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
|
|---|---|---|---|---|---|---|---|---|
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | quickSort_sorted | [371, 1] | [379, 42] | let sr := quickSort'_sortedRange (arr : Vec α arr.size) | α : Type
inst✝ : Order α
arr : Array α
⊢ sorted (quickSort arr) | α : Type
inst✝ : Order α
arr : Array α
sr : sortedRange (quickSort' ⟨arr, ⋯⟩) 0 (arr.size - 1) := quickSort'_sortedRange ⟨arr, instCoeDepArrayVecSize.proof_1⟩
⊢ sorted (quickSort arr) |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | quickSort_sorted | [371, 1] | [379, 42] | intro i j ij | α : Type
inst✝ : Order α
arr : Array α
sr : sortedRange (quickSort' ⟨arr, ⋯⟩) 0 (arr.size - 1) := quickSort'_sortedRange ⟨arr, instCoeDepArrayVecSize.proof_1⟩
⊢ sorted (quickSort arr) | α : Type
inst✝ : Order α
arr : Array α
sr : sortedRange (quickSort' ⟨arr, ⋯⟩) 0 (arr.size - 1) := quickSort'_sortedRange ⟨arr, instCoeDepArrayVecSize.proof_1⟩
i j : Fin (quickSort arr).size
ij : i ≤ j
⊢ (compare (quickSort arr)[i] (quickSort arr)[j]).isLE = true |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | quickSort_sorted | [371, 1] | [379, 42] | have hi : 0 ≤ i.val := Nat.zero_le i.val | α : Type
inst✝ : Order α
arr : Array α
sr : sortedRange (quickSort' ⟨arr, ⋯⟩) 0 (arr.size - 1) := quickSort'_sortedRange ⟨arr, instCoeDepArrayVecSize.proof_1⟩
i j : Fin (quickSort arr).size
ij : i ≤ j
⊢ (compare (quickSort arr)[i] (quickSort arr)[j]).isLE = true | α : Type
inst✝ : Order α
arr : Array α
sr : sortedRange (quickSort' ⟨arr, ⋯⟩) 0 (arr.size - 1) := quickSort'_sortedRange ⟨arr, instCoeDepArrayVecSize.proof_1⟩
i j : Fin (quickSort arr).size
ij : i ≤ j
hi : 0 ≤ ↑i
⊢ (compare (quickSort arr)[i] (quickSort arr)[j]).isLE = true |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | quickSort_sorted | [371, 1] | [379, 42] | have s : (quickSort arr).size = arr.size := quickSort_size arr | α : Type
inst✝ : Order α
arr : Array α
sr : sortedRange (quickSort' ⟨arr, ⋯⟩) 0 (arr.size - 1) := quickSort'_sortedRange ⟨arr, instCoeDepArrayVecSize.proof_1⟩
i j : Fin (quickSort arr).size
ij : i ≤ j
hi : 0 ≤ ↑i
⊢ (compare (quickSort arr)[i] (quickSort arr)[j]).isLE = true | α : Type
inst✝ : Order α
arr : Array α
sr : sortedRange (quickSort' ⟨arr, ⋯⟩) 0 (arr.size - 1) := quickSort'_sortedRange ⟨arr, instCoeDepArrayVecSize.proof_1⟩
i j : Fin (quickSort arr).size
ij : i ≤ j
hi : 0 ≤ ↑i
s : (quickSort arr).size = arr.size
⊢ (compare (quickSort arr)[i] (quickSort arr)[j]).isLE = true |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | quickSort_sorted | [371, 1] | [379, 42] | have hj : j.val ≤ arr.size - 1 := Nat.le_sub_one_of_lt (s ▸ j.isLt) | α : Type
inst✝ : Order α
arr : Array α
sr : sortedRange (quickSort' ⟨arr, ⋯⟩) 0 (arr.size - 1) := quickSort'_sortedRange ⟨arr, instCoeDepArrayVecSize.proof_1⟩
i j : Fin (quickSort arr).size
ij : i ≤ j
hi : 0 ≤ ↑i
s : (quickSort arr).size = arr.size
⊢ (compare (quickSort arr)[i] (quickSort arr)[j]).isLE = true | α : Type
inst✝ : Order α
arr : Array α
sr : sortedRange (quickSort' ⟨arr, ⋯⟩) 0 (arr.size - 1) := quickSort'_sortedRange ⟨arr, instCoeDepArrayVecSize.proof_1⟩
i j : Fin (quickSort arr).size
ij : i ≤ j
hi : 0 ≤ ↑i
s : (quickSort arr).size = arr.size
hj : ↑j ≤ arr.size - 1
⊢ (compare (quickSort arr)[i] (quickSort arr)[j]).isLE = true |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Sorted.lean | quickSort_sorted | [371, 1] | [379, 42] | exact sr (i.cast s) (j.cast s) hi ij hj | α : Type
inst✝ : Order α
arr : Array α
sr : sortedRange (quickSort' ⟨arr, ⋯⟩) 0 (arr.size - 1) := quickSort'_sortedRange ⟨arr, instCoeDepArrayVecSize.proof_1⟩
i j : Fin (quickSort arr).size
ij : i ≤ j
hi : 0 ≤ ↑i
s : (quickSort arr).size = arr.size
hj : ↑j ≤ arr.size - 1
⊢ (compare (quickSort arr)[i] (quickSort arr)[j]).isLE = true | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/QuickSort.lean | Nat.sub_add_eq_add_sub | [32, 1] | [37, 64] | induction km with
| refl => simp [Nat.add_sub_cancel_left]
| @step m km ih =>
have : k ≤ m + n := Nat.le_trans km (Nat.le_add_right ..)
