mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Logic.Equiv.Fintype

Goal state before:
pp: α : Type u_1
β : Type u_2
s t : Set α
inst✝¹ : Fintype ↑s
inst✝ : Fintype ↑t
h : Fintype.card ↑s = Fintype.card ↑t
fs : Finset α := Finset.map (Function.Embedding.subtype fun x ↦ x ∈ s) Finset.univ
ft : Finset α := Finset.map (Function.Embedding.subtype fun x ↦ x ∈ t) Finset.univ
hs : ∀ (x : α), x ∈ fs ↔ x ∈ s
ht : ∀ (x : α), x ∈ ft ↔ x ∈ t
hst : ∀ (x : α), x ∈ fs \ ft ↔ x ∈ s \ t
hts : ∀ (x : α), x ∈ ft \ fs ↔ x ∈ t \ s
hc : Fintype.card { x // x ∈ s \ t } = Fintype.card { x // x ∈ t \ s }
⊢ ↑(s \ t) ≃ ↑(t \ s)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
s t : Set α
inst✝¹ : Fintype ↑s
inst✝ : Fintype ↑t
h : Fintype.card ↑s = Fintype.card ↑t
fs : Finset α := Finset.map (Function.Embedding.subtype fun x ↦ x ∈ s) Finset.univ
ft : Finset α := Finset.map (Function.Embedding.subtype fun x ↦ x ∈ t) Finset.univ
hs : ∀ (x : α), x ∈ fs ↔ x ∈ s
ht : ∀ (x : α), x ∈ ft ↔ x ∈ t
hst : ∀ (x : α), x ∈ fs \ ft ↔ x ∈ s \ t
hts : ∀ (x : α), x ∈ ft \ fs ↔ x ∈ t \ s
hc : Fintype.card { x // x ∈ s \ t } = Fintype.card { x // x ∈ t \ s }
⊢ ↑(s \ t) ≃ ↑(t \ s)
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: case cons.succ
α : Type u_1
n : ℕ
ns : List ℕ
IH :
  ∀ (l : List α) {i : ℕ} (hi : i < (l.splitWrtCompositionAux ns).length),
    (l.splitWrtCompositionAux ns)[i] = drop (take i ns).sum (take (take (i + 1) ns).sum l)
l : List α
i : ℕ
hi : i + 1 < (l.splitWrtCompositionAux (n :: ns)).length
⊢ take ((take (i + 1) ns).sum - (take i ns).sum) (drop (n + (take i ns).sum) l) =
    take ((take (i + 1) ns).sum - (take i ns).sum) (drop (n + (take i ns).sum) l)

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: case cons
α : Type u_1
n : ℕ
ns : List ℕ
IH : ∀ {l : List α}, ns.sum = l.length → (l.splitWrtCompositionAux ns).flatten = l
l : List α
h : n + ns.sum = l.length
⊢ (l.splitWrtCompositionAux (n :: ns)).flatten = l

Tactic:
]

Goal state after:
case cons
α : Type u_1
n : ℕ
ns : List ℕ
IH : ∀ {l : List α}, ns.sum = l.length → (l.splitWrtCompositionAux ns).flatten = l
l : List α
h : n + ns.sum = l.length
⊢ (l.splitWrtCompositionAux (n :: ns)).flatten = l
Module: Mathlib.GroupTheory.Perm.Support

Goal state before:
pp: case insert
α : Type u_2
β : Type u_3
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : α → Perm β
x : β
a : α
s : Finset α
ha : a ∉ s
ih : ∀ {comm : (↑s).Pairwise (Commute on f)}, (∀ a ∈ s, x ∉ (f a).support) → x ∉ (s.noncommProd f comm).support
comm : (↑(insert a s)).Pairwise (Commute on f)
hs : ∀ a_1 ∈ insert a s, x ∉ (f a_1).support
⊢ x ∉ (f a * s.noncommProd f ⋯).support

Tactic:
]

