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/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesRealLePowFour | [106, 1] | [116, 15] | apply prodPrimesLePowFour | x : β
hx : 1 β€ x
β’ prodPrimes βxββ β€ 4 ^ (βxββ - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : 1 β€ x
β’ prodPrimes βxββ β€ 4 ^ (βxββ - 1)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesRealLePowFour | [106, 1] | [116, 15] | norm_cast | x : β
hx : 1 β€ x
β’ β(4 ^ (βxββ - 1)) = 4 ^ β(βxββ - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : 1 β€ x
β’ β(4 ^ (βxββ - 1)) = 4 ^ β(βxββ - 1)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesRealLePowFour | [106, 1] | [116, 15] | rw [Real.rpow_le_rpow_left_iff (by norm_num)] | x : β
hx : 1 β€ x
β’ 4 ^ β(βxββ - 1) β€ 4 ^ (x - 1) | x : β
hx : 1 β€ x
β’ β(βxββ - 1) β€ x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : 1 β€ x
β’ 4 ^ β(βxββ - 1) β€ 4 ^ (x - 1)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesRealLePowFour | [106, 1] | [116, 15] | rw [β Nat.floor_sub_one] | x : β
hx : 1 β€ x
β’ β(βxββ - 1) β€ x - 1 | x : β
hx : 1 β€ x
β’ ββx - 1ββ β€ x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : 1 β€ x
β’ β(βxββ - 1) β€ x - 1
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesRealLePowFour | [106, 1] | [116, 15] | apply Nat.floor_le | x : β
hx : 1 β€ x
β’ ββx - 1ββ β€ x - 1 | case ha
x : β
hx : 1 β€ x
β’ 0 β€ x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : 1 β€ x
β’ ββx - 1ββ β€ x - 1
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesRealLePowFour | [106, 1] | [116, 15] | linarith | case ha
x : β
hx : 1 β€ x
β’ 0 β€ x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
x : β
hx : 1 β€ x
β’ 0 β€ x - 1
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/ProdPrimes.lean | prodPrimesRealLePowFour | [106, 1] | [116, 15] | norm_num | x : β
hx : 1 β€ x
β’ 1 < 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : 1 β€ x
β’ 1 < 4
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | allPrimeNums | [9, 1] | [16, 7] | simp (config := {decide := false}) [nums] | β’ List.Allβ Nat.Prime nums | β’ Nat.Prime 2 β§
Nat.Prime 3 β§
Nat.Prime 5 β§
Nat.Prime 7 β§
Nat.Prime 13 β§
Nat.Prime 23 β§
Nat.Prime 43 β§
Nat.Prime 83 β§
Nat.Prime 163 β§ Nat.Prime 317 β§ Nat.Prime 631 β§ Nat.Prime 1259 β§ Nat.Prime 2503 β§ Nat.Prime 4001 | Please generate a tactic in lean4 to solve the state.
STATE:
β’ List.Allβ Nat.Prime nums
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | allPrimeNums | [9, 1] | [16, 7] | norm_num1 | β’ Nat.Prime 2 β§
Nat.Prime 3 β§
Nat.Prime 5 β§
Nat.Prime 7 β§
Nat.Prime 13 β§
Nat.Prime 23 β§
Nat.Prime 43 β§
Nat.Prime 83 β§
Nat.Prime 163 β§ Nat.Prime 317 β§ Nat.Prime 631 β§ Nat.Prime 1259 β§ Nat.Prime 2503 β§ Nat.Prime 4001 | β’ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True | Please generate a tactic in lean4 to solve the state.
STATE:
β’ Nat.Prime 2 β§
Nat.Prime 3 β§
Nat.Prime 5 β§
Nat.Prime 7 β§
Nat.Prime 13 β§
Nat.Prime 23 β§
Nat.Prime 43 β§
Nat.Prime 83 β§
Nat.Prime 163 β§ Nat.Prime 317 β§ Nat.Prime 631 β§ Nat.Prime 1259 β§ Nat.Prime 2503 β§ Nat.Prime 4001
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | allPrimeNums | [9, 1] | [16, 7] | simp | β’ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β’ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True β§ True
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | cases l with
| nil => contradiction
| cons x l =>
induction l generalizing a x with
| nil =>
intro n hn
simp at *
apply And.intro <;> linarith
| cons y l h =>
rw [List.chain_cons] at hl_mono
rw [List.chain_cons] at hl_slow
simp at hb
specialize @h x y hl_mono.right hl_slow.right (List.cons_ne_nil y l) hb
intro n hn
specialize h n
have : β lα΅’, lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l := by simp
simp_rw [this]
simp only [or_and_right]
rw [exists_or]
cases le_or_lt x n with
| inl hx =>
apply Or.inr
simp only [Set.mem_Ico] at *
specialize h (And.intro hx hn.right)
exact h
| inr hx =>
apply Or.inl
simp at *
apply And.intro <;> linarith | a b : β
l : List β
hl_mono : List.Chain (fun u v => u < v) a l
hl_slow : List.Chain (fun u v => v β€ 2 * u) a l
hl_cons : l β []
hb : b < List.getLast l hl_cons
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β l β§ lα΅’ β Set.Ioc n (2 * n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : β
l : List β
hl_mono : List.Chain (fun u v => u < v) a l
hl_slow : List.Chain (fun u v => v β€ 2 * u) a l
hl_cons : l β []
hb : b < List.getLast l hl_cons
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β l β§ lα΅’ β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | contradiction | case nil
a b : β
hl_mono : List.Chain (fun u v => u < v) a []
hl_slow : List.Chain (fun u v => v β€ 2 * u) a []
hl_cons : [] β []
hb : b < List.getLast [] hl_cons
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β [] β§ lα΅’ β Set.Ioc n (2 * n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
a b : β
hl_mono : List.Chain (fun u v => u < v) a []
hl_slow : List.Chain (fun u v => v β€ 2 * u) a []
hl_cons : [] β []
hb : b < List.getLast [] hl_cons
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β [] β§ lα΅’ β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | induction l generalizing a x with
| nil =>
intro n hn
simp at *
apply And.intro <;> linarith
| cons y l h =>
rw [List.chain_cons] at hl_mono
rw [List.chain_cons] at hl_slow
simp at hb
specialize @h x y hl_mono.right hl_slow.right (List.cons_ne_nil y l) hb
intro n hn
specialize h n
have : β lα΅’, lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l := by simp
simp_rw [this]
simp only [or_and_right]
rw [exists_or]
cases le_or_lt x n with
| inl hx =>
apply Or.inr
simp only [Set.mem_Ico] at *
specialize h (And.intro hx hn.right)
exact h
| inr hx =>
apply Or.inl
simp at *
apply And.intro <;> linarith | case cons
a b x : β
l : List β
hl_mono : List.Chain (fun u v => u < v) a (x :: l)
hl_slow : List.Chain (fun u v => v β€ 2 * u) a (x :: l)
hl_cons : x :: l β []
