Menu

Rainbow docs menu

Reference · Lemmas behind operations-research checks

rainbow.math.lemmas.pcmax-lower-bound

Lemmas behind operations-research checks src/rainbow/math/lemmas/pcmax_lower_bound.clj0 definitions, 3 theorems

General lemmas: pcmax lower bound. 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 math.sums.two-terms math.linear.matrix.row-length-shape math.linear.matrix.row-length-decider math.rational.signed.structural-equality math.linear.vector.equivalence math.linear.vector.uniform-clamp math.linear.vector.pointwise-order math.linear.matrix.action-prepend math.linear.matrix.action-associative math.linear.matrix.transpose-multiply math.linear.matrix.transpose-shape math.linear.matrix.transpose-involution math.linear.matrix.transpose-involution-positive math.linear.matrix.transpose-multiply-reverse math.linear.matrix.composable-transpose math.linear.matrix.associativity-counterexample math.linear.matrix.signed-scale math.linear.matrix.signed-scale-right math.linear.matrix.add math.linear.matrix.add-shape math.linear.matrix.prepend-add math.linear.matrix.add-right math.linear.matrix.add-algebra math.linear.matrix.additive-group math.linear.matrix.negate-linear math.linear.matrix.subtract-linear math.linear.matrix.signed-scale-algebra 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/pcmax-job-bound

TheoremFor all inputsNo axiomsSource

signed/pcmax-job-bound :
  (Y : ℕ → ℕ → ℚ) (p : ℕ → ℚ) (Cp : ℚ) (m n : ℕ)
  (hY01 : ∀ (yi : ℕ), ∀ (yi-lt : nat/strict-below yi m), ∀ (yj : ℕ), ∀ (yj-lt : nat/strict-below yj n), Y yi yj · (1 + signed/negate (Y yi yj)) = 0)
  (hcol : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), signed/expr-sum-value (fun ci => Y ci idx) m = 1)
  (hload : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx m), signed/expr-sum-value (fun lj => Y idx lj · p lj) n ≤ Cp)
  (hp : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), 0 ≤ p idx)
  (j : ℕ) (j-lt : nat/strict-below j n)
  → p j ≤ Cp

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

signed/pcmax-average-bound

TheoremFor all inputsNo axiomsSource

signed/pcmax-average-bound :
  (Y : ℕ → ℕ → ℚ) (p : ℕ → ℚ) (Cp : ℚ) (m n : ℕ)
  (hY01 : ∀ (yi : ℕ), ∀ (yi-lt : nat/strict-below yi m), ∀ (yj : ℕ), ∀ (yj-lt : nat/strict-below yj n), Y yi yj · (1 + signed/negate (Y yi yj)) = 0)
  (hcol : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), signed/expr-sum-value (fun ci => Y ci idx) m = 1)
  (hload : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx m), signed/expr-sum-value (fun lj => Y idx lj · p lj) n ≤ Cp)
  → signed/expr-sum-value (fun ap => p ap) n ≤ Cp · signed/from-nnrat (nnrat/make m 0)

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

signed/at-most-cancel-positive-right

TheoremFor all inputsNo axiomsSource

signed/at-most-cancel-positive-right :
  (a b d : ℚ)
  (hd : 0 < d)
  (h : a · d ≤ b · d)
  → a ≤ b

0 = signed/zero< = signed/less-than· = signed/multiplyℚ = signed/SignedRational≤ = signed/at-most