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Proof step 27 from proof middle-coef-max_exp-big-primes (previous step)

2 * i < 3 * j, j ≤ i, prime(j), 2 < i, 0 < i, 2 * i < j ^ 3, j * j ≤ 2 * i, 0 ≤ i, j < 3, j * j < j ^ 3, 0 < j ⊦ (2 * i) / j + (2 * i) / (j * j) = 2 * i / j + 2 * i / (j * j), j = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-basic1 inst: name: lsrec seq: ⊦ i < j ↔ i +1 = j ∨ i +1 < j subst: [j, i] → [3, j] flags: rewrite

Additional information

Used simplifier rules: succ-01: ⊦ i +1 = i + 1; from specification int-basic2%numadd-01: ⊦ (* j) = i0 + (* j0) ↔ j - j0 = i0; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%numls-03: ⊦ i + (* j) < (* j0) ↔ i < j0 - j; from specification int-abs%two-ls: ⊦ 0 < j → (j < 2 ↔ j = 1); from specification int-basic1%one: ⊦ i / 1 = i; from specification int-div%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-mult-two: ⊦ i + i = 2 * i; from specification int-mult%eq-pos: ⊦ 0 < i → (i * j = i * j0 ↔ j = j0); from specification int-mult%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%div-mult-num: j ≠ 0, j0 % j = 0 ⊦ (i * (* j0)) / (* j) = i * j0 / j; from specification int-div%one: ⊦ 1 * i = i; from specification int-mult%div-mult-num-01: j % j0 = 0, j ≠ 0, j0 ≠ 0 ⊦ (i * (* j0)) / (* j) = i / j / j0; from specification int-div%numeq-02: ⊦ (* i2) * i + j = i + j0 ↔ (i2 - 1) * i + j = j0; from specification int-mult%nummult-le-02: ⊦ 0 < i ∧ 0 ≤ j → ((* j) ≤ (* i) * i0 ↔ (i + j - 1) / i ≤ i0); from specification int-div%numleledel-01: i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j; from specification int-abs%nummult-ls: ⊦ 0 < i ∧ 0 ≤ j → ((* i) * i0 < (* j) ↔ i0 < (i + j - 1) / i); from specification int-div%div-zero-03: 0 < j ⊦ 0 ≤ i ∧ i < j → i / (* j) = 0; from specification int-div%ax-04: ⊦ i + 0 = i; from specification int-basic2%div-eq-same-01: j ≠ 0 ⊦ i / j = i ↔ i = 0 ∨ j = 1; from specification int-div%numlelsdel-01: i0 ≠ 0, i0 ≤ i ⊦ (* i) < j → (* i0) < j; from specification int-abs%prime-two-01: ⊦ prime(2); from specification int-gcd%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%numlslsdel: j0 ≠ 0, j < j0 ⊦ i < (* j) → i < (* j0); from specification int-basic1%num-between: i + 1 = i0 ⊦ j < (* i0) → ¬ (* i) < j; from specification int-basic2 Used terms: 3, j