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

Goal state before:
pp: case a
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
S : Finset G
ε : ℝ
hε₀ : 0 < ε
hε₁ : ε ≤ 1
hS : S.Nonempty
K : ℝ := ⋯
hK : K < 1
ex : Finset G → ℝ := ⋯
κ : ℝ := ⋯
H : Subgroup G
w✝ : Fintype ↥H
hH : IsAtom K S (↑H).toFinset
Z : Finset G
hZS : Z ⊆ S
hHZS : ↑H * ↑Z = ↑H * ↑S
hZinj : Set.InjOn (fun x ↦ ↑H <• x) ↑Z
A : Finset G
hA₁ : #S ≤ #A
hA₂ : ↑(#(A * S)) ≤ (2 - ε) * ↑(#S)
calc₁ : ex (↑H).toFinset ≤ (1 - ε / 2) * ↑(#S)
⊢ S ⊆ (↑H).toFinset * S

Tactic:
·

Goal state after:
case a
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
S : Finset G
ε : ℝ
hε₀ : 0 < ε
hε₁ : ε ≤ 1
hS : S.Nonempty
K : ℝ := ⋯
hK : K < 1
ex : Finset G → ℝ := ⋯
κ : ℝ := ⋯
H : Subgroup G
w✝ : Fintype ↥H
hH : IsAtom K S (↑H).toFinset
Z : Finset G
hZS : Z ⊆ S
hHZS : ↑H * ↑Z = ↑H * ↑S
hZinj : Set.InjOn (fun x ↦ ↑H <• x) ↑Z
A : Finset G
hA₁ : #S ≤ #A
hA₂ : ↑(#(A * S)) ≤ (2 - ε) * ↑(#S)
calc₁ : ex (↑H).toFinset ≤ (1 - ε / 2) * ↑(#S)
⊢ S ⊆ (↑H).toFinset * S
Module: Mathlib.Combinatorics.Graph.Basic

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
y : α
e : β
G : Graph α β
x' y' : α
hy' : G.IsLink e y y'
hx' : G.IsLink e x' x'
h : ∀ (z : α), G.Inc e z → z = x' ∨ z = y
⊢ G.IsLink e x' y

pp: case inr
α : Type u_1
β : Type u_2
x : α
e : β
G : Graph α β
x' : α
hx' : G.IsLink e x x'
y' : α
h : ∀ (z : α), G.Inc e z → z = x ∨ z = x'
hy' : G.IsLink e x' y'
⊢ G.IsLink e x x'

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
y : α
e : β
G : Graph α β
x' y' : α
hy' : G.IsLink e y y'
hx' : G.IsLink e x' x'
h : ∀ (z : α), G.Inc e z → z = x' ∨ z = y
⊢ G.IsLink e x' y
Module: Mathlib.Combinatorics.Graph.Basic

Goal state before:
pp: case inl.inl
α : Type u_1
β : Type u_2
e : β
G : Graph α β
x' y' : α
hx' : G.IsLink e x' x'
hy' : G.IsLink e x' y'
h : ∀ (z : α), G.Inc e z → z = x' ∨ z = x'
⊢ G.IsLink e x' x'

pp: case inl.inr
α : Type u_1
β : Type u_2
y : α
e : β
G : Graph α β
x' : α
hx' : G.IsLink e x' x'
h : ∀ (z : α), G.Inc e z → z = x' ∨ z = y
hy' : G.IsLink e y x'
⊢ G.IsLink e x' y

Tactic:
·

Goal state after:
case inl.inl
α : Type u_1
β : Type u_2
e : β
G : Graph α β
x' y' : α
hx' : G.IsLink e x' x'
hy' : G.IsLink e x' y'
h : ∀ (z : α), G.Inc e z → z = x' ∨ z = x'
⊢ G.IsLink e x' x'
Module: Mathlib.Combinatorics.Graph.Delete

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
G H : Graph α β
h : H ≤ G
F : Set β
⊢ V(H.restrict F) ⊆ V(G.restrict F)

pp: case refine_2
α : Type u_1
β : Type u_2
G H : Graph α β
h : H ≤ G
F : Set β
⊢ E(H.restrict F) ⊆ E(G.restrict F)

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
G H : Graph α β
h : H ≤ G
F : Set β
⊢ V(H.restrict F) ⊆ V(G.restrict F)
Module: Mathlib.Combinatorics.Graph.Basic

