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/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Equivs.lean | piFinSuccCastSucc_apply | [11, 1] | [17, 6] | simp_rw [piFinSuccCastSucc, Equiv.instTrans_trans, Equiv.trans_apply, Equiv.prodCongr_apply,
Equiv.piFinSuccAbove_apply, extractNth, succAbove_last] | n : ℕ
α : Type u_1
v : Fin (n + 2) → α
x✝ : Fin n
⊢ (piFinSuccCastSucc v).2 x✝ = (v ∘ fun i => i.castSucc.succ) x✝ | n : ℕ
α : Type u_1
v : Fin (n + 2) → α
x✝ : Fin n
⊢ ((Equiv.prodAssoc α α (Fin n → α)).symm
(Prod.map (⇑(Equiv.refl α)) (fun f => (f (last n), fun j => f j.castSucc)) ((Equiv.piFinSucc (n + 1) α) v))).2
x✝ =
(v ∘ fun i => i.castSucc.succ) x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
v : Fin (n + 2) → α
x✝ : Fin n
⊢ (piFinSuccCastSucc v).2 x✝ = (v ∘ fun i => i.castSucc.succ) x✝
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Equivs.lean | piFinSuccCastSucc_apply | [11, 1] | [17, 6] | rfl | n : ℕ
α : Type u_1
v : Fin (n + 2) → α
x✝ : Fin n
⊢ ((Equiv.prodAssoc α α (Fin n → α)).symm
(Prod.map (⇑(Equiv.refl α)) (fun f => (f (last n), fun j => f j.castSucc)) ((Equiv.piFinSucc (n + 1) α) v))).2
x✝ =
(v ∘ fun i => i.castSucc.succ) x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
v : Fin (n + 2) → α
x✝ : Fin n
⊢ ((Equiv.prodAssoc α α (Fin n → α)).symm
(Prod.map (⇑(Equiv.refl α)) (fun f => (f (last n), fun j => f j.castSucc)) ((Equiv.piFinSucc (n + 1) α) v))).2
x✝ =
(v ∘ fun i => i.castSucc.succ) x✝
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Equivs.lean | piFinSuccCastSucc_apply_fst_fst | [19, 1] | [21, 48] | simp_rw [piFinSuccCastSucc_apply] | n : ℕ
α : Type u_1
v : Fin (n + 2) → α
⊢ (piFinSuccCastSucc v).1.1 = v 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
v : Fin (n + 2) → α
⊢ (piFinSuccCastSucc v).1.1 = v 0
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Equivs.lean | piFinSuccCastSucc_apply_fst_snd | [23, 1] | [25, 55] | simp_rw [piFinSuccCastSucc_apply] | n : ℕ
α : Type u_1
v : Fin (n + 2) → α
⊢ (piFinSuccCastSucc v).1.2 = v (last (n + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
v : Fin (n + 2) → α
⊢ (piFinSuccCastSucc v).1.2 = v (last (n + 1))
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Equivs.lean | piFinSuccCastSucc_apply_snd | [27, 1] | [29, 77] | simp only [piFinSuccCastSucc_apply] | n : ℕ
α : Type u_1
v : Fin (n + 2) → α
⊢ (piFinSuccCastSucc v).2 = v ∘ fun i => i.castSucc.succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
v : Fin (n + 2) → α
⊢ (piFinSuccCastSucc v).2 = v ∘ fun i => i.castSucc.succ
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Equivs.lean | piFinSuccCastSucc_symm_apply_castSucc_succ | [31, 1] | [37, 58] | simp only [piFinSuccCastSucc, Equiv.instTrans_trans, Equiv.symm_trans_apply, Equiv.symm_symm,
Equiv.prodAssoc_apply, Equiv.prodCongr_symm, Equiv.refl_symm, Equiv.prodCongr_apply,
Equiv.coe_refl, Equiv.piFinSuccAbove_symm_apply, insertNth_last', Prod_map, id_eq,
Equiv.piFinSucc_symm_apply, cons_succ, snoc_castSucc] | α : Type u_1
n : ℕ
a b : α
v : Fin n → α
i : Fin n
⊢ piFinSuccCastSucc.symm ((a, b), v) i.castSucc.succ = v i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
n : ℕ
a b : α
v : Fin n → α
i : Fin n
⊢ piFinSuccCastSucc.symm ((a, b), v) i.castSucc.succ = v i
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Equivs.lean | piFinSuccCastSucc_symm_apply_succ_castSucc | [39, 1] | [42, 68] | rw [<- succ_castSucc, piFinSuccCastSucc_symm_apply_castSucc_succ] | α : Type u_1
n : ℕ
a b : α
v : Fin n → α
i : Fin n
⊢ piFinSuccCastSucc.symm ((a, b), v) i.succ.castSucc = v i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
n : ℕ
a b : α
v : Fin n → α
i : Fin n
⊢ piFinSuccCastSucc.symm ((a, b), v) i.succ.castSucc = v i
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Equivs.lean | piFinSuccCastSucc_symm_apply_last | [48, 1] | [54, 67] | simp_rw [piFinSuccCastSucc, Equiv.instTrans_trans, Equiv.symm_trans_apply, Equiv.symm_symm,
Equiv.prodAssoc_apply, Equiv.prodCongr_symm, Equiv.refl_symm, Equiv.prodCongr_apply,
Equiv.coe_refl, Equiv.piFinSuccAbove_symm_apply, insertNth_last', Prod_map, id_eq,
Equiv.piFinSucc_symm_apply, cons_snoc_eq_snoc_cons, snoc_last] | α : Type u_1
n : ℕ
a b : α
v : Fin n → α
⊢ piFinSuccCastSucc.symm ((a, b), v) (last (n + 1)) = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
n : ℕ
a b : α
v : Fin n → α
⊢ piFinSuccCastSucc.symm ((a, b), v) (last (n + 1)) = b
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Nat.lean | Nat.eq_false_true_of_cond_succ_lt_of_cond_succ_lt | [12, 1] | [20, 86] | refine Order.eq_false_true_of_cond_succ_lt_of_cond_succ_lt ?_ ?_ | m : ℕ
bm : Bool
n : ℕ
bn : Bool
hmn : (m + bif bm then 1 else 0) < n + bif bn then 1 else 0
hnm : (n + bif bn then 0 else 1) < m + bif bm then 0 else 1
⊢ bm = false ∧ bn = true ∧ m = n | case refine_1
m : ℕ
bm : Bool
n : ℕ
bn : Bool
hmn : (m + bif bm then 1 else 0) < n + bif bn then 1 else 0
hnm : (n + bif bn then 0 else 1) < m + bif bm then 0 else 1
⊢ (bif bm then Order.succ else id) m < (bif bn then Order.succ else id) n
case refine_2
m : ℕ
bm : Bool
n : ℕ
bn : Bool
hmn : (m + bif bm then 1 else 0) < n + bif bn then 1 else 0
hnm : (n + bif bn then 0 else 1) < m + bif bm then 0 else 1
⊢ (bif bn then id else Order.succ) n < (bif bm then id else Order.succ) m | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
bm : Bool
n : ℕ
bn : Bool
hmn : (m + bif bm then 1 else 0) < n + bif bn then 1 else 0
hnm : (n + bif bn then 0 else 1) < m + bif bm then 0 else 1
⊢ bm = false ∧ bn = true ∧ m = n
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Nat.lean | Nat.eq_false_true_of_cond_succ_lt_of_cond_succ_lt | [12, 1] | [20, 86] | cases bm <;> cases bn <;>
simp only [succ_eq_succ, cond_true, cond_false, id_eq] at hnm hmn ⊢ <;> exact hmn | case refine_1
m : ℕ
bm : Bool
n : ℕ
bn : Bool
hmn : (m + bif bm then 1 else 0) < n + bif bn then 1 else 0
hnm : (n + bif bn then 0 else 1) < m + bif bm then 0 else 1
⊢ (bif bm then Order.succ else id) m < (bif bn then Order.succ else id) n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
m : ℕ
bm : Bool
n : ℕ
bn : Bool
hmn : (m + bif bm then 1 else 0) < n + bif bn then 1 else 0
hnm : (n + bif bn then 0 else 1) < m + bif bm then 0 else 1
⊢ (bif bm then Order.succ else id) m < (bif bn then Order.succ else id) n
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Nat.lean | Nat.eq_false_true_of_cond_succ_lt_of_cond_succ_lt | [12, 1] | [20, 86] | cases bm <;> cases bn <;>
simp only [succ_eq_succ, cond_true, cond_false, id_eq] at hnm hmn ⊢ <;> exact hnm | case refine_2
