Theorem: is_two_pretransitive_weak_jordan (Jordan/Jordan.lean)

State before:
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
n : ℕ
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
⊢ IsMultiplyPretransitive G α 2
Tactic:
revert α G
State after:
n : ℕ
⊢ ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
    [inst_3 : DecidableEq α],
    IsPreprimitive G α →
      ∀ {s : Set α},
        Set.ncard s = Nat.succ n →
          1 + Nat.succ n < Fintype.card α →
            IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2

State before:
n : ℕ
⊢ ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
    [inst_3 : DecidableEq α],
    IsPreprimitive G α →
      ∀ {s : Set α},
        Set.ncard s = Nat.succ n →
          1 + Nat.succ n < Fintype.card α →
            IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
Tactic:
induction' n using Nat.strong_induction_on with n hrec
State after:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
⊢ ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
    [inst_3 : DecidableEq α],
    IsPreprimitive G α →
      ∀ {s : Set α},
        Set.ncard s = Nat.succ n →
          1 + Nat.succ n < Fintype.card α →
            IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2

State before:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
⊢ ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
    [inst_3 : DecidableEq α],
    IsPreprimitive G α →
      ∀ {s : Set α},
        Set.ncard s = Nat.succ n →
          1 + Nat.succ n < Fintype.card α →
            IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
Tactic:
intro α G _ _ _ _ hG s hsn hsn' hs_trans
State after:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
⊢ IsMultiplyPretransitive G α 2

State before:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
⊢ IsMultiplyPretransitive G α 2
Tactic:
have hs_ne_top : s ≠ ⊤ := by
  intro hs
  rw [hs, Set.top_eq_univ, Set.ncard_univ] at hsn
  rw [← hsn, Nat.card_eq_fintype_card, add_lt_iff_neg_right] at hsn'
  contradiction
State after:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
⊢ IsMultiplyPretransitive G α 2

State before:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
⊢ IsMultiplyPretransitive G α 2
Tactic:
have hs_nonempty : s.Nonempty := by
  rw [← Set.ncard_pos, hsn]
  exact Nat.succ_pos n
State after:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
⊢ IsMultiplyPretransitive G α 2

State before:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
⊢ IsMultiplyPretransitive G α 2
Tactic:
cases' Nat.lt_or_ge n.succ 2 with hn hn
State after:
case h.inl
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ IsMultiplyPretransitive G α 2

case h.inr
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inr
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
⊢ IsMultiplyPretransitive G α 2
Tactic:
cases' Nat.lt_or_ge (2 * n.succ) (Fintype.card α) with hn1 hn2
State after:
case h.inr.inl
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
⊢ IsMultiplyPretransitive G α 2

case h.inr.inr
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
⊢ IsMultiplyPretransitive G α 2

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
⊢ s ≠ ⊤
Tactic:
intro hs
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs : s = ⊤
⊢ False

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs : s = ⊤
⊢ False
Tactic:
rw [hs, Set.top_eq_univ, Set.ncard_univ] at hsn
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Nat.card α = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs : s = ⊤
⊢ False

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Nat.card α = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs : s = ⊤
⊢ False
Tactic:
rw [← hsn, Nat.card_eq_fintype_card, add_lt_iff_neg_right] at hsn'
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Nat.card α = Nat.succ n
hsn' : 1 < 0
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs : s = ⊤
⊢ False

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Nat.card α = Nat.succ n
hsn' : 1 < 0
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs : s = ⊤
⊢ False
Tactic:
contradiction
State after:
no goals

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
⊢ Set.Nonempty s
Tactic:
rw [← Set.ncard_pos, hsn]
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
⊢ 0 < Nat.succ n

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
⊢ 0 < Nat.succ n
Tactic:
exact Nat.succ_pos n
State after:
no goals

