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 | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | apply Equiv.Perm.mem_cycleFactorsFinset_support_le hx' | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hx : x ∈ Equiv.Perm.support g
hx' : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g := Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hx
⊢ z x ∈ Equiv.Perm.support g | case mp.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hx : x ∈ Equiv.Perm.support g
hx' : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g := Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hx
⊢ z x ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g 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 z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hx : x ∈ Equiv.Perm.support g
hx' : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g := Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hx
⊢ z x ∈ Equiv.Perm.support g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | obtain ⟨Hz'⟩ := Hz (g.cycleOf x)
(Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hx) | case mp.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hx : x ∈ Equiv.Perm.support g
hx' : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g := Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hx
⊢ z x ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x) | case mp.a.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hx : x ∈ Equiv.Perm.support g
hx' : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g := Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hx
Hz' :
∀ (x_1 : α), x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x) ↔ z x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x)
h✝ : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z Hz') ∈ Subgroup.zpowers (Equiv.Perm.cycleOf g x)
⊢ z x ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hx : x ∈ Equiv.Perm.support g
hx' : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g := Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hx
⊢ z x ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rw [← Hz' x, Equiv.Perm.mem_support_cycleOf_iff] | case mp.a.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hx : x ∈ Equiv.Perm.support g
hx' : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g := Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hx
Hz' :
∀ (x_1 : α), x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x) ↔ z x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x)
h✝ : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z Hz') ∈ Subgroup.zpowers (Equiv.Perm.cycleOf g x)
⊢ z x ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x) | case mp.a.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hx : x ∈ Equiv.Perm.support g
hx' : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g := Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hx
Hz' :
∀ (x_1 : α), x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x) ↔ z x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x)
h✝ : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z Hz') ∈ Subgroup.zpowers (Equiv.Perm.cycleOf g x)
⊢ Equiv.Perm.SameCycle g x x ∧ x ∈ Equiv.Perm.support g | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.a.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hx : x ∈ Equiv.Perm.support g
hx' : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g := Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hx
Hz' :
∀ (x_1 : α), x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x) ↔ z x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x)
h✝ : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z Hz') ∈ Subgroup.zpowers (Equiv.Perm.cycleOf g x)
⊢ z x ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | exact ⟨Equiv.Perm.SameCycle.refl _ _, hx⟩ | case mp.a.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hx : x ∈ Equiv.Perm.support g
hx' : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g := Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hx
Hz' :
∀ (x_1 : α), x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x) ↔ z x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x)
h✝ : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z Hz') ∈ Subgroup.zpowers (Equiv.Perm.cycleOf g x)
⊢ Equiv.Perm.SameCycle g x x ∧ x ∈ Equiv.Perm.support g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.a.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hx : x ∈ Equiv.Perm.support g
hx' : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g := Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hx
Hz' :
∀ (x_1 : α), x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x) ↔ z x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g x)
h✝ : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z Hz') ∈ Subgroup.zpowers (Equiv.Perm.cycleOf g x)
⊢ Equiv.Perm.SameCycle g x x ∧ x ∈ Equiv.Perm.support g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | intro hzx | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
⊢ z x ∈ Equiv.Perm.support g → x ∈ Equiv.Perm.support g | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hzx : z x ∈ Equiv.Perm.support g
⊢ x ∈ Equiv.Perm.support g | 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 z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
⊢ z x ∈ Equiv.Perm.support g → x ∈ Equiv.Perm.support g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | let hzx' := Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hzx | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hzx : z x ∈ Equiv.Perm.support g
⊢ x ∈ Equiv.Perm.support g | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hzx : z x ∈ Equiv.Perm.support g
hzx' : Equiv.Perm.cycleOf g (z x) ∈ Equiv.Perm.cycleFactorsFinset g :=
Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hzx
⊢ x ∈ Equiv.Perm.support g | 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 z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hzx : z x ∈ Equiv.Perm.support g
⊢ x ∈ Equiv.Perm.support g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | apply Equiv.Perm.mem_cycleFactorsFinset_support_le hzx' | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hzx : z x ∈ Equiv.Perm.support g
hzx' : Equiv.Perm.cycleOf g (z x) ∈ Equiv.Perm.cycleFactorsFinset g :=
Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hzx
⊢ x ∈ Equiv.Perm.support g | case mpr.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hzx : z x ∈ Equiv.Perm.support g
hzx' : Equiv.Perm.cycleOf g (z x) ∈ Equiv.Perm.cycleFactorsFinset g :=
Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hzx
⊢ x ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z 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 z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hzx : z x ∈ Equiv.Perm.support g
hzx' : Equiv.Perm.cycleOf g (z x) ∈ Equiv.Perm.cycleFactorsFinset g :=
Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hzx
⊢ x ∈ Equiv.Perm.support g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | obtain ⟨Hz'⟩ := Hz (g.cycleOf (z x)) hzx' | case mpr.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hzx : z x ∈ Equiv.Perm.support g
hzx' : Equiv.Perm.cycleOf g (z x) ∈ Equiv.Perm.cycleFactorsFinset g :=
Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hzx
⊢ x ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x)) | case mpr.a.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hzx : z x ∈ Equiv.Perm.support g
hzx' : Equiv.Perm.cycleOf g (z x) ∈ Equiv.Perm.cycleFactorsFinset g :=
Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hzx
Hz' :
∀ (x_1 : α),
x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x)) ↔ z x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x))
h✝ : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z Hz') ∈ Subgroup.zpowers (Equiv.Perm.cycleOf g (z x))