simp [Nat.succ_sub km, Nat.succ_add, ih, Nat.succ_sub this] | m n k : Nat
km : k ≤ m
⊢ m - k + n = m + n - k | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/QuickSort.lean | Nat.sub_add_eq_add_sub | [32, 1] | [37, 64] | simp [Nat.add_sub_cancel_left] | case refl
m n k : Nat
⊢ k - k + n = k + n - k | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/QuickSort.lean | Nat.sub_add_eq_add_sub | [32, 1] | [37, 64] | have : k ≤ m + n := Nat.le_trans km (Nat.le_add_right ..) | case step
m✝ n k m : Nat
km : k.le m
ih : m - k + n = m + n - k
⊢ m.succ - k + n = m.succ + n - k | case step
m✝ n k m : Nat
km : k.le m
ih : m - k + n = m + n - k
this : k ≤ m + n
⊢ m.succ - k + n = m.succ + n - k |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/QuickSort.lean | Nat.sub_add_eq_add_sub | [32, 1] | [37, 64] | simp [Nat.succ_sub km, Nat.succ_add, ih, Nat.succ_sub this] | case step
m✝ n k m : Nat
km : k.le m
ih : m - k + n = m + n - k
this : k ≤ m + n
⊢ m.succ - k + n = m.succ + n - k | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/QuickSort.lean | Nat.lt_sub_right | [39, 1] | [43, 32] | show m - k + 1 ≤ n - k | m n k : Nat
mk : k ≤ m
mn : m < n
⊢ m - k < n - k | m n k : Nat
mk : k ≤ m
mn : m < n
⊢ m - k + 1 ≤ n - k |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/QuickSort.lean | Nat.lt_sub_right | [39, 1] | [43, 32] | have : m - k + 1 = m + 1 - k := Nat.sub_add_eq_add_sub mk | m n k : Nat
mk : k ≤ m
mn : m < n
⊢ m - k + 1 ≤ n - k | m n k : Nat
mk : k ≤ m
mn : m < n
this : m - k + 1 = m + 1 - k
⊢ m - k + 1 ≤ n - k |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/QuickSort.lean | Nat.lt_sub_right | [39, 1] | [43, 32] | rw [this] | m n k : Nat
mk : k ≤ m
mn : m < n
this : m - k + 1 = m + 1 - k
⊢ m - k + 1 ≤ n - k | m n k : Nat
mk : k ≤ m
mn : m < n
this : m - k + 1 = m + 1 - k
⊢ m + 1 - k ≤ n - k |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/QuickSort.lean | Nat.lt_sub_right | [39, 1] | [43, 32] | apply Nat.sub_le_sub_right mn | m n k : Nat
mk : k ≤ m
mn : m < n
this : m - k + 1 = m + 1 - k
⊢ m + 1 - k ≤ n - k | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Induction.lean | partitionImpl.simp_base | [39, 1] | [46, 29] | assumption | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first i j : Fin n
fi : first ≤ i
ij : i ≤ j
h : ¬first < i
⊢ ↑first ≤ ↑i | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Induction.lean | partitionImpl.simp_base | [39, 1] | [46, 29] | unfold partitionImpl | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first i j : Fin n
fi : first ≤ i
ij : i ≤ j
h : ¬first < i
⊢ partitionImpl arr first i j fi ij = (⟨j, ⋯⟩, arr) | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first i j : Fin n
fi : first ≤ i
ij : i ≤ j
h : ¬first < i
⊢ (if h : first < i then
let_fun this := ⋯;
if arr[i] < arr[first] then
let_fun this_1 := ⋯;
partitionImpl arr first i.prev j this this_1
else
let arr := (dbgTraceIfShared "swap1" arr).swap i j;
match partitionImpl arr first i.prev j.prev this ⋯ with
| (⟨mid, hm⟩, arr) => (⟨mid, ⋯⟩, arr)
else (⟨j, ⋯⟩, arr)) =
(⟨j, ⋯⟩, arr) |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Induction.lean | partitionImpl.simp_base | [39, 1] | [46, 29] | simp [*, dbgTraceIfShared] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first i j : Fin n
fi : first ≤ i
ij : i ≤ j
h : ¬first < i
⊢ (if h : first < i then
let_fun this := ⋯;
if arr[i] < arr[first] then
let_fun this_1 := ⋯;
partitionImpl arr first i.prev j this this_1
else
let arr := (dbgTraceIfShared "swap1" arr).swap i j;
match partitionImpl arr first i.prev j.prev this ⋯ with
| (⟨mid, hm⟩, arr) => (⟨mid, ⋯⟩, arr)
else (⟨j, ⋯⟩, arr)) =
(⟨j, ⋯⟩, arr) | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Induction.lean | partitionImpl.simp_step_lt | [49, 1] | [55, 11] | rw [partitionImpl] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first i j : Fin n
fi : first ≤ i
ij : i ≤ j
h : first < i
x✝ : arr[i] < arr[first]
⊢ partitionImpl arr first i j fi ij = partitionImpl arr first i.prev j ⋯ ⋯ | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first i j : Fin n
fi : first ≤ i
ij : i ≤ j
h : first < i
x✝ : arr[i] < arr[first]
⊢ (if h : first < i then
let_fun this := ⋯;
if arr[i] < arr[first] then
let_fun this_1 := ⋯;
partitionImpl arr first i.prev j this this_1
else
let arr := (dbgTraceIfShared "swap1" arr).swap i j;
match partitionImpl arr first i.prev j.prev this ⋯ with