hb : b < List.getLast (x :: l) hl_cons
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: l β§ lα΅’ β Set.Ioc n (2 * n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
a b x : β
l : List β
hl_mono : List.Chain (fun u v => u < v) a (x :: l)
hl_slow : List.Chain (fun u v => v β€ 2 * u) a (x :: l)
hl_cons : x :: l β []
hb : b < List.getLast (x :: l) hl_cons
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: l β§ lα΅’ β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | intro n hn | case cons.nil
b a x : β
hl_mono : List.Chain (fun u v => u < v) a [x]
hl_slow : List.Chain (fun u v => v β€ 2 * u) a [x]
hl_cons : [x] β []
hb : b < List.getLast [x] hl_cons
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β [x] β§ lα΅’ β Set.Ioc n (2 * n) | case cons.nil
b a x : β
hl_mono : List.Chain (fun u v => u < v) a [x]
hl_slow : List.Chain (fun u v => v β€ 2 * u) a [x]
hl_cons : [x] β []
hb : b < List.getLast [x] hl_cons
n : β
hn : n β Set.Icc a b
β’ β lα΅’, lα΅’ β [x] β§ lα΅’ β Set.Ioc n (2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.nil
b a x : β
hl_mono : List.Chain (fun u v => u < v) a [x]
hl_slow : List.Chain (fun u v => v β€ 2 * u) a [x]
hl_cons : [x] β []
hb : b < List.getLast [x] hl_cons
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β [x] β§ lα΅’ β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | simp at * | case cons.nil
b a x : β
hl_mono : List.Chain (fun u v => u < v) a [x]
hl_slow : List.Chain (fun u v => v β€ 2 * u) a [x]
hl_cons : [x] β []
hb : b < List.getLast [x] hl_cons
n : β
hn : n β Set.Icc a b
β’ β lα΅’, lα΅’ β [x] β§ lα΅’ β Set.Ioc n (2 * n) | case cons.nil
b a x : β
hl_cons : [x] β []
hb : b < x
n : β
hl_mono : a < x
hl_slow : x β€ 2 * a
hn : a β€ n β§ n β€ b
β’ n < x β§ x β€ 2 * n | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.nil
b a x : β
hl_mono : List.Chain (fun u v => u < v) a [x]
hl_slow : List.Chain (fun u v => v β€ 2 * u) a [x]
hl_cons : [x] β []
hb : b < List.getLast [x] hl_cons
n : β
hn : n β Set.Icc a b
β’ β lα΅’, lα΅’ β [x] β§ lα΅’ β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | apply And.intro <;> linarith | case cons.nil
b a x : β
hl_cons : [x] β []
hb : b < x
n : β
hl_mono : a < x
hl_slow : x β€ 2 * a
hn : a β€ n β§ n β€ b
β’ n < x β§ x β€ 2 * n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.nil
b a x : β
hl_cons : [x] β []
hb : b < x
n : β
hl_mono : a < x
hl_slow : x β€ 2 * a
hn : a β€ n β§ n β€ b
β’ n < x β§ x β€ 2 * n
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | rw [List.chain_cons] at hl_mono | case cons.cons
b y : β
l : List β
h :
β {a : β} (x : β),
List.Chain (fun u v => u < v) a (x :: l) β
List.Chain (fun u v => v β€ 2 * u) a (x :: l) β
β (hl_cons : x :: l β []),
b < List.getLast (x :: l) hl_cons β β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: l β§ lα΅’ β Set.Ioc n (2 * n)
a x : β
hl_mono : List.Chain (fun u v => u < v) a (x :: y :: l)
hl_slow : List.Chain (fun u v => v β€ 2 * u) a (x :: y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (x :: y :: l) hl_cons
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n) | case cons.cons
b y : β
l : List β
h :
β {a : β} (x : β),
List.Chain (fun u v => u < v) a (x :: l) β
List.Chain (fun u v => v β€ 2 * u) a (x :: l) β
β (hl_cons : x :: l β []),
b < List.getLast (x :: l) hl_cons β β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: l β§ lα΅’ β Set.Ioc n (2 * n)
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : List.Chain (fun u v => v β€ 2 * u) a (x :: y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (x :: y :: l) hl_cons
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons
b y : β
l : List β
h :
β {a : β} (x : β),
List.Chain (fun u v => u < v) a (x :: l) β
List.Chain (fun u v => v β€ 2 * u) a (x :: l) β
β (hl_cons : x :: l β []),
b < List.getLast (x :: l) hl_cons β β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: l β§ lα΅’ β Set.Ioc n (2 * n)
a x : β
hl_mono : List.Chain (fun u v => u < v) a (x :: y :: l)
hl_slow : List.Chain (fun u v => v β€ 2 * u) a (x :: y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (x :: y :: l) hl_cons
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | rw [List.chain_cons] at hl_slow | case cons.cons
b y : β
l : List β
h :
β {a : β} (x : β),
List.Chain (fun u v => u < v) a (x :: l) β
List.Chain (fun u v => v β€ 2 * u) a (x :: l) β
β (hl_cons : x :: l β []),
b < List.getLast (x :: l) hl_cons β β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: l β§ lα΅’ β Set.Ioc n (2 * n)
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : List.Chain (fun u v => v β€ 2 * u) a (x :: y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (x :: y :: l) hl_cons
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n) | case cons.cons
b y : β
l : List β
h :
β {a : β} (x : β),
List.Chain (fun u v => u < v) a (x :: l) β
List.Chain (fun u v => v β€ 2 * u) a (x :: l) β
β (hl_cons : x :: l β []),
b < List.getLast (x :: l) hl_cons β β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: l β§ lα΅’ β Set.Ioc n (2 * n)
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (x :: y :: l) hl_cons
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons
b y : β
l : List β
h :
β {a : β} (x : β),
List.Chain (fun u v => u < v) a (x :: l) β
List.Chain (fun u v => v β€ 2 * u) a (x :: l) β
β (hl_cons : x :: l β []),
b < List.getLast (x :: l) hl_cons β β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: l β§ lα΅’ β Set.Ioc n (2 * n)
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : List.Chain (fun u v => v β€ 2 * u) a (x :: y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (x :: y :: l) hl_cons
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | simp at hb | case cons.cons
b y : β
l : List β
h :
β {a : β} (x : β),
List.Chain (fun u v => u < v) a (x :: l) β
List.Chain (fun u v => v β€ 2 * u) a (x :: l) β
β (hl_cons : x :: l β []),
b < List.getLast (x :: l) hl_cons β β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: l β§ lα΅’ β Set.Ioc n (2 * n)
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (x :: y :: l) hl_cons
