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theorem proved term proof

pivot_identity

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formal statement (Lean)

 199theorem pivot_identity (v₀ v₁ : ℝ) :
 200    -- If agent 0 wins (v₀ ≥ v₁), they pay v₁.
 201    (v₀ ≥ v₁ → payment₀ v₀ v₁ = v₁) ∧
 202    -- If agent 1 wins (v₀ < v₁), they pay v₀.
 203    (v₀ < v₁ → payment₁ v₀ v₁ = v₀) := by

proof body

Term-mode proof.

 204  refine ⟨?_, ?_⟩
 205  · intro h
 206    unfold payment₀
 207    rw [if_pos h]
 208  · intro h
 209    unfold payment₁
 210    rw [if_neg (by linarith)]
 211
 212/-- **σ-CONSERVATION (single-item case).** Under truthful bidding, the
 213total surplus on the utility ledger (winner's utility) plus the
 214payment equals the winner's valuation. Equivalently: the auction
 215extracts a payment exactly equal to the social-welfare externality. -/

used by (3)

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depends on (4)

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