| (⟨mid, hm⟩, arr) => (⟨mid, ⋯⟩, arr)
else (⟨j, ⋯⟩, arr)) =
partitionImpl arr first i.prev j ⋯ ⋯ |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Induction.lean | partitionImpl.simp_step_lt | [49, 1] | [55, 11] | simp [*] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first i j : Fin n
fi : first ≤ i
ij : i ≤ j
h : first < i
x✝ : arr[i] < arr[first]
⊢ (if h : first < i then
let_fun this := ⋯;
if arr[i] < arr[first] then
let_fun this_1 := ⋯;
partitionImpl arr first i.prev j this this_1
else
let arr := (dbgTraceIfShared "swap1" arr).swap i j;
match partitionImpl arr first i.prev j.prev this ⋯ with
| (⟨mid, hm⟩, arr) => (⟨mid, ⋯⟩, arr)
else (⟨j, ⋯⟩, arr)) =
partitionImpl arr first i.prev j ⋯ ⋯ | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Induction.lean | partitionImpl.simp_step_ge | [58, 1] | [67, 29] | rw [partitionImpl] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first i j : Fin n
fi : first ≤ i
ij : i ≤ j
h : first < i
x✝ : ¬arr[i] < arr[first]
⊢ partitionImpl arr first i j fi ij =
match partitionImpl (arr.swap i j) first i.prev j.prev ⋯ ⋯ with
| (⟨mid, hm⟩, arr) => (⟨mid, ⋯⟩, arr) | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first i j : Fin n
fi : first ≤ i
ij : i ≤ j
h : first < i
x✝ : ¬arr[i] < arr[first]
⊢ (if h : first < i then
let_fun this := ⋯;
if arr[i] < arr[first] then
let_fun this_1 := ⋯;
partitionImpl arr first i.prev j this this_1
else
let arr := (dbgTraceIfShared "swap1" arr).swap i j;
match partitionImpl arr first i.prev j.prev this ⋯ with
| (⟨mid, hm⟩, arr) => (⟨mid, ⋯⟩, arr)
else (⟨j, ⋯⟩, arr)) =
match partitionImpl (arr.swap i j) first i.prev j.prev ⋯ ⋯ with
| (⟨mid, hm⟩, arr) => (⟨mid, ⋯⟩, arr) |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Induction.lean | partitionImpl.simp_step_ge | [58, 1] | [67, 29] | simp [*, dbgTraceIfShared] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first i j : Fin n
fi : first ≤ i
ij : i ≤ j
h : first < i
x✝ : ¬arr[i] < arr[first]
⊢ (if h : first < i then
let_fun this := ⋯;
if arr[i] < arr[first] then
let_fun this_1 := ⋯;
partitionImpl arr first i.prev j this this_1
else
let arr := (dbgTraceIfShared "swap1" arr).swap i j;
match partitionImpl arr first i.prev j.prev this ⋯ with
| (⟨mid, hm⟩, arr) => (⟨mid, ⋯⟩, arr)
else (⟨j, ⋯⟩, arr)) =
match partitionImpl (arr.swap i j) first i.prev j.prev ⋯ ⋯ with
| (⟨mid, hm⟩, arr) => (⟨mid, ⋯⟩, arr) | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Induction.lean | quickSortImpl.simp_base | [102, 1] | [107, 11] | unfold quickSortImpl | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
h : ¬first < ↑last
⊢ quickSortImpl arr first last = arr | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
h : ¬first < ↑last
⊢ (if lt : first < ↑last then
let parted := partition arr ⟨first, ⋯⟩ last ⋯;
let mid := parted.fst.val;
let hm := ⋯;
let arr_1 := parted.snd;
let_fun this := ⋯;
let_fun this_1 := ⋯;
let arr_2 := quickSortImpl arr_1 first mid.prev;
quickSortImpl arr_2 (↑mid + 1) last
else arr) =
arr |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Induction.lean | quickSortImpl.simp_base | [102, 1] | [107, 11] | simp [*] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
h : ¬first < ↑last
⊢ (if lt : first < ↑last then
let parted := partition arr ⟨first, ⋯⟩ last ⋯;
let mid := parted.fst.val;
let hm := ⋯;
let arr_1 := parted.snd;
let_fun this := ⋯;
let_fun this_1 := ⋯;
let arr_2 := quickSortImpl arr_1 first mid.prev;
quickSortImpl arr_2 (↑mid + 1) last
else arr) =
arr | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Induction.lean | quickSortImpl.simp_step | [110, 1] | [117, 11] | rw [quickSortImpl] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
⊢ let parted := partition arr ⟨first, ⋯⟩ last ⋯;
let mid := parted.fst.val;
quickSortImpl arr first last = quickSortImpl (quickSortImpl parted.snd first mid.prev) (↑mid + 1) last | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
⊢ let parted := partition arr ⟨first, ⋯⟩ last ⋯;
let mid := parted.fst.val;
(if lt : first < ↑last then
let parted := partition arr ⟨first, ⋯⟩ last ⋯;
let mid := parted.fst.val;
let hm := ⋯;
let arr_1 := parted.snd;
let_fun this := ⋯;
let_fun this_1 := ⋯;
let arr_2 := quickSortImpl arr_1 first mid.prev;
quickSortImpl arr_2 (↑mid + 1) last
else arr) =
quickSortImpl (quickSortImpl parted.snd first mid.prev) (↑mid + 1) last |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Induction.lean | quickSortImpl.simp_step | [110, 1] | [117, 11] | simp [*] | α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
