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Reference · Lemmas behind operations-research checks

rainbow.math.lemmas.cea-rule

Lemmas behind operations-research checks src/rainbow/math/lemmas/cea_rule.clj0 definitions, 5 theorems

General lemmas: cea rule. 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/decided-min-at-most-left

TheoremFor all inputsNo axiomsSource

signed/decided-min-at-most-left :
  (c l : ℚ)
  → data/Bool.rec (fun x₄ => ℚ) l c (signed/decide-at-most c l) ≤ c

ℚ = signed/SignedRational≤ = signed/at-most

signed/decided-min-monotone-level

TheoremFor all inputsNo axiomsSource

signed/decided-min-monotone-level :
  (c l k : ℚ)
  (h : l ≤ k)
  → data/Bool.rec (fun x₄ => ℚ) l c (signed/decide-at-most c l) ≤ data/Bool.rec (fun x₈ => ℚ) k c (signed/decide-at-most c k)

ℚ = signed/SignedRational≤ = signed/at-most

signed/decided-min-monotone-claim

TheoremFor all inputsNo axiomsSource

signed/decided-min-monotone-claim :
  (c d l : ℚ)
  (h : c ≤ d)
  → data/Bool.rec (fun x₄ => ℚ) l c (signed/decide-at-most c l) ≤ data/Bool.rec (fun x₈ => ℚ) l d (signed/decide-at-most d l)

ℚ = signed/SignedRational≤ = signed/at-most

signed/termwise-equal-of-equal-sums

TheoremFor all inputsNo axiomsSource

signed/termwise-equal-of-equal-sums :
  (F G : ℕ → ℚ) (n : ℕ)
  (hle : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), F idx ≤ G idx)
  (hsum : signed/expr-sum-value F n = signed/expr-sum-value G n)
  (idx : ℕ) (bound : nat/strict-below idx n)
  → F idx = G idx

= = signed/equivalent: equality of rational valuesℕ = data/Natℚ = signed/SignedRational≤ = signed/at-most

signed/cea-award-unique

TheoremFor all inputsNo axiomsSource

signed/cea-award-unique :
  (c d x y : ℕ → ℚ) (l k E Ey : ℚ) (n : ℕ)
  (hx : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), x idx = data/Bool.rec (fun x₂ => ℚ) l (c idx) (signed/decide-at-most (c idx) l))
  (hy : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), y idx = data/Bool.rec (fun x₄ => ℚ) k (d idx) (signed/decide-at-most (d idx) k))
  (hEx : signed/expr-sum-value (fun sx => x sx) n = E)
  (hEy : signed/expr-sum-value (fun sy => y sy) n = Ey)
  (hc : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), c idx = d idx)
  (hE : E = Ey)
  (idx : ℕ) (bound : nat/strict-below idx n)
  → x idx = y idx

= = signed/equivalent: equality of rational valuesℕ = data/Natℚ = signed/SignedRational