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/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | simp only [Subgroup.mem_zpowers_iff] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
⊢ subtypePerm g hc' ∈ Subgroup.zpowers (subtypePermOfSupport c) ↔ ofSubtype (subtypePerm g hc') ∈ Subgroup.zpowers c | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
⊢ (∃ k, subtypePermOfSupport c ^ k = subtypePerm g hc') ↔ ∃ k, c ^ k = ofSubtype (subtypePerm g hc') | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
⊢ subtypePerm g hc' ∈ Subgroup.zpowers (subtypePermOfSupport c) ↔ ofSubtype (subtypePerm g hc') ∈ Subgroup.zpowers c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | apply exists_congr | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
⊢ (∃ k, subtypePermOfSupport c ^ k = subtypePerm g hc') ↔ ∃ k, c ^ k = ofSubtype (subtypePerm g hc') | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
⊢ ∀ (a : ℤ), subtypePermOfSupport c ^ a = subtypePerm g hc' ↔ c ^ a = ofSubtype (subtypePerm g hc') | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
⊢ (∃ k, subtypePermOfSupport c ^ k = subtypePerm g hc') ↔ ∃ k, c ^ k = ofSubtype (subtypePerm g hc')
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | intro k | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
⊢ ∀ (a : ℤ), subtypePermOfSupport c ^ a = subtypePerm g hc' ↔ c ^ a = ofSubtype (subtypePerm g hc') | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
⊢ subtypePermOfSupport c ^ k = subtypePerm g hc' ↔ c ^ k = ofSubtype (subtypePerm g hc') | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
⊢ ∀ (a : ℤ), subtypePermOfSupport c ^ a = subtypePerm g hc' ↔ c ^ a = ofSubtype (subtypePerm g hc')
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | unfold subtypePermOfSupport | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
⊢ subtypePermOfSupport c ^ k = subtypePerm g hc' ↔ c ^ k = ofSubtype (subtypePerm g hc') | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
⊢ subtypePerm c ⋯ ^ k = subtypePerm g hc' ↔ c ^ k = ofSubtype (subtypePerm g hc') | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
⊢ subtypePermOfSupport c ^ k = subtypePerm g hc' ↔ c ^ k = ofSubtype (subtypePerm g hc')
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | rw [Equiv.Perm.subtypePerm_zpow] | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
⊢ subtypePerm c ⋯ ^ k = subtypePerm g hc' ↔ c ^ k = ofSubtype (subtypePerm g hc') | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
⊢ subtypePerm (c ^ k) ⋯ = subtypePerm g hc' ↔ c ^ k = ofSubtype (subtypePerm g hc') | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
⊢ subtypePerm c ⋯ ^ k = subtypePerm g hc' ↔ c ^ k = ofSubtype (subtypePerm g hc')
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | simp only [Equiv.ext_iff, subtypePerm_apply, Subtype.mk.injEq, Subtype.forall] | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
⊢ subtypePerm (c ^ k) ⋯ = subtypePerm g hc' ↔ c ^ k = ofSubtype (subtypePerm g hc') | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
⊢ (∀ a ∈ support c, (c ^ k) a = g a) ↔ ∀ (x : α), (c ^ k) x = (ofSubtype (subtypePerm g hc')) x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
⊢ subtypePerm (c ^ k) ⋯ = subtypePerm g hc' ↔ c ^ k = ofSubtype (subtypePerm g hc')
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | apply forall_congr' | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
⊢ (∀ a ∈ support c, (c ^ k) a = g a) ↔ ∀ (x : α), (c ^ k) x = (ofSubtype (subtypePerm g hc')) x | case h.h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
⊢ ∀ (a : α), a ∈ support c → (c ^ k) a = g a ↔ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
⊢ (∀ a ∈ support c, (c ^ k) a = g a) ↔ ∀ (x : α), (c ^ k) x = (ofSubtype (subtypePerm g hc')) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | intro a | case h.h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
⊢ ∀ (a : α), a ∈ support c → (c ^ k) a = g a ↔ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a | case h.h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
⊢ a ∈ support c → (c ^ k) a = g a ↔ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
⊢ ∀ (a : α), a ∈ support c → (c ^ k) a = g a ↔ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | by_cases ha : a ∈ c.support | case h.h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
⊢ a ∈ support c → (c ^ k) a = g a ↔ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ a ∈ support c → (c ^ k) a = g a ↔ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a ↔ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
⊢ a ∈ support c → (c ^ k) a = g a ↔ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | rw [imp_iff_right ha] | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ a ∈ support c → (c ^ k) a = g a ↔ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ (c ^ k) a = g a ↔ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ a ∈ support c → (c ^ k) a = g a ↔ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | apply Eq.congr rfl | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ (c ^ k) a = g a ↔ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ g a = (ofSubtype (subtypePerm g hc')) a | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ (c ^ k) a = g a ↔ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | rw [Equiv.Perm.ofSubtype_apply_of_mem] | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ g a = (ofSubtype (subtypePerm g hc')) a | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ g a = ↑((subtypePerm g hc') { val := a, property := ?pos.ha✝ })
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ a ∈ support c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ g a = (ofSubtype (subtypePerm g hc')) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | rfl | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ g a = ↑((subtypePerm g hc') { val := a, property := ?pos.ha✝ })
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ a ∈ support c | case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ a ∈ support c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ g a = ↑((subtypePerm g hc') { val := a, property := ?pos.ha✝ })
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ a ∈ support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | exact ha | case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ a ∈ support c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∈ support c
⊢ a ∈ support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | rw [iff_true_left _] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a ↔ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a ↔ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | rw [Equiv.Perm.ofSubtype_apply_of_not_mem] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ (c ^ k) a = a
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ (c ^ k) a = (ofSubtype (subtypePerm g hc')) a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | rw [←Equiv.Perm.not_mem_support] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ (c ^ k) a = a
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support (c ^ k)
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ (c ^ k) a = a
