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/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | exact Subgroup.mem_comap.mp hy | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
y : G := Quotient.out' x
hy : y ∈ comap (QuotientGroup.mk' N) (MonoidHom.range φ)
⊢ ↑(Quotient.out' x) ∈ MonoidHom.range φ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
y : G := Quotient.out' x
hy : y ∈ comap (QuotientGroup.mk' N) (MonoidHom.range φ)
⊢ ↑(Quotient.out' x) ∈ MonoidHom.range φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | rw [S_top] | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
y : G := Quotient.out' x
⊢ y ∈ comap (QuotientGroup.mk' N) (MonoidHom.range φ) | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
y : G := Quotient.out' x
⊢ y ∈ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
y : G := Quotient.out' x
⊢ y ∈ comap (QuotientGroup.mk' N) (MonoidHom.range φ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | exact Subgroup.mem_top y | G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
y : G := Quotient.out' x
⊢ y ∈ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
S_top : comap (QuotientGroup.mk' N) (MonoidHom.range φ) = ⊤
x : G ⧸ N
a✝ : x ∈ ⊤
y : G := Quotient.out' x
⊢ y ∈ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | intro g hg | case left
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ N ≤ comap (QuotientGroup.mk' N) (MonoidHom.range φ) | case left
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
g : G
hg : g ∈ N
⊢ g ∈ comap (QuotientGroup.mk' N) (MonoidHom.range φ) | Please generate a tactic in lean4 to solve the state.
STATE:
case left
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ N ≤ comap (QuotientGroup.mk' N) (MonoidHom.range φ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | rw [Subgroup.mem_comap] | case left
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
g : G
hg : g ∈ N
⊢ g ∈ comap (QuotientGroup.mk' N) (MonoidHom.range φ) | case left
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
g : G
hg : g ∈ N
⊢ (QuotientGroup.mk' N) g ∈ MonoidHom.range φ | Please generate a tactic in lean4 to solve the state.
STATE:
case left
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
g : G
hg : g ∈ N
⊢ g ∈ comap (QuotientGroup.mk' N) (MonoidHom.range φ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | convert (MonoidHom.range φ).one_mem | case left
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
g : G
hg : g ∈ N
⊢ (QuotientGroup.mk' N) g ∈ MonoidHom.range φ | case h.e'_4
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
g : G
hg : g ∈ N
⊢ (QuotientGroup.mk' N) g = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case left
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
g : G
hg : g ∈ N
⊢ (QuotientGroup.mk' N) g ∈ MonoidHom.range φ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | simp only [hg, QuotientGroup.mk'_apply, QuotientGroup.eq_one_iff] | case h.e'_4
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
g : G
hg : g ∈ N
⊢ (QuotientGroup.mk' N) g = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
g : G
hg : g ∈ N
⊢ (QuotientGroup.mk' N) g = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | intro h hh | case right
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ H ≤ comap (QuotientGroup.mk' N) (MonoidHom.range φ) | case right
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
h : G
hh : h ∈ H
⊢ h ∈ comap (QuotientGroup.mk' N) (MonoidHom.range φ) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
⊢ H ≤ comap (QuotientGroup.mk' N) (MonoidHom.range φ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | simp only [mem_comap, QuotientGroup.mk'_apply, MonoidHom.mem_range, MonoidHom.coe_comp,
QuotientGroup.coe_mk', coeSubtype, Function.comp_apply, Subtype.exists, exists_prop, φ] | case right
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
h : G
hh : h ∈ H
⊢ h ∈ comap (QuotientGroup.mk' N) (MonoidHom.range φ) | case right
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
h : G
hh : h ∈ H
⊢ ∃ a ∈ H, ↑a = ↑h | Please generate a tactic in lean4 to solve the state.
STATE:
case right
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
h : G
hh : h ∈ H
⊢ h ∈ comap (QuotientGroup.mk' N) (MonoidHom.range φ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Mathlib/Commutators.lean | contains_commutators_of | [86, 1] | [122, 10] | use h | case right
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
h : G
hh : h ∈ H
⊢ ∃ a ∈ H, ↑a = ↑h | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
G : Type u_1
inst✝ : Group G
N : Subgroup G
nN : Normal N
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : Subgroup.IsCommutative H
φ : ↥H →* G ⧸ N := MonoidHom.comp (QuotientGroup.mk' N) (Subgroup.subtype H)
h : G
hh : h ∈ H
⊢ ∃ a ∈ H, ↑a = ↑h
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.fixingSubgroup_conj | [44, 1] | [53, 100] | ext k | α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g : G
⊢ fixingSubgroup G (g • s) = MulAut.conj g • fixingSubgroup G s | case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
⊢ k ∈ fixingSubgroup G (g • s) ↔ k ∈ MulAut.conj g • fixingSubgroup G s | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g : G
⊢ fixingSubgroup G (g • s) = MulAut.conj g • fixingSubgroup G s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.fixingSubgroup_conj | [44, 1] | [53, 100] | simp only [mem_fixingSubgroup_iff, Subgroup.mem_pointwise_smul_iff_inv_smul_mem] | case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
⊢ k ∈ fixingSubgroup G (g • s) ↔ k ∈ MulAut.conj g • fixingSubgroup G s | case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
⊢ (∀ y ∈ g • s, k • y = y) ↔ ∀ y ∈ s, ((MulAut.conj g)⁻¹ • k) • y = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
⊢ k ∈ fixingSubgroup G (g • s) ↔ k ∈ MulAut.conj g • fixingSubgroup G s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.fixingSubgroup_conj | [44, 1] | [53, 100] | rw [Equiv.forall_congr (toPerm g⁻¹)] | case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
