internlm/Lean-Github huggingface dataset:
Theorem: Usa2002P1.usa2002_p1_generalized
Proof state before:
case h.left.intro.intro.inr.inr
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
s1 s2 : Finset S
hs12 : f s1 = f s2
h4 : f s1 = Color.blue → s ∉ s1 ∧ f' (Finset.subtype (fun a => a ≠ s) s1) = Color.blue
mpr✝¹ : s ∉ s1 ∧ f' (Finset.subtype (fun a => a ≠ s) s1) = Color.blue → f s1 = Color.blue
h4' : f s2 = Color.blue → s ∉ s2 ∧ f' (Finset.subtype (fun a => a ≠ s) s2) = Color.blue
mpr✝ : s ∉ s2 ∧ f' (Finset.subtype (fun a => a ≠ s) s2) = Color.blue → f s2 = Color.blue
hfs1 : ¬f s1 = Color.blue
hss : s ∉ s1 ∧ s ∉ s2
⊢ (if False then Color.red else f' (Finset.subtype (fun a => a ≠ s) (s1 ∪ s2))) =
    if s ∈ s1 then Color.red else f' (Finset.subtype (fun a => a ≠ s) s1)
Tactic:
simp only [hss.1, Finset.subtype_union, ite_false]
Proof state after:
case h.left.intro.intro.inr.inr
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
s1 s2 : Finset S
hs12 : f s1 = f s2
h4 : f s1 = Color.blue → s ∉ s1 ∧ f' (Finset.subtype (fun a => a ≠ s) s1) = Color.blue
mpr✝¹ : s ∉ s1 ∧ f' (Finset.subtype (fun a => a ≠ s) s1) = Color.blue → f s1 = Color.blue
h4' : f s2 = Color.blue → s ∉ s2 ∧ f' (Finset.subtype (fun a => a ≠ s) s2) = Color.blue
mpr✝ : s ∉ s2 ∧ f' (Finset.subtype (fun a => a ≠ s) s2) = Color.blue → f s2 = Color.blue
hfs1 : ¬f s1 = Color.blue
hss : s ∉ s1 ∧ s ∉ s2
⊢ f' (Finset.subtype (fun a => a ≠ s) s1 ∪ Finset.subtype (fun a => a ≠ s) s2) = f' (Finset.subtype (fun a => a ≠ s) s1)
Theorem: Usa2002P1.usa2002_p1_generalized
Proof state before:
case h.left.intro.intro.inr.inr
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
s1 s2 : Finset S
hs12 : f s1 = f s2
h4 : f s1 = Color.blue → s ∉ s1 ∧ f' (Finset.subtype (fun a => a ≠ s) s1) = Color.blue
mpr✝¹ : s ∉ s1 ∧ f' (Finset.subtype (fun a => a ≠ s) s1) = Color.blue → f s1 = Color.blue
h4' : f s2 = Color.blue → s ∉ s2 ∧ f' (Finset.subtype (fun a => a ≠ s) s2) = Color.blue
mpr✝ : s ∉ s2 ∧ f' (Finset.subtype (fun a => a ≠ s) s2) = Color.blue → f s2 = Color.blue
hfs1 : ¬f s1 = Color.blue
hss : s ∉ s1 ∧ s ∉ s2
⊢ f' (Finset.subtype (fun a => a ≠ s) s1 ∪ Finset.subtype (fun a => a ≠ s) s2) = f' (Finset.subtype (fun a => a ≠ s) s1)
Tactic:
apply hf1'
Proof state after:
case h.left.intro.intro.inr.inr.a
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
s1 s2 : Finset S
hs12 : f s1 = f s2
h4 : f s1 = Color.blue → s ∉ s1 ∧ f' (Finset.subtype (fun a => a ≠ s) s1) = Color.blue
mpr✝¹ : s ∉ s1 ∧ f' (Finset.subtype (fun a => a ≠ s) s1) = Color.blue → f s1 = Color.blue
h4' : f s2 = Color.blue → s ∉ s2 ∧ f' (Finset.subtype (fun a => a ≠ s) s2) = Color.blue
mpr✝ : s ∉ s2 ∧ f' (Finset.subtype (fun a => a ≠ s) s2) = Color.blue → f s2 = Color.blue
hfs1 : ¬f s1 = Color.blue
hss : s ∉ s1 ∧ s ∉ s2
⊢ f' (Finset.subtype (fun a => a ≠ s) s1) = f' (Finset.subtype (fun a => a ≠ s) s2)