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | intro ha' | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support (c ^ k)
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
ha' : a ∈ support (c ^ k)
⊢ False
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support (c ^ k)
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | apply ha | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
ha' : a ∈ support (c ^ k)
⊢ False
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
ha' : a ∈ support (c ^ k)
⊢ a ∈ support c
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
ha' : a ∈ support (c ^ k)
⊢ False
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | apply Equiv.Perm.support_zpow_le | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
ha' : a ∈ support (c ^ k)
⊢ a ∈ support c
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a | case neg.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
ha' : a ∈ support (c ^ k)
⊢ a ∈ support (c ^ ?neg.n✝)
case neg.n
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
ha' : a ∈ support (c ^ k)
⊢ ℤ
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
ha' : a ∈ support (c ^ k)
⊢ a ∈ support c
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | exact ha' | case neg.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
ha' : a ∈ support (c ^ k)
⊢ a ∈ support (c ^ ?neg.n✝)
case neg.n
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
ha' : a ∈ support (c ^ k)
⊢ ℤ
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a | case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
ha' : a ∈ support (c ^ k)
⊢ a ∈ support (c ^ ?neg.n✝)
case neg.n
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
ha' : a ∈ support (c ^ k)
⊢ ℤ
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | exact ha | case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∉ support c
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.commute_iff | [941, 1] | [969, 37] | exact fun b => False.elim (ha b) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : IsCycle c
hc' : ∀ (x : α), x ∈ support c ↔ g x ∈ support c
k : ℤ
a : α
ha : a ∉ support c
⊢ a ∈ support c → (c ^ k) a = g a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.forall_commute_iff | [971, 1] | [980, 87] | apply forall_congr' | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Perm α
⊢ (∀ c ∈ cycleFactorsFinset g, Commute z c) ↔
∀ c ∈ cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ support c ↔ z x ∈ support c), ofSubtype (subtypePerm z hc) ∈ Subgroup.zpowers c | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Perm α
⊢ ∀ (a : Perm α),
a ∈ cycleFactorsFinset g → Commute z a ↔
a ∈ cycleFactorsFinset g →
∃ (hc : ∀ (x : α), x ∈ support a ↔ z x ∈ support a), ofSubtype (subtypePerm z hc) ∈ Subgroup.zpowers a | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Perm α
⊢ (∀ c ∈ cycleFactorsFinset g, Commute z c) ↔
∀ c ∈ cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ support c ↔ z x ∈ support c), ofSubtype (subtypePerm z hc) ∈ Subgroup.zpowers c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.forall_commute_iff | [971, 1] | [980, 87] | intro c | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Perm α
⊢ ∀ (a : Perm α),
a ∈ cycleFactorsFinset g → Commute z a ↔
a ∈ cycleFactorsFinset g →
∃ (hc : ∀ (x : α), x ∈ support a ↔ z x ∈ support a), ofSubtype (subtypePerm z hc) ∈ Subgroup.zpowers a | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z c : Perm α
⊢ c ∈ cycleFactorsFinset g → Commute z c ↔
c ∈ cycleFactorsFinset g →
∃ (hc : ∀ (x : α), x ∈ support c ↔ z x ∈ support c), ofSubtype (subtypePerm z hc) ∈ Subgroup.zpowers c | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Perm α
⊢ ∀ (a : Perm α),
a ∈ cycleFactorsFinset g → Commute z a ↔
a ∈ cycleFactorsFinset g →
∃ (hc : ∀ (x : α), x ∈ support a ↔ z x ∈ support a), ofSubtype (subtypePerm z hc) ∈ Subgroup.zpowers a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.forall_commute_iff | [971, 1] | [980, 87] | apply imp_congr_right | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z c : Perm α
⊢ c ∈ cycleFactorsFinset g → Commute z c ↔
c ∈ cycleFactorsFinset g →
∃ (hc : ∀ (x : α), x ∈ support c ↔ z x ∈ support c), ofSubtype (subtypePerm z hc) ∈ Subgroup.zpowers c | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z c : Perm α
⊢ c ∈ cycleFactorsFinset g →
(Commute z c ↔
∃ (hc : ∀ (x : α), x ∈ support c ↔ z x ∈ support c), ofSubtype (subtypePerm z hc) ∈ Subgroup.zpowers c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z c : Perm α
⊢ c ∈ cycleFactorsFinset g → Commute z c ↔
c ∈ cycleFactorsFinset g →
∃ (hc : ∀ (x : α), x ∈ support c ↔ z x ∈ support c), ofSubtype (subtypePerm z hc) ∈ Subgroup.zpowers c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.forall_commute_iff | [971, 1] | [980, 87] | intro hc | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z c : Perm α
⊢ c ∈ cycleFactorsFinset g →
(Commute z c ↔
∃ (hc : ∀ (x : α), x ∈ support c ↔ z x ∈ support c), ofSubtype (subtypePerm z hc) ∈ Subgroup.zpowers c) | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z c : Perm α
hc : c ∈ cycleFactorsFinset g
⊢ Commute z c ↔ ∃ (hc : ∀ (x : α), x ∈ support c ↔ z x ∈ support c), ofSubtype (subtypePerm z hc) ∈ Subgroup.zpowers c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z c : Perm α
⊢ c ∈ cycleFactorsFinset g →
(Commute z c ↔
∃ (hc : ∀ (x : α), x ∈ support c ↔ z x ∈ support c), ofSubtype (subtypePerm z hc) ∈ Subgroup.zpowers c)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.IsCycle.forall_commute_iff | [971, 1] | [980, 87] | exact Equiv.Perm.IsCycle.commute_iff (Equiv.Perm.mem_cycleFactorsFinset_iff.mp hc).1 | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z c : Perm α
hc : c ∈ cycleFactorsFinset g
⊢ Commute z c ↔ ∃ (hc : ∀ (x : α), x ∈ support c ↔ z x ∈ support c), ofSubtype (subtypePerm z hc) ∈ Subgroup.zpowers c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z c : Perm α
hc : c ∈ cycleFactorsFinset g
⊢ Commute z c ↔ ∃ (hc : ∀ (x : α), x ∈ support c ↔ z x ∈ support c), ofSubtype (subtypePerm z hc) ∈ Subgroup.zpowers c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.subtypePerm_on_cycleFactorsFinset | [1045, 1] | [1051, 57] | ext ⟨x, hx⟩ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : c ∈ cycleFactorsFinset g
⊢ subtypePerm g ⋯ = subtypePermOfSupport c | case H.mk.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : c ∈ cycleFactorsFinset g
x : α
hx : x ∈ support c
⊢ ↑((subtypePerm g ⋯) { val := x, property := hx }) = ↑((subtypePermOfSupport c) { val := x, property := hx }) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : c ∈ cycleFactorsFinset g
⊢ subtypePerm g ⋯ = subtypePermOfSupport c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.subtypePerm_on_cycleFactorsFinset | [1045, 1] | [1051, 57] | simp only [subtypePerm_apply, Subtype.coe_mk, subtypePermOfSupport] | case H.mk.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : c ∈ cycleFactorsFinset g
x : α
hx : x ∈ support c
⊢ ↑((subtypePerm g ⋯) { val := x, property := hx }) = ↑((subtypePermOfSupport c) { val := x, property := hx }) | case H.mk.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : c ∈ cycleFactorsFinset g
x : α
hx : x ∈ support c
⊢ g x = c x | Please generate a tactic in lean4 to solve the state.