State before:
case h.inl
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ IsMultiplyPretransitive G α 2
Tactic:
have hn : n = 0 := by
  rw [← le_zero_iff]
  apply Nat.le_of_succ_le_succ
  apply Nat.le_of_lt_succ
  exact hn
State after:
case h.inl
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inl
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
⊢ IsMultiplyPretransitive G α 2
Tactic:
simp only [hn, Set.ncard_eq_one] at hsn
State after:
case h.inl
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
hsn : ∃ a, s = {a}
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inl
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
hsn : ∃ a, s = {a}
⊢ IsMultiplyPretransitive G α 2
Tactic:
obtain ⟨a, hsa⟩ := hsn
State after:
case h.inl.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
a : α
hsa : s = {a}
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inl.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
a : α
hsa : s = {a}
⊢ IsMultiplyPretransitive G α 2
Tactic:
rw [hsa] at hs_trans
State after:
case h.inl.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn' : 1 + Nat.succ n < Fintype.card α
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
a : α
hs_trans : IsPretransitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inl.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn' : 1 + Nat.succ n < Fintype.card α
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
a : α
hs_trans : IsPretransitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ IsMultiplyPretransitive G α 2
Tactic:
rw [stabilizer.isMultiplyPretransitive G α hG.toIsPretransitive]
State after:
case h.inl.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn' : 1 + Nat.succ n < Fintype.card α
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
a : α
hs_trans : IsPretransitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ IsMultiplyPretransitive (↥(stabilizer G ?m.143299)) (↥(SubMulAction.ofStabilizer G ?m.143299)) 1

n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn' : 1 + Nat.succ n < Fintype.card α
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
a : α
hs_trans : IsPretransitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ α

State before:
case h.inl.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn' : 1 + Nat.succ n < Fintype.card α
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
a : α
hs_trans : IsPretransitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ IsMultiplyPretransitive (↥(stabilizer G ?m.143299)) (↥(SubMulAction.ofStabilizer G ?m.143299)) 1

n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn' : 1 + Nat.succ n < Fintype.card α
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
a : α
hs_trans : IsPretransitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ α
Tactic:
rw [← isPretransitive_iff_is_one_pretransitive]
State after:
case h.inl.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn' : 1 + Nat.succ n < Fintype.card α
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
a : α
hs_trans : IsPretransitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ IsPretransitive ↥(stabilizer G ?m.143299) ↥(SubMulAction.ofStabilizer G ?m.143299)

n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn' : 1 + Nat.succ n < Fintype.card α
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
a : α
hs_trans : IsPretransitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ α

n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn' : 1 + Nat.succ n < Fintype.card α
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
a : α
hs_trans : IsPretransitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ α

State before:
case h.inl.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn' : 1 + Nat.succ n < Fintype.card α
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
a : α
hs_trans : IsPretransitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ IsPretransitive ↥(stabilizer G ?m.143299) ↥(SubMulAction.ofStabilizer G ?m.143299)

n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn' : 1 + Nat.succ n < Fintype.card α
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
a : α
hs_trans : IsPretransitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ α

n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn' : 1 + Nat.succ n < Fintype.card α
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
a : α
hs_trans : IsPretransitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ α
Tactic:
apply isPretransitive.of_surjective_map
  (SubMulAction.OfFixingSubgroupOfSingleton.map_bijective G a).surjective hs_trans
State after:
no goals

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ n = 0
Tactic:
rw [← le_zero_iff]
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ n ≤ 0

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ n ≤ 0
Tactic:
apply Nat.le_of_succ_le_succ
State after:
case a
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ Nat.succ n ≤ Nat.succ 0

State before:
case a
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ Nat.succ n ≤ Nat.succ 0
Tactic:
apply Nat.le_of_lt_succ
State after:
case a.a
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ Nat.succ n < Nat.succ (Nat.succ 0)

State before:
case a.a
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ Nat.succ n < Nat.succ (Nat.succ 0)
Tactic:
exact hn
State after:
no goals