⊢ x ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hzx : z x ∈ Equiv.Perm.support g
hzx' : Equiv.Perm.cycleOf g (z x) ∈ Equiv.Perm.cycleFactorsFinset g :=
Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hzx
⊢ x ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rw [Hz' x, Equiv.Perm.mem_support_cycleOf_iff] | case mpr.a.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hzx : z x ∈ Equiv.Perm.support g
hzx' : Equiv.Perm.cycleOf g (z x) ∈ Equiv.Perm.cycleFactorsFinset g :=
Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hzx
Hz' :
∀ (x_1 : α),
x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x)) ↔ z x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x))
h✝ : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z Hz') ∈ Subgroup.zpowers (Equiv.Perm.cycleOf g (z x))
⊢ x ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x)) | case mpr.a.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hzx : z x ∈ Equiv.Perm.support g
hzx' : Equiv.Perm.cycleOf g (z x) ∈ Equiv.Perm.cycleFactorsFinset g :=
Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hzx
Hz' :
∀ (x_1 : α),
x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x)) ↔ z x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x))
h✝ : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z Hz') ∈ Subgroup.zpowers (Equiv.Perm.cycleOf g (z x))
⊢ Equiv.Perm.SameCycle g (z x) (z x) ∧ z x ∈ Equiv.Perm.support g | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.a.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hzx : z x ∈ Equiv.Perm.support g
hzx' : Equiv.Perm.cycleOf g (z x) ∈ Equiv.Perm.cycleFactorsFinset g :=
Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hzx
Hz' :
∀ (x_1 : α),
x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x)) ↔ z x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x))
h✝ : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z Hz') ∈ Subgroup.zpowers (Equiv.Perm.cycleOf g (z x))
⊢ x ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | exact ⟨Equiv.Perm.SameCycle.refl _ _, hzx⟩ | case mpr.a.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hzx : z x ∈ Equiv.Perm.support g
hzx' : Equiv.Perm.cycleOf g (z x) ∈ Equiv.Perm.cycleFactorsFinset g :=
Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hzx
Hz' :
∀ (x_1 : α),
x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x)) ↔ z x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x))
h✝ : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z Hz') ∈ Subgroup.zpowers (Equiv.Perm.cycleOf g (z x))
⊢ Equiv.Perm.SameCycle g (z x) (z x) ∧ z x ∈ Equiv.Perm.support g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.a.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
x : α
hzx : z x ∈ Equiv.Perm.support g
hzx' : Equiv.Perm.cycleOf g (z x) ∈ Equiv.Perm.cycleFactorsFinset g :=
Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff.mpr hzx
Hz' :
∀ (x_1 : α),
x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x)) ↔ z x_1 ∈ Equiv.Perm.support (Equiv.Perm.cycleOf g (z x))
h✝ : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z Hz') ∈ Subgroup.zpowers (Equiv.Perm.cycleOf g (z x))
⊢ Equiv.Perm.SameCycle g (z x) (z x) ∧ z x ∈ Equiv.Perm.support g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rw [hθ_1 _ x] | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : Equiv.Perm.cycleOf g x = 1
⊢ ((θ g) (u, v)) x = z x | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : Equiv.Perm.cycleOf g x = 1
⊢ ↑((u, v).1 { val := x, property := ?pos✝ }) = z x
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : Equiv.Perm.cycleOf g x = 1
⊢ x ∈ Function.fixedPoints ⇑g | 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 z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : Equiv.Perm.cycleOf g x = 1
⊢ ((θ g) (u, v)) x = z x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | simp only [u, v, Equiv.Perm.subtypePerm_apply] | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : Equiv.Perm.cycleOf g x = 1
⊢ ↑((u, v).1 { val := x, property := ?pos✝ }) = z x
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : Equiv.Perm.cycleOf g x = 1
⊢ x ∈ Function.fixedPoints ⇑g | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : Equiv.Perm.cycleOf g x = 1
⊢ x ∈ Function.fixedPoints ⇑g | 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 z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : Equiv.Perm.cycleOf g x = 1
⊢ ↑((u, v).1 { val := x, property := ?pos✝ }) = z x
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : Equiv.Perm.cycleOf g x = 1
⊢ x ∈ Function.fixedPoints ⇑g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | simp only [Equiv.Perm.cycleOf_eq_one_iff] at hx | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : Equiv.Perm.cycleOf g x = 1
⊢ x ∈ Function.fixedPoints ⇑g | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : g x = x
⊢ x ∈ Function.fixedPoints ⇑g | 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 z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : Equiv.Perm.cycleOf g x = 1
⊢ x ∈ Function.fixedPoints ⇑g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | exact hx | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : g x = x
⊢ x ∈ Function.fixedPoints ⇑g | no goals | 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 z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : g x = x
⊢ x ∈ Function.fixedPoints ⇑g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rw [hθ_2 _ x ⟨g.cycleOf x, (Equiv.Perm.cycleOf_ne_one_iff_mem g).mp hx⟩ rfl] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ ((θ g) (u, v)) x = z x | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ ↑((u, v).2 { val := Equiv.Perm.cycleOf g x, property := ⋯ }) x = z x | 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 z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ ((θ g) (u, v)) x = z x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | change (v _ : Equiv.Perm α) x = _ | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ ↑((u, v).2 { val := Equiv.Perm.cycleOf g x, property := ⋯ }) x = z x | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ ↑(v { val := Equiv.Perm.cycleOf g x, property := ⋯ }) x = z x | 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 z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ ↑((u, v).2 { val := Equiv.Perm.cycleOf g x, property := ⋯ }) x = z x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rw [Equiv.Perm.ofSubtype_apply_of_mem] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ ↑(v { val := Equiv.Perm.cycleOf g x, property := ⋯ }) x = z x | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ ↑((Equiv.Perm.subtypePerm z ⋯) { val := x, property := ?neg.ha✝ }) = z x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ x ∈ Equiv.Perm.support ↑{ val := Equiv.Perm.cycleOf g x, property := ⋯ } | 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 z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ ↑(v { val := Equiv.Perm.cycleOf g x, property := ⋯ }) x = z x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rfl | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ ↑((Equiv.Perm.subtypePerm z ⋯) { val := x, property := ?neg.ha✝ }) = z x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ x ∈ Equiv.Perm.support ↑{ val := Equiv.Perm.cycleOf g x, property := ⋯ } | case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ x ∈ Equiv.Perm.support ↑{ val := Equiv.Perm.cycleOf g x, property := ⋯ } | 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 z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ ↑((Equiv.Perm.subtypePerm z ⋯) { val := x, property := ?neg.ha✝ }) = z x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ x ∈ Equiv.Perm.support ↑{ val := Equiv.Perm.cycleOf g x, property := ⋯ }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | simp only [Equiv.Perm.mem_support, Equiv.Perm.cycleOf_apply_self, ne_eq] | case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ x ∈ Equiv.Perm.support ↑{ val := Equiv.Perm.cycleOf g x, property := ⋯ } | case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ ¬g x = x | 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 z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ x ∈ Equiv.Perm.support ↑{ val := Equiv.Perm.cycleOf g x, property := ⋯ }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | simp only [Equiv.Perm.cycleOf_eq_one_iff] at hx | case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ ¬g x = x | case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬g x = x
