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/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallCoversSelf | [90, 1] | [97, 2] | have h2 : ball x r ⊆ ⋃ x ∈ a, ball x r := by
simp [a]
rfl | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r✝ r' r₁ r₂ r₃ : ℝ
x : X
r : ℝ
a : Finset X := {x}
h : a.card ≤ 1
⊢ CoveredByBalls (ball x r) 1 r | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r✝ r' r₁ r₂ r₃ : ℝ
x : X
r : ℝ
a : Finset X := {x}
h : a.card ≤ 1
h2 : ball x r ⊆ ⋃ x ∈ a, ball x r
⊢ CoveredByBalls (ball x r) 1 r | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r✝ r' r₁ r₂ r₃ : ℝ
x : X
r : ℝ
a : Finset X := {x}
h : a.card ≤ 1
⊢ CoveredByBalls (ball x r) 1 r
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallCoversSelf | [90, 1] | [97, 2] | exact ⟨a, h, h2⟩ | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r✝ r' r₁ r₂ r₃ : ℝ
x : X
r : ℝ
a : Finset X := {x}
h : a.card ≤ 1
h2 : ball x r ⊆ ⋃ x ∈ a, ball x r
⊢ CoveredByBalls (ball x r) 1 r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r✝ r' r₁ r₂ r₃ : ℝ
x : X
r : ℝ
a : Finset X := {x}
h : a.card ≤ 1
h2 : ball x r ⊆ ⋃ x ∈ a, ball x r
⊢ CoveredByBalls (ball x r) 1 r
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallCoversSelf | [90, 1] | [97, 2] | rfl | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r✝ r' r₁ r₂ r₃ : ℝ
x : X
r : ℝ
a : Finset X := {x}
⊢ a.card ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r✝ r' r₁ r₂ r₃ : ℝ
x : X
r : ℝ
a : Finset X := {x}
⊢ a.card ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallCoversSelf | [90, 1] | [97, 2] | simp [a] | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r✝ r' r₁ r₂ r₃ : ℝ
x : X
r : ℝ
a : Finset X := {x}
h : a.card ≤ 1
⊢ ball x r ⊆ ⋃ x ∈ a, ball x r | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r✝ r' r₁ r₂ r₃ : ℝ
x : X
r : ℝ
a : Finset X := {x}
h : a.card ≤ 1
⊢ ball x r ⊆ ball x r | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r✝ r' r₁ r₂ r₃ : ℝ
x : X
r : ℝ
a : Finset X := {x}
h : a.card ≤ 1
⊢ ball x r ⊆ ⋃ x ∈ a, ball x r
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallCoversSelf | [90, 1] | [97, 2] | rfl | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r✝ r' r₁ r₂ r₃ : ℝ
x : X
r : ℝ
a : Finset X := {x}
h : a.card ≤ 1
⊢ ball x r ⊆ ball x r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r✝ r' r₁ r₂ r₃ : ℝ
x : X
r : ℝ
a : Finset X := {x}
h : a.card ≤ 1
⊢ ball x r ⊆ ball x r
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallsCoverBalls.pow_mul | [99, 1] | [114, 37] | induction k | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
k : ℕ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
⊢ BallsCoverBalls X (a ^ k * r) r (n ^ k) | case zero
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
⊢ BallsCoverBalls X (a ^ 0 * r) r (n ^ 0)
case succ
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
n✝ : ℕ
a✝ : BallsCoverBalls X (a ^ n✝ * r) r (n ^ n✝)
⊢ BallsCoverBalls X (a ^ (n✝ + 1) * r) r (n ^ (n✝ + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
k : ℕ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
⊢ BallsCoverBalls X (a ^ k * r) r (n ^ k)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallsCoverBalls.pow_mul | [99, 1] | [114, 37] | case zero =>
simp
intro x
exact BallCoversSelf x r | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
⊢ BallsCoverBalls X (a ^ 0 * r) r (n ^ 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
⊢ BallsCoverBalls X (a ^ 0 * r) r (n ^ 0)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallsCoverBalls.pow_mul | [99, 1] | [114, 37] | case succ m h2 =>
specialize h (r * a^m)
rw[<- mul_assoc, mul_comm, <- mul_assoc] at h
norm_cast
ring_nf
rw[mul_comm a]
rw[mul_comm] at h2
norm_cast at h2
exact BallsCoverBalls.trans h h2 | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
m : ℕ
h2 : BallsCoverBalls X (a ^ m * r) r (n ^ m)
⊢ BallsCoverBalls X (a ^ (m + 1) * r) r (n ^ (m + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
m : ℕ
h2 : BallsCoverBalls X (a ^ m * r) r (n ^ m)
⊢ BallsCoverBalls X (a ^ (m + 1) * r) r (n ^ (m + 1))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallsCoverBalls.pow_mul | [99, 1] | [114, 37] | simp | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
⊢ BallsCoverBalls X (a ^ 0 * r) r (n ^ 0) | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
⊢ BallsCoverBalls X r r 1 | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
⊢ BallsCoverBalls X (a ^ 0 * r) r (n ^ 0)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallsCoverBalls.pow_mul | [99, 1] | [114, 37] | intro x | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
⊢ BallsCoverBalls X r r 1 | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
x : X
⊢ CoveredByBalls (ball x r) 1 r | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
⊢ BallsCoverBalls X r r 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallsCoverBalls.pow_mul | [99, 1] | [114, 37] | exact BallCoversSelf x r | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
x : X
⊢ CoveredByBalls (ball x r) 1 r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
x : X
⊢ CoveredByBalls (ball x r) 1 r
