mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Combinatorics.Colex

Goal state before:
pp: case inr.inl
α : Type u_1
inst✝ : LinearOrder α
𝒜₁ 𝒜₂ : Finset (Finset α)
r : ℕ
h₁ : IsInitSeg 𝒜₁ r
h₂ : IsInitSeg 𝒜₂ r
h : (𝒜₁ \ 𝒜₂).Nonempty ∧ (𝒜₂ \ 𝒜₁).Nonempty
s : Finset α
hs : s ∈ 𝒜₁ ∧ s ∉ 𝒜₂
t : Finset α
ht : t ∈ 𝒜₂ ∧ t ∉ 𝒜₁
hst : toColex s = toColex t
⊢ False

pp: case inr.inr
α : Type u_1
inst✝ : LinearOrder α
𝒜₁ 𝒜₂ : Finset (Finset α)
r : ℕ
h₁ : IsInitSeg 𝒜₁ r
h₂ : IsInitSeg 𝒜₂ r
h : (𝒜₁ \ 𝒜₂).Nonempty ∧ (𝒜₂ \ 𝒜₁).Nonempty
s : Finset α
hs : s ∈ 𝒜₁ ∧ s ∉ 𝒜₂
t : Finset α
ht : t ∈ 𝒜₂ ∧ t ∉ 𝒜₁
hts : toColex t < toColex s
⊢ False

Tactic:
·

Goal state after:
case inr.inl
α : Type u_1
inst✝ : LinearOrder α
𝒜₁ 𝒜₂ : Finset (Finset α)
r : ℕ
h₁ : IsInitSeg 𝒜₁ r
h₂ : IsInitSeg 𝒜₂ r
h : (𝒜₁ \ 𝒜₂).Nonempty ∧ (𝒜₂ \ 𝒜₁).Nonempty
s : Finset α
hs : s ∈ 𝒜₁ ∧ s ∉ 𝒜₂
t : Finset α
ht : t ∈ 𝒜₂ ∧ t ∉ 𝒜₁
hst : toColex s = toColex t
⊢ False
Module: Mathlib.LinearAlgebra.CrossProduct

Goal state before:
pp: case «0»
R : Type u_1
inst✝ : CommRing R
u v w : Fin 3 → R
⊢ ![![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 * w 2 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 * w 1,
        ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 * w 0 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 * w 2,
        ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 * w 1 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 * w 0]
      ((fun i ↦ i) ⟨0, ⋯⟩) =
    ((u 0 * w 0 + u 1 * w 1 + u 2 * w 2) • v - (v 0 * w 0 + v 1 * w 1 + v 2 * w 2) • u) ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
·

Goal state after:
case «0»
R : Type u_1
inst✝ : CommRing R
u v w : Fin 3 → R
⊢ ![![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 * w 2 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 * w 1,
        ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 * w 0 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 * w 2,
        ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 * w 1 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 * w 0]
      ((fun i ↦ i) ⟨0, ⋯⟩) =
    ((u 0 * w 0 + u 1 * w 1 + u 2 * w 2) • v - (v 0 * w 0 + v 1 * w 1 + v 2 * w 2) • u) ((fun i ↦ i) ⟨0, ⋯⟩)
Module: Mathlib.Combinatorics.Colex

Goal state before:
pp: case inr.inr
α : Type u_1
inst✝ : LinearOrder α
𝒜₁ 𝒜₂ : Finset (Finset α)
r : ℕ
h₁ : IsInitSeg 𝒜₁ r
h₂ : IsInitSeg 𝒜₂ r
h : (𝒜₁ \ 𝒜₂).Nonempty ∧ (𝒜₂ \ 𝒜₁).Nonempty
s : Finset α
hs : s ∈ 𝒜₁ ∧ s ∉ 𝒜₂
t : Finset α
ht : t ∈ 𝒜₂ ∧ t ∉ 𝒜₁
hts : toColex t < toColex s
⊢ False