⊢ ¬g x = x | 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 z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬Equiv.Perm.cycleOf g x = 1
⊢ ¬g x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | exact hx | case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬g x = x
⊢ ¬g x = x | no goals | 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 z : Equiv.Perm α
Hz :
∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
hu : ∀ (x : α), x ∈ Function.fixedPoints ⇑g ↔ z x ∈ Function.fixedPoints ⇑g
u : Equiv.Perm { x // x ∈ Function.fixedPoints ⇑g } := Equiv.Perm.subtypePerm z hu
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c) :=
fun c => { val := Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z ⋯), property := ⋯ }
x : α
hx : ¬g x = x
⊢ ¬g x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rintro ⟨⟨u, v⟩, h⟩ | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
⊢ z ∈ Set.range ⇑(θ g) →
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g)) | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
⊢ ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g)) | 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 z : Equiv.Perm α
⊢ z ∈ Set.range ⇑(θ g) →
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rw [hφ_mem_ker_iff] | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
⊢ ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g)) | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
⊢ ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
⊢ ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rw [Equiv.Perm.IsCycle.forall_commute_iff] | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
⊢ ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
⊢ ∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
⊢ ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | intro c hc | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
⊢ ∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
⊢ ∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
⊢ ∀ c ∈ Equiv.Perm.cycleFactorsFinset g,
∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | intro x | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
⊢ ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
⊢ ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | simp only [← Equiv.Perm.eq_cycleOf_of_mem_cycleFactorsFinset_iff g c hc] | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ c = Equiv.Perm.cycleOf g x ↔ c = Equiv.Perm.cycleOf g (z x) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rw [← h] | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ c = Equiv.Perm.cycleOf g x ↔ c = Equiv.Perm.cycleOf g (z x) | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ c = Equiv.Perm.cycleOf g x ↔ c = Equiv.Perm.cycleOf g (((θ g) (u, v)) x) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ c = Equiv.Perm.cycleOf g x ↔ c = Equiv.Perm.cycleOf g (z x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | unfold θ | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ c = Equiv.Perm.cycleOf g x ↔ c = Equiv.Perm.cycleOf g (((θ g) (u, v)) x) | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ c = Equiv.Perm.cycleOf g x ↔
c =
Equiv.Perm.cycleOf g
(({ toOneHom := { toFun := fun kv => θFun g kv.1 kv.2, map_one' := ⋯ }, map_mul' := ⋯ } (u, v)) x) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ c = Equiv.Perm.cycleOf g x ↔ c = Equiv.Perm.cycleOf g (((θ g) (u, v)) x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | unfold θFun | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ c = Equiv.Perm.cycleOf g x ↔
c =
Equiv.Perm.cycleOf g
(({ toOneHom := { toFun := fun kv => θFun g kv.1 kv.2, map_one' := ⋯ }, map_mul' := ⋯ } (u, v)) x) | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ c = Equiv.Perm.cycleOf g x ↔
c =
Equiv.Perm.cycleOf g
(({
toOneHom :=
{
toFun := fun kv =>
{ toFun := θAux g kv.1 kv.2, invFun := θAux g kv.1⁻¹ kv.2⁻¹, left_inv := ⋯, right_inv := ⋯ },
map_one' := ⋯ },
map_mul' := ⋯ }
(u, v))
x) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ c = Equiv.Perm.cycleOf g x ↔
c =
Equiv.Perm.cycleOf g
(({ toOneHom := { toFun := fun kv => θFun g kv.1 kv.2, map_one' := ⋯ }, map_mul' := ⋯ } (u, v)) x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | dsimp only [MonoidHom.coe_mk, OneHom.coe_mk, Equiv.coe_fn_mk] | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ c = Equiv.Perm.cycleOf g x ↔
c =
Equiv.Perm.cycleOf g
(({
toOneHom :=
{
toFun := fun kv =>
{ toFun := θAux g kv.1 kv.2, invFun := θAux g kv.1⁻¹ kv.2⁻¹, left_inv := ⋯, right_inv := ⋯ },
map_one' := ⋯ },
map_mul' := ⋯ }
(u, v))
x) | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ c = Equiv.Perm.cycleOf g x ↔ c = Equiv.Perm.cycleOf g (θAux g u v x) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ c = Equiv.Perm.cycleOf g x ↔
c =
Equiv.Perm.cycleOf g
(({
toOneHom :=
{
toFun := fun kv =>
{ toFun := θAux g kv.1 kv.2, invFun := θAux g kv.1⁻¹ kv.2⁻¹, left_inv := ⋯, right_inv := ⋯ },
map_one' := ⋯ },
map_mul' := ⋯ }
(u, v))
x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rw [θAux_cycleOf_apply_eq] | case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ c = Equiv.Perm.cycleOf g x ↔ c = Equiv.Perm.cycleOf g (θAux g u v x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
x : α
⊢ c = Equiv.Perm.cycleOf g x ↔ c = Equiv.Perm.cycleOf g (θAux g u v x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | use hc' | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ?m.373134
⊢ ∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
⊢ Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc') ∈ Subgroup.zpowers c | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ?m.373134
⊢ ∃ (hc : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c),
Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc) ∈ Subgroup.zpowers c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | suffices Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc') = v ⟨c, hc⟩ by
rw [this]
exact (v _).prop | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
⊢ Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc') ∈ Subgroup.zpowers c | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
⊢ Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc') = ↑(v { val := c, property := 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 z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
⊢ Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc') ∈ Subgroup.zpowers c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | ext x | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
⊢ Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc') = ↑(v { val := c, property := hc }) | case h.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
⊢ (Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc')) x = ↑(v { val := c, property := hc }) x | 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 : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
⊢ Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc') = ↑(v { val := c, property := hc })
