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Reference · Lemmas behind operations-research checks

rainbow.math.lemmas.one-center

Lemmas behind operations-research checks src/rainbow/math/lemmas/one_center.clj0 definitions, 6 theorems

General lemmas: one center. 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.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

signed/le-indicator-cases

TheoremFor all inputsNo axiomsSource

signed/le-indicator-cases :
  (p q : ℚ)
  → proposition/disjunction (data/Bool.rec (fun x₁₆ => ℚ) 0 1 (signed/decide-at-most p q) = 0) (proposition/conjunction (data/Bool.rec (fun x₂₄ => ℚ) 0 1 (signed/decide-at-most p q) = 1) (p ≤ q))

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

signed/eq-indicator-cases

TheoremFor all inputsNo axiomsSource

signed/eq-indicator-cases :
  (p c : ℚ)
  → proposition/disjunction (data/Bool.rec (fun x₂₈ => ℚ) 0 1 (signed/decide-at-most p c) · data/Bool.rec (fun x₃₂ => ℚ) 0 1 (signed/decide-at-most c p) = 0) (proposition/conjunction (data/Bool.rec (fun x₄₄ => ℚ) 0 1 (signed/decide-at-most p c) · data/Bool.rec (fun x₄₈ => ℚ) 0 1 (signed/decide-at-most c p) = 1) (p = c))

0 = signed/zero1 = signed/one= = signed/equivalent: equality of rational values· = signed/multiplyℚ = signed/SignedRational

signed/indicator-attained-bound

TheoremFor all inputsNo axiomsSource

signed/indicator-attained-bound :
  (I F : ℕ → ℚ) (c R : ℚ) (n : ℕ)
  (hI : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), proposition/disjunction (I idx = 0) (proposition/conjunction (I idx = 1) (F idx = c)))
  (hsum : 1 ≤ signed/expr-sum-value (fun si => I si) n)
  (hb : ∀ (idx : ℕ), ∀ (bound : nat/strict-below idx n), F idx ≤ R)
  → c ≤ R

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

signed/two-times

TheoremFor all inputsNo axiomsSource

signed/two-times :
  (x : ℚ)
  → signed/from-nnrat 2 · x = x + x

+ = signed/add= = signed/equivalent: equality of rational values· = signed/multiplyℚ = signed/SignedRational

signed/at-most-halve

TheoremFor all inputsNo axiomsSource

signed/at-most-halve :
  (u v : ℚ)
  (h : u + u ≤ v + v)
  → u ≤ v

+ = signed/addℚ = signed/SignedRational≤ = signed/at-most

signed/interval-midpoint-covers

TheoremFor all inputsNo axiomsSource

signed/interval-midpoint-covers :
  (ai x r lo hi : ℚ)
  (lo-a : lo ≤ ai)
  (a-hi : ai ≤ hi)
  (xx : x + x = lo + hi)
  (rr : r + r = hi + signed/negate lo)
  → proposition/conjunction (ai + signed/negate x ≤ r) (x + signed/negate ai ≤ r)

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