-- Mathlib/Combinatorics/SetFamily/FourFunctions.lean
/-- The **Four Functions Theorem**, aka **Ahlswede-Daykin Inequality**. -/
lemma four_functions_theorem [DecidableEq α] (h₁ : 0 ≤ f₁) (h₂ : 0 ≤ f₂) (h₃ : 0 ≤ f₃) (h₄ : 0 ≤ f₄)
    (h : ∀ a b, f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f₄ (a ⊔ b)) (s t : Finset α) :
    (∑ a ∈ s, f₁ a) * ∑ a ∈ t, f₂ a ≤ (∑ a ∈ s ⊼ t, f₃ a) * ∑ a ∈ s ⊻ t, f₄ a

Proof state:
α : Type u_1
β : Type u_2
inst✝³ : DistribLattice α
inst✝² : LinearOrderedCommSemiring β
inst✝¹ : ExistsAddOfLE β
f₁ f₂ f₃ f₄ : α → β
inst✝ : DecidableEq α
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ (a b : α), f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f₄ (a ⊔ b)
s t : Finset α
L : Sublattice α := { carrier := latticeClosure (↑s ∪ ↑t), supClosed' := ⋯, infClosed' := ⋯ }
this : Finite ↥L
⊢ (∑ a ∈ s, f₁ a) * ∑ a ∈ t, f₂ a ≤ (∑ a ∈ s ⊼ t, f₃ a) * ∑ a ∈ s ⊻ t, f₄ a

Tactic:
set s' : Finset L := s.preimage (↑) Subtype.coe_injective.injOn

Result:
α : Type u_1
β : Type u_2
inst✝³ : DistribLattice α
inst✝² : LinearOrderedCommSemiring β
inst✝¹ : ExistsAddOfLE β
f₁ f₂ f₃ f₄ : α → β
inst✝ : DecidableEq α
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ (a b : α), f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f₄ (a ⊔ b)
s t : Finset α
L : Sublattice α := { carrier := latticeClosure (↑s ∪ ↑t), supClosed' := ⋯, infClosed' := ⋯ }
this : Finite ↥L
s' : Finset ↥L := s.preimage Subtype.val ⋯
⊢ (∑ a ∈ s, f₁ a) * ∑ a ∈ t, f₂ a ≤ (∑ a ∈ s ⊼ t, f₃ a) * ∑ a ∈ s ⊻ t, f₄ a

Proof state:
case intro.intro.intro.intro
α : Type u_1
β✝ : Type u_2
inst✝³ : DistribLattice α
inst✝² : LinearOrderedCommSemiring β✝
inst✝¹ : ExistsAddOfLE β✝
f₁ f₂ f₃ f₄ : α → β✝
inst✝ : DecidableEq α
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ (a b : α), f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f₄ (a ⊔ b)
s t : Finset α
L : Sublattice α := { carrier := latticeClosure (↑s ∪ ↑t), supClosed' := ⋯, infClosed' := ⋯ }
this : Finite ↥L
t' : Finset ↥L
ht' : map { toFun := ⇑L.subtype, inj' := ⋯ } t' = t
s' : Finset ↥L
hs' : map { toFun := ⇑L.subtype, inj' := ⋯ } s' = s
β : Type u_1
w✝¹ : DecidableEq β
w✝ : Fintype β
g : LatticeHom (↥L) (Finset β)
hg : Injective ⇑g
⊢ (∑ a ∈ s, f₁ a) * ∑ a ∈ t, f₂ a ≤ (∑ a ∈ s ⊼ t, f₃ a) * ∑ a ∈ s ⊻ t, f₄ a

Tactic:
have := four_functions_theorem_aux (extend g (f₁ ∘ (↑)) 0) (extend g (f₂ ∘ (↑)) 0)
  (extend g (f₃ ∘ (↑)) 0) (extend g (f₄ ∘ (↑)) 0) (extend_nonneg (fun _ ↦ h₁ _) le_rfl)
  (extend_nonneg (fun _ ↦ h₂ _) le_rfl) (extend_nonneg (fun _ ↦ h₃ _) le_rfl)
  (extend_nonneg (fun _ ↦ h₄ _) le_rfl) ?_ (s'.map ⟨g, hg⟩) (t'.map ⟨g, hg⟩)