⊢ let parted := partition arr ⟨first, ⋯⟩ last ⋯;
let mid := parted.fst.val;
(if lt : first < ↑last then
let parted := partition arr ⟨first, ⋯⟩ last ⋯;
let mid := parted.fst.val;
let hm := ⋯;
let arr_1 := parted.snd;
let_fun this := ⋯;
let_fun this_1 := ⋯;
let arr_2 := quickSortImpl arr_1 first mid.prev;
quickSortImpl arr_2 (↑mid + 1) last
else arr) =
quickSortImpl (quickSortImpl parted.snd first mid.prev) (↑mid + 1) last | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Util.lean | prodPrimesSplit | [11, 1] | [20, 46] | rw [← Nat.Ico_succ_succ] | b a : ℕ
ha : a ≤ b
⊢ prodPrimes b = prodPrimes a * ∏ p in primes (Finset.Ioc a b), p | b a : ℕ
ha : a ≤ b
⊢ prodPrimes b = prodPrimes a * ∏ p in primes (Finset.Ico (Nat.succ a) (Nat.succ b)), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Util.lean | prodPrimesSplit | [11, 1] | [20, 46] | simp [prodPrimes, primes] | b a : ℕ
ha : a ≤ b
⊢ prodPrimes b = prodPrimes a * ∏ p in primes (Finset.Ico (Nat.succ a) (Nat.succ b)), p | b a : ℕ
ha : a ≤ b
⊢ ∏ p in Finset.filter Nat.Prime (Finset.range (Nat.succ b)), p =
(∏ p in Finset.filter Nat.Prime (Finset.range (Nat.succ a)), p) *
∏ p in Finset.filter Nat.Prime (Finset.Ico (Nat.succ a) (Nat.succ b)), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Util.lean | prodPrimesSplit | [11, 1] | [20, 46] | rw [Finset.range_eq_Ico] | b a : ℕ
ha : a ≤ b
⊢ ∏ p in Finset.filter Nat.Prime (Finset.range (Nat.succ b)), p =
(∏ p in Finset.filter Nat.Prime (Finset.range (Nat.succ a)), p) *
∏ p in Finset.filter Nat.Prime (Finset.Ico (Nat.succ a) (Nat.succ b)), p | b a : ℕ
ha : a ≤ b
⊢ ∏ p in Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ b)), p =
(∏ p in Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ a)), p) *
∏ p in Finset.filter Nat.Prime (Finset.Ico (Nat.succ a) (Nat.succ b)), p |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Util.lean | prodPrimesSplit | [11, 1] | [20, 46] | rw [← Finset.prod_union] | b a : ℕ
ha : a ≤ b
⊢ ∏ p in Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ b)), p =
(∏ p in Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ a)), p) *
∏ p in Finset.filter Nat.Prime (Finset.Ico (Nat.succ a) (Nat.succ b)), p | b a : ℕ
ha : a ≤ b
⊢ ∏ p in Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ b)), p =
∏ x in
Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ a)) ∪
Finset.filter Nat.Prime (Finset.Ico (Nat.succ a) (Nat.succ b)),
x
b a : ℕ
ha : a ≤ b
⊢ Disjoint (Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ a)))
(Finset.filter Nat.Prime (Finset.Ico (Nat.succ a) (Nat.succ b))) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Util.lean | prodPrimesSplit | [11, 1] | [20, 46] | . rw [← Finset.filter_union]
rw [Finset.Ico_union_Ico_eq_Ico (Nat.zero_le a.succ) (Nat.succ_le_succ ha)] | b a : ℕ
ha : a ≤ b
⊢ ∏ p in Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ b)), p =
∏ x in
Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ a)) ∪
Finset.filter Nat.Prime (Finset.Ico (Nat.succ a) (Nat.succ b)),
x
b a : ℕ
ha : a ≤ b
⊢ Disjoint (Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ a)))
(Finset.filter Nat.Prime (Finset.Ico (Nat.succ a) (Nat.succ b))) | b a : ℕ
ha : a ≤ b
⊢ Disjoint (Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ a)))
(Finset.filter Nat.Prime (Finset.Ico (Nat.succ a) (Nat.succ b))) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Util.lean | prodPrimesSplit | [11, 1] | [20, 46] | . apply Finset.disjoint_filter_filter
apply Finset.Ico_disjoint_Ico_consecutive | b a : ℕ
ha : a ≤ b
⊢ Disjoint (Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ a)))
(Finset.filter Nat.Prime (Finset.Ico (Nat.succ a) (Nat.succ b))) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Util.lean | prodPrimesSplit | [11, 1] | [20, 46] | rw [← Finset.filter_union] | b a : ℕ
ha : a ≤ b
⊢ ∏ p in Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ b)), p =
∏ x in
Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ a)) ∪
Finset.filter Nat.Prime (Finset.Ico (Nat.succ a) (Nat.succ b)),
x | b a : ℕ
ha : a ≤ b
⊢ ∏ p in Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ b)), p =
∏ x in Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ a) ∪ Finset.Ico (Nat.succ a) (Nat.succ b)), x |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Util.lean | prodPrimesSplit | [11, 1] | [20, 46] | rw [Finset.Ico_union_Ico_eq_Ico (Nat.zero_le a.succ) (Nat.succ_le_succ ha)] | b a : ℕ