Goal state after:
case insert
α : Type u_2
β : Type u_3
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : α → Perm β
x : β
a : α
s : Finset α
ha : a ∉ s
ih : ∀ {comm : (↑s).Pairwise (Commute on f)}, (∀ a ∈ s, x ∉ (f a).support) → x ∉ (s.noncommProd f comm).support
comm : (↑(insert a s)).Pairwise (Commute on f)
hs : ∀ a_1 ∈ insert a s, x ∉ (f a_1).support
⊢ x ∉ (f a * s.noncommProd f ⋯).support
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: case cons
α : Type u_1
n : ℕ
ns : List ℕ
IH : ∀ {l : List α}, ns.sum = l.length → (l.splitWrtCompositionAux ns).flatten = l
l : List α
h : n + ns.sum = l.length
⊢ take n l ++ drop n l = l

pp: case cons
α : Type u_1
n : ℕ
ns : List ℕ
IH : ∀ {l : List α}, ns.sum = l.length → (l.splitWrtCompositionAux ns).flatten = l
l : List α
h : n + ns.sum = l.length
⊢ ns.sum = (drop n l).length

Tactic:
]

Goal state after:
case cons
α : Type u_1
n : ℕ
ns : List ℕ
IH : ∀ {l : List α}, ns.sum = l.length → (l.splitWrtCompositionAux ns).flatten = l
l : List α
h : n + ns.sum = l.length
⊢ take n l ++ drop n l = l

case cons
α : Type u_1
n : ℕ
ns : List ℕ
IH : ∀ {l : List α}, ns.sum = l.length → (l.splitWrtCompositionAux ns).flatten = l
l : List α
h : n + ns.sum = l.length
⊢ ns.sum = (drop n l).length
Module: Mathlib.GroupTheory.Perm.Support

Goal state before:
pp: case insert
α : Type u_2
β : Type u_3
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : α → Perm β
x : β
a : α
s : Finset α
ha : a ∉ s
ih : ∀ {comm : (↑s).Pairwise (Commute on f)}, (∀ a ∈ s, x ∉ (f a).support) → x ∉ (s.noncommProd f comm).support
comm : (↑(insert a s)).Pairwise (Commute on f)
hs : ∀ a_1 ∈ insert a s, x ∉ (f a_1).support
⊢ x ∉ (f a).support ∧ x ∉ (s.noncommProd f ⋯).support

Tactic:
]

Goal state after:
case insert
α : Type u_2
β : Type u_3
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : α → Perm β
x : β
a : α
s : Finset α
ha : a ∉ s
ih : ∀ {comm : (↑s).Pairwise (Commute on f)}, (∀ a ∈ s, x ∉ (f a).support) → x ∉ (s.noncommProd f comm).support
comm : (↑(insert a s)).Pairwise (Commute on f)
hs : ∀ a_1 ∈ insert a s, x ∉ (f a_1).support
⊢ x ∉ (f a).support ∧ x ∉ (s.noncommProd f ⋯).support
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: case cons
α : Type u_1
n : ℕ
ns : List ℕ
IH : ∀ {l : List α}, ns.sum = l.length → (l.splitWrtCompositionAux ns).flatten = l
l : List α
h : n + ns.sum = l.length
⊢ ns.sum = ns.sum

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.Perm.Support

Goal state before:
pp: case succ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
k : ℕ
hk : ∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
x : α
hx : x ∈ f.support ∩ g.support
⊢ (g ^ k) (g x) = (g ^ k) (g x)

pp: case succ.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
k : ℕ
hk : ∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
x : α
hx : x ∈ f.support ∩ g.support
⊢ g x ∈ f.support ∩ g.support

Tactic:
]

Goal state after:
case succ.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
k : ℕ
hk : ∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
x : α
hx : x ∈ f.support ∩ g.support
⊢ g x ∈ f.support ∩ g.support
Module: Mathlib.GroupTheory.Perm.Support

Goal state before:
pp: case succ.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
k : ℕ
hk : ∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
x : α
hx : x ∈ f.support ∩ g.support
⊢ x ∈ f.support ∩ g.support

Tactic:
]

Goal state after:
case succ.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
k : ℕ
hk : ∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
x : α
hx : x ∈ f.support ∩ g.support
⊢ x ∈ f.support ∩ g.support
Module: Mathlib.GroupTheory.Perm.Support

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : f.Disjoint g
a : α
⊢ f (g a) = a → f a = a ∧ g a = a

