp: case pos
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → a = x → M
h : a ∈ s
⊢ b a ⋯ = b a ⋯

pp: case pos.h₀
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → a = x → M
h : a ∈ s
⊢ ∀ b_1 ∈ s, b_1 ≠ a → (if h : a = b_1 then b b_1 h else 1) = 1

pp: case pos.h₁
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → a = x → M
h : a ∈ s
⊢ a ∉ s → (if h : a = a then b a h else 1) = 1

Tactic:
]

Goal state after:
case pos.h₀
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → a = x → M
h : a ∈ s
⊢ ∀ b_1 ∈ s, b_1 ≠ a → (if h : a = b_1 then b b_1 h else 1) = 1

case pos.h₁
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → a = x → M
h : a ∈ s
⊢ a ∉ s → (if h : a = a then b a h else 1) = 1
Module: Mathlib.Algebra.BigOperators.Group.Finset.Piecewise

Goal state before:
pp: case pos.h₀
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → a = x → M
h✝ : a ∈ s
b✝ : ι
a✝ : b✝ ∈ s
h : b✝ ≠ a
⊢ 1 = 1

pp: case pos.h₀.hnc
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → a = x → M
h✝ : a ∈ s
b✝ : ι
a✝ : b✝ ∈ s
h : b✝ ≠ a
⊢ ¬a = b✝

Tactic:
]

Goal state after:
case pos.h₀.hnc
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → a = x → M
h✝ : a ∈ s
b✝ : ι
a✝ : b✝ ∈ s
h : b✝ ≠ a
⊢ ¬a = b✝
Module: Mathlib.Algebra.BigOperators.Group.Finset.Piecewise

Goal state before:
pp: case neg
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → a = x → M
h : a ∉ s
⊢ 1 = 1

pp: case neg
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → a = x → M
h : a ∉ s
⊢ ∀ x ∈ s, (if h : a = x then b x h else 1) = 1

Tactic:
]

Goal state after:
case neg
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → a = x → M
h : a ∉ s
⊢ ∀ x ∈ s, (if h : a = x then b x h else 1) = 1
Module: Mathlib.Data.Vector.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
σ : Type u_3
φ : Type u_4
n : ℕ
p✝ : α → Prop
i : Fin n
l : List α
p : l.length = n
⊢ l.length - 1 = n - 1

pp: α : Type u_1
β : Type u_2
σ : Type u_3
φ : Type u_4
n : ℕ
p✝ : α → Prop
i : Fin n
l : List α
p : l.length = n
⊢ ↑i < l.length

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
σ : Type u_3
φ : Type u_4
n : ℕ
p✝ : α → Prop
i : Fin n
l : List α
p : l.length = n
⊢ l.length - 1 = n - 1

α : Type u_1
β : Type u_2
σ : Type u_3
φ : Type u_4
n : ℕ
p✝ : α → Prop
i : Fin n
l : List α
p : l.length = n
⊢ ↑i < l.length
Module: Mathlib.Data.Vector.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
σ : Type u_3
φ : Type u_4
n : ℕ
p✝ : α → Prop
i : Fin n
l : List α
p : l.length = n
⊢ n - 1 = n - 1

Tactic:
]

Goal state after:
Module: Mathlib.Data.Vector.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
σ : Type u_3
φ : Type u_4
n : ℕ
p✝ : α → Prop
i : Fin n
l : List α
p : l.length = n
⊢ ↑i < n

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
σ : Type u_3
φ : Type u_4
n : ℕ
p✝ : α → Prop
i : Fin n
l : List α
p : l.length = n
⊢ ↑i < n
Module: Mathlib.Algebra.BigOperators.Group.Finset.Piecewise

Goal state before:
pp: case pos
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → x = a → M
h : a ∈ s
⊢ b a ⋯ = b a ⋯

pp: case pos.h₀
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → x = a → M
h : a ∈ s
⊢ ∀ b_1 ∈ s, b_1 ≠ a → (if h : b_1 = a then b b_1 h else 1) = 1

pp: case pos.h₁
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → x = a → M
h : a ∈ s
⊢ a ∉ s → (if h : a = a then b a h else 1) = 1

