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Reference · Lemmas behind operations-research checks

rainbow.math.lemmas.abc-class

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

General lemmas: abc class. 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

signed/strict-indicator-nonnegative

TheoremFor all inputsNo axiomsSource

signed/strict-indicator-nonnegative :
  (sn-x sn-y : ℚ)
  → 0 ≤ data/Bool.rec (fun x₄ => ℚ) 1 0 (signed/decide-at-most sn-y sn-x)

0 = signed/zero1 = signed/oneℚ = signed/SignedRational≤ = signed/at-most

signed/strict-indicator-zero-order

TheoremFor all inputsNo axiomsSource

signed/strict-indicator-zero-order :
  (sz-x sz-y : ℚ)
  (sz-h0 : data/Bool.rec (fun x₄ => ℚ) 1 0 (signed/decide-at-most sz-y sz-x) = 0)
  → sz-y ≤ sz-x

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

signed/abc-selected-dominates

TheoremFor all inputsNo axiomsSource

signed/abc-selected-dominates :
  (dm-v dm-z : ℕ → ℚ) (dm-n : ℕ)
  (dm-01 : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx dm-n), dm-z idx · (1 + signed/negate (dm-z idx)) = 0)
  (dm-top : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx dm-n), dm-z idx · signed/expr-sum-value (fun rw-j => (1 + signed/negate (dm-z rw-j)) · data/Bool.rec (fun x₃ => ℚ) 1 0 (signed/decide-at-most (dm-v rw-j) (dm-v idx))) dm-n = 0)
  (dm-i : ℕ) (dm-hi : nat/strict-below dm-i dm-n) (dm-j : ℕ)
  (dm-hj : nat/strict-below dm-j dm-n)
  (dm-zi : dm-z dm-i = 1)
  (dm-zj : dm-z dm-j = 0)
  → dm-v dm-j ≤ dm-v dm-i

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