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Reference · Lemmas behind operations-research checks

rainbow.math.lemmas.gilmore-pricing

Lemmas behind operations-research checks src/rainbow/math/lemmas/gilmore_pricing.clj0 definitions, 2 theorems

General lemmas: gilmore pricing. The mathematical facts the corresponding checks rest on, each checked by the kernel.

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signed/positive-difference-negative-reverse

TheoremFor all inputsNo axiomsSource

signed/positive-difference-negative-reverse :
  (value unit : ℚ)
  (positive : 0 < value + signed/negate unit)
  → unit + signed/negate value < 0

+ = signed/add0 = signed/zero< = signed/less-thanℚ = signed/SignedRational

signed/gilmore-complete-pricing-cover-bound

TheoremFor all inputsNo axiomsSource

signed/gilmore-complete-pricing-cover-bound :
  (A : ℕ → ℕ → ℚ) (d w : ℕ → ℚ) (W : ℚ) (prices x : ℕ → ℚ) (m n : ℕ)
  (prices-nonnegative : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx m), 0 ≤ prices idx)
  (all-pricing : ∀ (pattern : ℕ → ℚ), ∀ (counts : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx m), 0 ≤ pattern idx), ∀ (capacity : signed/expr-sum-value (fun z => pattern z · w z) m ≤ W), signed/expr-sum-value (fun z => pattern z · prices z) m ≤ 1)
  (pattern-counts : ∀ (column : ℕ), ∀ (hc : nat/strict-below column n), ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx m), 0 ≤ (fun item => A item column) idx)
  (pattern-capacity : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), signed/expr-sum-value (fun z => (fun item => A item idx) z · w z) m ≤ W)
  (cover : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx m), d idx ≤ signed/expr-sum-value (fun j => A idx j · x j) n)
  (nonnegative : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), 0 ≤ x idx)
  → signed/expr-sum-value (fun z => d z · prices z) m ≤ signed/expr-sum-value x n

0 = signed/zero1 = signed/one· = signed/multiplyℕ = data/Natℚ = signed/SignedRational≤ = signed/at-most