ha : a ≤ b
⊢ ∏ p in Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ b)), p =
∏ x in Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ a) ∪ Finset.Ico (Nat.succ a) (Nat.succ b)), x | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Util.lean | prodPrimesSplit | [11, 1] | [20, 46] | apply Finset.disjoint_filter_filter | b a : ℕ
ha : a ≤ b
⊢ Disjoint (Finset.filter Nat.Prime (Finset.Ico 0 (Nat.succ a)))
(Finset.filter Nat.Prime (Finset.Ico (Nat.succ a) (Nat.succ b))) | case a
b a : ℕ
ha : a ≤ b
⊢ Disjoint (Finset.Ico 0 (Nat.succ a)) (Finset.Ico (Nat.succ a) (Nat.succ b)) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Util.lean | prodPrimesSplit | [11, 1] | [20, 46] | apply Finset.Ico_disjoint_Ico_consecutive | case a
b a : ℕ
ha : a ≤ b
⊢ Disjoint (Finset.Ico 0 (Nat.succ a)) (Finset.Ico (Nat.succ a) (Nat.succ b)) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | natRealCoeLtCoe | [20, 1] | [20, 71] | simp | a b : ℕ
⊢ ↑a < ↑b ↔ a < b | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | natRealCoeLeCoe | [22, 1] | [22, 71] | simp | a b : ℕ
⊢ ↑a ≤ ↑b ↔ a ≤ b | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | rw [← natRealCoeLeCoe] at h | n : ℕ
hn : 0 < n
h : 4 ^ n ≤ (2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3)
⊢ 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | have : 4^(↑n/3 : ℝ) = 4^(↑n : ℝ) / 4^(2*↑n/3 : ℝ) | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (↑n / 3) = 4 ^ ↑n / 4 ^ (2 * ↑n / 3)
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
this : 4 ^ (↑n / 3) = 4 ^ ↑n / 4 ^ (2 * ↑n / 3)
⊢ 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | . rw [eq_div_iff]
swap; {apply ne_of_gt; apply Real.rpow_pos_of_pos (by norm_num)}
rw [← Real.rpow_add (by norm_num)]
simp [Real.rpow_def_of_pos] norm_num
push_cast
simp [div_add_div_same]
norm_cast
rw [(by linarith : n + 2*n = n*3)]
simp | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (↑n / 3) = 4 ^ ↑n / 4 ^ (2 * ↑n / 3)
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
this : 4 ^ (↑n / 3) = 4 ^ ↑n / 4 ^ (2 * ↑n / 3)
⊢ 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
this : 4 ^ (↑n / 3) = 4 ^ ↑n / 4 ^ (2 * ↑n / 3)
⊢ 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | rw [this] | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
this : 4 ^ (↑n / 3) = 4 ^ ↑n / 4 ^ (2 * ↑n / 3)
⊢ 4 ^ (↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
this : 4 ^ (↑n / 3) = 4 ^ ↑n / 4 ^ (2 * ↑n / 3)
⊢ 4 ^ ↑n / 4 ^ (2 * ↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | clear this | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
this : 4 ^ (↑n / 3) = 4 ^ ↑n / 4 ^ (2 * ↑n / 3)
⊢ 4 ^ ↑n / 4 ^ (2 * ↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ ↑n / 4 ^ (2 * ↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | rw [div_le_iff] | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ ↑n / 4 ^ (2 * ↑n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ ↑n ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) * 4 ^ (2 * ↑n / 3)
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 0 < 4 ^ (2 * ↑n / 3) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | swap | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ ↑n ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) * 4 ^ (2 * ↑n / 3)
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 0 < 4 ^ (2 * ↑n / 3) | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 0 < 4 ^ (2 * ↑n / 3)
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ ↑n ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) * 4 ^ (2 * ↑n / 3) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | {apply Real.rpow_pos_of_pos; norm_num} | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 0 < 4 ^ (2 * ↑n / 3)
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ ↑n ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) * 4 ^ (2 * ↑n / 3) | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ ↑n ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) * 4 ^ (2 * ↑n / 3) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | rw [Real.rpow_nat_cast] | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ ↑n ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) * 4 ^ (2 * ↑n / 3) | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ n ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) * 4 ^ (2 * ↑n / 3) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | norm_cast | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ n ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) * 4 ^ (2 * ↑n / 3) | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ ↑(4 ^ n) ≤ ↑(2 * n) ^ (1 + Real.sqrt ↑(2 * n)) * 4 ^ (↑(2 * n) / 3) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | apply le_trans h | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ ↑(4 ^ n) ≤ ↑(2 * n) ^ (1 + Real.sqrt ↑(2 * n)) * 4 ^ (↑(2 * n) / 3) | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3)) ≤