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallsCoverBalls.pow_mul | [99, 1] | [114, 37] | specialize h (r * a^m) | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
m : ℕ
h2 : BallsCoverBalls X (a ^ m * r) r (n ^ m)
⊢ BallsCoverBalls X (a ^ (m + 1) * r) r (n ^ (m + 1)) | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
m : ℕ
h2 : BallsCoverBalls X (a ^ m * r) r (n ^ m)
h : BallsCoverBalls X (a * (r * a ^ m)) (r * a ^ m) n
⊢ BallsCoverBalls X (a ^ (m + 1) * r) r (n ^ (m + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
m : ℕ
h2 : BallsCoverBalls X (a ^ m * r) r (n ^ m)
⊢ BallsCoverBalls X (a ^ (m + 1) * r) r (n ^ (m + 1))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallsCoverBalls.pow_mul | [99, 1] | [114, 37] | rw[<- mul_assoc, mul_comm, <- mul_assoc] at h | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
m : ℕ
h2 : BallsCoverBalls X (a ^ m * r) r (n ^ m)
h : BallsCoverBalls X (a * (r * a ^ m)) (r * a ^ m) n
⊢ BallsCoverBalls X (a ^ (m + 1) * r) r (n ^ (m + 1)) | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
m : ℕ
h2 : BallsCoverBalls X (a ^ m * r) r (n ^ m)
h : BallsCoverBalls X (a ^ m * a * r) (r * a ^ m) n
⊢ BallsCoverBalls X (a ^ (m + 1) * r) r (n ^ (m + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
m : ℕ
h2 : BallsCoverBalls X (a ^ m * r) r (n ^ m)
h : BallsCoverBalls X (a * (r * a ^ m)) (r * a ^ m) n
⊢ BallsCoverBalls X (a ^ (m + 1) * r) r (n ^ (m + 1))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallsCoverBalls.pow_mul | [99, 1] | [114, 37] | ring_nf | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
m : ℕ
h2 : BallsCoverBalls X (a ^ m * r) r (n ^ m)
h : BallsCoverBalls X (a ^ m * a * r) (r * a ^ m) n
⊢ BallsCoverBalls X (a ^ (m + 1) * r) r (n ^ (m + 1)) | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
m : ℕ
h2 : BallsCoverBalls X (a ^ m * r) r (n ^ m)
h : BallsCoverBalls X (a ^ m * a * r) (r * a ^ m) n
⊢ BallsCoverBalls X (a * a ^ m * r) r (n * n ^ m) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
m : ℕ
h2 : BallsCoverBalls X (a ^ m * r) r (n ^ m)
h : BallsCoverBalls X (a ^ m * a * r) (r * a ^ m) n
⊢ BallsCoverBalls X (a ^ (m + 1) * r) r (n ^ (m + 1))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallsCoverBalls.pow_mul | [99, 1] | [114, 37] | rw[mul_comm a] | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
m : ℕ
h2 : BallsCoverBalls X (a ^ m * r) r (n ^ m)
h : BallsCoverBalls X (a ^ m * a * r) (r * a ^ m) n
⊢ BallsCoverBalls X (a * a ^ m * r) r (n * n ^ m) | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
m : ℕ
h2 : BallsCoverBalls X (a ^ m * r) r (n ^ m)
h : BallsCoverBalls X (a ^ m * a * r) (r * a ^ m) n
⊢ BallsCoverBalls X (a ^ m * a * r) r (n * n ^ m) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
m : ℕ
h2 : BallsCoverBalls X (a ^ m * r) r (n ^ m)
h : BallsCoverBalls X (a ^ m * a * r) (r * a ^ m) n
⊢ BallsCoverBalls X (a * a ^ m * r) r (n * n ^ m)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallsCoverBalls.pow_mul | [99, 1] | [114, 37] | rw[mul_comm] at h2 | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
m : ℕ
h2 : BallsCoverBalls X (a ^ m * r) r (n ^ m)
h : BallsCoverBalls X (a ^ m * a * r) (r * a ^ m) n
⊢ BallsCoverBalls X (a ^ m * a * r) r (n * n ^ m) | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
m : ℕ
h2 : BallsCoverBalls X (r * a ^ m) r (n ^ m)
h : BallsCoverBalls X (a ^ m * a * r) (r * a ^ m) n
⊢ BallsCoverBalls X (a ^ m * a * r) r (n * n ^ m) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
m : ℕ
h2 : BallsCoverBalls X (a ^ m * r) r (n ^ m)
h : BallsCoverBalls X (a ^ m * a * r) (r * a ^ m) n
⊢ BallsCoverBalls X (a ^ m * a * r) r (n * n ^ m)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallsCoverBalls.pow_mul | [99, 1] | [114, 37] | exact BallsCoverBalls.trans h h2 | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
m : ℕ
h2 : BallsCoverBalls X (r * a ^ m) r (n ^ m)
h : BallsCoverBalls X (a ^ m * a * r) (r * a ^ m) n
⊢ BallsCoverBalls X (a ^ m * a * r) r (n * n ^ m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m✝ : ℕ
r r' r₁ r₂ r₃ a : ℝ
m : ℕ
h2 : BallsCoverBalls X (r * a ^ m) r (n ^ m)
h : BallsCoverBalls X (a ^ m * a * r) (r * a ^ m) n
⊢ BallsCoverBalls X (a ^ m * a * r) r (n * n ^ m)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallsCoverBalls.pow | [116, 1] | [119, 7] | convert BallsCoverBalls.pow_mul h using 1 | X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
k : ℕ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
⊢ BallsCoverBalls X (a ^ k) 1 (n ^ k) | case h.e'_3
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
k : ℕ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
⊢ a ^ k = a ^ k * 1 | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
k : ℕ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
⊢ BallsCoverBalls X (a ^ k) 1 (n ^ k)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/CoverByBalls.lean | BallsCoverBalls.pow | [116, 1] | [119, 7] | simp | case h.e'_3
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
k : ℕ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
⊢ a ^ k = a ^ k * 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
X : Type u_1
inst✝ : PseudoMetricSpace X
s t : Set X
n m : ℕ
r r' r₁ r₂ r₃ a : ℝ
k : ℕ