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n) | case cons.cons
b y : β
l : List β
h :
β {a : β} (x : β),
List.Chain (fun u v => u < v) a (x :: l) β
List.Chain (fun u v => v β€ 2 * u) a (x :: l) β
β (hl_cons : x :: l β []),
b < List.getLast (x :: l) hl_cons β β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: l β§ lα΅’ β Set.Ioc n (2 * n)
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons
b y : β
l : List β
h :
β {a : β} (x : β),
List.Chain (fun u v => u < v) a (x :: l) β
List.Chain (fun u v => v β€ 2 * u) a (x :: l) β
β (hl_cons : x :: l β []),
b < List.getLast (x :: l) hl_cons β β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: l β§ lα΅’ β Set.Ioc n (2 * n)
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (x :: y :: l) hl_cons
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | specialize @h x y hl_mono.right hl_slow.right (List.cons_ne_nil y l) hb | case cons.cons
b y : β
l : List β
h :
β {a : β} (x : β),
List.Chain (fun u v => u < v) a (x :: l) β
List.Chain (fun u v => v β€ 2 * u) a (x :: l) β
β (hl_cons : x :: l β []),
b < List.getLast (x :: l) hl_cons β β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: l β§ lα΅’ β Set.Ioc n (2 * n)
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n) | case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
h : β (n : β), n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons
b y : β
l : List β
h :
β {a : β} (x : β),
List.Chain (fun u v => u < v) a (x :: l) β
List.Chain (fun u v => v β€ 2 * u) a (x :: l) β
β (hl_cons : x :: l β []),
b < List.getLast (x :: l) hl_cons β β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: l β§ lα΅’ β Set.Ioc n (2 * n)
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | intro n hn | case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
h : β (n : β), n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n) | case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
h : β (n : β), n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
n : β
hn : n β Set.Icc a b
β’ β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
h : β (n : β), n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
β’ β (n : β), n β Set.Icc a b β β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | specialize h n | case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
h : β (n : β), n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
n : β
hn : n β Set.Icc a b
β’ β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n) | case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
β’ β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
h : β (n : β), n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
n : β
hn : n β Set.Icc a b
β’ β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | have : β lα΅’, lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l := by simp | case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
β’ β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n) | case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
β’ β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
β’ β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | simp_rw [this] | case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
β’ β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n) | case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
β’ β lα΅’, (lα΅’ = x β¨ lα΅’ β y :: l) β§ lα΅’ β Set.Ioc n (2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
β’ β lα΅’, lα΅’ β x :: y :: l β§ lα΅’ β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | simp only [or_and_right] | case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
β’ β lα΅’, (lα΅’ = x β¨ lα΅’ β y :: l) β§ lα΅’ β Set.Ioc n (2 * n) | case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
β’ β lα΅’, lα΅’ = x β§ lα΅’ β Set.Ioc n (2 * n) β¨ lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
β’ β lα΅’, (lα΅’ = x β¨ lα΅’ β y :: l) β§ lα΅’ β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | rw [exists_or] | case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
β’ β lα΅’, lα΅’ = x β§ lα΅’ β Set.Ioc n (2 * n) β¨ lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n) | case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
β’ (β x_1, x_1 = x β§ x_1 β Set.Ioc n (2 * n)) β¨ β x, x β y :: l β§ x β Set.Ioc n (2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
β’ β lα΅’, lα΅’ = x β§ lα΅’ β Set.Ioc n (2 * n) β¨ lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | cases le_or_lt x n with
| inl hx =>
apply Or.inr
simp only [Set.mem_Ico] at *
specialize h (And.intro hx hn.right)
exact h
| inr hx =>
apply Or.inl
simp at *
apply And.intro <;> linarith | case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
β’ (β x_1, x_1 = x β§ x_1 β Set.Ioc n (2 * n)) β¨ β x, x β y :: l β§ x β Set.Ioc n (2 * n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
β’ (β x_1, x_1 = x β§ x_1 β Set.Ioc n (2 * n)) β¨ β x, x β y :: l β§ x β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | simp | b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
β’ β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
β’ β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | apply Or.inr | case cons.cons.inl
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
hx : x β€ n
β’ (β x_1, x_1 = x β§ x_1 β Set.Ioc n (2 * n)) β¨ β x, x β y :: l β§ x β Set.Ioc n (2 * n) | case cons.cons.inl.h
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
hx : x β€ n
β’ β x, x β y :: l β§ x β Set.Ioc n (2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons.inl
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
hx : x β€ n
β’ (β x_1, x_1 = x β§ x_1 β Set.Ioc n (2 * n)) β¨ β x, x β y :: l β§ x β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | specialize h (And.intro hx hn.right) | case cons.cons.inl.h
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
hx : x β€ n
β’ β x, x β y :: l β§ x β Set.Ioc n (2 * n) | case cons.cons.inl.h
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