Tactic:
]

Goal state after:
case pos.h₀
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → x = a → M
h : a ∈ s
⊢ ∀ b_1 ∈ s, b_1 ≠ a → (if h : b_1 = a then b b_1 h else 1) = 1

case pos.h₁
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → x = a → M
h : a ∈ s
⊢ a ∉ s → (if h : a = a then b a h else 1) = 1
Module: Mathlib.Algebra.BigOperators.Group.Finset.Piecewise

Goal state before:
pp: case pos.h₀
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → x = a → M
h✝ : a ∈ s
b✝ : ι
a✝ : b✝ ∈ s
h : b✝ ≠ a
⊢ 1 = 1

pp: case pos.h₀.hnc
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → x = a → M
h✝ : a ∈ s
b✝ : ι
a✝ : b✝ ∈ s
h : b✝ ≠ a
⊢ ¬b✝ = a

Tactic:
]

Goal state after:
case pos.h₀.hnc
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → x = a → M
h✝ : a ∈ s
b✝ : ι
a✝ : b✝ ∈ s
h : b✝ ≠ a
⊢ ¬b✝ = a
Module: Mathlib.Algebra.BigOperators.Group.Finset.Piecewise

Goal state before:
pp: case neg
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → x = a → M
h : a ∉ s
⊢ 1 = 1

pp: case neg
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → x = a → M
h : a ∉ s
⊢ ∀ x ∈ s, (if h : x = a then b x h else 1) = 1

Tactic:
]

Goal state after:
case neg
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
a : ι
b : (x : ι) → x = a → M
h : a ∉ s
⊢ ∀ x ∈ s, (if h : x = a then b x h else 1) = 1
Module: Mathlib.Algebra.BigOperators.Group.Finset.Piecewise

Goal state before:
pp: ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s t : Finset ι
f g : ι → M
⊢ (∏ x ∈ s ∩ t, f x) * ∏ x ∈ s \ t, g x = (∏ x ∈ s ∩ t, f x) * ∏ x ∈ s \ t, g x

Tactic:
]

Goal state after:
Module: Mathlib.Control.Traversable.Basic

Goal state before:
pp: F : Type u → Type v
inst✝³ : Applicative F
G : Type u → Type w
inst✝² : Applicative G
η : ApplicativeTransformation F G
inst✝¹ : LawfulApplicative F
inst✝ : LawfulApplicative G
α β : Type u
x : α → β
y : F α
⊢ x <$> (fun {α} ↦ η.app α) y = x <$> (fun {α} ↦ η.app α) y

Tactic:
]

Goal state after:
Module: Mathlib.Data.Fintype.Sum

Goal state before:
pp: case a
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
k : β
b : α → β
⊢ ∀ x ∈ univ, b ↑x ∈ (image b univ).erase k

Tactic:
]

Goal state after:
case a
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
k : β
b : α → β
⊢ ∀ x ∈ univ, b ↑x ∈ (image b univ).erase k
Module: Mathlib.Data.Fintype.Sum

Goal state before:
pp: case a
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
k : β
b : α → β
i : β
hi : i ∈ (image b univ).erase k
⊢ ∃ a ∈ univ, b ↑a = i

Tactic:
]

Goal state after:
case a
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
k : β
b : α → β
i : β
hi : i ∈ (image b univ).erase k
⊢ ∃ a ∈ univ, b ↑a = i
Module: Mathlib.Algebra.BigOperators.Group.Finset.Piecewise

Goal state before:
pp: ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
i : ι
h : i ∈ s
f : ι → M
⊢ (∏ x ∈ s \ {i}, f x) * f i = (∏ x ∈ s \ {i}, f x) * f i

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.BigOperators.Group.Finset.Piecewise