h : ∀ (r : ℝ), BallsCoverBalls X (a * r) r n
⊢ a ^ k = a ^ k * 1
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/IEEE.lean | ieee_to_flformat | [30, 1] | [40, 31] | have h := fmt.prop | fmt : ieee_format
⊢ is_valid_flformat (mk_format fmt) | fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ is_valid_flformat (mk_format fmt) | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : ieee_format
⊢ is_valid_flformat (mk_format fmt)
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/IEEE.lean | ieee_to_flformat | [30, 1] | [40, 31] | apply And.intro | fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ is_valid_flformat (mk_format fmt) | case left
fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ 1 < (mk_format fmt).r
case right
fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ (mk_format fmt).r % 2 = 0 ∧ 1 < (mk_format fmt).p | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ is_valid_flformat (mk_format fmt)
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/IEEE.lean | ieee_to_flformat | [30, 1] | [40, 31] | case left =>
apply h.left | fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ 1 < (mk_format fmt).r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ 1 < (mk_format fmt).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/IEEE.lean | ieee_to_flformat | [30, 1] | [40, 31] | case right =>
apply And.intro
case left =>
apply h.right.left
case right =>
apply h.right.right.left | fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ (mk_format fmt).r % 2 = 0 ∧ 1 < (mk_format fmt).p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ (mk_format fmt).r % 2 = 0 ∧ 1 < (mk_format fmt).p
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/IEEE.lean | ieee_to_flformat | [30, 1] | [40, 31] | apply h.left | fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ 1 < (mk_format fmt).r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ 1 < (mk_format fmt).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/IEEE.lean | ieee_to_flformat | [30, 1] | [40, 31] | apply And.intro | fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ (mk_format fmt).r % 2 = 0 ∧ 1 < (mk_format fmt).p | case left
fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ (mk_format fmt).r % 2 = 0
case right
fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ 1 < (mk_format fmt).p | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ (mk_format fmt).r % 2 = 0 ∧ 1 < (mk_format fmt).p
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/IEEE.lean | ieee_to_flformat | [30, 1] | [40, 31] | case left =>
apply h.right.left | fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ (mk_format fmt).r % 2 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ (mk_format fmt).r % 2 = 0
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/IEEE.lean | ieee_to_flformat | [30, 1] | [40, 31] | case right =>
apply h.right.right.left | fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ 1 < (mk_format fmt).p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ 1 < (mk_format fmt).p
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/IEEE.lean | ieee_to_flformat | [30, 1] | [40, 31] | apply h.right.left | fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ (mk_format fmt).r % 2 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ (mk_format fmt).r % 2 = 0
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/IEEE.lean | ieee_to_flformat | [30, 1] | [40, 31] | apply h.right.right.left | fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ 1 < (mk_format fmt).p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : ieee_format
h : is_valid_ieee_format ↑fmt
⊢ 1 < (mk_format fmt).p
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_real_le_1_le | [11, 1] | [16, 11] | intro h₁ h₂ | x : ℝ
m : ℤ
⊢ 0 < x → 1 ≤ m → x ≤ ↑m * x | x : ℝ
m : ℤ
h₁ : 0 < x
h₂ : 1 ≤ m
⊢ x ≤ ↑m * x | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
m : ℤ
⊢ 0 < x → 1 ≤ m → x ≤ ↑m * x
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_real_le_1_le | [11, 1] | [16, 11] | rw [le_mul_iff_one_le_left] | x : ℝ
m : ℤ
h₁ : 0 < x
h₂ : 1 ≤ m
⊢ x ≤ ↑m * x | x : ℝ
m : ℤ
h₁ : 0 < x
h₂ : 1 ≤ m
⊢ 1 ≤ ↑m
x : ℝ
m : ℤ
h₁ : 0 < x
h₂ : 1 ≤ m
⊢ 0 < x | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
m : ℤ
h₁ : 0 < x
h₂ : 1 ≤ m
⊢ x ≤ ↑m * x
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_real_le_1_le | [11, 1] | [16, 11] | norm_cast | x : ℝ
m : ℤ
h₁ : 0 < x
h₂ : 1 ≤ m
⊢ 1 ≤ ↑m
x : ℝ
m : ℤ
h₁ : 0 < x
h₂ : 1 ≤ m
⊢ 0 < x | x : ℝ
m : ℤ
h₁ : 0 < x
h₂ : 1 ≤ m
⊢ 0 < x | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
m : ℤ
h₁ : 0 < x
h₂ : 1 ≤ m
⊢ 1 ≤ ↑m
x : ℝ
m : ℤ
h₁ : 0 < x
h₂ : 1 ≤ m
⊢ 0 < x
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_real_le_1_le | [11, 1] | [16, 11] | exact h₁ | x : ℝ
m : ℤ
h₁ : 0 < x
h₂ : 1 ≤ m
⊢ 0 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
m : ℤ
h₁ : 0 < x
h₂ : 1 ≤ m
⊢ 0 < x
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | flformat_radix_lt_0 | [19, 1] | [20, 27] | linarith [fmt.prop.left] | fmt : flformat
⊢ 0 < (↑fmt).r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
⊢ 0 < (↑fmt).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | flformat_radix_lt_1 | [23, 1] | [24, 27] | linarith [fmt.prop.left] | fmt : flformat