m : ℕ
bm : Bool
n : ℕ
bn : Bool
hmn : (m + bif bm then 1 else 0) < n + bif bn then 1 else 0
hnm : (n + bif bn then 0 else 1) < m + bif bm then 0 else 1
⊢ (bif bn then id else Order.succ) n < (bif bm then id else Order.succ) m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
m : ℕ
bm : Bool
n : ℕ
bn : Bool
hmn : (m + bif bm then 1 else 0) < n + bif bn then 1 else 0
hnm : (n + bif bn then 0 else 1) < m + bif bm then 0 else 1
⊢ (bif bn then id else Order.succ) n < (bif bm then id else Order.succ) m
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.rev_eq_zero_iff_last | [10, 1] | [12, 26] | convert rev_inj | m : ℕ
i : Fin (m + 1)
⊢ i.rev = 0 ↔ i = last m | case h.e'_1.h.e'_3.h
m : ℕ
i : Fin (m + 1)
⊢ 0 = (last m).rev | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
i : Fin (m + 1)
⊢ i.rev = 0 ↔ i = last m
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.rev_eq_zero_iff_last | [10, 1] | [12, 26] | exact (rev_last m).symm | case h.e'_1.h.e'_3.h
m : ℕ
i : Fin (m + 1)
⊢ 0 = (last m).rev | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.h.e'_3.h
m : ℕ
i : Fin (m + 1)
⊢ 0 = (last m).rev
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.rev_ne_zero_iff_ne_last | [14, 1] | [15, 40] | simp_rw [ne_eq, rev_eq_zero_iff_last] | m : ℕ
i : Fin (m + 1)
⊢ i.rev ≠ 0 ↔ i ≠ last m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
i : Fin (m + 1)
⊢ i.rev ≠ 0 ↔ i ≠ last m
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.rev_pos_iff_lt_last | [17, 1] | [19, 32] | simp_rw [lt_last_iff_ne_last, pos_iff_ne_zero] | m : ℕ
i : Fin (m + 1)
⊢ 0 < i.rev ↔ i < last m | m : ℕ
i : Fin (m + 1)
⊢ i.rev ≠ 0 ↔ i ≠ last m | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
i : Fin (m + 1)
⊢ 0 < i.rev ↔ i < last m
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.rev_pos_iff_lt_last | [17, 1] | [19, 32] | exact rev_ne_zero_iff_ne_last | m : ℕ
i : Fin (m + 1)
⊢ i.rev ≠ 0 ↔ i ≠ last m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
i : Fin (m + 1)
⊢ i.rev ≠ 0 ↔ i ≠ last m
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.eq_zero_iff_rev_eq_last | [21, 1] | [22, 43] | convert rev_rev i ▸ rev_eq_zero_iff_last | m : ℕ
i : Fin (m + 1)
⊢ i = 0 ↔ i.rev = last m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
i : Fin (m + 1)
⊢ i = 0 ↔ i.rev = last m
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.ne_zero_iff_rev_ne_last | [24, 1] | [25, 46] | convert rev_rev i ▸ rev_ne_zero_iff_ne_last | m : ℕ
i : Fin (m + 1)
⊢ i ≠ 0 ↔ i.rev ≠ last m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
i : Fin (m + 1)
⊢ i ≠ 0 ↔ i.rev ≠ last m
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.pos_iff_rev_lt_last | [27, 1] | [28, 42] | convert rev_rev i ▸ rev_pos_iff_lt_last | m : ℕ
i : Fin (m + 1)
⊢ 0 < i ↔ i.rev < last m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
i : Fin (m + 1)
⊢ 0 < i ↔ i.rev < last m
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.rev_castSucc_succ | [30, 1] | [31, 50] | simp_rw [rev_succ, rev_castSucc, succ_castSucc] | m : ℕ
i : Fin m
⊢ i.castSucc.succ.rev = i.rev.castSucc.succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
i : Fin m
⊢ i.castSucc.succ.rev = i.rev.castSucc.succ
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.rev_succ_castSucc | [33, 1] | [34, 47] | simp_rw [← succ_castSucc, rev_castSucc_succ] | m : ℕ
i : Fin m
⊢ i.succ.castSucc.rev = i.rev.succ.castSucc | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
i : Fin m
⊢ i.succ.castSucc.rev = i.rev.succ.castSucc
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.castSucc_rev_castSucc | [36, 1] | [37, 50] | simp_rw [rev_succ, rev_castSucc, succ_castSucc] | m : ℕ
i : Fin m
⊢ i.castSucc.rev.castSucc = i.succ.rev.succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
i : Fin m
⊢ i.castSucc.rev.castSucc = i.succ.rev.succ
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove | [51, 1] | [61, 12] | rcases lt_or_le (castSucc i) j with (h | h) | m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
⊢ (j.succAbove i).succAbove (i.predAbove j) = j | case inl
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : i.castSucc < j
⊢ (j.succAbove i).succAbove (i.predAbove j) = j
case inr
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : j ≤ i.castSucc
⊢ (j.succAbove i).succAbove (i.predAbove j) = j | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
⊢ (j.succAbove i).succAbove (i.predAbove j) = j
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove | [51, 1] | [61, 12] | rw [succAbove_of_castSucc_lt _ _ h, predAbove_of_castSucc_lt _ _ h,
succAbove_castSucc_of_le, succ_pred] | case inl
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : i.castSucc < j
⊢ (j.succAbove i).succAbove (i.predAbove j) = j | case inl.h
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : i.castSucc < j
⊢ i ≤ j.pred ⋯ | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : i.castSucc < j
⊢ (j.succAbove i).succAbove (i.predAbove j) = j
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove | [51, 1] | [61, 12] | rw [le_pred_iff, ← castSucc_lt_iff_succ_le] | case inl.h
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : i.castSucc < j
⊢ i ≤ j.pred ⋯ | case inl.h
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : i.castSucc < j
⊢ i.castSucc < j | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : i.castSucc < j
⊢ i ≤ j.pred ⋯
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove | [51, 1] | [61, 12] | exact h | case inl.h
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : i.castSucc < j
⊢ i.castSucc < j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : i.castSucc < j
⊢ i.castSucc < j
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove | [51, 1] | [61, 12] | rw [succAbove_of_le_castSucc _ _ h, predAbove_of_le_castSucc _ _ h,
succAbove_succ_of_le, castSucc_castPred] | case inr
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : j ≤ i.castSucc
⊢ (j.succAbove i).succAbove (i.predAbove j) = j | case inr.h
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : j ≤ i.castSucc
⊢ j.castPred ⋯ ≤ i | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : j ≤ i.castSucc
⊢ (j.succAbove i).succAbove (i.predAbove j) = j
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove | [51, 1] | [61, 12] | rw [castPred_le_iff] | case inr.h
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : j ≤ i.castSucc
⊢ j.castPred ⋯ ≤ i | case inr.h
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : j ≤ i.castSucc
⊢ j ≤ i.castSucc | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : j ≤ i.castSucc