STATE:
case H.mk.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : c ∈ cycleFactorsFinset g
x : α
hx : x ∈ support c
⊢ ↑((subtypePerm g ⋯) { val := x, property := hx }) = ↑((subtypePermOfSupport c) { val := x, property := hx })
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.subtypePerm_on_cycleFactorsFinset | [1045, 1] | [1051, 57] | exact ((mem_cycleFactorsFinset_iff.mp hc).2 x hx).symm | case H.mk.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : c ∈ cycleFactorsFinset g
x : α
hx : x ∈ support c
⊢ g x = c x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H.mk.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
hc : c ∈ cycleFactorsFinset g
x : α
hx : x ∈ support c
⊢ g x = c x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.commute_of_mem_cycleFactorsFinset_iff | [1054, 1] | [1068, 6] | rw [Equiv.Perm.IsCycle.commute_iff' (mem_cycleFactorsFinset_iff.mp hc).1] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
⊢ Commute k c ↔
∃ (hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c), subtypePerm k hc' ∈ Subgroup.zpowers (subtypePerm g ⋯) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
⊢ (∃ (hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c),
subtypePerm k hc' ∈ Subgroup.zpowers (subtypePermOfSupport c)) ↔
∃ (hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c), subtypePerm k hc' ∈ Subgroup.zpowers (subtypePerm g ⋯) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
⊢ Commute k c ↔
∃ (hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c), subtypePerm k hc' ∈ Subgroup.zpowers (subtypePerm g ⋯)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.commute_of_mem_cycleFactorsFinset_iff | [1054, 1] | [1068, 6] | apply exists_congr | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
⊢ (∃ (hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c),
subtypePerm k hc' ∈ Subgroup.zpowers (subtypePermOfSupport c)) ↔
∃ (hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c), subtypePerm k hc' ∈ Subgroup.zpowers (subtypePerm g ⋯) | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
⊢ ∀ (a : ∀ (x : α), x ∈ support c ↔ k x ∈ support c),
subtypePerm k a ∈ Subgroup.zpowers (subtypePermOfSupport c) ↔ subtypePerm k a ∈ Subgroup.zpowers (subtypePerm g ⋯) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
⊢ (∃ (hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c),
subtypePerm k hc' ∈ Subgroup.zpowers (subtypePermOfSupport c)) ↔
∃ (hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c), subtypePerm k hc' ∈ Subgroup.zpowers (subtypePerm g ⋯)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.commute_of_mem_cycleFactorsFinset_iff | [1054, 1] | [1068, 6] | intro hc' | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
⊢ ∀ (a : ∀ (x : α), x ∈ support c ↔ k x ∈ support c),
subtypePerm k a ∈ Subgroup.zpowers (subtypePermOfSupport c) ↔ subtypePerm k a ∈ Subgroup.zpowers (subtypePerm g ⋯) | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
⊢ subtypePerm k hc' ∈ Subgroup.zpowers (subtypePermOfSupport c) ↔ subtypePerm k hc' ∈ Subgroup.zpowers (subtypePerm g ⋯) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
⊢ ∀ (a : ∀ (x : α), x ∈ support c ↔ k x ∈ support c),
subtypePerm k a ∈ Subgroup.zpowers (subtypePermOfSupport c) ↔ subtypePerm k a ∈ Subgroup.zpowers (subtypePerm g ⋯)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.commute_of_mem_cycleFactorsFinset_iff | [1054, 1] | [1068, 6] | simp only [Subgroup.mem_zpowers_iff] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
⊢ subtypePerm k hc' ∈ Subgroup.zpowers (subtypePermOfSupport c) ↔ subtypePerm k hc' ∈ Subgroup.zpowers (subtypePerm g ⋯) | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
⊢ (∃ k_1, subtypePermOfSupport c ^ k_1 = subtypePerm k hc') ↔ ∃ k_1, subtypePerm g ⋯ ^ k_1 = subtypePerm k hc' | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
⊢ subtypePerm k hc' ∈ Subgroup.zpowers (subtypePermOfSupport c) ↔ subtypePerm k hc' ∈ Subgroup.zpowers (subtypePerm g ⋯)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.commute_of_mem_cycleFactorsFinset_iff | [1054, 1] | [1068, 6] | apply exists_congr | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
⊢ (∃ k_1, subtypePermOfSupport c ^ k_1 = subtypePerm k hc') ↔ ∃ k_1, subtypePerm g ⋯ ^ k_1 = subtypePerm k hc' | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
⊢ ∀ (a : ℤ), subtypePermOfSupport c ^ a = subtypePerm k hc' ↔ subtypePerm g ⋯ ^ a = subtypePerm k hc' | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
⊢ (∃ k_1, subtypePermOfSupport c ^ k_1 = subtypePerm k hc') ↔ ∃ k_1, subtypePerm g ⋯ ^ k_1 = subtypePerm k hc'
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.commute_of_mem_cycleFactorsFinset_iff | [1054, 1] | [1068, 6] | intro n | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
⊢ ∀ (a : ℤ), subtypePermOfSupport c ^ a = subtypePerm k hc' ↔ subtypePerm g ⋯ ^ a = subtypePerm k hc' | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
n : ℤ
⊢ subtypePermOfSupport c ^ n = subtypePerm k hc' ↔ subtypePerm g ⋯ ^ n = subtypePerm k hc' | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
⊢ ∀ (a : ℤ), subtypePermOfSupport c ^ a = subtypePerm k hc' ↔ subtypePerm g ⋯ ^ a = subtypePerm k hc'
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.commute_of_mem_cycleFactorsFinset_iff | [1054, 1] | [1068, 6] | unfold subtypePermOfSupport | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
n : ℤ
⊢ subtypePermOfSupport c ^ n = subtypePerm k hc' ↔ subtypePerm g ⋯ ^ n = subtypePerm k hc' | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
n : ℤ