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : f.Disjoint g
a : α
⊢ f (g a) = a → f a = a ∧ g a = a
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: n : ℕ
this : Fintype.card (Finset (Fin (n - 1))) = 2 ^ (n - 1)
⊢ Fintype.card (CompositionAsSet n) = Fintype.card (Finset (Fin (n - 1)))

Tactic:
]

Goal state after:
n : ℕ
this : Fintype.card (Finset (Fin (n - 1))) = 2 ^ (n - 1)
⊢ Fintype.card (CompositionAsSet n) = Fintype.card (Finset (Fin (n - 1)))
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: n : ℕ
c : CompositionAsSet n
⊢ c.length < c.length + 1

Tactic:
]

Goal state after:
n : ℕ
c : CompositionAsSet n
⊢ c.length < c.length + 1
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: n : ℕ
c : CompositionAsSet n
⊢ c.boundaries.min' ⋯ = 0

Tactic:
]

Goal state after:
n : ℕ
c : CompositionAsSet n
⊢ c.boundaries.min' ⋯ = 0
Module: Mathlib.GroupTheory.Perm.Support

Goal state before:
pp: case cons
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hd : Perm α
tl : List (Perm α)
hl : List.Pairwise Disjoint tl → tl.prod.support = List.foldr (fun x1 x2 ↦ x1 ⊔ x2) ⊥ (List.map support tl)
h : (∀ a' ∈ tl, hd.Disjoint a') ∧ List.Pairwise Disjoint tl
⊢ (hd :: tl).prod.support = List.foldr (fun x1 x2 ↦ x1 ⊔ x2) ⊥ (List.map support (hd :: tl))

Tactic:
]

Goal state after:
case cons
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hd : Perm α
tl : List (Perm α)
hl : List.Pairwise Disjoint tl → tl.prod.support = List.foldr (fun x1 x2 ↦ x1 ⊔ x2) ⊥ (List.map support tl)
h : (∀ a' ∈ tl, hd.Disjoint a') ∧ List.Pairwise Disjoint tl
⊢ (hd :: tl).prod.support = List.foldr (fun x1 x2 ↦ x1 ⊔ x2) ⊥ (List.map support (hd :: tl))
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: n : ℕ
c : CompositionAsSet n
i : ℕ
IH : ∀ (h : i < c.boundaries.card), (take i c.blocks).sum = ↑(c.boundary ⟨i, h⟩)
h : i + 1 < c.length + 1
⊢ i < c.blocks.length

Tactic:
]

Goal state after:
n : ℕ
c : CompositionAsSet n
i : ℕ
IH : ∀ (h : i < c.boundaries.card), (take i c.blocks).sum = ↑(c.boundary ⟨i, h⟩)
h : i + 1 < c.length + 1
⊢ i < c.blocks.length
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: n : ℕ
c : CompositionAsSet n
i : ℕ
IH : ∀ (h : i < c.boundaries.card), (take i c.blocks).sum = ↑(c.boundary ⟨i, h⟩)
h : i + 1 < c.boundaries.card
A : i < c.blocks.length
⊢ c.blocks.length ≤ c.boundaries.card

Tactic:
by simp [blocks, length]

Goal state after:
Module: Mathlib.GroupTheory.Perm.Finite

Goal state before:
pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ τ : Perm α
f : { x // x ∈ ↑σ.support } ≃ { x // x ∈ ↑τ.support }
hf : ∀ (x : α) (hx : x ∈ ↑σ.support), ↑(f ⟨σ x, ⋯⟩) = τ ↑(f ⟨x, hx⟩)
⊢ f.extendSubtype * σ = τ * f.extendSubtype

Tactic:
]

Goal state after:
α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ τ : Perm α
f : { x // x ∈ ↑σ.support } ≃ { x // x ∈ ↑τ.support }
hf : ∀ (x : α) (hx : x ∈ ↑σ.support), ↑(f ⟨σ x, ⋯⟩) = τ ↑(f ⟨x, hx⟩)
⊢ f.extendSubtype * σ = τ * f.extendSubtype
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: case succ
n : ℕ
c : CompositionAsSet n
i : ℕ
IH : ∀ (h : i < c.boundaries.card), (take i c.blocks).sum = ↑(c.boundary ⟨i, h⟩)
h : i + 1 < c.boundaries.card
A : i < c.blocks.length
B : i < c.boundaries.card
⊢ ↑(c.boundary ⟨i, B⟩) + c.blocks[i] = ↑(c.boundary ⟨i + 1, h⟩)