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | by_cases hx : x ∈ c.support | case h.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
⊢ (Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc')) x = ↑(v { val := c, property := hc }) x | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ (Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc')) x = ↑(v { val := c, property := hc }) x
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ (Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc')) x = ↑(v { val := c, property := 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 z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
⊢ (Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc')) x = ↑(v { val := c, property := hc }) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rw [this] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
this : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc') = ↑(v { val := c, property := hc })
⊢ Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc') ∈ Subgroup.zpowers c | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
this : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc') = ↑(v { val := c, property := hc })
⊢ ↑(v { val := c, property := hc }) ∈ Subgroup.zpowers c | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
this : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc') = ↑(v { val := c, property := hc })
⊢ Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc') ∈ Subgroup.zpowers c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | exact (v _).prop | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
this : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc') = ↑(v { val := c, property := hc })
⊢ ↑(v { val := c, property := hc }) ∈ Subgroup.zpowers c | 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 z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
this : Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc') = ↑(v { val := c, property := hc })
⊢ ↑(v { val := c, property := hc }) ∈ Subgroup.zpowers c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rw [Equiv.Perm.ofSubtype_apply_of_mem, Equiv.Perm.subtypePerm_apply] | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ (Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc')) x = ↑(v { val := c, property := hc }) x | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ ↑{ val := z ↑{ val := x, property := ?pos.ha✝ }, property := ⋯ } = ↑(v { val := c, property := hc }) x
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.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 z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ (Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc')) x = ↑(v { val := c, property := hc }) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | dsimp | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ ↑{ val := z ↑{ val := x, property := ?pos.ha✝ }, property := ⋯ } = ↑(v { val := c, property := hc }) x
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ z x = ↑(v { val := c, property := hc }) x
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.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 z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ ↑{ val := z ↑{ val := x, property := ?pos.ha✝ }, property := ⋯ } = ↑(v { val := c, property := hc }) x
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rw [← h, hθ_2 _ x ⟨c, hc⟩] | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ z x = ↑(v { val := c, property := hc }) x
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ Equiv.Perm.cycleOf g x = ↑{ val := c, property := hc }
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.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 z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ z x = ↑(v { val := c, property := hc }) x
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | exact (Equiv.Perm.cycle_is_cycleOf hx hc).symm | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ Equiv.Perm.cycleOf g x = ↑{ val := c, property := hc }
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c | case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.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 z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ Equiv.Perm.cycleOf g x = ↑{ val := c, property := hc }
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | exact hx | case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.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 z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∈ Equiv.Perm.support c
⊢ x ∈ Equiv.Perm.support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rw [Equiv.Perm.ofSubtype_apply_of_not_mem] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ (Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc')) x = ↑(v { val := c, property := hc }) x | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x = ↑(v { val := c, property := hc }) x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | 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 z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ (Equiv.Perm.ofSubtype (Equiv.Perm.subtypePerm z hc')) x = ↑(v { val := c, property := hc }) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | obtain ⟨m, hm⟩ := (v ⟨c, hc⟩).prop | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x = ↑(v { val := c, property := hc }) x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | case neg.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : (fun x => ↑{ val := c, property := hc } ^ x) m = ↑(v { val := c, property := hc })
⊢ x = ↑(v { val := c, property := hc }) x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | 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 z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x = ↑(v { val := c, property := hc }) x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | dsimp only at hm | case neg.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : (fun x => ↑{ val := c, property := hc } ^ x) m = ↑(v { val := c, property := hc })
⊢ x = ↑(v { val := c, property := hc }) x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | case neg.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
⊢ x = ↑(v { val := c, property := hc }) x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : (fun x => ↑{ val := c, property := hc } ^ x) m = ↑(v { val := c, property := hc })
⊢ x = ↑(v { val := c, property := hc }) x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rw [← hm] | case neg.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
⊢ x = ↑(v { val := c, property := hc }) x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | case neg.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
⊢ x = (c ^ m) x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
⊢ x = ↑(v { val := c, property := hc }) x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | apply symm | case neg.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
⊢ x = (c ^ m) x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | case neg.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
⊢ (c ^ m) x = x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
⊢ x = (c ^ m) x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | rw [← Equiv.Perm.not_mem_support] | case neg.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
⊢ (c ^ m) x = x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | case neg.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
⊢ x ∉ Equiv.Perm.support (c ^ m)
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
⊢ (c ^ m) x = x
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | intro hx' | case neg.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
⊢ x ∉ Equiv.Perm.support (c ^ m)
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | case neg.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