Tactic:
·

Goal state after:
case inr.inr
α : Type u_1
inst✝ : LinearOrder α
𝒜₁ 𝒜₂ : Finset (Finset α)
r : ℕ
h₁ : IsInitSeg 𝒜₁ r
h₂ : IsInitSeg 𝒜₂ r
h : (𝒜₁ \ 𝒜₂).Nonempty ∧ (𝒜₂ \ 𝒜₁).Nonempty
s : Finset α
hs : s ∈ 𝒜₁ ∧ s ∉ 𝒜₂
t : Finset α
ht : t ∈ 𝒜₂ ∧ t ∉ 𝒜₁
hts : toColex t < toColex s
⊢ False
Module: Mathlib.LinearAlgebra.CrossProduct

Goal state before:
pp: case «1»
R : Type u_1
inst✝ : CommRing R
u v w : Fin 3 → R
⊢ ![![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 * w 2 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 * w 1,
        ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 * w 0 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 * w 2,
        ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 * w 1 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 * w 0]
      ((fun i ↦ i) ⟨1, ⋯⟩) =
    ((u 0 * w 0 + u 1 * w 1 + u 2 * w 2) • v - (v 0 * w 0 + v 1 * w 1 + v 2 * w 2) • u) ((fun i ↦ i) ⟨1, ⋯⟩)

Tactic:
·

Goal state after:
case «1»
R : Type u_1
inst✝ : CommRing R
u v w : Fin 3 → R
⊢ ![![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 * w 2 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 * w 1,
        ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 * w 0 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 * w 2,
        ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 * w 1 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 * w 0]
      ((fun i ↦ i) ⟨1, ⋯⟩) =
    ((u 0 * w 0 + u 1 * w 1 + u 2 * w 2) • v - (v 0 * w 0 + v 1 * w 1 + v 2 * w 2) • u) ((fun i ↦ i) ⟨1, ⋯⟩)
Module: Mathlib.Combinatorics.Colex

Goal state before:
pp: case refine_1
α : Type u_1
inst✝¹ : LinearOrder α
𝒜 : Finset (Finset α)
r : ℕ
inst✝ : Fintype α
h𝒜 : IsInitSeg 𝒜 r
h𝒜₀ : 𝒜.Nonempty
hs : 𝒜.sup' h𝒜₀ ⇑toColex ∈ ⇑ofColex ⁻¹' ↑𝒜
t : Finset α
p : t ∈ 𝒜
⊢ #(𝒜.sup' h𝒜₀ ⇑toColex) = #t ∧ toColex t ≤ toColex (𝒜.sup' h𝒜₀ ⇑toColex)

pp: case refine_2
α : Type u_1
inst✝¹ : LinearOrder α
𝒜 : Finset (Finset α)
r : ℕ
inst✝ : Fintype α
h𝒜 : IsInitSeg 𝒜 r
h𝒜₀ : 𝒜.Nonempty
hs : 𝒜.sup' h𝒜₀ ⇑toColex ∈ ⇑ofColex ⁻¹' ↑𝒜
t : Finset α
⊢ #(𝒜.sup' h𝒜₀ ⇑toColex) = #t ∧ toColex t ≤ toColex (𝒜.sup' h𝒜₀ ⇑toColex) → t ∈ 𝒜

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
inst✝¹ : LinearOrder α
𝒜 : Finset (Finset α)
r : ℕ
inst✝ : Fintype α
h𝒜 : IsInitSeg 𝒜 r
h𝒜₀ : 𝒜.Nonempty
hs : 𝒜.sup' h𝒜₀ ⇑toColex ∈ ⇑ofColex ⁻¹' ↑𝒜
t : Finset α
p : t ∈ 𝒜
⊢ #(𝒜.sup' h𝒜₀ ⇑toColex) = #t ∧ toColex t ≤ toColex (𝒜.sup' h𝒜₀ ⇑toColex)
Module: Mathlib.LinearAlgebra.CrossProduct

