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/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sdiff_late | [32, 1] | [38, 17] | exact nAB.2 nB | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
nr : n ∈ Finset.range m
nB : n ∈ B
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
nr : n ∈ Finset.range m
nB : n ∈ B
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | partial_geometric_bound | [41, 1] | [44, 57] | intro n _ | a : ℝ
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
⊢ ∀ n ∉ N, 0 ≤ a ^ n | a : ℝ
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
n : ℕ
a✝ : n ∉ N
⊢ 0 ≤ a ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
⊢ ∀ n ∉ N, 0 ≤ a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | partial_geometric_bound | [41, 1] | [44, 57] | bound | a : ℝ
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
n : ℕ
a✝ : n ∉ N
⊢ 0 ≤ a ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
n : ℕ
a✝ : n ∉ N
⊢ 0 ≤ a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | partial_scaled_geometric_bound | [46, 1] | [49, 42] | rw [←Finset.mul_sum] | a : ℝ
c : ℝ≥0
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
⊢ (N.sum fun n => ↑c * a ^ n) ≤ ↑c * (1 - a)⁻¹ | a : ℝ
c : ℝ≥0
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
⊢ (↑c * N.sum fun i => a ^ i) ≤ ↑c * (1 - a)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
c : ℝ≥0
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
⊢ (N.sum fun n => ↑c * a ^ n) ≤ ↑c * (1 - a)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | partial_scaled_geometric_bound | [46, 1] | [49, 42] | bound [partial_geometric_bound N a0 a1] | a : ℝ
c : ℝ≥0
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
⊢ (↑c * N.sum fun i => a ^ i) ≤ ↑c * (1 - a)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
c : ℝ≥0
N : Finset ℕ
a0 : 0 ≤ a
a1 : a < 1
⊢ (↑c * N.sum fun i => a ^ i) ≤ ↑c * (1 - a)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | set Ns := Finset.image (fun n ↦ n - m) N | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
⊢ N.sum f = (Finset.image (fun n => n - m) N).sum fun n => f (n + m) | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
⊢ N.sum f = Ns.sum fun n => f (n + m) | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
⊢ N.sum f = (Finset.image (fun n => n - m) N).sum fun n => f (n + m)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | rw [NNs] | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
⊢ N.sum f = Ns.sum fun n => f (n + m) | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
⊢ (Finset.image (fun n => n + m) Ns).sum f = Ns.sum fun n => f (n + m) | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
⊢ N.sum f = Ns.sum fun n => f (n + m)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | apply Finset.sum_image | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
⊢ (Finset.image (fun n => n + m) Ns).sum f = Ns.sum fun n => f (n + m) | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
⊢ ∀ x ∈ Ns, ∀ y ∈ Ns, x + m = y + m → x = y | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
⊢ (Finset.image (fun n => n + m) Ns).sum f = Ns.sum fun n => f (n + m)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | intro a _ b _ | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
⊢ ∀ x ∈ Ns, ∀ y ∈ Ns, x + m = y + m → x = y | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
a : ℕ
a✝¹ : a ∈ Ns
b : ℕ
a✝ : b ∈ Ns
⊢ a + m = b + m → a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
⊢ ∀ x ∈ Ns, ∀ y ∈ Ns, x + m = y + m → x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | exact Nat.add_right_cancel | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
a : ℕ
a✝¹ : a ∈ Ns
b : ℕ
a✝ : b ∈ Ns
⊢ a + m = b + m → a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
NNs : N = Finset.image (fun n => n + m) Ns
a : ℕ
a✝¹ : a ∈ Ns
b : ℕ
a✝ : b ∈ Ns
⊢ a + m = b + m → a = b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | apply Finset.ext | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
⊢ N = Finset.image (fun n => n + m) Ns | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
⊢ ∀ (a : ℕ), a ∈ N ↔ a ∈ Finset.image (fun n => n + m) Ns | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
⊢ N = Finset.image (fun n => n + m) Ns
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | intro k | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
⊢ ∀ (a : ℕ), a ∈ N ↔ a ∈ Finset.image (fun n => n + m) Ns | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ k ∈ Finset.image (fun n => n + m) Ns | Please generate a tactic in lean4 to solve the state.
STATE:
case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
⊢ ∀ (a : ℕ), a ∈ N ↔ a ∈ Finset.image (fun n => n + m) Ns
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | rw [Finset.image_image, Finset.mem_image] | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ k ∈ Finset.image (fun n => n + m) Ns | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ ∃ a ∈ N, ((fun n => n + m) ∘ fun n => n - m) a = k | Please generate a tactic in lean4 to solve the state.
STATE:
case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ k ∈ Finset.image (fun n => n + m) Ns
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | simp | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ ∃ a ∈ N, ((fun n => n + m) ∘ fun n => n - m) a = k | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ ∃ a ∈ N, a - m + m = k | Please generate a tactic in lean4 to solve the state.