⊢ 1 < (↑fmt).r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
⊢ 1 < (↑fmt).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | flformat_radix_le_2 | [27, 1] | [28, 27] | linarith [fmt.prop.left] | fmt : flformat
⊢ 2 ≤ (↑fmt).r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
⊢ 2 ≤ (↑fmt).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | flformat_radix_lt_0_real | [31, 1] | [33, 27] | norm_cast | fmt : flformat
⊢ 0 < ↑(↑fmt).r | fmt : flformat
⊢ 0 < (↑fmt).r | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
⊢ 0 < ↑(↑fmt).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | flformat_radix_lt_0_real | [31, 1] | [33, 27] | linarith [fmt.prop.left] | fmt : flformat
⊢ 0 < (↑fmt).r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
⊢ 0 < (↑fmt).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | flformat_radix_lt_1_real | [36, 1] | [38, 27] | norm_cast | fmt : flformat
⊢ 1 < ↑(↑fmt).r | fmt : flformat
⊢ 1 < (↑fmt).r | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
⊢ 1 < ↑(↑fmt).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | flformat_radix_lt_1_real | [36, 1] | [38, 27] | linarith [fmt.prop.left] | fmt : flformat
⊢ 1 < (↑fmt).r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
⊢ 1 < (↑fmt).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | flformat_radix_le_2_real | [45, 1] | [47, 63] | norm_cast | fmt : flformat
⊢ 2 ≤ ↑(↑fmt).r | fmt : flformat
⊢ 2 ≤ (↑fmt).r | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
⊢ 2 ≤ ↑(↑fmt).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | flformat_radix_le_2_real | [45, 1] | [47, 63] | linarith [Int.lt_add_one_iff.2 fmt.prop.left, fmt.prop.left] | fmt : flformat
⊢ 2 ≤ (↑fmt).r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
⊢ 2 ≤ (↑fmt).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_radix_ipow_lt_0 | [51, 1] | [54, 65] | have h : 0 < fmt.val.r := by apply Int.one_lt_zero_lt; exact fmt.prop.left | fmt : flformat
e : ℤ
⊢ 0 < ↑(↑fmt).r ^ e | fmt : flformat
e : ℤ
h : 0 < (↑fmt).r
⊢ 0 < ↑(↑fmt).r ^ e | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
e : ℤ
⊢ 0 < ↑(↑fmt).r ^ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_radix_ipow_lt_0 | [51, 1] | [54, 65] | apply ipow_lt_zero | fmt : flformat
e : ℤ
h : 0 < (↑fmt).r
⊢ 0 < ↑(↑fmt).r ^ e | case a
fmt : flformat
e : ℤ
h : 0 < (↑fmt).r
⊢ 0 < ↑(↑fmt).r | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
e : ℤ
h : 0 < (↑fmt).r
⊢ 0 < ↑(↑fmt).r ^ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_radix_ipow_lt_0 | [51, 1] | [54, 65] | simp_all only [Int.one_lt_zero_le_iff, zero_add, Int.cast_pos] | case a
fmt : flformat
e : ℤ
h : 0 < (↑fmt).r
⊢ 0 < ↑(↑fmt).r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
fmt : flformat
e : ℤ
h : 0 < (↑fmt).r
⊢ 0 < ↑(↑fmt).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_radix_ipow_lt_0 | [51, 1] | [54, 65] | apply Int.one_lt_zero_lt | fmt : flformat
e : ℤ
⊢ 0 < (↑fmt).r | case a
fmt : flformat
e : ℤ
⊢ 1 < (↑fmt).r | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
e : ℤ
⊢ 0 < (↑fmt).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_radix_ipow_lt_0 | [51, 1] | [54, 65] | exact fmt.prop.left | case a
fmt : flformat
e : ℤ
⊢ 1 < (↑fmt).r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
fmt : flformat
e : ℤ
⊢ 1 < (↑fmt).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_ipow_le_real | [57, 1] | [59, 29] | apply ipow_le_real | fmt : flformat
x : ℝ
⊢ ∃ e, x ≤ ↑(↑fmt).r ^ e | case a
fmt : flformat
x : ℝ
⊢ 2 ≤ ↑(↑fmt).r | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
⊢ ∃ e, x ≤ ↑(↑fmt).r ^ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_ipow_le_real | [57, 1] | [59, 29] | simp [flformat_radix_le_2] | case a
fmt : flformat
x : ℝ
⊢ 2 ≤ ↑(↑fmt).r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
fmt : flformat
x : ℝ
⊢ 2 ≤ ↑(↑fmt).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_ipow_le_real_2 | [63, 1] | [67, 43] | intros | fmt : flformat
x : ℝ
⊢ 0 < x → ∃ e, x ≤ ↑(↑fmt).r ^ e | fmt : flformat
x : ℝ
a✝ : 0 < x
⊢ ∃ e, x ≤ ↑(↑fmt).r ^ e | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
⊢ 0 < x → ∃ e, x ≤ ↑(↑fmt).r ^ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_ipow_le_real_2 | [63, 1] | [67, 43] | apply ipow_le_real | fmt : flformat
x : ℝ
a✝ : 0 < x
⊢ ∃ e, x ≤ ↑(↑fmt).r ^ e | case a
fmt : flformat
x : ℝ
a✝ : 0 < x
⊢ 2 ≤ ↑(↑fmt).r | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
a✝ : 0 < x
⊢ ∃ e, x ≤ ↑(↑fmt).r ^ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_ipow_le_real_2 | [63, 1] | [67, 43] | simp_all only [flformat_radix_le_2_real] | case a
fmt : flformat
x : ℝ
a✝ : 0 < x
⊢ 2 ≤ ↑(↑fmt).r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
fmt : flformat
x : ℝ
a✝ : 0 < x
⊢ 2 ≤ ↑(↑fmt).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | pow_le_real | [77, 1] | [87, 31] | simpa using hk | a x : ℝ
ha1 : a ≠ 1
hx0 : x ≠ 0
ha0 : ¬a ≤ 0
hal1 : a < 1
k : ℕ
hk : (abs x)⁻¹ < a⁻¹ ^ k
⊢ (abs x)⁻¹ < (a ^ k)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a x : ℝ
ha1 : a ≠ 1
hx0 : x ≠ 0
ha0 : ¬a ≤ 0
hal1 : a < 1
k : ℕ
hk : (abs x)⁻¹ < a⁻¹ ^ k
⊢ (abs x)⁻¹ < (a ^ k)⁻¹