Goal state before:
pp: ι : Type u_1
M : Type u_3
inst✝ : CommMonoid M
f : ι → M
a : ι
s : Finset ι
ha : a ∈ s
⊢ f a ∣ f a * ∏ x ∈ s \ {a}, f x

Tactic:
]

Goal state after:
ι : Type u_1
M : Type u_3
inst✝ : CommMonoid M
f : ι → M
a : ι
s : Finset ι
ha : a ∈ s
⊢ f a ∣ f a * ∏ x ∈ s \ {a}, f x
Module: Mathlib.Data.Fintype.Sum

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
t : Finset β
hαt : Fintype.card α = #t
f : α → β
hfst : image f ∅ ⊆ t
hfs : Set.InjOn f ↑∅
⊢ Fintype.card α = #t

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
t : Finset β
hαt : Fintype.card α = #t
f : α → β
hfst : image f ∅ ⊆ t
hfs : Set.InjOn f ↑∅
⊢ Fintype.card α = #t
Module: Mathlib.Algebra.BigOperators.Group.Finset.Piecewise

Goal state before:
pp: ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
i : ι
h : i ∈ s
f : ι → M
b : M
⊢ (∏ x ∈ s ∩ {i}, b) * ∏ x ∈ s \ {i}, f x = b * ∏ x ∈ s \ {i}, f x

Tactic:
]

Goal state after:
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : DecidableEq ι
s : Finset ι
i : ι
h : i ∈ s
f : ι → M
b : M
⊢ (∏ x ∈ s ∩ {i}, b) * ∏ x ∈ s \ {i}, f x = b * ∏ x ∈ s \ {i}, f x
Module: Mathlib.Algebra.BigOperators.Group.Finset.Piecewise

Goal state before:
pp: case pos
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
s : Finset ι
p : ι → Prop
inst✝ : DecidablePred p
h : ∀ i ∈ s, ∀ j ∈ s, p i → p j → i = j
a : M
i : ι
hi : i ∈ s
hpi : p i
⊢ a = a

pp: case pos
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
s : Finset ι
p : ι → Prop
inst✝ : DecidablePred p
h : ∀ i ∈ s, ∀ j ∈ s, p i → p j → i = j
a : M
i : ι
hi : i ∈ s
hpi : p i
⊢ ∀ b ∈ s, b ≠ i → (if p b then a else 1) = 1

Tactic:
]

Goal state after:
case pos
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
s : Finset ι
p : ι → Prop
inst✝ : DecidablePred p
h : ∀ i ∈ s, ∀ j ∈ s, p i → p j → i = j
a : M
i : ι
hi : i ∈ s
hpi : p i
⊢ ∀ b ∈ s, b ≠ i → (if p b then a else 1) = 1
Module: Mathlib.Algebra.BigOperators.Group.Finset.Piecewise

Goal state before:
pp: case neg
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
s : Finset ι
p : ι → Prop
inst✝ : DecidablePred p
h✝ : ∀ i ∈ s, ∀ j ∈ s, p i → p j → i = j
a : M
h : ∀ i ∈ s, ¬p i
⊢ 1 = 1

pp: case neg
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
s : Finset ι
p : ι → Prop
inst✝ : DecidablePred p
h✝ : ∀ i ∈ s, ∀ j ∈ s, p i → p j → i = j
a : M
h : ∀ i ∈ s, ¬p i
⊢ ∀ x ∈ s, (if p x then a else 1) = 1

Tactic:
]

Goal state after:
case neg
ι : Type u_1
M : Type u_3
inst✝¹ : CommMonoid M
s : Finset ι
p : ι → Prop
inst✝ : DecidablePred p
h✝ : ∀ i ∈ s, ∀ j ∈ s, p i → p j → i = j
a : M
h : ∀ i ∈ s, ¬p i
⊢ ∀ x ∈ s, (if p x then a else 1) = 1
Module: Mathlib.Algebra.BigOperators.Group.Finset.Piecewise