hx' : x ∈ Equiv.Perm.support (c ^ m)
⊢ False
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
⊢ x ∉ Equiv.Perm.support (c ^ m)
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | apply hx | case neg.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
hx' : x ∈ Equiv.Perm.support (c ^ m)
⊢ False
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | case neg.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
hx' : x ∈ Equiv.Perm.support (c ^ m)
⊢ x ∈ Equiv.Perm.support c
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
hx' : x ∈ Equiv.Perm.support (c ^ m)
⊢ False
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | exact (Equiv.Perm.support_zpow_le c m) hx' | case neg.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
hx' : x ∈ Equiv.Perm.support (c ^ m)
⊢ x ∈ Equiv.Perm.support c
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
m : ℤ
hm : c ^ m = ↑(v { val := c, property := hc })
hx' : x ∈ Equiv.Perm.support (c ^ m)
⊢ x ∈ Equiv.Perm.support c
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_ker_eq_θ_range | [2732, 1] | [2812, 31] | exact hx | case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c | no goals | 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 z : Equiv.Perm α
u : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
h : (θ g) (u, v) = z
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
hc' : ∀ (x : α), x ∈ Equiv.Perm.support c ↔ z x ∈ Equiv.Perm.support c
x : α
hx : x ∉ Equiv.Perm.support c
⊢ x ∉ Equiv.Perm.support c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | let u : (Set.range (θ g)) → (φ g).ker := fun ⟨z, hz⟩ => by
rw [← hφ_ker_eq_θ_range] at hz
suffices ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g by
use ⟨ConjAct.toConjAct z, this⟩
have hK : Function.Injective (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g).subtype := by
apply Subgroup.subtype_injective
rw [← Subgroup.mem_map_iff_mem hK]
simp only [Subgroup.coeSubtype, Subgroup.coe_mk]
exact hz
obtain ⟨u, ⟨_, hu'⟩⟩ := hz
rw [← hu']
exact u.prop | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(Set.range ⇑(θ g)) = Fintype.card ↥(MonoidHom.ker (φ g)) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
⊢ Fintype.card ↑(Set.range ⇑(θ g)) = Fintype.card ↥(MonoidHom.ker (φ g)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(Set.range ⇑(θ g)) = Fintype.card ↥(MonoidHom.ker (φ g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | suffices Function.Bijective u by
exact Fintype.card_of_bijective this | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
⊢ Fintype.card ↑(Set.range ⇑(θ g)) = Fintype.card ↥(MonoidHom.ker (φ g)) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
⊢ Function.Bijective u | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
⊢ Fintype.card ↑(Set.range ⇑(θ g)) = Fintype.card ↥(MonoidHom.ker (φ g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | constructor | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
⊢ Function.Bijective u | case left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
⊢ Function.Injective u
case right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
⊢ Function.Surjective u | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
⊢ Function.Bijective u
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | rw [← hφ_ker_eq_θ_range] at hz | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz : z ∈ Set.range ⇑(θ g)
⊢ ↥(MonoidHom.ker (φ g)) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
⊢ ↥(MonoidHom.ker (φ g)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz : z ∈ Set.range ⇑(θ g)
⊢ ↥(MonoidHom.ker (φ g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | suffices ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g by
use ⟨ConjAct.toConjAct z, this⟩
have hK : Function.Injective (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g).subtype := by
apply Subgroup.subtype_injective
rw [← Subgroup.mem_map_iff_mem hK]
simp only [Subgroup.coeSubtype, Subgroup.coe_mk]
exact hz | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
⊢ ↥(MonoidHom.ker (φ g)) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
⊢ ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
⊢ ↥(MonoidHom.ker (φ g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | obtain ⟨u, ⟨_, hu'⟩⟩ := hz | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
⊢ ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g | case intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
u : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
left✝ : u ∈ ↑(MonoidHom.ker (φ g))
hu' : (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) u = ConjAct.toConjAct z
⊢ ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
⊢ ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | rw [← hu'] | case intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
u : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
left✝ : u ∈ ↑(MonoidHom.ker (φ g))
hu' : (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) u = ConjAct.toConjAct z
⊢ ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g | case intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
u : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
left✝ : u ∈ ↑(MonoidHom.ker (φ g))
hu' : (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) u = ConjAct.toConjAct z
⊢ (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) u ∈
MulAction.stabilizer (ConjAct (Equiv.Perm α)) g | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
u : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
left✝ : u ∈ ↑(MonoidHom.ker (φ g))
hu' : (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) u = ConjAct.toConjAct z
⊢ ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | exact u.prop | case intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
u : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
left✝ : u ∈ ↑(MonoidHom.ker (φ g))
hu' : (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) u = ConjAct.toConjAct z
⊢ (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) u ∈
MulAction.stabilizer (ConjAct (Equiv.Perm α)) g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
u : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
left✝ : u ∈ ↑(MonoidHom.ker (φ g))
hu' : (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) u = ConjAct.toConjAct z
⊢ (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) u ∈
MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | use ⟨ConjAct.toConjAct z, this⟩ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
this : ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ ↥(MonoidHom.ker (φ g)) | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
this : ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ { val := ConjAct.toConjAct z, property := this } ∈ MonoidHom.ker (φ g) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
this : ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ ↥(MonoidHom.ker (φ g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | have hK : Function.Injective (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g).subtype := by
apply Subgroup.subtype_injective | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
this : ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ { val := ConjAct.toConjAct z, property := this } ∈ MonoidHom.ker (φ g) | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
this : ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hK : Function.Injective ⇑(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g))
⊢ { val := ConjAct.toConjAct z, property := this } ∈ MonoidHom.ker (φ g) | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
this : ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ { val := ConjAct.toConjAct z, property := this } ∈ MonoidHom.ker (φ g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | rw [← Subgroup.mem_map_iff_mem hK] | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
this : ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hK : Function.Injective ⇑(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g))
⊢ { val := ConjAct.toConjAct z, property := this } ∈ MonoidHom.ker (φ g) | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
this : ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hK : Function.Injective ⇑(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g))
⊢ (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g))
{ val := ConjAct.toConjAct z, property := this } ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g)) | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
this : ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hK : Function.Injective ⇑(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g))
⊢ { val := ConjAct.toConjAct z, property := this } ∈ MonoidHom.ker (φ g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | simp only [Subgroup.coeSubtype, Subgroup.coe_mk] | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
this : ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hK : Function.Injective ⇑(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g))
⊢ (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g))
{ val := ConjAct.toConjAct z, property := this } ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g)) | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
this : ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hK : Function.Injective ⇑(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g))
⊢ ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g)) | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
this : ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hK : Function.Injective ⇑(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g))
⊢ (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g))
{ val := ConjAct.toConjAct z, property := this } ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | exact hz | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
this : ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hK : Function.Injective ⇑(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g))
⊢ ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
this : ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hK : Function.Injective ⇑(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g))
⊢ ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | apply Subgroup.subtype_injective | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
this : ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ Function.Injective ⇑(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) | 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✝ g : Equiv.Perm α
x✝ : ↑(Set.range ⇑(θ g))
z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
this : ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ Function.Injective ⇑(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | exact Fintype.card_of_bijective this | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
this : Function.Bijective u
⊢ Fintype.card ↑(Set.range ⇑(θ g)) = Fintype.card ↥(MonoidHom.ker (φ g)) | 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✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
this : Function.Bijective u
⊢ Fintype.card ↑(Set.range ⇑(θ g)) = Fintype.card ↥(MonoidHom.ker (φ g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | rintro ⟨z, hz⟩ ⟨w, hw⟩ hzw | case left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
⊢ Function.Injective u | case left.mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
z : Equiv.Perm α
hz : z ∈ Set.range ⇑(θ g)
w : Equiv.Perm α
hw : w ∈ Set.range ⇑(θ g)
hzw : u { val := z, property := hz } = u { val := w, property := hw }
⊢ { val := z, property := hz } = { val := w, property := hw } | 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✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
⊢ Function.Injective u
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | simpa only [u, Subtype.mk_eq_mk, MulEquiv.apply_eq_iff_eq] using hzw | case left.mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
z : Equiv.Perm α
hz : z ∈ Set.range ⇑(θ g)
w : Equiv.Perm α
hw : w ∈ Set.range ⇑(θ g)
hzw : u { val := z, property := hz } = u { val := w, property := hw }
⊢ { val := z, property := hz } = { val := w, property := hw } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
z : Equiv.Perm α
hz : z ∈ Set.range ⇑(θ g)
w : Equiv.Perm α
hw : w ∈ Set.range ⇑(θ g)
hzw : u { val := z, property := hz } = u { val := w, property := hw }
⊢ { val := z, property := hz } = { val := w, property := hw }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | rintro ⟨w, hw⟩ | case right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
⊢ Function.Surjective u | case right.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ ∃ a, u a = { val := w, property := hw } | 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✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
⊢ Function.Surjective u
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | use! ConjAct.ofConjAct ((MulAction.stabilizer (ConjAct (Equiv.Perm α)) g).subtype w) | case right.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ ∃ a, u a = { val := w, property := hw } | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ ConjAct.ofConjAct ((Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) w) ∈ Set.range ⇑(θ g)
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ u
{ val := ConjAct.ofConjAct ((Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) w),
property := ?property } =
{ val := w, property := hw } | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ ∃ a, u a = { val := w, property := hw }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | rw [← hφ_ker_eq_θ_range] | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ ConjAct.ofConjAct ((Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) w) ∈ Set.range ⇑(θ g)
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ u
{ val := ConjAct.ofConjAct ((Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) w),
property := ?property } =
{ val := w, property := hw } | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ ConjAct.toConjAct (ConjAct.ofConjAct ((Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) w)) ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ u
{ val := ConjAct.ofConjAct ((Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) w),
property := ⋯ } =
{ val := w, property := hw } | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ ConjAct.ofConjAct ((Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) w) ∈ Set.range ⇑(θ g)
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ u
{ val := ConjAct.ofConjAct ((Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) w),
property := ?property } =
{ val := w, property := hw }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | simp only [Subgroup.coeSubtype, ConjAct.toConjAct_ofConjAct, Subgroup.mem_map,
SetLike.coe_eq_coe, exists_prop, exists_eq_right, hw] | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ ConjAct.toConjAct (ConjAct.ofConjAct ((Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) w)) ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ u
{ val := ConjAct.ofConjAct ((Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) w),
property := ⋯ } =
{ val := w, property := hw } | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ u
{ val := ConjAct.ofConjAct ((Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) w),
property := ⋯ } =
