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Reference · Lemmas behind operations-research checks

rainbow.math.lemmas.auc-readings

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

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

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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 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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/auc-weight-bounds

TheoremFor all inputsNo axiomsSource

signed/auc-weight-bounds :
  (aw-b1 aw-b2 : data/Bool)
  → proposition/conjunction (0 ≤ signed/from-nnrat 2 · data/Bool.rec (fun x₄₄ => ℚ) 1 0 aw-b1 + data/Bool.rec (fun x₅₆ => ℚ) 0 (data/Bool.rec (fun x₆₀ => ℚ) 0 1 aw-b2) aw-b1) (signed/from-nnrat 2 · data/Bool.rec (fun x₆₄ => ℚ) 1 0 aw-b1 + data/Bool.rec (fun x₇₆ => ℚ) 0 (data/Bool.rec (fun x₈₀ => ℚ) 0 1 aw-b2) aw-b1 ≤ signed/from-nnrat 2)

+ = signed/add0 = signed/zero1 = signed/one· = signed/multiplyℚ = signed/SignedRational≤ = signed/at-most

signed/complement-nonnegative

TheoremFor all inputsNo axiomsSource

signed/complement-nonnegative :
  (cn-y : ℚ)
  (cn-h : cn-y · (1 + signed/negate cn-y) = 0)
  → 0 ≤ 1 + signed/negate cn-y

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

signed/auc-unit-interval

TheoremFor all inputsNo axiomsSource

signed/auc-unit-interval :
  (au-a au-p au-n au-w : ℚ)
  (au-h : signed/from-nnrat 2 · (au-a · (au-p · au-n)) = au-w)
  (au-h0 : 0 ≤ au-w)
  (au-h1 : au-w ≤ signed/from-nnrat 2 · (au-p · au-n))
  (au-hp : 1 ≤ au-p)
  (au-hn : 1 ≤ au-n)
  → proposition/conjunction (0 ≤ au-a) (au-a ≤ 1)

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