⊢ (∀ y ∈ g • s, k • y = y) ↔ ∀ y ∈ s, ((MulAut.conj g)⁻¹ • k) • y = y | case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
⊢ ∀ {x : α}, x ∈ g • s → k • x = x ↔ (toPerm g⁻¹) x ∈ s → ((MulAut.conj g)⁻¹ • k) • (toPerm g⁻¹) x = (toPerm g⁻¹) x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
⊢ (∀ y ∈ g • s, k • y = y) ↔ ∀ y ∈ s, ((MulAut.conj g)⁻¹ • k) • y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.fixingSubgroup_conj | [44, 1] | [53, 100] | intro x | case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
⊢ ∀ {x : α}, x ∈ g • s → k • x = x ↔ (toPerm g⁻¹) x ∈ s → ((MulAut.conj g)⁻¹ • k) • (toPerm g⁻¹) x = (toPerm g⁻¹) x | case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
x : α
⊢ x ∈ g • s → k • x = x ↔ (toPerm g⁻¹) x ∈ s → ((MulAut.conj g)⁻¹ • k) • (toPerm g⁻¹) x = (toPerm g⁻¹) x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
⊢ ∀ {x : α}, x ∈ g • s → k • x = x ↔ (toPerm g⁻¹) x ∈ s → ((MulAut.conj g)⁻¹ • k) • (toPerm g⁻¹) x = (toPerm g⁻¹) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.fixingSubgroup_conj | [44, 1] | [53, 100] | simp only [toPerm_apply] | case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
x : α
⊢ x ∈ g • s → k • x = x ↔ (toPerm g⁻¹) x ∈ s → ((MulAut.conj g)⁻¹ • k) • (toPerm g⁻¹) x = (toPerm g⁻¹) x | case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
x : α
⊢ x ∈ g • s → k • x = x ↔ g⁻¹ • x ∈ s → ((MulAut.conj g)⁻¹ • k) • g⁻¹ • x = g⁻¹ • x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
x : α
⊢ x ∈ g • s → k • x = x ↔ (toPerm g⁻¹) x ∈ s → ((MulAut.conj g)⁻¹ • k) • (toPerm g⁻¹) x = (toPerm g⁻¹) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.fixingSubgroup_conj | [44, 1] | [53, 100] | rw [← Set.mem_smul_set_iff_inv_smul_mem] | case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
x : α
⊢ x ∈ g • s → k • x = x ↔ g⁻¹ • x ∈ s → ((MulAut.conj g)⁻¹ • k) • g⁻¹ • x = g⁻¹ • x | case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
x : α
⊢ x ∈ g • s → k • x = x ↔ x ∈ g • s → ((MulAut.conj g)⁻¹ • k) • g⁻¹ • x = g⁻¹ • x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
x : α
⊢ x ∈ g • s → k • x = x ↔ g⁻¹ • x ∈ s → ((MulAut.conj g)⁻¹ • k) • g⁻¹ • x = g⁻¹ • x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.fixingSubgroup_conj | [44, 1] | [53, 100] | simp only [MulAut.smul_def, MulAut.conj_inv_apply, mul_smul, smul_inv_smul, smul_left_cancel_iff] | case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
x : α
⊢ x ∈ g • s → k • x = x ↔ x ∈ g • s → ((MulAut.conj g)⁻¹ • k) • g⁻¹ • x = g⁻¹ • x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝ : Type ?u.1776
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g k : G
x : α
⊢ x ∈ g • s → k • x = x ↔ x ∈ g • s → ((MulAut.conj g)⁻¹ • k) • g⁻¹ • x = g⁻¹ • x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT'_isConj | [55, 1] | [67, 35] | ext k | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ IwasawaT' (g • s) = MulAut.conj g • IwasawaT' s | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g k : Perm α
⊢ k ∈ IwasawaT' (g • s) ↔ k ∈ MulAut.conj g • IwasawaT' s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ IwasawaT' (g • s) = MulAut.conj g • IwasawaT' s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT'_isConj | [55, 1] | [67, 35] | unfold IwasawaT' | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g k : Perm α
⊢ k ∈ IwasawaT' (g • s) ↔ k ∈ MulAut.conj g • IwasawaT' s | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g k : Perm α
⊢ k ∈ fixingSubgroup (Perm α) (↑(g • s))ᶜ ↔ k ∈ MulAut.conj g • fixingSubgroup (Perm α) (↑s)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
s : Finset α
g k : Perm α
⊢ k ∈ IwasawaT' (g • s) ↔ k ∈ MulAut.conj g • IwasawaT' s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT'_isConj | [55, 1] | [67, 35] | simp only [Finset.coe_smul_finset, mem_fixingSubgroup_iff,
Subgroup.mem_pointwise_smul_iff_inv_smul_mem] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g k : Perm α
⊢ k ∈ fixingSubgroup (Perm α) (↑(g • s))ᶜ ↔ k ∈ MulAut.conj g • fixingSubgroup (Perm α) (↑s)ᶜ | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g k : Perm α
⊢ (∀ y ∈ (g • ↑s)ᶜ, k • y = y) ↔ ∀ y ∈ (↑s)ᶜ, ((MulAut.conj g)⁻¹ • k) • y = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g k : Perm α
⊢ k ∈ fixingSubgroup (Perm α) (↑(g • s))ᶜ ↔ k ∈ MulAut.conj g • fixingSubgroup (Perm α) (↑s)ᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT'_isConj | [55, 1] | [67, 35] | rw [Equiv.forall_congr (toPerm g⁻¹)] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g k : Perm α
⊢ (∀ y ∈ (g • ↑s)ᶜ, k • y = y) ↔ ∀ y ∈ (↑s)ᶜ, ((MulAut.conj g)⁻¹ • k) • y = y | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g k : Perm α
⊢ ∀ {x : α},
x ∈ (g • ↑s)ᶜ → k • x = x ↔ (toPerm g⁻¹) x ∈ (↑s)ᶜ → ((MulAut.conj g)⁻¹ • k) • (toPerm g⁻¹) x = (toPerm g⁻¹) 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
s : Finset α
g k : Perm α
⊢ (∀ y ∈ (g • ↑s)ᶜ, k • y = y) ↔ ∀ y ∈ (↑s)ᶜ, ((MulAut.conj g)⁻¹ • k) • y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT'_isConj | [55, 1] | [67, 35] | intro x | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g k : Perm α
⊢ ∀ {x : α},
x ∈ (g • ↑s)ᶜ → k • x = x ↔ (toPerm g⁻¹) x ∈ (↑s)ᶜ → ((MulAut.conj g)⁻¹ • k) • (toPerm g⁻¹) x = (toPerm g⁻¹) x | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g k : Perm α
x : α
⊢ x ∈ (g • ↑s)ᶜ → k • x = x ↔ (toPerm g⁻¹) x ∈ (↑s)ᶜ → ((MulAut.conj g)⁻¹ • k) • (toPerm g⁻¹) x = (toPerm g⁻¹) 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
s : Finset α
g k : Perm α
⊢ ∀ {x : α},
x ∈ (g • ↑s)ᶜ → k • x = x ↔ (toPerm g⁻¹) x ∈ (↑s)ᶜ → ((MulAut.conj g)⁻¹ • k) • (toPerm g⁻¹) x = (toPerm g⁻¹) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT'_isConj | [55, 1] | [67, 35] | simp only [toPerm_apply] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g k : Perm α
x : α
⊢ x ∈ (g • ↑s)ᶜ → k • x = x ↔ (toPerm g⁻¹) x ∈ (↑s)ᶜ → ((MulAut.conj g)⁻¹ • k) • (toPerm g⁻¹) x = (toPerm g⁻¹) x | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g k : Perm α