STATE:
case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ ∃ a ∈ N, ((fun n => n + m) ∘ fun n => n - m) a = k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | apply Iff.intro | case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ ∃ a ∈ N, a - m + m = k | case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N → ∃ a ∈ N, a - m + m = k
case a.mpr
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ (∃ a ∈ N, a - m + m = k) → k ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case a
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N ↔ ∃ a ∈ N, a - m + m = k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | intro kN | case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N → ∃ a ∈ N, a - m + m = k | case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ ∃ a ∈ N, a - m + m = k | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ k ∈ N → ∃ a ∈ N, a - m + m = k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | exists k | case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ ∃ a ∈ N, a - m + m = k | case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k ∈ N ∧ k - m + m = k | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ ∃ a ∈ N, a - m + m = k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | apply And.intro | case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k ∈ N ∧ k - m + m = k | case a.mp.left
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k ∈ N
case a.mp.right
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k - m + m = k | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k ∈ N ∧ k - m + m = k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | assumption | case a.mp.left
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k ∈ N
case a.mp.right
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k - m + m = k | case a.mp.right
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k - m + m = k | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp.left
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k ∈ N
case a.mp.right
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k - m + m = k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | exact Nat.sub_add_cancel (h k kN) | case a.mp.right
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k - m + m = k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp.right
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
kN : k ∈ N
⊢ k - m + m = k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | intro ha | case a.mpr
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ (∃ a ∈ N, a - m + m = k) → k ∈ N | case a.mpr
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
ha : ∃ a ∈ N, a - m + m = k
⊢ k ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
⊢ (∃ a ∈ N, a - m + m = k) → k ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | rcases ha with ⟨a, aN, ak⟩ | case a.mpr
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
ha : ∃ a ∈ N, a - m + m = k
⊢ k ∈ N | case a.mpr.intro.intro
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a - m + m = k
⊢ k ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k : ℕ
ha : ∃ a ∈ N, a - m + m = k
⊢ k ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | rw [Nat.sub_add_cancel (h a aN)] at ak | case a.mpr.intro.intro
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a - m + m = k
⊢ k ∈ N | case a.mpr.intro.intro
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a = k
⊢ k ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.intro.intro
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a - m + m = k
⊢ k ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | rw [← ak] | case a.mpr.intro.intro
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a = k
⊢ k ∈ N | case a.mpr.intro.intro
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a = k
⊢ a ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.intro.intro
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a = k
⊢ k ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum | [52, 1] | [66, 44] | assumption | case a.mpr.intro.intro
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a = k
⊢ a ∈ N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.intro.intro
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
Ns : Finset ℕ := Finset.image (fun n => n - m) N
k a : ℕ
aN : a ∈ N
ak : a = k
⊢ a ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum' | [69, 1] | [72, 28] | exists Finset.image (fun n ↦ n - m) N | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
⊢ ∃ M, N.sum f = M.sum fun n => f (n + m) | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
⊢ N.sum f = (Finset.image (fun n => n - m) N).sum fun n => f (n + m) | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
⊢ ∃ M, N.sum f = M.sum fun n => f (n + m)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_series_sum' | [69, 1] | [72, 28] | exact late_series_sum h f | m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
