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/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | rw [← hs] at hmt | case inr.hG.intro.intro
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα : Fintype.card α ≥ 2
s : Set α
left✝ : s ⊆ Set.univ
hs : Set.ncard s = Fintype.card α - 2
⊢ (Fintype.card α)! ≤ 2 * Fintype.card ↥G | case inr.hG.intro.intro
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≥ 2
s : Set α
hmt : IsMultiplyPretransitive (↥G) α (Set.ncard s)
left✝ : s ⊆ Set.univ
hs : Set.ncard s = Fintype.card α - 2
⊢ (Fintype.card α)! ≤ 2 * Fintype.card ↥G | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.hG.intro.intro
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα : Fintype.card α ≥ 2
s : Set α
left✝ : s ⊆ Set.univ
hs : Set.ncard s = Fintype.card α - 2
⊢ (Fintype.card α)! ≤ 2 * Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | rw [← hmt.index_of_fixingSubgroup G α s, hs, Nat.sub_sub_self hα,
Nat.factorial_two, mul_comm] | case inr.hG.intro.intro
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≥ 2
s : Set α
hmt : IsMultiplyPretransitive (↥G) α (Set.ncard s)
left✝ : s ⊆ Set.univ
hs : Set.ncard s = Fintype.card α - 2
⊢ (Fintype.card α)! ≤ 2 * Fintype.card ↥G | case inr.hG.intro.intro
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≥ 2
s : Set α
hmt : IsMultiplyPretransitive (↥G) α (Set.ncard s)
left✝ : s ⊆ Set.univ
hs : Set.ncard s = Fintype.card α - 2
⊢ 2 * Subgroup.index (fixingSubgroup (↥G) s) ≤ 2 * Fintype.card ↥G | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.hG.intro.intro
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≥ 2
s : Set α
hmt : IsMultiplyPretransitive (↥G) α (Set.ncard s)
left✝ : s ⊆ Set.univ
hs : Set.ncard s = Fintype.card α - 2
⊢ (Fintype.card α)! ≤ 2 * Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | apply Nat.mul_le_mul_left | case inr.hG.intro.intro
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≥ 2
s : Set α
hmt : IsMultiplyPretransitive (↥G) α (Set.ncard s)
left✝ : s ⊆ Set.univ
hs : Set.ncard s = Fintype.card α - 2
⊢ 2 * Subgroup.index (fixingSubgroup (↥G) s) ≤ 2 * Fintype.card ↥G | case inr.hG.intro.intro.h
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≥ 2
s : Set α
hmt : IsMultiplyPretransitive (↥G) α (Set.ncard s)
left✝ : s ⊆ Set.univ
hs : Set.ncard s = Fintype.card α - 2
⊢ Subgroup.index (fixingSubgroup (↥G) s) ≤ Fintype.card ↥G | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.hG.intro.intro
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≥ 2
s : Set α
hmt : IsMultiplyPretransitive (↥G) α (Set.ncard s)
left✝ : s ⊆ Set.univ
hs : Set.ncard s = Fintype.card α - 2
⊢ 2 * Subgroup.index (fixingSubgroup (↥G) s) ≤ 2 * Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | apply Nat.le_of_dvd (Fintype.card_pos) | case inr.hG.intro.intro.h
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≥ 2
s : Set α
hmt : IsMultiplyPretransitive (↥G) α (Set.ncard s)
left✝ : s ⊆ Set.univ
hs : Set.ncard s = Fintype.card α - 2
⊢ Subgroup.index (fixingSubgroup (↥G) s) ≤ Fintype.card ↥G | case inr.hG.intro.intro.h
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≥ 2
s : Set α
hmt : IsMultiplyPretransitive (↥G) α (Set.ncard s)
left✝ : s ⊆ Set.univ
hs : Set.ncard s = Fintype.card α - 2
⊢ Subgroup.index (fixingSubgroup (↥G) s) ∣ Fintype.card ↥G | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.hG.intro.intro.h
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≥ 2
s : Set α
hmt : IsMultiplyPretransitive (↥G) α (Set.ncard s)
left✝ : s ⊆ Set.univ
hs : Set.ncard s = Fintype.card α - 2
⊢ Subgroup.index (fixingSubgroup (↥G) s) ≤ Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | apply Subgroup.index_dvd_card | case inr.hG.intro.intro.h
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≥ 2
s : Set α
hmt : IsMultiplyPretransitive (↥G) α (Set.ncard s)
left✝ : s ⊆ Set.univ
hs : Set.ncard s = Fintype.card α - 2
⊢ Subgroup.index (fixingSubgroup (↥G) s) ∣ Fintype.card ↥G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.hG.intro.intro.h
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hα : Fintype.card α ≥ 2
s : Set α
hmt : IsMultiplyPretransitive (↥G) α (Set.ncard s)
left✝ : s ⊆ Set.univ
hs : Set.ncard s = Fintype.card α - 2
⊢ Subgroup.index (fixingSubgroup (↥G) s) ∣ Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | rw [Nat.lt_succ_iff] at hα1 | case inl
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α < 2
⊢ alternatingGroup α ≤ G | case inl
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
⊢ alternatingGroup α ≤ G | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α < 2
⊢ alternatingGroup α ≤ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | suffices alternatingGroup α = ⊥ by
rw [this]; exact bot_le | case inl
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
⊢ alternatingGroup α ≤ G | case inl
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
⊢ alternatingGroup α = ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
⊢ alternatingGroup α ≤ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | rw [← Subgroup.card_le_one_iff_eq_bot] | case inl
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
⊢ alternatingGroup α = ⊥ | case inl
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