{ val := w, property := hw } | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ ConjAct.toConjAct (ConjAct.ofConjAct ((Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) w)) ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ u
{ val := ConjAct.ofConjAct ((Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) w),
property := ⋯ } =
{ val := w, property := hw }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card' | [2896, 1] | [2923, 99] | simp only [u, Subgroup.coeSubtype, ConjAct.toConjAct_ofConjAct, Subtype.mk_eq_mk, SetLike.eta] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ u
{ val := ConjAct.ofConjAct ((Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) w),
property := ⋯ } =
{ val := w, property := hw } | no goals | 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✝ g : Equiv.Perm α
u : ↑(Set.range ⇑(θ g)) → ↥(MonoidHom.ker (φ g)) :=
fun x =>
match x with
| { val := z, property := hz } =>
let_fun this := ⋯;
{ val := { val := ConjAct.toConjAct z, property := this }, property := ⋯ }
w : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
hw : w ∈ MonoidHom.ker (φ g)
⊢ u
{ val := ConjAct.ofConjAct ((Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) w),
property := ⋯ } =
{ val := w, property := hw }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_fixedBy | [2926, 1] | [2935, 79] | rw [Equiv.Perm.sum_cycleType, ← Finset.card_compl] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(MulAction.fixedBy α g) = Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(MulAction.fixedBy α g) = (Equiv.Perm.support g)ᶜ.card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(MulAction.fixedBy α g) = Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_fixedBy | [2926, 1] | [2935, 79] | simp only [Fintype.card_ofFinset, Set.mem_compl_iff, Finset.mem_coe,
Equiv.Perm.mem_support, Classical.not_not] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(MulAction.fixedBy α g) = (Equiv.Perm.support g)ᶜ.card | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ (Finset.filter (fun x => x ∈ MulAction.fixedBy α g) Finset.univ).card = (Equiv.Perm.support g)ᶜ.card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(MulAction.fixedBy α g) = (Equiv.Perm.support g)ᶜ.card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_fixedBy | [2926, 1] | [2935, 79] | apply congr_arg | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ (Finset.filter (fun x => x ∈ MulAction.fixedBy α g) Finset.univ).card = (Equiv.Perm.support g)ᶜ.card | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Finset.filter (fun x => x ∈ MulAction.fixedBy α g) Finset.univ = (Equiv.Perm.support g)ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ (Finset.filter (fun x => x ∈ MulAction.fixedBy α g) Finset.univ).card = (Equiv.Perm.support g)ᶜ.card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_fixedBy | [2926, 1] | [2935, 79] | ext x | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Finset.filter (fun x => x ∈ MulAction.fixedBy α g) Finset.univ = (Equiv.Perm.support g)ᶜ | case h.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x : α
⊢ x ∈ Finset.filter (fun x => x ∈ MulAction.fixedBy α g) Finset.univ ↔ x ∈ (Equiv.Perm.support g)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✝ g : Equiv.Perm α
⊢ Finset.filter (fun x => x ∈ MulAction.fixedBy α g) Finset.univ = (Equiv.Perm.support g)ᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_fixedBy | [2926, 1] | [2935, 79] | simp only [MulAction.mem_fixedBy, Equiv.Perm.smul_def, Finset.mem_filter, Finset.mem_univ,
true_and_iff, Finset.mem_compl, Equiv.Perm.mem_support, Classical.not_not] | case h.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x : α
⊢ x ∈ Finset.filter (fun x => x ∈ MulAction.fixedBy α g) Finset.univ ↔ x ∈ (Equiv.Perm.support g)ᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x : α
⊢ x ∈ Finset.filter (fun x => x ∈ MulAction.fixedBy α g) Finset.univ ↔ x ∈ (Equiv.Perm.support g)ᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Fintype.card_pfun | [2938, 1] | [2947, 40] | by_cases hp : p | α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
⊢ Fintype.card ((hp : p) → β hp) = if h : p then Fintype.card (β h) else 1 | case pos
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : p
⊢ Fintype.card ((hp : p) → β hp) = if h : p then Fintype.card (β h) else 1
case neg
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
⊢ Fintype.card ((hp : p) → β hp) = if h : p then Fintype.card (β h) else 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
⊢ Fintype.card ((hp : p) → β hp) = if h : p then Fintype.card (β h) else 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Fintype.card_pfun | [2938, 1] | [2947, 40] | simp only [dif_pos hp] | case pos
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : p
⊢ Fintype.card ((hp : p) → β hp) = if h : p then Fintype.card (β h) else 1 | case pos
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : p
⊢ Fintype.card ((hp : p) → β hp) = Fintype.card (β hp) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : p
⊢ Fintype.card ((hp : p) → β hp) = if h : p then Fintype.card (β h) else 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Fintype.card_pfun | [2938, 1] | [2947, 40] | rw [Fintype.card_eq] | case pos
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : p
⊢ Fintype.card ((hp : p) → β hp) = Fintype.card (β hp) | case pos
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : p
⊢ Nonempty (((hp : p) → β hp) ≃ β hp) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : p
⊢ Fintype.card ((hp : p) → β hp) = Fintype.card (β hp)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Fintype.card_pfun | [2938, 1] | [2947, 40] | exact ⟨@Equiv.funUnique p (β hp) (uniqueProp hp)⟩ | case pos
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : p
⊢ Nonempty (((hp : p) → β hp) ≃ β hp) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : p
⊢ Nonempty (((hp : p) → β hp) ≃ β hp)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Fintype.card_pfun | [2938, 1] | [2947, 40] | simp only [dif_neg hp] | case neg
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
⊢ Fintype.card ((hp : p) → β hp) = if h : p then Fintype.card (β h) else 1 | case neg
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
⊢ Fintype.card ((hp : p) → β hp) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
⊢ Fintype.card ((hp : p) → β hp) = if h : p then Fintype.card (β h) else 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Fintype.card_pfun | [2938, 1] | [2947, 40] | rw [Fintype.card_eq_one_iff] | case neg
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
⊢ Fintype.card ((hp : p) → β hp) = 1 | case neg
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
⊢ ∃ x, ∀ (y : (hp : p) → β hp), y = x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
⊢ Fintype.card ((hp : p) → β hp) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Fintype.card_pfun | [2938, 1] | [2947, 40] | use (fun h => False.elim (hp h)) | case neg
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
⊢ ∃ x, ∀ (y : (hp : p) → β hp), y = x | case h
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
⊢ ∀ (y : (hp : p) → β hp), y = fun h => ⋯.elim | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
⊢ ∃ x, ∀ (y : (hp : p) → β hp), y = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Fintype.card_pfun | [2938, 1] | [2947, 40] | intro u | case h
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
⊢ ∀ (y : (hp : p) → β hp), y = fun h => ⋯.elim | case h
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
u : (hp : p) → β hp
⊢ u = fun h => ⋯.elim | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
⊢ ∀ (y : (hp : p) → β hp), y = fun h => ⋯.elim
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Fintype.card_pfun | [2938, 1] | [2947, 40] | ext h | case h
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