x : α
⊢ x ∈ (g • ↑s)ᶜ → k • x = x ↔ g⁻¹ • x ∈ (↑s)ᶜ → ((MulAut.conj g)⁻¹ • k) • g⁻¹ • x = g⁻¹ • 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
s : Finset α
g k : Perm α
x : α
⊢ x ∈ (g • ↑s)ᶜ → k • x = x ↔ (toPerm g⁻¹) x ∈ (↑s)ᶜ → ((MulAut.conj g)⁻¹ • k) • (toPerm g⁻¹) x = (toPerm g⁻¹) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT'_isConj | [55, 1] | [67, 35] | rw [← Set.mem_smul_set_iff_inv_smul_mem] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g k : Perm α
x : α
⊢ x ∈ (g • ↑s)ᶜ → k • x = x ↔ g⁻¹ • x ∈ (↑s)ᶜ → ((MulAut.conj g)⁻¹ • k) • g⁻¹ • x = g⁻¹ • x | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g k : Perm α
x : α
⊢ x ∈ (g • ↑s)ᶜ → k • x = x ↔ x ∈ g • (↑s)ᶜ → ((MulAut.conj g)⁻¹ • k) • g⁻¹ • x = g⁻¹ • 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
s : Finset α
g k : Perm α
x : α
⊢ x ∈ (g • ↑s)ᶜ → k • x = x ↔ g⁻¹ • x ∈ (↑s)ᶜ → ((MulAut.conj g)⁻¹ • k) • g⁻¹ • x = g⁻¹ • x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT'_isConj | [55, 1] | [67, 35] | simp only [Set.mem_compl_iff, Perm.smul_def, smul_compl_set, MulAut.smul_def,
MulAut.conj_inv_apply, coe_mul, Function.comp_apply, apply_inv_self,
EmbeddingLike.apply_eq_iff_eq] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g k : Perm α
x : α
⊢ x ∈ (g • ↑s)ᶜ → k • x = x ↔ x ∈ g • (↑s)ᶜ → ((MulAut.conj g)⁻¹ • k) • g⁻¹ • x = g⁻¹ • x | 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
s : Finset α
g k : Perm α
x : α
⊢ x ∈ (g • ↑s)ᶜ → k • x = x ↔ x ∈ g • (↑s)ᶜ → ((MulAut.conj g)⁻¹ • k) • g⁻¹ • x = g⁻¹ • x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | ext k | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
⊢ MonoidHom.range ofSubtype = fixingSubgroup (Perm α) sᶜ | case h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
⊢ k ∈ MonoidHom.range ofSubtype ↔ k ∈ fixingSubgroup (Perm α) sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
⊢ MonoidHom.range ofSubtype = fixingSubgroup (Perm α) sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | simp only [mem_fixingSubgroup_iff, Set.mem_compl_iff, Finset.mem_coe, Perm.smul_def, Finset.coe_sort_coe,
MonoidHom.mem_range] | case h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
⊢ k ∈ MonoidHom.range ofSubtype ↔ k ∈ fixingSubgroup (Perm α) sᶜ | case h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
⊢ (∃ x, ofSubtype x = k) ↔ ∀ y ∉ s, k y = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
⊢ k ∈ MonoidHom.range ofSubtype ↔ k ∈ fixingSubgroup (Perm α) sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | constructor | case h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
⊢ (∃ x, ofSubtype x = k) ↔ ∀ y ∉ s, k y = y | case h.mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
⊢ (∃ x, ofSubtype x = k) → ∀ y ∉ s, k y = y
case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
⊢ (∀ y ∉ s, k y = y) → ∃ x, ofSubtype x = k | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
⊢ (∃ x, ofSubtype x = k) ↔ ∀ y ∉ s, k y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | rintro ⟨k, rfl⟩ | case h.mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
⊢ (∃ x, ofSubtype x = k) → ∀ y ∉ s, k y = y | case h.mp.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm { x // x ∈ s }
⊢ ∀ y ∉ s, (ofSubtype k) y = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
⊢ (∃ x, ofSubtype x = k) → ∀ y ∉ s, k y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | intro y hy | case h.mp.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm { x // x ∈ s }
⊢ ∀ y ∉ s, (ofSubtype k) y = y | case h.mp.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm { x // x ∈ s }
y : α
hy : y ∉ s
⊢ (ofSubtype k) y = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm { x // x ∈ s }
⊢ ∀ y ∉ s, (ofSubtype k) y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | rw [ofSubtype_apply_of_not_mem k hy] | case h.mp.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm { x // x ∈ s }
y : α
hy : y ∉ s
⊢ (ofSubtype k) y = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm { x // x ∈ s }
y : α
hy : y ∉ s
⊢ (ofSubtype k) y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | intro h | case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
⊢ (∀ y ∉ s, k y = y) → ∃ x, ofSubtype x = k | case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
⊢ ∃ x, ofSubtype x = k | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
⊢ (∀ y ∉ s, k y = y) → ∃ x, ofSubtype x = k
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | suffices hk' : _ by
use Equiv.Perm.subtypePerm k hk'
rw [Equiv.Perm.ofSubtype_subtypePerm]
simp only [ne_eq, Finset.mem_coe]
intro x
rw [not_imp_comm]
exact h x | case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
⊢ ∃ x, ofSubtype x = k | case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
⊢ ∀ (x : α), x ∈ s ↔ k x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
⊢ ∃ x, ofSubtype x = k
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | intro x | case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
⊢ ∀ (x : α), x ∈ s ↔ k x ∈ s | case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
x : α
⊢ x ∈ s ↔ k x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
⊢ ∀ (x : α), x ∈ s ↔ k x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | rw [← Equiv.Perm.smul_def] | case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
x : α
⊢ x ∈ s ↔ k x ∈ s | case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
x : α
⊢ x ∈ s ↔ k • x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
x : α
⊢ x ∈ s ↔ k x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | nth_rewrite 2 [hks] | case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
x : α
⊢ x ∈ s ↔ k • x ∈ s | case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
x : α
⊢ x ∈ s ↔ k • x ∈ k • s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
x : α
⊢ x ∈ s ↔ k • x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | rw [Set.smul_mem_smul_set_iff] | case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