⊢ subtypePerm c ⋯ ^ n = subtypePerm k hc' ↔ subtypePerm g ⋯ ^ n = subtypePerm k hc' | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
n : ℤ
⊢ subtypePermOfSupport c ^ n = subtypePerm k hc' ↔ subtypePerm g ⋯ ^ n = subtypePerm k hc'
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.commute_of_mem_cycleFactorsFinset_iff | [1054, 1] | [1068, 6] | rw [Equiv.Perm.subtypePerm_on_cycleFactorsFinset hc] | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
n : ℤ
⊢ subtypePerm c ⋯ ^ n = subtypePerm k hc' ↔ subtypePerm g ⋯ ^ n = subtypePerm k hc' | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
n : ℤ
⊢ subtypePerm c ⋯ ^ n = subtypePerm k hc' ↔ subtypePermOfSupport c ^ n = subtypePerm k hc' | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
n : ℤ
⊢ subtypePerm c ⋯ ^ n = subtypePerm k hc' ↔ subtypePerm g ⋯ ^ n = subtypePerm k hc'
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.commute_of_mem_cycleFactorsFinset_iff | [1054, 1] | [1068, 6] | rfl | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
n : ℤ
⊢ subtypePerm c ⋯ ^ n = subtypePerm k hc' ↔ subtypePermOfSupport c ^ n = subtypePerm k hc' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
hc : c ∈ cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ support c ↔ k x ∈ support c
n : ℤ
⊢ subtypePerm c ⋯ ^ n = subtypePerm k hc' ↔ subtypePermOfSupport c ^ n = subtypePerm k hc'
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.zpow_mod_card_support_cycleOf_self_apply | [1082, 1] | [1090, 58] | by_cases hx : f x = x | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ inst✝ : Fintype α
f : Perm α
n : ℤ
x : α
⊢ (f ^ (n % ↑(support (cycleOf f x)).card)) x = (f ^ n) x | case pos
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ inst✝ : Fintype α
f : Perm α
n : ℤ
x : α
hx : f x = x
⊢ (f ^ (n % ↑(support (cycleOf f x)).card)) x = (f ^ n) x
case neg
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ inst✝ : Fintype α
f : Perm α
n : ℤ
x : α
hx : ¬f x = x
⊢ (f ^ (n % ↑(support (cycleOf f x)).card)) x = (f ^ n) x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ inst✝ : Fintype α
f : Perm α
n : ℤ
x : α
⊢ (f ^ (n % ↑(support (cycleOf f x)).card)) x = (f ^ n) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.zpow_mod_card_support_cycleOf_self_apply | [1082, 1] | [1090, 58] | rw [Equiv.Perm.zpow_apply_eq_self_of_apply_eq_self hx,
Equiv.Perm.zpow_apply_eq_self_of_apply_eq_self hx] | case pos
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ inst✝ : Fintype α
f : Perm α
n : ℤ
x : α
hx : f x = x
⊢ (f ^ (n % ↑(support (cycleOf f x)).card)) x = (f ^ n) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ inst✝ : Fintype α
f : Perm α
n : ℤ
x : α
hx : f x = x
⊢ (f ^ (n % ↑(support (cycleOf f x)).card)) x = (f ^ n) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.zpow_mod_card_support_cycleOf_self_apply | [1082, 1] | [1090, 58] | rw [← f.cycleOf_zpow_apply_self, ← f.cycleOf_zpow_apply_self,
← (f.isCycle_cycleOf hx).orderOf, zpow_mod_orderOf] | case neg
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ inst✝ : Fintype α
f : Perm α
n : ℤ
x : α
hx : ¬f x = x
⊢ (f ^ (n % ↑(support (cycleOf f x)).card)) x = (f ^ n) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ inst✝ : Fintype α
f : Perm α
n : ℤ
x : α
hx : ¬f x = x
⊢ (f ^ (n % ↑(support (cycleOf f x)).card)) x = (f ^ n) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | let q := n / g.support.card | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
⊢ (g ^ n) x = x ↔ n % ↑(support g).card = 0 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
⊢ (g ^ n) x = x ↔ n % ↑(support g).card = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
⊢ (g ^ n) x = x ↔ n % ↑(support g).card = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | let r := n % g.support.card | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
⊢ (g ^ n) x = x ↔ n % ↑(support g).card = 0 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ (g ^ n) x = x ↔ n % ↑(support g).card = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
⊢ (g ^ n) x = x ↔ n % ↑(support g).card = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | change _ ↔ r = 0 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ (g ^ n) x = x ↔ n % ↑(support g).card = 0 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ (g ^ n) x = x ↔ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ (g ^ n) x = x ↔ n % ↑(support g).card = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | have div_euc : r + g.support.card * q = n ∧ 0 ≤ r ∧ r < g.support.card := by
rw [← Int.ediv_emod_unique _]
constructor; rfl; rfl
simp only [Int.coe_nat_pos]
apply lt_of_lt_of_le _ (Equiv.Perm.IsCycle.two_le_card_support hg); norm_num | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ (g ^ n) x = x ↔ r = 0 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
div_euc : r + ↑(support g).card * q = n ∧ 0 ≤ r ∧ r < ↑(support g).card
⊢ (g ^ n) x = x ↔ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ (g ^ n) x = x ↔ r = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | simp only [← hg.orderOf] at div_euc | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
div_euc : r + ↑(support g).card * q = n ∧ 0 ≤ r ∧ r < ↑(support g).card
⊢ (g ^ n) x = x ↔ r = 0 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
div_euc : r + ↑(orderOf g) * q = n ∧ 0 ≤ r ∧ r < ↑(orderOf g)
⊢ (g ^ n) x = x ↔ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
div_euc : r + ↑(support g).card * q = n ∧ 0 ≤ r ∧ r < ↑(support g).card
⊢ (g ^ n) x = x ↔ r = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | obtain ⟨m, hm⟩ := Int.eq_ofNat_of_zero_le div_euc.2.1 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