Goal state before:
pp: case «2»
R : Type u_1
inst✝ : CommRing R
u v w : Fin 3 → R
⊢ ![![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 * w 2 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 * w 1,
        ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 * w 0 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 * w 2,
        ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 * w 1 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 * w 0]
      ((fun i ↦ i) ⟨2, ⋯⟩) =
    ((u 0 * w 0 + u 1 * w 1 + u 2 * w 2) • v - (v 0 * w 0 + v 1 * w 1 + v 2 * w 2) • u) ((fun i ↦ i) ⟨2, ⋯⟩)

Tactic:
·

Goal state after:
case «2»
R : Type u_1
inst✝ : CommRing R
u v w : Fin 3 → R
⊢ ![![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 * w 2 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 * w 1,
        ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 * w 0 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 * w 2,
        ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 * w 1 -
          ![u 1 * v 2 - u 2 * v 1, u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 * w 0]
      ((fun i ↦ i) ⟨2, ⋯⟩) =
    ((u 0 * w 0 + u 1 * w 1 + u 2 * w 2) • v - (v 0 * w 0 + v 1 * w 1 + v 2 * w 2) • u) ((fun i ↦ i) ⟨2, ⋯⟩)
Module: Mathlib.LinearAlgebra.Projectivization.Constructions

Goal state before:
pp: case inl
F : Type u_1
inst✝¹ : Field F
inst✝ : DecidableEq F
v : ℙ F (Fin 3 → F)
⊢ v.cross v = v.cross v

pp: case inr
F : Type u_1
inst✝¹ : Field F
inst✝ : DecidableEq F
v w : ℙ F (Fin 3 → F)
h : v ≠ w
⊢ v.cross w = w.cross v

Tactic:
·

Goal state after:
case inl
F : Type u_1
inst✝¹ : Field F
inst✝ : DecidableEq F
v : ℙ F (Fin 3 → F)
⊢ v.cross v = v.cross v
Module: Mathlib.LinearAlgebra.Projectivization.Constructions

Goal state before:
pp: case inr
F : Type u_1
inst✝¹ : Field F
inst✝ : DecidableEq F
v w : ℙ F (Fin 3 → F)
h : v ≠ w
⊢ v.cross w = w.cross v

Tactic:
·

Goal state after:
case inr
F : Type u_1
inst✝¹ : Field F
inst✝ : DecidableEq F
v w : ℙ F (Fin 3 → F)
h : v ≠ w
⊢ v.cross w = w.cross v
Module: Mathlib.Combinatorics.Colex

Goal state before:
pp: case refine_1
α : Type u_1
inst✝¹ : LinearOrder α
𝒜 : Finset (Finset α)
r : ℕ
inst✝ : Fintype α
h𝒜 : IsInitSeg 𝒜 r
h𝒜₀ : 𝒜.Nonempty
hs : 𝒜.sup' h𝒜₀ ⇑toColex ∈ ⇑ofColex ⁻¹' ↑𝒜
t : Finset α
p : t ∈ 𝒜
⊢ #(𝒜.sup' h𝒜₀ ⇑toColex) = #t ∧ toColex t ≤ toColex (𝒜.sup' h𝒜₀ ⇑toColex)

pp: case refine_2
α : Type u_1
inst✝¹ : LinearOrder α
𝒜 : Finset (Finset α)
r : ℕ
inst✝ : Fintype α
h𝒜 : IsInitSeg 𝒜 r
h𝒜₀ : 𝒜.Nonempty
hs : 𝒜.sup' h𝒜₀ ⇑toColex ∈ ⇑ofColex ⁻¹' ↑𝒜
t : Finset α
⊢ #(𝒜.sup' h𝒜₀ ⇑toColex) = #t ∧ toColex t ≤ toColex (𝒜.sup' h𝒜₀ ⇑toColex) → t ∈ 𝒜