⊢ Fintype.card ↥(alternatingGroup α) ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
⊢ alternatingGroup α = ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | suffices Fintype.card (alternatingGroup α) ≤ Fintype.card (Equiv.Perm α) by
apply le_trans this
rw [Fintype.card_perm]
exact Nat.factorial_le hα1 | case inl
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
⊢ Fintype.card ↥(alternatingGroup α) ≤ 1 | case inl
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
⊢ Fintype.card ↥(alternatingGroup α) ≤ Fintype.card (Equiv.Perm α) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
⊢ Fintype.card ↥(alternatingGroup α) ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | convert Fintype.card_subtype_le (fun x ↦ x ∈ alternatingGroup α) | case inl
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
⊢ Fintype.card ↥(alternatingGroup α) ≤ Fintype.card (Equiv.Perm α) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
⊢ Fintype.card ↥(alternatingGroup α) ≤ Fintype.card (Equiv.Perm α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | rw [this] | M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
this : alternatingGroup α = ⊥
⊢ alternatingGroup α ≤ G | M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
this : alternatingGroup α = ⊥
⊢ ⊥ ≤ G | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
this : alternatingGroup α = ⊥
⊢ alternatingGroup α ≤ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | exact bot_le | M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
this : alternatingGroup α = ⊥
⊢ ⊥ ≤ G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
this : alternatingGroup α = ⊥
⊢ ⊥ ≤ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | apply le_trans this | M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
this : Fintype.card ↥(alternatingGroup α) ≤ Fintype.card (Equiv.Perm α)
⊢ Fintype.card ↥(alternatingGroup α) ≤ 1 | M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
this : Fintype.card ↥(alternatingGroup α) ≤ Fintype.card (Equiv.Perm α)
⊢ Fintype.card (Equiv.Perm α) ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
this : Fintype.card ↥(alternatingGroup α) ≤ Fintype.card (Equiv.Perm α)
⊢ Fintype.card ↥(alternatingGroup α) ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | rw [Fintype.card_perm] | M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
this : Fintype.card ↥(alternatingGroup α) ≤ Fintype.card (Equiv.Perm α)
⊢ Fintype.card (Equiv.Perm α) ≤ 1 | M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
this : Fintype.card ↥(alternatingGroup α) ≤ Fintype.card (Equiv.Perm α)
⊢ (Fintype.card α)! ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
this : Fintype.card ↥(alternatingGroup α) ≤ Fintype.card (Equiv.Perm α)
⊢ Fintype.card (Equiv.Perm α) ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | exact Nat.factorial_le hα1 | M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
this : Fintype.card ↥(alternatingGroup α) ≤ Fintype.card (Equiv.Perm α)
⊢ (Fintype.card α)! ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα1 : Fintype.card α ≤ 1
this : Fintype.card ↥(alternatingGroup α) ≤ Fintype.card (Equiv.Perm α)
⊢ (Fintype.card α)! ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | rw [Set.ncard_univ, Nat.card_eq_fintype_card] | M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα : Fintype.card α ≥ 2
⊢ Fintype.card α - 2 ≤ Set.ncard Set.univ | M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα : Fintype.card α ≥ 2
⊢ Fintype.card α - 2 ≤ Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα : Fintype.card α ≥ 2
⊢ Fintype.card α - 2 ≤ Set.ncard Set.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.IsMultiplyPretransitive.alternatingGroup_le_of_sub_two | [1140, 1] | [1167, 32] | exact sub_le (Fintype.card α) 2 | M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα : Fintype.card α ≥ 2
⊢ Fintype.card α - 2 ≤ Fintype.card α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.274186
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 2)
hα : Fintype.card α ≥ 2
⊢ Fintype.card α - 2 ≤ Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.isPretransitive | [1171, 1] | [1180, 11] | rw [isPretransitive_iff_is_one_pretransitive] | M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ IsPretransitive (↥(alternatingGroup α)) α | M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ IsMultiplyPretransitive (↥(alternatingGroup α)) α 1 | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ IsPretransitive (↥(alternatingGroup α)) α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.isPretransitive | [1171, 1] | [1180, 11] | apply isMultiplyPretransitive_of_higher | M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ IsMultiplyPretransitive (↥(alternatingGroup α)) α 1 | case hn
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ IsMultiplyPretransitive (↥(alternatingGroup α)) α ?n
case hmn
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ 1 ≤ ?n
case hα
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ ↑?n ≤ PartENat.card α
case n
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ ℕ | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ IsMultiplyPretransitive (↥(alternatingGroup α)) α 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.isPretransitive | [1171, 1] | [1180, 11] | apply IsMultiplyPretransitive.alternatingGroup_of_sub_two | case hn