⊢ N.sum f = (Finset.image (fun n => n - m) N).sum fun n => f (n + m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
N : Finset ℕ
h : Late N m
f : ℕ → ℝ
⊢ N.sum f = (Finset.image (fun n => n - m) N).sum fun n => f (n + m)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | rcases late_series_sum' h (fun n ↦ a^n) with ⟨M,L⟩ | m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
⊢ (N.sum fun n => a ^ n) ≤ a ^ m * (1 - a)⁻¹ | case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m)
⊢ (N.sum fun n => a ^ n) ≤ a ^ m * (1 - a)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
⊢ (N.sum fun n => a ^ n) ≤ a ^ m * (1 - a)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | rw [L] | case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m)
⊢ (N.sum fun n => a ^ n) ≤ a ^ m * (1 - a)⁻¹ | case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m)
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m)
⊢ (N.sum fun n => a ^ n) ≤ a ^ m * (1 - a)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | clear L | case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m)
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹ | case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m)
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | have pa : (fun n ↦ a^(n + m)) = (fun n ↦ a^n * a^m) := by apply funext; intro n; rw [pow_add] | case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹ | case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | calc
M.sum (fun n ↦ a^(n + m)) = M.sum (fun n ↦ a^n * a^m) := by rw [ pa ]
_ = M.sum (fun n ↦ a^n) * a^m := (Finset.sum_mul _ _ _).symm
_ ≤ (1 - a)⁻¹ * a^m := by bound [partial_geometric_bound M a0 a1]
_ = a^m * (1 - a)⁻¹ := by ring | case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | apply funext | m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
⊢ (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m | case h
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
⊢ ∀ (x : ℕ), a ^ (x + m) = a ^ x * a ^ m | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
⊢ (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | intro n | case h
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
⊢ ∀ (x : ℕ), a ^ (x + m) = a ^ x * a ^ m | case h
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
n : ℕ
⊢ a ^ (n + m) = a ^ n * a ^ m | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
⊢ ∀ (x : ℕ), a ^ (x + m) = a ^ x * a ^ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | rw [pow_add] | case h
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
n : ℕ
⊢ a ^ (n + m) = a ^ n * a ^ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
n : ℕ
⊢ a ^ (n + m) = a ^ n * a ^ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | rw [ pa ] | m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (M.sum fun n => a ^ (n + m)) = M.sum fun n => a ^ n * a ^ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (M.sum fun n => a ^ (n + m)) = M.sum fun n => a ^ n * a ^ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | bound [partial_geometric_bound M a0 a1] | m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (M.sum fun n => a ^ n) * a ^ m ≤ (1 - a)⁻¹ * a ^ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (M.sum fun n => a ^ n) * a ^ m ≤ (1 - a)⁻¹ * a ^ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_geometric_bound | [74, 1] | [83, 35] | ring | m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (1 - a)⁻¹ * a ^ m = a ^ m * (1 - a)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
a : ℝ
N : Finset ℕ
h : Late N m
a0 : 0 ≤ a
a1 : a < 1
M : Finset ℕ
pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
⊢ (1 - a)⁻¹ * a ^ m = a ^ m * (1 - a)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | apply Finset.ext | A B : Finset ℕ
⊢ A = A \ B ∪ A ∩ B | case a
A B : Finset ℕ
⊢ ∀ (a : ℕ), a ∈ A ↔ a ∈ A \ B ∪ A ∩ B | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
⊢ A = A \ B ∪ A ∩ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | simp only [Finset.mem_union, Finset.mem_sdiff, Finset.mem_inter] | case a
A B : Finset ℕ
⊢ ∀ (a : ℕ), a ∈ A ↔ a ∈ A \ B ∪ A ∩ B | case a
A B : Finset ℕ
⊢ ∀ (a : ℕ), a ∈ A ↔ a ∈ A ∧ a ∉ B ∨ a ∈ A ∧ a ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A B : Finset ℕ
⊢ ∀ (a : ℕ), a ∈ A ↔ a ∈ A \ B ∪ A ∩ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | intro x | case a
A B : Finset ℕ
⊢ ∀ (a : ℕ), a ∈ A ↔ a ∈ A ∧ a ∉ B ∨ a ∈ A ∧ a ∈ B | case a
A B : Finset ℕ
x : ℕ
⊢ x ∈ A ↔ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A B : Finset ℕ
⊢ ∀ (a : ℕ), a ∈ A ↔ a ∈ A ∧ a ∉ B ∨ a ∈ A ∧ a ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | constructor | case a
A B : Finset ℕ
x : ℕ
⊢ x ∈ A ↔ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B | case a.mp
A B : Finset ℕ
x : ℕ