Goal state before:
pp: ι : Type u_1
M : Type u_3
s : Finset ι
inst✝¹ : CommMonoid M
inst✝ : IsRightCancelMul M
f g : ι → M
i : ι
hi : i ∈ s
hfg : ∀ j ∈ s, j ≠ i → f j = g j
⊢ f i = g i ↔ f i = g i

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.BigOperators.Group.Finset.Piecewise

Goal state before:
pp: ι : Type u_1
M : Type u_3
inst✝² : CommMonoid M
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
i : ι
f : (j : ι) → i = j → M
⊢ f i ⋯ = f i ⋯

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.BigOperators.Group.Finset.Piecewise

Goal state before:
pp: ι : Type u_1
M : Type u_3
inst✝² : CommMonoid M
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
i : ι
f : (j : ι) → j = i → M
⊢ f i ⋯ = f i ⋯

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.BigOperators.Group.Finset.Piecewise

Goal state before:
pp: ι : Type u_1
M : Type u_3
inst✝² : CommMonoid M
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
i : ι
f : ι → M
⊢ f i = f i

Tactic:
]

Goal state after:
Module: Mathlib.Control.Applicative

Goal state before:
pp: F : Type u_1 → Type u_2
AF : Applicative F
inst✝ : LawfulApplicative F
α✝ β✝ : Type u_1
f : F (α✝ → β✝)
x : F α✝
⊢ f <*> x = f <*> x

Tactic:
]

Goal state after:
Module: Mathlib.Control.Applicative

Goal state before:
pp: F : Type u → Type w
G : Type v → Type u
inst✝⁷ : Applicative F
inst✝⁶ : Applicative G
inst✝⁵ : LawfulApplicative F
inst✝⁴ : LawfulApplicative G
α β γ : Type v
f : Type u → Type w
g : Type v → Type u
inst✝³ : Applicative f
inst✝² : Applicative g
inst✝¹ : CommApplicative f
inst✝ : CommApplicative g
α✝ β✝ : Type v
x✝¹ : Comp f g α✝
x✝ : Comp f g β✝
⊢ mk ((flip fun x1 x2 ↦ x1 <*> x2) <$> x✝ <*> mk ((fun x ↦ Prod.mk <$> x) <$> x✝¹)) =
    mk ((fun x1 x2 ↦ x1 <*> x2) <$> mk ((fun x ↦ (fun b a ↦ (a, b)) <$> x) <$> x✝) <*> x✝¹)

Tactic:
]

Goal state after:
F : Type u → Type w
G : Type v → Type u
inst✝⁷ : Applicative F
inst✝⁶ : Applicative G
inst✝⁵ : LawfulApplicative F
inst✝⁴ : LawfulApplicative G
α β γ : Type v
f : Type u → Type w
g : Type v → Type u
inst✝³ : Applicative f
inst✝² : Applicative g
inst✝¹ : CommApplicative f
inst✝ : CommApplicative g
α✝ β✝ : Type v
x✝¹ : Comp f g α✝
x✝ : Comp f g β✝
⊢ mk ((flip fun x1 x2 ↦ x1 <*> x2) <$> x✝ <*> mk ((fun x ↦ Prod.mk <$> x) <$> x✝¹)) =
    mk ((fun x1 x2 ↦ x1 <*> x2) <$> mk ((fun x ↦ (fun b a ↦ (a, b)) <$> x) <$> x✝) <*> x✝¹)
Module: Mathlib.Control.Applicative

Goal state before:
pp: case e_a.e_a
F : Type u → Type w
G : Type v → Type u
inst✝⁷ : Applicative F
inst✝⁶ : Applicative G
inst✝⁵ : LawfulApplicative F
inst✝⁴ : LawfulApplicative G
α β γ : Type v
f : Type u → Type w
g : Type v → Type u
inst✝³ : Applicative f
inst✝² : Applicative g
inst✝¹ : CommApplicative f
inst✝ : CommApplicative g
α✝ β✝ : Type v
x✝¹ : Comp f g α✝
x✝ : Comp f g β✝
x : g β✝
y : g α✝
⊢ flip Prod.mk <$> x <*> y = (fun b a ↦ (a, b)) <$> x <*> y

