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Reference · Lemmas behind operations-research checks

rainbow.math.lemmas.auction-assignment

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

General lemmas: auction assignment. 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 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math.linear.matrix.signed-scale-units math.linear.matrix.signed-scale-congruence math.linear.matrix.signed-scale-subtract math.rational.signed.distance-multiply-nonnegative math.rational.signed.vector-sup-distance algorithms.proofs.inner-product math.rational.signed.distance-triangle math.rational.signed.distance-translation math.rational.signed.distance-additive math.linear.vector.dot-distance-bound math.linear.vector.row-sum-lipschitz math.linear.matrix.action-sup-bound math.linear.matrix.signed-scale-order math.linear.matrix.pointwise-order-laws math.linear.matrix.signed-scale-antitone math.linear.matrix.add-order math.linear.matrix.signed-scale-positive math.linear.matrix.signed-scale-sign math.linear.matrix.positive-action math.linear.matrix.transpose-order math.linear.matrix.positive-multiply math.linear.matrix.gram-positive math.linear.matrix.gram-kernel math.linear.matrix.row-gram math.linear.matrix.row-gram-kernel math.linear.vector.norm-pythagorean math.linear.matrix.least-squares-normal math.linear.matrix.least-squares-equality math.linear.matrix.mixed-sign-action math.linear.matrix.mixed-sign-multiply math.linear.matrix.antitone math.linear.matrix.interval-order math.linear.matrix.affine-order math.linear.matrix.linear math.rational.signed.vector-sup-congruence math.finite.vector math.linear.vector.pointwise-order-laws math.linear.vector.add-order math.rational.signed.vector-sup-translation math.linear.matrix.affine-sup-distance math.linear.matrix.row-sum-lipschitz math.linear.matrix.row-sum-fixed-point-uniqueness math.linear.matrix.half-row-sum-fixed-point-uniqueness math.linear.matrix.affine-half-iterate-contraction math.linear.matrix.affine-half-iterate-rate math.rational.signed.geometric-bridge math.linear.matrix.affine-half-iterate-converges math.linear.matrix.affine-strong-row-contraction math.linear.matrix.affine-residual-certificate math.linear.matrix.affine-inverse-gap-certificate math.linear.matrix.affine-general-rate-certificate math.sums.of-nat

signed/assignment-net-sum

TheoremFor all inputsNo axiomsSource

signed/assignment-net-sum :
  (z a : ℕ → ℕ → ℚ) (p : ℕ → ℚ) (n : ℕ)
  (hcol : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), signed/expr-sum-value (fun ci => z ci idx) n = 1)
  → signed/expr-sum-value (fun ri => signed/expr-sum-value (fun nk => z ri nk · (a ri nk + signed/negate (p nk))) n) n = signed/expr-sum-value (fun vi => signed/expr-sum-value (fun vk => z vi vk · a vi vk) n) n + signed/negate (signed/expr-sum-value (fun ps => p ps) n)

+ = signed/add1 = signed/one= = signed/equivalent: equality of rational values· = signed/multiplyℕ = data/Natℚ = signed/SignedRational

signed/auction-near-optimal

TheoremFor all inputsNo axiomsSource

signed/auction-near-optimal :
  (a x : ℕ → ℕ → ℚ) (p : ℕ → ℚ) (eps : ℚ) (n : ℕ)
  (hxcol : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), signed/expr-sum-value (fun ci => x ci idx) n = 1)
  (hcs : ∀ (ci : ℕ), ∀ (ci-lt : nat/strict-below ci n), ∀ (ck : ℕ), ∀ (ck-lt : nat/strict-below ck n), a ci ck + signed/negate (p ck) ≤ signed/expr-sum-value (fun nk => x ci nk · (a ci nk + signed/negate (p nk))) n + eps)
  (y : ℕ → ℕ → ℚ)
  (hy01 : ∀ (yi : ℕ), ∀ (yi-lt : nat/strict-below yi n), ∀ (yj : ℕ), ∀ (yj-lt : nat/strict-below yj n), y yi yj · (1 + signed/negate (y yi yj)) = 0)
  (hrow : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), signed/expr-sum-value (fun rj => y idx rj) n = 1)
  (hcol : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), signed/expr-sum-value (fun ci => y ci idx) n = 1)
  → signed/expr-sum-value (fun vi => signed/expr-sum-value (fun vk => y vi vk · a vi vk) n) n ≤ signed/expr-sum-value (fun vi => signed/expr-sum-value (fun vk => x vi vk · a vi vk) n) n + eps · signed/from-nnrat (nnrat/make n 0)

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