Result:
case intro.intro.intro.intro.refine_2
α : Type u_1
β✝ : Type u_2
inst✝³ : DistribLattice α
inst✝² : LinearOrderedCommSemiring β✝
inst✝¹ : ExistsAddOfLE β✝
f₁ f₂ f₃ f₄ : α → β✝
inst✝ : DecidableEq α
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ (a b : α), f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f₄ (a ⊔ b)
s t : Finset α
L : Sublattice α := { carrier := latticeClosure (↑s ∪ ↑t), supClosed' := ⋯, infClosed' := ⋯ }
this✝ : Finite ↥L
t' : Finset ↥L
ht' : map { toFun := ⇑L.subtype, inj' := ⋯ } t' = t
s' : Finset ↥L
hs' : map { toFun := ⇑L.subtype, inj' := ⋯ } s' = s
β : Type u_1
w✝¹ : DecidableEq β
w✝ : Fintype β
g : LatticeHom (↥L) (Finset β)
hg : Injective ⇑g
this :
  (∑ s ∈ map { toFun := ⇑g, inj' := hg } s', extend (⇑g) (f₁ ∘ Subtype.val) 0 s) *
      ∑ s ∈ map { toFun := ⇑g, inj' := hg } t', extend (⇑g) (f₂ ∘ Subtype.val) 0 s ≤
    (∑ s ∈ map { toFun := ⇑g, inj' := hg } s' ⊼ map { toFun := ⇑g, inj' := hg } t',
        extend (⇑g) (f₃ ∘ Subtype.val) 0 s) *
      ∑ s ∈ map { toFun := ⇑g, inj' := hg } s' ⊻ map { toFun := ⇑g, inj' := hg } t', extend (⇑g) (f₄ ∘ Subtype.val) 0 s
⊢ (∑ a ∈ s, f₁ a) * ∑ a ∈ t, f₂ a ≤ (∑ a ∈ s ⊼ t, f₃ a) * ∑ a ∈ s ⊻ t, f₄ a

case intro.intro.intro.intro.refine_1
α : Type u_1
β✝ : Type u_2
inst✝³ : DistribLattice α
inst✝² : LinearOrderedCommSemiring β✝
inst✝¹ : ExistsAddOfLE β✝
f₁ f₂ f₃ f₄ : α → β✝
inst✝ : DecidableEq α
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ (a b : α), f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f₄ (a ⊔ b)
s t : Finset α
L : Sublattice α := { carrier := latticeClosure (↑s ∪ ↑t), supClosed' := ⋯, infClosed' := ⋯ }
this : Finite ↥L
t' : Finset ↥L
ht' : map { toFun := ⇑L.subtype, inj' := ⋯ } t' = t
s' : Finset ↥L
hs' : map { toFun := ⇑L.subtype, inj' := ⋯ } s' = s
β : Type u_1
w✝¹ : DecidableEq β
w✝ : Fintype β
g : LatticeHom (↥L) (Finset β)
hg : Injective ⇑g
⊢ ∀ (s t : Finset β),
    extend (⇑g) (f₁ ∘ Subtype.val) 0 s * extend (⇑g) (f₂ ∘ Subtype.val) 0 t ≤
      extend (⇑g) (f₃ ∘ Subtype.val) 0 (s ∩ t) * extend (⇑g) (f₄ ∘ Subtype.val) 0 (s ∪ t)

