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Proof step 5 from proof modeq-pos-03 (previous step)

j0 = i1 * j + j1, j0 + 1 < i0, j0 < 0, 0 < j, j0 + 1 < j, j0 < j, 0 < i0, j0 < i0, abs(j1) < j, 0 ≤ i0 * j + j0, j1 ≤ 0 ⊦ (i0 * j + j0) % j = j1 + j, j1 = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
j1 < i1 + i0, i1 * j + j1 < i0, i1 * j + j1 < j, 0 < i1 + i0, j1 < i0, j1 < j, j1 + 1 < i0, j1 + 1 < j, 1 < j, (i1 * j + j1) + 1 < j, i1 * j + j1 < 0, (i1 * j + j1) + 1 < i0, j1 < 0, 0 < j + j1, i1 < i0, 0 < i0, i1 < j, i1 < j + j1, 0 < j, (i1 * j + j1) + 1 < i1 + i0, i1 * j + j1 < i1 + i0, j1 + 1 < i1 + i0, i1 ≤ 0, 0 ≤ j * (i1 + i0) + j1, i1 ≤ j * (i1 + i0) + j1, 0 ≤ j, i1 ≤ j ⊦ (j * (i1 + i0) + j1) % j = j1 + j, j1 = 0

Rule argument

Additional information

Used simplifier rules: numlsf-01: ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs% ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); (s)% ⊦ i < 1 ↔ i ≤ 0; (s)% ⊦ 0 < i → (i * j < i ↔ j < 1); (s)% ⊦ 0 ≤ j1 ∧ j1 < j → (i0 * j + j1 < j ↔ i0 ≤ 0); (s)% ⊦ 0 < i → (0 < i * j ↔ 0 < j); (s)% ⊦ i < i + j ↔ 0 < j; (s)% ⊦ i + j < i + i0 ↔ j < i0; (s)% ⊦ i +1 = i + 1; (s)%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%trans: ⊦ i < j ∧ j < j0 → i < j0; (forward) from specification int-basic1%lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1% ⊦ ~ i < j ↔ 0 < j + i; (s)% ⊦ i < 0 → abs(i) = ~ i; (s)% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)%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% ⊦ i * j + i * j0 = i * (j + j0); (s)