Theorem: Usa2002P1.usa2002_p1_generalized
Proof state before:
case h.left.intro.intro.inr.inr.a
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
s1 s2 : Finset S
hs12 : f s1 = f s2
h4 : f s1 = Color.blue → s ∉ s1 ∧ f' (Finset.subtype (fun a => a ≠ s) s1) = Color.blue
mpr✝¹ : s ∉ s1 ∧ f' (Finset.subtype (fun a => a ≠ s) s1) = Color.blue → f s1 = Color.blue
h4' : f s2 = Color.blue → s ∉ s2 ∧ f' (Finset.subtype (fun a => a ≠ s) s2) = Color.blue
mpr✝ : s ∉ s2 ∧ f' (Finset.subtype (fun a => a ≠ s) s2) = Color.blue → f s2 = Color.blue
hfs1 : ¬f s1 = Color.blue
hss : s ∉ s1 ∧ s ∉ s2
⊢ f' (Finset.subtype (fun a => a ≠ s) s1) = f' (Finset.subtype (fun a => a ≠ s) s2)
Tactic:
simp only [hss.1, hss.2, f, S'] at hs12
Proof state after:
case h.left.intro.intro.inr.inr.a
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
s1 s2 : Finset S
h4 : f s1 = Color.blue → s ∉ s1 ∧ f' (Finset.subtype (fun a => a ≠ s) s1) = Color.blue
mpr✝¹ : s ∉ s1 ∧ f' (Finset.subtype (fun a => a ≠ s) s1) = Color.blue → f s1 = Color.blue
h4' : f s2 = Color.blue → s ∉ s2 ∧ f' (Finset.subtype (fun a => a ≠ s) s2) = Color.blue
mpr✝ : s ∉ s2 ∧ f' (Finset.subtype (fun a => a ≠ s) s2) = Color.blue → f s2 = Color.blue
hfs1 : ¬f s1 = Color.blue
hss : s ∉ s1 ∧ s ∉ s2
hs12 :
  (if False then Color.red else f' (Finset.subtype (fun a => a ≠ s) s1)) =
    if False then Color.red else f' (Finset.subtype (fun a => a ≠ s) s2)
⊢ f' (Finset.subtype (fun a => a ≠ s) s1) = f' (Finset.subtype (fun a => a ≠ s) s2)
Theorem: Usa2002P1.usa2002_p1_generalized
Proof state before:
case h.left.intro.intro.inr.inr.a
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
s1 s2 : Finset S
h4 : f s1 = Color.blue → s ∉ s1 ∧ f' (Finset.subtype (fun a => a ≠ s) s1) = Color.blue
mpr✝¹ : s ∉ s1 ∧ f' (Finset.subtype (fun a => a ≠ s) s1) = Color.blue → f s1 = Color.blue
h4' : f s2 = Color.blue → s ∉ s2 ∧ f' (Finset.subtype (fun a => a ≠ s) s2) = Color.blue
mpr✝ : s ∉ s2 ∧ f' (Finset.subtype (fun a => a ≠ s) s2) = Color.blue → f s2 = Color.blue
hfs1 : ¬f s1 = Color.blue
hss : s ∉ s1 ∧ s ∉ s2
hs12 :
  (if False then Color.red else f' (Finset.subtype (fun a => a ≠ s) s1)) =
    if False then Color.red else f' (Finset.subtype (fun a => a ≠ s) s2)
⊢ f' (Finset.subtype (fun a => a ≠ s) s1) = f' (Finset.subtype (fun a => a ≠ s) s2)
Tactic:
exact hs12
Proof state after:
no goals
Theorem: Usa2002P1.usa2002_p1_generalized
Proof state before:
case h.right
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
⊢ Fintype.card { a // f a = Color.red } = N
Tactic:
have h2' : ∀ (a : Finset S),
    f a = Color.red ↔ s ∈ a ∨
         f' (Finset.subtype (fun a => a ≠ s) a) = Color.red := by
  intro a
  simp only [ne_eq, dite_eq_ite, ite_eq_left_iff, f, S']
  tauto
Proof state after:
case h.right
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