div_euc : r + ↑(orderOf g) * q = n ∧ 0 ≤ r ∧ r < ↑(orderOf g)
⊢ (g ^ n) x = x ↔ r = 0 | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
div_euc : r + ↑(orderOf g) * q = n ∧ 0 ≤ r ∧ r < ↑(orderOf g)
m : ℕ
hm : r = ↑m
⊢ (g ^ n) x = x ↔ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
div_euc : r + ↑(orderOf g) * q = n ∧ 0 ≤ r ∧ r < ↑(orderOf g)
⊢ (g ^ n) x = x ↔ r = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | simp only [hm, Nat.cast_nonneg, Nat.cast_lt, true_and_iff] at div_euc | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
div_euc : r + ↑(orderOf g) * q = n ∧ 0 ≤ r ∧ r < ↑(orderOf g)
m : ℕ
hm : r = ↑m
⊢ (g ^ n) x = x ↔ r = 0 | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ n) x = x ↔ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
div_euc : r + ↑(orderOf g) * q = n ∧ 0 ≤ r ∧ r < ↑(orderOf g)
m : ℕ
hm : r = ↑m
⊢ (g ^ n) x = x ↔ r = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | simp only [hm, Nat.cast_eq_zero] | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ n) x = x ↔ r = 0 | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ n) x = x ↔ m = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ n) x = x ↔ r = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | rw [← div_euc.1, zpow_add g, zpow_mul] | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ n) x = x ↔ m = 0 | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ ↑m * (g ^ ↑(orderOf g)) ^ q) x = x ↔ m = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ n) x = x ↔ m = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | simp only [zpow_natCast, coe_mul, Function.comp_apply] | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ ↑m * (g ^ ↑(orderOf g)) ^ q) x = x ↔ m = 0 | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ m) (((g ^ orderOf g) ^ q) x) = x ↔ m = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ ↑m * (g ^ ↑(orderOf g)) ^ q) x = x ↔ m = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | simp only [pow_orderOf_eq_one, one_zpow, coe_one, id_eq] | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ m) (((g ^ orderOf g) ^ q) x) = x ↔ m = 0 | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ m) x = x ↔ m = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ m) (((g ^ orderOf g) ^ q) x) = x ↔ m = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | simp only [this] | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
this : (g ^ m) x = x ↔ g ^ m = 1
⊢ (g ^ m) x = x ↔ m = 0 | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
this : (g ^ m) x = x ↔ g ^ m = 1
⊢ g ^ m = 1 ↔ m = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
this : (g ^ m) x = x ↔ g ^ m = 1
⊢ (g ^ m) x = x ↔ m = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | cases' dec_em (m = 0) with hm0 hm0' | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
this : (g ^ m) x = x ↔ g ^ m = 1
⊢ g ^ m = 1 ↔ m = 0 | case intro.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
this : (g ^ m) x = x ↔ g ^ m = 1
hm0 : m = 0
⊢ g ^ m = 1 ↔ m = 0
case intro.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
this : (g ^ m) x = x ↔ g ^ m = 1
hm0' : ¬m = 0
⊢ g ^ m = 1 ↔ m = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
this : (g ^ m) x = x ↔ g ^ m = 1
⊢ g ^ m = 1 ↔ m = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | rw [← Int.ediv_emod_unique _] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ r + ↑(support g).card * q = n ∧ 0 ≤ r ∧ r < ↑(support g).card | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ n / ↑(support g).card = q ∧ n % ↑(support g).card = r
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < ↑(support g).card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ r + ↑(support g).card * q = n ∧ 0 ≤ r ∧ r < ↑(support g).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | constructor | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ n / ↑(support g).card = q ∧ n % ↑(support g).card = r
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < ↑(support g).card | case left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ n / ↑(support g).card = q
case right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ n % ↑(support g).card = r
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < ↑(support g).card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ n / ↑(support g).card = q ∧ n % ↑(support g).card = r
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < ↑(support g).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | rfl | case left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ n / ↑(support g).card = q
case right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ n % ↑(support g).card = r
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < ↑(support g).card | case right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ n % ↑(support g).card = r
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < ↑(support g).card | Please generate a tactic in lean4 to solve the state.
STATE:
case left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ n / ↑(support g).card = q
case right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ n % ↑(support g).card = r
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < ↑(support g).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | rfl | case right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ n % ↑(support g).card = r
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < ↑(support g).card | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < ↑(support g).card | Please generate a tactic in lean4 to solve the state.