Tactic:
]

Goal state after:
case succ
n : ℕ
c : CompositionAsSet n
i : ℕ
IH : ∀ (h : i < c.boundaries.card), (take i c.blocks).sum = ↑(c.boundary ⟨i, h⟩)
h : i + 1 < c.boundaries.card
A : i < c.blocks.length
B : i < c.boundaries.card
⊢ ↑(c.boundary ⟨i, B⟩) + c.blocks[i] = ↑(c.boundary ⟨i + 1, h⟩)
Module: Mathlib.GroupTheory.Perm.Support

Goal state before:
pp: case insert
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
ι : Type u_2
k : ι → Perm α
i : ι
s : Finset ι
hi : i ∉ s
hrec :
  ∀ (hs : (↑s).Pairwise fun i j ↦ (k i).Disjoint (k j)), (s.noncommProd k ⋯).support = s.biUnion fun i ↦ (k i).support
hs : (↑(insert i s)).Pairwise fun i j ↦ (k i).Disjoint (k j)
hs' : (↑s).Pairwise fun i j ↦ (k i).Disjoint (k j)
⊢ (k i * s.noncommProd k ⋯).support = (k i).support ∪ s.biUnion fun i ↦ (k i).support

Tactic:
]

Goal state after:
case insert
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
ι : Type u_2
k : ι → Perm α
i : ι
s : Finset ι
hi : i ∉ s
hrec :
  ∀ (hs : (↑s).Pairwise fun i j ↦ (k i).Disjoint (k j)), (s.noncommProd k ⋯).support = s.biUnion fun i ↦ (k i).support
hs : (↑(insert i s)).Pairwise fun i j ↦ (k i).Disjoint (k j)
hs' : (↑s).Pairwise fun i j ↦ (k i).Disjoint (k j)
⊢ (k i * s.noncommProd k ⋯).support = (k i).support ∪ s.biUnion fun i ↦ (k i).support
Module: Mathlib.GroupTheory.Perm.Finite

Goal state before:
pp: case pos
α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ τ : Perm α
f : { x // x ∈ ↑σ.support } ≃ { x // x ∈ ↑τ.support }
hf : ∀ (x : α) (hx : x ∈ ↑σ.support), ↑(f ⟨σ x, ⋯⟩) = τ ↑(f ⟨x, hx⟩)
x : α
hx : x ∈ σ.support
⊢ ↑(f ⟨σ x, ?pos.hx✝⟩) = τ ↑(f ⟨x, ?pos.hx✝¹⟩)

pp: case pos.hx
α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ τ : Perm α
f : { x // x ∈ ↑σ.support } ≃ { x // x ∈ ↑τ.support }
hf : ∀ (x : α) (hx : x ∈ ↑σ.support), ↑(f ⟨σ x, ⋯⟩) = τ ↑(f ⟨x, hx⟩)
x : α
hx : x ∈ σ.support
⊢ x ∈ ↑σ.support

pp: case pos.hx
α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ τ : Perm α
f : { x // x ∈ ↑σ.support } ≃ { x // x ∈ ↑τ.support }
hf : ∀ (x : α) (hx : x ∈ ↑σ.support), ↑(f ⟨σ x, ⋯⟩) = τ ↑(f ⟨x, hx⟩)
x : α
hx : x ∈ σ.support
⊢ σ x ∈ ↑σ.support

Tactic:
]