Tactic:
]

Goal state after:
case e_a.e_a
F : Type u → Type w
G : Type v → Type u
inst✝⁷ : Applicative F
inst✝⁶ : Applicative G
inst✝⁵ : LawfulApplicative F
inst✝⁴ : LawfulApplicative G
α β γ : Type v
f : Type u → Type w
g : Type v → Type u
inst✝³ : Applicative f
inst✝² : Applicative g
inst✝¹ : CommApplicative f
inst✝ : CommApplicative g
α✝ β✝ : Type v
x✝¹ : Comp f g α✝
x✝ : Comp f g β✝
x : g β✝
y : g α✝
⊢ flip Prod.mk <$> x <*> y = (fun b a ↦ (a, b)) <$> x <*> y
Module: Mathlib.Data.Fintype.Sum

Goal state before:
pp: case h.inr
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
t : Finset β
hαt : Fintype.card α = #t
a : α
s : Finset α
has : a ∉ s
H : ∀ {f : α → β}, image f s ⊆ t → Set.InjOn f ↑s → ∃ g, ∀ i ∈ s, ↑(g i) = f i
f : α → β
hfst : image f (insert a s) ⊆ t
hfs : Set.InjOn f ↑(insert a s)
hfst' : image f s ⊆ t
hfs' : Set.InjOn f ↑s
g' : α ≃ ↥t
hg' : ∀ i ∈ s, ↑(g' i) = f i
hfat : f a ∈ t
i : α
hi : i ∈ s
⊢ f i = f i

pp: case h.inr.a
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
t : Finset β
hαt : Fintype.card α = #t
a : α
s : Finset α
has : a ∉ s
H : ∀ {f : α → β}, image f s ⊆ t → Set.InjOn f ↑s → ∃ g, ∀ i ∈ s, ↑(g i) = f i
f : α → β
hfst : image f (insert a s) ⊆ t
hfs : Set.InjOn f ↑(insert a s)
hfst' : image f s ⊆ t
hfs' : Set.InjOn f ↑s
g' : α ≃ ↥t
hg' : ∀ i ∈ s, ↑(g' i) = f i
hfat : f a ∈ t
i : α
hi : i ∈ s
⊢ g' i ≠ ⟨f a, hfat⟩

pp: case h.inr.a
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
t : Finset β
hαt : Fintype.card α = #t
a : α
s : Finset α
has : a ∉ s
H : ∀ {f : α → β}, image f s ⊆ t → Set.InjOn f ↑s → ∃ g, ∀ i ∈ s, ↑(g i) = f i
f : α → β
hfst : image f (insert a s) ⊆ t
hfs : Set.InjOn f ↑(insert a s)
hfst' : image f s ⊆ t
hfs' : Set.InjOn f ↑s
g' : α ≃ ↥t
hg' : ∀ i ∈ s, ↑(g' i) = f i
hfat : f a ∈ t
i : α
hi : i ∈ s
⊢ g' i ≠ g' a

Tactic:
]

Goal state after:
case h.inr.a
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
t : Finset β
hαt : Fintype.card α = #t
a : α
s : Finset α
has : a ∉ s
H : ∀ {f : α → β}, image f s ⊆ t → Set.InjOn f ↑s → ∃ g, ∀ i ∈ s, ↑(g i) = f i
f : α → β
hfst : image f (insert a s) ⊆ t
hfs : Set.InjOn f ↑(insert a s)
hfst' : image f s ⊆ t
hfs' : Set.InjOn f ↑s
g' : α ≃ ↥t
hg' : ∀ i ∈ s, ↑(g' i) = f i
hfat : f a ∈ t
i : α
hi : i ∈ s
⊢ g' i ≠ ⟨f a, hfat⟩