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ IsMultiplyPretransitive (↥(alternatingGroup α)) α ?n
case hmn
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ 1 ≤ ?n
case hα
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ ↑?n ≤ PartENat.card α
case n
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ ℕ | case hmn
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ 1 ≤ Fintype.card α - 2
case hα
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α | Please generate a tactic in lean4 to solve the state.
STATE:
case hn
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ IsMultiplyPretransitive (↥(alternatingGroup α)) α ?n
case hmn
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ 1 ≤ ?n
case hα
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ ↑?n ≤ PartENat.card α
case n
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ ℕ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.isPretransitive | [1171, 1] | [1180, 11] | apply le_trans _ (Nat.sub_le_sub_right h 2) | case hmn
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ 1 ≤ Fintype.card α - 2
case hα
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α | M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ 1 ≤ 3 - 2
case hα
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α | Please generate a tactic in lean4 to solve the state.
STATE:
case hmn
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ 1 ≤ Fintype.card α - 2
case hα
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.isPretransitive | [1171, 1] | [1180, 11] | norm_num | M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ 1 ≤ 3 - 2
case hα
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α | case hα
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ 1 ≤ 3 - 2
case hα
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.isPretransitive | [1171, 1] | [1180, 11] | simp only [ge_iff_le, PartENat.card_eq_coe_fintype_card, PartENat.coe_le_coe,
tsub_le_iff_right, le_add_iff_nonneg_right] | case hα
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α | case hα
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ 0 ≤ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case hα
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.isPretransitive | [1171, 1] | [1180, 11] | norm_num | case hα
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ 0 ≤ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hα
M : Type ?u.287770
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ 0 ≤ 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | cases' le_or_lt (Fintype.card α) 2 with h2 h2 | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
⊢ IsTrivialBlock B | case inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : Fintype.card α ≤ 2
⊢ IsTrivialBlock B
case inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | cases' le_or_lt (Fintype.card α) 3 with h3 h4 | case inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
⊢ IsTrivialBlock B | case inr.inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
⊢ IsTrivialBlock B
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | apply IsPreprimitive.has_trivial_blocks ?_ hB | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ IsTrivialBlock B | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ IsPreprimitive (↥(alternatingGroup α)) α | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | apply IsMultiplyPretransitive.isPreprimitive_of_two | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ IsPreprimitive (↥(alternatingGroup α)) α | case h2
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ IsMultiplyPretransitive (↥(alternatingGroup α)) α 2 | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ IsPreprimitive (↥(alternatingGroup α)) α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | apply isMultiplyPretransitive_of_higher | case h2
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ IsMultiplyPretransitive (↥(alternatingGroup α)) α 2 | case h2.hn
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ IsMultiplyPretransitive (↥(alternatingGroup α)) α ?h2.n
case h2.hmn
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ 2 ≤ ?h2.n
case h2.hα
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ ↑?h2.n ≤ PartENat.card α
case h2.n
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ ℕ | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ IsMultiplyPretransitive (↥(alternatingGroup α)) α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | apply IsMultiplyPretransitive.alternatingGroup_of_sub_two | case h2.hn
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ IsMultiplyPretransitive (↥(alternatingGroup α)) α ?h2.n
case h2.hmn
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ 2 ≤ ?h2.n
case h2.hα
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ ↑?h2.n ≤ PartENat.card α
case h2.n
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ ℕ | case h2.hmn
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ 2 ≤ Fintype.card α - 2
case h2.hα
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α | Please generate a tactic in lean4 to solve the state.