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | pow_le_real | [77, 1] | [87, 31] | simpa | a x : ℝ
ha1 : a ≠ 1
hx0 : x ≠ 0
ha0 : ¬a ≤ 0
hal1 : ¬a < 1
k : ℕ
hk : (abs x)⁻¹ < a ^ k
⊢ (abs x)⁻¹ < (a ^ (-↑k))⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a x : ℝ
ha1 : a ≠ 1
hx0 : x ≠ 0
ha0 : ¬a ≤ 0
hal1 : ¬a < 1
k : ℕ
hk : (abs x)⁻¹ < a ^ k
⊢ (abs x)⁻¹ < (a ^ (-↑k))⁻¹
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_int_bounded | [91, 1] | [96, 8] | intro hn ha | α✝ : Type u_1
s : Set α✝
S : ℤ → Prop
b : ℤ
⊢ Set.Nonempty s → (∀ (e : ℤ), e ∈ setOf S → e ≤ b) → ∃ e', is_sup_int S e' | α✝ : Type u_1
s : Set α✝
S : ℤ → Prop
b : ℤ
hn : Set.Nonempty s
ha : ∀ (e : ℤ), e ∈ setOf S → e ≤ b
⊢ ∃ e', is_sup_int S e' | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
s : Set α✝
S : ℤ → Prop
b : ℤ
⊢ Set.Nonempty s → (∀ (e : ℤ), e ∈ setOf S → e ≤ b) → ∃ e', is_sup_int S e'
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_int_bounded | [91, 1] | [96, 8] | simp [setOf] at ha | α✝ : Type u_1
s : Set α✝
S : ℤ → Prop
b : ℤ
hn : Set.Nonempty s
ha : ∀ (e : ℤ), e ∈ setOf S → e ≤ b
⊢ ∃ e', is_sup_int S e' | α✝ : Type u_1
s : Set α✝
S : ℤ → Prop
b : ℤ
hn : Set.Nonempty s
ha : ∀ (e : ℤ), e ∈ S → e ≤ b
⊢ ∃ e', is_sup_int S e' | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
s : Set α✝
S : ℤ → Prop
b : ℤ
hn : Set.Nonempty s
ha : ∀ (e : ℤ), e ∈ setOf S → e ≤ b
⊢ ∃ e', is_sup_int S e'
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_int_bounded | [91, 1] | [96, 8] | simp [is_sup_int, IsLUB, IsLeast, lowerBounds, upperBounds] | α✝ : Type u_1
s : Set α✝
S : ℤ → Prop
b : ℤ
hn : Set.Nonempty s
ha : ∀ (e : ℤ), e ∈ S → e ≤ b
⊢ ∃ e', is_sup_int S e' | α✝ : Type u_1
s : Set α✝
S : ℤ → Prop
b : ℤ
hn : Set.Nonempty s
ha : ∀ (e : ℤ), e ∈ S → e ≤ b
⊢ ∃ e', (∀ ⦃a : ℤ⦄, a ∈ S → a ≤ e') ∧ ∀ ⦃a : ℤ⦄, (∀ ⦃a_1 : ℤ⦄, a_1 ∈ S → a_1 ≤ a) → e' ≤ a | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
s : Set α✝
S : ℤ → Prop
b : ℤ
hn : Set.Nonempty s
ha : ∀ (e : ℤ), e ∈ S → e ≤ b
⊢ ∃ e', is_sup_int S e'
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_int_bounded | [91, 1] | [96, 8] | sorry | α✝ : Type u_1
s : Set α✝
S : ℤ → Prop
b : ℤ
hn : Set.Nonempty s
ha : ∀ (e : ℤ), e ∈ S → e ≤ b
⊢ ∃ e', (∀ ⦃a : ℤ⦄, a ∈ S → a ≤ e') ∧ ∀ ⦃a : ℤ⦄, (∀ ⦃a_1 : ℤ⦄, a_1 ∈ S → a_1 ≤ a) → e' ≤ a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
s : Set α✝
S : ℤ → Prop
b : ℤ
hn : Set.Nonempty s
ha : ∀ (e : ℤ), e ∈ S → e ≤ b
⊢ ∃ e', (∀ ⦃a : ℤ⦄, a ∈ S → a ≤ e') ∧ ∀ ⦃a : ℤ⦄, (∀ ⦃a_1 : ℤ⦄, a_1 ∈ S → a_1 ≤ a) → e' ≤ a
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | have ⟨fmt_val, FMT⟩ := fmt | fmt : flformat
x : ℝ
e : ℤ
⊢ x ≠ 0 → e = greatest_e fmt x → is_greatest_e fmt x e | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
⊢ x ≠ 0 → e = greatest_e { val := fmt_val, property := FMT } x → is_greatest_e { val := fmt_val, property := FMT } x e | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
⊢ x ≠ 0 → e = greatest_e fmt x → is_greatest_e fmt x e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | intro hx he | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
⊢ x ≠ 0 → e = greatest_e { val := fmt_val, property := FMT } x → is_greatest_e { val := fmt_val, property := FMT } x e | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = greatest_e { val := fmt_val, property := FMT } x
⊢ is_greatest_e { val := fmt_val, property := FMT } x e | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
⊢ x ≠ 0 → e = greatest_e { val := fmt_val, property := FMT } x → is_greatest_e { val := fmt_val, property := FMT } x e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | simp only [greatest_e, is_greatest_e] at * | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = greatest_e { val := fmt_val, property := FMT } x
⊢ is_greatest_e { val := fmt_val, property := FMT } x e | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ↑fmt_val.r ^ e ≤ abs x ∧ ∀ (e' : ℤ), ↑fmt_val.r ^ e' ≤ abs x → e' ≤ e | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = greatest_e { val := fmt_val, property := FMT } x
⊢ is_greatest_e { val := fmt_val, property := FMT } x e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | apply And.intro | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ↑fmt_val.r ^ e ≤ abs x ∧ ∀ (e' : ℤ), ↑fmt_val.r ^ e' ≤ abs x → e' ≤ e | case left
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ↑fmt_val.r ^ e ≤ abs x
case right
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ∀ (e' : ℤ), ↑fmt_val.r ^ e' ≤ abs x → e' ≤ e | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ↑fmt_val.r ^ e ≤ abs x ∧ ∀ (e' : ℤ), ↑fmt_val.r ^ e' ≤ abs x → e' ≤ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | have R_GT_1 : 1 < fmt_val.r := FMT.left | case left
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ↑fmt_val.r ^ e ≤ abs x
case right
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ∀ (e' : ℤ), ↑fmt_val.r ^ e' ≤ abs x → e' ≤ e | case left
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ↑fmt_val.r ^ e ≤ abs x
case right
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ∀ (e' : ℤ), ↑fmt_val.r ^ e' ≤ abs x → e' ≤ e | Please generate a tactic in lean4 to solve the state.