hx : x β€ n
h : β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
β’ β x, x β y :: l β§ x β Set.Ioc n (2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons.inl.h
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
hx : x β€ n
β’ β x, x β y :: l β§ x β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | exact h | case cons.cons.inl.h
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
hx : x β€ n
h : β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
β’ β x, x β y :: l β§ x β Set.Ioc n (2 * n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons.inl.h
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
hx : x β€ n
h : β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
β’ β x, x β y :: l β§ x β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | apply Or.inl | case cons.cons.inr
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
hx : n < x
β’ (β x_1, x_1 = x β§ x_1 β Set.Ioc n (2 * n)) β¨ β x, x β y :: l β§ x β Set.Ioc n (2 * n) | case cons.cons.inr.h
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
hx : n < x
β’ β x_1, x_1 = x β§ x_1 β Set.Ioc n (2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons.inr
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
hx : n < x
β’ (β x_1, x_1 = x β§ x_1 β Set.Ioc n (2 * n)) β¨ β x, x β y :: l β§ x β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | simp at * | case cons.cons.inr.h
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
hx : n < x
β’ β x_1, x_1 = x β§ x_1 β Set.Ioc n (2 * n) | case cons.cons.inr.h
b y : β
l : List β
a x : β
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hx : n < x
hl_mono : a < x β§ x < y β§ List.Chain (fun u v => u < v) y l
hl_slow : x β€ 2 * a β§ y β€ 2 * x β§ List.Chain (fun u v => v β€ 2 * u) y l
hl_cons : True
hn : a β€ n β§ n β€ b
h : x β€ n β n β€ b β n < y β§ y β€ 2 * n β¨ β a, a β l β§ n < a β§ a β€ 2 * n
this : True
β’ n < x β§ x β€ 2 * n | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons.inr.h
b y : β
l : List β
a x : β
hl_mono : a < x β§ List.Chain (fun u v => u < v) x (y :: l)
hl_slow : x β€ 2 * a β§ List.Chain (fun u v => v β€ 2 * u) x (y :: l)
hl_cons : x :: y :: l β []
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hn : n β Set.Icc a b
h : n β Set.Icc x b β β lα΅’, lα΅’ β y :: l β§ lα΅’ β Set.Ioc n (2 * n)
this : β (lα΅’ : β), lα΅’ β x :: y :: l β lα΅’ = x β¨ lα΅’ β y :: l
hx : n < x
β’ β x_1, x_1 = x β§ x_1 β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | existsOfSlowMono | [18, 1] | [52, 37] | apply And.intro <;> linarith | case cons.cons.inr.h
b y : β
l : List β
a x : β
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hx : n < x
hl_mono : a < x β§ x < y β§ List.Chain (fun u v => u < v) y l
hl_slow : x β€ 2 * a β§ y β€ 2 * x β§ List.Chain (fun u v => v β€ 2 * u) y l
hl_cons : True
hn : a β€ n β§ n β€ b
h : x β€ n β n β€ b β n < y β§ y β€ 2 * n β¨ β a, a β l β§ n < a β§ a β€ 2 * n
this : True
β’ n < x β§ x β€ 2 * n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.cons.inr.h
b y : β
l : List β
a x : β
hb : b < List.getLast (y :: l) (_ : Β¬y :: l = [])
n : β
hx : n < x
hl_mono : a < x β§ x < y β§ List.Chain (fun u v => u < v) y l
hl_slow : x β€ 2 * a β§ y β€ 2 * x β§ List.Chain (fun u v => v β€ 2 * u) y l
hl_cons : True
hn : a β€ n β§ n β€ b
h : x β€ n β n β€ b β n < y β§ y β€ 2 * n β¨ β a, a β l β§ n < a β§ a β€ 2 * n
this : True
β’ n < x β§ x β€ 2 * n
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | landauTrick | [54, 1] | [67, 23] | have hp : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2*n) | n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
β’ β p, p β Set.Ioc n (2 * n) β§ Nat.Prime p | case hp
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
β’ n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hp : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β Set.Ioc n (2 * n) β§ Nat.Prime p | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
β’ β p, p β Set.Ioc n (2 * n) β§ Nat.Prime p
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | landauTrick | [54, 1] | [67, 23] | . apply existsOfSlowMono <;> simp [nums] | case hp
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
β’ n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hp : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β Set.Ioc n (2 * n) β§ Nat.Prime p | n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hp : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β Set.Ioc n (2 * n) β§ Nat.Prime p | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
β’ n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hp : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β Set.Ioc n (2 * n) β§ Nat.Prime p
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | landauTrick | [54, 1] | [67, 23] | have hp : β p, p β nums β§ p β Set.Ioc n (2*n) | n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hp : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β Set.Ioc n (2 * n) β§ Nat.Prime p | case hp
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hp : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β nums β§ p β Set.Ioc n (2 * n)
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
hp : β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β Set.Ioc n (2 * n) β§ Nat.Prime p | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hp : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β Set.Ioc n (2 * n) β§ Nat.Prime p
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | landauTrick | [54, 1] | [67, 23] | . apply hp
apply And.intro <;> linarith | case hp
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hp : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β nums β§ p β Set.Ioc n (2 * n)
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
hp : β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β Set.Ioc n (2 * n) β§ Nat.Prime p | n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
hp : β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β Set.Ioc n (2 * n) β§ Nat.Prime p | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hp : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β nums β§ p β Set.Ioc n (2 * n)