x : α
⊢ x ∈ s ↔ k • x ∈ k • s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
x : α
⊢ x ∈ s ↔ k • x ∈ k • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | apply le_antisymm | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
⊢ s = k • s | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
⊢ s ≤ k • s
case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
⊢ k • s ≤ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
⊢ s = k • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | . intro x hx
rw [Set.mem_smul_set_iff_inv_smul_mem, Perm.smul_def]
by_contra hx'
rw [← h _ hx', apply_inv_self] at hx'
exact hx' hx | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
⊢ s ≤ k • s
case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
⊢ k • s ≤ s | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
⊢ k • s ≤ s | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
⊢ s ≤ k • s
case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
⊢ k • s ≤ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | intro x hx | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
⊢ s ≤ k • s | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
x : α
hx : x ∈ s
⊢ x ∈ k • s | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
⊢ s ≤ k • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | rw [Set.mem_smul_set_iff_inv_smul_mem, Perm.smul_def] | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
x : α
hx : x ∈ s
⊢ x ∈ k • s | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
x : α
hx : x ∈ s
⊢ k⁻¹ x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
x : α
hx : x ∈ s
⊢ x ∈ k • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | by_contra hx' | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
x : α
hx : x ∈ s
⊢ k⁻¹ x ∈ s | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
x : α
hx : x ∈ s
hx' : k⁻¹ x ∉ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
x : α
hx : x ∈ s
⊢ k⁻¹ x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | rw [← h _ hx', apply_inv_self] at hx' | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
x : α
hx : x ∈ s
hx' : k⁻¹ x ∉ s
⊢ False | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
x : α
hx : x ∈ s
hx' : x ∉ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
x : α
hx : x ∈ s
hx' : k⁻¹ x ∉ s
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | exact hx' hx | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
x : α
hx : x ∈ s
hx' : x ∉ s
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
x : α
hx : x ∈ s
hx' : x ∉ s
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | intro a ha | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
⊢ k • s ≤ s | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
a : α
ha : a ∈ k • s
⊢ a ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
⊢ k • s ≤ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | by_contra ha' | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
a : α
ha : a ∈ k • s
⊢ a ∈ s | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
a : α
ha : a ∈ k • s
ha' : a ∉ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
a : α
ha : a ∈ k • s
⊢ a ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | rw [← h _ ha', ← Perm.smul_def, Set.smul_mem_smul_set_iff] at ha | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
a : α
ha : a ∈ k • s
ha' : a ∉ s
⊢ False | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
a : α
ha : a ∈ s
ha' : a ∉ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
a : α
ha : a ∈ k • s
ha' : a ∉ s
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | exact ha' ha | case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
a : α
ha : a ∈ s
ha' : a ∉ s
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
a : α
ha : a ∈ s
ha' : a ∉ s
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | use Equiv.Perm.subtypePerm k hk' | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ?m.15812
⊢ ∃ x, ofSubtype x = k | case h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ∀ (x : α), x ∈ s ↔ k x ∈ s
⊢ ofSubtype (subtypePerm k hk') = k | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ?m.15812
⊢ ∃ x, ofSubtype x = k
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | rw [Equiv.Perm.ofSubtype_subtypePerm] | case h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ∀ (x : α), x ∈ s ↔ k x ∈ s
⊢ ofSubtype (subtypePerm k hk') = k | case h.h₂
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ∀ (x : α), x ∈ s ↔ k x ∈ s
⊢ ∀ (x : α), k x ≠ x → x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ∀ (x : α), x ∈ s ↔ k x ∈ s
⊢ ofSubtype (subtypePerm k hk') = k
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | simp only [ne_eq, Finset.mem_coe] | case h.h₂
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ∀ (x : α), x ∈ s ↔ k x ∈ s
⊢ ∀ (x : α), k x ≠ x → x ∈ s | case h.h₂
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ∀ (x : α), x ∈ s ↔ k x ∈ s
⊢ ∀ (x : α), ¬k x = x → x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₂
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ∀ (x : α), x ∈ s ↔ k x ∈ s
⊢ ∀ (x : α), k x ≠ x → x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | intro x | case h.h₂
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ∀ (x : α), x ∈ s ↔ k x ∈ s
⊢ ∀ (x : α), ¬k x = x → x ∈ s | case h.h₂
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ∀ (x : α), x ∈ s ↔ k x ∈ s
x : α
⊢ ¬k x = x → x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₂
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ∀ (x : α), x ∈ s ↔ k x ∈ s
⊢ ∀ (x : α), ¬k x = x → x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | rw [not_imp_comm] | case h.h₂
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ∀ (x : α), x ∈ s ↔ k x ∈ s
x : α
⊢ ¬k x = x → x ∈ s | case h.h₂
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ∀ (x : α), x ∈ s ↔ k x ∈ s
x : α
⊢ x ∉ s → k x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₂