Tactic:
· rw [h𝒜.1 p, h𝒜.1 hs]
    exact ⟨rfl, le_sup' _ p⟩

Goal state after:
case refine_2
α : Type u_1
inst✝¹ : LinearOrder α
𝒜 : Finset (Finset α)
r : ℕ
inst✝ : Fintype α
h𝒜 : IsInitSeg 𝒜 r
h𝒜₀ : 𝒜.Nonempty
hs : 𝒜.sup' h𝒜₀ ⇑toColex ∈ ⇑ofColex ⁻¹' ↑𝒜
t : Finset α
⊢ #(𝒜.sup' h𝒜₀ ⇑toColex) = #t ∧ toColex t ≤ toColex (𝒜.sup' h𝒜₀ ⇑toColex) → t ∈ 𝒜
Module: Mathlib.Combinatorics.Colex

Goal state before:
pp: case refine_2.inl
α : Type u_1
inst✝¹ : LinearOrder α
𝒜 : Finset (Finset α)
r : ℕ
inst✝ : Fintype α
h𝒜 : IsInitSeg 𝒜 r
h𝒜₀ : 𝒜.Nonempty
hs : 𝒜.sup' h𝒜₀ ⇑toColex ∈ ⇑ofColex ⁻¹' ↑𝒜
t : Finset α
cards : #(𝒜.sup' h𝒜₀ ⇑toColex) = #t
le : toColex t ≤ toColex (𝒜.sup' h𝒜₀ ⇑toColex)
p : toColex t = toColex (𝒜.sup' h𝒜₀ ⇑toColex)
⊢ t ∈ 𝒜

pp: case refine_2.inr
α : Type u_1
inst✝¹ : LinearOrder α
𝒜 : Finset (Finset α)
r : ℕ
inst✝ : Fintype α
h𝒜 : IsInitSeg 𝒜 r
h𝒜₀ : 𝒜.Nonempty
hs : 𝒜.sup' h𝒜₀ ⇑toColex ∈ ⇑ofColex ⁻¹' ↑𝒜
t : Finset α
cards : #(𝒜.sup' h𝒜₀ ⇑toColex) = #t
le : toColex t ≤ toColex (𝒜.sup' h𝒜₀ ⇑toColex)
p : toColex t < toColex (𝒜.sup' h𝒜₀ ⇑toColex)
⊢ t ∈ 𝒜

Tactic:
·

Goal state after:
case refine_2.inl
α : Type u_1
inst✝¹ : LinearOrder α
𝒜 : Finset (Finset α)
r : ℕ
inst✝ : Fintype α
h𝒜 : IsInitSeg 𝒜 r
h𝒜₀ : 𝒜.Nonempty
hs : 𝒜.sup' h𝒜₀ ⇑toColex ∈ ⇑ofColex ⁻¹' ↑𝒜
t : Finset α
cards : #(𝒜.sup' h𝒜₀ ⇑toColex) = #t
le : toColex t ≤ toColex (𝒜.sup' h𝒜₀ ⇑toColex)
p : toColex t = toColex (𝒜.sup' h𝒜₀ ⇑toColex)
⊢ t ∈ 𝒜
Module: Mathlib.Combinatorics.Colex

Goal state before:
pp: case refine_2.inr
α : Type u_1
inst✝¹ : LinearOrder α
𝒜 : Finset (Finset α)
r : ℕ
inst✝ : Fintype α
h𝒜 : IsInitSeg 𝒜 r
h𝒜₀ : 𝒜.Nonempty
hs : 𝒜.sup' h𝒜₀ ⇑toColex ∈ ⇑ofColex ⁻¹' ↑𝒜
t : Finset α
cards : #(𝒜.sup' h𝒜₀ ⇑toColex) = #t
le : toColex t ≤ toColex (𝒜.sup' h𝒜₀ ⇑toColex)
p : toColex t < toColex (𝒜.sup' h𝒜₀ ⇑toColex)
⊢ t ∈ 𝒜