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
hp : β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β Set.Ioc n (2 * n) β§ Nat.Prime p
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | landauTrick | [54, 1] | [67, 23] | cases hp with
| intro p hp =>
exists p
apply And.intro hp.right
refine List.allβ_iff_forall.mp ?_ p hp.left
exact allPrimeNums | n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
hp : β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β Set.Ioc n (2 * n) β§ Nat.Prime p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
hp : β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β Set.Ioc n (2 * n) β§ Nat.Prime p
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | landauTrick | [54, 1] | [67, 23] | apply existsOfSlowMono <;> simp [nums] | case hp
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
β’ n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
β’ n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | landauTrick | [54, 1] | [67, 23] | apply hp | case hp
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hp : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β nums β§ p β Set.Ioc n (2 * n) | case hp
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hp : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ n β Set.Icc 1 4000 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hp : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β nums β§ p β Set.Ioc n (2 * n)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | landauTrick | [54, 1] | [67, 23] | apply And.intro <;> linarith | case hp
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hp : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ n β Set.Icc 1 4000 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hp : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
β’ n β Set.Icc 1 4000
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | landauTrick | [54, 1] | [67, 23] | exists p | case intro
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
p : β
hp : p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β Set.Ioc n (2 * n) β§ Nat.Prime p | case intro
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
p : β
hp : p β nums β§ p β Set.Ioc n (2 * n)
β’ p β Set.Ioc n (2 * n) β§ Nat.Prime p | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
p : β
hp : p β nums β§ p β Set.Ioc n (2 * n)
β’ β p, p β Set.Ioc n (2 * n) β§ Nat.Prime p
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | landauTrick | [54, 1] | [67, 23] | apply And.intro hp.right | case intro
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
p : β
hp : p β nums β§ p β Set.Ioc n (2 * n)
β’ p β Set.Ioc n (2 * n) β§ Nat.Prime p | case intro
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
p : β
hp : p β nums β§ p β Set.Ioc n (2 * n)
β’ Nat.Prime p | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
p : β
hp : p β nums β§ p β Set.Ioc n (2 * n)
β’ p β Set.Ioc n (2 * n) β§ Nat.Prime p
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | landauTrick | [54, 1] | [67, 23] | refine List.allβ_iff_forall.mp ?_ p hp.left | case intro
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
p : β
hp : p β nums β§ p β Set.Ioc n (2 * n)
β’ Nat.Prime p | case intro
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
p : β
hp : p β nums β§ p β Set.Ioc n (2 * n)
β’ List.Allβ Nat.Prime nums | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
p : β
hp : p β nums β§ p β Set.Ioc n (2 * n)
β’ Nat.Prime p
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/LandauTrick.lean | landauTrick | [54, 1] | [67, 23] | exact allPrimeNums | case intro
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
p : β
hp : p β nums β§ p β Set.Ioc n (2 * n)
β’ List.Allβ Nat.Prime nums | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : β
hn_pos : 0 < n
hn_lt : n β€ 4000
hpβ : n β Set.Icc 1 4000 β β p, p β nums β§ p β Set.Ioc n (2 * n)
p : β
hp : p β nums β§ p β Set.Ioc n (2 * n)
β’ List.Allβ Nat.Prime nums
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | PrimeRange.prodNeZero | [11, 1] | [12, 43] | simp [prod, PrimeRange, List.mem_filter] | k : β
β’ prod k β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k : β
β’ prod k β 0
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | PrimeRange.prodDvdOfLt | [17, 1] | [22, 17] | intro hk | k p : β
hp : Nat.Prime p
β’ p < k β p β£ prod k | k p : β
hp : Nat.Prime p
hk : p < k
β’ p β£ prod k | Please generate a tactic in lean4 to solve the state.
STATE:
k p : β
hp : Nat.Prime p
β’ p < k β p β£ prod k
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | PrimeRange.prodDvdOfLt | [17, 1] | [22, 17] | apply List.dvd_prod | k p : β
hp : Nat.Prime p
hk : p < k
β’ p β£ prod k | case ha
k p : β
hp : Nat.Prime p
hk : p < k
β’ p β PrimeRange k | Please generate a tactic in lean4 to solve the state.
STATE:
k p : β
hp : Nat.Prime p
hk : p < k
β’ p β£ prod k
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | PrimeRange.prodDvdOfLt | [17, 1] | [22, 17] | simp [PrimeRange, List.mem_filter] | case ha
k p : β
hp : Nat.Prime p
hk : p < k
β’ p β PrimeRange k | case ha
k p : β
hp : Nat.Prime p
hk : p < k
β’ p < k β§ Nat.Prime p | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
k p : β
hp : Nat.Prime p
hk : p < k
β’ p β PrimeRange k
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | PrimeRange.prodDvdOfLt | [17, 1] | [22, 17] | exact β¨hk, hpβ© | case ha
k p : β
hp : Nat.Prime p
hk : p < k
β’ p < k β§ Nat.Prime p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
k p : β
hp : Nat.Prime p
hk : p < k
β’ p < k β§ Nat.Prime p
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | intros | k u : β
β’ 2 β€ u β§ u β€ Nat.succ k β Β¬Nat.Prime (PrimeRange.prod (Nat.succ (Nat.succ k)) + u) | k u : β
aβ : 2 β€ u β§ u β€ Nat.succ k
β’ Β¬Nat.Prime (PrimeRange.prod (Nat.succ (Nat.succ k)) + u) | Please generate a tactic in lean4 to solve the state.