State before:
case h.inr.inl
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
⊢ IsMultiplyPretransitive G α 2
Tactic:
have : 1 < s.ncard := by rw [hsn]; exact hn
State after:
case h.inr.inl
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
this : 1 < Set.ncard s
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inr.inl
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
this : 1 < Set.ncard s
⊢ IsMultiplyPretransitive G α 2
Tactic:
rw [Set.one_lt_ncard] at this
State after:
case h.inr.inl
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
this : ∃ a ∈ s, ∃ b ∈ s, a ≠ b
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inr.inl
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
this : ∃ a ∈ s, ∃ b ∈ s, a ≠ b
⊢ IsMultiplyPretransitive G α 2
Tactic:
obtain ⟨a, ha, b, hb, hab⟩ := this
State after:
case h.inr.inl.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inr.inl.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
⊢ IsMultiplyPretransitive G α 2
Tactic:
obtain ⟨g, hga, hgb⟩ := Rudio hG s (Set.toFinite s) hs_nonempty hs_ne_top a b hab
State after:
case h.inr.inl.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inr.inl.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
⊢ IsMultiplyPretransitive G α 2
Tactic:
let t := s ∩ g • s
State after:
case h.inr.inl.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inr.inl.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
⊢ IsMultiplyPretransitive G α 2
Tactic:
have ht_trans : IsPretransitive (fixingSubgroup G t)
  (SubMulAction.ofFixingSubgroup G t) :=
    isPretransitive_ofFixingSubgroup_inter hs_trans (by
      apply Set.ncard_pigeonhole_compl'
      rw [smul_set_ncard_eq, hsn, ← two_mul]
      exact hn1)
State after:
case h.inr.inl.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inr.inl.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ IsMultiplyPretransitive G α 2
Tactic:
suffices ∃ m, m < n ∧ t.ncard = Nat.succ m by
  obtain ⟨m, hmn, htm⟩ := this
  apply hrec m hmn hG htm _ ht_trans
  apply lt_trans _ hsn'
  rw [add_lt_add_iff_left, Nat.succ_lt_succ_iff]
  exact hmn
State after:
case h.inr.inl.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ ∃ m < n, Set.ncard t = Nat.succ m

State before:
case h.inr.inl.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ ∃ m < n, Set.ncard t = Nat.succ m
Tactic:
use t.ncard.pred
State after:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ Nat.pred (Set.ncard t) < n ∧ Set.ncard t = Nat.succ (Nat.pred (Set.ncard t))

State before:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ Set.ncard t ≠ 0
Tactic:
apply Set.ncard_ne_zero_of_mem (a := a)
State after:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ a ∈ t

State before:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ a ∈ t
Tactic:
exact ⟨ha, hga⟩
State after:
no goals

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
⊢ 1 < Set.ncard s
Tactic:
rw [hsn]
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
⊢ 1 < Nat.succ n

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
⊢ 1 < Nat.succ n
Tactic:
exact hn
State after:
no goals

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
⊢ s ∪ g • s ≠ ⊤
Tactic:
apply Set.ncard_pigeonhole_compl'
State after:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
⊢ Set.ncard s + Set.ncard (g • s) < Fintype.card α

State before:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
⊢ Set.ncard s + Set.ncard (g • s) < Fintype.card α
Tactic:
rw [smul_set_ncard_eq, hsn, ← two_mul]
State after:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
⊢ 2 * Nat.succ n < Fintype.card α

State before:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
⊢ 2 * Nat.succ n < Fintype.card α
Tactic:
exact hn1
State after:
no goals

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : ∃ m < n, Set.ncard t = Nat.succ m
⊢ IsMultiplyPretransitive G α 2
Tactic:
obtain ⟨m, hmn, htm⟩ := this
State after:
case intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ IsMultiplyPretransitive G α 2

State before:
case intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ IsMultiplyPretransitive G α 2
Tactic:
apply hrec m hmn hG htm _ ht_trans
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ 1 + Nat.succ m < Fintype.card α

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ 1 + Nat.succ m < Fintype.card α
Tactic:
apply lt_trans _ hsn'
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ 1 + Nat.succ m < 1 + Nat.succ n

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ 1 + Nat.succ m < 1 + Nat.succ n
Tactic:
rw [add_lt_add_iff_left, Nat.succ_lt_succ_iff]
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ m < n

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ m < n
Tactic:
exact hmn
State after:
no goals

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Nat.pred (Set.ncard t) < n ∧ Set.ncard t = Nat.succ (Nat.pred (Set.ncard t))
Tactic:
rw [← Nat.succ_lt_succ_iff, ← hsn, Nat.succ_pred this]
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t < Set.ncard s ∧ Set.ncard t = Set.ncard t