STATE:
case h2.hn
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ IsMultiplyPretransitive (↥(alternatingGroup α)) α ?h2.n
case h2.hmn
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ 2 ≤ ?h2.n
case h2.hα
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ ↑?h2.n ≤ PartENat.card α
case h2.n
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ ℕ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | apply le_trans _ (Nat.sub_le_sub_right h4 2) | case h2.hmn
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ 2 ≤ Fintype.card α - 2
case h2.hα
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ 2 ≤ succ 3 - 2
case h2.hα
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α | Please generate a tactic in lean4 to solve the state.
STATE:
case h2.hmn
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ 2 ≤ Fintype.card α - 2
case h2.hα
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | norm_num | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ 2 ≤ succ 3 - 2
case h2.hα
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α | case h2.hα
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ 2 ≤ succ 3 - 2
case h2.hα
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | simp only [PartENat.card_eq_coe_fintype_card, cast_le, tsub_le_iff_right, le_add_iff_nonneg_right,
_root_.zero_le] | case h2.hα
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h2.hα
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h4 : 3 < Fintype.card α
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | exact IsTrivialBlock.of_card_le_2 h2 B | case inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : Fintype.card α ≤ 2
⊢ IsTrivialBlock B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : Fintype.card α ≤ 2
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | have h3' : Fintype.card α = 3 := le_antisymm h3 h2 | case inr.inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
⊢ IsTrivialBlock B | case inr.inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | cases' le_or_lt (Fintype.card B) 1 with h1 h2 | case inr.inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
⊢ IsTrivialBlock B | case inr.inl.inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h1 : Fintype.card ↑B ≤ 1
⊢ IsTrivialBlock B
case inr.inl.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2✝ : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h2 : 1 < Fintype.card ↑B
⊢ IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | apply Or.intro_left | case inr.inl.inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h1 : Fintype.card ↑B ≤ 1
⊢ IsTrivialBlock B | case inr.inl.inl.h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h1 : Fintype.card ↑B ≤ 1
⊢ Set.Subsingleton B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h1 : Fintype.card ↑B ≤ 1
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [← Set.subsingleton_coe, ← Fintype.card_le_one_iff_subsingleton] | case inr.inl.inl.h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h1 : Fintype.card ↑B ≤ 1
⊢ Set.Subsingleton B | case inr.inl.inl.h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h1 : Fintype.card ↑B ≤ 1
⊢ Fintype.card ↑B ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inl.h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h1 : Fintype.card ↑B ≤ 1
⊢ Set.Subsingleton B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | exact h1 | case inr.inl.inl.h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h1 : Fintype.card ↑B ≤ 1
⊢ Fintype.card ↑B ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inl.h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h1 : Fintype.card ↑B ≤ 1
⊢ Fintype.card ↑B ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | apply Or.intro_right | case inr.inl.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2✝ : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h2 : 1 < Fintype.card ↑B
⊢ IsTrivialBlock B | case inr.inl.inr.h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2✝ : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h2 : 1 < Fintype.card ↑B
⊢ B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2✝ : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h2 : 1 < Fintype.card ↑B
⊢ IsTrivialBlock B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [Fintype.one_lt_card_iff] at h2 | case inr.inl.inr.h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2✝ : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h2 : 1 < Fintype.card ↑B
⊢ B = ⊤ | case inr.inl.inr.h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2✝ : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h2 : ∃ a b, a ≠ b
⊢ B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inr.h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2✝ : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h2 : 1 < Fintype.card ↑B
⊢ B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | obtain ⟨⟨a, ha⟩, ⟨b, hb⟩, hab⟩ := h2 | case inr.inl.inr.h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2✝ : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h2 : ∃ a b, a ≠ b
⊢ B = ⊤ | case inr.inl.inr.h.intro.mk.intro.mk
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : { val := a, property := ha } ≠ { val := b, property := hb }
⊢ B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inr.h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2✝ : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
h2 : ∃ a b, a ≠ b
⊢ B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | simp only [Ne.def, Subtype.mk_eq_mk] at hab | case inr.inl.inr.h.intro.mk.intro.mk
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : { val := a, property := ha } ≠ { val := b, property := hb }
⊢ B = ⊤ | case inr.inl.inr.h.intro.mk.intro.mk
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
⊢ B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inr.h.intro.mk.intro.mk