STATE:
k u : β
β’ 2 β€ u β§ u β€ Nat.succ k β Β¬Nat.Prime (PrimeRange.prod (Nat.succ (Nat.succ k)) + u)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | rw [Nat.prime_def_lt'] | k u : β
aβ : 2 β€ u β§ u β€ Nat.succ k
β’ Β¬Nat.Prime (PrimeRange.prod (Nat.succ (Nat.succ k)) + u) | k u : β
aβ : 2 β€ u β§ u β€ Nat.succ k
β’ Β¬(2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§
β (m : β),
2 β€ m β m < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β Β¬m β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u) | Please generate a tactic in lean4 to solve the state.
STATE:
k u : β
aβ : 2 β€ u β§ u β€ Nat.succ k
β’ Β¬Nat.Prime (PrimeRange.prod (Nat.succ (Nat.succ k)) + u)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | simp | k u : β
aβ : 2 β€ u β§ u β€ Nat.succ k
β’ Β¬(2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§
β (m : β),
2 β€ m β m < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β Β¬m β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u) | k u : β
aβ : 2 β€ u β§ u β€ Nat.succ k
β’ 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u β
β x, x < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ x β§ x β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | Please generate a tactic in lean4 to solve the state.
STATE:
k u : β
aβ : 2 β€ u β§ u β€ Nat.succ k
β’ Β¬(2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§
β (m : β),
2 β€ m β m < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β Β¬m β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | intros | k u : β
aβ : 2 β€ u β§ u β€ Nat.succ k
β’ 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u β
β x, x < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ x β§ x β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
β’ β x, x < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ x β§ x β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | Please generate a tactic in lean4 to solve the state.
STATE:
k u : β
aβ : 2 β€ u β§ u β€ Nat.succ k
β’ 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u β
β x, x < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ x β§ x β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | have : u β 1 | k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
β’ β x, x < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ x β§ x β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | case this
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
β’ u β 1
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
β’ β x, x < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ x β§ x β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | Please generate a tactic in lean4 to solve the state.
STATE:
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
β’ β x, x < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ x β§ x β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | . linarith | case this
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
β’ u β 1
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
β’ β x, x < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ x β§ x β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
β’ β x, x < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ x β§ x β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | Please generate a tactic in lean4 to solve the state.
STATE:
case this
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
β’ u β 1
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
β’ β x, x < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ x β§ x β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | cases Nat.exists_prime_and_dvd this with
| intro p hp =>
exists p
have hu_pos : 0 < u
. linarith
have hpu : p β€ u := Nat.le_of_dvd hu_pos hp.right
apply And.intro
. apply lt_of_le_of_lt hpu
apply Nat.lt_add_of_pos_left
apply PrimeRange.prodPos
. apply And.intro
. exact Nat.Prime.two_le hp.left
. apply Nat.dvd_add _ hp.right
apply PrimeRange.prodDvdOfLt hp.left
linarith | k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
β’ β x, x < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ x β§ x β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
β’ β x, x < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ x β§ x β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | linarith | case this
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
β’ u β 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
β’ u β 1
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | exists p | case intro
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
β’ β x, x < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ x β§ x β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | case intro
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
β’ β x, x < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ x β§ x β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | have hu_pos : 0 < u | case intro
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | case hu_pos
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
β’ 0 < u
case intro
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | . linarith | case hu_pos
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
β’ 0 < u
case intro
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | case intro
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | Please generate a tactic in lean4 to solve the state.
STATE:
case hu_pos
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
β’ 0 < u
case intro
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | have hpu : p β€ u := Nat.le_of_dvd hu_pos hp.right | case intro
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | case intro
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | apply And.intro | case intro
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | case intro.left
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u
case intro.right
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u β§ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | . apply lt_of_le_of_lt hpu
apply Nat.lt_add_of_pos_left
apply PrimeRange.prodPos | case intro.left
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u
case intro.right
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | case intro.right
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.left
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u
case intro.right
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | . apply And.intro
. exact Nat.Prime.two_le hp.left
. apply Nat.dvd_add _ hp.right
apply PrimeRange.prodDvdOfLt hp.left
linarith | case intro.right
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.right
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | linarith | case hu_pos
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
β’ 0 < u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hu_pos
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
β’ 0 < u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | apply lt_of_le_of_lt hpu | case intro.left
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u | case intro.left
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ u < PrimeRange.prod (Nat.succ (Nat.succ k)) + u | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.left
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p < PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | apply Nat.lt_add_of_pos_left | case intro.left
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ u < PrimeRange.prod (Nat.succ (Nat.succ k)) + u | case intro.left.h
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ 0 < PrimeRange.prod (Nat.succ (Nat.succ k)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.left
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ u < PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | apply PrimeRange.prodPos | case intro.left.h
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ 0 < PrimeRange.prod (Nat.succ (Nat.succ k)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.left.h
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ 0 < PrimeRange.prod (Nat.succ (Nat.succ k))
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | apply And.intro | case intro.right
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | case intro.right.left
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ 2 β€ p
case intro.right.right
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.right
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ 2 β€ p β§ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | . exact Nat.Prime.two_le hp.left | case intro.right.left
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ 2 β€ p
case intro.right.right
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | case intro.right.right
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.right.left
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ 2 β€ p
case intro.right.right
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | . apply Nat.dvd_add _ hp.right
apply PrimeRange.prodDvdOfLt hp.left
linarith | case intro.right.right
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.right.right
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | exact Nat.Prime.two_le hp.left | case intro.right.left
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ 2 β€ p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.right.left
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ 2 β€ p
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | apply Nat.dvd_add _ hp.right | case intro.right.right
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u | k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.right.right
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | apply PrimeRange.prodDvdOfLt hp.left | k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p β£ PrimeRange.prod (Nat.succ (Nat.succ k)) | k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p < Nat.succ (Nat.succ k) | Please generate a tactic in lean4 to solve the state.