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t < Set.ncard s ∧ Set.ncard t = Set.ncard t
Tactic:
constructor
State after:
case left
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t < Set.ncard s

case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t = Set.ncard t

State before:
case left
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t < Set.ncard s
Tactic:
apply Set.ncard_lt_ncard _ (Set.toFinite s)
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ t ⊂ s

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ t ⊂ s
Tactic:
constructor
State after:
case left
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ t ⊆ s

case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t

State before:
case left
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ t ⊆ s

case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t
Tactic:
apply Set.inter_subset_left
State after:
case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t

State before:
case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t
Tactic:
intro h
State after:
case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ False

State before:
case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ False
Tactic:
apply hgb
State after:
case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ b ∈ g • s

State before:
case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ b ∈ g • s
Tactic:
apply Set.inter_subset_right
State after:
case right.a
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ b ∈ ?right.s ∩ g • s

case right.s
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ Set α

State before:
case right.a
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ b ∈ ?right.s ∩ g • s

case right.s
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ Set α
Tactic:
apply h
State after:
case right.a.a
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ b ∈ s

State before:
case right.a.a
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ b ∈ s
Tactic:
exact hb
State after:
no goals

State before:
case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn1 : 2 * Nat.succ n < Fintype.card α
a : α
ha : a ∈ s
b : α
hb : b ∈ s
hab : a ≠ b
g : G
hga : a ∈ g • s
hgb : b ∉ g • s
t : Set α := s ∩ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t = Set.ncard t
Tactic:
rfl
State after:
no goals

State before:
case h.inr.inr
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
⊢ IsMultiplyPretransitive G α 2
Tactic:
have : Set.Nontrivial sᶜ := by
  rw [← Set.one_lt_ncard_iff_nontrivial]
  rw [← Nat.add_lt_add_iff_left, Set.ncard_add_ncard_compl]
  rw [Nat.card_eq_fintype_card, add_comm, hsn]
  exact hsn'
State after:
case h.inr.inr
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
this : Set.Nontrivial sᶜ
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inr.inr
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
this : Set.Nontrivial sᶜ
⊢ IsMultiplyPretransitive G α 2
Tactic:
obtain ⟨a, ha : a ∈ sᶜ, b, hb : b ∈ sᶜ, hab⟩ := this
State after:
case h.inr.inr.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inr.inr.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ IsMultiplyPretransitive G α 2
Tactic:
obtain ⟨g, hga, hgb⟩ := Rudio hG sᶜ (Set.toFinite sᶜ)
  (Set.nonempty_of_mem ha)
  (by intro h
      simp only [Set.top_eq_univ, Set.compl_univ_iff] at h
      simp only [h, Set.not_nonempty_empty] at hs_nonempty)
  a b hab
State after:
case h.inr.inr.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inr.inr.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
⊢ IsMultiplyPretransitive G α 2
Tactic:
let t := s ∩ g • s
State after:
case h.inr.inr.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inr.inr.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
⊢ IsMultiplyPretransitive G α 2
Tactic:
have : a ∉ s ∪ g • s := by
  rw [Set.mem_union]
  intro h
  cases' h with h h
  exact ha h
  rw [Set.mem_smul_set_iff_inv_smul_mem] at hga h
  exact hga h
State after:
case h.inr.inr.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inr.inr.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
⊢ IsMultiplyPretransitive G α 2
Tactic:
have ht_trans : IsPretransitive (fixingSubgroup G t)
  (SubMulAction.ofFixingSubgroup G t) :=
    isPretransitive_ofFixingSubgroup_inter hs_trans
      (by intro h; apply this; rw [h]; trivial)
State after:
case h.inr.inr.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ IsMultiplyPretransitive G α 2

State before:
case h.inr.inr.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ IsMultiplyPretransitive G α 2
Tactic:
suffices ∃ m : ℕ, m < n ∧ t.ncard = Nat.succ m by
  obtain ⟨m, hmn, htm⟩ := this
  refine' hrec m hmn hG htm (by
    apply lt_trans _ hsn'
    rw [add_lt_add_iff_left, Nat.succ_lt_succ_iff]
    exact hmn) ht_trans
State after:
case h.inr.inr.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ ∃ m < n, Set.ncard t = Nat.succ m

State before:
case h.inr.inr.intro.intro.intro.intro.intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ ∃ m < n, Set.ncard t = Nat.succ m
Tactic:
use t.ncard.pred
State after:
case h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ Nat.pred (Set.ncard t) < n ∧ Set.ncard t = Nat.succ (Nat.pred (Set.ncard t))

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
⊢ Set.Nontrivial sᶜ
Tactic:
rw [← Set.one_lt_ncard_iff_nontrivial]
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
⊢ 1 < Set.ncard sᶜ

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
⊢ 1 < Set.ncard sᶜ
Tactic:
rw [← Nat.add_lt_add_iff_left, Set.ncard_add_ncard_compl]
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
⊢ Set.ncard s + 1 < Nat.card α

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
⊢ Set.ncard s + 1 < Nat.card α
Tactic:
rw [Nat.card_eq_fintype_card, add_comm, hsn]
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
⊢ 1 + Nat.succ n < Fintype.card α

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
⊢ 1 + Nat.succ n < Fintype.card α
Tactic:
exact hsn'
State after:
no goals

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
⊢ sᶜ ≠ ⊤
Tactic:
intro h
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
h : sᶜ = ⊤
⊢ False

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
h : sᶜ = ⊤
⊢ False
Tactic:
simp only [Set.top_eq_univ, Set.compl_univ_iff] at h
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
h : s = ∅
⊢ False

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
h : s = ∅
⊢ False
Tactic:
simp only [h, Set.not_nonempty_empty] at hs_nonempty
State after:
no goals

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
⊢ a ∉ s ∪ g • s
Tactic:
rw [Set.mem_union]
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
⊢ ¬(a ∈ s ∨ a ∈ g • s)

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
⊢ ¬(a ∈ s ∨ a ∈ g • s)
Tactic:
intro h
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
h : a ∈ s ∨ a ∈ g • s
⊢ False

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
h : a ∈ s ∨ a ∈ g • s
⊢ False
Tactic:
cases' h with h h
State after:
case inl
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
h : a ∈ s
⊢ False

case inr
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
h : a ∈ g • s
⊢ False

State before:
case inl
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
h : a ∈ s
⊢ False

case inr
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
h : a ∈ g • s
⊢ False
Tactic:
exact ha h
State after:
case inr
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
h : a ∈ g • s
⊢ False

State before:
case inr
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
h : a ∈ g • s
⊢ False
Tactic:
rw [Set.mem_smul_set_iff_inv_smul_mem] at hga h
State after:
case inr
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : g⁻¹ • a ∈ sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
h : g⁻¹ • a ∈ s
⊢ False

State before:
case inr
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : g⁻¹ • a ∈ sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
h : g⁻¹ • a ∈ s
⊢ False
Tactic:
exact hga h
State after:
no goals

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
⊢ s ∪ g • s ≠ ⊤
Tactic:
intro h
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
h : s ∪ g • s = ⊤
⊢ False

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
h : s ∪ g • s = ⊤
⊢ False
Tactic:
apply this
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
h : s ∪ g • s = ⊤
⊢ a ∈ s ∪ g • s

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
h : s ∪ g • s = ⊤
⊢ a ∈ s ∪ g • s
Tactic:
rw [h]
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
h : s ∪ g • s = ⊤
⊢ a ∈ ⊤

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
h : s ∪ g • s = ⊤
⊢ a ∈ ⊤
Tactic:
trivial
State after:
no goals

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : ∃ m < n, Set.ncard t = Nat.succ m
⊢ IsMultiplyPretransitive G α 2
Tactic:
obtain ⟨m, hmn, htm⟩ := this
State after:
case intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ IsMultiplyPretransitive G α 2

State before:
case intro.intro
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ IsMultiplyPretransitive G α 2
Tactic:
refine' hrec m hmn hG htm (by
  apply lt_trans _ hsn'
  rw [add_lt_add_iff_left, Nat.succ_lt_succ_iff]
  exact hmn) ht_trans
State after:
no goals

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ 1 + Nat.succ m < Fintype.card α
Tactic:
apply lt_trans _ hsn'
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ 1 + Nat.succ m < 1 + Nat.succ n

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ 1 + Nat.succ m < 1 + Nat.succ n
Tactic:
rw [add_lt_add_iff_left, Nat.succ_lt_succ_iff]
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ m < n

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ m < n
Tactic:
exact hmn
State after:
no goals

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Nat.pred (Set.ncard t) < n ∧ Set.ncard t = Nat.succ (Nat.pred (Set.ncard t))
Tactic:
rw [← Nat.succ_lt_succ_iff, ← hsn, Nat.succ_pred this]
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t < Set.ncard s ∧ Set.ncard t = Set.ncard t

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t < Set.ncard s ∧ Set.ncard t = Set.ncard t
Tactic:
constructor
State after:
case left
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t < Set.ncard s

case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t = Set.ncard t

State before:
case left
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t < Set.ncard s
Tactic:
apply Set.ncard_lt_ncard _ (Set.toFinite s)
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ t ⊂ s

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ t ⊂ s
Tactic:
constructor
State after:
case left
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ t ⊆ s

case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t

State before:
case left
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ t ⊆ s

case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t
Tactic:
apply Set.inter_subset_left
State after:
case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t

State before:
case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t
Tactic:
intro h
State after:
case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ False

State before:
case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ False
Tactic:
suffices s = g • s by
  apply hb
  rw [this]
  simp only [smul_compl_set, Set.mem_compl_iff, Set.not_not_mem] at hgb
  exact hgb
State after:
case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ s = g • s

State before:
case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ s = g • s
Tactic:
apply Set.eq_of_subset_of_ncard_le
State after:
case right.h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ s ⊆ g • s

case right.h'
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ Set.ncard (g • s) ≤ Set.ncard s

case right.ht
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ autoParam (Set.Finite (g • s)) _auto✝

State before:
case right.ht
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ autoParam (Set.Finite (g • s)) _auto✝
Tactic:
exact Set.toFinite (g • s)
State after:
no goals

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝¹ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this✝ : Set.ncard t ≠ 0
h : s ⊆ t
this : s = g • s
⊢ False
Tactic:
apply hb
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝¹ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this✝ : Set.ncard t ≠ 0
h : s ⊆ t
this : s = g • s
⊢ b ∈ s

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝¹ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this✝ : Set.ncard t ≠ 0
h : s ⊆ t
this : s = g • s
⊢ b ∈ s
Tactic:
rw [this]
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝¹ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this✝ : Set.ncard t ≠ 0
h : s ⊆ t
this : s = g • s
⊢ b ∈ g • s

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝¹ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this✝ : Set.ncard t ≠ 0
h : s ⊆ t
this : s = g • s
⊢ b ∈ g • s
Tactic:
simp only [smul_compl_set, Set.mem_compl_iff, Set.not_not_mem] at hgb
State after:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
t : Set α := s ∩ g • s
this✝¹ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this✝ : Set.ncard t ≠ 0
h : s ⊆ t
this : s = g • s
hgb : b ∈ g • s
⊢ b ∈ g • s

State before:
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
t : Set α := s ∩ g • s
this✝¹ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this✝ : Set.ncard t ≠ 0
h : s ⊆ t
this : s = g • s
hgb : b ∈ g • s
⊢ b ∈ g • s
Tactic:
exact hgb
State after:
no goals

State before:
case right.h
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ s ⊆ g • s
Tactic:
exact subset_trans h (Set.inter_subset_right _ _)
State after:
no goals

State before:
case right.h'
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ Set.ncard (g • s) ≤ Set.ncard s
Tactic:
rw [smul_set_ncard_eq]
State after:
no goals

State before:
case right
n : ℕ
hrec :
  ∀ m < n,
    ∀ {α : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : Fintype α]
      [inst_3 : DecidableEq α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          Set.ncard s = Nat.succ m →
            1 + Nat.succ m < Fintype.card α →
              IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
hG : IsPreprimitive G α
s : Set α
hsn : Set.ncard s = Nat.succ n
hsn' : 1 + Nat.succ n < Fintype.card α
hs_trans : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
hn2 : 2 * Nat.succ n ≥ Fintype.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
this✝ : a ∉ s ∪ g • s
ht_trans : IsPretransitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t = Set.ncard t
Tactic:
rfl
State after:
no goals