Goal state after:
case pos
α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ τ : Perm α
f : { x // x ∈ ↑σ.support } ≃ { x // x ∈ ↑τ.support }
hf : ∀ (x : α) (hx : x ∈ ↑σ.support), ↑(f ⟨σ x, ⋯⟩) = τ ↑(f ⟨x, hx⟩)
x : α
hx : x ∈ σ.support
⊢ ↑(f ⟨σ x, ?pos.hx✝⟩) = τ ↑(f ⟨x, ?pos.hx✝¹⟩)

case pos.hx
α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ τ : Perm α
f : { x // x ∈ ↑σ.support } ≃ { x // x ∈ ↑τ.support }
hf : ∀ (x : α) (hx : x ∈ ↑σ.support), ↑(f ⟨σ x, ⋯⟩) = τ ↑(f ⟨x, hx⟩)
x : α
hx : x ∈ σ.support
⊢ x ∈ ↑σ.support

case pos.hx
α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ τ : Perm α
f : { x // x ∈ ↑σ.support } ≃ { x // x ∈ ↑τ.support }
hf : ∀ (x : α) (hx : x ∈ ↑σ.support), ↑(f ⟨σ x, ⋯⟩) = τ ↑(f ⟨x, hx⟩)
x : α
hx : x ∈ σ.support
⊢ σ x ∈ ↑σ.support
Module: Mathlib.GroupTheory.Perm.Support

Goal state before:
pp: case insert
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
ι : Type u_2
k : ι → Perm α
i : ι
s : Finset ι
hi : i ∉ s
hrec :
  ∀ (hs : (↑s).Pairwise fun i j ↦ (k i).Disjoint (k j)), (s.noncommProd k ⋯).support = s.biUnion fun i ↦ (k i).support
hs : (↑(insert i s)).Pairwise fun i j ↦ (k i).Disjoint (k j)
hs' : (↑s).Pairwise fun i j ↦ (k i).Disjoint (k j)
⊢ ((k i).support ∪ s.biUnion fun i ↦ (k i).support) = (k i).support ∪ s.biUnion fun i ↦ (k i).support

pp: case insert
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
ι : Type u_2
k : ι → Perm α
i : ι
s : Finset ι
hi : i ∉ s
hrec :
  ∀ (hs : (↑s).Pairwise fun i j ↦ (k i).Disjoint (k j)), (s.noncommProd k ⋯).support = s.biUnion fun i ↦ (k i).support
hs : (↑(insert i s)).Pairwise fun i j ↦ (k i).Disjoint (k j)
hs' : (↑s).Pairwise fun i j ↦ (k i).Disjoint (k j)
⊢ (k i).Disjoint (s.noncommProd k ⋯)

Tactic:
]

Goal state after:
case insert
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
ι : Type u_2
k : ι → Perm α
i : ι
s : Finset ι
hi : i ∉ s
hrec :
  ∀ (hs : (↑s).Pairwise fun i j ↦ (k i).Disjoint (k j)), (s.noncommProd k ⋯).support = s.biUnion fun i ↦ (k i).support
hs : (↑(insert i s)).Pairwise fun i j ↦ (k i).Disjoint (k j)
hs' : (↑s).Pairwise fun i j ↦ (k i).Disjoint (k j)
⊢ (k i).Disjoint (s.noncommProd k ⋯)
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: case mp
n : ℕ
c : CompositionAsSet n
j : Fin (n + 1)
hj : j ∈ c.boundaries
i : Fin c.boundaries.card
hi : ↑((c.boundaries.orderIsoOfFin ⋯) i) = ↑⟨j, hj⟩
⊢ ∃ i < c.boundaries.card, (take i c.blocks).sum = ↑j

Tactic:
]

Goal state after:
case mp
n : ℕ
c : CompositionAsSet n
j : Fin (n + 1)
hj : j ∈ c.boundaries
i : Fin c.boundaries.card
hi : ↑((c.boundaries.orderIsoOfFin ⋯) i) = ↑⟨j, hj⟩
⊢ ∃ i < c.boundaries.card, (take i c.blocks).sum = ↑j
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: case mp
n : ℕ
c : CompositionAsSet n
j : Fin (n + 1)
hj : j ∈ c.boundaries
i : Fin c.boundaries.card
hi : (c.boundaries.orderEmbOfFin ⋯) i = j
⊢ ↑(c.boundary ⟨↑i, ⋯⟩) = ↑((c.boundaries.orderEmbOfFin ⋯) i)

Tactic:
]

Goal state after:
case mp
n : ℕ
c : CompositionAsSet n
j : Fin (n + 1)
hj : j ∈ c.boundaries
i : Fin c.boundaries.card
hi : (c.boundaries.orderEmbOfFin ⋯) i = j
⊢ ↑(c.boundary ⟨↑i, ⋯⟩) = ↑((c.boundaries.orderEmbOfFin ⋯) i)
Module: Mathlib.GroupTheory.Perm.Finite