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ∀ (x : α), x ∈ s ↔ k x ∈ s
x : α
⊢ ¬k x = x → x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.ofSubtype_range_eq | [69, 1] | [101, 35] | exact h x | case h.h₂
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ∀ (x : α), x ∈ s ↔ k x ∈ s
x : α
⊢ x ∉ s → k x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₂
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
s : Set α
inst✝ : DecidablePred fun a => a ∈ s
k : Perm α
h : ∀ y ∉ s, k y = y
hks : s = k • s
hk' : ∀ (x : α), x ∈ s ↔ k x ∈ s
x : α
⊢ x ∉ s → k x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | MulAction.smul_compl_set_eq | [103, 1] | [107, 48] | ext k | α✝ : Type ?u.17129
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g : G
⊢ (g • s)ᶜ = g • sᶜ | case h
α✝ : Type ?u.17129
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g : G
k : α
⊢ k ∈ (g • s)ᶜ ↔ k ∈ g • sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.17129
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g : G
⊢ (g • s)ᶜ = g • sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | MulAction.smul_compl_set_eq | [103, 1] | [107, 48] | simp only [Set.mem_compl_iff, smul_compl_set] | case h
α✝ : Type ?u.17129
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g : G
k : α
⊢ k ∈ (g • s)ᶜ ↔ k ∈ g • sᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝ : Type ?u.17129
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
g : G
k : α
⊢ k ∈ (g • s)ᶜ ↔ k ∈ g • sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.this1 | [110, 1] | [113, 6] | simp only [← MonoidHom.map_range] | α : Type ?u.19573
inst✝³ : DecidableEq α
inst✝² : Fintype α
G : Type u_1
H : Type u_2
inst✝¹ : Group H
inst✝ : Group G
f : H →* G
g : G
⊢ MulAut.conj g • MonoidHom.range f = MonoidHom.range (MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj g)) f) | α : Type ?u.19573
inst✝³ : DecidableEq α
inst✝² : Fintype α
G : Type u_1
H : Type u_2
inst✝¹ : Group H
inst✝ : Group G
f : H →* G
g : G
⊢ MulAut.conj g • MonoidHom.range f = Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (MonoidHom.range f) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.19573
inst✝³ : DecidableEq α
inst✝² : Fintype α
G : Type u_1
H : Type u_2
inst✝¹ : Group H
inst✝ : Group G
f : H →* G
g : G
⊢ MulAut.conj g • MonoidHom.range f = MonoidHom.range (MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj g)) f)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.this1 | [110, 1] | [113, 6] | rfl | α : Type ?u.19573
inst✝³ : DecidableEq α
inst✝² : Fintype α
G : Type u_1
H : Type u_2
inst✝¹ : Group H
inst✝ : Group G
f : H →* G
g : G
⊢ MulAut.conj g • MonoidHom.range f = Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (MonoidHom.range f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.19573
inst✝³ : DecidableEq α
inst✝² : Fintype α
G : Type u_1
H : Type u_2
inst✝¹ : Group H
inst✝ : Group G
f : H →* G
g : G
⊢ MulAut.conj g • MonoidHom.range f = Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (MonoidHom.range f)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj' | [120, 1] | [126, 39] | unfold IwasawaT | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ IwasawaT (g • s) = MulAut.conj g • IwasawaT s | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ IwasawaT (g • s) = MulAut.conj g • IwasawaT s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj' | [120, 1] | [126, 39] | unfold Iwt | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ MonoidHom.range ofSubtype = MulAut.conj g • MonoidHom.range ofSubtype | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj' | [120, 1] | [126, 39] | simp only [Equiv.Perm.ofSubtype_range_eq] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ MonoidHom.range ofSubtype = MulAut.conj g • MonoidHom.range ofSubtype | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ fixingSubgroup (Perm α) (↑(g • s))ᶜ = MulAut.conj g • fixingSubgroup (Perm α) (↑s)ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ MonoidHom.range ofSubtype = MulAut.conj g • MonoidHom.range ofSubtype
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj' | [120, 1] | [126, 39] | simp only [Finset.coe_smul_finset, ← smul_compl_set] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ fixingSubgroup (Perm α) (↑(g • s))ᶜ = MulAut.conj g • fixingSubgroup (Perm α) (↑s)ᶜ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ fixingSubgroup (Perm α) (g • (↑s)ᶜ) = MulAut.conj g • fixingSubgroup (Perm α) (↑s)ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ fixingSubgroup (Perm α) (↑(g • s))ᶜ = MulAut.conj g • fixingSubgroup (Perm α) (↑s)ᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj' | [120, 1] | [126, 39] | apply Equiv.Perm.fixingSubgroup_conj | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ fixingSubgroup (Perm α) (g • (↑s)ᶜ) = MulAut.conj g • fixingSubgroup (Perm α) (↑s)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
s : Finset α
g : Perm α
⊢ fixingSubgroup (Perm α) (g • (↑s)ᶜ) = MulAut.conj g • fixingSubgroup (Perm α) (↑s)ᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | unfold IwasawaT | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ IwasawaT (g • s) = MulAut.conj g • IwasawaT s | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ IwasawaT (g • s) = MulAut.conj g • IwasawaT s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | have hkg : ∀ a, a ∈ s ↔ g a ∈ g • s := by
intro a
rw [← Equiv.Perm.smul_def, Finset.smul_mem_smul_finset_iff] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | let kg : s ≃ (g • s : Finset α) := Equiv.subtypeEquiv g hkg | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | let kg' := Equiv.permCongrMul kg | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | suffices (Iwt (g • s)).comp kg'.toMonoidHom
= ((MulAut.conj g).toMonoidHom.comp (Iwt s)) by
rw [this1, ← this, ← SetLike.coe_set_eq]
simp only [Finset.coe_sort_coe, MonoidHom.coe_range, MonoidHom.coe_comp, MulEquiv.coe_toMonoidHom,
EquivLike.surjective_comp, EquivLike.range_comp] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
⊢ MonoidHom.comp (Iwt (g • s)) (MulEquiv.toMonoidHom kg') =
MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj g)) (Iwt s) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | ext h x | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
⊢ MonoidHom.comp (Iwt (g • s)) (MulEquiv.toMonoidHom kg') =
MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj g)) (Iwt s) | case h.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
⊢ ((MonoidHom.comp (Iwt (g • s)) (MulEquiv.toMonoidHom kg')) h) x =
((MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj g)) (Iwt s)) h) x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
⊢ MonoidHom.comp (Iwt (g • s)) (MulEquiv.toMonoidHom kg') =
MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj g)) (Iwt s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | unfold Iwt | case h.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
⊢ ((MonoidHom.comp (Iwt (g • s)) (MulEquiv.toMonoidHom kg')) h) x =
((MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj g)) (Iwt s)) h) x | case h.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
⊢ ((MonoidHom.comp ofSubtype (MulEquiv.toMonoidHom kg')) h) x =
((MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj g)) ofSubtype) h) 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
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
⊢ ((MonoidHom.comp (Iwt (g • s)) (MulEquiv.toMonoidHom kg')) h) x =
((MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj g)) (Iwt s)) h) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | simp only [Finset.coe_sort_coe, MonoidHom.coe_comp, MulEquiv.coe_toMonoidHom, Function.comp_apply,
MulAut.conj_apply, coe_mul] | case h.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
⊢ ((MonoidHom.comp ofSubtype (MulEquiv.toMonoidHom kg')) h) x =
((MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj g)) ofSubtype) h) x | case h.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
⊢ (ofSubtype (kg' h)) x = g ((ofSubtype h) (g⁻¹ 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
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
⊢ ((MonoidHom.comp ofSubtype (MulEquiv.toMonoidHom kg')) h) x =
((MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj g)) ofSubtype) h) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | by_cases hx : g⁻¹ x ∈ s | case h.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
⊢ (ofSubtype (kg' h)) x = g ((ofSubtype h) (g⁻¹ x)) | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
⊢ (ofSubtype (kg' h)) x = g ((ofSubtype h) (g⁻¹ x))
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
⊢ (ofSubtype (kg' h)) x = g ((ofSubtype h) (g⁻¹ 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
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
⊢ (ofSubtype (kg' h)) x = g ((ofSubtype h) (g⁻¹ x))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | intro a | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ ∀ (a : α), a ∈ s ↔ g a ∈ g • s | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
a : α
⊢ a ∈ s ↔ g a ∈ g • s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
⊢ ∀ (a : α), a ∈ s ↔ g a ∈ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | rw [← Equiv.Perm.smul_def, Finset.smul_mem_smul_finset_iff] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
a : α
⊢ a ∈ s ↔ g a ∈ g • s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
a : α
⊢ a ∈ s ↔ g a ∈ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | rw [this1, ← this, ← SetLike.coe_set_eq] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
this :
MonoidHom.comp (Iwt (g • s)) (MulEquiv.toMonoidHom kg') =
MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj g)) (Iwt s)
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
this :
MonoidHom.comp (Iwt (g • s)) (MulEquiv.toMonoidHom kg') =
MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj g)) (Iwt s)
⊢ ↑(MonoidHom.range (Iwt (g • s))) = ↑(MonoidHom.range (MonoidHom.comp (Iwt (g • s)) (MulEquiv.toMonoidHom kg'))) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
this :
MonoidHom.comp (Iwt (g • s)) (MulEquiv.toMonoidHom kg') =
MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj g)) (Iwt s)
⊢ MonoidHom.range (Iwt (g • s)) = MulAut.conj g • MonoidHom.range (Iwt s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | simp only [Finset.coe_sort_coe, MonoidHom.coe_range, MonoidHom.coe_comp, MulEquiv.coe_toMonoidHom,
EquivLike.surjective_comp, EquivLike.range_comp] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
this :
MonoidHom.comp (Iwt (g • s)) (MulEquiv.toMonoidHom kg') =
MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj g)) (Iwt s)
⊢ ↑(MonoidHom.range (Iwt (g • s))) = ↑(MonoidHom.range (MonoidHom.comp (Iwt (g • s)) (MulEquiv.toMonoidHom kg'))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
this :
MonoidHom.comp (Iwt (g • s)) (MulEquiv.toMonoidHom kg') =
MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj g)) (Iwt s)
⊢ ↑(MonoidHom.range (Iwt (g • s))) = ↑(MonoidHom.range (MonoidHom.comp (Iwt (g • s)) (MulEquiv.toMonoidHom kg')))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | suffices hx' : x ∈ g • s by
rw [ofSubtype_apply_of_mem h hx]
rw [ofSubtype_apply_of_mem ((Equiv.permCongrMul kg) h) hx']
unfold Equiv.permCongrMul
simp only [toFun_as_coe, invFun_as_coe, permCongr_symm, subtypeEquiv_symm, MulEquiv.coe_mk,
coe_fn_mk, permCongr_apply, subtypeEquiv_apply, EmbeddingLike.apply_eq_iff_eq]
rfl | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
⊢ (ofSubtype (kg' h)) x = g ((ofSubtype h) (g⁻¹ x)) | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
⊢ x ∈ g • s | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
⊢ (ofSubtype (kg' h)) x = g ((ofSubtype h) (g⁻¹ x))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | exact Finset.inv_smul_mem_iff.mp hx | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
⊢ x ∈ g • s | 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
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
⊢ x ∈ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | rw [ofSubtype_apply_of_mem h hx] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
hx' : x ∈ g • s
⊢ (ofSubtype (kg' h)) x = g ((ofSubtype h) (g⁻¹ x)) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
hx' : x ∈ g • s
⊢ (ofSubtype (kg' h)) x = g ↑(h { val := g⁻¹ x, property := hx }) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
hx' : x ∈ g • s
⊢ (ofSubtype (kg' h)) x = g ((ofSubtype h) (g⁻¹ x))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | rw [ofSubtype_apply_of_mem ((Equiv.permCongrMul kg) h) hx'] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
hx' : x ∈ g • s
⊢ (ofSubtype (kg' h)) x = g ↑(h { val := g⁻¹ x, property := hx }) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
hx' : x ∈ g • s
⊢ ↑(((Equiv.permCongrMul kg) h) { val := x, property := hx' }) = g ↑(h { val := g⁻¹ x, property := hx }) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
hx' : x ∈ g • s
⊢ (ofSubtype (kg' h)) x = g ↑(h { val := g⁻¹ x, property := hx })
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | unfold Equiv.permCongrMul | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
hx' : x ∈ g • s
⊢ ↑(((Equiv.permCongrMul kg) h) { val := x, property := hx' }) = g ↑(h { val := g⁻¹ x, property := hx }) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
hx' : x ∈ g • s
⊢ ↑(((let __src := permCongr kg;
{ toEquiv := __src, map_mul' := ⋯ })
h)
{ val := x, property := hx' }) =
g ↑(h { val := g⁻¹ x, property := hx }) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
hx' : x ∈ g • s
⊢ ↑(((Equiv.permCongrMul kg) h) { val := x, property := hx' }) = g ↑(h { val := g⁻¹ x, property := hx })
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | simp only [toFun_as_coe, invFun_as_coe, permCongr_symm, subtypeEquiv_symm, MulEquiv.coe_mk,
coe_fn_mk, permCongr_apply, subtypeEquiv_apply, EmbeddingLike.apply_eq_iff_eq] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
hx' : x ∈ g • s
⊢ ↑(((let __src := permCongr kg;
{ toEquiv := __src, map_mul' := ⋯ })
h)
{ val := x, property := hx' }) =
g ↑(h { val := g⁻¹ x, property := hx }) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
hx' : x ∈ g • s
⊢ ↑(kg (h (kg.symm { val := x, property := hx' }))) = g ↑(h { val := g⁻¹ x, property := hx }) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
hx' : x ∈ g • s
⊢ ↑(((let __src := permCongr kg;
{ toEquiv := __src, map_mul' := ⋯ })
h)
{ val := x, property := hx' }) =
g ↑(h { val := g⁻¹ x, property := hx })
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | rfl | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
hx' : x ∈ g • s
⊢ ↑(kg (h (kg.symm { val := x, property := hx' }))) = g ↑(h { val := g⁻¹ x, property := hx }) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∈ s
hx' : x ∈ g • s
⊢ ↑(kg (h (kg.symm { val := x, property := hx' }))) = g ↑(h { val := g⁻¹ x, property := hx })
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | suffices hx' : x ∉ g • s by
rw [ofSubtype_apply_of_not_mem h hx]
rw [ofSubtype_apply_of_not_mem ((Equiv.permCongrMul kg) h) hx']
simp only [apply_inv_self] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
⊢ (ofSubtype (kg' h)) x = g ((ofSubtype h) (g⁻¹ x)) | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
⊢ x ∉ g • s | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
⊢ (ofSubtype (kg' h)) x = g ((ofSubtype h) (g⁻¹ x))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | rw [← Finset.inv_smul_mem_iff] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
⊢ x ∉ g • s | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
⊢ g⁻¹ • x ∉ s | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
⊢ x ∉ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | exact hx | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
⊢ g⁻¹ • x ∉ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
⊢ g⁻¹ • x ∉ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | rw [ofSubtype_apply_of_not_mem h hx] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
hx' : x ∉ g • s
⊢ (ofSubtype (kg' h)) x = g ((ofSubtype h) (g⁻¹ x)) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
hx' : x ∉ g • s
⊢ (ofSubtype (kg' h)) x = g (g⁻¹ x) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
hx' : x ∉ g • s
⊢ (ofSubtype (kg' h)) x = g ((ofSubtype h) (g⁻¹ x))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | rw [ofSubtype_apply_of_not_mem ((Equiv.permCongrMul kg) h) hx'] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
hx' : x ∉ g • s
⊢ (ofSubtype (kg' h)) x = g (g⁻¹ x) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
hx' : x ∉ g • s
⊢ x = g (g⁻¹ x) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
hx' : x ∉ g • s
⊢ (ofSubtype (kg' h)) x = g (g⁻¹ x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.IwasawaT_is_conj | [128, 1] | [178, 13] | simp only [apply_inv_self] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
hx' : x ∉ g • s
⊢ x = g (g⁻¹ x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Finset α
g : Perm α
hkg : ∀ (a : α), a ∈ s ↔ g a ∈ g • s
kg : { x // x ∈ s } ≃ { x // x ∈ g • s } := subtypeEquiv g hkg
kg' : Perm { x // x ∈ s } ≃* Perm { x // x ∈ g • s } := Equiv.permCongrMul kg
h : Perm { x // x ∈ s }
x : α
hx : g⁻¹ x ∉ s
hx' : x ∉ g • s
⊢ x = g (g⁻¹ x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.normal_subgroups | [215, 1] | [241, 21] | rw [← alternatingGroup.commutator_group_eq hα] | α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ alternatingGroup α ≤ N | α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ commutator (Perm α) ≤ N | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ alternatingGroup α ≤ N
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.normal_subgroups | [215, 1] | [241, 21] | refine' commutator_le_iwasawa _ iwasawa_two hnN _ | α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ commutator (Perm α) ≤ N | case refine'_1