STATE:
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p β£ PrimeRange.prod (Nat.succ (Nat.succ k))
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | gapAfterProdPrimeRange | [27, 1] | [49, 17] | linarith | k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p < Nat.succ (Nat.succ k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k u : β
aβΒΉ : 2 β€ u β§ u β€ Nat.succ k
aβ : 2 β€ PrimeRange.prod (Nat.succ (Nat.succ k)) + u
this : u β 1
p : β
hp : Nat.Prime p β§ p β£ u
hu_pos : 0 < u
hpu : p β€ u
β’ p < Nat.succ (Nat.succ k)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | existsGap | [51, 1] | [57, 11] | exists PrimeRange.prod k.succ.succ + 2 | k : β
β’ β m, β (x : β), x < k β Β¬Nat.Prime (m + x) | k : β
β’ β (x : β), x < k β Β¬Nat.Prime (PrimeRange.prod (Nat.succ (Nat.succ k)) + 2 + x) | Please generate a tactic in lean4 to solve the state.
STATE:
k : β
β’ β m, β (x : β), x < k β Β¬Nat.Prime (m + x)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | existsGap | [51, 1] | [57, 11] | intros x hx | k : β
β’ β (x : β), x < k β Β¬Nat.Prime (PrimeRange.prod (Nat.succ (Nat.succ k)) + 2 + x) | k x : β
hx : x < k
β’ Β¬Nat.Prime (PrimeRange.prod (Nat.succ (Nat.succ k)) + 2 + x) | Please generate a tactic in lean4 to solve the state.
STATE:
k : β
β’ β (x : β), x < k β Β¬Nat.Prime (PrimeRange.prod (Nat.succ (Nat.succ k)) + 2 + x)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | existsGap | [51, 1] | [57, 11] | rw [add_assoc] | k x : β
hx : x < k
β’ Β¬Nat.Prime (PrimeRange.prod (Nat.succ (Nat.succ k)) + 2 + x) | k x : β
hx : x < k
β’ Β¬Nat.Prime (PrimeRange.prod (Nat.succ (Nat.succ k)) + (2 + x)) | Please generate a tactic in lean4 to solve the state.
STATE:
k x : β
hx : x < k
β’ Β¬Nat.Prime (PrimeRange.prod (Nat.succ (Nat.succ k)) + 2 + x)
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | existsGap | [51, 1] | [57, 11] | apply gapAfterProdPrimeRange | k x : β
hx : x < k
β’ Β¬Nat.Prime (PrimeRange.prod (Nat.succ (Nat.succ k)) + (2 + x)) | case a
k x : β
hx : x < k
β’ 2 β€ 2 + x β§ 2 + x β€ Nat.succ k | Please generate a tactic in lean4 to solve the state.
STATE:
k x : β
hx : x < k
β’ Β¬Nat.Prime (PrimeRange.prod (Nat.succ (Nat.succ k)) + (2 + x))
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | existsGap | [51, 1] | [57, 11] | simp | case a
k x : β
hx : x < k
β’ 2 β€ 2 + x β§ 2 + x β€ Nat.succ k | case a
k x : β
hx : x < k
β’ 2 + x β€ Nat.succ k | Please generate a tactic in lean4 to solve the state.
STATE:
case a
k x : β
hx : x < k
β’ 2 β€ 2 + x β§ 2 + x β€ Nat.succ k
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/UnboundedGaps.lean | existsGap | [51, 1] | [57, 11] | linarith | case a
k x : β
hx : x < k
β’ 2 + x β€ Nat.succ k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
k x : β
hx : x < k
β’ 2 + x β€ Nat.succ k
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | powNeOneIff' | [23, 1] | [30, 14] | cases i with
| zero => simp
| succ i =>
simp [not_iff_not]
simp [pow_succ] intro hx
simp [hx] | x i : β
β’ x ^ i β 1 β i β 0 β§ x β 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x i : β
β’ x ^ i β 1 β i β 0 β§ x β 1
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | powNeOneIff' | [23, 1] | [30, 14] | simp | case zero
x : β
β’ x ^ Nat.zero β 1 β Nat.zero β 0 β§ x β 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
x : β
β’ x ^ Nat.zero β 1 β Nat.zero β 0 β§ x β 1
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | powNeOneIff' | [23, 1] | [30, 14] | simp [not_iff_not] | case succ
x i : β
β’ x ^ Nat.succ i β 1 β Nat.succ i β 0 β§ x β 1 | case succ
x i : β
β’ x ^ Nat.succ i = 1 β x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x i : β
β’ x ^ Nat.succ i β 1 β Nat.succ i β 0 β§ x β 1
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | powNeOneIff' | [23, 1] | [30, 14] | simp [pow_succ] | case succ
x i : β
β’ x ^ Nat.succ i = 1 β x = 1 | case succ
x i : β
β’ x = 1 β x ^ i = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x i : β
β’ x ^ Nat.succ i = 1 β x = 1
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | powNeOneIff' | [23, 1] | [30, 14] | intro hx | case succ
x i : β
β’ x = 1 β x ^ i = 1 | case succ
x i : β
hx : x = 1
β’ x ^ i = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x i : β
β’ x = 1 β x ^ i = 1
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | powNeOneIff' | [23, 1] | [30, 14] | simp [hx] | case succ
x i : β
hx : x = 1
β’ x ^ i = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x i : β
hx : x = 1
β’ x ^ i = 1
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | primeOfPowFactorizationNeOne | [34, 1] | [41, 52] | intro hx | x n p : β
β’ p ^ β(Nat.factorization n) x β 1 β Nat.Prime x | x n p : β
hx : p ^ β(Nat.factorization n) x β 1
β’ Nat.Prime x | Please generate a tactic in lean4 to solve the state.
