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/Cycles.lean | fastCycleMin_eq_min_cycleAtTo | [285, 1] | [297, 76] | exact apply_pow_mem_cycleAtTo_of_lt ((add_lt_add_iff_right _).mpr hb) | case succ.refine'_4
α : Type u
inst✝² : DecidableEq α
π : Perm α
inst✝¹ : Fintype α
inst✝ : LinearOrder α
i : ℕ
hi : ∀ {x : α}, FastCycleMin i π x = (CycleAtTo π (2 ^ i) x).min' ⋯
x : α
b : ℕ
hb : b < 2 ^ i
⊢ (π ^ (b + 2 ^ i)) x ∈ CycleAtTo π (2 ^ i + 2 ^ i) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.refine'_4
α : Type u
inst✝² : DecidableEq α
π : Perm α
inst✝¹ : Fintype α
inst✝ : LinearOrder α
i : ℕ
hi : ∀ {x : α}, FastCycleMin i π x = (CycleAtTo π (2 ^ i) x).min' ⋯
x : α
b : ℕ
hb : b < 2 ^ i
⊢ (π ^ (b + 2 ^ i)) x ∈ CycleAtTo π (2 ^ i + 2 ^ i) x
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleMin_eq_fastCycleMin | [301, 1] | [303, 69] | rw [fastCycleMin_eq_min_cycleAtTo, cycleMin_eq_min_cycleAtTo_ge h] | α : Type u
inst✝² : DecidableEq α
π : Perm α
inst✝¹ : Fintype α
inst✝ : LinearOrder α
x : α
i : ℕ
h : orderOf (π.cycleOf x) ≤ 2 ^ i
⊢ FastCycleMin i π x = CycleMin π x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝² : DecidableEq α
π : Perm α
inst✝¹ : Fintype α
inst✝ : LinearOrder α
x : α
i : ℕ
h : orderOf (π.cycleOf x) ≤ 2 ^ i
⊢ FastCycleMin i π x = CycleMin π x
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleMin_eq_cycleMin_apply | [306, 1] | [307, 51] | simp_rw [cycleMin_def, cycleAt_apply_eq_cycleAt] | α : Type u
inst✝² : DecidableEq α
π : Perm α
inst✝¹ : Fintype α
inst✝ : LinearOrder α
x : α
⊢ CycleMin π x = CycleMin π (π x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝² : DecidableEq α
π : Perm α
inst✝¹ : Fintype α
inst✝ : LinearOrder α
x : α
⊢ CycleMin π x = CycleMin π (π x)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Cycles.lean | cycleMin_eq_cycleMin_apply_inv | [309, 1] | [310, 74] | rw [cycleMin_eq_cycleMin_apply (x := (π⁻¹ x)), Equiv.Perm.apply_inv_self] | α : Type u
inst✝² : DecidableEq α
π : Perm α
inst✝¹ : Fintype α
inst✝ : LinearOrder α
x : α
⊢ CycleMin π x = CycleMin π (π⁻¹ x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝² : DecidableEq α
π : Perm α
inst✝¹ : Fintype α
inst✝ : LinearOrder α
x : α
⊢ CycleMin π x = CycleMin π (π⁻¹ x)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_inv_mul_eq_mul_inv_cmtr | [13, 1] | [14, 75] | simp_rw [commutatorElement_inv, commutatorElement_def, inv_inv, mul_assoc] | G : Type u
inst✝ : Group G
x y : G
⊢ ⁅x, y⁆⁻¹ * y = y * ⁅x, y⁻¹⁆ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u
inst✝ : Group G
x y : G
⊢ ⁅x, y⁆⁻¹ * y = y * ⁅x, y⁻¹⁆
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_mul_eq_mul_inv_cmtr_inv | [16, 1] | [18, 43] | simp_rw [commutatorElement_inv, commutatorElement_def, inv_mul_cancel_right,
mul_assoc, mul_inv_cancel_left, inv_inv] | G : Type u
inst✝ : Group G
x y : G
⊢ ⁅x, y⁆ * y = y * ⁅x, y⁻¹⁆⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u
inst✝ : Group G
x y : G
⊢ ⁅x, y⁆ * y = y * ⁅x, y⁻¹⁆⁻¹
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_pow_inv_mul_eq_mul_inv_cmtr_pow | [20, 1] | [24, 34] | induction' k with n hn | G : Type u
inst✝ : Group G
x y : G
k : ℕ
⊢ (⁅x, y⁆ ^ k)⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ k | case zero
G : Type u
inst✝ : Group G
x y : G
⊢ (⁅x, y⁆ ^ 0)⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ 0
case succ
G : Type u
inst✝ : Group G
x y : G
n : ℕ
hn : (⁅x, y⁆ ^ n)⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ n
⊢ (⁅x, y⁆ ^ (n + 1))⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u
inst✝ : Group G
x y : G
k : ℕ
⊢ (⁅x, y⁆ ^ k)⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ k
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_pow_inv_mul_eq_mul_inv_cmtr_pow | [20, 1] | [24, 34] | simp_rw [pow_zero, inv_one, mul_one, one_mul] | case zero
G : Type u
inst✝ : Group G
x y : G
⊢ (⁅x, y⁆ ^ 0)⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
G : Type u
inst✝ : Group G
x y : G
⊢ (⁅x, y⁆ ^ 0)⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ 0
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_pow_inv_mul_eq_mul_inv_cmtr_pow | [20, 1] | [24, 34] | simp_rw [pow_succ ⁅x, y⁻¹⁆, pow_succ' ⁅x, y⁆, ← mul_assoc, hn.symm, mul_inv_rev, mul_assoc,
cmtr_inv_mul_eq_mul_inv_cmtr] | case succ
G : Type u
inst✝ : Group G
x y : G
n : ℕ
hn : (⁅x, y⁆ ^ n)⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ n
⊢ (⁅x, y⁆ ^ (n + 1))⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
G : Type u
inst✝ : Group G
x y : G
n : ℕ
hn : (⁅x, y⁆ ^ n)⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ n
⊢ (⁅x, y⁆ ^ (n + 1))⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ (n + 1)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | pow_mul_eq_mul_inv_cmtr_pow_inv | [26, 1] | [28, 99] | rw [eq_mul_inv_iff_mul_eq, mul_assoc, ← cmtr_pow_inv_mul_eq_mul_inv_cmtr_pow, mul_inv_cancel_left] | G : Type u
inst✝ : Group G
x y : G
k : ℕ
⊢ ⁅x, y⁆ ^ k * y = y * (⁅x, y⁻¹⁆ ^ k)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u
inst✝ : Group G
x y : G
k : ℕ
⊢ ⁅x, y⁆ ^ k * y = y * (⁅x, y⁻¹⁆ ^ k)⁻¹
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_zpow_inv_mul_eq_mul_inv_cmtr_zpow | [30, 1] | [33, 79] | cases k | G : Type u
inst✝ : Group G
x y : G
k : ℤ
⊢ (⁅x, y⁆ ^ k)⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ k | case ofNat
G : Type u
inst✝ : Group G
x y : G
a✝ : ℕ
⊢ (⁅x, y⁆ ^ Int.ofNat a✝)⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ Int.ofNat a✝
case negSucc
G : Type u
inst✝ : Group G
x y : G
a✝ : ℕ
⊢ (⁅x, y⁆ ^ Int.negSucc a✝)⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ Int.negSucc a✝ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u
inst✝ : Group G
x y : G
k : ℤ
⊢ (⁅x, y⁆ ^ k)⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ k
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_zpow_inv_mul_eq_mul_inv_cmtr_zpow | [30, 1] | [33, 79] | simp only [Int.ofNat_eq_coe, zpow_natCast, zpow_neg, cmtr_pow_inv_mul_eq_mul_inv_cmtr_pow] | case ofNat
G : Type u
inst✝ : Group G
x y : G
a✝ : ℕ
⊢ (⁅x, y⁆ ^ Int.ofNat a✝)⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ Int.ofNat a✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ofNat
G : Type u
inst✝ : Group G
x y : G
a✝ : ℕ
⊢ (⁅x, y⁆ ^ Int.ofNat a✝)⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ Int.ofNat a✝
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_zpow_inv_mul_eq_mul_inv_cmtr_zpow | [30, 1] | [33, 79] | simp only [zpow_negSucc, zpow_neg, inv_inv, pow_mul_eq_mul_inv_cmtr_pow_inv] | case negSucc
G : Type u
inst✝ : Group G
x y : G
a✝ : ℕ
⊢ (⁅x, y⁆ ^ Int.negSucc a✝)⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ Int.negSucc a✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case negSucc
G : Type u
inst✝ : Group G
x y : G
a✝ : ℕ
⊢ (⁅x, y⁆ ^ Int.negSucc a✝)⁻¹ * y = y * ⁅x, y⁻¹⁆ ^ Int.negSucc a✝
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_zpow_mul_eq_mul_inv_cmtr_zpow_inv | [35, 1] | [37, 77] | rw [← zpow_neg, ← cmtr_zpow_inv_mul_eq_mul_inv_cmtr_zpow, zpow_neg, inv_inv] | G : Type u
inst✝ : Group G
x y : G
k : ℤ
⊢ ⁅x, y⁆ ^ k * y = y * (⁅x, y⁻¹⁆ ^ k)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u
inst✝ : Group G
x y : G
k : ℤ
⊢ ⁅x, y⁆ ^ k * y = y * (⁅x, y⁻¹⁆ ^ k)⁻¹
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_inv_eq_cmtr_iff_cmtr_square_id | [39, 1] | [43, 36] | simp_rw [pow_two, commutatorElement_eq_one_iff_mul_comm, eq_comm (a := (x * (y * y))),
commutatorElement_def, mul_assoc, mul_left_cancel_iff, ← inv_mul_eq_one (a := y * (x⁻¹ * y⁻¹)),
mul_eq_one_iff_eq_inv, mul_inv_rev, inv_inv, mul_assoc, ← eq_inv_mul_iff_mul_eq (b := y),
mul_inv_eq_iff_eq_mul, mul_assoc] | G : Type u
inst✝ : Group G
x y : G
⊢ ⁅x, y⁆ = ⁅x, y⁻¹⁆ ↔ ⁅x, y ^ 2⁆ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u
inst✝ : Group G
x y : G
⊢ ⁅x, y⁆ = ⁅x, y⁻¹⁆ ↔ ⁅x, y ^ 2⁆ = 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | mul_cmtr_unfix_of_unfix | [57, 1] | [61, 34] | simp_rw [Perm.mul_apply, cmtr_apply,
