Harmonic.lean`

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5 declarations, 0 contain sorry, 0 declared as axioms

Block harmonic sums

Mathlib's harmonic n is the initial harmonic sum ∑_{1 ≤ k ≤ n} 1 / k. Several problems instead need the block sum ∑_{n ≤ k ≤ m} 1 / k, which is what harmonicBlock gives. Like harmonic it is valued in ℚ; cast at the use site when a real value is wanted.

Main definitions

harmonicBlock: the block harmonic sum ∑_{n ≤ k ≤ m} 1 / k.

Main results

harmonicBlock_eq_zero_of_lt: the block sum is empty when m < n.
harmonicBlock_one: harmonicBlock 1 n is Mathlib's harmonic n.

Imports (4)

Imported by 1 file

FormalConjecturesForMathlib.lean

def harmonicBlock The block harmonic sum $\sum_{n\leq k\leq m}\frac{1}{k}$ line 45 source
Source documentation
The block harmonic sum $\sum_{n\leq k\leq m}\frac{1}{k}$. For n = 1 this is Mathlib's harmonic, see harmonicBlock_one.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Rat
Proof uses: Finset.sum, Finset.Icc, Inv.inv
6 instances, coercions and auxiliary constants left out.
```
def harmonicBlock (n m : ℕ) : ℚ := ∑ k ∈ Finset.Icc n m, (k : ℚ)⁻¹
…
```
theorem harmonicBlock_eq_sum_one_div line 49 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Rat, harmonicBlock, Finset.sum, Finset.Icc, HDiv.hDiv, OfNat.ofNat
Proof uses: of_eq_true, Inv.inv, Eq.trans, True, congrArg, Finset.sum_congr, Eq.refl, Finset, Membership.mem, one_div, eq_self
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem harmonicBlock_eq_zero_of_lt line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, Eq, Rat, harmonicBlock, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Finset.sum_congr, Finset.Icc, EmptyCollection.emptyCollection, Finset, Inv.inv, Finset.Icc_eq_empty_of_lt, Membership.mem, Eq.refl, eq_self
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem harmonicBlock_succ_top line 57 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, HAdd.hAdd, OfNat.ofNat, Eq, Rat, harmonicBlock, Inv.inv
Proof uses: of_eq_true, Finset.sum, Finset.Icc, Eq.trans, True, congrFun', congrArg, AddCommMonoid, Finset.sum_Icc_succ_top, Nat.cast_add, Nat.cast_one, eq_self
35 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem harmonicBlock_one The block sum starting at 1 is Mathlib's harmonic number line 62 source
Source documentation
The block sum starting at 1 is Mathlib's harmonic number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Rat, harmonicBlock, OfNat.ofNat, harmonic
Proof uses: Nat.recAux, of_eq_true, Eq.trans, True, congrFun', congrArg, Finset.sum_congr, Finset.Icc, EmptyCollection.emptyCollection, Finset, Inv.inv, Finset.Icc_eq_empty_of_lt, LT.lt, eq_self, Membership.mem, Eq.refl, Eq.mpr, HAdd.hAdd, id, harmonicBlock_succ_top and 22 more
70 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem harmonicBlock_one (n : ℕ) : harmonicBlock 1 n = harmonic n := by
…
```
Mentions: harmonicBlock