Theorem: Finset.card_mul_le_card_shadow_mul
File: Mathlib/Combinatorics/SetFamily/LYM.lean

tactic: refine le_trans ?_ tsub_tsub_le_tsub_add
state_before: case refine_2
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
⊢ (bipartiteAbove (fun x x_1 => x ⊆ x_1) 𝒜 s).card ≤ Fintype.card α - r + 1
state_after: case refine_2
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
⊢ (bipartiteAbove (fun x x_1 => x ⊆ x_1) 𝒜 s).card ≤ Fintype.card α - (r - 1)

tactic: rw [← (Set.Sized.shadow h𝒜) hs, ← card_compl, ← card_image_of_injOn (insert_inj_on' _)]
state_before: case refine_2
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
⊢ (bipartiteAbove (fun x x_1 => x ⊆ x_1) 𝒜 s).card ≤ Fintype.card α - (r - 1)
state_after: case refine_2
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
⊢ (bipartiteAbove (fun x x_1 => x ⊆ x_1) 𝒜 s).card ≤ (image (fun a => insert a s) sᶜ).card

tactic: refine card_le_card fun t ht => ?_
state_before: case refine_2
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
⊢ (bipartiteAbove (fun x x_1 => x ⊆ x_1) 𝒜 s).card ≤ (image (fun a => insert a s) sᶜ).card
state_after: case refine_2
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
t : Finset α
ht : t ∈ bipartiteAbove (fun x x_1 => x ⊆ x_1) 𝒜 s
⊢ t ∈ image (fun a => insert a s) sᶜ

tactic: rw [mem_bipartiteAbove] at ht
state_before: case refine_2
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
t : Finset α
ht : t ∈ bipartiteAbove (fun x x_1 => x ⊆ x_1) 𝒜 s
⊢ t ∈ image (fun a => insert a s) sᶜ
state_after: case refine_2
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
t : Finset α
ht : t ∈ 𝒜 ∧ s ⊆ t
⊢ t ∈ image (fun a => insert a s) sᶜ

tactic: have : ∅ ∉ 𝒜 := by
  rw [← mem_coe, h𝒜.empty_mem_iff, coe_eq_singleton]
  rintro rfl
  rw [shadow_singleton_empty] at hs
  exact not_mem_empty s hs
state_before: case refine_2
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
t : Finset α
ht : t ∈ 𝒜 ∧ s ⊆ t
⊢ t ∈ image (fun a => insert a s) sᶜ
state_after: case refine_2
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
t : Finset α
ht : t ∈ 𝒜 ∧ s ⊆ t
this : ∅ ∉ 𝒜
⊢ t ∈ image (fun a => insert a s) sᶜ

tactic: have h := exists_eq_insert_iff.2 ⟨ht.2, by
  rw [(sized_shadow_iff this).1 (Set.Sized.shadow h𝒜) ht.1, (Set.Sized.shadow h𝒜) hs]⟩
state_before: case refine_2
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
t : Finset α
ht : t ∈ 𝒜 ∧ s ⊆ t
this : ∅ ∉ 𝒜
⊢ t ∈ image (fun a => insert a s) sᶜ
state_after: case refine_2
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
t : Finset α
ht : t ∈ 𝒜 ∧ s ⊆ t
this : ∅ ∉ 𝒜
h : ∃ a ∉ s, insert a s = t
⊢ t ∈ image (fun a => insert a s) sᶜ

tactic: rcases h with ⟨a, ha, rfl⟩
state_before: case refine_2
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
t : Finset α
ht : t ∈ 𝒜 ∧ s ⊆ t
this : ∅ ∉ 𝒜
h : ∃ a ∉ s, insert a s = t
⊢ t ∈ image (fun a => insert a s) sᶜ
state_after: case refine_2.intro.intro
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
this : ∅ ∉ 𝒜
a : α
ha : a ∉ s
ht : insert a s ∈ 𝒜 ∧ s ⊆ insert a s
⊢ insert a s ∈ image (fun a => insert a s) sᶜ