← Perm.eq_inv_iff_eq (f := y).not, ← Perm.eq_inv_iff_eq (f := x).not] | α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁆) q ≠ q | α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), ¬y (x⁻¹ (y⁻¹ q)) = x⁻¹ (y⁻¹ q) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁆) q ≠ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | mul_cmtr_unfix_of_unfix | [57, 1] | [61, 34] | exact fun q => hy (x⁻¹ (y⁻¹ q)) | α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), ¬y (x⁻¹ (y⁻¹ q)) = x⁻¹ (y⁻¹ q) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), ¬y (x⁻¹ (y⁻¹ q)) = x⁻¹ (y⁻¹ q)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_mul_unfix_of_unfix | [63, 1] | [66, 28] | simp_rw [Perm.mul_apply, cmtr_apply, Perm.inv_apply_self, ← Perm.eq_inv_iff_eq (f := x).not] | α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ * y) q ≠ q | α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), ¬y (x⁻¹ q) = x⁻¹ q | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ * y) q ≠ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_mul_unfix_of_unfix | [63, 1] | [66, 28] | exact fun q => hy (x⁻¹ q) | α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), ¬y (x⁻¹ q) = x⁻¹ q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), ¬y (x⁻¹ q) = x⁻¹ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | mul_inv_cmtr_inv_unfix_of_unfix | [68, 1] | [71, 35] | simp_rw [← cmtr_mul_eq_mul_inv_cmtr_inv] | α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁻¹⁆⁻¹) q ≠ q | α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ * y) q ≠ q | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁻¹⁆⁻¹) q ≠ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | mul_inv_cmtr_inv_unfix_of_unfix | [68, 1] | [71, 35] | exact cmtr_mul_unfix_of_unfix hy | α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ * y) q ≠ q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ * y) q ≠ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_inv_apply_ne_apply_of_unfix | [73, 1] | [77, 56] | simp_rw [Perm.inv_eq_iff_eq.not] | α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ⁅x, y⁆⁻¹ q ≠ y q | α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ¬q = ⁅x, y⁆ (y q) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ⁅x, y⁆⁻¹ q ≠ y q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_inv_apply_ne_apply_of_unfix | [73, 1] | [77, 56] | exact Ne.symm (cmtr_mul_unfix_of_unfix (x := x) hy q) | α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ¬q = ⁅x, y⁆ (y q) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
hy : ∀ (q : α), y q ≠ q
⊢ ¬q = ⁅x, y⁆ (y q)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | mul_cmtr_pow_unfix | [79, 1] | [90, 26] | induction' k using Nat.twoStepInduction with k IH | α : Type u
x y : Perm α
q : α
k : ℕ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁆ ^ k) q ≠ q | case H1
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁆ ^ 0) q ≠ q
case H2
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁆ ^ 1) q ≠ q
case H3
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (y * ⁅x, y⁆ ^ k) q ≠ q
_IH2✝ : ∀ (q : α), (y * ⁅x, y⁆ ^ k.succ) q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁆ ^ k.succ.succ) q ≠ q | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
k : ℕ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁆ ^ k) q ≠ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | mul_cmtr_pow_unfix | [79, 1] | [90, 26] | rw [pow_zero, mul_one] | case H1
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁆ ^ 0) q ≠ q | case H1
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), y q ≠ q | Please generate a tactic in lean4 to solve the state.
STATE:
case H1
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁆ ^ 0) q ≠ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | mul_cmtr_pow_unfix | [79, 1] | [90, 26] | exact hy | case H1
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), y q ≠ q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H1
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), y q ≠ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | mul_cmtr_pow_unfix | [79, 1] | [90, 26] | rw [pow_one] | case H2
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁆ ^ 1) q ≠ q | case H2
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁆) q ≠ q | Please generate a tactic in lean4 to solve the state.
STATE:
case H2
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁆ ^ 1) q ≠ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | mul_cmtr_pow_unfix | [79, 1] | [90, 26] | exact mul_cmtr_unfix_of_unfix hy | case H2
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁆) q ≠ q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H2
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁆) q ≠ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | mul_cmtr_pow_unfix | [79, 1] | [90, 26] | intros q h | case H3
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (y * ⁅x, y⁆ ^ k) q ≠ q
_IH2✝ : ∀ (q : α), (y * ⁅x, y⁆ ^ k.succ) q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁆ ^ k.succ.succ) q ≠ q | case H3
α : Type u
x y : Perm α
q✝ : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (y * ⁅x, y⁆ ^ k) q ≠ q
_IH2✝ : ∀ (q : α), (y * ⁅x, y⁆ ^ k.succ) q ≠ q
q : α
h : (y * ⁅x, y⁆ ^ k.succ.succ) q = q
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case H3
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (y * ⁅x, y⁆ ^ k) q ≠ q
_IH2✝ : ∀ (q : α), (y * ⁅x, y⁆ ^ k.succ) q ≠ q
⊢ ∀ (q : α), (y * ⁅x, y⁆ ^ k.succ.succ) q ≠ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | mul_cmtr_pow_unfix | [79, 1] | [90, 26] | simp_rw [pow_succ (n := k.succ), pow_succ' (n := k), ← mul_assoc, ← hxy,
← cmtr_inv_mul_eq_mul_inv_cmtr, hxy, mul_assoc,
Perm.mul_apply, Perm.inv_eq_iff_eq] at h | case H3
α : Type u
x y : Perm α
q✝ : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (y * ⁅x, y⁆ ^ k) q ≠ q
_IH2✝ : ∀ (q : α), (y * ⁅x, y⁆ ^ k.succ) q ≠ q
q : α
h : (y * ⁅x, y⁆ ^ k.succ.succ) q = q
⊢ False | case H3
α : Type u
x y : Perm α
q✝ : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (y * ⁅x, y⁆ ^ k) q ≠ q
_IH2✝ : ∀ (q : α), (y * ⁅x, y⁆ ^ k.succ) q ≠ q
q : α
h : y ((⁅x, y⁆ ^ k) (⁅x, y⁆ q)) = ⁅x, y⁆ q
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case H3
α : Type u
x y : Perm α
q✝ : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (y * ⁅x, y⁆ ^ k) q ≠ q
_IH2✝ : ∀ (q : α), (y * ⁅x, y⁆ ^ k.succ) q ≠ q
q : α
h : (y * ⁅x, y⁆ ^ k.succ.succ) q = q
⊢ False
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | mul_cmtr_pow_unfix | [79, 1] | [90, 26] | exact IH (⁅x, y⁆ q) h | case H3
α : Type u
x y : Perm α
q✝ : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (y * ⁅x, y⁆ ^ k) q ≠ q
_IH2✝ : ∀ (q : α), (y * ⁅x, y⁆ ^ k.succ) q ≠ q
q : α
h : y ((⁅x, y⁆ ^ k) (⁅x, y⁆ q)) = ⁅x, y⁆ q
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H3
α : Type u
x y : Perm α
q✝ : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (y * ⁅x, y⁆ ^ k) q ≠ q
_IH2✝ : ∀ (q : α), (y * ⁅x, y⁆ ^ k.succ) q ≠ q
q : α
h : y ((⁅x, y⁆ ^ k) (⁅x, y⁆ q)) = ⁅x, y⁆ q
⊢ False
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_pow_apply_ne_apply | [92, 1] | [95, 46] | simp_rw [← Perm.eq_inv_iff_eq.not, ← Perm.mul_apply, cmtr_pow_inv_mul_eq_mul_inv_cmtr_pow, hxy] | α : Type u
x y : Perm α
q : α
k : ℕ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ (⁅x, y⁆ ^ k) q ≠ y q | α : Type u
x y : Perm α
q : α
k : ℕ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ¬q = (y * ⁅x, y⁆ ^ k) q | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
k : ℕ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ (⁅x, y⁆ ^ k) q ≠ y q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_pow_apply_ne_apply | [92, 1] | [95, 46] | exact Ne.symm (mul_cmtr_pow_unfix hxy hy _) | α : Type u
x y : Perm α
q : α
k : ℕ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ¬q = (y * ⁅x, y⁆ ^ k) q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
k : ℕ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ¬q = (y * ⁅x, y⁆ ^ k) q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_pow_mul_unfix | [97, 1] | [109, 28] | induction' k using Nat.twoStepInduction with k IH | α : Type u
x y : Perm α
q : α
k : ℕ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ ^ k * y) q ≠ q | case H1
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ ^ 0 * y) q ≠ q
case H2
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ ^ 1 * y) q ≠ q
case H3
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (⁅x, y⁆ ^ k * y) q ≠ q
_IH2✝ : ∀ (q : α), (⁅x, y⁆ ^ k.succ * y) q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ ^ k.succ.succ * y) q ≠ q | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
k : ℕ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ ^ k * y) q ≠ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_pow_mul_unfix | [97, 1] | [109, 28] | rw [pow_zero, one_mul] | case H1
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ ^ 0 * y) q ≠ q | case H1
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), y q ≠ q | Please generate a tactic in lean4 to solve the state.