⊢ x ∈ A → x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
case a.mpr
A B : Finset ℕ
x : ℕ
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B → x ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A B : Finset ℕ
x : ℕ
⊢ x ∈ A ↔ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | intro a | case a.mp
A B : Finset ℕ
x : ℕ
⊢ x ∈ A → x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B | case a.mp
A B : Finset ℕ
x : ℕ
a : x ∈ A
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
A B : Finset ℕ
x : ℕ
⊢ x ∈ A → x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | by_cases b : x ∈ B | case a.mp
A B : Finset ℕ
x : ℕ
a : x ∈ A
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B | case pos
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∈ B
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
case neg
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∉ B
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
A B : Finset ℕ
x : ℕ
a : x ∈ A
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | right | case pos
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∈ B
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B | case pos.h
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∈ B
⊢ x ∈ A ∧ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∈ B
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | use a,b | case pos.h
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∈ B
⊢ x ∈ A ∧ x ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∈ B
⊢ x ∈ A ∧ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | left | case neg
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∉ B
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B | case neg.h
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∉ B
⊢ x ∈ A ∧ x ∉ B | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∉ B
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | use a,b | case neg.h
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∉ B
⊢ x ∈ A ∧ x ∉ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
A B : Finset ℕ
x : ℕ
a : x ∈ A
b : x ∉ B
⊢ x ∈ A ∧ x ∉ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | intro h | case a.mpr
A B : Finset ℕ
x : ℕ
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B → x ∈ A | case a.mpr
A B : Finset ℕ
x : ℕ
h : x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
⊢ x ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
A B : Finset ℕ
x : ℕ
⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B → x ∈ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | cases' h with m m | case a.mpr
A B : Finset ℕ
x : ℕ
h : x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
⊢ x ∈ A | case a.mpr.inl
A B : Finset ℕ
x : ℕ
m : x ∈ A ∧ x ∉ B
⊢ x ∈ A
case a.mpr.inr
A B : Finset ℕ
x : ℕ
m : x ∈ A ∧ x ∈ B
⊢ x ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
A B : Finset ℕ
x : ℕ
h : x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
⊢ x ∈ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | repeat exact m.1 | case a.mpr.inl
A B : Finset ℕ
x : ℕ
m : x ∈ A ∧ x ∉ B
⊢ x ∈ A
case a.mpr.inr
A B : Finset ℕ
x : ℕ
m : x ∈ A ∧ x ∈ B
⊢ x ∈ A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.inl
A B : Finset ℕ
x : ℕ
m : x ∈ A ∧ x ∉ B
⊢ x ∈ A
case a.mpr.inr
A B : Finset ℕ
x : ℕ
m : x ∈ A ∧ x ∈ B
⊢ x ∈ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_partition | [85, 1] | [95, 21] | exact m.1 | case a.mpr.inr
A B : Finset ℕ
x : ℕ
m : x ∈ A ∧ x ∈ B
⊢ x ∈ A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.inr
A B : Finset ℕ
x : ℕ
m : x ∈ A ∧ x ∈ B
⊢ x ∈ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_sum_partition | [97, 1] | [101, 59] | have ha : A = A \ B ∪ A ∩ B := finset_partition A B | A B : Finset ℕ
f : ℕ → ℂ
⊢ A.sum f = (A \ B).sum f + (A ∩ B).sum f | A B : Finset ℕ
f : ℕ → ℂ
ha : A = A \ B ∪ A ∩ B
⊢ A.sum f = (A \ B).sum f + (A ∩ B).sum f | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
f : ℕ → ℂ
⊢ A.sum f = (A \ B).sum f + (A ∩ B).sum f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_sum_partition | [97, 1] | [101, 59] | nth_rw 1 [ha] | A B : Finset ℕ
f : ℕ → ℂ
ha : A = A \ B ∪ A ∩ B
⊢ A.sum f = (A \ B).sum f + (A ∩ B).sum f | A B : Finset ℕ
f : ℕ → ℂ
ha : A = A \ B ∪ A ∩ B
⊢ (A \ B ∪ A ∩ B).sum f = (A \ B).sum f + (A ∩ B).sum f | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
f : ℕ → ℂ
ha : A = A \ B ∪ A ∩ B
⊢ A.sum f = (A \ B).sum f + (A ∩ B).sum f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | finset_sum_partition | [97, 1] | [101, 59] | exact Finset.sum_union (Finset.disjoint_sdiff_inter A B) | A B : Finset ℕ
f : ℕ → ℂ
ha : A = A \ B ∪ A ∩ B
⊢ (A \ B ∪ A ∩ B).sum f = (A \ B).sum f + (A ∩ B).sum f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
f : ℕ → ℂ
ha : A = A \ B ∪ A ∩ B
⊢ (A \ B ∪ A ∩ B).sum f = (A \ B).sum f + (A ∩ B).sum f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_union | [106, 1] | [107, 41] | rw [symmDiff_def, Finset.sup_eq_union] | A B : Finset ℕ