Proof state:
α : Type u_1
β : Type u_2
inst✝³ : DistribLattice α
inst✝² : LinearOrderedCommSemiring β
inst✝¹ : ExistsAddOfLE β
f₁ f₂ f₃ f₄ : α → β
inst✝ : DecidableEq α
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ (a b : α), f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f₄ (a ⊔ b)
s t : Finset α
L : Sublattice α := { carrier := latticeClosure (↑s ∪ ↑t), supClosed' := ⋯, infClosed' := ⋯ }
this : Finite ↥L
s' : Finset ↥L := s.preimage Subtype.val ⋯
t' : Finset ↥L := t.preimage Subtype.val ⋯
⊢ (∑ a ∈ s, f₁ a) * ∑ a ∈ t, f₂ a ≤ (∑ a ∈ s ⊼ t, f₃ a) * ∑ a ∈ s ⊻ t, f₄ a

Tactic:
have hs' : s'.map ⟨L.subtype, Subtype.coe_injective⟩ = s := by
  simp [s', map_eq_image, image_preimage, filter_eq_self]
  exact fun a ha ↦ subset_latticeClosure <| Set.subset_union_left ha

Result:
α : Type u_1
β : Type u_2
inst✝³ : DistribLattice α
inst✝² : LinearOrderedCommSemiring β
inst✝¹ : ExistsAddOfLE β
f₁ f₂ f₃ f₄ : α → β
inst✝ : DecidableEq α
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ (a b : α), f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f₄ (a ⊔ b)
s t : Finset α
L : Sublattice α := { carrier := latticeClosure (↑s ∪ ↑t), supClosed' := ⋯, infClosed' := ⋯ }
this : Finite ↥L
s' : Finset ↥L := s.preimage Subtype.val ⋯
t' : Finset ↥L := t.preimage Subtype.val ⋯
hs' : map { toFun := ⇑L.subtype, inj' := ⋯ } s' = s
⊢ (∑ a ∈ s, f₁ a) * ∑ a ∈ t, f₂ a ≤ (∑ a ∈ s ⊼ t, f₃ a) * ∑ a ∈ s ⊻ t, f₄ a

Proof state:
α : Type u_1
β : Type u_2
inst✝³ : DistribLattice α
inst✝² : LinearOrderedCommSemiring β
inst✝¹ : ExistsAddOfLE β
f₁ f₂ f₃ f₄ : α → β
inst✝ : DecidableEq α
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ (a b : α), f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f₄ (a ⊔ b)
s t : Finset α
L : Sublattice α := { carrier := latticeClosure (↑s ∪ ↑t), supClosed' := ⋯, infClosed' := ⋯ }
this : Finite ↥L
t' : Finset ↥L
ht' : map { toFun := ⇑L.subtype, inj' := ⋯ } t' = t
s' : Finset ↥L
hs' : map { toFun := ⇑L.subtype, inj' := ⋯ } s' = s
⊢ (∑ a ∈ s, f₁ a) * ∑ a ∈ t, f₂ a ≤ (∑ a ∈ s ⊼ t, f₃ a) * ∑ a ∈ s ⊻ t, f₄ a

Tactic:
obtain ⟨β, _, _, g, hg⟩ := exists_birkhoff_representation L

Result:
case intro.intro.intro.intro
α : Type u_1
β✝ : Type u_2
inst✝³ : DistribLattice α
inst✝² : LinearOrderedCommSemiring β✝
inst✝¹ : ExistsAddOfLE β✝
f₁ f₂ f₃ f₄ : α → β✝
inst✝ : DecidableEq α
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ (a b : α), f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f₄ (a ⊔ b)
s t : Finset α
L : Sublattice α := { carrier := latticeClosure (↑s ∪ ↑t), supClosed' := ⋯, infClosed' := ⋯ }
this : Finite ↥L
t' : Finset ↥L
ht' : map { toFun := ⇑L.subtype, inj' := ⋯ } t' = t
s' : Finset ↥L
hs' : map { toFun := ⇑L.subtype, inj' := ⋯ } s' = s
β : Type u_1
w✝¹ : DecidableEq β
w✝ : Fintype β
g : LatticeHom (↥L) (Finset β)
hg : Injective ⇑g
⊢ (∑ a ∈ s, f₁ a) * ∑ a ∈ t, f₂ a ≤ (∑ a ∈ s ⊼ t, f₃ a) * ∑ a ∈ s ⊻ t, f₄ a