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : { val := a, property := ha } ≠ { val := b, property := hb }
⊢ B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | obtain ⟨c, hc⟩ := this | case inr.inl.inr.h.intro.mk.intro.mk
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
this : ∃ c, c ∉ {a, b}
⊢ B = ⊤ | case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : c ∉ {a, b}
⊢ B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inr.h.intro.mk.intro.mk
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
this : ∃ c, c ∉ {a, b}
⊢ B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | simp only [Finset.mem_insert, Finset.mem_singleton, not_or] at hc | case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : c ∉ {a, b}
⊢ B = ⊤ | case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ B = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : c ∉ {a, b}
⊢ B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | suffices ({a, b, c} : Finset α) = Finset.univ by
rw [eq_top_iff]
rw [Set.top_eq_univ, ← Finset.coe_univ, ← this]
intro x hx
simp only [Finset.coe_insert, Finset.coe_singleton, Set.mem_insert_iff,
Set.mem_singleton_iff] at hx
cases' hx with hxa hx
rw [hxa]; exact ha
cases' hx with hxb hxc
rw [hxb]; exact hb
rw [hxc]
let g : alternatingGroup α :=
⟨Equiv.swap a b * Equiv.swap c b,by rw [Equiv.Perm.mem_alternatingGroup]
rw [Equiv.Perm.sign_mul]
rw [Equiv.Perm.sign_swap hab]
rw [Equiv.Perm.sign_swap hc.right]
simp only [Int.units_mul_self]⟩
suffices g • B = B by
rw [← this]
use b
apply And.intro hb
change (Equiv.swap a b * Equiv.swap c b) • b = c
simp only [Equiv.Perm.smul_def, Equiv.Perm.coe_mul, Function.comp_apply]
rw [Equiv.swap_apply_right]
rw [Equiv.swap_apply_of_ne_of_ne hc.left hc.right]
apply hB.def_mem ha
change (Equiv.swap a b * Equiv.swap c b) • a ∈ B
simp only [Equiv.Perm.smul_def, Equiv.Perm.coe_mul, Function.comp_apply]
rw [Equiv.swap_apply_of_ne_of_ne (ne_comm.mp hc.left) hab]
rw [Equiv.swap_apply_left]
exact hb | case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ B = ⊤ | case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ {a, b, c} = Finset.univ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [← Finset.card_eq_iff_eq_univ, h3'] | case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ {a, b, c} = Finset.univ | case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ {a, b, c}.card = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ {a, b, c} = Finset.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [Finset.card_insert_of_not_mem] | case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ {a, b, c}.card = 3 | case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ {b, c}.card + 1 = 3
case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ a ∉ {b, c} | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ {a, b, c}.card = 3
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [Finset.card_pair (ne_comm.mp hc.right)] | case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ {b, c}.card + 1 = 3
case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ a ∉ {b, c} | case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ a ∉ {b, c} | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ {b, c}.card + 1 = 3
case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ a ∉ {b, c}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | simp only [Finset.mem_insert, Finset.mem_singleton, not_or] | case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ a ∉ {b, c} | case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ ¬a = b ∧ ¬a = c | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ a ∉ {b, c}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | apply And.intro hab | case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ ¬a = b ∧ ¬a = c | case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ ¬a = c | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ ¬a = b ∧ ¬a = c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | exact ne_comm.mp hc.left | case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ ¬a = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inr.h.intro.mk.intro.mk.intro
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
⊢ ¬a = c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | by_contra h | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
⊢ ∃ c, c ∉ {a, b} | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ¬∃ c, c ∉ {a, b}
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
⊢ ∃ c, c ∉ {a, b}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | push_neg at h | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ¬∃ c, c ∉ {a, b}
⊢ False | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ¬∃ c, c ∉ {a, b}
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [lt_iff_not_ge] at h2 | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b} = Finset.univ
⊢ False | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : ¬2 ≥ Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b} = Finset.univ
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b} = Finset.univ
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | apply h2 | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : ¬2 ≥ Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b} = Finset.univ
⊢ False | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : ¬2 ≥ Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b} = Finset.univ
⊢ 2 ≥ Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : ¬2 ≥ Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b} = Finset.univ
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [ge_iff_le] | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : ¬2 ≥ Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b} = Finset.univ
⊢ 2 ≥ Fintype.card α | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : ¬2 ≥ Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b} = Finset.univ
