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Reference · Lemmas behind operations-research checks

rainbow.math.lemmas.lp-farkas

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

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

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signed/expr-sum-zero-objective

TheoremFor all inputsNo axiomsSource

signed/expr-sum-zero-objective :
  (x : ℕ → ℚ) (n : ℕ)
  → signed/expr-sum-value (fun z => (fun x₂ => 0) z · x z) n = 0

0 = signed/zero= = signed/equivalent: equality of rational values· = signed/multiplyℕ = data/Natℚ = signed/SignedRational

signed/expr-sum-lp-farkas-nonnegative

TheoremFor all inputsNo axiomsSource

signed/expr-sum-lp-farkas-nonnegative :
  (A : ℕ → ℕ → ℚ) (b x y : ℕ → ℚ) (m n : ℕ)
  (primal : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx m), signed/expr-sum-value (fun j => A idx j · x j) n ≤ b idx)
  (xn : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), 0 ≤ x idx)
  (yn : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx m), 0 ≤ y idx)
  (dual : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), (fun x₁ => 0) idx ≤ signed/expr-sum-value (fun i => A i idx · y i) m)
  → 0 ≤ signed/expr-sum-value (fun z => b z · y z) m

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