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Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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8 declarations, 0 contain sorry, 0 declared as axioms

Numerical growth lemmas for the dimension theorem

Purely arithmetic facts about functions ℕ → ℕ; no ring theory in this file. We say f has growth degree at most d (DimensionTheorem.GrowthLE f d) if f n ≤ C * (n + 1) ^ d for some constant C.

Main results

DimensionTheorem.GrowthLE.of_le_sub — the telescoping lemma: if g is monotone, g n + f (n - c) ≤ f n for all n ≥ c, and f has growth degree at most d + 1, then g has growth degree at most d.
DimensionTheorem.card_sym_le — there are at most (n + 1) ^ (k - 1) monomials of degree n in k ≥ 1 variables: Fintype.card (Sym (Fin k) n) ≤ (n + 1) ^ (k - 1).

Sources

The telescoping lemma has no direct textbook counterpart. Classical treatments (e.g. Atiyah–Macdonald, Introduction to Commutative Algebra, Ch. 11) define d(R) as the degree of the Hilbert–Samuel polynomial, for which "the difference P(n) - P(n - c) has degree one less" is immediate. This development defines d(R) by growth order alone (avoiding the Hilbert–Serre theorem), and that step is replaced here: for monotone g dominated by the c-step difference of f, summing along an arithmetic progression of step c gives n * g n ≤ f (n * (c + 1)), which forces the degree drop. Used in FLT/Slop/DimensionTheorem/DimLeGrowth.lean for the inductive step of dim R ≤ d(R).

The counting bound is Nat.multichoose k n ≤ (n + 1) ^ (k - 1), by induction from the Pascal-type recurrence Nat.multichoose_succ_succ. Used in FLT/Slop/DimensionTheorem/GrowthLeDelta.lean to bound the number of degree-n monomials in the generators of an ideal of definition.

Imports (3)

Imported by 2 files

def DimensionTheorem.GrowthLE f : ℕ → ℕ grows at most like n ^ d line 52 source
Source documentation
f : ℕ → ℕ grows at most like n ^ d. We use (n + 1) ^ d to avoid degenerate behavior at n = 0.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: Exists, LE.le, HMul.hMul, HPow.hPow, HAdd.hAdd, OfNat.ofNat
10 instances, coercions and auxiliary constants left out.
```
def GrowthLE (f : ℕ → ℕ) (d : ℕ) : Prop :=
…
```
lemma DimensionTheorem.GrowthLE.of_le line 57 source
standard axioms: propext what this means
Statement uses: Nat, DimensionTheorem.GrowthLE, LE.le
Proof uses: HMul.hMul, HPow.hPow, HAdd.hAdd, OfNat.ofNat, Exists.intro, LE.le.trans
13 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma DimensionTheorem.GrowthLE.mono line 62 source
standard axioms: propext what this means
Statement uses: Nat, DimensionTheorem.GrowthLE, LE.le
Proof uses: HMul.hMul, HPow.hPow, HAdd.hAdd, OfNat.ofNat, Exists.intro, LE.le.trans, Nat.mul_le_mul_left, Nat.pow_le_pow_right, Nat.succ, Nat.succ_le_succ, Nat.zero_le
13 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma DimensionTheorem.GrowthLE.of_bounded line 68 source
standard axioms: propext what this means
Statement uses: Nat, LE.le, DimensionTheorem.GrowthLE, OfNat.ofNat
Proof uses: Exists.intro, HMul.hMul, HPow.hPow, HAdd.hAdd, Eq.mpr, id, Eq, congrArg, Eq.trans, pow_zero, mul_one
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma DimensionTheorem.GrowthLE.telescope_aux Telescoping lemma line 78 source
Source documentation
Telescoping lemma. If g is monotone and, for n ≥ c, the value g n is dominated by the difference f n - f (n - c) (stated additively as g n + f (n - c) ≤ f n), and f has growth degree at most d + 1, then g has growth degree at most d.
This is the numerical heart of the inductive step dim ≤ d(R) in the dimension theorem: passing to R/xR drops the growth degree of the Hilbert–Samuel function.
```
private lemma telescope_aux {f g : ℕ → ℕ} {c : ℕ}
    (h : ∀ n, c ≤ n → g n + f (n - c) ≤ f n) (n : ℕ) :
    ∀ m : ℕ, f n + ∑ j ∈ Finset.range m, g (n + (j + 1) * c) ≤ f (n + m * c) := by
…
```
lemma DimensionTheorem.GrowthLE.of_le_sub hc is kept for interface stability but is not actually needed: each application of h is at a point n + (j + 1) * c ≥ c regardless line 100 source
Source comment
`hc` is kept for interface stability but is not actually needed: each application of `h` is at a point `n + (j + 1) * c ≥ c` regardless.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, DimensionTheorem.GrowthLE, HAdd.hAdd, Monotone, LE.le, HSub.hSub
Proof uses: HMul.hMul, HPow.hPow, Exists.intro, Eq, GT.gt, Or, Nat.eq_zero_or_pos, of_eq_true, Eq.trans, True, congrArg, congrFun', zero_add, one_pow, mul_one, Function.swap, Eq.symm, Finset.sum, Finset.range, HSMul.hSMul and 55 more
89 instances, coercions and auxiliary constants left out.

Tactic trace: 25 steps
Open to load the checked before-and-after goal states.
```
lemma of_le_sub {f g : ℕ → ℕ} {c d : ℕ} (hc : 0 < c) (hf : GrowthLE f (d + 1))
    (hg : Monotone g) (h : ∀ n, c ≤ n → g n + f (n - c) ≤ f n) :
    GrowthLE g d := by
…
```
Mentions: DimensionTheorem.GrowthLE
lemma DimensionTheorem.multichoose_le_pow Documented declaration: multichoose_le_pow line 147 source
Source documentation
The number of monomials of degree n in k variables is at most (n + 1) ^ (k - 1) (for k ≥ 1): a multiset of size n over Fin k is determined by the multiplicities of the first k - 1 elements, each of which lies in {0, ..., n}.
```
private lemma multichoose_le_pow (k : ℕ) :
    ∀ n : ℕ, Nat.multichoose (k + 1) n ≤ (n + 1) ^ k := by
…
```
lemma DimensionTheorem.card_sym_le line 165 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Fintype.card, Sym, Fin, HPow.hPow, HAdd.hAdd, HSub.hSub
Proof uses: Eq, Exists, Exists.intro, Decidable.byContradiction, Not, Eq.mpr, Nat.multichoose, id, congrArg, Sym.card_sym_fin_eq_multichoose, add_tsub_cancel_right, Eq.symm
47 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.