⊢ j.castPred ⋯ ≤ i
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove | [51, 1] | [61, 12] | exact h | case inr.h
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : j ≤ i.castSucc
⊢ j ≤ i.castSucc | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
m : ℕ
i : Fin (m + 1)
j : Fin (m + 2)
h : j ≤ i.castSucc
⊢ j ≤ i.castSucc
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.castSucc_le_succAbove | [66, 1] | [68, 32] | obtain h | h := succAbove_eq_castSucc_or_succ p i <;> rw [h] | n : ℕ
p : Fin (n + 1)
i : Fin n
⊢ i.castSucc ≤ p.succAbove i | case inr
n : ℕ
p : Fin (n + 1)
i : Fin n
h : p.succAbove i = i.succ
⊢ i.castSucc ≤ i.succ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
p : Fin (n + 1)
i : Fin n
⊢ i.castSucc ≤ p.succAbove i
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.castSucc_le_succAbove | [66, 1] | [68, 32] | exact (castSucc_lt_succ _).le | case inr
n : ℕ
p : Fin (n + 1)
i : Fin n
h : p.succAbove i = i.succ
⊢ i.castSucc ≤ i.succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
n : ℕ
p : Fin (n + 1)
i : Fin n
h : p.succAbove i = i.succ
⊢ i.castSucc ≤ i.succ
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_le_succ | [70, 1] | [72, 32] | obtain h | h := succAbove_eq_castSucc_or_succ p i <;> rw [h] | n : ℕ
p : Fin (n + 1)
i : Fin n
⊢ p.succAbove i ≤ i.succ | case inl
n : ℕ
p : Fin (n + 1)
i : Fin n
h : p.succAbove i = i.castSucc
⊢ i.castSucc ≤ i.succ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
p : Fin (n + 1)
i : Fin n
⊢ p.succAbove i ≤ i.succ
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_le_succ | [70, 1] | [72, 32] | exact (castSucc_lt_succ _).le | case inl
n : ℕ
p : Fin (n + 1)
i : Fin n
h : p.succAbove i = i.castSucc
⊢ i.castSucc ≤ i.succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
n : ℕ
p : Fin (n + 1)
i : Fin n
h : p.succAbove i = i.castSucc
⊢ i.castSucc ≤ i.succ
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove_succAbove | [74, 1] | [93, 46] | rcases lt_or_le (castSucc i) j with (hij | hij) | m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
⊢ (j.succAbove i).succAbove ((i.predAbove j).succAbove k) = j.succAbove (i.succAbove k) | case inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
⊢ (j.succAbove i).succAbove ((i.predAbove j).succAbove k) = j.succAbove (i.succAbove k)
case inr
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
⊢ (j.succAbove i).succAbove ((i.predAbove j).succAbove k) = j.succAbove (i.succAbove k) | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
⊢ (j.succAbove i).succAbove ((i.predAbove j).succAbove k) = j.succAbove (i.succAbove k)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove_succAbove | [74, 1] | [93, 46] | rw [succAbove_of_castSucc_lt _ _ hij, predAbove_of_castSucc_lt _ _ hij] | case inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
⊢ (j.succAbove i).succAbove ((i.predAbove j).succAbove k) = j.succAbove (i.succAbove k) | case inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
⊢ i.castSucc.succAbove ((j.pred ⋯).succAbove k) = j.succAbove (i.succAbove k) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
⊢ (j.succAbove i).succAbove ((i.predAbove j).succAbove k) = j.succAbove (i.succAbove k)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove_succAbove | [74, 1] | [93, 46] | rcases lt_or_le (castSucc k) i with (hik | hik) | case inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
⊢ i.castSucc.succAbove ((j.pred ⋯).succAbove k) = j.succAbove (i.succAbove k) | case inl.inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
hik : k.castSucc < i
⊢ i.castSucc.succAbove ((j.pred ⋯).succAbove k) = j.succAbove (i.succAbove k)
case inl.inr
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
hik : i ≤ k.castSucc
⊢ i.castSucc.succAbove ((j.pred ⋯).succAbove k) = j.succAbove (i.succAbove k) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
⊢ i.castSucc.succAbove ((j.pred ⋯).succAbove k) = j.succAbove (i.succAbove k)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove_succAbove | [74, 1] | [93, 46] | have H := (castSucc_lt_iff_succ_le.mp
(castSucc_lt_castSucc_iff.mpr hik)).trans_lt hij | case inl.inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
hik : k.castSucc < i
⊢ i.castSucc.succAbove ((j.pred ⋯).succAbove k) = j.succAbove (i.succAbove k) | case inl.inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
hik : k.castSucc < i
H : k.castSucc.succ < j
⊢ i.castSucc.succAbove ((j.pred ⋯).succAbove k) = j.succAbove (i.succAbove k) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
hik : k.castSucc < i
⊢ i.castSucc.succAbove ((j.pred ⋯).succAbove k) = j.succAbove (i.succAbove k)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove_succAbove | [74, 1] | [93, 46] | rw [succAbove_of_castSucc_lt _ _ hik, succAbove_of_succ_le _ _ H.le,
succAbove_of_castSucc_lt _ k ((lt_pred_iff _).mpr H), succAbove_castSucc_of_lt _ _ hik] | case inl.inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
hik : k.castSucc < i
H : k.castSucc.succ < j
⊢ i.castSucc.succAbove ((j.pred ⋯).succAbove k) = j.succAbove (i.succAbove k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
hik : k.castSucc < i
H : k.castSucc.succ < j
⊢ i.castSucc.succAbove ((j.pred ⋯).succAbove k) = j.succAbove (i.succAbove k)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove_succAbove | [74, 1] | [93, 46] | rw [succAbove_of_le_castSucc _ _ hik,
succAbove_castSucc_of_le, ← succ_succAbove_succ, succ_pred] | case inl.inr
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
hik : i ≤ k.castSucc
⊢ i.castSucc.succAbove ((j.pred ⋯).succAbove k) = j.succAbove (i.succAbove k) | case inl.inr.h
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
hik : i ≤ k.castSucc
⊢ i ≤ (j.pred ⋯).succAbove k | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
hik : i ≤ k.castSucc
⊢ i.castSucc.succAbove ((j.pred ⋯).succAbove k) = j.succAbove (i.succAbove k)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove_succAbove | [74, 1] | [93, 46] | exact hik.trans (castSucc_le_succAbove _ _) | case inl.inr.h
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
hik : i ≤ k.castSucc
⊢ i ≤ (j.pred ⋯).succAbove k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr.h
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : i.castSucc < j
hik : i ≤ k.castSucc
⊢ i ≤ (j.pred ⋯).succAbove k
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove_succAbove | [74, 1] | [93, 46] | rw [succAbove_of_le_castSucc _ _ hij, predAbove_of_le_castSucc _ _ hij] | case inr
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
