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Reference · Lemmas behind operations-research checks

rainbow.math.lemmas.lp-ray

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

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

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signed/expr-sum-ray-objective

TheoremFor all inputsNo axiomsSource

signed/expr-sum-ray-objective :
  (c x d : ℕ → ℚ) (t : ℚ) (n : ℕ)
  → signed/expr-sum-value (fun z => c z · (fun ray-index => x ray-index + t · d ray-index) z) n = signed/expr-sum-value (fun z => c z · x z) n + t · signed/expr-sum-value (fun z => c z · d z) n

+ = signed/add= = signed/equivalent: equality of rational values· = signed/multiplyℕ = data/Natℚ = signed/SignedRational

signed/expr-sum-lp-feasible-ray

TheoremFor all inputsNo axiomsSource

signed/expr-sum-lp-feasible-ray :
  (A : ℕ → ℕ → ℚ) (b c x d : ℕ → ℚ) (t : 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)
  (direction : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx m), signed/expr-sum-value (fun j => A idx j · d j) n ≤ (fun x₁ => 0) idx)
  (dn : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), 0 ≤ d idx)
  (tn : 0 ≤ t)
  → proposition/conjunction (∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), 0 ≤ (fun ray-index => x ray-index + t · d ray-index) idx) (conjunction)
  where conjunction := proposition/conjunction (∀ (idx : ℕ), ∀ (bound : nat/strict-below idx m), signed/expr-sum-value (fun j => A idx j · (fun ray-index => x ray-index + t · d ray-index) j) n ≤ b idx) (signed/expr-sum-value (fun z => c z · (fun ray-index => x ray-index + t · d ray-index) z) n = signed/expr-sum-value (fun z => c z · x z) n + t · signed/expr-sum-value (fun z => c z · d z) n)

+ = signed/add0 = signed/zero= = signed/equivalent: equality of rational values· = signed/multiplyℕ = data/Natℚ = signed/SignedRational≤ = signed/at-most