⊢ Fintype.card α ≤ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : ¬2 ≥ Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b} = Finset.univ
⊢ 2 ≥ Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [← Finset.card_eq_iff_eq_univ] at this | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : ¬2 ≥ Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b} = Finset.univ
⊢ Fintype.card α ≤ 2 | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : ¬2 ≥ Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b}.card = Fintype.card α
⊢ Fintype.card α ≤ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : ¬2 ≥ Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b} = Finset.univ
⊢ Fintype.card α ≤ 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [← this] | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : ¬2 ≥ Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b}.card = Fintype.card α
⊢ Fintype.card α ≤ 2 | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : ¬2 ≥ Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b}.card = Fintype.card α
⊢ {a, b}.card ≤ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : ¬2 ≥ Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b}.card = Fintype.card α
⊢ Fintype.card α ≤ 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [Finset.card_pair hab] | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : ¬2 ≥ Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b}.card = Fintype.card α
⊢ {a, b}.card ≤ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : ¬2 ≥ Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
this : {a, b}.card = Fintype.card α
⊢ {a, b}.card ≤ 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | ext c | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
⊢ {a, b} = Finset.univ | case a
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
c : α
⊢ c ∈ {a, b} ↔ c ∈ Finset.univ | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
⊢ {a, b} = Finset.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | constructor | case a
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
c : α
⊢ c ∈ {a, b} ↔ c ∈ Finset.univ | case a.mp
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
c : α
⊢ c ∈ {a, b} → c ∈ Finset.univ
case a.mpr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
c : α
⊢ c ∈ Finset.univ → c ∈ {a, b} | Please generate a tactic in lean4 to solve the state.
STATE:
case a
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
c : α
⊢ c ∈ {a, b} ↔ c ∈ Finset.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | intro _ | case a.mp
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
c : α
⊢ c ∈ {a, b} → c ∈ Finset.univ | case a.mp
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
c : α
a✝ : c ∈ {a, b}
⊢ c ∈ Finset.univ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
c : α
⊢ c ∈ {a, b} → c ∈ Finset.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | exact Finset.mem_univ c | case a.mp
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
c : α
a✝ : c ∈ {a, b}
⊢ c ∈ Finset.univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
c : α
a✝ : c ∈ {a, b}
⊢ c ∈ Finset.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | intro _ | case a.mpr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
c : α
⊢ c ∈ Finset.univ → c ∈ {a, b} | case a.mpr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
c : α
a✝ : c ∈ Finset.univ
⊢ c ∈ {a, b} | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
c : α
⊢ c ∈ Finset.univ → c ∈ {a, b}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | exact h c | case a.mpr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
c : α
a✝ : c ∈ Finset.univ
⊢ c ∈ {a, b} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
h : ∀ (c : α), c ∈ {a, b}
c : α
a✝ : c ∈ Finset.univ
⊢ c ∈ {a, b}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [eq_top_iff] | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
⊢ B = ⊤ | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
⊢ ⊤ ≤ B | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
⊢ B = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [Set.top_eq_univ, ← Finset.coe_univ, ← this] | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
⊢ ⊤ ≤ B | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
⊢ ↑{a, b, c} ≤ B | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
⊢ ⊤ ≤ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | intro x hx | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
⊢ ↑{a, b, c} ≤ B | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hx : x ∈ ↑{a, b, c}
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
⊢ ↑{a, b, c} ≤ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | simp only [Finset.coe_insert, Finset.coe_singleton, Set.mem_insert_iff,
Set.mem_singleton_iff] at hx | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hx : x ∈ ↑{a, b, c}
⊢ x ∈ B | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hx : x = a ∨ x = b ∨ x = c
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hx : x ∈ ↑{a, b, c}
⊢ x ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | cases' hx with hxa hx | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hx : x = a ∨ x = b ∨ x = c
⊢ x ∈ B | case inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxa : x = a
⊢ x ∈ B
case inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hx : x = b ∨ x = c
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hx : x = a ∨ x = b ∨ x = c
⊢ x ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [hxa] | case inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxa : x = a
⊢ x ∈ B
case inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hx : x = b ∨ x = c
⊢ x ∈ B | case inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxa : x = a
⊢ a ∈ B
case inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hx : x = b ∨ x = c