↑(2 * n) ^ (1 + Real.sqrt ↑(2 * n)) * 4 ^ (↑(2 * n) / 3) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | clear h | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3)) ≤
↑(2 * n) ^ (1 + Real.sqrt ↑(2 * n)) * 4 ^ (↑(2 * n) / 3) | n : ℕ
hn : 0 < n
⊢ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3)) ≤
↑(2 * n) ^ (1 + Real.sqrt ↑(2 * n)) * 4 ^ (↑(2 * n) / 3) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | push_cast | n : ℕ
hn : 0 < n
⊢ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3)) ≤
↑(2 * n) ^ (1 + Real.sqrt ↑(2 * n)) * 4 ^ (↑(2 * n) / 3) | n : ℕ
hn : 0 < n
⊢ (2 * ↑n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) * 4 ^ (2 * ↑n / 3) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | apply mul_le_mul | n : ℕ
hn : 0 < n
⊢ (2 * ↑n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) * 4 ^ (2 * ↑n / 3) | case h₁
n : ℕ
hn : 0 < n
⊢ (2 * ↑n) ^ (1 + Nat.sqrt (2 * n)) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
case h₂
n : ℕ
hn : 0 < n
⊢ 4 ^ Nat.pred (2 * n / 3) ≤ 4 ^ (2 * ↑n / 3)
case c0
n : ℕ
hn : 0 < n
⊢ 0 ≤ 4 ^ Nat.pred (2 * n / 3)
case b0
n : ℕ
hn : 0 < n
⊢ 0 ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | . rw [← Real.rpow_nat_cast]
rw [Real.rpow_le_rpow_left_iff (by norm_cast; linarith)]
simp [-Real.sqrt_mul']
norm_cast
apply Real.nat_sqrt_le_real_sqrt | case h₁
n : ℕ
hn : 0 < n
⊢ (2 * ↑n) ^ (1 + Nat.sqrt (2 * n)) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n))
case h₂
n : ℕ
hn : 0 < n
⊢ 4 ^ Nat.pred (2 * n / 3) ≤ 4 ^ (2 * ↑n / 3)
case c0
n : ℕ
hn : 0 < n
⊢ 0 ≤ 4 ^ Nat.pred (2 * n / 3)
case b0
n : ℕ
hn : 0 < n
⊢ 0 ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) | case h₂
n : ℕ
hn : 0 < n
⊢ 4 ^ Nat.pred (2 * n / 3) ≤ 4 ^ (2 * ↑n / 3)
case c0
n : ℕ
hn : 0 < n
⊢ 0 ≤ 4 ^ Nat.pred (2 * n / 3)
case b0
n : ℕ
hn : 0 < n
⊢ 0 ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | . rw [← Real.rpow_nat_cast]
rw [Real.rpow_le_rpow_left_iff (by norm_cast)]
rw [le_div_iff (by norm_num)]
norm_cast
apply Nat.le_trans (Nat.mul_le_mul_right 3 (Nat.pred_le _))
apply Nat.div_mul_le_self | case h₂
n : ℕ
hn : 0 < n
⊢ 4 ^ Nat.pred (2 * n / 3) ≤ 4 ^ (2 * ↑n / 3)
case c0
n : ℕ
hn : 0 < n
⊢ 0 ≤ 4 ^ Nat.pred (2 * n / 3)
case b0
n : ℕ
hn : 0 < n
⊢ 0 ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) | case c0
n : ℕ
hn : 0 < n
⊢ 0 ≤ 4 ^ Nat.pred (2 * n / 3)
case b0
n : ℕ
hn : 0 < n
⊢ 0 ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | . apply pow_nonneg (by norm_num) | case c0
n : ℕ
hn : 0 < n
⊢ 0 ≤ 4 ^ Nat.pred (2 * n / 3)
case b0
n : ℕ
hn : 0 < n
⊢ 0 ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) | case b0
n : ℕ
hn : 0 < n
⊢ 0 ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | . apply Real.rpow_nonneg_of_nonneg (by norm_cast; linarith) | case b0
n : ℕ
hn : 0 < n
⊢ 0 ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | rw [eq_div_iff] | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (↑n / 3) = 4 ^ ↑n / 4 ^ (2 * ↑n / 3) | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (↑n / 3) * 4 ^ (2 * ↑n / 3) = 4 ^ ↑n
case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (2 * ↑n / 3) ≠ 0 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | swap | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (↑n / 3) * 4 ^ (2 * ↑n / 3) = 4 ^ ↑n
case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (2 * ↑n / 3) ≠ 0 | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (2 * ↑n / 3) ≠ 0
case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (↑n / 3) * 4 ^ (2 * ↑n / 3) = 4 ^ ↑n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | {apply ne_of_gt; apply Real.rpow_pos_of_pos (by norm_num)} | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (2 * ↑n / 3) ≠ 0