Goal state before:
pp: case mp
α : Type u_1
β : Type u_2
e f : β
G₁ G₂ : Graph α β
h : G₁.Inc e = G₂.Inc f
⊢ G₁.IsLink e = G₂.IsLink f

pp: case mpr
α : Type u_1
β : Type u_2
e f : β
G₁ G₂ : Graph α β
h : G₁.IsLink e = G₂.IsLink f
⊢ G₁.Inc e = G₂.Inc f

Tactic:
·

Goal state after:
case mp
α : Type u_1
β : Type u_2
e f : β
G₁ G₂ : Graph α β
h : G₁.Inc e = G₂.Inc f
⊢ G₁.IsLink e = G₂.IsLink f
Module: Mathlib.Combinatorics.Graph.Basic

Goal state before:
pp: case mpr
α : Type u_1
β : Type u_2
e f : β
G₁ G₂ : Graph α β
h : G₁.IsLink e = G₂.IsLink f
⊢ G₁.Inc e = G₂.Inc f

Tactic:
·

Goal state after:
case mpr
α : Type u_1
β : Type u_2
e f : β
G₁ G₂ : Graph α β
h : G₁.IsLink e = G₂.IsLink f
⊢ G₁.Inc e = G₂.Inc f
Module: Mathlib.Combinatorics.Graph.Basic

Goal state before:
pp: case e'_3.e'_3
α : Type u_1
β : Type u_2
G₁ G₂ : Graph α β
hV : V(G₁) = V(G₂)
h : ∀ (e : β) (x y : α), G₁.IsLink e x y ↔ G₂.IsLink e x y
⊢ V(G₂) = V(G₁)

pp: case e'_3.e'_4
α : Type u_1
β : Type u_2
G₁ G₂ : Graph α β
hV : V(G₁) = V(G₂)
h : ∀ (e : β) (x y : α), G₁.IsLink e x y ↔ G₂.IsLink e x y
⊢ G₂.IsLink = G₁.IsLink

pp: case e'_3.e'_5
α : Type u_1
β : Type u_2
G₁ G₂ : Graph α β
hV : V(G₁) = V(G₂)
h : ∀ (e : β) (x y : α), G₁.IsLink e x y ↔ G₂.IsLink e x y
⊢ E(G₂) = E(G₁)

Tactic:
·

Goal state after:
case e'_3.e'_3
α : Type u_1
β : Type u_2
G₁ G₂ : Graph α β
hV : V(G₁) = V(G₂)
h : ∀ (e : β) (x y : α), G₁.IsLink e x y ↔ G₂.IsLink e x y
⊢ V(G₂) = V(G₁)
Module: Mathlib.Combinatorics.Graph.Basic

Goal state before:
pp: case e'_3.e'_4
α : Type u_1
β : Type u_2
G₁ G₂ : Graph α β
hV : V(G₁) = V(G₂)
h : ∀ (e : β) (x y : α), G₁.IsLink e x y ↔ G₂.IsLink e x y
⊢ G₂.IsLink = G₁.IsLink

pp: case e'_3.e'_5
α : Type u_1
β : Type u_2
G₁ G₂ : Graph α β
hV : V(G₁) = V(G₂)
h : ∀ (e : β) (x y : α), G₁.IsLink e x y ↔ G₂.IsLink e x y
⊢ E(G₂) = E(G₁)

Tactic:
·

Goal state after:
case e'_3.e'_4
α : Type u_1
β : Type u_2
G₁ G₂ : Graph α β
hV : V(G₁) = V(G₂)
h : ∀ (e : β) (x y : α), G₁.IsLink e x y ↔ G₂.IsLink e x y
⊢ G₂.IsLink = G₁.IsLink
Module: Mathlib.Combinatorics.Graph.Basic

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
v : α
G : Graph α β
hv : v ∈ V(G)
hss : V(G).Subsingleton
hrw : V(G) = {v}
e : β
x : α
h : G.Inc e x
⊢ (bouquet v E(G)).Inc e x

pp: case refine_2
α : Type u_1
β : Type u_2
v : α
G : Graph α β
hv : v ∈ V(G)
hss : V(G).Subsingleton
hrw : V(G) = {v}
e : β
x : α
h : (bouquet v E(G)).Inc e x
⊢ G.Inc e x

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
v : α
G : Graph α β
hv : v ∈ V(G)
hss : V(G).Subsingleton
hrw : V(G) = {v}
e : β
x : α
h : G.Inc e x
⊢ (bouquet v E(G)).Inc e x
Module: Mathlib.Topology.Algebra.Semigroup