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxa : x = a
⊢ x ∈ B
case inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hx : x = b ∨ x = c
⊢ x ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | exact ha | case inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxa : x = a
⊢ a ∈ B
case inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hx : x = b ∨ x = c
⊢ x ∈ B | case inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hx : x = b ∨ x = c
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxa : x = a
⊢ a ∈ B
case inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hx : x = b ∨ x = c
⊢ x ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | cases' hx with hxb hxc | case inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hx : x = b ∨ x = c
⊢ x ∈ B | case inr.inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxb : x = b
⊢ x ∈ B
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hx : x = b ∨ x = c
⊢ x ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [hxb] | case inr.inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxb : x = b
⊢ x ∈ B
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ x ∈ B | case inr.inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxb : x = b
⊢ b ∈ B
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxb : x = b
⊢ x ∈ B
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ x ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | exact hb | case inr.inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxb : x = b
⊢ b ∈ B
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ x ∈ B | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxb : x = b
⊢ b ∈ B
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ x ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [hxc] | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ x ∈ B | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ c ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ x ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | let g : alternatingGroup α :=
⟨Equiv.swap a b * Equiv.swap c b,by rw [Equiv.Perm.mem_alternatingGroup]
rw [Equiv.Perm.sign_mul]
rw [Equiv.Perm.sign_swap hab]
rw [Equiv.Perm.sign_swap hc.right]
simp only [Int.units_mul_self]⟩ | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ c ∈ B | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ c ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ c ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | suffices g • B = B by
rw [← this]
use b
apply And.intro hb
change (Equiv.swap a b * Equiv.swap c b) • b = c
simp only [Equiv.Perm.smul_def, Equiv.Perm.coe_mul, Function.comp_apply]
rw [Equiv.swap_apply_right]
rw [Equiv.swap_apply_of_ne_of_ne hc.left hc.right] | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ c ∈ B | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ g • B = B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ c ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | apply hB.def_mem ha | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ g • B = B | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ g • a ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ g • B = B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | change (Equiv.swap a b * Equiv.swap c b) • a ∈ B | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ g • a ∈ B | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ (Equiv.swap a b * Equiv.swap c b) • a ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ g • a ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | simp only [Equiv.Perm.smul_def, Equiv.Perm.coe_mul, Function.comp_apply] | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ (Equiv.swap a b * Equiv.swap c b) • a ∈ B | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ (Equiv.swap a b) ((Equiv.swap c b) a) ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ (Equiv.swap a b * Equiv.swap c b) • a ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [Equiv.swap_apply_of_ne_of_ne (ne_comm.mp hc.left) hab] | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ (Equiv.swap a b) ((Equiv.swap c b) a) ∈ B | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ (Equiv.swap a b) a ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ (Equiv.swap a b) ((Equiv.swap c b) a) ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [Equiv.swap_apply_left] | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ (Equiv.swap a b) a ∈ B | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ b ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ (Equiv.swap a b) a ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | exact hb | case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ b ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
⊢ b ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [Equiv.Perm.mem_alternatingGroup] | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ Equiv.swap a b * Equiv.swap c b ∈ alternatingGroup α | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ Equiv.Perm.sign (Equiv.swap a b * Equiv.swap c b) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ Equiv.swap a b * Equiv.swap c b ∈ alternatingGroup α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [Equiv.Perm.sign_mul] | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ Equiv.Perm.sign (Equiv.swap a b * Equiv.swap c b) = 1 | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ Equiv.Perm.sign (Equiv.swap a b) * Equiv.Perm.sign (Equiv.swap c b) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ Equiv.Perm.sign (Equiv.swap a b * Equiv.swap c b) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [Equiv.Perm.sign_swap hab] | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ Equiv.Perm.sign (Equiv.swap a b) * Equiv.Perm.sign (Equiv.swap c b) = 1 | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ -1 * Equiv.Perm.sign (Equiv.swap c b) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ Equiv.Perm.sign (Equiv.swap a b) * Equiv.Perm.sign (Equiv.swap c b) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [Equiv.Perm.sign_swap hc.right] | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ -1 * Equiv.Perm.sign (Equiv.swap c b) = 1 | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ -1 * -1 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ -1 * Equiv.Perm.sign (Equiv.swap c b) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | simp only [Int.units_mul_self] | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ -1 * -1 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this : {a, b, c} = Finset.univ