⊢ (j.succAbove i).succAbove ((i.predAbove j).succAbove k) = j.succAbove (i.succAbove k) | case inr
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
⊢ i.succ.succAbove ((j.castPred ⋯).succAbove k) = j.succAbove (i.succAbove k) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
⊢ (j.succAbove i).succAbove ((i.predAbove j).succAbove k) = j.succAbove (i.succAbove k)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove_succAbove | [74, 1] | [93, 46] | rcases lt_or_le i (succ k) with (hik | hik) | case inr
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
⊢ i.succ.succAbove ((j.castPred ⋯).succAbove k) = j.succAbove (i.succAbove k) | case inr.inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
hik : i < k.succ
⊢ i.succ.succAbove ((j.castPred ⋯).succAbove k) = j.succAbove (i.succAbove k)
case inr.inr
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
hik : k.succ ≤ i
⊢ i.succ.succAbove ((j.castPred ⋯).succAbove k) = j.succAbove (i.succAbove k) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
⊢ i.succ.succAbove ((j.castPred ⋯).succAbove k) = j.succAbove (i.succAbove k)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove_succAbove | [74, 1] | [93, 46] | have H := ((hij.trans_lt (castSucc_lt_castSucc_iff.mpr hik))) | case inr.inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
hik : i < k.succ
⊢ i.succ.succAbove ((j.castPred ⋯).succAbove k) = j.succAbove (i.succAbove k) | case inr.inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
hik : i < k.succ
H : j < k.succ.castSucc
⊢ i.succ.succAbove ((j.castPred ⋯).succAbove k) = j.succAbove (i.succAbove k) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
hik : i < k.succ
⊢ i.succ.succAbove ((j.castPred ⋯).succAbove k) = j.succAbove (i.succAbove k)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove_succAbove | [74, 1] | [93, 46] | rw [succAbove_of_lt_succ _ _ hik, succAbove_of_le_castSucc _ _ H.le,
succAbove_of_lt_succ _ k ((castPred_lt_iff _).mpr H), succAbove_succ_of_lt _ _ hik] | case inr.inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
hik : i < k.succ
H : j < k.succ.castSucc
⊢ i.succ.succAbove ((j.castPred ⋯).succAbove k) = j.succAbove (i.succAbove k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
hik : i < k.succ
H : j < k.succ.castSucc
⊢ i.succ.succAbove ((j.castPred ⋯).succAbove k) = j.succAbove (i.succAbove k)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove_succAbove | [74, 1] | [93, 46] | rw [succAbove_of_succ_le _ _ hik, succAbove_succ_of_le,
← castSucc_succAbove_castSucc, castSucc_castPred] | case inr.inr
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
hik : k.succ ≤ i
⊢ i.succ.succAbove ((j.castPred ⋯).succAbove k) = j.succAbove (i.succAbove k) | case inr.inr.h
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
hik : k.succ ≤ i
⊢ (j.castPred ⋯).succAbove k ≤ i | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
hik : k.succ ≤ i
⊢ i.succ.succAbove ((j.castPred ⋯).succAbove k) = j.succAbove (i.succAbove k)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Fin.lean | Fin.succAbove_succAbove_predAbove_succAbove | [74, 1] | [93, 46] | exact (succAbove_le_succ _ _).trans hik | case inr.inr.h
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
hik : k.succ ≤ i
⊢ (j.castPred ⋯).succAbove k ≤ i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.h
m : ℕ
i : Fin (m + 1)
k : Fin m
j : Fin (m + 2)
hij : j ≤ i.castSucc
hik : k.succ ≤ i
⊢ (j.castPred ⋯).succAbove k ≤ i
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | Equiv.Perm.cycleOf_pow_apply | [7, 1] | [15, 30] | induction' a with a IH generalizing y | α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x y : β
a : ℕ
⊢ (f.cycleOf x ^ a) y = if f.SameCycle x y then (f ^ a) y else y | case zero
α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x y : β
⊢ (f.cycleOf x ^ 0) y = if f.SameCycle x y then (f ^ 0) y else y
case succ
α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x : β
a : ℕ
IH : ∀ (y : β), (f.cycleOf x ^ a) y = if f.SameCycle x y then (f ^ a) y else y
y : β
⊢ (f.cycleOf x ^ (a + 1)) y = if f.SameCycle x y then (f ^ (a + 1)) y else y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x y : β
a : ℕ
⊢ (f.cycleOf x ^ a) y = if f.SameCycle x y then (f ^ a) y else y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | Equiv.Perm.cycleOf_pow_apply | [7, 1] | [15, 30] | simp_rw [pow_zero, one_apply, ite_self] | case zero
α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x y : β
⊢ (f.cycleOf x ^ 0) y = if f.SameCycle x y then (f ^ 0) y else y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x y : β
⊢ (f.cycleOf x ^ 0) y = if f.SameCycle x y then (f ^ 0) y else y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | Equiv.Perm.cycleOf_pow_apply | [7, 1] | [15, 30] | simp_rw [pow_succ', mul_apply, IH, cycleOf_apply] | case succ
α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x : β
a : ℕ
IH : ∀ (y : β), (f.cycleOf x ^ a) y = if f.SameCycle x y then (f ^ a) y else y
y : β
⊢ (f.cycleOf x ^ (a + 1)) y = if f.SameCycle x y then (f ^ (a + 1)) y else y | case succ
α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x : β
a : ℕ
IH : ∀ (y : β), (f.cycleOf x ^ a) y = if f.SameCycle x y then (f ^ a) y else y
y : β
⊢ (if f.SameCycle x (if f.SameCycle x y then (f ^ a) y else y) then f (if f.SameCycle x y then (f ^ a) y else y)
else if f.SameCycle x y then (f ^ a) y else y) =
if f.SameCycle x y then f ((f ^ a) y) else y | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x : β
a : ℕ
IH : ∀ (y : β), (f.cycleOf x ^ a) y = if f.SameCycle x y then (f ^ a) y else y
y : β
⊢ (f.cycleOf x ^ (a + 1)) y = if f.SameCycle x y then (f ^ (a + 1)) y else y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | Equiv.Perm.cycleOf_pow_apply | [7, 1] | [15, 30] | by_cases h : f.SameCycle x y | case succ
α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x : β
a : ℕ
IH : ∀ (y : β), (f.cycleOf x ^ a) y = if f.SameCycle x y then (f ^ a) y else y
y : β
⊢ (if f.SameCycle x (if f.SameCycle x y then (f ^ a) y else y) then f (if f.SameCycle x y then (f ^ a) y else y)
else if f.SameCycle x y then (f ^ a) y else y) =
if f.SameCycle x y then f ((f ^ a) y) else y | case pos
α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x : β
a : ℕ
IH : ∀ (y : β), (f.cycleOf x ^ a) y = if f.SameCycle x y then (f ^ a) y else y
y : β
h : f.SameCycle x y
⊢ (if f.SameCycle x (if f.SameCycle x y then (f ^ a) y else y) then f (if f.SameCycle x y then (f ^ a) y else y)
else if f.SameCycle x y then (f ^ a) y else y) =