u : (hp : p) → β hp
⊢ u = fun h => ⋯.elim | case h.h
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
u : (hp : p) → β hp
h : p
⊢ u h = ⋯.elim | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
u : (hp : p) → β hp
⊢ u = fun h => ⋯.elim
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Fintype.card_pfun | [2938, 1] | [2947, 40] | exfalso | case h.h
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
u : (hp : p) → β hp
h : p
⊢ u h = ⋯.elim | case h.h
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
u : (hp : p) → β hp
h : p
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
u : (hp : p) → β hp
h : p
⊢ u h = ⋯.elim
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Fintype.card_pfun | [2938, 1] | [2947, 40] | exact hp h | case h.h
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
u : (hp : p) → β hp
h : p
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type ?u.398702
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
p : Prop
inst✝¹ : Decidable p
β : p → Type u_1
inst✝ : (hp : p) → Fintype (β hp)
hp : ¬p
u : (hp : p) → β hp
h : p
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card | [2950, 1] | [2970, 18] | rw [Set.card_range_of_injective (hθ_injective g)] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(Set.range ⇑(θ g)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (Equiv.Perm.cycleType g) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card
(Equiv.Perm ↑(Function.fixedPoints ⇑g) ×
((c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c))) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (Equiv.Perm.cycleType g) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(Set.range ⇑(θ g)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (Equiv.Perm.cycleType g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card | [2950, 1] | [2970, 18] | rw [Fintype.card_prod] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card
(Equiv.Perm ↑(Function.fixedPoints ⇑g) ×
((c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c))) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (Equiv.Perm.cycleType g) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card (Equiv.Perm ↑(Function.fixedPoints ⇑g)) *
Fintype.card ((c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (Equiv.Perm.cycleType g) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card
(Equiv.Perm ↑(Function.fixedPoints ⇑g) ×
((c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c))) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (Equiv.Perm.cycleType g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card | [2950, 1] | [2970, 18] | rw [Fintype.card_perm] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card (Equiv.Perm ↑(Function.fixedPoints ⇑g)) *
Fintype.card ((c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (Equiv.Perm.cycleType g) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Nat.factorial (Fintype.card ↑(Function.fixedPoints ⇑g)) *
Fintype.card ((c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (Equiv.Perm.cycleType g) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card (Equiv.Perm ↑(Function.fixedPoints ⇑g)) *
Fintype.card ((c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (Equiv.Perm.cycleType g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card | [2950, 1] | [2970, 18] | rw [Fintype.card_pi] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Nat.factorial (Fintype.card ↑(Function.fixedPoints ⇑g)) *
Fintype.card ((c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (Equiv.Perm.cycleType g) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ (Nat.factorial (Fintype.card ↑(Function.fixedPoints ⇑g)) *
Finset.prod Finset.univ fun a => Fintype.card ↥(Subgroup.zpowers ↑a)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (Equiv.Perm.cycleType g) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Nat.factorial (Fintype.card ↑(Function.fixedPoints ⇑g)) *
Fintype.card ((c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (Equiv.Perm.cycleType g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card | [2950, 1] | [2970, 18] | apply congr_arg | case hx
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Nat.factorial (Fintype.card ↑(Function.fixedPoints ⇑g)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) | case hx.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(Function.fixedPoints ⇑g) = Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g) | 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✝ g : Equiv.Perm α
⊢ Nat.factorial (Fintype.card ↑(Function.fixedPoints ⇑g)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card | [2950, 1] | [2970, 18] | exact Equiv.Perm.card_fixedBy g | case hx.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(Function.fixedPoints ⇑g) = Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hx.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(Function.fixedPoints ⇑g) = Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card | [2950, 1] | [2970, 18] | rw [Equiv.Perm.cycleType] | case hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ (Finset.prod Finset.univ fun a => Fintype.card ↥(Subgroup.zpowers ↑a)) = Multiset.prod (Equiv.Perm.cycleType g) | case hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ (Finset.prod Finset.univ fun a => Fintype.card ↥(Subgroup.zpowers ↑a)) =
Multiset.prod (Multiset.map (Finset.card ∘ Equiv.Perm.support) (Equiv.Perm.cycleFactorsFinset g).val) | Please generate a tactic in lean4 to solve the state.
STATE:
case hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ (Finset.prod Finset.univ fun a => Fintype.card ↥(Subgroup.zpowers ↑a)) = Multiset.prod (Equiv.Perm.cycleType g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card | [2950, 1] | [2970, 18] | simp only [Finset.univ_eq_attach, Finset.attach_val, Function.comp_apply] | case hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ (Finset.prod Finset.univ fun a => Fintype.card ↥(Subgroup.zpowers ↑a)) =
Multiset.prod (Multiset.map (Finset.card ∘ Equiv.Perm.support) (Equiv.Perm.cycleFactorsFinset g).val) | case hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ (Finset.prod (Finset.attach (Equiv.Perm.cycleFactorsFinset g)) fun a => Fintype.card ↥(Subgroup.zpowers ↑a)) =
Multiset.prod (Multiset.map (fun x => (Equiv.Perm.support x).card) (Equiv.Perm.cycleFactorsFinset g).val) | Please generate a tactic in lean4 to solve the state.
STATE:
case hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ (Finset.prod Finset.univ fun a => Fintype.card ↥(Subgroup.zpowers ↑a)) =
Multiset.prod (Multiset.map (Finset.card ∘ Equiv.Perm.support) (Equiv.Perm.cycleFactorsFinset g).val)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card | [2950, 1] | [2970, 18] | rw [Finset.prod_attach (s := g.cycleFactorsFinset) (f := fun a ↦ Fintype.card (Subgroup.zpowers (a : Equiv.Perm α)))] | case hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ (Finset.prod (Finset.attach (Equiv.Perm.cycleFactorsFinset g)) fun a => Fintype.card ↥(Subgroup.zpowers ↑a)) =
Multiset.prod (Multiset.map (fun x => (Equiv.Perm.support x).card) (Equiv.Perm.cycleFactorsFinset g).val) | case hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ (Finset.prod (Equiv.Perm.cycleFactorsFinset g) fun x => Fintype.card ↥(Subgroup.zpowers x)) =
Multiset.prod (Multiset.map (fun x => (Equiv.Perm.support x).card) (Equiv.Perm.cycleFactorsFinset g).val) | Please generate a tactic in lean4 to solve the state.
STATE:
case hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ (Finset.prod (Finset.attach (Equiv.Perm.cycleFactorsFinset g)) fun a => Fintype.card ↥(Subgroup.zpowers ↑a)) =
Multiset.prod (Multiset.map (fun x => (Equiv.Perm.support x).card) (Equiv.Perm.cycleFactorsFinset g).val)
TACTIC:
|
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