STATE:
case left
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ↑fmt_val.r ^ e ≤ abs x
case right
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ∀ (e' : ℤ), ↑fmt_val.r ^ e' ≤ abs x → e' ≤ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | suffices e ∈ { z : ℤ | (fmt_val.r : ℝ) ^ z ≤ abs x } by {
have H := this.out;
have R_GT_1 : 1 < fmt_val.r := FMT.left;
norm_cast at H;
} | case left
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ↑fmt_val.r ^ e ≤ abs x
case right
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ∀ (e' : ℤ), ↑fmt_val.r ^ e' ≤ abs x → e' ≤ e | case left
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ e ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
case right
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ∀ (e' : ℤ), ↑fmt_val.r ^ e' ≤ abs x → e' ≤ e | Please generate a tactic in lean4 to solve the state.
STATE:
case left
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ↑fmt_val.r ^ e ≤ abs x
case right
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ∀ (e' : ℤ), ↑fmt_val.r ^ e' ≤ abs x → e' ≤ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | case left =>
norm_cast
rw[he];
apply Int.csupₛ_mem;
case h1 => {
simp[Set.Nonempty];
apply pow_le_real
norm_cast
aesop_subst he
simp_all only [ne_eq, Int.one_lt_zero_le_iff, Int.one_lt_ne_one, not_false_iff]
aesop_subst he
simp_all only [ne_eq, Int.one_lt_zero_le_iff, not_false_iff]
}
case h2 => {
simp [BddAbove]
simp [Set.Nonempty]
simp [upperBounds]
existsi e
intro a ha
have hpa : a ∈ {z : ℤ | (fmt_val.r : ℝ) ^ z ≤ |x| } → a ≤ e := by
simp
norm_cast at *
intro A_IN_SET
sorry
aesop_subst he
simp_all only [ne_eq, Int.one_lt_zero_le_iff, Int.cast_eq_zero, Set.mem_setOf_eq, forall_true_left]
} | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ e ∈ { z | ↑fmt_val.r ^ z ≤ abs x } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ e ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | case right =>
intro e1 hp
aesop_subst he
have he1: e1 ∈ { z | (fmt_val.r: ℝ) ^ z ≤ abs x } := by
simp_all only [ne_eq, Int.cast_eq_zero, Set.mem_setOf_eq]
apply le_csupₛ
case h₂ =>
simp_all only [ne_eq, Int.cast_eq_zero, Set.mem_setOf_eq]
case h₁ =>
sorry | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ∀ (e' : ℤ), ↑fmt_val.r ^ e' ≤ abs x → e' ≤ e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ∀ (e' : ℤ), ↑fmt_val.r ^ e' ≤ abs x → e' ≤ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | have H := this.out | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
this : e ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ↑fmt_val.r ^ e ≤ abs x | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
this : e ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
H : ↑fmt_val.r ^ e ≤ abs x
⊢ ↑fmt_val.r ^ e ≤ abs x | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
this : e ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ↑fmt_val.r ^ e ≤ abs x
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | have R_GT_1 : 1 < fmt_val.r := FMT.left | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
this : e ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
H : ↑fmt_val.r ^ e ≤ abs x
⊢ ↑fmt_val.r ^ e ≤ abs x | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1✝ : 1 < fmt_val.r
this : e ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
H : ↑fmt_val.r ^ e ≤ abs x
R_GT_1 : 1 < fmt_val.r
⊢ ↑fmt_val.r ^ e ≤ abs x | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
this : e ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
H : ↑fmt_val.r ^ e ≤ abs x
⊢ ↑fmt_val.r ^ e ≤ abs x
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | norm_cast at H | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1✝ : 1 < fmt_val.r
this : e ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
H : ↑fmt_val.r ^ e ≤ abs x
R_GT_1 : 1 < fmt_val.r
⊢ ↑fmt_val.r ^ e ≤ abs x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1✝ : 1 < fmt_val.r
this : e ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
H : ↑fmt_val.r ^ e ≤ abs x
R_GT_1 : 1 < fmt_val.r
⊢ ↑fmt_val.r ^ e ≤ abs x
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | rw[he] | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ e ∈ { z | ↑fmt_val.r ^ z ≤ abs x } | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ supₛ { z | ↑fmt_val.r ^ z ≤ abs x } ∈ { z | ↑fmt_val.r ^ z ≤ abs x } | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ e ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | apply Int.csupₛ_mem | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ supₛ { z | ↑fmt_val.r ^ z ≤ abs x } ∈ { z | ↑fmt_val.r ^ z ≤ abs x } | case h1
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ Set.Nonempty { z | ↑fmt_val.r ^ z ≤ abs x }
case h2
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ BddAbove { z | ↑fmt_val.r ^ z ≤ abs x } | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ supₛ { z | ↑fmt_val.r ^ z ≤ abs x } ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | case h1 => {
simp[Set.Nonempty];
apply pow_le_real
norm_cast
aesop_subst he
simp_all only [ne_eq, Int.one_lt_zero_le_iff, Int.one_lt_ne_one, not_false_iff]
aesop_subst he
simp_all only [ne_eq, Int.one_lt_zero_le_iff, not_false_iff]
} | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ Set.Nonempty { z | ↑fmt_val.r ^ z ≤ abs x } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ Set.Nonempty { z | ↑fmt_val.r ^ z ≤ abs x }
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | case h2 => {
simp [BddAbove]
simp [Set.Nonempty]
simp [upperBounds]
existsi e
intro a ha
have hpa : a ∈ {z : ℤ | (fmt_val.r : ℝ) ^ z ≤ |x| } → a ≤ e := by
simp
norm_cast at *
intro A_IN_SET
sorry
aesop_subst he
simp_all only [ne_eq, Int.one_lt_zero_le_iff, Int.cast_eq_zero, Set.mem_setOf_eq, forall_true_left]
} | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ BddAbove { z | ↑fmt_val.r ^ z ≤ abs x } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ BddAbove { z | ↑fmt_val.r ^ z ≤ abs x }