Goal state before:
pp: case neg
α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ τ : Perm α
f : { x // x ∈ ↑σ.support } ≃ { x // x ∈ ↑τ.support }
hf : ∀ (x : α) (hx : x ∈ ↑σ.support), ↑(f ⟨σ x, ⋯⟩) = τ ↑(f ⟨x, hx⟩)
x : α
hx : x ∉ σ.support
⊢ f.extendSubtype x = f.extendSubtype x

pp: α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ τ : Perm α
f : { x // x ∈ ↑σ.support } ≃ { x // x ∈ ↑τ.support }
hf : ∀ (x : α) (hx : x ∈ ↑σ.support), ↑(f ⟨σ x, ⋯⟩) = τ ↑(f ⟨x, hx⟩)
x : α
hx : x ∉ σ.support
⊢ x ∉ ↑σ.support

Tactic:
]

Goal state after:
α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ τ : Perm α
f : { x // x ∈ ↑σ.support } ≃ { x // x ∈ ↑τ.support }
hf : ∀ (x : α) (hx : x ∈ ↑σ.support), ↑(f ⟨σ x, ⋯⟩) = τ ↑(f ⟨x, hx⟩)
x : α
hx : x ∉ σ.support
⊢ x ∉ ↑σ.support
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: n : ℕ
c : CompositionAsSet n
j : Fin (n + 1)
i : ℕ
hi : i < c.boundaries.card
H : (take i c.blocks).sum = ↑j
⊢ (take i c.blocks).sum = ↑j

Tactic:
]

Goal state after:
n : ℕ
c : CompositionAsSet n
j : Fin (n + 1)
i : ℕ
hi : i < c.boundaries.card
H : (take i c.blocks).sum = ↑j
⊢ (take i c.blocks).sum = ↑j
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: n : ℕ
c : CompositionAsSet n
this : take c.length c.blocks = c.blocks
⊢ ↑(Fin.last n) = n

Tactic:
]

Goal state after:
n : ℕ
c : CompositionAsSet n
this : take c.length c.blocks = c.blocks
⊢ ↑(Fin.last n) = n
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: n : ℕ
c : Composition n
⊢ c.toCompositionAsSet.length = c.length

Tactic:
simp [Composition.toCompositionAsSet, CompositionAsSet.length, c.card_boundaries_eq_succ_length]

Goal state after:
Module: Mathlib.GroupTheory.Perm.Support

Goal state before:
pp: case cons
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hd : Perm α
tl : List (Perm α)
hl : tl.prod.support ⊆ List.foldr (fun x1 x2 ↦ x1 ⊔ x2) ⊥ (List.map support tl)
⊢ (hd * tl.prod).support ⊆ hd.support ⊔ List.foldr (fun x1 x2 ↦ x1 ⊔ x2) ⊥ (List.map support tl)

Tactic:
]

Goal state after:
case cons
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hd : Perm α
tl : List (Perm α)
hl : tl.prod.support ⊆ List.foldr (fun x1 x2 ↦ x1 ⊔ x2) ⊥ (List.map support tl)
⊢ (hd * tl.prod).support ⊆ hd.support ⊔ List.foldr (fun x1 x2 ↦ x1 ⊔ x2) ⊥ (List.map support tl)
Module: Mathlib.GroupTheory.Perm.Finite

Goal state before:
pp: case mp
m : Type u_1
n : Type u_2
inst✝¹ : Finite m
inst✝ : Finite n
σ : Perm (m ⊕ n)
h : Set.MapsTo (⇑(Equiv.symm σ)) (Set.range Sum.inl) (Set.range Sum.inl)
⊢ Set.MapsTo (⇑σ) (Set.range Sum.inr) (Set.range Sum.inr)

Tactic:
]