Tactic:
·

Goal state after:
case refine_2.inr
α : Type u_1
inst✝¹ : LinearOrder α
𝒜 : Finset (Finset α)
r : ℕ
inst✝ : Fintype α
h𝒜 : IsInitSeg 𝒜 r
h𝒜₀ : 𝒜.Nonempty
hs : 𝒜.sup' h𝒜₀ ⇑toColex ∈ ⇑ofColex ⁻¹' ↑𝒜
t : Finset α
cards : #(𝒜.sup' h𝒜₀ ⇑toColex) = #t
le : toColex t ≤ toColex (𝒜.sup' h𝒜₀ ⇑toColex)
p : toColex t < toColex (𝒜.sup' h𝒜₀ ⇑toColex)
⊢ t ∈ 𝒜
Module: Mathlib.Combinatorics.Colex

Goal state before:
pp: n : ℕ
hn : 2 ≤ n
s t : Colex (Finset ℕ)
a : ℕ
hat : a ∈ ofColex t
has : a ∉ ofColex s
ha : ∀ b ∈ ofColex s, b ∉ ofColex t → b < a
x✝¹ : ℕ
x✝ : x✝¹ ∈ ofColex t \ ofColex s
⊢ 0 ≤ n ^ x✝¹

Tactic:
by lia

Goal state after:
Module: Mathlib.Combinatorics.Configuration

Goal state before:
pp: case pos
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : Nondegenerate P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card L ≤ Fintype.card P
t : L → Finset P := fun l ↦ {p | p ∉ l}.toFinset
s : Finset L
hs₀ : #s = 0
⊢ #s ≤ #(s.biUnion t)

pp: case neg
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : Nondegenerate P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card L ≤ Fintype.card P
t : L → Finset P := fun l ↦ {p | p ∉ l}.toFinset
s : Finset L
hs₀ : ¬#s = 0
⊢ #s ≤ #(s.biUnion t)

Tactic:
·

Goal state after:
case pos
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : Nondegenerate P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card L ≤ Fintype.card P
t : L → Finset P := fun l ↦ {p | p ∉ l}.toFinset
s : Finset L
hs₀ : #s = 0
⊢ #s ≤ #(s.biUnion t)
Module: Mathlib.Combinatorics.Configuration

Goal state before:
pp: case pos
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : Nondegenerate P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card L ≤ Fintype.card P
t : L → Finset P := fun l ↦ {p | p ∉ l}.toFinset
s : Finset L
hs₀ : ¬#s = 0
hs₁ : #s = 1
⊢ #s ≤ #(s.biUnion t)

pp: case neg
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : Nondegenerate P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card L ≤ Fintype.card P
t : L → Finset P := fun l ↦ {p | p ∉ l}.toFinset
s : Finset L
hs₀ : ¬#s = 0
hs₁ : ¬#s = 1
⊢ #s ≤ #(s.biUnion t)

Tactic:
·

Goal state after:
case pos
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : Nondegenerate P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card L ≤ Fintype.card P
t : L → Finset P := fun l ↦ {p | p ∉ l}.toFinset
s : Finset L
hs₀ : ¬#s = 0
hs₁ : #s = 1
⊢ #s ≤ #(s.biUnion t)
Module: Mathlib.Combinatorics.Derangements.Basic

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
p : α → Prop
inst✝ : DecidablePred p
f : Perm (Subtype p)
hf : f ∈ derangements (Subtype p)
a : α
hfa : a ∈ fixedPoints ⇑↑((Perm.subtypeEquivSubtypePerm p) f)
ha : p a
⊢ False

pp: case refine_2
α : Type u_1
β : Type u_2
p : α → Prop
inst✝ : DecidablePred p
f : Perm (Subtype p)
⊢ (∀ a ∈ fixedPoints ⇑↑((Perm.subtypeEquivSubtypePerm p) f), ¬p a) → f ∈ derangements (Subtype p)

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
p : α → Prop
inst✝ : DecidablePred p
f : Perm (Subtype p)
hf : f ∈ derangements (Subtype p)
a : α
hfa : a ∈ fixedPoints ⇑↑((Perm.subtypeEquivSubtypePerm p) f)
ha : p a
⊢ False
Module: Mathlib.Combinatorics.Derangements.Basic