Goal state before:
pp: M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
⊢ ∃ m, m * m = m

pp: M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := ⋯
⊢ ∃ N', Minimal (fun x ↦ x ∈ S) N'

Tactic:
·

Goal state after:
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
⊢ ∃ m, m * m = m
Module: Mathlib.Topology.Algebra.Semigroup

Goal state before:
pp: case ht
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := ⋯
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
⊢ (fun x ↦ x * m) '' N ∈ S

pp: case hts
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := ⋯
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
⊢ (fun x ↦ x * m) '' N ⊆ N

Tactic:
·

Goal state after:
case ht
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := ⋯
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
⊢ (fun x ↦ x * m) '' N ∈ S
Module: Mathlib.Topology.Algebra.Semigroup

Goal state before:
pp: case hts
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := ⋯
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
⊢ (fun x ↦ x * m) '' N ⊆ N

Tactic:
·

Goal state after:
case hts
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := ⋯
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
⊢ (fun x ↦ x * m) '' N ⊆ N
Module: Mathlib.Topology.Algebra.Semigroup

Goal state before:
pp: case ht
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := ⋯
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun x ↦ x * m) '' N = N
⊢ N ∩ {m' | m' * m = m} ∈ S

pp: case hts
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := ⋯
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun x ↦ x * m) '' N = N
⊢ N ∩ {m' | m' * m = m} ⊆ N

Tactic:
·

Goal state after:
case ht
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := ⋯
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun x ↦ x * m) '' N = N
⊢ N ∩ {m' | m' * m = m} ∈ S
Module: Mathlib.Topology.Algebra.Semigroup

Goal state before:
pp: case ht.refine_1
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun x ↦ x * m) '' N = N
⊢ (N ∩ {m' | m' * m = m}).Nonempty

pp: case ht.refine_2
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun x ↦ x * m) '' N = N
⊢ ∀ m_1 ∈ N ∩ {m' | m' * m = m}, ∀ m' ∈ N ∩ {m' | m' * m = m}, m_1 * m' ∈ N ∩ {m' | m' * m = m}

Tactic:
·

Goal state after:
case ht.refine_1
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun x ↦ x * m) '' N = N
⊢ (N ∩ {m' | m' * m = m}).Nonempty
Module: Mathlib.Topology.Algebra.Semigroup

Goal state before:
pp: case ht.refine_2
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun x ↦ x * m) '' N = N
⊢ ∀ m_1 ∈ N ∩ {m' | m' * m = m}, ∀ m' ∈ N ∩ {m' | m' * m = m}, m_1 * m' ∈ N ∩ {m' | m' * m = m}

Tactic:
·

Goal state after:
case ht.refine_2
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
scaling_eq_self : (fun x ↦ x * m) '' N = N
⊢ ∀ m_1 ∈ N ∩ {m' | m' * m = m}, ∀ m' ∈ N ∩ {m' | m' * m = m}, m_1 * m' ∈ N ∩ {m' | m' * m = m}
Module: Mathlib.Topology.Algebra.Semigroup

Goal state before:
pp: case refine_1
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
⊢ (⋂₀ c).Nonempty

pp: case refine_2
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
m : M
hm : m ∈ ⋂₀ c
m' : M
hm' : m' ∈ ⋂₀ c
⊢ m * m' ∈ ⋂₀ c

Tactic:
·

Goal state after:
case refine_1
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
⊢ (⋂₀ c).Nonempty
Module: Mathlib.Topology.Algebra.Semigroup

Goal state before:
pp: case refine_1
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
⊢ (⋂₀ c).Nonempty

Tactic:
obtain rfl | hcnemp := c.eq_empty_or_nonempty

Goal state after:
case refine_1.inl
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
hcs : ∅ ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) ∅
⊢ (⋂₀ ∅).Nonempty

case refine_1.inr
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ (⋂₀ c).Nonempty
Module: Mathlib.Topology.Algebra.Semigroup

Goal state before:
pp: case refine_1.inl
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
hcs : ∅ ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) ∅
⊢ (⋂₀ ∅).Nonempty

pp: case refine_1.inr
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ (⋂₀ c).Nonempty

Tactic:
·

Goal state after:
case refine_1.inl
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
hcs : ∅ ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) ∅
⊢ (⋂₀ ∅).Nonempty
Module: Mathlib.Topology.Algebra.Semigroup