STATE:
x n p : β
β’ p ^ β(Nat.factorization n) x β 1 β Nat.Prime x
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | primeOfPowFactorizationNeOne | [34, 1] | [41, 52] | rw [powNeOneIff'] at hx | x n p : β
hx : p ^ β(Nat.factorization n) x β 1
β’ Nat.Prime x | x n p : β
hx : β(Nat.factorization n) x β 0 β§ p β 1
β’ Nat.Prime x | Please generate a tactic in lean4 to solve the state.
STATE:
x n p : β
hx : p ^ β(Nat.factorization n) x β 1
β’ Nat.Prime x
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | primeOfPowFactorizationNeOne | [34, 1] | [41, 52] | by_contra hp | x n p : β
hx : β(Nat.factorization n) x β 0 β§ p β 1
β’ Nat.Prime x | x n p : β
hx : β(Nat.factorization n) x β 0 β§ p β 1
hp : Β¬Nat.Prime x
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
x n p : β
hx : β(Nat.factorization n) x β 0 β§ p β 1
β’ Nat.Prime x
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | primeOfPowFactorizationNeOne | [34, 1] | [41, 52] | cases hx with | intro h hx =>
apply h
simp
exact Nat.factorization_eq_zero_of_non_prime _ hp | x n p : β
hx : β(Nat.factorization n) x β 0 β§ p β 1
hp : Β¬Nat.Prime x
β’ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x n p : β
hx : β(Nat.factorization n) x β 0 β§ p β 1
hp : Β¬Nat.Prime x
β’ False
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | primeOfPowFactorizationNeOne | [34, 1] | [41, 52] | apply h | case intro
x n p : β
hp : Β¬Nat.Prime x
h : β(Nat.factorization n) x β 0
hx : p β 1
β’ False | case intro
x n p : β
hp : Β¬Nat.Prime x
h : β(Nat.factorization n) x β 0
hx : p β 1
β’ β(Nat.factorization n) x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
x n p : β
hp : Β¬Nat.Prime x
h : β(Nat.factorization n) x β 0
hx : p β 1
β’ False
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | primeOfPowFactorizationNeOne | [34, 1] | [41, 52] | exact Nat.factorization_eq_zero_of_non_prime _ hp | case intro
x n p : β
hp : Β¬Nat.Prime x
h : β(Nat.factorization n) x β 0
hx : p β 1
β’ β(Nat.factorization n) x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
x n p : β
hp : Β¬Nat.Prime x
h : β(Nat.factorization n) x β 0
hx : p β 1
β’ β(Nat.factorization n) x = 0
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.leOne | [47, 1] | [59, 13] | intro hnp | n p : β
hp : Nat.Prime p
β’ 2 * n < p ^ 2 β β(Nat.factorization (Nat.centralBinom n)) p β€ 1 | n p : β
hp : Nat.Prime p
hnp : 2 * n < p ^ 2
β’ β(Nat.factorization (Nat.centralBinom n)) p β€ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
n p : β
hp : Nat.Prime p
β’ 2 * n < p ^ 2 β β(Nat.factorization (Nat.centralBinom n)) p β€ 1
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.leOne | [47, 1] | [59, 13] | apply le_trans (Nat.factorization_choose_le_log) | n p : β
hp : Nat.Prime p
hnp : 2 * n < p ^ 2
β’ β(Nat.factorization (Nat.centralBinom n)) p β€ 1 | n p : β
hp : Nat.Prime p
hnp : 2 * n < p ^ 2
β’ Nat.log p (2 * n) β€ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
n p : β
hp : Nat.Prime p
hnp : 2 * n < p ^ 2
β’ β(Nat.factorization (Nat.centralBinom n)) p β€ 1
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.leOne | [47, 1] | [59, 13] | have hnp := Nat.le_pred_of_lt hnp | n p : β
hp : Nat.Prime p
hnp : 2 * n < p ^ 2
β’ Nat.log p (2 * n) β€ 1 | n p : β
hp : Nat.Prime p
hnpβ : 2 * n < p ^ 2
hnp : 2 * n β€ p ^ 2 - 1
β’ Nat.log p (2 * n) β€ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
n p : β
hp : Nat.Prime p
hnp : 2 * n < p ^ 2
β’ Nat.log p (2 * n) β€ 1
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.leOne | [47, 1] | [59, 13] | apply le_trans (Nat.log_mono_right hnp) | n p : β
hp : Nat.Prime p
hnpβ : 2 * n < p ^ 2
hnp : 2 * n β€ p ^ 2 - 1
β’ Nat.log p (2 * n) β€ 1 | n p : β
hp : Nat.Prime p
hnpβ : 2 * n < p ^ 2
hnp : 2 * n β€ p ^ 2 - 1
β’ Nat.log p (p ^ 2 - 1) β€ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
n p : β
hp : Nat.Prime p
hnpβ : 2 * n < p ^ 2
hnp : 2 * n β€ p ^ 2 - 1
β’ Nat.log p (2 * n) β€ 1
TACTIC:
|
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/CentralBinom.lean | FactorizationCentralBinom.leOne | [47, 1] | [59, 13] | rw [β Nat.lt_succ_iff] | n p : β
hp : Nat.Prime p
hnpβ : 2 * n < p ^ 2
hnp : 2 * n β€ p ^ 2 - 1
β’ Nat.log p (p ^ 2 - 1) β€ 1 | n p : β
hp : Nat.Prime p
hnpβ : 2 * n < p ^ 2
hnp : 2 * n β€ p ^ 2 - 1
β’ Nat.log p (p ^ 2 - 1) < Nat.succ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
n p : β
hp : Nat.Prime p
hnpβ : 2 * n < p ^ 2
hnp : 2 * n β€ p ^ 2 - 1
β’ Nat.log p (p ^ 2 - 1) β€ 1
TACTIC:
|
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