case h.inr.a
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
t : Finset β
hαt : Fintype.card α = #t
a : α
s : Finset α
has : a ∉ s
H : ∀ {f : α → β}, image f s ⊆ t → Set.InjOn f ↑s → ∃ g, ∀ i ∈ s, ↑(g i) = f i
f : α → β
hfst : image f (insert a s) ⊆ t
hfs : Set.InjOn f ↑(insert a s)
hfst' : image f s ⊆ t
hfs' : Set.InjOn f ↑s
g' : α ≃ ↥t
hg' : ∀ i ∈ s, ↑(g' i) = f i
hfat : f a ∈ t
i : α
hi : i ∈ s
⊢ g' i ≠ g' a
Module: Mathlib.Data.Fintype.Sum

Goal state before:
pp: α : Type u_1
p q : α → Prop
inst✝² : Fintype { x // p x }
inst✝¹ : Fintype { x // q x }
inst✝ : Fintype { x // p x ∨ q x }
⊢ card { x // p x } + card { x // q x } = card { x // p x } + card { x // q x }

Tactic:
]

Goal state after:
Module: Mathlib.Data.Fintype.Powerset

Goal state before:
pp: α : Type u_1
inst✝ : Fintype α
s : Finset α
⊢ s = univ ↔ s = univ

Tactic:
]

Goal state after:
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
n : ℕ
v : List α
hv : v.length = n
w : List α
hw : w.length = n
h : ∀ (m : Fin n), get ⟨v, hv⟩ m = get ⟨w, hw⟩ m
⊢ n = n

Tactic:
]

Goal state after:
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
n : ℕ
a : α
v : Vector α n.succ
v' : Vector α n
h : v.head = a ∧ v.tail = v'
⊢ a ::ᵥ v' = a ::ᵥ v'

Tactic:
]

Goal state after:
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
n : ℕ
a : α
v : Vector α n.succ
v' : Vector α n
⊢ ¬v.head = a ∨ ¬v.tail = v' ↔ v.head ≠ a ∨ v.tail ≠ v'

Tactic:
]

Goal state after:
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
f : Fin 0 → α
⊢ ↑⟨[], ⋯⟩ = []

Tactic:
]

Goal state after:
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
n : ℕ
f : Fin (n + 1) → α
⊢ (f 0 :: List.ofFn fun i ↦ f i.succ) = f 0 :: List.ofFn fun i ↦ f i.succ

Tactic:
]

Goal state after:
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
n : ℕ
β : Type u_6
v : Vector α (n + 1)
f : α → β
a : α
v' : Vector α n
h : v = a ::ᵥ v'
⊢ f a = f a

Tactic:
]

Goal state after:
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
n : ℕ
β : Type u_6
v : Vector α (n + 1)
f : α → β
a : α
v' : Vector α n
h : v = a ::ᵥ v'
⊢ map f v' = map f v'

Tactic:
]

Goal state after:
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
σ : Type u_4
φ : Type u_5
m n : ℕ
p : α → Prop
f : (a : α) → p a → β
v : Vector α (n + 1)
hp : ∀ (x : α), x ∈ v.toList → p x
⊢ v.head = v.head ∨ v.head ∈ v.tail.toList

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
σ : Type u_4
φ : Type u_5
m n : ℕ
p : α → Prop
f : (a : α) → p a → β
v : Vector α (n + 1)
hp : ∀ (x : α), x ∈ v.toList → p x
⊢ v.head = v.head ∨ v.head ∈ v.tail.toList
Module: Mathlib.Data.Sym.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
n n' m : ℕ
s✝ : Sym α n
a✝ b a : α
s : Sym α n
⊢ n + 1 = n.succ

Tactic:
]

Goal state after:
Module: Mathlib.Data.Countable.Basic

Goal state before:
pp: α : Sort u
⊢ IsEmpty (α ↪ ℕ) ↔ IsEmpty (α ↪ ℕ)