h2' : ∀ (a : Finset S), f a = Color.red ↔ s ∈ a ∨ f' (Finset.subtype (fun a => a ≠ s) a) = Color.red
⊢ Fintype.card { a // f a = Color.red } = N
Theorem: Usa2002P1.usa2002_p1_generalized
Proof state before:
case h.right
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
h2' : ∀ (a : Finset S), f a = Color.red ↔ s ∈ a ∨ f' (Finset.subtype (fun a => a ≠ s) a) = Color.red
⊢ Fintype.card { a // f a = Color.red } = N
Tactic:
let b : { a // f a = Color.red } →
       { a : Finset S // s ∈ a } ⊕ { a' // f' a' = Color.red } :=
  fun ⟨a, ha⟩ ↦
    if h : s ∈ a
    then Sum.inl ⟨a, h⟩
    else Sum.inr ⟨Finset.subtype _ a, by
      match (h2' a).mp ha with
      | .inl h' => contradiction
      | .inr hh => exact hh ⟩
Proof state after:
case h.right
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
h2' : ∀ (a : Finset S), f a = Color.red ↔ s ∈ a ∨ f' (Finset.subtype (fun a => a ≠ s) a) = Color.red
b : { a // f a = Color.red } → { a // s ∈ a } ⊕ { a' // f' a' = Color.red } :=
  fun x =>
    match x with
    | ⟨a, ha⟩ => if h : s ∈ a then Sum.inl ⟨a, h⟩ else Sum.inr ⟨Finset.subtype (fun a => a ≠ s) a, ⋯⟩
⊢ Fintype.card { a // f a = Color.red } = N
Theorem: Usa2002P1.usa2002_p1_generalized
Proof state before:
case h.right
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
h2' : ∀ (a : Finset S), f a = Color.red ↔ s ∈ a ∨ f' (Finset.subtype (fun a => a ≠ s) a) = Color.red
b : { a // f a = Color.red } → { a // s ∈ a } ⊕ { a' // f' a' = Color.red } :=
  fun x =>
    match x with
    | ⟨a, ha⟩ => if h : s ∈ a then Sum.inl ⟨a, h⟩ else Sum.inr ⟨Finset.subtype (fun a => a ≠ s) a, ⋯⟩
h3 : Function.Bijective b
⊢ Fintype.card { a // f a = Color.red } = N
Tactic:
rw [Fintype.card_of_bijective h3]
Proof state after:
case h.right
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
h2' : ∀ (a : Finset S), f a = Color.red ↔ s ∈ a ∨ f' (Finset.subtype (fun a => a ≠ s) a) = Color.red
b : { a // f a = Color.red } → { a // s ∈ a } ⊕ { a' // f' a' = Color.red } :=
  fun x =>
    match x with
    | ⟨a, ha⟩ => if h : s ∈ a then Sum.inl ⟨a, h⟩ else Sum.inr ⟨Finset.subtype (fun a => a ≠ s) a, ⋯⟩
h3 : Function.Bijective b
⊢ Fintype.card ({ a // s ∈ a } ⊕ { a' // f' a' = Color.red }) = N
Theorem: Usa2002P1.usa2002_p1_generalized
Proof state before:
case h.right
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
h2' : ∀ (a : Finset S), f a = Color.red ↔ s ∈ a ∨ f' (Finset.subtype (fun a => a ≠ s) a) = Color.red
b : { a // f a = Color.red } → { a // s ∈ a } ⊕ { a' // f' a' = Color.red } :=
  fun x =>
    match x with
    | ⟨a, ha⟩ => if h : s ∈ a then Sum.inl ⟨a, h⟩ else Sum.inr ⟨Finset.subtype (fun a => a ≠ s) a, ⋯⟩
h3 : Function.Bijective b
h4 : Fintype.card { a // s ∈ a } = 2 ^ k
⊢ Fintype.card ({ a // s ∈ a } ⊕ { a' // f' a' = Color.red }) = N
Tactic:
simp only [Fintype.card_sum, h4, hf2']
Proof state after:
