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Reference · Lemmas behind operations-research checks

rainbow.math.lemmas.headway-wait

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

General lemmas: headway wait. 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.affine-general-rate-certificate math.sums.of-nat math.lemmas.one-center math.rational.signed.multiply-associative

signed/count-positive-of-sum

TheoremFor all inputsNo axiomsSource

signed/count-positive-of-sum :
  (cp-h : ℕ → ℚ) (cp-n : ℕ)
  (cp-pos : 0 < signed/expr-sum-value cp-h cp-n)
  → 0 < signed/from-nnrat (nnrat/make cp-n 0)

0 = signed/zero< = signed/less-thanℕ = data/Natℚ = signed/SignedRational

signed/square-deviation-general

TheoremFor all inputsNo axiomsSource

signed/square-deviation-general :
  (dg-h : ℕ → ℚ) (dg-n : ℕ) (dg-s1 dg-s2 : ℚ)
  (dg-h1 : dg-s1 = signed/expr-sum-value dg-h dg-n)
  (dg-h2 : dg-s2 = signed/expr-sum-value (fun sq-i => dg-h sq-i · dg-h sq-i) dg-n)
  (dg-a : ℚ)
  → signed/expr-sum-value (fun dj => (dg-a + signed/negate (dg-h dj)) · (dg-a + signed/negate (dg-h dj))) dg-n = dg-a · dg-a · signed/from-nnrat (nnrat/make dg-n 0) + dg-s2 + signed/negate (dg-a · dg-s1 + dg-a · dg-s1)

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

signed/cauchy-schwarz-sum

TheoremFor all inputsNo axiomsSource

signed/cauchy-schwarz-sum :
  (cs-h : ℕ → ℚ) (cs-n : ℕ) (cs-s1 cs-s2 : ℚ)
  (cs-h1 : cs-s1 = signed/expr-sum-value cs-h cs-n)
  (cs-h2 : cs-s2 = signed/expr-sum-value (fun sq-i => cs-h sq-i · cs-h sq-i) cs-n)
  → cs-s1 · cs-s1 ≤ signed/from-nnrat (nnrat/make cs-n 0) · cs-s2

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