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Proof step 41 from proof gcd-mult-pos (previous step)

0 < i1, 0 < j, 0 < i, j * i1 + i * i1 ≤ abs(i % j * i1) + abs(j * i1), 0 ≤ i, j ≤ i, 0 ≤ i % j * i1, 0 ≤ j * i1, ∀ j0, i0, i2. i→n(j0 * i0 + i2 * i0) < i→n(j * i1 + i * i1) → 0 < i2 ∧ 0 < i0 ∧ 0 ≤ j0 → gcd(i2 * i0, j0 * i0) = gcd(i2, j0) * i0 ⊦ gcd(j * i1, i % j * i1) = gcd(j, i % j) * i1, i1 = 0, j = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 42:
0 < i1, 0 < j, 0 < i, j ≤ abs(i % j), i ≤ abs(i % j), 0 ≤ i, j ≤ i, ∀ j0, i0, i2. i→n(j0 * i0 + i2 * i0) < i→n(j * i1 + i * i1) → 0 < i2 ∧ 0 < i0 ∧ 0 ≤ j0 → gcd(i2 * i0, j0 * i0) = gcd(i2, j0) * i0 ⊦ i % j = 0, gcd(j * i1, i % j * i1) = gcd(j, i % j) * i1, i1 = 0, j = 0

Rule argument

Additional information

Used simplifier rules: ⊦ 0 ≤ abs(i); (s)% ⊦ 0 < abs(i) ↔ i ≠ 0; (s)%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-le: ⊦ i ≤ j ∧ 0 < i → 0 < j; (forward) from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2% ⊦ 0 < i → (i * j ≤ i * i0 ↔ j ≤ i0); (s)% ⊦ i + j ≤ i0 + j ↔ i ≤ i0; (s)% ⊦ 0 < j → abs(j) = j; (s)% ⊦ abs(i * j) = abs(i) * abs(j); (s)% ⊦ 0 < j ∧ 0 ≤ i → 0 ≤ i % j; (s)%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); (s)