Goal state after:
case mp
m : Type u_1
n : Type u_2
inst✝¹ : Finite m
inst✝ : Finite n
σ : Perm (m ⊕ n)
h : Set.MapsTo (⇑(Equiv.symm σ)) (Set.range Sum.inl) (Set.range Sum.inl)
⊢ Set.MapsTo (⇑σ) (Set.range Sum.inr) (Set.range Sum.inr)
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: n : ℕ
c : Composition n
d : CompositionAsSet n := c.toCompositionAsSet
length_eq : d.blocks.length = c.blocks.length
i : ℕ
hi : i ≤ d.blocks.length
i_lt : i < d.boundaries.card
⊢ (take i d.blocks).sum = (take i c.blocks).sum

Tactic:
have i_lt' : i < c.boundaries.card := i_lt

Goal state after:
n : ℕ
c : Composition n
d : CompositionAsSet n := c.toCompositionAsSet
length_eq : d.blocks.length = c.blocks.length
i : ℕ
hi : i ≤ d.blocks.length
i_lt : i < d.boundaries.card
i_lt' : i < c.boundaries.card
⊢ (take i d.blocks).sum = (take i c.blocks).sum
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: n : ℕ
c : Composition n
d : CompositionAsSet n := c.toCompositionAsSet
length_eq : d.blocks.length = c.blocks.length
i : ℕ
hi : i ≤ d.blocks.length
i_lt : i < d.boundaries.card
i_lt' : i < c.length + 1
⊢ i < c.length + 1

Tactic:
]

Goal state after:
n : ℕ
c : Composition n
d : CompositionAsSet n := c.toCompositionAsSet
length_eq : d.blocks.length = c.blocks.length
i : ℕ
hi : i ≤ d.blocks.length
i_lt : i < d.boundaries.card
i_lt' : i < c.length + 1
⊢ i < c.length + 1
Module: Mathlib.GroupTheory.Perm.Finite

Goal state before:
pp: case mpr
m : Type u_1
n : Type u_2
inst✝¹ : Finite m
inst✝ : Finite n
σ : Perm (m ⊕ n)
h : Set.MapsTo (⇑(Equiv.symm σ)) (Set.range Sum.inr) (Set.range Sum.inr)
⊢ Set.MapsTo (⇑σ) (Set.range Sum.inl) (Set.range Sum.inl)

Tactic:
]

Goal state after:
case mpr
m : Type u_1
n : Type u_2
inst✝¹ : Finite m
inst✝ : Finite n
σ : Perm (m ⊕ n)
h : Set.MapsTo (⇑(Equiv.symm σ)) (Set.range Sum.inr) (Set.range Sum.inr)
⊢ Set.MapsTo (⇑σ) (Set.range Sum.inl) (Set.range Sum.inl)
Module: Mathlib.GroupTheory.Perm.Support

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
x y z : α
h : (x ≠ y ∧ x ≠ z ∧ True) ∧ y ≠ z ∧ True
⊢ {x, y, z} = {x, y} ⊔ {y, z}

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
x y z : α
h : (x ≠ y ∧ x ≠ z ∧ True) ∧ y ≠ z ∧ True
⊢ {x, y, z} = {x, y} ⊔ {y, z}
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: n : ℕ
c : Composition n
d : CompositionAsSet n := c.toCompositionAsSet
length_eq : d.blocks.length = c.blocks.length
i : ℕ
hi : i ≤ d.blocks.length
i_lt : i < d.boundaries.card
i_lt' : i < c.boundaries.card
i_lt'' : i < c.length + 1
A : (d.boundaries.orderEmbOfFin ⋯) ⟨i, i_lt⟩ = (c.boundaries.orderEmbOfFin ⋯) ⟨i, i_lt''⟩
B : c.sizeUpTo i = ↑(c.boundary ⟨i, i_lt''⟩)
⊢ ↑(c.boundary ⟨i, i_lt''⟩) = ↑(c.boundary ⟨i, i_lt''⟩)

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.Composition

Goal state before:
pp: n : ℕ
⊢ Fintype.card (Composition n) = Fintype.card (CompositionAsSet n)

Tactic:
]

Goal state after:
n : ℕ
⊢ Fintype.card (Composition n) = Fintype.card (CompositionAsSet n)
Module: Mathlib.GroupTheory.Perm.List