Goal state before:
pp: M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ ⋂₀ c = ⋂ i, ↑i

pp: case refine_1.inr.convert_1
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ Directed (fun x1 x2 ↦ x1 ⊇ x2) Subtype.val

pp: case refine_1.inr.convert_2
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ ∀ (i : ↑c), (↑i).Nonempty

pp: case refine_1.inr.convert_3
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ ∀ (i : ↑c), IsCompact ↑i

pp: case refine_1.inr.convert_4
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ ∀ (i : ↑c), IsClosed ↑i

Tactic:
·

Goal state after:
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ ⋂₀ c = ⋂ i, ↑i
Module: Mathlib.Topology.Algebra.Semigroup

Goal state before:
pp: case refine_1.inr.convert_1
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ Directed (fun x1 x2 ↦ x1 ⊇ x2) Subtype.val

pp: case refine_1.inr.convert_2
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ ∀ (i : ↑c), (↑i).Nonempty

pp: case refine_1.inr.convert_3
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ ∀ (i : ↑c), IsCompact ↑i

pp: case refine_1.inr.convert_4
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ ∀ (i : ↑c), IsClosed ↑i

Tactic:
·

Goal state after:
case refine_1.inr.convert_1
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcnemp : c.Nonempty
⊢ Directed (fun x1 x2 ↦ x1 ⊇ x2) Subtype.val
Module: Mathlib.Topology.Algebra.Semigroup

Goal state before:
pp: case refine_2
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
m : M
hm : m ∈ ⋂₀ c
m' : M
hm' : m' ∈ ⋂₀ c
⊢ m * m' ∈ ⋂₀ c

Tactic:
·

Goal state after:
case refine_2
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
c : Set (Set M)
hcs : c ⊆ S
hc : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
m : M
hm : m ∈ ⋂₀ c
m' : M
hm' : m' ∈ ⋂₀ c
⊢ m * m' ∈ ⋂₀ c
Module: Mathlib.Combinatorics.HalesJewett

Goal state before:
pp: case refine_1
α : Type u_2
ι : Type u_3
inst✝ : Nontrivial α
l m : Line α ι
i : ι
a b : α
hba : b ≠ a
hlm : ∀ (x : α) (x_1 : ι), ↑l x x_1 = ↑m x x_1
h : l.idxFun i = some a
⊢ m.idxFun i = some a

pp: case refine_2
α : Type u_2
ι : Type u_3
inst✝ : Nontrivial α
l m : Line α ι
i : ι
a b : α
hba : b ≠ a
hlm : ∀ (x : α) (x_1 : ι), ↑l x x_1 = ↑m x x_1
h : m.idxFun i = some a
⊢ l.idxFun i = some a

Tactic:
·

Goal state after:
case refine_1
α : Type u_2
ι : Type u_3
inst✝ : Nontrivial α
l m : Line α ι
i : ι
a b : α
hba : b ≠ a
hlm : ∀ (x : α) (x_1 : ι), ↑l x x_1 = ↑m x x_1
h : l.idxFun i = some a
⊢ m.idxFun i = some a
Module: Mathlib.Combinatorics.HalesJewett

Goal state before:
pp: case refine_2
α : Type u_2
ι : Type u_3
inst✝ : Nontrivial α
l m : Line α ι
i : ι
a b : α
hba : b ≠ a
hlm : ∀ (x : α) (x_1 : ι), ↑l x x_1 = ↑m x x_1
h : m.idxFun i = some a
⊢ l.idxFun i = some a

Tactic:
·

Goal state after:
case refine_2
α : Type u_2
ι : Type u_3
inst✝ : Nontrivial α
l m : Line α ι
i : ι
a b : α
hba : b ≠ a
hlm : ∀ (x : α) (x_1 : ι), ↑l x x_1 = ↑m x x_1
h : m.idxFun i = some a
⊢ l.idxFun i = some a
Module: Mathlib.Combinatorics.Hindman

Goal state before:
pp: case refine_5
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
h : ∃ m ∈ S, m * m = m
⊢ ∃ U, U * U = U ∧ ∀ᶠ (m : M) in ↑U, m ∈ FP a

pp: case refine_1
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
⊢ ∀ (r : Ultrafilter M), Continuous fun x ↦ x * r

pp: case refine_2
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
⊢ S.Nonempty

pp: case refine_3
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
⊢ IsCompact S

pp: case refine_4
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
⊢ ∀ x ∈ S, ∀ y ∈ S, x * y ∈ S

Tactic:
·

Goal state after:
case refine_5
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
h : ∃ m ∈ S, m * m = m
⊢ ∃ U, U * U = U ∧ ∀ᶠ (m : M) in ↑U, m ∈ FP a