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | simp[Set.Nonempty] | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ Set.Nonempty { z | ↑fmt_val.r ^ z ≤ abs x } | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ∃ z, ↑fmt_val.r ^ z ≤ abs x | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ Set.Nonempty { z | ↑fmt_val.r ^ z ≤ abs x }
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | apply pow_le_real | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ∃ z, ↑fmt_val.r ^ z ≤ abs x | case ha1
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ↑fmt_val.r ≠ 1
case hx0
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ x ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ∃ z, ↑fmt_val.r ^ z ≤ abs x
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | norm_cast | case ha1
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ↑fmt_val.r ≠ 1
case hx0
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ x ≠ 0 | case ha1
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ¬fmt_val.r = 1
case hx0
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ x ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case ha1
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ↑fmt_val.r ≠ 1
case hx0
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ x ≠ 0
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | aesop_subst he | case ha1
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ¬fmt_val.r = 1
case hx0
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ x ≠ 0 | case ha1
fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
R_GT_1 : 1 < fmt_val.r
⊢ ¬fmt_val.r = 1
case hx0
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ x ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case ha1
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ¬fmt_val.r = 1
case hx0
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ x ≠ 0
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | simp_all only [ne_eq, Int.one_lt_zero_le_iff, Int.one_lt_ne_one, not_false_iff] | case ha1
fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
R_GT_1 : 1 < fmt_val.r
⊢ ¬fmt_val.r = 1
case hx0
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ x ≠ 0 | case hx0
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ x ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case ha1
fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
R_GT_1 : 1 < fmt_val.r
⊢ ¬fmt_val.r = 1
case hx0
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ x ≠ 0
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | aesop_subst he | case hx0
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ x ≠ 0 | case hx0
fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
R_GT_1 : 1 < fmt_val.r
⊢ x ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hx0
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ x ≠ 0
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | simp_all only [ne_eq, Int.one_lt_zero_le_iff, not_false_iff] | case hx0
fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
R_GT_1 : 1 < fmt_val.r
⊢ x ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hx0
fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
R_GT_1 : 1 < fmt_val.r
⊢ x ≠ 0
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | simp [BddAbove] | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ BddAbove { z | ↑fmt_val.r ^ z ≤ abs x } | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ Set.Nonempty (upperBounds { z | ↑fmt_val.r ^ z ≤ abs x }) | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ BddAbove { z | ↑fmt_val.r ^ z ≤ abs x }
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | simp [Set.Nonempty] | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ Set.Nonempty (upperBounds { z | ↑fmt_val.r ^ z ≤ abs x }) | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ∃ x_1, x_1 ∈ upperBounds { z | ↑fmt_val.r ^ z ≤ abs x } | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ Set.Nonempty (upperBounds { z | ↑fmt_val.r ^ z ≤ abs x })
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | simp [upperBounds] | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ∃ x_1, x_1 ∈ upperBounds { z | ↑fmt_val.r ^ z ≤ abs x } | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ∃ x_1, ∀ ⦃a : ℤ⦄, ↑fmt_val.r ^ a ≤ abs x → a ≤ x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ∃ x_1, x_1 ∈ upperBounds { z | ↑fmt_val.r ^ z ≤ abs x }
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | existsi e | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ∃ x_1, ∀ ⦃a : ℤ⦄, ↑fmt_val.r ^ a ≤ abs x → a ≤ x_1 | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ∀ ⦃a : ℤ⦄, ↑fmt_val.r ^ a ≤ abs x → a ≤ e | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ∃ x_1, ∀ ⦃a : ℤ⦄, ↑fmt_val.r ^ a ≤ abs x → a ≤ x_1
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | intro a ha | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ∀ ⦃a : ℤ⦄, ↑fmt_val.r ^ a ≤ abs x → a ≤ e | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
⊢ a ≤ e | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
⊢ ∀ ⦃a : ℤ⦄, ↑fmt_val.r ^ a ≤ abs x → a ≤ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | have hpa : a ∈ {z : ℤ | (fmt_val.r : ℝ) ^ z ≤ |x| } → a ≤ e := by
simp
norm_cast at *
intro A_IN_SET
sorry | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
⊢ a ≤ e | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
hpa : a ∈ { z | ↑fmt_val.r ^ z ≤ abs x } → a ≤ e
⊢ a ≤ e | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
⊢ a ≤ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | aesop_subst he | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
hpa : a ∈ { z | ↑fmt_val.r ^ z ≤ abs x } → a ≤ e
⊢ a ≤ e | fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
R_GT_1 : 1 < fmt_val.r
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