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
a : α
⊢ { f // a ∈ fixedPoints ⇑↑f } ≃ { f // fixedPoints ⇑f ⊆ {a} ∧ a ∈ fixedPoints ⇑f }

pp: case refine_2
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
a : α
⊢ { f // a ∉ fixedPoints ⇑↑f } ≃ { f // fixedPoints ⇑f ⊆ {a} ∧ a ∉ fixedPoints ⇑f }

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
a : α
⊢ { f // a ∈ fixedPoints ⇑↑f } ≃ { f // fixedPoints ⇑f ⊆ {a} ∧ a ∈ fixedPoints ⇑f }
Module: Mathlib.Combinatorics.Derangements.Basic

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
a : α
⊢ { f // a ∉ fixedPoints ⇑↑f } ≃ { f // fixedPoints ⇑f ⊆ {a} ∧ a ∉ fixedPoints ⇑f }

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
a : α
⊢ { f // a ∉ fixedPoints ⇑↑f } ≃ { f // fixedPoints ⇑f ⊆ {a} ∧ a ∉ fixedPoints ⇑f }
Module: Mathlib.Combinatorics.Derangements.Basic

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
a : α
f : Perm α
⊢ fixedPoints ⇑f ⊆ {a} ∧ a ∈ fixedPoints ⇑f ↔ fixedPoints ⇑f = {a}

pp: case refine_2
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
a : α
f : Perm α
⊢ fixedPoints ⇑f ⊆ {a} ∧ a ∉ fixedPoints ⇑f ↔ fixedPoints ⇑f = ∅

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
a : α
f : Perm α
⊢ fixedPoints ⇑f ⊆ {a} ∧ a ∈ fixedPoints ⇑f ↔ fixedPoints ⇑f = {a}
Module: Mathlib.LinearAlgebra.CrossProduct

Goal state before:
pp: case «0»
R : Type u_1
inst✝ : CommRing R
u v w : Fin 3 → R
⊢ ![u 1 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2 -
          u 2 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1,
        u 2 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 -
          u 0 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2,
        u 0 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 -
          u 1 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0]
      ((fun i ↦ i) ⟨0, ⋯⟩) =
    ((u 0 * w 0 + u 1 * w 1 + u 2 * w 2) • v - (v 0 * u 0 + v 1 * u 1 + v 2 * u 2) • w) ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
·

Goal state after:
case «0»
R : Type u_1
inst✝ : CommRing R
u v w : Fin 3 → R
⊢ ![u 1 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2 -
          u 2 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1,
        u 2 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 -
          u 0 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2,
        u 0 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 -
          u 1 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0]
      ((fun i ↦ i) ⟨0, ⋯⟩) =
    ((u 0 * w 0 + u 1 * w 1 + u 2 * w 2) • v - (v 0 * u 0 + v 1 * u 1 + v 2 * u 2) • w) ((fun i ↦ i) ⟨0, ⋯⟩)
Module: Mathlib.Combinatorics.Derangements.Basic

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
a : α
f : Perm α
⊢ fixedPoints ⇑f ⊆ {a} ∧ a ∉ fixedPoints ⇑f ↔ fixedPoints ⇑f = ∅

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
a : α
f : Perm α
⊢ fixedPoints ⇑f ⊆ {a} ∧ a ∉ fixedPoints ⇑f ↔ fixedPoints ⇑f = ∅
Module: Mathlib.LinearAlgebra.CrossProduct

Goal state before:
pp: case «1»
R : Type u_1
inst✝ : CommRing R
u v w : Fin 3 → R
⊢ ![u 1 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2 -
          u 2 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1,
        u 2 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 -
          u 0 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2,
        u 0 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 -
          u 1 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0]
      ((fun i ↦ i) ⟨1, ⋯⟩) =
    ((u 0 * w 0 + u 1 * w 1 + u 2 * w 2) • v - (v 0 * u 0 + v 1 * u 1 + v 2 * u 2) • w) ((fun i ↦ i) ⟨1, ⋯⟩)

