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Proof step 4 from proof mod-rec (previous step)

i = i0 * j + j0, 0 < j, 0 < i, j0 < j, j ≤ i, 0 ≤ j0 ⊦ j0 = (i - j) % j

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
0 < j, j0 < j, 0 < i0 * j + j0, 0 < i0, j ≤ i0 * j + j0, 0 ≤ j0 ⊦ j0 = (i0 * j + j0 - j) % j

Rule argument

Additional information

Used simplifier rules: forward-le-05: ⊦ 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%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult% ⊦ 0 < i → (0 < i * j ↔ 0 < j); (s)% ⊦ i < i + j ↔ 0 < j; (s)%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs