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Duplicated from  ricspse/deepseek-ai-DeepSeek-Prover-V2-671B

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DeepSeek-Prover-V2-671B

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😃: Complete the following Lean 4 code: lean4 { import Mathlib import Aesop set_option maxHeartbeats 0 open BigOperators /-- AMC 12A 2015 Problem 10 (x y : ℤ) 0 < y < x ∧ x + y + x *y = 80 Show x = 26. -/ theorem amc12a_2015_p10 (x y : ℤ) (h₀ : 0 < y) (h₁ : y < x) (h₂ : x + y + (x * y) = 80) : x = 26 := by sorry } Before producing the Lean 4 code to formally prove the given theorem, provide a detailed proof plan outlining the main proof steps and strategies. The plan should highlight key ideas, intermediate lemmas, and proof structures that will guide the construction of the final formal proof.
🤖: ### Detailed Proof

Problem Analysis:
We are given two integers x and y such that:

1. 0 < y,
2. y < x, and
3. x + y + x * y = 80.

We need to prove that x = 26.

Key Observations:

1. Since y is a positive integer and y < x, x must be at least y + 1.
2. The equation x + y + x * y = 80 can be rewritten as x * (y + 1) + y = 80 or (x + 1)(y + 1) = 81.
   • The second form is derived by adding 1 to both sides: x + y + x * y + 1 = 81, which factors as (x + 1)(y + 1) = 81.
3. The number 81 has positive factor pairs (1, 81), (3, 27), (9, 9), (27, 3), and (81, 1) in the integers. However, since y > 0 and x > y, we can exclude some of these possibilities.

Approach:

1. We will use the factored form (x + 1)(y + 1) = 81 to find possible values of x and y.
2. We will consider all positive integer factor pairs of 81 and ...