STATE:
case right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ n % ↑(support g).card = r
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < ↑(support g).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | simp only [Int.coe_nat_pos] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < ↑(support g).card | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < (support g).card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < ↑(support g).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | apply lt_of_lt_of_le _ (Equiv.Perm.IsCycle.two_le_card_support hg) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < (support g).card | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < (support g).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | norm_num | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
⊢ 0 < 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | constructor | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ m) x = x ↔ g ^ m = 1 | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ m) x = x → g ^ m = 1
case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ g ^ m = 1 → (g ^ m) x = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ m) x = x ↔ g ^ m = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | intro hgm | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ m) x = x → g ^ m = 1 | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
hgm : (g ^ m) x = x
⊢ g ^ m = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ m) x = x → g ^ m = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | simp [Equiv.Perm.IsCycle.pow_eq_one_iff hg] | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
hgm : (g ^ m) x = x
⊢ g ^ m = 1 | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
hgm : (g ^ m) x = x
⊢ ∃ x, ¬g x = x ∧ (g ^ m) x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
hgm : (g ^ m) x = x
⊢ g ^ m = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | use x | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
hgm : (g ^ m) x = x
⊢ ∃ x, ¬g x = x ∧ (g ^ m) x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
hgm : (g ^ m) x = x
⊢ ∃ x, ¬g x = x ∧ (g ^ m) x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | intro hgm | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ g ^ m = 1 → (g ^ m) x = x | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
hgm : g ^ m = 1
⊢ (g ^ m) x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ g ^ m = 1 → (g ^ m) x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | rw [hgm] | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
hgm : g ^ m = 1
⊢ (g ^ m) x = x | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
hgm : g ^ m = 1
⊢ 1 x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
hgm : g ^ m = 1
⊢ (g ^ m) x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | simp only [Equiv.Perm.coe_one, id.def] | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
hgm : g ^ m = 1
⊢ 1 x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
hgm : g ^ m = 1
⊢ 1 x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | simp only [hm0, pow_zero, Nat.cast_zero] | case intro.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
this : (g ^ m) x = x ↔ g ^ m = 1
hm0 : m = 0
⊢ g ^ m = 1 ↔ m = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
this : (g ^ m) x = x ↔ g ^ m = 1
hm0 : m = 0
⊢ g ^ m = 1 ↔ m = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | simp only [Nat.cast_eq_zero, hm0', iff_false] | case intro.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
this : (g ^ m) x = x ↔ g ^ m = 1
hm0' : ¬m = 0
⊢ g ^ m = 1 ↔ m = 0 | case intro.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
this : (g ^ m) x = x ↔ g ^ m = 1
hm0' : ¬m = 0
⊢ ¬g ^ m = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
this : (g ^ m) x = x ↔ g ^ m = 1
hm0' : ¬m = 0
⊢ g ^ m = 1 ↔ m = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycle_zpow_mem_support_iff | [1093, 1] | [1120, 51] | exact pow_ne_one_of_lt_orderOf' hm0' div_euc.2 | case intro.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
this : (g ^ m) x = x ↔ g ^ m = 1
hm0' : ¬m = 0
⊢ ¬g ^ m = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
hg : IsCycle g
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(support g).card
r : ℤ := n % ↑(support g).card
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
this : (g ^ m) x = x ↔ g ^ m = 1
hm0' : ¬m = 0
⊢ ¬g ^ m = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.zpow_eq_zpow_on_iff | [1124, 1] | [1136, 80] | rw [Int.emod_eq_emod_iff_emod_sub_eq_zero] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (g ^ m) x = (g ^ n) x ↔ m % ↑(support (cycleOf g x)).card = n % ↑(support (cycleOf g x)).card | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (g ^ m) x = (g ^ n) x ↔ (m - n) % ↑(support (cycleOf g x)).card = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (g ^ m) x = (g ^ n) x ↔ m % ↑(support (cycleOf g x)).card = n % ↑(support (cycleOf g x)).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.zpow_eq_zpow_on_iff | [1124, 1] | [1136, 80] | conv_lhs => rw [← Int.sub_add_cancel m n, Int.add_comm, zpow_add] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (g ^ m) x = (g ^ n) x ↔ (m - n) % ↑(support (cycleOf g x)).card = 0 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (g ^ n * g ^ (m - n)) x = (g ^ n) x ↔ (m - n) % ↑(support (cycleOf g x)).card = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (g ^ m) x = (g ^ n) x ↔ (m - n) % ↑(support (cycleOf g x)).card = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.zpow_eq_zpow_on_iff | [1124, 1] | [1136, 80] | simp only [coe_mul, Function.comp_apply, EmbeddingLike.apply_eq_iff_eq,
EuclideanDomain.mod_eq_zero] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (g ^ n * g ^ (m - n)) x = (g ^ n) x ↔ (m - n) % ↑(support (cycleOf g x)).card = 0 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (g ^ (m - n)) x = x ↔ ↑(support (cycleOf g x)).card ∣ m - n | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (g ^ n * g ^ (m - n)) x = (g ^ n) x ↔ (m - n) % ↑(support (cycleOf g x)).card = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.zpow_eq_zpow_on_iff | [1124, 1] | [1136, 80] | rw [← Equiv.Perm.cycleOf_zpow_apply_self g x] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (g ^ (m - n)) x = x ↔ ↑(support (cycleOf g x)).card ∣ m - n | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (cycleOf g x ^ (m - n)) x = x ↔ ↑(support (cycleOf g x)).card ∣ m - n | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (g ^ (m - n)) x = x ↔ ↑(support (cycleOf g x)).card ∣ m - n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.zpow_eq_zpow_on_iff | [1124, 1] | [1136, 80] | rw [Equiv.Perm.cycle_zpow_mem_support_iff] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (cycleOf g x ^ (m - n)) x = x ↔ ↑(support (cycleOf g x)).card ∣ m - n | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (m - n) % ↑(support (cycleOf g x)).card = 0 ↔ ↑(support (cycleOf g x)).card ∣ m - n
case hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ IsCycle (cycleOf g x)
case hx
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (cycleOf g x) x ≠ x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (cycleOf g x ^ (m - n)) x = x ↔ ↑(support (cycleOf g x)).card ∣ m - n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.zpow_eq_zpow_on_iff | [1124, 1] | [1136, 80] | simp only [EuclideanDomain.mod_eq_zero] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (m - n) % ↑(support (cycleOf g x)).card = 0 ↔ ↑(support (cycleOf g x)).card ∣ m - n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (m - n) % ↑(support (cycleOf g x)).card = 0 ↔ ↑(support (cycleOf g x)).card ∣ m - n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.zpow_eq_zpow_on_iff | [1124, 1] | [1136, 80] | exact Equiv.Perm.isCycle_cycleOf g hx | case hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ IsCycle (cycleOf g x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ IsCycle (cycleOf g x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.zpow_eq_zpow_on_iff | [1124, 1] | [1136, 80] | simp only [Equiv.Perm.mem_support, Equiv.Perm.cycleOf_apply_self] | case hx
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (cycleOf g x) x ≠ x | case hx
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ g x ≠ x | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (cycleOf g x) x ≠ x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.zpow_eq_zpow_on_iff | [1124, 1] | [1136, 80] | exact hx | case hx
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ g x ≠ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ g x ≠ x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | commute_ofSubtype_disjoint | [1174, 1] | [1184, 29] | apply Equiv.Perm.Disjoint.commute | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
⊢ Commute (Equiv.Perm.ofSubtype x) (Equiv.Perm.ofSubtype y) | case h
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
⊢ Equiv.Perm.Disjoint (Equiv.Perm.ofSubtype x) (Equiv.Perm.ofSubtype y) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
⊢ Commute (Equiv.Perm.ofSubtype x) (Equiv.Perm.ofSubtype y)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | commute_ofSubtype_disjoint | [1174, 1] | [1184, 29] | intro a | case h
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
⊢ Equiv.Perm.Disjoint (Equiv.Perm.ofSubtype x) (Equiv.Perm.ofSubtype y) | case h
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
⊢ (Equiv.Perm.ofSubtype x) a = a ∨ (Equiv.Perm.ofSubtype y) a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
⊢ Equiv.Perm.Disjoint (Equiv.Perm.ofSubtype x) (Equiv.Perm.ofSubtype y)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | commute_ofSubtype_disjoint | [1174, 1] | [1184, 29] | by_cases hx : p a | case h
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
⊢ (Equiv.Perm.ofSubtype x) a = a ∨ (Equiv.Perm.ofSubtype y) a = a | case pos
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ (Equiv.Perm.ofSubtype x) a = a ∨ (Equiv.Perm.ofSubtype y) a = a
case neg
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : ¬p a
⊢ (Equiv.Perm.ofSubtype x) a = a ∨ (Equiv.Perm.ofSubtype y) a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
⊢ (Equiv.Perm.ofSubtype x) a = a ∨ (Equiv.Perm.ofSubtype y) a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | commute_ofSubtype_disjoint | [1174, 1] | [1184, 29] | rw [Equiv.Perm.ofSubtype_apply_of_not_mem y] | case pos
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ (Equiv.Perm.ofSubtype x) a = a ∨ (Equiv.Perm.ofSubtype y) a = a | case pos
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ (Equiv.Perm.ofSubtype x) a = a ∨ a = a
case pos
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ ¬q a | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ (Equiv.Perm.ofSubtype x) a = a ∨ (Equiv.Perm.ofSubtype y) a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | commute_ofSubtype_disjoint | [1174, 1] | [1184, 29] | apply Or.intro_right | case pos
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ (Equiv.Perm.ofSubtype x) a = a ∨ a = a
case pos
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ ¬q a | case pos.h
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ a = a
case pos
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ ¬q a | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ (Equiv.Perm.ofSubtype x) a = a ∨ a = a
case pos
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ ¬q a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | commute_ofSubtype_disjoint | [1174, 1] | [1184, 29] | rfl | case pos.h
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ a = a
case pos
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ ¬q a | case pos
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ ¬q a | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ a = a
case pos
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ ¬q a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | commute_ofSubtype_disjoint | [1174, 1] | [1184, 29] | exact not_and.mp (hpq a) hx | case pos
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ ¬q a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : p a
⊢ ¬q a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | commute_ofSubtype_disjoint | [1174, 1] | [1184, 29] | rw [Equiv.Perm.ofSubtype_apply_of_not_mem x hx] | case neg
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : ¬p a
⊢ (Equiv.Perm.ofSubtype x) a = a ∨ (Equiv.Perm.ofSubtype y) a = a | case neg
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : ¬p a
⊢ a = a ∨ (Equiv.Perm.ofSubtype y) a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : ¬p a
⊢ (Equiv.Perm.ofSubtype x) a = a ∨ (Equiv.Perm.ofSubtype y) a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | commute_ofSubtype_disjoint | [1174, 1] | [1184, 29] | apply Or.intro_left | case neg
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : ¬p a
⊢ a = a ∨ (Equiv.Perm.ofSubtype y) a = a | case neg.h
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : ¬p a
⊢ a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : ¬p a
⊢ a = a ∨ (Equiv.Perm.ofSubtype y) a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | commute_ofSubtype_disjoint | [1174, 1] | [1184, 29] | rfl | case neg.h
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : ¬p a
⊢ a = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Fintype α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hpq : ∀ (a : α), ¬(p a ∧ q a)
x : Equiv.Perm (Subtype p)
y : Equiv.Perm (Subtype q)
a : α
hx : ¬p a
⊢ a = a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.of_partition_card | [1348, 1] | [1370, 57] | suffices ∀ x : Equiv.Perm α, (x ∈ MulAction.stabilizer (Equiv.Perm α) p ↔ p ∘ x = p) by
simp_rw [← this]
suffices Fintype.card {f | f ∈ MulAction.stabilizer (Equiv.Perm α) p} = Fintype.card (MulAction.stabilizer (Equiv.Perm α) p) by
rw [this, Fintype.card_congr (Φ p).toEquiv]
simp only [Set.coe_setOf, Set.mem_setOf_eq, Fintype.card_pi]
apply Finset.prod_congr rfl
intro i _
exact Fintype.card_perm