hpa : a ∈ { z | ↑fmt_val.r ^ z ≤ abs x } → a ≤ supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ a ≤ supₛ { z | ↑fmt_val.r ^ z ≤ abs x } | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
hpa : a ∈ { z | ↑fmt_val.r ^ z ≤ abs x } → a ≤ e
⊢ a ≤ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | simp_all only [ne_eq, Int.one_lt_zero_le_iff, Int.cast_eq_zero, Set.mem_setOf_eq, forall_true_left] | fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
R_GT_1 : 1 < fmt_val.r
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
hpa : a ∈ { z | ↑fmt_val.r ^ z ≤ abs x } → a ≤ supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ a ≤ supₛ { z | ↑fmt_val.r ^ z ≤ abs x } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
R_GT_1 : 1 < fmt_val.r
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
hpa : a ∈ { z | ↑fmt_val.r ^ z ≤ abs x } → a ≤ supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ a ≤ supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | simp | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
⊢ a ∈ { z | ↑fmt_val.r ^ z ≤ abs x } → a ≤ e | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
⊢ ↑fmt_val.r ^ a ≤ abs x → a ≤ e | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
⊢ a ∈ { z | ↑fmt_val.r ^ z ≤ abs x } → a ≤ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | norm_cast at * | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
⊢ ↑fmt_val.r ^ a ≤ abs x → a ≤ e | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
hx : ¬x = 0
R_GT_1 : 1 < fmt_val.r
⊢ ↑fmt_val.r ^ a ≤ abs x → a ≤ e | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
R_GT_1 : 1 < fmt_val.r
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
⊢ ↑fmt_val.r ^ a ≤ abs x → a ≤ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | intro A_IN_SET | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
hx : ¬x = 0
R_GT_1 : 1 < fmt_val.r
⊢ ↑fmt_val.r ^ a ≤ abs x → a ≤ e | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
hx : ¬x = 0
R_GT_1 : 1 < fmt_val.r
A_IN_SET : ↑fmt_val.r ^ a ≤ abs x
⊢ a ≤ e | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
hx : ¬x = 0
R_GT_1 : 1 < fmt_val.r
⊢ ↑fmt_val.r ^ a ≤ abs x → a ≤ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | sorry | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
hx : ¬x = 0
R_GT_1 : 1 < fmt_val.r
A_IN_SET : ↑fmt_val.r ^ a ≤ abs x
⊢ a ≤ e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
a : ℤ
ha : ↑fmt_val.r ^ a ≤ abs x
hx : ¬x = 0
R_GT_1 : 1 < fmt_val.r
A_IN_SET : ↑fmt_val.r ^ a ≤ abs x
⊢ a ≤ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | intro e1 hp | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ∀ (e' : ℤ), ↑fmt_val.r ^ e' ≤ abs x → e' ≤ e | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
⊢ e1 ≤ e | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ ∀ (e' : ℤ), ↑fmt_val.r ^ e' ≤ abs x → e' ≤ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | aesop_subst he | fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
⊢ e1 ≤ e | fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
⊢ e1 ≤ supₛ { z | ↑fmt_val.r ^ z ≤ abs x } | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
e : ℤ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
he : e = supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
⊢ e1 ≤ e
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | have he1: e1 ∈ { z | (fmt_val.r: ℝ) ^ z ≤ abs x } := by
simp_all only [ne_eq, Int.cast_eq_zero, Set.mem_setOf_eq] | fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
⊢ e1 ≤ supₛ { z | ↑fmt_val.r ^ z ≤ abs x } | fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
he1 : e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ e1 ≤ supₛ { z | ↑fmt_val.r ^ z ≤ abs x } | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
⊢ e1 ≤ supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | apply le_csupₛ | fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
he1 : e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ e1 ≤ supₛ { z | ↑fmt_val.r ^ z ≤ abs x } | case h₁
fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
he1 : e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ BddAbove { z | ↑fmt_val.r ^ z ≤ abs x }
case h₂
fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
he1 : e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x } | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
he1 : e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ e1 ≤ supₛ { z | ↑fmt_val.r ^ z ≤ abs x }
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | case h₂ =>
simp_all only [ne_eq, Int.cast_eq_zero, Set.mem_setOf_eq] | fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
he1 : e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
he1 : e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | case h₁ =>
sorry | fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
he1 : e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ BddAbove { z | ↑fmt_val.r ^ z ≤ abs x } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
he1 : e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ BddAbove { z | ↑fmt_val.r ^ z ≤ abs x }
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | simp_all only [ne_eq, Int.cast_eq_zero, Set.mem_setOf_eq] | fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
⊢ e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
⊢ e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Float_theorem.lean | float_greatest_e_exists | [99, 1] | [155, 12] | simp_all only [ne_eq, Int.cast_eq_zero, Set.mem_setOf_eq] | fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
he1 : e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : flformat
x : ℝ
fmt_val : format
FMT : is_valid_flformat fmt_val
hx : x ≠ 0
e1 : ℤ
hp : ↑fmt_val.r ^ e1 ≤ abs x
he1 : e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
⊢ e1 ∈ { z | ↑fmt_val.r ^ z ≤ abs x }
TACTIC:
|
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