Tactic:
]

Goal state after:
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
σ : Type u_4
φ : Type u_5
m n : ℕ
p : α → Prop
f : (a : α) → p a → β
v : Vector α (n + 1)
hp : ∀ (x : α), x ∈ v.toList → p x
x : α
hx : x ∈ v.tail.toList
⊢ x = v.head ∨ x ∈ v.tail.toList

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
σ : Type u_4
φ : Type u_5
m n : ℕ
p : α → Prop
f : (a : α) → p a → β
v : Vector α (n + 1)
hp : ∀ (x : α), x ∈ v.toList → p x
x : α
hx : x ∈ v.tail.toList
⊢ x = v.head ∨ x ∈ v.tail.toList
Module: Mathlib.Data.Sym.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
n n' m✝ : ℕ
s : Sym α n
a✝ b : α
inst✝ : DecidableEq α
m : Multiset α
hc : m.card = n + 1
a : α
h : a ∈ m
⊢ n = n

Tactic:
]

Goal state after:
Module: Mathlib.Data.Sym.Basic

Goal state before:
pp: α : Type u_1
n : ℕ
a : α
⊢ Multiset.replicate n a = Multiset.replicate n a

Tactic:
]

Goal state after:
Module: Mathlib.Data.Sym.Basic

Goal state before:
pp: α : Type u_1
n : ℕ
s : Sym α n
a : α
⊢ ((↑s).card = n ∧ ∀ b ∈ ↑s, b = a) ↔ ∀ b ∈ s, b = a

Tactic:
]

Goal state after:
α : Type u_1
n : ℕ
s : Sym α n
a : α
⊢ ((↑s).card = n ∧ ∀ b ∈ ↑s, b = a) ↔ ∀ b ∈ s, b = a
Module: Mathlib.Data.Sym.Basic

Goal state before:
pp: α : Type u_1
n : ℕ
s : Sym α (n + 1)
a : α
h✝ : a ∈ s
m : Multiset α
h : n = m.card
⊢ m.card = n

Tactic:
]

Goal state after:
α : Type u_1
n : ℕ
s : Sym α (n + 1)
a : α
h✝ : a ∈ s
m : Multiset α
h : n = m.card
⊢ m.card = n
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: case nil
α : Type u_1
n i : ℕ
ih : i < n - 1
h✝ : [].length = n
h : 0 = n
⊢ (tail ⟨[], h✝⟩).get ⟨i, ih⟩ = get ⟨[], h✝⟩ ⟨i + 1, ⋯⟩

Tactic:
]

Goal state after:
case nil
α : Type u_1
n i : ℕ
ih : i < n - 1
h✝ : [].length = n
h : 0 = n
⊢ (tail ⟨[], h✝⟩).get ⟨i, ih⟩ = get ⟨[], h✝⟩ ⟨i + 1, ⋯⟩
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: case nil
α : Type u_1
n i : ℕ
ih✝ : i < n - 1
ih : i < 0 - 1
h✝ : [].length = n
h : 0 = n
⊢ (tail ⟨[], h✝⟩).get ⟨i, ih✝⟩ = get ⟨[], h✝⟩ ⟨i + 1, ⋯⟩

Tactic:
]

Goal state after:
case nil
α : Type u_1
n i : ℕ
ih✝ : i < n - 1
ih : i < 0 - 1
h✝ : [].length = n
h : 0 = n
⊢ (tail ⟨[], h✝⟩).get ⟨i, ih✝⟩ = get ⟨[], h✝⟩ ⟨i + 1, ⋯⟩
Module: Mathlib.Data.Sym.Basic

Goal state before:
pp: case succ
α : Type u_1
β : Type u_2
n n' m : ℕ
s✝ : Sym α n
a b✝ : α
inst✝ : Subsingleton α
n✝ : ℕ
s s' : Sym α (n✝ + 1)
b : α
⊢ replicate (n✝ + 1) b = replicate (n✝ + 1) b