⊢ A ∆ B = A \ B ∪ B \ A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
⊢ A ∆ B = A \ B ∪ B \ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_bound | [109, 1] | [121, 70] | rw [finset_sum_partition A B f, finset_sum_partition B A f, Finset.inter_comm B A] | A B : Finset ℕ
f : ℕ → ℂ
⊢ dist (A.sum f) (B.sum f) ≤ (A ∆ B).sum fun n => Complex.abs (f n) | A B : Finset ℕ
f : ℕ → ℂ
⊢ dist ((A \ B).sum f + (A ∩ B).sum f) ((B \ A).sum f + (A ∩ B).sum f) ≤ (A ∆ B).sum fun n => Complex.abs (f n) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
f : ℕ → ℂ
⊢ dist (A.sum f) (B.sum f) ≤ (A ∆ B).sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_bound | [109, 1] | [121, 70] | rw [dist_add_right ((A \ B).sum f) ((B \ A).sum f) ((A ∩ B).sum f)] | A B : Finset ℕ
f : ℕ → ℂ
⊢ dist ((A \ B).sum f + (A ∩ B).sum f) ((B \ A).sum f + (A ∩ B).sum f) ≤ (A ∆ B).sum fun n => Complex.abs (f n) | A B : Finset ℕ
f : ℕ → ℂ
⊢ dist ((A \ B).sum f) ((B \ A).sum f) ≤ (A ∆ B).sum fun n => Complex.abs (f n) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
f : ℕ → ℂ
⊢ dist ((A \ B).sum f + (A ∩ B).sum f) ((B \ A).sum f + (A ∩ B).sum f) ≤ (A ∆ B).sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_bound | [109, 1] | [121, 70] | rw [Complex.dist_eq] | A B : Finset ℕ
f : ℕ → ℂ
⊢ dist ((A \ B).sum f) ((B \ A).sum f) ≤ (A ∆ B).sum fun n => Complex.abs (f n) | A B : Finset ℕ
f : ℕ → ℂ
⊢ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≤ (A ∆ B).sum fun n => Complex.abs (f n) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
f : ℕ → ℂ
⊢ dist ((A \ B).sum f) ((B \ A).sum f) ≤ (A ∆ B).sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_bound | [109, 1] | [121, 70] | trans (A \ B).sum (fun n ↦ abs (f n)) + (B \ A).sum (fun n ↦ abs (f n)) | A B : Finset ℕ
f : ℕ → ℂ
⊢ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≤ (A ∆ B).sum fun n => Complex.abs (f n) | A B : Finset ℕ
f : ℕ → ℂ
⊢ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≤
((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)
A B : Finset ℕ
f : ℕ → ℂ
⊢ (((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)) ≤
(A ∆ B).sum fun n => Complex.abs (f n) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
f : ℕ → ℂ
⊢ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≤ (A ∆ B).sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_bound | [109, 1] | [121, 70] | have ha := finset_complex_abs_sum_le (A \ B) f | A B : Finset ℕ
f : ℕ → ℂ
⊢ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≤
((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n) | A B : Finset ℕ
f : ℕ → ℂ
ha : Complex.abs ((A \ B).sum fun n => f n) ≤ (A \ B).sum fun n => Complex.abs (f n)
⊢ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≤
((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
f : ℕ → ℂ
⊢ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≤
((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_bound | [109, 1] | [121, 70] | have hb := finset_complex_abs_sum_le (B \ A) f | A B : Finset ℕ
f : ℕ → ℂ
ha : Complex.abs ((A \ B).sum fun n => f n) ≤ (A \ B).sum fun n => Complex.abs (f n)
⊢ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≤
((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n) | A B : Finset ℕ
f : ℕ → ℂ
ha : Complex.abs ((A \ B).sum fun n => f n) ≤ (A \ B).sum fun n => Complex.abs (f n)
hb : Complex.abs ((B \ A).sum fun n => f n) ≤ (B \ A).sum fun n => Complex.abs (f n)
⊢ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≤
((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
f : ℕ → ℂ
ha : Complex.abs ((A \ B).sum fun n => f n) ≤ (A \ B).sum fun n => Complex.abs (f n)
⊢ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≤
((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_bound | [109, 1] | [121, 70] | calc abs ((A \ B).sum f - (B \ A).sum f)
_ ≤ abs ((A \ B).sum f) + abs ((B \ A).sum f) := by bound
_ ≤ (A \ B).sum (fun n ↦ abs (f n)) + (B \ A).sum (fun n ↦ abs (f n)) := by bound | A B : Finset ℕ
f : ℕ → ℂ
ha : Complex.abs ((A \ B).sum fun n => f n) ≤ (A \ B).sum fun n => Complex.abs (f n)
hb : Complex.abs ((B \ A).sum fun n => f n) ≤ (B \ A).sum fun n => Complex.abs (f n)
⊢ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≤
((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
f : ℕ → ℂ
ha : Complex.abs ((A \ B).sum fun n => f n) ≤ (A \ B).sum fun n => Complex.abs (f n)
hb : Complex.abs ((B \ A).sum fun n => f n) ≤ (B \ A).sum fun n => Complex.abs (f n)
⊢ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≤
((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_bound | [109, 1] | [121, 70] | bound | A B : Finset ℕ
f : ℕ → ℂ
ha : Complex.abs ((A \ B).sum fun n => f n) ≤ (A \ B).sum fun n => Complex.abs (f n)
hb : Complex.abs ((B \ A).sum fun n => f n) ≤ (B \ A).sum fun n => Complex.abs (f n)
⊢ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≤ Complex.abs ((A \ B).sum f) + Complex.abs ((B \ A).sum f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
f : ℕ → ℂ
ha : Complex.abs ((A \ B).sum fun n => f n) ≤ (A \ B).sum fun n => Complex.abs (f n)
hb : Complex.abs ((B \ A).sum fun n => f n) ≤ (B \ A).sum fun n => Complex.abs (f n)
⊢ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≤ Complex.abs ((A \ B).sum f) + Complex.abs ((B \ A).sum f)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_bound | [109, 1] | [121, 70] | bound | A B : Finset ℕ
f : ℕ → ℂ
ha : Complex.abs ((A \ B).sum fun n => f n) ≤ (A \ B).sum fun n => Complex.abs (f n)
hb : Complex.abs ((B \ A).sum fun n => f n) ≤ (B \ A).sum fun n => Complex.abs (f n)
⊢ Complex.abs ((A \ B).sum f) + Complex.abs ((B \ A).sum f) ≤
((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
f : ℕ → ℂ
ha : Complex.abs ((A \ B).sum fun n => f n) ≤ (A \ B).sum fun n => Complex.abs (f n)
hb : Complex.abs ((B \ A).sum fun n => f n) ≤ (B \ A).sum fun n => Complex.abs (f n)
⊢ Complex.abs ((A \ B).sum f) + Complex.abs ((B \ A).sum f) ≤
((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_bound | [109, 1] | [121, 70] | apply le_of_eq | A B : Finset ℕ
f : ℕ → ℂ
⊢ (((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)) ≤
(A ∆ B).sum fun n => Complex.abs (f n) | case a
A B : Finset ℕ
f : ℕ → ℂ
⊢ (((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)) =
(A ∆ B).sum fun n => Complex.abs (f n) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
f : ℕ → ℂ
⊢ (((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)) ≤
(A ∆ B).sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_bound | [109, 1] | [121, 70] | rw [←Finset.sum_union (sdiff_sdiff_disjoint A B), symmDiff_union] | case a
A B : Finset ℕ
f : ℕ → ℂ
⊢ (((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)) =
(A ∆ B).sum fun n => Complex.abs (f n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A B : Finset ℕ
f : ℕ → ℂ
⊢ (((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)) =
(A ∆ B).sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_late | [124, 1] | [135, 16] | intro n ab | A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
⊢ Late (A ∆ B) m | A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
ab : n ∈ A ∆ B
⊢ n ≥ m | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
⊢ Late (A ∆ B) m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_late | [124, 1] | [135, 16] | rw [symmDiff_def, Finset.sup_eq_union, Finset.mem_union] at ab | A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
ab : n ∈ A ∆ B
⊢ n ≥ m | A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
ab : n ∈ A \ B ∨ n ∈ B \ A
⊢ n ≥ m | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
ab : n ∈ A ∆ B
⊢ n ≥ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_late | [124, 1] | [135, 16] | by_contra h | A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
ab : n ∈ A \ B ∨ n ∈ B \ A
⊢ n ≥ m | A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
ab : n ∈ A \ B ∨ n ∈ B \ A
h : ¬n ≥ m
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
ab : n ∈ A \ B ∨ n ∈ B \ A
⊢ n ≥ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_late | [124, 1] | [135, 16] | simp at h | A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
ab : n ∈ A \ B ∨ n ∈ B \ A
h : ¬n ≥ m
⊢ False | A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
ab : n ∈ A \ B ∨ n ∈ B \ A
h : n < m
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
ab : n ∈ A \ B ∨ n ∈ B \ A
h : ¬n ≥ m
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_late | [124, 1] | [135, 16] | cases' ab with a b | A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
ab : n ∈ A \ B ∨ n ∈ B \ A
h : n < m
⊢ False | case inl
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h : n < m
a : n ∈ A \ B
⊢ False
case inr
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h : n < m
b : n ∈ B \ A
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
ab : n ∈ A \ B ∨ n ∈ B \ A
h : n < m
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_late | [124, 1] | [135, 16] | rw [Finset.mem_sdiff] at a | case inl
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h : n < m
a : n ∈ A \ B
⊢ False | case inl
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h : n < m
a : n ∈ A ∧ n ∉ B
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h : n < m
a : n ∈ A \ B
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_late | [124, 1] | [135, 16] | have h := Finset.mem_of_subset hb (Finset.mem_range.mpr h) | case inl
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h : n < m
a : n ∈ A ∧ n ∉ B
⊢ False | case inl
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h✝ : n < m
a : n ∈ A ∧ n ∉ B
h : n ∈ B
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h : n < m
a : n ∈ A ∧ n ∉ B
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_late | [124, 1] | [135, 16] | exact a.2 h | case inl
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h✝ : n < m
a : n ∈ A ∧ n ∉ B
h : n ∈ B
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h✝ : n < m
a : n ∈ A ∧ n ∉ B
h : n ∈ B
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_late | [124, 1] | [135, 16] | rw [Finset.mem_sdiff] at b | case inr
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h : n < m
b : n ∈ B \ A
⊢ False | case inr
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h : n < m
b : n ∈ B ∧ n ∉ A
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h : n < m
b : n ∈ B \ A
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_late | [124, 1] | [135, 16] | have h := Finset.mem_of_subset ha (Finset.mem_range.mpr h) | case inr
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h : n < m
b : n ∈ B ∧ n ∉ A
⊢ False | case inr
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h✝ : n < m
b : n ∈ B ∧ n ∉ A
h : n ∈ A
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h : n < m
b : n ∈ B ∧ n ∉ A
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | symmDiff_late | [124, 1] | [135, 16] | exact b.2 h | case inr
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h✝ : n < m
b : n ∈ B ∧ n ∉ A
h : n ∈ A
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
A B : Finset ℕ
m : ℕ
ha : A ≥ Finset.range m
hb : B ≥ Finset.range m
n : ℕ
h✝ : n < m
b : n ∈ B ∧ n ∉ A
h : n ∈ A
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sub_near | [138, 1] | [147, 54] | rw [abs_le] | a z : ℂ
⊢ |Complex.abs (a - z) - Complex.abs a| ≤ Complex.abs z | a z : ℂ
⊢ -Complex.abs z ≤ Complex.abs (a - z) - Complex.abs a ∧ Complex.abs (a - z) - Complex.abs a ≤ Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
a z : ℂ
⊢ |Complex.abs (a - z) - Complex.abs a| ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sub_near | [138, 1] | [147, 54] | constructor | a z : ℂ
⊢ -Complex.abs z ≤ Complex.abs (a - z) - Complex.abs a ∧ Complex.abs (a - z) - Complex.abs a ≤ Complex.abs z | case left
a z : ℂ
⊢ -Complex.abs z ≤ Complex.abs (a - z) - Complex.abs a
case right
a z : ℂ
⊢ Complex.abs (a - z) - Complex.abs a ≤ Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
a z : ℂ
⊢ -Complex.abs z ≤ Complex.abs (a - z) - Complex.abs a ∧ Complex.abs (a - z) - Complex.abs a ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sub_near | [138, 1] | [147, 54] | simp only [neg_le_sub_iff_le_add] | case left
a z : ℂ
⊢ -Complex.abs z ≤ Complex.abs (a - z) - Complex.abs a | case left
a z : ℂ
⊢ Complex.abs a ≤ Complex.abs (a - z) + Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
case left
a z : ℂ
⊢ -Complex.abs z ≤ Complex.abs (a - z) - Complex.abs a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sub_near | [138, 1] | [147, 54] | calc abs (a - z) + abs z
_ ≥ |abs a - abs z| + abs z := by bound
_ ≥ abs a - abs z + abs z := by bound
_ = abs a := by simp only [sub_add_cancel] | case left
a z : ℂ
⊢ Complex.abs a ≤ Complex.abs (a - z) + Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
a z : ℂ
⊢ Complex.abs a ≤ Complex.abs (a - z) + Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sub_near | [138, 1] | [147, 54] | bound | a z : ℂ
⊢ Complex.abs (a - z) + Complex.abs z ≥ |Complex.abs a - Complex.abs z| + Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a z : ℂ
⊢ Complex.abs (a - z) + Complex.abs z ≥ |Complex.abs a - Complex.abs z| + Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sub_near | [138, 1] | [147, 54] | bound | a z : ℂ
⊢ |Complex.abs a - Complex.abs z| + Complex.abs z ≥ Complex.abs a - Complex.abs z + Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a z : ℂ
⊢ |Complex.abs a - Complex.abs z| + Complex.abs z ≥ Complex.abs a - Complex.abs z + Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sub_near | [138, 1] | [147, 54] | simp only [sub_add_cancel] | a z : ℂ
⊢ Complex.abs a - Complex.abs z + Complex.abs z = Complex.abs a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a z : ℂ
⊢ Complex.abs a - Complex.abs z + Complex.abs z = Complex.abs a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sub_near | [138, 1] | [147, 54] | calc
abs (a - z) - abs a ≤ abs a + abs z - abs a := by bound
_ = abs z := by simp only [add_sub_cancel_left] | case right
a z : ℂ
⊢ Complex.abs (a - z) - Complex.abs a ≤ Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
a z : ℂ
⊢ Complex.abs (a - z) - Complex.abs a ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sub_near | [138, 1] | [147, 54] | bound | a z : ℂ
⊢ Complex.abs (a - z) - Complex.abs a ≤ Complex.abs a + Complex.abs z - Complex.abs a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a z : ℂ
⊢ Complex.abs (a - z) - Complex.abs a ≤ Complex.abs a + Complex.abs z - Complex.abs a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sub_near | [138, 1] | [147, 54] | simp only [add_sub_cancel_left] | a z : ℂ
⊢ Complex.abs a + Complex.abs z - Complex.abs a = Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a z : ℂ
⊢ Complex.abs a + Complex.abs z - Complex.abs a = Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | add_near | [149, 1] | [152, 13] | have h := sub_near a (-z) | a z : ℂ
⊢ |Complex.abs (a + z) - Complex.abs a| ≤ Complex.abs z | a z : ℂ
h : |Complex.abs (a - -z) - Complex.abs a| ≤ Complex.abs (-z)
⊢ |Complex.abs (a + z) - Complex.abs a| ≤ Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
a z : ℂ
⊢ |Complex.abs (a + z) - Complex.abs a| ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | add_near | [149, 1] | [152, 13] | simp only [sub_neg_eq_add, map_neg_eq_map] at h | a z : ℂ
h : |Complex.abs (a - -z) - Complex.abs a| ≤ Complex.abs (-z)
⊢ |Complex.abs (a + z) - Complex.abs a| ≤ Complex.abs z | a z : ℂ
h : |Complex.abs (a + z) - Complex.abs a| ≤ Complex.abs z
⊢ |Complex.abs (a + z) - Complex.abs a| ≤ Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
a z : ℂ
h : |Complex.abs (a - -z) - Complex.abs a| ≤ Complex.abs (-z)
⊢ |Complex.abs (a + z) - Complex.abs a| ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | add_near | [149, 1] | [152, 13] | assumption | a z : ℂ
h : |Complex.abs (a + z) - Complex.abs a| ≤ Complex.abs z
⊢ |Complex.abs (a + z) - Complex.abs a| ≤ Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a z : ℂ
h : |Complex.abs (a + z) - Complex.abs a| ≤ Complex.abs z
⊢ |Complex.abs (a + z) - Complex.abs a| ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | mem_slitPlane_of_near_one | [154, 1] | [163, 13] | intro h | z : ℂ
⊢ Complex.abs (z - 1) < 1 → z ∈ slitPlane | z : ℂ
h : Complex.abs (z - 1) < 1
⊢ z ∈ slitPlane | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ Complex.abs (z - 1) < 1 → z ∈ slitPlane
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | mem_slitPlane_of_near_one | [154, 1] | [163, 13] | apply Or.inl | z : ℂ
h : Complex.abs (z - 1) < 1
⊢ z ∈ slitPlane | case h
z : ℂ
h : Complex.abs (z - 1) < 1
⊢ 0 < z.re | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h : Complex.abs (z - 1) < 1
⊢ z ∈ slitPlane
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | mem_slitPlane_of_near_one | [154, 1] | [163, 13] | have hr : (1 - z).re < 1 := by
calc
(1 - z).re ≤ |(1 - z).re| := le_abs_self (1 - z).re
_ ≤ abs (1 - z) := (Complex.abs_re_le_abs _)
_ = abs (z - 1) := by rw [←Complex.abs.map_neg (1 - z)]; simp only [neg_sub]
_ < 1 := h | case h
z : ℂ
h : Complex.abs (z - 1) < 1
⊢ 0 < z.re | case h
z : ℂ
h : Complex.abs (z - 1) < 1
hr : (1 - z).re < 1
⊢ 0 < z.re | Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : ℂ
h : Complex.abs (z - 1) < 1
⊢ 0 < z.re
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | mem_slitPlane_of_near_one | [154, 1] | [163, 13] | simp only [Complex.sub_re, Complex.one_re, sub_lt_self_iff] at hr | case h
z : ℂ
h : Complex.abs (z - 1) < 1
hr : (1 - z).re < 1
⊢ 0 < z.re | case h
z : ℂ
h : Complex.abs (z - 1) < 1
hr : 0 < z.re
⊢ 0 < z.re | Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : ℂ
h : Complex.abs (z - 1) < 1
hr : (1 - z).re < 1
⊢ 0 < z.re
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | mem_slitPlane_of_near_one | [154, 1] | [163, 13] | assumption | case h
z : ℂ
h : Complex.abs (z - 1) < 1
hr : 0 < z.re
⊢ 0 < z.re | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : ℂ
h : Complex.abs (z - 1) < 1
hr : 0 < z.re
⊢ 0 < z.re
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | mem_slitPlane_of_near_one | [154, 1] | [163, 13] | calc
(1 - z).re ≤ |(1 - z).re| := le_abs_self (1 - z).re
_ ≤ abs (1 - z) := (Complex.abs_re_le_abs _)
_ = abs (z - 1) := by rw [←Complex.abs.map_neg (1 - z)]; simp only [neg_sub]
_ < 1 := h | z : ℂ
h : Complex.abs (z - 1) < 1
⊢ (1 - z).re < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h : Complex.abs (z - 1) < 1
⊢ (1 - z).re < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | mem_slitPlane_of_near_one | [154, 1] | [163, 13] | rw [←Complex.abs.map_neg (1 - z)] | z : ℂ
h : Complex.abs (z - 1) < 1
⊢ Complex.abs (1 - z) = Complex.abs (z - 1) | z : ℂ
h : Complex.abs (z - 1) < 1
⊢ Complex.abs (-(1 - z)) = Complex.abs (z - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h : Complex.abs (z - 1) < 1
⊢ Complex.abs (1 - z) = Complex.abs (z - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | mem_slitPlane_of_near_one | [154, 1] | [163, 13] | simp only [neg_sub] | z : ℂ
h : Complex.abs (z - 1) < 1
⊢ Complex.abs (-(1 - z)) = Complex.abs (z - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h : Complex.abs (z - 1) < 1
⊢ Complex.abs (-(1 - z)) = Complex.abs (z - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | near_one_avoids_zero | [165, 1] | [166, 73] | intro h | z : ℂ
⊢ Complex.abs (z - 1) < 1 → z ≠ 0 | z : ℂ
h : Complex.abs (z - 1) < 1
⊢ z ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ Complex.abs (z - 1) < 1 → z ≠ 0
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.