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Reference · Lemmas behind operations-research checks

rainbow.math.lemmas.mnl-assortment

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

General lemmas: mnl assortment. 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-associative math.rational.signed.multiply-ring math.rational.signed.distributive

signed/mnl-term-bound

TheoremFor all inputsNo axiomsSource

signed/mnl-term-bound :
  (r v s t R : ℚ)
  (hv : 0 ≤ v)
  (hs : s · (1 + signed/negate s) = 0)
  (ht : t · (1 + signed/negate t) = 0)
  (hoff : s · (R + signed/negate r) ≤ 0)
  (hleft : (1 + signed/negate s) · (r + signed/negate R) ≤ 0)
  → t · (r + signed/negate R) · v ≤ s · (r + signed/negate R) · v

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

signed/mnl-weighted-gap

TheoremFor all inputsNo axiomsSource

signed/mnl-weighted-gap :
  (u r v : ℕ → ℚ) (R : ℚ) (n : ℕ)
  → signed/expr-sum-value (fun gs => u gs · (r gs + signed/negate R) · v gs) n = signed/expr-sum-value (fun ru => u ru · r ru · v ru) n + signed/negate (R · signed/expr-sum-value (fun wu => u wu · v wu) n)

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

signed/mnl-assortment-optimal

TheoremFor all inputsNo axiomsSource

signed/mnl-assortment-optimal :
  (r v s t : ℕ → ℚ) (R R2 : ℚ) (n : ℕ)
  (hv : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), 0 ≤ v idx)
  (hs : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), s idx · (1 + signed/negate (s idx)) = 0)
  (hrev : R · (1 + signed/expr-sum-value (fun wu => s wu · v wu) n) = signed/expr-sum-value (fun ru => s ru · r ru · v ru) n)
  (hoff : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), s idx · (R + signed/negate (r idx)) ≤ 0)
  (hleft : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), (1 + signed/negate (s idx)) · (r idx + signed/negate R) ≤ 0)
  (ht : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), t idx · (1 + signed/negate (t idx)) = 0)
  (hrev2 : R2 · (1 + signed/expr-sum-value (fun wu => t wu · v wu) n) = signed/expr-sum-value (fun ru => t ru · r ru · v ru) n)
  → R2 ≤ R

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