x : α
hxc : x = c
⊢ -1 * -1 = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [← this] | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ c ∈ B | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ c ∈ g • B | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ c ∈ B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | use b | M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ c ∈ g • B | case h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ b ∈ B ∧ (fun x => (Submonoid.subtype (alternatingGroup α).toSubmonoid) g • x) b = c | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ c ∈ g • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | apply And.intro hb | case h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ b ∈ B ∧ (fun x => (Submonoid.subtype (alternatingGroup α).toSubmonoid) g • x) b = c | case h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ (fun x => (Submonoid.subtype (alternatingGroup α).toSubmonoid) g • x) b = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ b ∈ B ∧ (fun x => (Submonoid.subtype (alternatingGroup α).toSubmonoid) g • x) b = c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | change (Equiv.swap a b * Equiv.swap c b) • b = c | case h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ (fun x => (Submonoid.subtype (alternatingGroup α).toSubmonoid) g • x) b = c | case h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ (Equiv.swap a b * Equiv.swap c b) • b = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ (fun x => (Submonoid.subtype (alternatingGroup α).toSubmonoid) g • x) b = c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | simp only [Equiv.Perm.smul_def, Equiv.Perm.coe_mul, Function.comp_apply] | case h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ (Equiv.swap a b * Equiv.swap c b) • b = c | case h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ (Equiv.swap a b) ((Equiv.swap c b) b) = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ (Equiv.swap a b * Equiv.swap c b) • b = c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [Equiv.swap_apply_right] | case h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ (Equiv.swap a b) ((Equiv.swap c b) b) = c | case h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ (Equiv.swap a b) c = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ (Equiv.swap a b) ((Equiv.swap c b) b) = c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.alternatingGroup.has_trivial_blocks | [1186, 1] | [1266, 20] | rw [Equiv.swap_apply_of_ne_of_ne hc.left hc.right] | case h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ (Equiv.swap a b) c = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type ?u.291196
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Fintype.card α
h3 : Fintype.card α ≤ 3
h3' : Fintype.card α = 3
a : α
ha : a ∈ B
b : α
hb : b ∈ B
hab : ¬a = b
c : α
hc : ¬c = a ∧ ¬c = b
this✝ : {a, b, c} = Finset.univ
x : α
hxc : x = c
g : ↥(alternatingGroup α) := { val := Equiv.swap a b * Equiv.swap c b, property := ⋯ }
this : g • B = B
⊢ (Equiv.swap a b) c = c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.AlternatingGroup.isPreprimitive | [1270, 1] | [1274, 44] | have := alternatingGroup.isPretransitive h | M : Type ?u.309192
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ IsPreprimitive (↥(alternatingGroup α)) α | M : Type ?u.309192
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
this : IsPretransitive (↥(alternatingGroup α)) α
⊢ IsPreprimitive (↥(alternatingGroup α)) α | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.309192
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
⊢ IsPreprimitive (↥(alternatingGroup α)) α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.AlternatingGroup.isPreprimitive | [1270, 1] | [1274, 44] | apply IsPreprimitive.mk | M : Type ?u.309192
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
this : IsPretransitive (↥(alternatingGroup α)) α
⊢ IsPreprimitive (↥(alternatingGroup α)) α | case has_trivial_blocks'
M : Type ?u.309192
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
this : IsPretransitive (↥(alternatingGroup α)) α
⊢ ∀ {B : Set α}, IsBlock (↥(alternatingGroup α)) B → IsTrivialBlock B | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type ?u.309192
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
this : IsPretransitive (↥(alternatingGroup α)) α
⊢ IsPreprimitive (↥(alternatingGroup α)) α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultipleTransitivity.lean | MulAction.AlternatingGroup.isPreprimitive | [1270, 1] | [1274, 44] | apply alternatingGroup.has_trivial_blocks | case has_trivial_blocks'
M : Type ?u.309192
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
this : IsPretransitive (↥(alternatingGroup α)) α
⊢ ∀ {B : Set α}, IsBlock (↥(alternatingGroup α)) B → IsTrivialBlock B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case has_trivial_blocks'
M : Type ?u.309192
α : Type u_1
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Fintype.card α
this : IsPretransitive (↥(alternatingGroup α)) α
⊢ ∀ {B : Set α}, IsBlock (↥(alternatingGroup α)) B → IsTrivialBlock B
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.