Tactic:
·

Goal state after:
case «1»
R : Type u_1
inst✝ : CommRing R
u v w : Fin 3 → R
⊢ ![u 1 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2 -
          u 2 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1,
        u 2 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 -
          u 0 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2,
        u 0 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 -
          u 1 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0]
      ((fun i ↦ i) ⟨1, ⋯⟩) =
    ((u 0 * w 0 + u 1 * w 1 + u 2 * w 2) • v - (v 0 * u 0 + v 1 * u 1 + v 2 * u 2) • w) ((fun i ↦ i) ⟨1, ⋯⟩)
Module: Mathlib.Combinatorics.Derangements.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
a : α
x : Perm α
⊢ (∀ (a_1 : α), a_1 ∉ {a}ᶜ ↔ a_1 ∈ fixedPoints ⇑x) ↔ fixedPoints ⇑x = {a}

Tactic:
rw [eq_comm, Set.ext_iff]

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
a : α
x : Perm α
⊢ (∀ (a_1 : α), a_1 ∉ {a}ᶜ ↔ a_1 ∈ fixedPoints ⇑x) ↔ ∀ (x_1 : α), x_1 ∈ {a} ↔ x_1 ∈ fixedPoints ⇑x
Module: Mathlib.Combinatorics.Configuration

Goal state before:
pp: case pos
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : Nondegenerate P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card L ≤ Fintype.card P
t : L → Finset P := fun l ↦ {p | p ∉ l}.toFinset
s : Finset L
hs₀ : ¬#s = 0
hs₁ : ¬#s = 1
hs₂ : #(s.biUnion t)ᶜ ≤ 1
hs₃ : #sᶜ = 0
⊢ #(s.biUnion t)ᶜ ≤ #sᶜ

pp: case neg
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : Nondegenerate P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card L ≤ Fintype.card P
t : L → Finset P := fun l ↦ {p | p ∉ l}.toFinset
s : Finset L
hs₀ : ¬#s = 0
hs₁ : ¬#s = 1
hs₂ : #(s.biUnion t)ᶜ ≤ 1
hs₃ : ¬#sᶜ = 0
⊢ #(s.biUnion t)ᶜ ≤ #sᶜ

Tactic:
·

Goal state after:
case pos
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : Nondegenerate P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card L ≤ Fintype.card P
t : L → Finset P := fun l ↦ {p | p ∉ l}.toFinset
s : Finset L
hs₀ : ¬#s = 0
hs₁ : ¬#s = 1
hs₂ : #(s.biUnion t)ᶜ ≤ 1
hs₃ : #sᶜ = 0
⊢ #(s.biUnion t)ᶜ ≤ #sᶜ
Module: Mathlib.LinearAlgebra.CrossProduct

Goal state before:
pp: case «2»
R : Type u_1
inst✝ : CommRing R
u v w : Fin 3 → R
⊢ ![u 1 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2 -
          u 2 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1,
        u 2 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 -
          u 0 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2,
        u 0 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 -
          u 1 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0]
      ((fun i ↦ i) ⟨2, ⋯⟩) =
    ((u 0 * w 0 + u 1 * w 1 + u 2 * w 2) • v - (v 0 * u 0 + v 1 * u 1 + v 2 * u 2) • w) ((fun i ↦ i) ⟨2, ⋯⟩)

Tactic:
·

Goal state after:
case «2»
R : Type u_1
inst✝ : CommRing R
u v w : Fin 3 → R
⊢ ![u 1 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2 -
          u 2 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1,
        u 2 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 -
          u 0 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2,
        u 0 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 -
          u 1 * ![v 1 * w 2 - v 2 * w 1, v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0]
      ((fun i ↦ i) ⟨2, ⋯⟩) =
    ((u 0 * w 0 + u 1 * w 1 + u 2 * w 2) • v - (v 0 * u 0 + v 1 * u 1 + v 2 * u 2) • w) ((fun i ↦ i) ⟨2, ⋯⟩)
Module: Mathlib.Combinatorics.Derangements.Basic