Tactic:
]

Goal state after:
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
n : ℕ
f : Fin n.succ → α
i : Fin (n.succ - 1)
⊢ f ⟨↑i + 1, ⋯⟩ = f i.succ

Tactic:
]

Goal state after:
α : Type u_1
n : ℕ
f : Fin n.succ → α
i : Fin (n.succ - 1)
⊢ f ⟨↑i + 1, ⋯⟩ = f i.succ
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
v : Vector α 1
⊢ (v.head ::ᵥ v.tail).toList = [(v.head ::ᵥ v.tail).head]

Tactic:
]

Goal state after:
α : Type u_1
v : Vector α 1
⊢ (v.head ::ᵥ v.tail).toList = [(v.head ::ᵥ v.tail).head]
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
n : ℕ
v : Vector α n
⊢ v.toList.Nodup ↔ Function.Injective v.get

Tactic:
obtain ⟨l, rfl⟩ := v
  exact List.nodup_iff_injective_get

Goal state after:
Module: Mathlib.Data.Sym.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
n n' m : ℕ
s : Sym α n
a b : α
e : α ≃ β
x : Sym α n
⊢ x = x

Tactic:
]

Goal state after:
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
n : ℕ
f : Fin n.succ → α
⊢ f 0 = f 0

Tactic:
]

Goal state after:
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
n : ℕ
a : α
v : Vector α n
i : Fin n
⊢ v.get i = v.get i

Tactic:
]

Goal state after:
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
n : ℕ
v : Vector α (n + 1)
⊢ v.reverse.toList.get (Fin.cast ⋯ 0) = v.toList.get (Fin.cast ⋯ (Fin.last n))

Tactic:
]

Goal state after:
α : Type u_1
n : ℕ
v : Vector α (n + 1)
⊢ v.reverse.toList.get (Fin.cast ⋯ 0) = v.toList.get (Fin.cast ⋯ (Fin.last n))
Module: Mathlib.Data.Sym.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
n n' m : ℕ
s : Sym α n
a b : α
e : α ≃ β
x : Sym β n
⊢ x = x

Tactic:
]

Goal state after:
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
β✝ : Type u_2
γ : Type u_3
σ : Type u_4
φ : Type u_5
m n : ℕ
β : Type u_6
f : β → α → β
b : β
v : Vector α n
⊢ n + 1 = n + 1

Tactic:
]

Goal state after:
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
n : ℕ
β : Type u_6
f : β → α → β
b : β
v : Vector α n
x : α
⊢ (scanl f b (x ::ᵥ v)).toList = (b ::ᵥ scanl f (f b x) v).toList

Tactic:
simp [scanl]

Goal state after:
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: α : Type u_1
β : Type u_6
f : β → α → β
b : β
v : Vector α 1
⊢ scanl f b (v.head ::ᵥ v.tail) = b ::ᵥ f b (v.head ::ᵥ v.tail).head ::ᵥ nil

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_6
f : β → α → β
b : β
v : Vector α 1
⊢ scanl f b (v.head ::ᵥ v.tail) = b ::ᵥ f b (v.head ::ᵥ v.tail).head ::ᵥ nil
Module: Mathlib.Data.Vector.Basic

Goal state before:
pp: case succ
α : Type u_1
β : Type u_6
f : β → α → β
b : β
n✝ : ℕ
v : Vector α (n✝ + 1)
⊢ (scanl f b (v.head ::ᵥ v.tail)).head = b

Tactic:
]

Goal state after:
case succ
α : Type u_1
β : Type u_6
f : β → α → β
b : β
n✝ : ℕ
v : Vector α (n✝ + 1)
⊢ (scanl f b (v.head ::ᵥ v.tail)).head = b
Module: Mathlib.Data.Sym.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
n n' m : ℕ
s✝ : Sym α n
a b : α
s : Sym α n
s' : Sym α n'
⊢ n + n' = n + n'

Tactic:
]

Goal state after: