url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | induction' n using Nat.strong_induction_on with n hrec | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | intro α G _ _ _ _ hG s hsn hsn' hs_prim | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | 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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | have hs_nonempty : s.Nonempty := by
rw [← Set.ncard_pos, hsn]
exact Nat.succ_pos n | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | cases' Nat.lt_or_ge n.succ 2 with hn hn | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ IsMultiplyPreprimitive 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | cases' Nat.lt_or_ge (2 * n.succ) (Fintype.card α) with hn1 hn2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n ≥ 2
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | intro hs | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
⊢ 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs : s = ⊤
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
⊢ s ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [hs, Set.top_eq_univ, Set.ncard_univ] at hsn | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs : s = ⊤
⊢ False | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs : s = ⊤
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs : s = ⊤
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [← hsn, Nat.card_eq_fintype_card, add_lt_iff_neg_right] at hsn' | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs : s = ⊤
⊢ False | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs : s = ⊤
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs : s = ⊤
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | contradiction | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs : s = ⊤
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs : s = ⊤
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [← Set.ncard_pos, hsn] | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
⊢ Set.Nonempty 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
⊢ 0 < Nat.succ n | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
⊢ Set.Nonempty s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | exact Nat.succ_pos n | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
⊢ 0 < Nat.succ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
⊢ 0 < Nat.succ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | have hn : n = 0 := by
rw [← le_zero_iff]
apply Nat.le_of_succ_le_succ
apply Nat.le_of_lt_succ
exact hn | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | simp only [hn, Set.ncard_eq_one] at hsn | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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}
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn✝ : Nat.succ n < 2
hn : n = 0
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | obtain ⟨a, hsa⟩ := hsn | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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}
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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}
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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}
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [hsa] at hs_prim | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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}
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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}
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [stabilizer.isMultiplyPreprimitive G α (Nat.le_refl 1) hG.toIsPretransitive] | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ IsMultiplyPreprimitive (↥(stabilizer G ?m.183750)) (↥(SubMulAction.ofStabilizer G ?m.183750)) 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ α | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [← isPreprimitive_iff_is_one_preprimitive] | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ IsMultiplyPreprimitive (↥(stabilizer G ?m.183750)) (↥(SubMulAction.ofStabilizer G ?m.183750)) 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ α | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ IsPreprimitive ↥(stabilizer G ?m.183750) ↥(SubMulAction.ofStabilizer G ?m.183750)
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ α | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ IsMultiplyPreprimitive (↥(stabilizer G ?m.183750)) (↥(SubMulAction.ofStabilizer G ?m.183750)) 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply isPreprimitive_of_surjective_map
(SubMulAction.OfFixingSubgroupOfSingleton.map_bijective G a).surjective hs_prim | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ IsPreprimitive ↥(stabilizer G ?m.183750) ↥(SubMulAction.ofStabilizer G ?m.183750)
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ IsPreprimitive ↥(stabilizer G ?m.183750) ↥(SubMulAction.ofStabilizer G ?m.183750)
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G {a}) ↥(SubMulAction.ofFixingSubgroup G {a})
hsa : s = {a}
⊢ α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [← le_zero_iff] | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ n = 0 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ n ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ n = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply Nat.le_of_succ_le_succ | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ n ≤ 0 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
hs_ne_top : s ≠ ⊤
hs_nonempty : Set.Nonempty s
hn : Nat.succ n < 2
⊢ n ≤ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply Nat.le_of_lt_succ | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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) | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | exact hn | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | have : 1 < s.ncard := by rw [hsn]; exact hn | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [Set.one_lt_ncard] at this | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | obtain ⟨a, ha, b, hb, hab⟩ := this | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | obtain ⟨g, hga, hgb⟩ := Rudio hG s (Set.toFinite s) hs_nonempty hs_ne_top a b hab | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | let t := s ∩ g • s | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | have ht_prim : IsPreprimitive (fixingSubgroup G t)
(SubMulAction.ofFixingSubgroup G t) :=
hs_prim.isPreprimitive_ofFixingSubgroup_inter (by
apply Set.ncard_pigeonhole_compl'
rw [smul_set_ncard_eq, hsn, ← two_mul]
exact hn1) | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | suffices ∃ m, m < n ∧ t.ncard = Nat.succ m by
obtain ⟨m, hmn, htm⟩ := this
apply hrec m hmn hG htm _ ht_prim
. apply lt_trans _ hsn'
rw [add_lt_add_iff_left, Nat.succ_lt_succ_iff]
exact hmn | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ ∃ m < n, Set.ncard t = Nat.succ m | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | use t.ncard.pred | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ ∃ m < n, Set.ncard t = Nat.succ m | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ Nat.pred (Set.ncard t) < n ∧ Set.ncard t = Nat.succ (Nat.pred (Set.ncard t)) | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ ∃ m < n, Set.ncard t = Nat.succ m
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [hsn] | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | exact hn | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply Set.ncard_pigeonhole_compl' | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 ≠ ⊤ | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [smul_set_ncard_eq, hsn, ← two_mul] | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | exact hn1 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | obtain ⟨m, hmn, htm⟩ := this | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : ∃ m < n, Set.ncard t = Nat.succ m
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : ∃ m < n, Set.ncard t = Nat.succ m
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply hrec m hmn hG htm _ ht_prim | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | . apply lt_trans _ hsn'
rw [add_lt_add_iff_left, Nat.succ_lt_succ_iff]
exact hmn | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply lt_trans _ hsn' | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [add_lt_add_iff_left, Nat.succ_lt_succ_iff] | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ m < n | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | exact hmn | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ m < n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ m < n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [← Nat.succ_lt_succ_iff, ← hsn, Nat.succ_pred this] | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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)) | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t < Set.ncard s ∧ Set.ncard t = Set.ncard t | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | constructor | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t < Set.ncard s ∧ Set.ncard t = Set.ncard t | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t = Set.ncard t | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply Set.ncard_lt_ncard _ (Set.toFinite s) | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t < Set.ncard 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ t ⊂ s | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t < Set.ncard s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | constructor | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ t ⊂ s | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ t ⊂ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply Set.inter_subset_left | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | intro h | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply hgb | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ False | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ b ∈ g • s | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply Set.inter_subset_right | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ b ∈ g • s | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ Set α | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ b ∈ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply h | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ Set α | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ b ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ Set α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | exact hb | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ b ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ b ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rfl | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t = Set.ncard t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t = Set.ncard t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply Set.ncard_ne_zero_of_mem (a := a) | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ Set.ncard t ≠ 0 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ a ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ Set.ncard t ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | exact ⟨ha, hga⟩ | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ a ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ a ∈ t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | 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' | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | obtain ⟨a, ha : a ∈ sᶜ, b, hb : b ∈ sᶜ, hab⟩ := this | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | 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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | let t := s ∩ g • s | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | 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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | have ht_prim : IsPreprimitive (fixingSubgroup G t)
(SubMulAction.ofFixingSubgroup G t) :=
hs_prim.isPreprimitive_ofFixingSubgroup_inter
(by intro h; apply this; rw [h]; trivial) | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | suffices ∃ m : ℕ, m < n ∧ t.ncard = Nat.succ m by
obtain ⟨m, hmn, htm⟩ := this
exact hrec m hmn hG htm (by
apply lt_trans _ hsn'
rw [add_lt_add_iff_left, Nat.succ_lt_succ_iff]
exact hmn) ht_prim | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ ∃ m < n, Set.ncard t = Nat.succ m | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | use t.ncard.pred | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ ∃ m < n, Set.ncard t = Nat.succ m | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ Nat.pred (Set.ncard t) < n ∧ Set.ncard t = Nat.succ (Nat.pred (Set.ncard t)) | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
⊢ ∃ m < n, Set.ncard t = Nat.succ m
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [← Set.one_lt_ncard_iff_nontrivial] | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [← Nat.add_lt_add_iff_left, Set.ncard_add_ncard_compl] | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [Nat.card_eq_fintype_card, add_comm, hsn] | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | exact hsn' | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | intro 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 ≠ ⊤ | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | simp only [Set.top_eq_univ, Set.compl_univ_iff] at 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | simp only [h, Set.not_nonempty_empty] at hs_nonempty | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [Set.mem_union] | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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) | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | intro 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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) | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | cases' h with h 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | exact ha h | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [Set.mem_smul_set_iff_inv_smul_mem] at hga h | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | exact hga h | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | intro 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 ≠ ⊤ | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply this | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 ∈ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | trivial | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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 ∈ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | obtain ⟨m, hmn, htm⟩ := this | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : ∃ m < n, Set.ncard t = Nat.succ m
⊢ IsMultiplyPreprimitive G α 2 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ IsMultiplyPreprimitive G α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : ∃ m < n, Set.ncard t = Nat.succ m
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | exact hrec m hmn hG htm (by
apply lt_trans _ hsn'
rw [add_lt_add_iff_left, Nat.succ_lt_succ_iff]
exact hmn) ht_prim | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ IsMultiplyPreprimitive G α 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ IsMultiplyPreprimitive G α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply lt_trans _ hsn' | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [add_lt_add_iff_left, Nat.succ_lt_succ_iff] | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ m < n | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | exact hmn | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ m < n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
m : ℕ
hmn : m < n
htm : Set.ncard t = Nat.succ m
⊢ m < n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [← Nat.succ_lt_succ_iff, ← hsn, Nat.succ_pred this] | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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)) | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t < Set.ncard s ∧ Set.ncard t = Set.ncard t | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | constructor | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t < Set.ncard s ∧ Set.ncard t = Set.ncard t | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t = Set.ncard t | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply Set.ncard_lt_ncard _ (Set.toFinite s) | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t < Set.ncard 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ t ⊂ s | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ Set.ncard t < Set.ncard s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | constructor | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ t ⊂ s | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ t ⊂ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply Set.inter_subset_left | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | intro h | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
⊢ ¬s ⊆ t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | 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 | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ False | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ s = g • s | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply Set.eq_of_subset_of_ncard_le | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ autoParam (Set.Finite (g • s)) _auto✝ | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ s = g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | exact Set.toFinite (g • 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ autoParam (Set.Finite (g • s)) _auto✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ autoParam (Set.Finite (g • s)) _auto✝
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | apply hb | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this✝ : Set.ncard t ≠ 0
h : s ⊆ t
this : s = g • s
⊢ False | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this✝ : Set.ncard t ≠ 0
h : s ⊆ t
this : s = g • s
⊢ b ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this✝ : Set.ncard t ≠ 0
h : s ⊆ t
this : s = g • s
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [this] | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this✝ : Set.ncard t ≠ 0
h : s ⊆ t
this : s = g • s
⊢ b ∈ 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this✝ : Set.ncard t ≠ 0
h : s ⊆ t
this : s = g • s
⊢ b ∈ g • s | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this✝ : Set.ncard t ≠ 0
h : s ⊆ t
this : s = g • s
⊢ b ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | simp only [smul_compl_set, Set.mem_compl_iff, Set.not_not_mem] at hgb | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this✝ : Set.ncard t ≠ 0
h : s ⊆ t
this : s = g • s
⊢ b ∈ g • 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this✝ : Set.ncard t ≠ 0
h : s ⊆ t
this : s = g • s
⊢ b ∈ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | exact hgb | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | exact subset_trans h (Set.inter_subset_right _ _) | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ s ⊆ g • s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ s ⊆ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | is_two_preprimitive_weak_jordan | [455, 1] | [603, 14] | rw [smul_set_ncard_eq] | 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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ Set.ncard (g • s) ≤ Set.ncard s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 →
IsPreprimitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s) → IsMultiplyPreprimitive 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_prim : IsPreprimitive ↥(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_prim : IsPreprimitive ↥(fixingSubgroup G t) ↥(SubMulAction.ofFixingSubgroup G t)
this : Set.ncard t ≠ 0
h : s ⊆ t
⊢ Set.ncard (g • s) ≤ Set.ncard s
TACTIC:
|
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