The Erdős–Turán conjecture on additive bases

I am proving a theorem in Lean 4 and submitting it to Conjectura.

## Problem CB007 — The Erdős–Turán conjecture on additive bases

If every number is a sum of two members of a set, must some number have many such representations?

## Environment (fixed — do not assume anything newer)

- Lean toolchain: `leanprover/lean4:v4.33.0-rc1`
- Mathlib: `v4.33.0-rc1`

If a lemma you want does not exist in that Mathlib, prove it inline instead of
importing something newer.

## The exact statement I must prove

```lean
theorem solution : ∀ (A : Set ℕ), (∀ n : ℕ, ∃ a ∈ A, ∃ b ∈ A, a + b = n) →
    ∀ C : ℕ, ∃ n : ℕ, C < Nat.card {p : ℕ × ℕ // p.1 ∈ A ∧ p.2 ∈ A ∧ p.1 + p.2 = n} := by
  sorry
```

## The file I submit

```lean
import Conjectura.Problems.CB007.Statement

namespace Submission

theorem solution : Conjectura.CB007.goal := by
  sorry

end Submission

```

## The Lean definitions of every term in this problem

These are the actual definitions your proof will be checked against. Do not
substitute your own version of any of them.

### Set

abbrev

```lean
Set
```
Defined in Mathlib.

### card

theorem

```lean
Nat.card
```
Defined in Mathlib.

## Rules — submissions violating these are rejected automatically

1. **Do not change the name or type of `solution`.** It must satisfy the
   statement above exactly.
2. **Do not redefine or shadow anything from the problem's Statement module.**
   Declaring your own `goal`, or redefining a name it depends on, produces a
   proof of a *different* statement and is rejected. This is the single most
   common rejection.
3. **No `sorry`** anywhere, including in helper lemmas. It surfaces as the
   axiom `sorryAx` and is detected transitively through imports.
4. **No `native_decide`** — it surfaces as `Lean.ofReduceBool` and is not
   accepted, because it trusts compiled code rather than the kernel.
5. Only these axioms are permitted: `propext`, `Classical.choice`,
   `Quot.sound`.
6. Follow Mathlib style: hypotheses left of the colon, explicit types,
   `snake_case` theorem names, `UpperCamelCase` types.

## What I want from you

Here is my argument in informal mathematics:

> [PASTE YOUR PROOF SKETCH HERE]

Turn it into Lean 4 that compiles under the environment above and satisfies the
statement exactly. Where you are unsure a lemma exists in this Mathlib version,
say so explicitly rather than guessing a name.