tactic: exact mem_image_of_mem _ (mem_compl.2 ha)
state_before: case refine_2.intro.intro
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
this : ∅ ∉ 𝒜
a : α
ha : a ∉ s
ht : insert a s ∈ 𝒜 ∧ s ⊆ insert a s
⊢ insert a s ∈ image (fun a => insert a s) sᶜ
state_after: no goals

tactic: rw [← h𝒜 hs, ← card_image_of_injOn s.erase_injOn]
state_before: case refine_1
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ 𝒜
⊢ r ≤ (bipartiteBelow (fun x x_1 => x ⊆ x_1) (∂ 𝒜) s).card
state_after: case refine_1
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ 𝒜
⊢ (image s.erase s).card ≤ (bipartiteBelow (fun x x_1 => x ⊆ x_1) (∂ 𝒜) s).card

tactic: refine card_le_card ?_
state_before: case refine_1
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ 𝒜
⊢ (image s.erase s).card ≤ (bipartiteBelow (fun x x_1 => x ⊆ x_1) (∂ 𝒜) s).card
state_after: case refine_1
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ 𝒜
⊢ image s.erase s ⊆ bipartiteBelow (fun x x_1 => x ⊆ x_1) (∂ 𝒜) s

tactic: simp_rw [image_subset_iff, mem_bipartiteBelow]
state_before: case refine_1
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ 𝒜
⊢ image s.erase s ⊆ bipartiteBelow (fun x x_1 => x ⊆ x_1) (∂ 𝒜) s
state_after: case refine_1
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ 𝒜
⊢ ∀ x ∈ s, s.erase x ∈ ∂ 𝒜 ∧ s.erase x ⊆ s

tactic: exact fun a ha => ⟨erase_mem_shadow hs ha, erase_subset _ _⟩
state_before: case refine_1
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ 𝒜
⊢ ∀ x ∈ s, s.erase x ∈ ∂ 𝒜 ∧ s.erase x ⊆ s
state_after: no goals

tactic: rw [← mem_coe, h𝒜.empty_mem_iff, coe_eq_singleton]
state_before: 𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
t : Finset α
ht : t ∈ 𝒜 ∧ s ⊆ t
⊢ ∅ ∉ 𝒜
state_after: 𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
t : Finset α
ht : t ∈ 𝒜 ∧ s ⊆ t
⊢ ¬𝒜 = {∅}

tactic: rintro rfl
state_before: 𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
t : Finset α
ht : t ∈ 𝒜 ∧ s ⊆ t
⊢ ¬𝒜 = {∅}
state_after: 𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
r : ℕ
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s t : Finset α
h𝒜 : Set.Sized r ↑{∅}
hs : s ∈ ∂ {∅}
ht : t ∈ {∅} ∧ s ⊆ t
⊢ False

tactic: rw [shadow_singleton_empty] at hs
state_before: 𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
r : ℕ
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s t : Finset α
h𝒜 : Set.Sized r ↑{∅}
hs : s ∈ ∂ {∅}
ht : t ∈ {∅} ∧ s ⊆ t
⊢ False
state_after: 𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
r : ℕ
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s t : Finset α
h𝒜 : Set.Sized r ↑{∅}
hs : s ∈ ∅
ht : t ∈ {∅} ∧ s ⊆ t
⊢ False

tactic: exact not_mem_empty s hs
state_before: 𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
r : ℕ
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s t : Finset α
h𝒜 : Set.Sized r ↑{∅}
hs : s ∈ ∅
ht : t ∈ {∅} ∧ s ⊆ t
⊢ False
state_after: no goals

tactic: rw [(sized_shadow_iff this).1 (Set.Sized.shadow h𝒜) ht.1, (Set.Sized.shadow h𝒜) hs]
state_before: 𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)
s : Finset α
hs : s ∈ ∂ 𝒜
t : Finset α
ht : t ∈ 𝒜 ∧ s ⊆ t
this : ∅ ∉ 𝒜
⊢ s.card + 1 = t.card
state_after: no goals