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Reference · Lemmas behind operations-research checks

rainbow.math.lemmas.max-closure

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

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

Depends onmath.sums.value-append math.nat.infinite math.rational.nnrat.identities math.nat.at-most-cancellation math.rational.nnrat.order-equivalence math.rational.nnrat.archimedean math.rational.nnrat.add-monotone math.rational.nnrat.add-cancellation math.rational.nnrat.additive-reassociation math.rational.signed.order-transitivity math.rational.signed.convergence math.rational.signed.order-respects math.rational.signed.limit-uniqueness math.rational.signed.strict-order math.rational.signed.multiply math.rational.nnrat.multiply-commute math.rational.nnrat.product-reassociation math.rational.nnrat.distribute-right math.rational.nnrat.ring math.rational.nnrat.multiply-monotone math.rational.nnrat.order-difference math.rational.signed.multiply-order math.rational.signed.equivalence-laws math.rational.nnrat.minmax math.rational.nnrat.farkas-elim-list math.rational.signed.no-zero-divisors math.rational.signed.decide-order math.rational.signed.decide-reflexive math.logic.boolean-and math.rational.signed.decide-equivalence math.rational.signed.decide-completeness math.rational.signed.unit-laws math.nat.strict-below-split math.nat.strict-below-more math.checks math.nat.at-most-add math.rational.nnrat.harmonic-convergence math.nat.divmod math.rational.signed.minmax math.rational.signed.minmax-algebra math.rational.signed.clamp math.rational.signed.deviation-rows math.rational.signed.absolute math.rational.signed.absolute-algebra math.rational.signed.distance math.nat.linear-order math.rational.signed.add-associative math.nat.semiring math.nat.multiply-zero-split math.rational.nnrat.multiply-zero-split math.nat.monus math.rational.nnrat.monus math.rational.nnrat.add-zero-split math.rational.nnrat.add-cancel-zero math.finite.rational-argmax math.rational.nnrat.order-total math.rational.nnrat.multiply-difference-zero math.linear.vector.norm-nonnegative math.linear.vector.norm-zero-reflection algorithms.proofs.counterfactual-distance-separation algorithms.proofs.counterfactual-distance-laws math.finite.matrix math.linear.matrix.equivalence math.linear.matrix.transpose math.linear.matrix.transpose-equivalence math.linear.matrix.multiply math.linear.matrix.multiply-shape math.linear.matrix.composable math.linear.matrix.composable-multiply math.linear.vector.additive-units math.linear.vector.shape-cancellation math.linear.vector.shape-map math.linear.vector.shape-add math.linear.vector.dot-additive math.linear.vector.dot-bilinear math.linear.vector.dot-negate math.linear.vector.dot-subtract math.linear.vector.dot-scale math.linear.matrix.action-scale math.rational.signed.multiply-nonzero math.rational.signed.order-monotone math.rational.signed.list-sum math.rational.signed.multiply-monotone math.rational.signed.map-sum-algebra math.rational.signed.negate-additive math.rational.signed.lp-objective-negate math.nat.pair-index-inverse math.sums.flow-cut math.sums.nonneg-zero math.rational.signed.multiply-ring math.rational.signed.distributive

signed/closure-arc-term

TheoremFor all inputsNo axiomsSource

signed/closure-arc-term :
  (a w xi xj : ℚ)
  (ha : a · (1 + signed/negate a) = 0)
  (hw : 0 ≤ w)
  (hwa : w ≤ a · w)
  (hxi : xi · (1 + signed/negate xi) = 0)
  (hxj : xj · (1 + signed/negate xj) = 0)
  (hrow : a · xi · (1 + signed/negate xj) = 0)
  → w · (xi + signed/negate xj) ≤ 0

+ = signed/add0 = signed/zero1 = signed/one= = signed/equivalent: equality of rational values· = signed/multiplyℚ = signed/SignedRational≤ = signed/at-most

signed/closure-weak-duality

TheoremFor all inputsNo axiomsSource

signed/closure-weak-duality :
  (A W : ℕ → ℕ → ℚ) (p u x : ℕ → ℚ) (n : ℕ)
  (hA : ∀ (ai : ℕ), ∀ (ai-lt : nat/strict-below ai n), ∀ (aj : ℕ), ∀ (aj-lt : nat/strict-below aj n), A ai aj · (1 + signed/negate (A ai aj)) = 0)
  (hu : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), 0 ≤ u idx)
  (hw : ∀ (wi : ℕ), ∀ (wi-lt : nat/strict-below wi n), ∀ (wj : ℕ), ∀ (wj-lt : nat/strict-below wj n), 0 ≤ W wi wj)
  (hwA : ∀ (vi : ℕ), ∀ (vi-lt : nat/strict-below vi n), ∀ (vj : ℕ), ∀ (vj-lt : nat/strict-below vj n), W vi vj ≤ A vi vj · W vi vj)
  (hdual : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), p idx + signed/expr-sum-value (fun ik => W ik idx) n ≤ u idx + signed/expr-sum-value (fun oj => W idx oj) n)
  (hx01 : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), x idx · (1 + signed/negate (x idx)) = 0)
  (hclosed : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), signed/expr-sum-value (fun rj => A idx rj · x idx · (1 + signed/negate (x rj))) n ≤ 0)
  → signed/expr-sum-value (fun pi => p pi · x pi) n ≤ signed/expr-sum-value (fun us => u us) n

+ = signed/add0 = signed/zero1 = signed/one= = signed/equivalent: equality of rational values· = signed/multiplyℕ = data/Natℚ = signed/SignedRational≤ = signed/at-most