if f.SameCycle x y then f ((f ^ a) y) else y
case neg
α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x : β
a : ℕ
IH : ∀ (y : β), (f.cycleOf x ^ a) y = if f.SameCycle x y then (f ^ a) y else y
y : β
h : ¬f.SameCycle x y
⊢ (if f.SameCycle x (if f.SameCycle x y then (f ^ a) y else y) then f (if f.SameCycle x y then (f ^ a) y else y)
else if f.SameCycle x y then (f ^ a) y else y) =
if f.SameCycle x y then f ((f ^ a) y) else y | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x : β
a : ℕ
IH : ∀ (y : β), (f.cycleOf x ^ a) y = if f.SameCycle x y then (f ^ a) y else y
y : β
⊢ (if f.SameCycle x (if f.SameCycle x y then (f ^ a) y else y) then f (if f.SameCycle x y then (f ^ a) y else y)
else if f.SameCycle x y then (f ^ a) y else y) =
if f.SameCycle x y then f ((f ^ a) y) else y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | Equiv.Perm.cycleOf_pow_apply | [7, 1] | [15, 30] | simp only [h, ↓reduceIte, sameCycle_pow_right] | case pos
α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x : β
a : ℕ
IH : ∀ (y : β), (f.cycleOf x ^ a) y = if f.SameCycle x y then (f ^ a) y else y
y : β
h : f.SameCycle x y
⊢ (if f.SameCycle x (if f.SameCycle x y then (f ^ a) y else y) then f (if f.SameCycle x y then (f ^ a) y else y)
else if f.SameCycle x y then (f ^ a) y else y) =
if f.SameCycle x y then f ((f ^ a) y) else y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x : β
a : ℕ
IH : ∀ (y : β), (f.cycleOf x ^ a) y = if f.SameCycle x y then (f ^ a) y else y
y : β
h : f.SameCycle x y
⊢ (if f.SameCycle x (if f.SameCycle x y then (f ^ a) y else y) then f (if f.SameCycle x y then (f ^ a) y else y)
else if f.SameCycle x y then (f ^ a) y else y) =
if f.SameCycle x y then f ((f ^ a) y) else y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | Equiv.Perm.cycleOf_pow_apply | [7, 1] | [15, 30] | simp only [h, ↓reduceIte] | case neg
α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x : β
a : ℕ
IH : ∀ (y : β), (f.cycleOf x ^ a) y = if f.SameCycle x y then (f ^ a) y else y
y : β
h : ¬f.SameCycle x y
⊢ (if f.SameCycle x (if f.SameCycle x y then (f ^ a) y else y) then f (if f.SameCycle x y then (f ^ a) y else y)
else if f.SameCycle x y then (f ^ a) y else y) =
if f.SameCycle x y then f ((f ^ a) y) else y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type ?u.39
π : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
β : Type v
inst✝¹ : DecidableEq β
inst✝ : Fintype β
f : Perm β
x : β
a : ℕ
IH : ∀ (y : β), (f.cycleOf x ^ a) y = if f.SameCycle x y then (f ^ a) y else y
y : β
h : ¬f.SameCycle x y
⊢ (if f.SameCycle x (if f.SameCycle x y then (f ^ a) y else y) then f (if f.SameCycle x y then (f ^ a) y else y)
else if f.SameCycle x y then (f ^ a) y else y) =
if f.SameCycle x y then f ((f ^ a) y) else y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | Equiv.Perm.pow_apply_injOn_Iio_orderOf_cycleOf | [17, 1] | [26, 27] | rintro a ha b hb hab | α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
⊢ Set.InjOn (fun t => (π ^ t) x) (Set.Iio (orderOf (π.cycleOf x))) | α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
⊢ Set.InjOn (fun t => (π ^ t) x) (Set.Iio (orderOf (π.cycleOf x)))
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | Equiv.Perm.pow_apply_injOn_Iio_orderOf_cycleOf | [17, 1] | [26, 27] | refine' pow_injOn_Iio_orderOf ha hb (ext (fun y => _)) | α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
⊢ a = b | α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
y : α
⊢ ((fun x_1 => π.cycleOf x ^ x_1) a) y = ((fun x_1 => π.cycleOf x ^ x_1) b) y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
⊢ a = b
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | Equiv.Perm.pow_apply_injOn_Iio_orderOf_cycleOf | [17, 1] | [26, 27] | simp_rw [cycleOf_pow_apply] | α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
y : α
⊢ ((fun x_1 => π.cycleOf x ^ x_1) a) y = ((fun x_1 => π.cycleOf x ^ x_1) b) y | α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
y : α
⊢ (if π.SameCycle x y then (π ^ a) y else y) = if π.SameCycle x y then (π ^ b) y else y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
y : α
⊢ ((fun x_1 => π.cycleOf x ^ x_1) a) y = ((fun x_1 => π.cycleOf x ^ x_1) b) y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | Equiv.Perm.pow_apply_injOn_Iio_orderOf_cycleOf | [17, 1] | [26, 27] | by_cases h : SameCycle π x y | α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
y : α
⊢ (if π.SameCycle x y then (π ^ a) y else y) = if π.SameCycle x y then (π ^ b) y else y | case pos
α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
y : α
h : π.SameCycle x y
⊢ (if π.SameCycle x y then (π ^ a) y else y) = if π.SameCycle x y then (π ^ b) y else y
case neg
α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
y : α
h : ¬π.SameCycle x y
⊢ (if π.SameCycle x y then (π ^ a) y else y) = if π.SameCycle x y then (π ^ b) y else y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
y : α
⊢ (if π.SameCycle x y then (π ^ a) y else y) = if π.SameCycle x y then (π ^ b) y else y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | Equiv.Perm.pow_apply_injOn_Iio_orderOf_cycleOf | [17, 1] | [26, 27] | simp_rw [h, ite_true] | case pos
α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
y : α
h : π.SameCycle x y
⊢ (if π.SameCycle x y then (π ^ a) y else y) = if π.SameCycle x y then (π ^ b) y else y | case pos
α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
y : α
h : π.SameCycle x y
⊢ (π ^ a) y = (π ^ b) y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
y : α
h : π.SameCycle x y
⊢ (if π.SameCycle x y then (π ^ a) y else y) = if π.SameCycle x y then (π ^ b) y else y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | Equiv.Perm.pow_apply_injOn_Iio_orderOf_cycleOf | [17, 1] | [26, 27] | rcases h with ⟨c, rfl⟩ | case pos
α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
y : α
h : π.SameCycle x y
⊢ (π ^ a) y = (π ^ b) y | case pos.intro
α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
c : ℤ
⊢ (π ^ a) ((π ^ c) x) = (π ^ b) ((π ^ c) x) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
y : α
h : π.SameCycle x y
⊢ (π ^ a) y = (π ^ b) y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | Equiv.Perm.pow_apply_injOn_Iio_orderOf_cycleOf | [17, 1] | [26, 27] | simp_rw [← zpow_natCast, zpow_apply_comm, zpow_natCast, hab] | case pos.intro
α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
c : ℤ
⊢ (π ^ a) ((π ^ c) x) = (π ^ b) ((π ^ c) x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
c : ℤ