case h.right
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
h2' : ∀ (a : Finset S), f a = Color.red ↔ s ∈ a ∨ f' (Finset.subtype (fun a => a ≠ s) a) = Color.red
b : { a // f a = Color.red } → { a // s ∈ a } ⊕ { a' // f' a' = Color.red } :=
  fun x =>
    match x with
    | ⟨a, ha⟩ => if h : s ∈ a then Sum.inl ⟨a, h⟩ else Sum.inr ⟨Finset.subtype (fun a => a ≠ s) a, ⋯⟩
h3 : Function.Bijective b
h4 : Fintype.card { a // s ∈ a } = 2 ^ k
⊢ 2 ^ k + (N - 2 ^ k) = N
Theorem: Usa2002P1.usa2002_p1_generalized
Proof state before:
case h.right
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
h2' : ∀ (a : Finset S), f a = Color.red ↔ s ∈ a ∨ f' (Finset.subtype (fun a => a ≠ s) a) = Color.red
b : { a // f a = Color.red } → { a // s ∈ a } ⊕ { a' // f' a' = Color.red } :=
  fun x =>
    match x with
    | ⟨a, ha⟩ => if h : s ∈ a then Sum.inl ⟨a, h⟩ else Sum.inr ⟨Finset.subtype (fun a => a ≠ s) a, ⋯⟩
h3 : Function.Bijective b
h4 : Fintype.card { a // s ∈ a } = 2 ^ k
⊢ 2 ^ k + (N - 2 ^ k) = N
Tactic:
rw [add_tsub_cancel_iff_le]
Proof state after:
case h.right
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
h2' : ∀ (a : Finset S), f a = Color.red ↔ s ∈ a ∨ f' (Finset.subtype (fun a => a ≠ s) a) = Color.red
b : { a // f a = Color.red } → { a // s ∈ a } ⊕ { a' // f' a' = Color.red } :=
  fun x =>
    match x with
    | ⟨a, ha⟩ => if h : s ∈ a then Sum.inl ⟨a, h⟩ else Sum.inr ⟨Finset.subtype (fun a => a ≠ s) a, ⋯⟩
h3 : Function.Bijective b
h4 : Fintype.card { a // s ∈ a } = 2 ^ k
⊢ 2 ^ k ≤ N
Theorem: Usa2002P1.usa2002_p1_generalized
Proof state before:
case h.right
k : ℕ
ih :
  ∀ {α : Type} [inst : DecidableEq α] [inst_1 : Fintype α],
    Fintype.card α = k →
      ∀ N ≤ 2 ^ k,
        ∃ f, (∀ (s1 s2 : Finset α), f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧ Fintype.card { a // f a = Color.red } = N
S : Type
hde : DecidableEq S
hft : Fintype S
hs : Fintype.card S = k + 1
N : ℕ
hN : N ≤ 2 ^ (k + 1)
s : S
S' : Type := { a // a ≠ s }
hs' : Fintype.card S' = k
hg : N > 2 ^ k
hl : N - 2 ^ k ≤ 2 ^ k
f' : Finset S' → Color
hf1' : ∀ (s1 s2 : Finset S'), f' s1 = f' s2 → f' (s1 ∪ s2) = f' s1
hf2' : Fintype.card { a // f' a = Color.red } = N - 2 ^ k
f : Finset S → Color := fun x => if s ∈ x then Color.red else f' (Finset.subtype (fun a => a ≠ s) x)
h2 : ∀ (a : Finset S), f a = Color.blue ↔ s ∉ a ∧ f' (Finset.subtype (fun a => a ≠ s) a) = Color.blue
h2' : ∀ (a : Finset S), f a = Color.red ↔ s ∈ a ∨ f' (Finset.subtype (fun a => a ≠ s) a) = Color.red
b : { a // f a = Color.red } → { a // s ∈ a } ⊕ { a' // f' a' = Color.red } :=
  fun x =>
    match x with
    | ⟨a, ha⟩ => if h : s ∈ a then Sum.inl ⟨a, h⟩ else Sum.inr ⟨Finset.subtype (fun a => a ≠ s) a, ⋯⟩
h3 : Function.Bijective b
h4 : Fintype.card { a // s ∈ a } = 2 ^ k
⊢ 2 ^ k ≤ N
Tactic:
exact Nat.le_of_lt hg
Proof state after:
no goals