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Reference · Lemmas behind operations-research checks

rainbow.math.lemmas.bottleneck-assignment

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

General lemmas: bottleneck assignment. 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

signed/cheaper-indicator-one

TheoremFor all inputsNo axiomsSource

signed/cheaper-indicator-one :
  (b x B : ℚ)
  (hxB : x ≤ B)
  (hnot : b ≤ B → proposition/falsity)
  → data/Bool.rec (fun x₄ => ℚ) 1 0 (signed/decide-at-most b x) = 1

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

signed/not-successor-at-most-self

TheoremFor all inputsNo axiomsSource

signed/not-successor-at-most-self :
  (X : ℚ)
  (h : X + 1 ≤ X)
  → proposition/falsity

+ = signed/add1 = signed/oneℚ = signed/SignedRational≤ = signed/at-most

signed/bottleneck-assignment-lower-bound

TheoremFor all inputsNo axiomsSource

signed/bottleneck-assignment-lower-bound :
  (A : ℕ → ℕ → ℚ) (r c : ℕ → ℚ) (b : ℚ) (n : ℕ)
  (hcov : ∀ (ki : ℕ), ∀ (ki-lt : nat/strict-below ki n), ∀ (kj : ℕ), ∀ (kj-lt : nat/strict-below kj n), data/Bool.rec (fun x₃ => ℚ) 1 0 (signed/decide-at-most b (A ki kj)) ≤ r ki + c kj)
  (hsize : signed/expr-sum-value (fun sr => r sr) n + signed/expr-sum-value (fun sc => c sc) n + 1 ≤ signed/from-nnrat (nnrat/make n 0))
  (y : ℕ → ℕ → ℚ) (B : ℚ)
  (hy01 : ∀ (yi : ℕ), ∀ (yi-lt : nat/strict-below yi n), ∀ (yj : ℕ), ∀ (yj-lt : nat/strict-below yj n), y yi yj · (1 + signed/negate (y yi yj)) = 0)
  (hrow : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), signed/expr-sum-value (fun rj => y idx rj) n = 1)
  (hcol : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), signed/expr-sum-value (fun ci => y ci idx) n = 1)
  (hbot : ∀ (bi : ℕ), ∀ (bi-lt : nat/strict-below bi n), ∀ (bj : ℕ), ∀ (bj-lt : nat/strict-below bj n), y bi bj · (A bi bj + signed/negate B) ≤ 0)
  → b ≤ B

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