⊢ (π ^ a) ((π ^ c) x) = (π ^ b) ((π ^ c) x)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | Equiv.Perm.pow_apply_injOn_Iio_orderOf_cycleOf | [17, 1] | [26, 27] | simp_rw [h, ite_false] | case neg
α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
y : α
h : ¬π.SameCycle x y
⊢ (if π.SameCycle x y then (π ^ a) y else y) = if π.SameCycle x y then (π ^ b) y else y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
π✝ : Perm α
inst✝³ : DecidableEq α
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
π : Perm α
x : α
a : ℕ
ha : a ∈ Set.Iio (orderOf (π.cycleOf x))
b : ℕ
hb : b ∈ Set.Iio (orderOf (π.cycleOf x))
hab : (fun t => (π ^ t) x) a = (fun t => (π ^ t) x) b
y : α
h : ¬π.SameCycle x y
⊢ (if π.SameCycle x y then (π ^ a) y else y) = if π.SameCycle x y then (π ^ b) y else y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | mem_cycleAt_iff | [43, 1] | [45, 52] | simp_rw [CycleAt, mem_filter, mem_univ, true_and] | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ y ∈ CycleAt π x ↔ π.SameCycle x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ y ∈ CycleAt π x ↔ π.SameCycle x y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | mem_cycleAt_iff_zpow | [47, 1] | [49, 6] | simp_rw [mem_cycleAt_iff] | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ y ∈ CycleAt π x ↔ ∃ k, (π ^ k) x = y | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ π.SameCycle x y ↔ ∃ k, (π ^ k) x = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ y ∈ CycleAt π x ↔ ∃ k, (π ^ k) x = y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | mem_cycleAt_iff_zpow | [47, 1] | [49, 6] | rfl | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ π.SameCycle x y ↔ ∃ k, (π ^ k) x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ π.SameCycle x y ↔ ∃ k, (π ^ k) x = y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAt_of_fixed | [69, 1] | [71, 39] | simp_rw [Finset.ext_iff, mem_cycleAt_iff_zpow, mem_singleton, (fun k => (h.perm_zpow k).eq),
exists_const, eq_comm, implies_true] | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
h : Function.IsFixedPt (⇑π) x
⊢ CycleAt π x = {x} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
h : Function.IsFixedPt (⇑π) x
⊢ CycleAt π x = {x}
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | card_cycleAt_eq_one_iff_fixedPt | [82, 1] | [87, 25] | rw [Finset.card_eq_one, fixedPt_iff_cycleAt_singleton] | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
⊢ Function.IsFixedPt (⇑π) x ↔ (CycleAt π x).card = 1 | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
⊢ CycleAt π x = {x} ↔ ∃ a, CycleAt π x = {a} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
⊢ Function.IsFixedPt (⇑π) x ↔ (CycleAt π x).card = 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | card_cycleAt_eq_one_iff_fixedPt | [82, 1] | [87, 25] | refine ⟨fun hx => ⟨_, hx⟩, ?_⟩ | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
⊢ CycleAt π x = {x} ↔ ∃ a, CycleAt π x = {a} | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
⊢ (∃ a, CycleAt π x = {a}) → CycleAt π x = {x} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
⊢ CycleAt π x = {x} ↔ ∃ a, CycleAt π x = {a}
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | card_cycleAt_eq_one_iff_fixedPt | [82, 1] | [87, 25] | rintro ⟨_, hx⟩ | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
⊢ (∃ a, CycleAt π x = {a}) → CycleAt π x = {x} | case intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x w✝ : α
hx : CycleAt π x = {w✝}
⊢ CycleAt π x = {x} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
⊢ (∃ a, CycleAt π x = {a}) → CycleAt π x = {x}
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | card_cycleAt_eq_one_iff_fixedPt | [82, 1] | [87, 25] | rw [hx, singleton_inj, eq_comm, ← mem_singleton, ← hx] | case intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x w✝ : α
hx : CycleAt π x = {w✝}
⊢ CycleAt π x = {x} | case intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x w✝ : α
hx : CycleAt π x = {w✝}
⊢ x ∈ CycleAt π x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x w✝ : α
hx : CycleAt π x = {w✝}
⊢ CycleAt π x = {x}
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | card_cycleAt_eq_one_iff_fixedPt | [82, 1] | [87, 25] | exact self_mem_cycleAt | case intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x w✝ : α
hx : CycleAt π x = {w✝}
⊢ x ∈ CycleAt π x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x w✝ : α
hx : CycleAt π x = {w✝}
⊢ x ∈ CycleAt π x
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAt_apply_eq_cycleAt | [89, 1] | [90, 85] | simp_rw [Finset.ext_iff, mem_cycleAt_iff, Perm.sameCycle_apply_left, implies_true] | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
⊢ CycleAt π (π x) = CycleAt π x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
⊢ CycleAt π (π x) = CycleAt π x
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | mem_cycleAt_iff_lt | [92, 1] | [100, 19] | rw [mem_cycleAt_iff] | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ y ∈ CycleAt π x ↔ ∃ b < orderOf (π.cycleOf x), (π ^ b) x = y | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ π.SameCycle x y ↔ ∃ b < orderOf (π.cycleOf x), (π ^ b) x = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ y ∈ CycleAt π x ↔ ∃ b < orderOf (π.cycleOf x), (π ^ b) x = y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | mem_cycleAt_iff_lt | [92, 1] | [100, 19] | refine ⟨?_, ?_⟩ | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ π.SameCycle x y ↔ ∃ b < orderOf (π.cycleOf x), (π ^ b) x = y | case refine_1
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ π.SameCycle x y → ∃ b < orderOf (π.cycleOf x), (π ^ b) x = y
case refine_2
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ (∃ b < orderOf (π.cycleOf x), (π ^ b) x = y) → π.SameCycle x y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ π.SameCycle x y ↔ ∃ b < orderOf (π.cycleOf x), (π ^ b) x = y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | mem_cycleAt_iff_lt | [92, 1] | [100, 19] | rintro hb | case refine_1
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ π.SameCycle x y → ∃ b < orderOf (π.cycleOf x), (π ^ b) x = y | case refine_1
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
hb : π.SameCycle x y
⊢ ∃ b < orderOf (π.cycleOf x), (π ^ b) x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ π.SameCycle x y → ∃ b < orderOf (π.cycleOf x), (π ^ b) x = y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | mem_cycleAt_iff_lt | [92, 1] | [100, 19] | rcases (hb.exists_pow_eq π) with ⟨b, _, _, rfl⟩ | case refine_1
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
hb : π.SameCycle x y
⊢ ∃ b < orderOf (π.cycleOf x), (π ^ b) x = y | case refine_1.intro.intro.intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
b : ℕ
left✝¹ : 0 < b
left✝ : b ≤ (π.cycleOf x).support.card + 1
hb : π.SameCycle x ((π ^ b) x)
⊢ ∃ b_1 < orderOf (π.cycleOf x), (π ^ b_1) x = (π ^ b) x | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
hb : π.SameCycle x y
⊢ ∃ b < orderOf (π.cycleOf x), (π ^ b) x = y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | mem_cycleAt_iff_lt | [92, 1] | [100, 19] | refine ⟨b % orderOf (π.cycleOf x), Nat.mod_lt _ (orderOf_pos _),
(π.pow_mod_orderOf_cycleOf_apply _ _)⟩ | case refine_1.intro.intro.intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
b : ℕ
left✝¹ : 0 < b
left✝ : b ≤ (π.cycleOf x).support.card + 1
hb : π.SameCycle x ((π ^ b) x)
⊢ ∃ b_1 < orderOf (π.cycleOf x), (π ^ b_1) x = (π ^ b) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.intro.intro.intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
b : ℕ
left✝¹ : 0 < b
left✝ : b ≤ (π.cycleOf x).support.card + 1
hb : π.SameCycle x ((π ^ b) x)
⊢ ∃ b_1 < orderOf (π.cycleOf x), (π ^ b_1) x = (π ^ b) x
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | mem_cycleAt_iff_lt | [92, 1] | [100, 19] | rintro ⟨b, _, rfl⟩ | case refine_2
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ (∃ b < orderOf (π.cycleOf x), (π ^ b) x = y) → π.SameCycle x y | case refine_2.intro.intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
b : ℕ
left✝ : b < orderOf (π.cycleOf x)
⊢ π.SameCycle x ((π ^ b) x) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ (∃ b < orderOf (π.cycleOf x), (π ^ b) x = y) → π.SameCycle x y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | mem_cycleAt_iff_lt | [92, 1] | [100, 19] | exact ⟨b, rfl⟩ | case refine_2.intro.intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
b : ℕ
left✝ : b < orderOf (π.cycleOf x)
⊢ π.SameCycle x ((π ^ b) x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
b : ℕ
left✝ : b < orderOf (π.cycleOf x)
⊢ π.SameCycle x ((π ^ b) x)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | mem_cycleAt_iff_le | [102, 1] | [103, 67] | simp_rw [mem_cycleAt_iff_lt, Nat.lt_iff_le_pred (orderOf_pos _)] | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ y ∈ CycleAt π x ↔ ∃ b ≤ orderOf (π.cycleOf x) - 1, (π ^ b) x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y x : α
⊢ y ∈ CycleAt π x ↔ ∃ b ≤ orderOf (π.cycleOf x) - 1, (π ^ b) x = y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | mem_cycleAtTo_iff | [108, 1] | [110, 54] | simp_rw [CycleAtTo, Finset.mem_image, Finset.mem_Iio] | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y : α
a : ℕ
x : α
⊢ y ∈ CycleAtTo π a x ↔ ∃ b < a, (π ^ b) x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
y : α
a : ℕ
x : α
⊢ y ∈ CycleAtTo π a x ↔ ∃ b < a, (π ^ b) x = y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | apply_pow_mem_cycleAtTo_apply_pow_of_ge_of_lt | [115, 1] | [118, 73] | rw [← tsub_add_cancel_of_le hcb, pow_add, Perm.mul_apply] | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
b a c : ℕ
x : α
hba : b < a + c
hcb : c ≤ b
⊢ (π ^ b) x ∈ CycleAtTo π a ((π ^ c) x) | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
b a c : ℕ
x : α
hba : b < a + c
hcb : c ≤ b
⊢ (π ^ (b - c)) ((π ^ c) x) ∈ CycleAtTo π a ((π ^ c) x) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
b a c : ℕ
x : α
hba : b < a + c
hcb : c ≤ b
⊢ (π ^ b) x ∈ CycleAtTo π a ((π ^ c) x)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | apply_pow_mem_cycleAtTo_apply_pow_of_ge_of_lt | [115, 1] | [118, 73] | exact apply_pow_mem_cycleAtTo_of_lt (Nat.sub_lt_right_of_lt_add hcb hba) | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
b a c : ℕ
x : α
hba : b < a + c
hcb : c ≤ b
⊢ (π ^ (b - c)) ((π ^ c) x) ∈ CycleAtTo π a ((π ^ c) x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
b a c : ℕ
x : α
hba : b < a + c
hcb : c ≤ b
⊢ (π ^ (b - c)) ((π ^ c) x) ∈ CycleAtTo π a ((π ^ c) x)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAtTo_zero | [126, 1] | [128, 33] | simp_rw [Finset.ext_iff, mem_cycleAtTo_iff, not_lt_zero', false_and, exists_false,
not_mem_empty, implies_true] | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
⊢ CycleAtTo π 0 x = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
⊢ CycleAtTo π 0 x = ∅
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAtTo_one | [130, 1] | [132, 58] | simp_rw [Finset.ext_iff, mem_cycleAtTo_iff, Nat.lt_one_iff, exists_eq_left, pow_zero,
Perm.one_apply, mem_singleton, eq_comm, implies_true] | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
⊢ CycleAtTo π 1 x = {x} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
⊢ CycleAtTo π 1 x = {x}
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAtTo_singleton_of_fixedPt | [134, 1] | [138, 59] | simp_rw [Finset.ext_iff, mem_singleton, mem_cycleAtTo_iff,
π.pow_apply_eq_self_of_apply_eq_self h, eq_comm (a := x)] | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
ha : 0 < a
h : Function.IsFixedPt (⇑π) x
⊢ CycleAtTo π a x = {x} | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
ha : 0 < a
h : Function.IsFixedPt (⇑π) x
⊢ ∀ (a_1 : α), (∃ b < a, a_1 = x) ↔ a_1 = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
ha : 0 < a
h : Function.IsFixedPt (⇑π) x
⊢ CycleAtTo π a x = {x}
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAtTo_singleton_of_fixedPt | [134, 1] | [138, 59] | exact fun _ => ⟨fun ⟨_, _, h⟩ => h, fun h => ⟨0, ha, h⟩⟩ | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
ha : 0 < a
h : Function.IsFixedPt (⇑π) x
⊢ ∀ (a_1 : α), (∃ b < a, a_1 = x) ↔ a_1 = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
ha : 0 < a
h : Function.IsFixedPt (⇑π) x
⊢ ∀ (a_1 : α), (∃ b < a, a_1 = x) ↔ a_1 = x
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAtTo_mono | [140, 1] | [144, 40] | intros a b hab x y h | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
⊢ Monotone fun x => CycleAtTo π x | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a b : ℕ
hab : a ≤ b
x y : α
h : y ∈ (fun x => CycleAtTo π x) a x
⊢ y ∈ (fun x => CycleAtTo π x) b x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
⊢ Monotone fun x => CycleAtTo π x
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAtTo_mono | [140, 1] | [144, 40] | rw [mem_cycleAtTo_iff] at h ⊢ | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a b : ℕ
hab : a ≤ b
x y : α
h : y ∈ (fun x => CycleAtTo π x) a x
⊢ y ∈ (fun x => CycleAtTo π x) b x | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a b : ℕ
hab : a ≤ b
x y : α
h : ∃ b < a, (π ^ b) x = y
⊢ ∃ b_1 < b, (π ^ b_1) x = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a b : ℕ