case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (↑n / 3) * 4 ^ (2 * ↑n / 3) = 4 ^ ↑n | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (↑n / 3) * 4 ^ (2 * ↑n / 3) = 4 ^ ↑n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | rw [← Real.rpow_add (by norm_num)] | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (↑n / 3) * 4 ^ (2 * ↑n / 3) = 4 ^ ↑n | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (↑n / 3 + 2 * ↑n / 3) = 4 ^ ↑n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | simp [Real.rpow_def_of_pos] | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (↑n / 3 + 2 * ↑n / 3) = 4 ^ ↑n | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ ↑n / 3 + 2 * ↑n / 3 = ↑n ∨ 4 = -1 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | norm_num | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ ↑n / 3 + 2 * ↑n / 3 = ↑n ∨ 4 = -1 | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ ↑n / 3 + 2 * ↑n / 3 = ↑n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | simp [div_add_div_same] | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ ↑n / 3 + 2 * ↑n / 3 = ↑n | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ (↑n + 2 * ↑n) / 3 = ↑n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | norm_cast | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ (↑n + 2 * ↑n) / 3 = ↑n | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ ↑(n + 2 * n) / 3 = ↑n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | rw [(by linarith : n + 2*n = n*3)] | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ ↑(n + 2 * n) / 3 = ↑n | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ ↑(n * 3) / 3 = ↑n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | simp | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ ↑(n * 3) / 3 = ↑n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | apply ne_of_gt | case this
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 4 ^ (2 * ↑n / 3) ≠ 0 | case this.h
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 0 < 4 ^ (2 * ↑n / 3) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | apply Real.rpow_pos_of_pos (by norm_num) | case this.h
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 0 < 4 ^ (2 * ↑n / 3) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | norm_num | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 0 < 4 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | linarith | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ n + 2 * n = n * 3 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | apply Real.rpow_pos_of_pos | n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 0 < 4 ^ (2 * ↑n / 3) | case hx
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 0 < 4 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | norm_num | case hx
n : ℕ
hn : 0 < n
h : ↑(4 ^ n) ≤ ↑((2 * n) ^ (1 + Nat.sqrt (2 * n)) * 4 ^ Nat.pred (2 * n / 3))
⊢ 0 < 4 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | rw [← Real.rpow_nat_cast] | case h₁
n : ℕ
hn : 0 < n
⊢ (2 * ↑n) ^ (1 + Nat.sqrt (2 * n)) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) | case h₁
n : ℕ
hn : 0 < n
⊢ (2 * ↑n) ^ ↑(1 + Nat.sqrt (2 * n)) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | rw [Real.rpow_le_rpow_left_iff (by norm_cast; linarith)] | case h₁
n : ℕ
hn : 0 < n
⊢ (2 * ↑n) ^ ↑(1 + Nat.sqrt (2 * n)) ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) | case h₁
n : ℕ
hn : 0 < n
⊢ ↑(1 + Nat.sqrt (2 * n)) ≤ 1 + Real.sqrt (2 * ↑n) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | simp [-Real.sqrt_mul'] | case h₁
n : ℕ
hn : 0 < n
⊢ ↑(1 + Nat.sqrt (2 * n)) ≤ 1 + Real.sqrt (2 * ↑n) | case h₁
n : ℕ
hn : 0 < n
⊢ ↑(Nat.sqrt (2 * n)) ≤ Real.sqrt (2 * ↑n) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | norm_cast | case h₁
n : ℕ
hn : 0 < n
⊢ ↑(Nat.sqrt (2 * n)) ≤ Real.sqrt (2 * ↑n) | case h₁
n : ℕ
hn : 0 < n
⊢ ↑(Nat.sqrt (2 * n)) ≤ Real.sqrt ↑(2 * n) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | apply Real.nat_sqrt_le_real_sqrt | case h₁
n : ℕ
hn : 0 < n
⊢ ↑(Nat.sqrt (2 * n)) ≤ Real.sqrt ↑(2 * n) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | norm_cast | n : ℕ
hn : 0 < n
⊢ 1 < 2 * ↑n | n : ℕ
hn : 0 < n
⊢ 1 < 2 * n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | linarith | n : ℕ
hn : 0 < n
⊢ 1 < 2 * n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | rw [← Real.rpow_nat_cast] | case h₂
n : ℕ
hn : 0 < n
⊢ 4 ^ Nat.pred (2 * n / 3) ≤ 4 ^ (2 * ↑n / 3) | case h₂
n : ℕ
hn : 0 < n
⊢ 4 ^ ↑(Nat.pred (2 * n / 3)) ≤ 4 ^ (2 * ↑n / 3) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | rw [Real.rpow_le_rpow_left_iff (by norm_cast)] | case h₂