Goal state before:
pp: case mp
α : Type u_1
inst✝ : DecidableEq α
a : α
f : Perm α
⊢ f ∈ fiber (some a) → f ∈ {f | fixedPoints ⇑f ⊆ {a}}

pp: case mpr
α : Type u_1
inst✝ : DecidableEq α
a : α
f : Perm α
⊢ f ∈ {f | fixedPoints ⇑f ⊆ {a}} → f ∈ fiber (some a)

Tactic:
·

Goal state after:
case mp
α : Type u_1
inst✝ : DecidableEq α
a : α
f : Perm α
⊢ f ∈ fiber (some a) → f ∈ {f | fixedPoints ⇑f ⊆ {a}}
Module: Mathlib.Combinatorics.Derangements.Basic

Goal state before:
pp: case mp.none
α : Type u_1
inst✝ : DecidableEq α
a : α
F : Perm (Option α)
F_derangement : F ∈ derangements (Option α)
F_none : F none = some a
x : α
x_fixed : some ((removeNone F) x) = some x
Fx : F (some x) = none
⊢ x ∈ {a}

pp: case mp.some
α : Type u_1
inst✝ : DecidableEq α
a : α
F : Perm (Option α)
F_derangement : F ∈ derangements (Option α)
F_none : F none = some a
x : α
x_fixed : some ((removeNone F) x) = some x
y : α
Fx : F (some x) = some y
⊢ x ∈ {a}

Tactic:
·

Goal state after:
case mp.none
α : Type u_1
inst✝ : DecidableEq α
a : α
F : Perm (Option α)
F_derangement : F ∈ derangements (Option α)
F_none : F none = some a
x : α
x_fixed : some ((removeNone F) x) = some x
Fx : F (some x) = none
⊢ x ∈ {a}
Module: Mathlib.Combinatorics.Configuration

Goal state before:
pp: case neg
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : Nondegenerate P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card L ≤ Fintype.card P
t : L → Finset P := fun l ↦ {p | p ∉ l}.toFinset
s : Finset L
hs₀ : ¬#s = 0
hs₁ : ¬#s = 1
hs₂ : #(s.biUnion t)ᶜ ≤ 1
hs₃ : ¬#sᶜ = 0
⊢ #(s.biUnion t)ᶜ ≤ #sᶜ

Tactic:
·

Goal state after:
case neg
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : Nondegenerate P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card L ≤ Fintype.card P
t : L → Finset P := fun l ↦ {p | p ∉ l}.toFinset
s : Finset L
hs₀ : ¬#s = 0
hs₁ : ¬#s = 1
hs₂ : #(s.biUnion t)ᶜ ≤ 1
hs₃ : ¬#sᶜ = 0
⊢ #(s.biUnion t)ᶜ ≤ #sᶜ
Module: Mathlib.Combinatorics.Derangements.Basic

Goal state before:
pp: case mp.some
α : Type u_1
inst✝ : DecidableEq α
a : α
F : Perm (Option α)
F_derangement : F ∈ derangements (Option α)
F_none : F none = some a
x : α
x_fixed : some ((removeNone F) x) = some x
y : α
Fx : F (some x) = some y
⊢ x ∈ {a}

Tactic:
·

Goal state after:
case mp.some
α : Type u_1
inst✝ : DecidableEq α
a : α
F : Perm (Option α)
F_derangement : F ∈ derangements (Option α)
F_none : F none = some a
x : α
x_fixed : some ((removeNone F) x) = some x
y : α
Fx : F (some x) = some y
⊢ x ∈ {a}
Module: Mathlib.Combinatorics.Derangements.Basic

Goal state before:
pp: case mpr
α : Type u_1
inst✝ : DecidableEq α
a : α
f : Perm α
⊢ f ∈ {f | fixedPoints ⇑f ⊆ {a}} → f ∈ fiber (some a)

Tactic:
·

Goal state after:
case mpr
α : Type u_1
inst✝ : DecidableEq α
a : α
f : Perm α
⊢ f ∈ {f | fixedPoints ⇑f ⊆ {a}} → f ∈ fiber (some a)