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ IsQuasipreprimitive (Perm α) ↑(Nat.Combination α 2)
case refine'_2
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ fixedPoints ↥N ↑(Nat.Combination α 2) ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ commutator (Perm α) ≤ N
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.normal_subgroups | [215, 1] | [241, 21] | intro h | case refine'_2
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ fixedPoints ↥N ↑(Nat.Combination α 2) ≠ ⊤ | case refine'_2
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ fixedPoints ↥N ↑(Nat.Combination α 2) ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.normal_subgroups | [215, 1] | [241, 21] | obtain ⟨g, hgN, hg_ne⟩ := N.nontrivial_iff_exists_ne_one.mp ntN | case refine'_2
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
⊢ False | case refine'_2.intro.intro
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
g : Perm α
hgN : g ∈ N
hg_ne : g ≠ 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.normal_subgroups | [215, 1] | [241, 21] | obtain ⟨s, hs⟩ := Nat.combination.mulAction_faithful
(G := Equiv.Perm α) (α := α) (g := g) 2
(by norm_num)
(by rw [PartENat.card_eq_coe_fintype_card, PartENat.coe_le_coe]
apply le_trans (by norm_num) hα)
(by exact hg_ne) | case refine'_2.intro.intro
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
g : Perm α
hgN : g ∈ N
hg_ne : g ≠ 1
⊢ False | case refine'_2.intro.intro.intro
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
g : Perm α
hgN : g ∈ N
hg_ne : g ≠ 1
s : ↑(Nat.Combination α 2)
hs : g • s ≠ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2.intro.intro
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
g : Perm α
hgN : g ∈ N
hg_ne : g ≠ 1
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.normal_subgroups | [215, 1] | [241, 21] | apply hs | case refine'_2.intro.intro.intro
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
g : Perm α
hgN : g ∈ N
hg_ne : g ≠ 1
s : ↑(Nat.Combination α 2)
hs : g • s ≠ s
⊢ False | case refine'_2.intro.intro.intro
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
g : Perm α
hgN : g ∈ N
hg_ne : g ≠ 1
s : ↑(Nat.Combination α 2)
hs : g • s ≠ s
⊢ g • s = s | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2.intro.intro.intro
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
g : Perm α
hgN : g ∈ N
hg_ne : g ≠ 1
s : ↑(Nat.Combination α 2)
hs : g • s ≠ s
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.normal_subgroups | [215, 1] | [241, 21] | suffices s ∈ fixedPoints N (Nat.Combination α 2) by
rw [mem_fixedPoints] at this
exact this ⟨g, hgN⟩ | case refine'_2.intro.intro.intro
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
g : Perm α
hgN : g ∈ N
hg_ne : g ≠ 1
s : ↑(Nat.Combination α 2)
hs : g • s ≠ s
⊢ g • s = s | case refine'_2.intro.intro.intro
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
g : Perm α
hgN : g ∈ N
hg_ne : g ≠ 1
s : ↑(Nat.Combination α 2)
hs : g • s ≠ s
⊢ s ∈ fixedPoints ↥N ↑(Nat.Combination α 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2.intro.intro.intro
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
g : Perm α
hgN : g ∈ N
hg_ne : g ≠ 1
s : ↑(Nat.Combination α 2)
hs : g • s ≠ s
⊢ g • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.normal_subgroups | [215, 1] | [241, 21] | rw [h, Set.top_eq_univ] | case refine'_2.intro.intro.intro
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
g : Perm α
hgN : g ∈ N
hg_ne : g ≠ 1
s : ↑(Nat.Combination α 2)
hs : g • s ≠ s
⊢ s ∈ fixedPoints ↥N ↑(Nat.Combination α 2) | case refine'_2.intro.intro.intro
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
g : Perm α
hgN : g ∈ N
hg_ne : g ≠ 1
s : ↑(Nat.Combination α 2)
hs : g • s ≠ s
⊢ s ∈ _root_.Set.univ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2.intro.intro.intro
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
g : Perm α
hgN : g ∈ N
hg_ne : g ≠ 1
s : ↑(Nat.Combination α 2)
hs : g • s ≠ s
⊢ s ∈ fixedPoints ↥N ↑(Nat.Combination α 2)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.normal_subgroups | [215, 1] | [241, 21] | apply Set.mem_univ | case refine'_2.intro.intro.intro
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
g : Perm α
hgN : g ∈ N
hg_ne : g ≠ 1
s : ↑(Nat.Combination α 2)
hs : g • s ≠ s
⊢ s ∈ _root_.Set.univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2.intro.intro.intro
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
h : fixedPoints ↥N ↑(Nat.Combination α 2) = ⊤
g : Perm α
hgN : g ∈ N
hg_ne : g ≠ 1
s : ↑(Nat.Combination α 2)
hs : g • s ≠ s
⊢ s ∈ _root_.Set.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.normal_subgroups | [215, 1] | [241, 21] | apply IsPreprimitive.isQuasipreprimitive | case refine'_1
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ IsQuasipreprimitive (Perm α) ↑(Nat.Combination α 2) | case refine'_1.hGX
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ IsPreprimitive (Perm α) ↑(Nat.Combination α 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ IsQuasipreprimitive (Perm α) ↑(Nat.Combination α 2)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermIwasawa.lean | Equiv.Perm.Equiv.Perm.normal_subgroups | [215, 1] | [241, 21] | apply Nat.Combination_isPreprimitive | case refine'_1.hGX
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ IsPreprimitive (Perm α) ↑(Nat.Combination α 2) | case refine'_1.hGX.h_one_le
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ 1 ≤ 2
case refine'_1.hGX.hn
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ 2 < Fintype.card α
case refine'_1.hGX.hα
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ Fintype.card α ≠ 2 * 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1.hGX
α✝ : Type ?u.55854
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα : 5 ≤ Fintype.card α
N : Subgroup (Perm α)
hnN : Subgroup.Normal N
ntN : Nontrivial ↥N
⊢ IsPreprimitive (Perm α) ↑(Nat.Combination α 2)
TACTIC:
|
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