STATE:
case H1
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ ^ 0 * y) q ≠ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_pow_mul_unfix | [97, 1] | [109, 28] | exact hy | case H1
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), y q ≠ q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H1
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), y q ≠ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_pow_mul_unfix | [97, 1] | [109, 28] | rw [pow_one] | case H2
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ ^ 1 * y) q ≠ q | case H2
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ * y) q ≠ q | Please generate a tactic in lean4 to solve the state.
STATE:
case H2
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ ^ 1 * y) q ≠ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_pow_mul_unfix | [97, 1] | [109, 28] | exact cmtr_mul_unfix_of_unfix hy | case H2
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ * y) q ≠ q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H2
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ * y) q ≠ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_pow_mul_unfix | [97, 1] | [109, 28] | intros q h | case H3
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (⁅x, y⁆ ^ k * y) q ≠ q
_IH2✝ : ∀ (q : α), (⁅x, y⁆ ^ k.succ * y) q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ ^ k.succ.succ * y) q ≠ q | case H3
α : Type u
x y : Perm α
q✝ : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (⁅x, y⁆ ^ k * y) q ≠ q
_IH2✝ : ∀ (q : α), (⁅x, y⁆ ^ k.succ * y) q ≠ q
q : α
h : (⁅x, y⁆ ^ k.succ.succ * y) q = q
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case H3
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (⁅x, y⁆ ^ k * y) q ≠ q
_IH2✝ : ∀ (q : α), (⁅x, y⁆ ^ k.succ * y) q ≠ q
⊢ ∀ (q : α), (⁅x, y⁆ ^ k.succ.succ * y) q ≠ q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_pow_mul_unfix | [97, 1] | [109, 28] | simp_rw [pow_succ (n := k.succ), pow_succ' (n := k), mul_assoc,
cmtr_mul_eq_mul_inv_cmtr_inv, hxy, Perm.mul_apply,
← Perm.eq_inv_iff_eq (f := ⁅x, y⁆)] at h | case H3
α : Type u
x y : Perm α
q✝ : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (⁅x, y⁆ ^ k * y) q ≠ q
_IH2✝ : ∀ (q : α), (⁅x, y⁆ ^ k.succ * y) q ≠ q
q : α
h : (⁅x, y⁆ ^ k.succ.succ * y) q = q
⊢ False | case H3
α : Type u
x y : Perm α
q✝ : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (⁅x, y⁆ ^ k * y) q ≠ q
_IH2✝ : ∀ (q : α), (⁅x, y⁆ ^ k.succ * y) q ≠ q
q : α
h : (⁅x, y⁆ ^ k) (y (⁅x, y⁆⁻¹ q)) = ⁅x, y⁆⁻¹ q
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case H3
α : Type u
x y : Perm α
q✝ : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (⁅x, y⁆ ^ k * y) q ≠ q
_IH2✝ : ∀ (q : α), (⁅x, y⁆ ^ k.succ * y) q ≠ q
q : α
h : (⁅x, y⁆ ^ k.succ.succ * y) q = q
⊢ False
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_pow_mul_unfix | [97, 1] | [109, 28] | exact IH (⁅x, y⁆⁻¹ q) h | case H3
α : Type u
x y : Perm α
q✝ : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (⁅x, y⁆ ^ k * y) q ≠ q
_IH2✝ : ∀ (q : α), (⁅x, y⁆ ^ k.succ * y) q ≠ q
q : α
h : (⁅x, y⁆ ^ k) (y (⁅x, y⁆⁻¹ q)) = ⁅x, y⁆⁻¹ q
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H3
α : Type u
x y : Perm α
q✝ : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
k : ℕ
IH : ∀ (q : α), (⁅x, y⁆ ^ k * y) q ≠ q
_IH2✝ : ∀ (q : α), (⁅x, y⁆ ^ k.succ * y) q ≠ q
q : α
h : (⁅x, y⁆ ^ k) (y (⁅x, y⁆⁻¹ q)) = ⁅x, y⁆⁻¹ q
⊢ False
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_pow_inv_apply_ne_apply | [111, 1] | [114, 46] | simp_rw [Perm.inv_eq_iff_eq.not] | α : Type u
x y : Perm α
q : α
k : ℕ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ (⁅x, y⁆ ^ k)⁻¹ q ≠ y q | α : Type u
x y : Perm α
q : α
k : ℕ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ¬q = (⁅x, y⁆ ^ k) (y q) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
k : ℕ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ (⁅x, y⁆ ^ k)⁻¹ q ≠ y q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_pow_inv_apply_ne_apply | [111, 1] | [114, 46] | exact Ne.symm (cmtr_pow_mul_unfix hxy hy _) | α : Type u
x y : Perm α
q : α
k : ℕ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ¬q = (⁅x, y⁆ ^ k) (y q) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
k : ℕ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ ¬q = (⁅x, y⁆ ^ k) (y q)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_zpow_apply_ne_apply | [116, 1] | [122, 35] | cases k | α : Type u
x y : Perm α
q : α
k : ℤ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ (⁅x, y⁆ ^ k) q ≠ y q | case ofNat
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
a✝ : ℕ
⊢ (⁅x, y⁆ ^ Int.ofNat a✝) q ≠ y q
case negSucc
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
a✝ : ℕ
⊢ (⁅x, y⁆ ^ Int.negSucc a✝) q ≠ y q | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
k : ℤ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ (⁅x, y⁆ ^ k) q ≠ y q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_zpow_apply_ne_apply | [116, 1] | [122, 35] | simp only [Int.ofNat_eq_coe, zpow_natCast, ne_eq, hxy, hy, not_false_eq_true, implies_true,
cmtr_pow_apply_ne_apply] | case ofNat
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
a✝ : ℕ
⊢ (⁅x, y⁆ ^ Int.ofNat a✝) q ≠ y q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ofNat
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
a✝ : ℕ
⊢ (⁅x, y⁆ ^ Int.ofNat a✝) q ≠ y q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_zpow_apply_ne_apply | [116, 1] | [122, 35] | simp only [zpow_negSucc, ne_eq, hxy, hy, not_false_eq_true, implies_true,
cmtr_pow_inv_apply_ne_apply] | case negSucc
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
a✝ : ℕ
⊢ (⁅x, y⁆ ^ Int.negSucc a✝) q ≠ y q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case negSucc
α : Type u
x y : Perm α
q : α
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
a✝ : ℕ
⊢ (⁅x, y⁆ ^ Int.negSucc a✝) q ≠ y q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_zpow_apply_ne_apply_cmtr_pow_apply | [124, 1] | [127, 40] | rw [← sub_add_cancel j k, zpow_add, Perm.mul_apply] | α : Type u
x y : Perm α
q : α
j k : ℤ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ (⁅x, y⁆ ^ j) q ≠ y ((⁅x, y⁆ ^ k) q) | α : Type u
x y : Perm α
q : α
j k : ℤ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ (⁅x, y⁆ ^ (j - k)) ((⁅x, y⁆ ^ k) q) ≠ y ((⁅x, y⁆ ^ k) q) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
j k : ℤ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ (⁅x, y⁆ ^ j) q ≠ y ((⁅x, y⁆ ^ k) q)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/Commutator.lean | cmtr_zpow_apply_ne_apply_cmtr_pow_apply | [124, 1] | [127, 40] | exact cmtr_zpow_apply_ne_apply hxy hy | α : Type u
x y : Perm α
q : α
j k : ℤ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ (⁅x, y⁆ ^ (j - k)) ((⁅x, y⁆ ^ k) q) ≠ y ((⁅x, y⁆ ^ k) q) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
x y : Perm α
q : α
j k : ℤ
hxy : ⁅x, y⁻¹⁆ = ⁅x, y⁆
hy : ∀ (q : α), y q ≠ q
⊢ (⁅x, y⁆ ^ (j - k)) ((⁅x, y⁆ ^ k) q) ≠ y ((⁅x, y⁆ ^ k) q)
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.perm_fin_two | [6, 1] | [13, 87] | rw [Equiv.ext_iff, forall_fin_two] | π : Equiv.Perm (Fin 2)
⊢ π = if π 0 = 1 then Equiv.swap 0 1 else 1 | π : Equiv.Perm (Fin 2)
⊢ π 0 = (if π 0 = 1 then Equiv.swap 0 1 else 1) 0 ∧ π 1 = (if π 0 = 1 then Equiv.swap 0 1 else 1) 1 | Please generate a tactic in lean4 to solve the state.
STATE:
π : Equiv.Perm (Fin 2)
⊢ π = if π 0 = 1 then Equiv.swap 0 1 else 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.perm_fin_two | [6, 1] | [13, 87] | rcases (exists_fin_two.mp ⟨π 0, rfl⟩) with (h0 | h0) <;>
rcases (exists_fin_two.mp ⟨π 1, rfl⟩) with (h1 | h1) <;>
simp only [h0, ite_true, Equiv.swap_apply_left, h1, Equiv.swap_apply_right, one_eq_zero_iff, id_eq,
OfNat.ofNat_ne_one, and_false, zero_eq_one_iff, ite_false, Equiv.Perm.coe_one, and_self] <;>
exact (zero_ne_one ((EmbeddingLike.apply_eq_iff_eq _).mp (h0.trans (h1.symm)))).elim | π : Equiv.Perm (Fin 2)
⊢ π 0 = (if π 0 = 1 then Equiv.swap 0 1 else 1) 0 ∧ π 1 = (if π 0 = 1 then Equiv.swap 0 1 else 1) 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Equiv.Perm (Fin 2)
⊢ π 0 = (if π 0 = 1 then Equiv.swap 0 1 else 1) 0 ∧ π 1 = (if π 0 = 1 then Equiv.swap 0 1 else 1) 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.perm_fin_two_mul_self | [15, 1] | [19, 17] | rw [perm_fin_two π] | π : Equiv.Perm (Fin 2)
⊢ π * π = 1 | π : Equiv.Perm (Fin 2)
⊢ ((if π 0 = 1 then Equiv.swap 0 1 else 1) * if π 0 = 1 then Equiv.swap 0 1 else 1) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
π : Equiv.Perm (Fin 2)
⊢ π * π = 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.perm_fin_two_mul_self | [15, 1] | [19, 17] | split_ifs | π : Equiv.Perm (Fin 2)
⊢ ((if π 0 = 1 then Equiv.swap 0 1 else 1) * if π 0 = 1 then Equiv.swap 0 1 else 1) = 1 | case pos
π : Equiv.Perm (Fin 2)
h✝ : π 0 = 1
⊢ Equiv.swap 0 1 * Equiv.swap 0 1 = 1
case neg
π : Equiv.Perm (Fin 2)
h✝ : ¬π 0 = 1
⊢ 1 * 1 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
π : Equiv.Perm (Fin 2)
⊢ ((if π 0 = 1 then Equiv.swap 0 1 else 1) * if π 0 = 1 then Equiv.swap 0 1 else 1) = 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.perm_fin_two_mul_self | [15, 1] | [19, 17] | rw [Equiv.swap_mul_self] | case pos
π : Equiv.Perm (Fin 2)
h✝ : π 0 = 1
⊢ Equiv.swap 0 1 * Equiv.swap 0 1 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
π : Equiv.Perm (Fin 2)
h✝ : π 0 = 1
⊢ Equiv.swap 0 1 * Equiv.swap 0 1 = 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.perm_fin_two_mul_self | [15, 1] | [19, 17] | rw [mul_one] | case neg
π : Equiv.Perm (Fin 2)
h✝ : ¬π 0 = 1
⊢ 1 * 1 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
π : Equiv.Perm (Fin 2)
h✝ : ¬π 0 = 1
⊢ 1 * 1 = 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.perm_fin_two_apply_apply | [21, 1] | [22, 75] | rw [← Equiv.Perm.mul_apply, perm_fin_two_mul_self, Equiv.Perm.one_apply] | q : Fin 2
π : Equiv.Perm (Fin 2)
⊢ π (π q) = q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
q : Fin 2
π : Equiv.Perm (Fin 2)
⊢ π (π q) = q
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.perm_fin_two_of_fix_zero | [24, 1] | [26, 62] | rw [perm_fin_two π] | π : Equiv.Perm (Fin 2)
h : π 0 = 0
⊢ π = 1 | π : Equiv.Perm (Fin 2)
h : π 0 = 0
⊢ (if π 0 = 1 then Equiv.swap 0 1 else 1) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
π : Equiv.Perm (Fin 2)
h : π 0 = 0
⊢ π = 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.perm_fin_two_of_fix_zero | [24, 1] | [26, 62] | simp_rw [h, zero_eq_one_iff, OfNat.ofNat_ne_one, ite_false] | π : Equiv.Perm (Fin 2)
h : π 0 = 0
⊢ (if π 0 = 1 then Equiv.swap 0 1 else 1) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Equiv.Perm (Fin 2)
h : π 0 = 0
⊢ (if π 0 = 1 then Equiv.swap 0 1 else 1) = 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.perm_fin_two_of_fix_one | [28, 1] | [30, 92] | rw [perm_fin_two π, ← h] | π : Equiv.Perm (Fin 2)
h : π 1 = 1
⊢ π = 1 | π : Equiv.Perm (Fin 2)
h : π 1 = 1
⊢ (if π 0 = π 1 then Equiv.swap 0 (π 1) else 1) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
π : Equiv.Perm (Fin 2)
h : π 1 = 1
⊢ π = 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.perm_fin_two_of_fix_one | [28, 1] | [30, 92] | simp only [EmbeddingLike.apply_eq_iff_eq, zero_eq_one_iff, OfNat.ofNat_ne_one, ite_false] | π : Equiv.Perm (Fin 2)
h : π 1 = 1
⊢ (if π 0 = π 1 then Equiv.swap 0 (π 1) else 1) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Equiv.Perm (Fin 2)
h : π 1 = 1
⊢ (if π 0 = π 1 then Equiv.swap 0 (π 1) else 1) = 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.perm_fin_two_of_unfix_zero | [32, 1] | [34, 24] | rw [perm_fin_two π] | π : Equiv.Perm (Fin 2)
h : π 0 = 1
⊢ π = Equiv.swap 0 1 | π : Equiv.Perm (Fin 2)
h : π 0 = 1
⊢ (if π 0 = 1 then Equiv.swap 0 1 else 1) = Equiv.swap 0 1 | Please generate a tactic in lean4 to solve the state.
STATE:
π : Equiv.Perm (Fin 2)
h : π 0 = 1
⊢ π = Equiv.swap 0 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.perm_fin_two_of_unfix_zero | [32, 1] | [34, 24] | simp_rw [h, ite_true] | π : Equiv.Perm (Fin 2)
h : π 0 = 1
⊢ (if π 0 = 1 then Equiv.swap 0 1 else 1) = Equiv.swap 0 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Equiv.Perm (Fin 2)
h : π 0 = 1
⊢ (if π 0 = 1 then Equiv.swap 0 1 else 1) = Equiv.swap 0 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.perm_fin_two_of_unfix_one | [36, 1] | [38, 24] | rw [perm_fin_two π, ← perm_fin_two_apply_apply (π := π) (q := 1)] | π : Equiv.Perm (Fin 2)
h : π 1 = 0
⊢ π = Equiv.swap 0 1 | π : Equiv.Perm (Fin 2)
h : π 1 = 0
⊢ (if π 0 = π (π 1) then Equiv.swap 0 (π (π 1)) else 1) = Equiv.swap 0 (π (π 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
π : Equiv.Perm (Fin 2)
h : π 1 = 0
⊢ π = Equiv.swap 0 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.perm_fin_two_of_unfix_one | [36, 1] | [38, 24] | simp_rw [h, ite_true] | π : Equiv.Perm (Fin 2)
h : π 1 = 0
⊢ (if π 0 = π (π 1) then Equiv.swap 0 (π (π 1)) else 1) = Equiv.swap 0 (π (π 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Equiv.Perm (Fin 2)
h : π 1 = 0
⊢ (if π 0 = π (π 1) then Equiv.swap 0 (π (π 1)) else 1) = Equiv.swap 0 (π (π 1))
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.cmtr_fin_two | [40, 1] | [46, 46] | rw [perm_fin_two x, perm_fin_two y] | x y : Equiv.Perm (Fin 2)
⊢ ⁅x, y⁆ = 1 | x y : Equiv.Perm (Fin 2)
⊢ ⁅if x 0 = 1 then Equiv.swap 0 1 else 1, if y 0 = 1 then Equiv.swap 0 1 else 1⁆ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Equiv.Perm (Fin 2)
⊢ ⁅x, y⁆ = 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.cmtr_fin_two | [40, 1] | [46, 46] | by_cases h : (x 0 = 1) | x y : Equiv.Perm (Fin 2)
⊢ ⁅if x 0 = 1 then Equiv.swap 0 1 else 1, if y 0 = 1 then Equiv.swap 0 1 else 1⁆ = 1 | case pos
x y : Equiv.Perm (Fin 2)
h : x 0 = 1
⊢ ⁅if x 0 = 1 then Equiv.swap 0 1 else 1, if y 0 = 1 then Equiv.swap 0 1 else 1⁆ = 1
case neg
x y : Equiv.Perm (Fin 2)
h : ¬x 0 = 1
⊢ ⁅if x 0 = 1 then Equiv.swap 0 1 else 1, if y 0 = 1 then Equiv.swap 0 1 else 1⁆ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Equiv.Perm (Fin 2)
⊢ ⁅if x 0 = 1 then Equiv.swap 0 1 else 1, if y 0 = 1 then Equiv.swap 0 1 else 1⁆ = 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.cmtr_fin_two | [40, 1] | [46, 46] | by_cases h₂ : (y 0 = 1) | case pos
x y : Equiv.Perm (Fin 2)
h : x 0 = 1
⊢ ⁅if x 0 = 1 then Equiv.swap 0 1 else 1, if y 0 = 1 then Equiv.swap 0 1 else 1⁆ = 1 | case pos
x y : Equiv.Perm (Fin 2)
h : x 0 = 1
h₂ : y 0 = 1
⊢ ⁅if x 0 = 1 then Equiv.swap 0 1 else 1, if y 0 = 1 then Equiv.swap 0 1 else 1⁆ = 1
case neg
x y : Equiv.Perm (Fin 2)
h : x 0 = 1
h₂ : ¬y 0 = 1
⊢ ⁅if x 0 = 1 then Equiv.swap 0 1 else 1, if y 0 = 1 then Equiv.swap 0 1 else 1⁆ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x y : Equiv.Perm (Fin 2)
h : x 0 = 1
⊢ ⁅if x 0 = 1 then Equiv.swap 0 1 else 1, if y 0 = 1 then Equiv.swap 0 1 else 1⁆ = 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.cmtr_fin_two | [40, 1] | [46, 46] | rw [if_pos h, if_pos h₂, commutatorElement_self] | case pos
x y : Equiv.Perm (Fin 2)
h : x 0 = 1
h₂ : y 0 = 1
⊢ ⁅if x 0 = 1 then Equiv.swap 0 1 else 1, if y 0 = 1 then Equiv.swap 0 1 else 1⁆ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x y : Equiv.Perm (Fin 2)
h : x 0 = 1
h₂ : y 0 = 1
⊢ ⁅if x 0 = 1 then Equiv.swap 0 1 else 1, if y 0 = 1 then Equiv.swap 0 1 else 1⁆ = 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.cmtr_fin_two | [40, 1] | [46, 46] | rw [if_neg h₂, commutatorElement_one_right] | case neg
x y : Equiv.Perm (Fin 2)
h : x 0 = 1
h₂ : ¬y 0 = 1
⊢ ⁅if x 0 = 1 then Equiv.swap 0 1 else 1, if y 0 = 1 then Equiv.swap 0 1 else 1⁆ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y : Equiv.Perm (Fin 2)
h : x 0 = 1
h₂ : ¬y 0 = 1
⊢ ⁅if x 0 = 1 then Equiv.swap 0 1 else 1, if y 0 = 1 then Equiv.swap 0 1 else 1⁆ = 1
TACTIC:
|
https://github.com/linesthatinterlace/controlbits.git | 4a0d924f7bd9e6dcc6719ef05314fdfd702c6a01 | Controlbits/PermFintwo.lean | Fin.cmtr_fin_two | [40, 1] | [46, 46] | rw [if_neg h, commutatorElement_one_left] | case neg
x y : Equiv.Perm (Fin 2)
h : ¬x 0 = 1
⊢ ⁅if x 0 = 1 then Equiv.swap 0 1 else 1, if y 0 = 1 then Equiv.swap 0 1 else 1⁆ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y : Equiv.Perm (Fin 2)
h : ¬x 0 = 1
⊢ ⁅if x 0 = 1 then Equiv.swap 0 1 else 1, if y 0 = 1 then Equiv.swap 0 1 else 1⁆ = 1
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Exercises1.lean | Category.imageQuotientFunctor_faithful | [29, 1] | [36, 10] | intro A B f g | α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
⊢ Faithful (imageQuotientFunctor F) | α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
A B : ImageQuotient F
f g : A ⟶ B
⊢ Functor.map (imageQuotientFunctor F) f = Functor.map (imageQuotientFunctor F) g → f = g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
⊢ Faithful (imageQuotientFunctor F)
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Exercises1.lean | Category.imageQuotientFunctor_faithful | [29, 1] | [36, 10] | refine HomQuotient.inductionOn f ?_ | α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
A B : ImageQuotient F
f g : A ⟶ B
⊢ Functor.map (imageQuotientFunctor F) f = Functor.map (imageQuotientFunctor F) g → f = g | α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
A B : ImageQuotient F
f g : A ⟶ B
⊢ ∀ (f : A.unquot ⟶ B.unquot),
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) f) =
Functor.map (imageQuotientFunctor F) g →
HomQuotient.quotHom (imageCongruence F) f = g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
A B : ImageQuotient F
f g : A ⟶ B
⊢ Functor.map (imageQuotientFunctor F) f = Functor.map (imageQuotientFunctor F) g → f = g
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Exercises1.lean | Category.imageQuotientFunctor_faithful | [29, 1] | [36, 10] | refine HomQuotient.inductionOn g ?_ | α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
A B : ImageQuotient F
f g : A ⟶ B
⊢ ∀ (f : A.unquot ⟶ B.unquot),
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) f) =
Functor.map (imageQuotientFunctor F) g →
HomQuotient.quotHom (imageCongruence F) f = g | α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
A B : ImageQuotient F
f g : A ⟶ B
⊢ ∀ (f f_1 : A.unquot ⟶ B.unquot),
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) f_1) =
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) f) →
HomQuotient.quotHom (imageCongruence F) f_1 = HomQuotient.quotHom (imageCongruence F) f | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
A B : ImageQuotient F
f g : A ⟶ B
⊢ ∀ (f : A.unquot ⟶ B.unquot),
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) f) =
Functor.map (imageQuotientFunctor F) g →
HomQuotient.quotHom (imageCongruence F) f = g
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Exercises1.lean | Category.imageQuotientFunctor_faithful | [29, 1] | [36, 10] | intro f g h | α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
A B : ImageQuotient F
f g : A ⟶ B
⊢ ∀ (f f_1 : A.unquot ⟶ B.unquot),
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) f_1) =
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) f) →
HomQuotient.quotHom (imageCongruence F) f_1 = HomQuotient.quotHom (imageCongruence F) f | α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
A B : ImageQuotient F
f✝ g✝ : A ⟶ B
f g : A.unquot ⟶ B.unquot
h :
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) g) =
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) f)
⊢ HomQuotient.quotHom (imageCongruence F) g = HomQuotient.quotHom (imageCongruence F) f | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
A B : ImageQuotient F
f g : A ⟶ B
⊢ ∀ (f f_1 : A.unquot ⟶ B.unquot),
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) f_1) =
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) f) →
HomQuotient.quotHom (imageCongruence F) f_1 = HomQuotient.quotHom (imageCongruence F) f
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Exercises1.lean | Category.imageQuotientFunctor_faithful | [29, 1] | [36, 10] | rw [HomQuotient.quotHom_eq_iff] | α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
A B : ImageQuotient F
f✝ g✝ : A ⟶ B
f g : A.unquot ⟶ B.unquot
h :
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) g) =
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) f)
⊢ HomQuotient.quotHom (imageCongruence F) g = HomQuotient.quotHom (imageCongruence F) f | α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
A B : ImageQuotient F
f✝ g✝ : A ⟶ B
f g : A.unquot ⟶ B.unquot
h :
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) g) =
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) f)
⊢ Congruence.rel (imageCongruence F) A.unquot B.unquot g f | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
A B : ImageQuotient F
f✝ g✝ : A ⟶ B
f g : A.unquot ⟶ B.unquot
h :
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) g) =
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) f)
⊢ HomQuotient.quotHom (imageCongruence F) g = HomQuotient.quotHom (imageCongruence F) f
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Exercises1.lean | Category.imageQuotientFunctor_faithful | [29, 1] | [36, 10] | exact h | α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
A B : ImageQuotient F
f✝ g✝ : A ⟶ B
f g : A.unquot ⟶ B.unquot
h :
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) g) =
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) f)
⊢ Congruence.rel (imageCongruence F) A.unquot B.unquot g f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type ?u.1282
δ : Type ?u.1285
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
A B : ImageQuotient F
f✝ g✝ : A ⟶ B
f g : A.unquot ⟶ B.unquot
h :
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) g) =
Functor.map (imageQuotientFunctor F) (HomQuotient.quotHom (imageCongruence F) f)
⊢ Congruence.rel (imageCongruence F) A.unquot B.unquot g f
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.whisker_eq | [36, 1] | [38, 12] | rw [w] | α : Type u_2
β : Type ?u.23211
γ : Type ?u.23214
δ : Type ?u.23217
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
h : B ⟶ C
f g : A ⟶ B
w : f = g
⊢ h ∘ f = h ∘ g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
β : Type ?u.23211
γ : Type ?u.23214
δ : Type ?u.23217
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
h : B ⟶ C
f g : A ⟶ B
w : f = g
⊢ h ∘ f = h ∘ g
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.eq_whisker | [40, 1] | [42, 12] | rw [w] | α : Type u_2
β : Type ?u.23648
γ : Type ?u.23651
δ : Type ?u.23654
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
f g : B ⟶ C
w : f = g
h : A ⟶ B
⊢ f ∘ h = g ∘ h | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_2
β : Type ?u.23648
γ : Type ?u.23651
δ : Type ?u.23654
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
f g : B ⟶ C
w : f = g
h : A ⟶ B
⊢ f ∘ h = g ∘ h
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.Functor.id_comp | [154, 1] | [155, 11] | aesop | α : Type u_1
β : Type u_2
γ : Type ?u.59997
δ : Type ?u.60000
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
⊢ 𝟭 β ◌ F = F | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type ?u.59997
δ : Type ?u.60000
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
⊢ 𝟭 β ◌ F = F
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.Functor.comp_id | [158, 1] | [159, 11] | aesop | α : Type u_1
β : Type u_2
γ : Type ?u.60194
δ : Type ?u.60197
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
⊢ F ◌ 𝟭 α = F | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type ?u.60194
δ : Type ?u.60197
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F : α ⥤ β
⊢ F ◌ 𝟭 α = F
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.Functor.comp_assoc | [162, 1] | [164, 11] | aesop | α : Type u_7
β : Type u_5
γ : Type u_1
δ : Type u_2
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
H : γ ⥤ δ
G : β ⥤ γ
F : α ⥤ β
⊢ (H ◌ G) ◌ F = H ◌ G ◌ F | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_7
β : Type u_5
γ : Type u_1
δ : Type u_2
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
H : γ ⥤ δ
G : β ⥤ γ
F : α ⥤ β
⊢ (H ◌ G) ◌ F = H ◌ G ◌ F
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.HomQuotient.comp_rel | [255, 1] | [263, 57] | have h₁ := r.whisker_rel f₁ g₁ g₂ hg | α : Type u_1
β : Type ?u.68744
γ : Type ?u.68747
δ : Type ?u.68750
A✝ B✝ C✝ D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B C : HomQuotient r
f₁ : B.unquot ⟶ C.unquot
g₁ : A.unquot ⟶ B.unquot
f₂ : B.unquot ⟶ C.unquot
g₂ : A.unquot ⟶ B.unquot
hf : Congruence.rel r B.unquot C.unquot f₁ f₂
hg : Congruence.rel r A.unquot B.unquot g₁ g₂
⊢ Quotient.mk (Congruence.setoid r A.unquot C.unquot) (f₁ ∘ g₁) =
Quotient.mk (Congruence.setoid r A.unquot C.unquot) (f₂ ∘ g₂) | α : Type u_1
β : Type ?u.68744
γ : Type ?u.68747
δ : Type ?u.68750
A✝ B✝ C✝ D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B C : HomQuotient r
f₁ : B.unquot ⟶ C.unquot
g₁ : A.unquot ⟶ B.unquot
f₂ : B.unquot ⟶ C.unquot
g₂ : A.unquot ⟶ B.unquot
hf : Congruence.rel r B.unquot C.unquot f₁ f₂
hg : Congruence.rel r A.unquot B.unquot g₁ g₂
h₁ : Congruence.rel r A.unquot C.unquot (f₁ ∘ g₁) (f₁ ∘ g₂)
⊢ Quotient.mk (Congruence.setoid r A.unquot C.unquot) (f₁ ∘ g₁) =
Quotient.mk (Congruence.setoid r A.unquot C.unquot) (f₂ ∘ g₂) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type ?u.68744
γ : Type ?u.68747
δ : Type ?u.68750
A✝ B✝ C✝ D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B C : HomQuotient r
f₁ : B.unquot ⟶ C.unquot
g₁ : A.unquot ⟶ B.unquot
f₂ : B.unquot ⟶ C.unquot
g₂ : A.unquot ⟶ B.unquot
hf : Congruence.rel r B.unquot C.unquot f₁ f₂
hg : Congruence.rel r A.unquot B.unquot g₁ g₂
⊢ Quotient.mk (Congruence.setoid r A.unquot C.unquot) (f₁ ∘ g₁) =
Quotient.mk (Congruence.setoid r A.unquot C.unquot) (f₂ ∘ g₂)
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.HomQuotient.comp_rel | [255, 1] | [263, 57] | have h₂ := r.rel_whisker f₁ f₂ g₂ hf | α : Type u_1
β : Type ?u.68744
γ : Type ?u.68747
δ : Type ?u.68750
A✝ B✝ C✝ D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B C : HomQuotient r
f₁ : B.unquot ⟶ C.unquot
g₁ : A.unquot ⟶ B.unquot
f₂ : B.unquot ⟶ C.unquot
g₂ : A.unquot ⟶ B.unquot
hf : Congruence.rel r B.unquot C.unquot f₁ f₂
hg : Congruence.rel r A.unquot B.unquot g₁ g₂
h₁ : Congruence.rel r A.unquot C.unquot (f₁ ∘ g₁) (f₁ ∘ g₂)
⊢ Quotient.mk (Congruence.setoid r A.unquot C.unquot) (f₁ ∘ g₁) =
Quotient.mk (Congruence.setoid r A.unquot C.unquot) (f₂ ∘ g₂) | α : Type u_1
β : Type ?u.68744
γ : Type ?u.68747
δ : Type ?u.68750
A✝ B✝ C✝ D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B C : HomQuotient r
f₁ : B.unquot ⟶ C.unquot
g₁ : A.unquot ⟶ B.unquot
f₂ : B.unquot ⟶ C.unquot
g₂ : A.unquot ⟶ B.unquot
hf : Congruence.rel r B.unquot C.unquot f₁ f₂
hg : Congruence.rel r A.unquot B.unquot g₁ g₂
h₁ : Congruence.rel r A.unquot C.unquot (f₁ ∘ g₁) (f₁ ∘ g₂)
h₂ : Congruence.rel r A.unquot C.unquot (f₁ ∘ g₂) (f₂ ∘ g₂)
⊢ Quotient.mk (Congruence.setoid r A.unquot C.unquot) (f₁ ∘ g₁) =
Quotient.mk (Congruence.setoid r A.unquot C.unquot) (f₂ ∘ g₂) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type ?u.68744
γ : Type ?u.68747
δ : Type ?u.68750
A✝ B✝ C✝ D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B C : HomQuotient r
f₁ : B.unquot ⟶ C.unquot
g₁ : A.unquot ⟶ B.unquot
f₂ : B.unquot ⟶ C.unquot
g₂ : A.unquot ⟶ B.unquot
hf : Congruence.rel r B.unquot C.unquot f₁ f₂
hg : Congruence.rel r A.unquot B.unquot g₁ g₂
h₁ : Congruence.rel r A.unquot C.unquot (f₁ ∘ g₁) (f₁ ∘ g₂)
⊢ Quotient.mk (Congruence.setoid r A.unquot C.unquot) (f₁ ∘ g₁) =
Quotient.mk (Congruence.setoid r A.unquot C.unquot) (f₂ ∘ g₂)
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.HomQuotient.comp_rel | [255, 1] | [263, 57] | exact Quotient.sound ((r.equivalence _ _).trans h₁ h₂) | α : Type u_1
β : Type ?u.68744
γ : Type ?u.68747
δ : Type ?u.68750
A✝ B✝ C✝ D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B C : HomQuotient r
f₁ : B.unquot ⟶ C.unquot
g₁ : A.unquot ⟶ B.unquot
f₂ : B.unquot ⟶ C.unquot
g₂ : A.unquot ⟶ B.unquot
hf : Congruence.rel r B.unquot C.unquot f₁ f₂
hg : Congruence.rel r A.unquot B.unquot g₁ g₂
h₁ : Congruence.rel r A.unquot C.unquot (f₁ ∘ g₁) (f₁ ∘ g₂)
h₂ : Congruence.rel r A.unquot C.unquot (f₁ ∘ g₂) (f₂ ∘ g₂)
⊢ Quotient.mk (Congruence.setoid r A.unquot C.unquot) (f₁ ∘ g₁) =
Quotient.mk (Congruence.setoid r A.unquot C.unquot) (f₂ ∘ g₂) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type ?u.68744
γ : Type ?u.68747
δ : Type ?u.68750
A✝ B✝ C✝ D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B C : HomQuotient r
f₁ : B.unquot ⟶ C.unquot
g₁ : A.unquot ⟶ B.unquot
f₂ : B.unquot ⟶ C.unquot
g₂ : A.unquot ⟶ B.unquot
hf : Congruence.rel r B.unquot C.unquot f₁ f₂
hg : Congruence.rel r A.unquot B.unquot g₁ g₂
h₁ : Congruence.rel r A.unquot C.unquot (f₁ ∘ g₁) (f₁ ∘ g₂)
h₂ : Congruence.rel r A.unquot C.unquot (f₁ ∘ g₂) (f₂ ∘ g₂)
⊢ Quotient.mk (Congruence.setoid r A.unquot C.unquot) (f₁ ∘ g₁) =
Quotient.mk (Congruence.setoid r A.unquot C.unquot) (f₂ ∘ g₂)
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.HomQuotient.quotHom_eq_iff | [291, 1] | [298, 27] | constructor | α : Type u_1
β : Type ?u.72067
γ : Type ?u.72070
δ : Type ?u.72073
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
f g : A ⟶ B
⊢ quotHom r f = quotHom r g ↔ Congruence.rel r A B f g | case mp
α : Type u_1
β : Type ?u.72067
γ : Type ?u.72070
δ : Type ?u.72073
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
f g : A ⟶ B
⊢ quotHom r f = quotHom r g → Congruence.rel r A B f g
case mpr
α : Type u_1
β : Type ?u.72067
γ : Type ?u.72070
δ : Type ?u.72073
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
f g : A ⟶ B
⊢ Congruence.rel r A B f g → quotHom r f = quotHom r g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type ?u.72067
γ : Type ?u.72070
δ : Type ?u.72073
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
f g : A ⟶ B
⊢ quotHom r f = quotHom r g ↔ Congruence.rel r A B f g
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.HomQuotient.quotHom_eq_iff | [291, 1] | [298, 27] | intro h | case mp
α : Type u_1
β : Type ?u.72067
γ : Type ?u.72070
δ : Type ?u.72073
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
f g : A ⟶ B
⊢ quotHom r f = quotHom r g → Congruence.rel r A B f g | case mp
α : Type u_1
β : Type ?u.72067
γ : Type ?u.72070
δ : Type ?u.72073
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
f g : A ⟶ B
h : quotHom r f = quotHom r g
⊢ Congruence.rel r A B f g | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
β : Type ?u.72067
γ : Type ?u.72070
δ : Type ?u.72073
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
f g : A ⟶ B
⊢ quotHom r f = quotHom r g → Congruence.rel r A B f g
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.HomQuotient.quotHom_eq_iff | [291, 1] | [298, 27] | exact Quotient.exact h | case mp
α : Type u_1
β : Type ?u.72067
γ : Type ?u.72070
δ : Type ?u.72073
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
f g : A ⟶ B
h : quotHom r f = quotHom r g
⊢ Congruence.rel r A B f g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
β : Type ?u.72067
γ : Type ?u.72070
δ : Type ?u.72073
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
f g : A ⟶ B
h : quotHom r f = quotHom r g
⊢ Congruence.rel r A B f g
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.HomQuotient.quotHom_eq_iff | [291, 1] | [298, 27] | intro h | case mpr
α : Type u_1
β : Type ?u.72067
γ : Type ?u.72070
δ : Type ?u.72073
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
f g : A ⟶ B
⊢ Congruence.rel r A B f g → quotHom r f = quotHom r g | case mpr
α : Type u_1
β : Type ?u.72067
γ : Type ?u.72070
δ : Type ?u.72073
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
f g : A ⟶ B
h : Congruence.rel r A B f g
⊢ quotHom r f = quotHom r g | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
β : Type ?u.72067
γ : Type ?u.72070
δ : Type ?u.72073
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
f g : A ⟶ B
⊢ Congruence.rel r A B f g → quotHom r f = quotHom r g
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.HomQuotient.quotHom_eq_iff | [291, 1] | [298, 27] | exact Quotient.sound h | case mpr
α : Type u_1
β : Type ?u.72067
γ : Type ?u.72070
δ : Type ?u.72073
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
f g : A ⟶ B
h : Congruence.rel r A B f g
⊢ quotHom r f = quotHom r g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
β : Type ?u.72067
γ : Type ?u.72070
δ : Type ?u.72073
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
f g : A ⟶ B
h : Congruence.rel r A B f g
⊢ quotHom r f = quotHom r g
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.NatTrans.id_comp | [355, 1] | [357, 15] | ext | α : Type u_1
β : Type u_2
γ : Type ?u.76228
δ : Type ?u.76231
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F G : α ⥤ β
η : NatTrans F G
⊢ comp (id G) η = η | case app.h
α : Type u_1
β : Type u_2
γ : Type ?u.76228
δ : Type ?u.76231
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F G : α ⥤ β
η : NatTrans F G
x✝ : α
⊢ app (comp (id G) η) x✝ = app η x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type ?u.76228
δ : Type ?u.76231
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F G : α ⥤ β
η : NatTrans F G
⊢ comp (id G) η = η
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.NatTrans.id_comp | [355, 1] | [357, 15] | simp | case app.h
α : Type u_1
β : Type u_2
γ : Type ?u.76228
δ : Type ?u.76231
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F G : α ⥤ β
η : NatTrans F G
x✝ : α
⊢ app (comp (id G) η) x✝ = app η x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case app.h
α : Type u_1
β : Type u_2
γ : Type ?u.76228
δ : Type ?u.76231
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F G : α ⥤ β
η : NatTrans F G
x✝ : α
⊢ app (comp (id G) η) x✝ = app η x✝
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.NatTrans.comp_id | [359, 1] | [361, 15] | ext | α : Type u_1
β : Type u_2
γ : Type ?u.76679
δ : Type ?u.76682
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F G : α ⥤ β
η : NatTrans F G
⊢ comp η (id F) = η | case app.h
α : Type u_1
β : Type u_2
γ : Type ?u.76679
δ : Type ?u.76682
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F G : α ⥤ β
η : NatTrans F G
x✝ : α
⊢ app (comp η (id F)) x✝ = app η x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type ?u.76679
δ : Type ?u.76682
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F G : α ⥤ β
η : NatTrans F G
⊢ comp η (id F) = η
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.NatTrans.comp_id | [359, 1] | [361, 15] | simp | case app.h
α : Type u_1
β : Type u_2
γ : Type ?u.76679
δ : Type ?u.76682
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F G : α ⥤ β
η : NatTrans F G
x✝ : α
⊢ app (comp η (id F)) x✝ = app η x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case app.h
α : Type u_1
β : Type u_2
γ : Type ?u.76679
δ : Type ?u.76682
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F G : α ⥤ β
η : NatTrans F G
x✝ : α
⊢ app (comp η (id F)) x✝ = app η x✝
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.NatTrans.comp_assoc | [363, 1] | [366, 15] | ext | α : Type u_1
β : Type u_2
γ : Type ?u.77130
δ : Type ?u.77133
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F G H K : α ⥤ β
η₁ : NatTrans H K
η₂ : NatTrans G H
η₃ : NatTrans F G
⊢ comp (comp η₁ η₂) η₃ = comp η₁ (comp η₂ η₃) | case app.h
α : Type u_1
β : Type u_2
γ : Type ?u.77130
δ : Type ?u.77133
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F G H K : α ⥤ β
η₁ : NatTrans H K
η₂ : NatTrans G H
η₃ : NatTrans F G
x✝ : α
⊢ app (comp (comp η₁ η₂) η₃) x✝ = app (comp η₁ (comp η₂ η₃)) x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type ?u.77130
δ : Type ?u.77133
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F G H K : α ⥤ β
η₁ : NatTrans H K
η₂ : NatTrans G H
η₃ : NatTrans F G
⊢ comp (comp η₁ η₂) η₃ = comp η₁ (comp η₂ η₃)
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.NatTrans.comp_assoc | [363, 1] | [366, 15] | simp | case app.h
α : Type u_1
β : Type u_2
γ : Type ?u.77130
δ : Type ?u.77133
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F G H K : α ⥤ β
η₁ : NatTrans H K
η₂ : NatTrans G H
η₃ : NatTrans F G
x✝ : α
⊢ app (comp (comp η₁ η₂) η₃) x✝ = app (comp η₁ (comp η₂ η₃)) x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case app.h
α : Type u_1
β : Type u_2
γ : Type ?u.77130
δ : Type ?u.77133
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
F G H K : α ⥤ β
η₁ : NatTrans H K
η₂ : NatTrans G H
η₃ : NatTrans F G
x✝ : α
⊢ app (comp (comp η₁ η₂) η₃) x✝ = app (comp η₁ (comp η₂ η₃)) x✝
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.HomQuotient.quotient_full | [434, 1] | [439, 17] | intro A B g | α : Type u_1
β : Type ?u.92190
γ : Type ?u.92193
δ : Type ?u.92196
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
⊢ Full (quotient r) | α : Type u_1
β : Type ?u.92190
γ : Type ?u.92193
δ : Type ?u.92196
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
g : Functor.obj (quotient r) A ⟶ Functor.obj (quotient r) B
⊢ ∃ f, Functor.map (quotient r) f = g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type ?u.92190
γ : Type ?u.92193
δ : Type ?u.92196
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
⊢ Full (quotient r)
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.HomQuotient.quotient_full | [434, 1] | [439, 17] | refine Quotient.inductionOn g ?_ | α : Type u_1
β : Type ?u.92190
γ : Type ?u.92193
δ : Type ?u.92196
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
g : Functor.obj (quotient r) A ⟶ Functor.obj (quotient r) B
⊢ ∃ f, Functor.map (quotient r) f = g | α : Type u_1
β : Type ?u.92190
γ : Type ?u.92193
δ : Type ?u.92196
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
g : Functor.obj (quotient r) A ⟶ Functor.obj (quotient r) B
⊢ ∀ (a : (Functor.obj (quotient r) A).unquot ⟶ (Functor.obj (quotient r) B).unquot),
∃ f,
Functor.map (quotient r) f =
Quotient.mk (Congruence.setoid r (Functor.obj (quotient r) A).unquot (Functor.obj (quotient r) B).unquot) a | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type ?u.92190
γ : Type ?u.92193
δ : Type ?u.92196
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
g : Functor.obj (quotient r) A ⟶ Functor.obj (quotient r) B
⊢ ∃ f, Functor.map (quotient r) f = g
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.HomQuotient.quotient_full | [434, 1] | [439, 17] | intro g | α : Type u_1
β : Type ?u.92190
γ : Type ?u.92193
δ : Type ?u.92196
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
g : Functor.obj (quotient r) A ⟶ Functor.obj (quotient r) B
⊢ ∀ (a : (Functor.obj (quotient r) A).unquot ⟶ (Functor.obj (quotient r) B).unquot),
∃ f,
Functor.map (quotient r) f =
Quotient.mk (Congruence.setoid r (Functor.obj (quotient r) A).unquot (Functor.obj (quotient r) B).unquot) a | α : Type u_1
β : Type ?u.92190
γ : Type ?u.92193
δ : Type ?u.92196
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
g✝ : Functor.obj (quotient r) A ⟶ Functor.obj (quotient r) B
g : (Functor.obj (quotient r) A).unquot ⟶ (Functor.obj (quotient r) B).unquot
⊢ ∃ f,
Functor.map (quotient r) f =
Quotient.mk (Congruence.setoid r (Functor.obj (quotient r) A).unquot (Functor.obj (quotient r) B).unquot) g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type ?u.92190
γ : Type ?u.92193
δ : Type ?u.92196
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
g : Functor.obj (quotient r) A ⟶ Functor.obj (quotient r) B
⊢ ∀ (a : (Functor.obj (quotient r) A).unquot ⟶ (Functor.obj (quotient r) B).unquot),
∃ f,
Functor.map (quotient r) f =
Quotient.mk (Congruence.setoid r (Functor.obj (quotient r) A).unquot (Functor.obj (quotient r) B).unquot) a
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.HomQuotient.quotient_full | [434, 1] | [439, 17] | exact ⟨g, rfl⟩ | α : Type u_1
β : Type ?u.92190
γ : Type ?u.92193
δ : Type ?u.92196
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
g✝ : Functor.obj (quotient r) A ⟶ Functor.obj (quotient r) B
g : (Functor.obj (quotient r) A).unquot ⟶ (Functor.obj (quotient r) B).unquot
⊢ ∃ f,
Functor.map (quotient r) f =
Quotient.mk (Congruence.setoid r (Functor.obj (quotient r) A).unquot (Functor.obj (quotient r) B).unquot) g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type ?u.92190
γ : Type ?u.92193
δ : Type ?u.92196
A✝ B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
A B : α
g✝ : Functor.obj (quotient r) A ⟶ Functor.obj (quotient r) B
g : (Functor.obj (quotient r) A).unquot ⟶ (Functor.obj (quotient r) B).unquot
⊢ ∃ f,
Functor.map (quotient r) f =
Quotient.mk (Congruence.setoid r (Functor.obj (quotient r) A).unquot (Functor.obj (quotient r) B).unquot) g
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.HomQuotient.quotient_essSurjective | [441, 1] | [444, 33] | intro B | α : Type u_1
β : Type ?u.92381
γ : Type ?u.92384
δ : Type ?u.92387
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
⊢ EssSurjective (quotient r) | α : Type u_1
β : Type ?u.92381
γ : Type ?u.92384
δ : Type ?u.92387
A B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
B : HomQuotient r
⊢ Nonempty ((A : α) ×' (Functor.obj (quotient r) A ≃ B)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type ?u.92381
γ : Type ?u.92384
δ : Type ?u.92387
A B C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
⊢ EssSurjective (quotient r)
TACTIC:
|
https://github.com/zeramorphic/category-theory.git | 6fd505ce3a3f8ed01594999fdac36bf0d1939c7c | CategoryTheory/Basic.lean | Category.HomQuotient.quotient_essSurjective | [441, 1] | [444, 33] | exact ⟨⟨B.unquot, Iso.refl _⟩⟩ | α : Type u_1
β : Type ?u.92381
γ : Type ?u.92384
δ : Type ?u.92387
A B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
B : HomQuotient r
⊢ Nonempty ((A : α) ×' (Functor.obj (quotient r) A ≃ B)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type ?u.92381
γ : Type ?u.92384
δ : Type ?u.92387
A B✝ C D : α
inst✝³ : Category α
inst✝² : Category β
inst✝¹ : Category γ
inst✝ : Category δ
r : Congruence α
B : HomQuotient r
⊢ Nonempty ((A : α) ×' (Functor.obj (quotient r) A ≃ B))
TACTIC:
|
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