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Proof step 55 from proof lagrange-lemma (previous step)

2 ≠ 0 ∧ (j - 1) % 2 = 0, j * i = (j1 ^2 + j0 ^2) + 1, j1 ^2 ≤ (j - 1) / 2 * (j - 1) / 2, j0 ^2 ≤ (j - 1) / 2 * (j - 1) / 2, 0 ≤ j1, 0 ≤ j, prime(j), j0 ≤ (j - 1) / 2, j1 ≤ (j - 1) / 2, 0 ≤ j0, 0 < j, 0 < i, 0 ≤ i, j * j ≤ (j - 1) * (j - 1) / 2 + 1, j ≤ i, j * i ≤ (j - 1) * (j - 1) / 2 + 1 ⊦ (j - 1) * (j - 1) / 2 < (j - 1) * (j - 1), j = 2

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 56:
⊦ 0 < i → (i * j < i * j0 ↔ j < j0)
Proof step 57:
0 < j - 1, 2 ≠ 0 ∧ (j - 1) % 2 = 0, j * i = (j1 ^2 + j0 ^2) + 1, j1 ^2 ≤ (j - 1) / 2 * (j - 1) / 2, j0 ^2 ≤ (j - 1) / 2 * (j - 1) / 2, 0 ≤ j1, 0 ≤ j, prime(j), j0 ≤ (j - 1) / 2, j1 ≤ (j - 1) / 2, 0 ≤ j0, 0 < j, 0 < i, 0 ≤ i, j * j ≤ (j - 1) * (j - 1) / 2 + 1, j ≤ i, j * i ≤ (j - 1) * (j - 1) / 2 + 1 ⊦ (j - 1) / 2 < j - 1, j = 2

Rule argument

spec: int-mult inst: name: ls-pos seq: ⊦ 0 < i → (i * j < i * j0 ↔ j < j0) subst: [j0, j, i] → [j - 1, (j - 1) / 2, j - 1] flags: rewrite

Additional information

Used simplifier rules: prime-odd-01: ⊦ prime(j) ∧ j ≠ 2 → (j - 1) % 2 = 0; from specification int-gcd%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%sub-ls-01: ⊦ i < j - i0 ↔ i + i0 < j; 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%bet: ⊦ 0 < i → (i ≤ 1 ↔ i = 1); from specification int-abs%one-01: ⊦ i = 1 → i * j = j; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%sub-eq-01: ⊦ i = j → i - j = 0; from specification int-abs%anti: ⊦ i ≤ j → (j ≤ i ↔ i = j); from specification int-abs%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%zero-i: ⊦ 0 ^2 = 0; from specification int-square%prime-one: ⊦ i ≤ 1 → ¬ prime(i); from specification int-gcd Used terms: j - 1, (j - 1) / 2, j - 1