n : ℕ
hn : 0 < n
⊢ 4 ^ ↑(Nat.pred (2 * n / 3)) ≤ 4 ^ (2 * ↑n / 3) | case h₂
n : ℕ
hn : 0 < n
⊢ ↑(Nat.pred (2 * n / 3)) ≤ 2 * ↑n / 3 |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | rw [le_div_iff (by norm_num)] | case h₂
n : ℕ
hn : 0 < n
⊢ ↑(Nat.pred (2 * n / 3)) ≤ 2 * ↑n / 3 | case h₂
n : ℕ
hn : 0 < n
⊢ ↑(Nat.pred (2 * n / 3)) * 3 ≤ 2 * ↑n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | norm_cast | case h₂
n : ℕ
hn : 0 < n
⊢ ↑(Nat.pred (2 * n / 3)) * 3 ≤ 2 * ↑n | case h₂
n : ℕ
hn : 0 < n
⊢ Nat.pred (2 * n / 3) * 3 ≤ 2 * n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | apply Nat.le_trans (Nat.mul_le_mul_right 3 (Nat.pred_le _)) | case h₂
n : ℕ
hn : 0 < n
⊢ Nat.pred (2 * n / 3) * 3 ≤ 2 * n | case h₂
n : ℕ
hn : 0 < n
⊢ 2 * n / 3 * 3 ≤ 2 * n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | apply Nat.div_mul_le_self | case h₂
n : ℕ
hn : 0 < n
⊢ 2 * n / 3 * 3 ≤ 2 * n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | norm_cast | n : ℕ
hn : 0 < n
⊢ 1 < 4 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | norm_num | n : ℕ
hn : 0 < n
⊢ 0 < 3 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | apply pow_nonneg (by norm_num) | case c0
n : ℕ
hn : 0 < n
⊢ 0 ≤ 4 ^ Nat.pred (2 * n / 3) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | norm_num | n : ℕ
hn : 0 < n
⊢ 0 ≤ 4 | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | apply Real.rpow_nonneg_of_nonneg (by norm_cast; linarith) | case b0
n : ℕ
hn : 0 < n
⊢ 0 ≤ (2 * ↑n) ^ (1 + Real.sqrt (2 * ↑n)) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | norm_cast | n : ℕ
hn : 0 < n
⊢ 0 ≤ 2 * ↑n | n : ℕ
hn : 0 < n
⊢ 0 ≤ 2 * n |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | rpowDivThreeLe | [24, 1] | [58, 62] | linarith | n : ℕ
hn : 0 < n
⊢ 0 ≤ 2 * n | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | succLtTwoPow | [60, 1] | [71, 38] | replace h2 := Nat.exists_eq_add_of_le h2 | a : ℕ
h2 : 2 ≤ a
⊢ Nat.succ a < 2 ^ a | a : ℕ
h2 : ∃ k, a = 2 + k
⊢ Nat.succ a < 2 ^ a |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | succLtTwoPow | [60, 1] | [71, 38] | cases h2 with | intro u hu =>
rw [hu]
clear a hu
induction u with
| zero => simp
| succ u h =>
rw [Nat.add_succ, Nat.pow_succ, Nat.mul_two]
rw [← Nat.add_one]
apply Nat.add_lt_add h
apply Nat.one_lt_pow <;> linarith | a : ℕ
h2 : ∃ k, a = 2 + k
⊢ Nat.succ a < 2 ^ a | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | succLtTwoPow | [60, 1] | [71, 38] | rw [hu] | case intro
a u : ℕ
hu : a = 2 + u
⊢ Nat.succ a < 2 ^ a | case intro
a u : ℕ
hu : a = 2 + u
⊢ Nat.succ (2 + u) < 2 ^ (2 + u) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | succLtTwoPow | [60, 1] | [71, 38] | clear a hu | case intro
a u : ℕ
hu : a = 2 + u
⊢ Nat.succ (2 + u) < 2 ^ (2 + u) | case intro
u : ℕ
⊢ Nat.succ (2 + u) < 2 ^ (2 + u) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | succLtTwoPow | [60, 1] | [71, 38] | induction u with
| zero => simp
| succ u h =>
rw [Nat.add_succ, Nat.pow_succ, Nat.mul_two]
rw [← Nat.add_one]
apply Nat.add_lt_add h
apply Nat.one_lt_pow <;> linarith | case intro
u : ℕ
⊢ Nat.succ (2 + u) < 2 ^ (2 + u) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | succLtTwoPow | [60, 1] | [71, 38] | simp | case intro.zero
⊢ Nat.succ (2 + Nat.zero) < 2 ^ (2 + Nat.zero) | no goals |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | succLtTwoPow | [60, 1] | [71, 38] | rw [Nat.add_succ, Nat.pow_succ, Nat.mul_two] | case intro.succ
u : ℕ
h : Nat.succ (2 + u) < 2 ^ (2 + u)
⊢ Nat.succ (2 + Nat.succ u) < 2 ^ (2 + Nat.succ u) | case intro.succ
u : ℕ
h : Nat.succ (2 + u) < 2 ^ (2 + u)
⊢ Nat.succ (Nat.succ (2 + u)) < 2 ^ (2 + u) + 2 ^ (2 + u) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | succLtTwoPow | [60, 1] | [71, 38] | rw [← Nat.add_one] | case intro.succ
u : ℕ
h : Nat.succ (2 + u) < 2 ^ (2 + u)
⊢ Nat.succ (Nat.succ (2 + u)) < 2 ^ (2 + u) + 2 ^ (2 + u) | case intro.succ
u : ℕ
h : Nat.succ (2 + u) < 2 ^ (2 + u)
⊢ 2 + u + 1 + 1 < 2 ^ (2 + u) + 2 ^ (2 + u) |
https://github.com/jvlmdr/from_the_book.git | 9fb6080539a2f32bb24719600a9e7531abf2328d | FromTheBook/Ch02/Bertrand/Bertrand.lean | succLtTwoPow | [60, 1] | [71, 38] | apply Nat.add_lt_add h | case intro.succ
u : ℕ
h : Nat.succ (2 + u) < 2 ^ (2 + u)
⊢ 2 + u + 1 + 1 < 2 ^ (2 + u) + 2 ^ (2 + u) | case intro.succ
u : ℕ
h : Nat.succ (2 + u) < 2 ^ (2 + u)
⊢ 1 < 2 ^ (2 + u) |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.