rw [Fintype.card_ofFinset, ← Fintype.subtype_card]
intro x
simp only [MulAction.mem_stabilizer_iff, Set.mem_setOf_eq, Finset.mem_univ, forall_true_left, ne_eq,
Finset.mem_filter, true_and] | α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
⊢ Fintype.card ↑{f | p ∘ ⇑f = p} = Finset.prod Finset.univ fun i => Nat.factorial (Fintype.card ↑{a | p a = i}) | α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
⊢ ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
⊢ Fintype.card ↑{f | p ∘ ⇑f = p} = Finset.prod Finset.univ fun i => Nat.factorial (Fintype.card ↑{a | p a = i})
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.of_partition_card | [1348, 1] | [1370, 57] | simp_rw [← this] | α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
⊢ Fintype.card ↑{f | p ∘ ⇑f = p} = Finset.prod Finset.univ fun i => Nat.factorial (Fintype.card ↑{a | p a = i}) | α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
⊢ Fintype.card ↑{f | f ∈ MulAction.stabilizer (Perm α) p} =
Finset.prod Finset.univ fun i => Nat.factorial (Fintype.card ↑{a | p a = i}) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
⊢ Fintype.card ↑{f | p ∘ ⇑f = p} = Finset.prod Finset.univ fun i => Nat.factorial (Fintype.card ↑{a | p a = i})
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.of_partition_card | [1348, 1] | [1370, 57] | suffices Fintype.card {f | f ∈ MulAction.stabilizer (Equiv.Perm α) p} = Fintype.card (MulAction.stabilizer (Equiv.Perm α) p) by
rw [this, Fintype.card_congr (Φ p).toEquiv]
simp only [Set.coe_setOf, Set.mem_setOf_eq, Fintype.card_pi]
apply Finset.prod_congr rfl
intro i _
exact Fintype.card_perm | α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
⊢ Fintype.card ↑{f | f ∈ MulAction.stabilizer (Perm α) p} =
Finset.prod Finset.univ fun i => Nat.factorial (Fintype.card ↑{a | p a = i}) | α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
⊢ Fintype.card ↑{f | f ∈ MulAction.stabilizer (Perm α) p} = Fintype.card ↥(MulAction.stabilizer (Perm α) p) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
⊢ Fintype.card ↑{f | f ∈ MulAction.stabilizer (Perm α) p} =
Finset.prod Finset.univ fun i => Nat.factorial (Fintype.card ↑{a | p a = i})
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.of_partition_card | [1348, 1] | [1370, 57] | rw [Fintype.card_ofFinset, ← Fintype.subtype_card] | α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
⊢ Fintype.card ↑{f | f ∈ MulAction.stabilizer (Perm α) p} = Fintype.card ↥(MulAction.stabilizer (Perm α) p) | case H
α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
⊢ ∀ (x : Perm α),
x ∈ Finset.filter (fun x => x ∈ {f | f ∈ MulAction.stabilizer (Perm α) p}) Finset.univ ↔
x ∈ MulAction.stabilizer (Perm α) p | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
⊢ Fintype.card ↑{f | f ∈ MulAction.stabilizer (Perm α) p} = Fintype.card ↥(MulAction.stabilizer (Perm α) p)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.of_partition_card | [1348, 1] | [1370, 57] | intro x | case H
α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
⊢ ∀ (x : Perm α),
x ∈ Finset.filter (fun x => x ∈ {f | f ∈ MulAction.stabilizer (Perm α) p}) Finset.univ ↔
x ∈ MulAction.stabilizer (Perm α) p | case H
α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
x : Perm α
⊢ x ∈ Finset.filter (fun x => x ∈ {f | f ∈ MulAction.stabilizer (Perm α) p}) Finset.univ ↔
x ∈ MulAction.stabilizer (Perm α) p | Please generate a tactic in lean4 to solve the state.
STATE:
case H
α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
⊢ ∀ (x : Perm α),
x ∈ Finset.filter (fun x => x ∈ {f | f ∈ MulAction.stabilizer (Perm α) p}) Finset.univ ↔
x ∈ MulAction.stabilizer (Perm α) p
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.of_partition_card | [1348, 1] | [1370, 57] | simp only [MulAction.mem_stabilizer_iff, Set.mem_setOf_eq, Finset.mem_univ, forall_true_left, ne_eq,
Finset.mem_filter, true_and] | case H
α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
x : Perm α
⊢ x ∈ Finset.filter (fun x => x ∈ {f | f ∈ MulAction.stabilizer (Perm α) p}) Finset.univ ↔
x ∈ MulAction.stabilizer (Perm α) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H
α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
x : Perm α
⊢ x ∈ Finset.filter (fun x => x ∈ {f | f ∈ MulAction.stabilizer (Perm α) p}) Finset.univ ↔
x ∈ MulAction.stabilizer (Perm α) p
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.of_partition_card | [1348, 1] | [1370, 57] | rw [this, Fintype.card_congr (Φ p).toEquiv] | α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this✝ : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
this : Fintype.card ↑{f | f ∈ MulAction.stabilizer (Perm α) p} = Fintype.card ↥(MulAction.stabilizer (Perm α) p)
⊢ Fintype.card ↑{f | f ∈ MulAction.stabilizer (Perm α) p} =
Finset.prod Finset.univ fun i => Nat.factorial (Fintype.card ↑{a | p a = i}) | α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this✝ : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
this : Fintype.card ↑{f | f ∈ MulAction.stabilizer (Perm α) p} = Fintype.card ↥(MulAction.stabilizer (Perm α) p)
⊢ Fintype.card ((i : ι) → Perm ↑{a | p a = i}) =
Finset.prod Finset.univ fun i => Nat.factorial (Fintype.card ↑{a | p a = i}) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this✝ : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
this : Fintype.card ↑{f | f ∈ MulAction.stabilizer (Perm α) p} = Fintype.card ↥(MulAction.stabilizer (Perm α) p)
⊢ Fintype.card ↑{f | f ∈ MulAction.stabilizer (Perm α) p} =
Finset.prod Finset.univ fun i => Nat.factorial (Fintype.card ↑{a | p a = i})
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.of_partition_card | [1348, 1] | [1370, 57] | simp only [Set.coe_setOf, Set.mem_setOf_eq, Fintype.card_pi] | α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this✝ : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
this : Fintype.card ↑{f | f ∈ MulAction.stabilizer (Perm α) p} = Fintype.card ↥(MulAction.stabilizer (Perm α) p)
⊢ Fintype.card ((i : ι) → Perm ↑{a | p a = i}) =
Finset.prod Finset.univ fun i => Nat.factorial (Fintype.card ↑{a | p a = i}) | α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this✝ : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
this : Fintype.card ↑{f | f ∈ MulAction.stabilizer (Perm α) p} = Fintype.card ↥(MulAction.stabilizer (Perm α) p)
⊢ (Finset.prod Finset.univ fun a => Fintype.card (Perm { x // p x = a })) =
Finset.prod Finset.univ fun x => Nat.factorial (Fintype.card { x_1 // p x_1 = x }) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
inst✝³ : DecidableEq α
inst✝² : Fintype α
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : α → ι
this✝ : ∀ (x : Perm α), x ∈ MulAction.stabilizer (Perm α) p ↔ p ∘ ⇑x = p
this : Fintype.card ↑{f | f ∈ MulAction.stabilizer (Perm α) p} = Fintype.card ↥(MulAction.stabilizer (Perm α) p)
⊢ Fintype.card ((i : ι) → Perm ↑{a | p a = i}) =
Finset.prod Finset.univ fun i => Nat.factorial (Fintype.card ↑{a | p a = i})
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.