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Reference · Lemmas behind operations-research checks

rainbow.math.lemmas.kstwo-range

Lemmas behind operations-research checks src/rainbow/math/lemmas/kstwo_range.clj0 definitions, 4 theorems

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

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math.linear.matrix.affine-general-rate-certificate math.sums.of-nat math.lemmas.one-center math.rational.signed.multiply-positive

signed/indicator-box

TheoremFor all inputsNo axiomsSource

signed/indicator-box :
  (ib-l ib-r : ℚ)
  → proposition/conjunction (0 ≤ data/Bool.rec (fun x₁₂ => ℚ) 0 1 (signed/decide-at-most ib-l ib-r)) (data/Bool.rec (fun x₁₆ => ℚ) 0 1 (signed/decide-at-most ib-l ib-r) ≤ 1)

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

signed/indicator-count-bounds

TheoremFor all inputsNo axiomsSource

signed/indicator-count-bounds :
  (ic-s : ℕ → ℚ) (ic-t : ℚ) (ic-n : ℕ)
  → proposition/conjunction (0 ≤ signed/expr-sum-value (fun ic-i => data/Bool.rec (fun x₁₈ => ℚ) 0 1 (signed/decide-at-most (ic-s ic-i) ic-t)) ic-n) (signed/expr-sum-value (fun ic-i => data/Bool.rec (fun x₂₄ => ℚ) 0 1 (signed/decide-at-most (ic-s ic-i) ic-t)) ic-n ≤ signed/from-nnrat (nnrat/make ic-n 0))

0 = signed/zero1 = signed/oneℕ = data/Natℚ = signed/SignedRational≤ = signed/at-most

signed/abs-reading

TheoremFor all inputsNo axiomsSource

signed/abs-reading :
  (ab-a ab-d : ℚ)
  (ab-1 : ab-d ≤ ab-a)
  (ab-2 : signed/negate ab-d ≤ ab-a)
  (ab-p : (ab-a + signed/negate ab-d) · (ab-a + ab-d) = 0)
  → proposition/conjunction (disjunction) (0 ≤ ab-a)
  where disjunction := proposition/disjunction (ab-a = ab-d) (ab-a = signed/negate ab-d)

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

signed/ks-abs-bound

TheoremFor all inputsNo axiomsSource

signed/ks-abs-bound :
  (kb-a kb-f kb-g kb-n kb-m : ℚ)
  (kb-e : proposition/disjunction (kb-a = kb-m · kb-f + signed/negate (kb-n · kb-g)) (kb-a = signed/negate (kb-m · kb-f + signed/negate (kb-n · kb-g))))
  (kb-f0 : 0 ≤ kb-f)
  (kb-fn : kb-f ≤ kb-n)
  (kb-g0 : 0 ≤ kb-g)
  (kb-gm : kb-g ≤ kb-m)
  (kb-hn : 0 ≤ kb-n)
  (kb-hm : 0 ≤ kb-m)
  → kb-a ≤ kb-n · kb-m

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