hab : a ≤ b
x y : α
h : y ∈ (fun x => CycleAtTo π x) a x
⊢ y ∈ (fun x => CycleAtTo π x) b x
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAtTo_mono | [140, 1] | [144, 40] | rcases h with ⟨c, hca, hc⟩ | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a b : ℕ
hab : a ≤ b
x y : α
h : ∃ b < a, (π ^ b) x = y
⊢ ∃ b_1 < b, (π ^ b_1) x = y | case intro.intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a b : ℕ
hab : a ≤ b
x y : α
c : ℕ
hca : c < a
hc : (π ^ c) x = y
⊢ ∃ b_1 < b, (π ^ b_1) x = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a b : ℕ
hab : a ≤ b
x y : α
h : ∃ b < a, (π ^ b) x = y
⊢ ∃ b_1 < b, (π ^ b_1) x = y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAtTo_mono | [140, 1] | [144, 40] | exact ⟨c, lt_of_lt_of_le hca hab, hc⟩ | case intro.intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a b : ℕ
hab : a ≤ b
x y : α
c : ℕ
hca : c < a
hc : (π ^ c) x = y
⊢ ∃ b_1 < b, (π ^ b_1) x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a b : ℕ
hab : a ≤ b
x y : α
c : ℕ
hca : c < a
hc : (π ^ c) x = y
⊢ ∃ b_1 < b, (π ^ b_1) x = y
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | card_cycleAtTo_le | [149, 1] | [151, 30] | convert Finset.card_image_le | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
⊢ (CycleAtTo π a x).card ≤ a | case h.e'_4
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
⊢ a = (Iio a).card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
⊢ (CycleAtTo π a x).card ≤ a
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | card_cycleAtTo_le | [149, 1] | [151, 30] | exact (Nat.card_Iio _).symm | case h.e'_4
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
⊢ a = (Iio a).card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
⊢ a = (Iio a).card
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAtTo_card_eq_of_le_orderOf_cycleOf | [153, 1] | [159, 94] | nth_rewrite 2 [← Nat.card_Iio a] | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
h : a ≤ orderOf (π.cycleOf x)
⊢ (CycleAtTo π a x).card = a | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
h : a ≤ orderOf (π.cycleOf x)
⊢ (CycleAtTo π a x).card = (Iio a).card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
h : a ≤ orderOf (π.cycleOf x)
⊢ (CycleAtTo π a x).card = a
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAtTo_card_eq_of_le_orderOf_cycleOf | [153, 1] | [159, 94] | apply Finset.card_image_iff.mpr | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
h : a ≤ orderOf (π.cycleOf x)
⊢ (CycleAtTo π a x).card = (Iio a).card | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
h : a ≤ orderOf (π.cycleOf x)
⊢ Set.InjOn (fun k => (π ^ k) x) ↑(Iio a) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
h : a ≤ orderOf (π.cycleOf x)
⊢ (CycleAtTo π a x).card = (Iio a).card
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAtTo_card_eq_of_le_orderOf_cycleOf | [153, 1] | [159, 94] | intros b hb c hc hbc | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
h : a ≤ orderOf (π.cycleOf x)
⊢ Set.InjOn (fun k => (π ^ k) x) ↑(Iio a) | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
h : a ≤ orderOf (π.cycleOf x)
b : ℕ
hb : b ∈ ↑(Iio a)
c : ℕ
hc : c ∈ ↑(Iio a)
hbc : (fun k => (π ^ k) x) b = (fun k => (π ^ k) x) c
⊢ b = c | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
h : a ≤ orderOf (π.cycleOf x)
⊢ Set.InjOn (fun k => (π ^ k) x) ↑(Iio a)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAtTo_card_eq_of_le_orderOf_cycleOf | [153, 1] | [159, 94] | simp_rw [coe_Iio, Set.mem_Iio] at hb hc | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
h : a ≤ orderOf (π.cycleOf x)
b : ℕ
hb : b ∈ ↑(Iio a)
c : ℕ
hc : c ∈ ↑(Iio a)
hbc : (fun k => (π ^ k) x) b = (fun k => (π ^ k) x) c
⊢ b = c | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
h : a ≤ orderOf (π.cycleOf x)
b c : ℕ
hbc : (fun k => (π ^ k) x) b = (fun k => (π ^ k) x) c
hb : b < a
hc : c < a
⊢ b = c | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
h : a ≤ orderOf (π.cycleOf x)
b : ℕ
hb : b ∈ ↑(Iio a)
c : ℕ
hc : c ∈ ↑(Iio a)
hbc : (fun k => (π ^ k) x) b = (fun k => (π ^ k) x) c
⊢ b = c
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAtTo_card_eq_of_le_orderOf_cycleOf | [153, 1] | [159, 94] | exact π.pow_apply_injOn_Iio_orderOf_cycleOf (lt_of_lt_of_le hb h) (lt_of_lt_of_le hc h) hbc | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
h : a ≤ orderOf (π.cycleOf x)
b c : ℕ
hbc : (fun k => (π ^ k) x) b = (fun k => (π ^ k) x) c
hb : b < a
hc : c < a
⊢ b = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
h : a ≤ orderOf (π.cycleOf x)
b c : ℕ
hbc : (fun k => (π ^ k) x) b = (fun k => (π ^ k) x) c
hb : b < a
hc : c < a
⊢ b = c
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAtTo_subset_cycleAt | [161, 1] | [164, 36] | rintro y hy | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
⊢ CycleAtTo π a x ⊆ CycleAt π x | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x y : α
hy : y ∈ CycleAtTo π a x
⊢ y ∈ CycleAt π x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x : α
⊢ CycleAtTo π a x ⊆ CycleAt π x
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAtTo_subset_cycleAt | [161, 1] | [164, 36] | rcases (mem_cycleAtTo_iff.mp hy) with ⟨b, _, hb⟩ | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x y : α
hy : y ∈ CycleAtTo π a x
⊢ y ∈ CycleAt π x | case intro.intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x y : α
hy : y ∈ CycleAtTo π a x
b : ℕ
left✝ : b < a
hb : (π ^ b) x = y
⊢ y ∈ CycleAt π x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x y : α
hy : y ∈ CycleAtTo π a x
⊢ y ∈ CycleAt π x
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAtTo_subset_cycleAt | [161, 1] | [164, 36] | exact mem_cycleAt_iff.mpr ⟨b, hb⟩ | case intro.intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x y : α
hy : y ∈ CycleAtTo π a x
b : ℕ
left✝ : b < a
hb : (π ^ b) x = y
⊢ y ∈ CycleAt π x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
a : ℕ
x y : α
hy : y ∈ CycleAtTo π a x
b : ℕ
left✝ : b < a
hb : (π ^ b) x = y
⊢ y ∈ CycleAt π x
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleAt_eq_cycleAtTo_orderOf_cycleOf | [166, 1] | [168, 80] | simp_rw [Finset.ext_iff, mem_cycleAtTo_iff, mem_cycleAt_iff_lt, implies_true] | α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
⊢ CycleAt π x = CycleAtTo π (orderOf (π.cycleOf x)) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : DecidableEq α
π : Perm α
inst✝ : Fintype α
x : α
⊢ CycleAt π x = CycleAtTo π (orderOf (π.cycleOf x)) x
TACTIC:
|
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