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Reference · Lemmas behind operations-research checks

rainbow.math.lemmas.fulfil-readings

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

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

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math.rational.signed.distributive 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 math.lemmas.one-center

signed/min-select-cases

TheoremFor all inputsNo axiomsSource

signed/min-select-cases :
  (ms-x ms-y : ℚ)
  → proposition/conjunction (data/Bool.rec (fun x₄₈ => ℚ) ms-y ms-x (signed/decide-at-most ms-x ms-y) ≤ ms-x) (proposition/conjunction (data/Bool.rec (fun x₇₂ => ℚ) ms-y ms-x (signed/decide-at-most ms-x ms-y) ≤ ms-y) (proposition/disjunction (data/Bool.rec (fun x₈₄ => ℚ) ms-y ms-x (signed/decide-at-most ms-x ms-y) = ms-x) (data/Bool.rec (fun x₈₈ => ℚ) ms-y ms-x (signed/decide-at-most ms-x ms-y) = ms-y)))

= = signed/equivalent: equality of rational valuesℚ = signed/SignedRational≤ = signed/at-most

signed/pick-line

TheoremFor all inputsNo axiomsSource

signed/pick-line :
  (pl-o pl-a pl-s pl-d : ℚ)
  (pl-ho : 0 ≤ pl-o)
  (pl-ha : 0 ≤ pl-a)
  (pl-hs : pl-s = data/Bool.rec (fun x₄ => ℚ) pl-a pl-o (signed/decide-at-most pl-o pl-a))
  (pl-hd : pl-d = pl-o + signed/negate pl-s)
  → proposition/conjunction (conjunction) (conjunction₁)
  where conjunction := proposition/conjunction (proposition/disjunction (pl-s = pl-o) (pl-s = pl-a)) (proposition/conjunction (pl-s ≤ pl-o) (proposition/conjunction (pl-s ≤ pl-a) (0 ≤ pl-s)))
  where conjunction₁ := proposition/conjunction (0 ≤ pl-d) (proposition/disjunction (pl-d = 0) (pl-s = pl-a))

+ = signed/add0 = signed/zero= = signed/equivalent: equality of rational valuesℚ = signed/SignedRational≤ = signed/at-most