Goal state before:
pp: case pos
α : Type u_1
inst✝ : DecidableEq α
z : α
l : List α
IH : ∀ {x : α} (y : α), x ∈ y :: l → (y :: l).formPerm x ∈ y :: l
x y : α
h : x ∈ y :: z :: l
hx : x ∈ z :: l
⊢ (if (z :: l).formPerm x = y then z else if (z :: l).formPerm x = z then y else (z :: l).formPerm x) ∈ y :: z :: l

Tactic:
]

Goal state after:
case pos
α : Type u_1
inst✝ : DecidableEq α
z : α
l : List α
IH : ∀ {x : α} (y : α), x ∈ y :: l → (y :: l).formPerm x ∈ y :: l
x y : α
h : x ∈ y :: z :: l
hx : x ∈ z :: l
⊢ (if (z :: l).formPerm x = y then z else if (z :: l).formPerm x = z then y else (z :: l).formPerm x) ∈ y :: z :: l
Module: Mathlib.GroupTheory.Perm.Support

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
x y x✝ : α
ha : f x✝ ≠ x✝
hx : x✝ ≠ x
hy : x✝ ≠ y
H : f x✝ = x✝
⊢ False

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
x y x✝ : α
ha : f x✝ ≠ x✝
hx : x✝ ≠ x
hy : x✝ ≠ y
H : f x✝ = x✝
⊢ False
Module: Mathlib.GroupTheory.Perm.List

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
l : List α
x : α
h : x ∉ l
⊢ x ∉ l

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
l : List α
x : α
h : x ∉ l
⊢ x ∉ l
Module: Mathlib.GroupTheory.Perm.List

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
x : α
xs : List α
⊢ x = x

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.Perm.Finite

Goal state before:
pp: m : Type u_1
n : Type u_2
inst✝¹ : Finite m
inst✝ : Finite n
σ : Perm (m ⊕ n)
h : Set.MapsTo (⇑σ) (Set.range Sum.inl) (Set.range Sum.inl)
h1 : ∀ (x : m ⊕ n), (∃ a, Sum.inl a = x) → ∃ a, Sum.inl a = σ x
h3 : ∀ (x : m ⊕ n), (∃ b, Sum.inr b = x) → ∃ b, Sum.inr b = σ x
σ₁' : Perm { x // ∃ a, Sum.inl a = x } := σ.subtypePermOfFintype h1
σ₂' : Perm { x // ∃ b, Sum.inr b = x } := σ.subtypePermOfFintype h3
σ₁ : Perm m := (ofInjective Sum.inl ⋯).symm.permCongr σ₁'
σ₂ : Perm n := (ofInjective Sum.inr ⋯).symm.permCongr σ₂'
⊢ ∃ a b, (sumCongrHom m n) (a, b) = σ

Tactic:
]

Goal state after:
m : Type u_1
n : Type u_2
inst✝¹ : Finite m
inst✝ : Finite n
σ : Perm (m ⊕ n)
h : Set.MapsTo (⇑σ) (Set.range Sum.inl) (Set.range Sum.inl)
h1 : ∀ (x : m ⊕ n), (∃ a, Sum.inl a = x) → ∃ a, Sum.inl a = σ x
h3 : ∀ (x : m ⊕ n), (∃ b, Sum.inr b = x) → ∃ b, Sum.inr b = σ x
σ₁' : Perm { x // ∃ a, Sum.inl a = x } := σ.subtypePermOfFintype h1
σ₂' : Perm { x // ∃ b, Sum.inr b = x } := σ.subtypePermOfFintype h3
σ₁ : Perm m := (ofInjective Sum.inl ⋯).symm.permCongr σ₁'
σ₂ : Perm n := (ofInjective Sum.inr ⋯).symm.permCongr σ₂'
⊢ ∃ a b, (sumCongrHom m n) (a, b) = σ
Module: Mathlib.GroupTheory.Perm.List

Goal state before:
pp: case cons.cons
α : Type u_1
inst✝ : DecidableEq α
x y : α
tl : List α
h : (x :: y :: tl).Nodup
hl : 1 < (x :: y :: tl).length
⊢ y = y

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.Perm.List

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
l : List α
x y : α
⊢ (y :: l).formPerm x = y ↔ (y :: l).formPerm x = y

Tactic:
]

Goal state after: