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Proof step 8 from proof mod-add1-posneg-pos (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ 0 < i → i1 < i1 + i; (s)% ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); (s)% ⊦ i + j < i + i0 ↔ j < i0; (s)%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 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%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs% ⊦ 0 < i → (0 < i * j ↔ 0 < j); (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); (s)%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2% ⊦ i + 0 = i; (s)% ⊦ i * j + i * j0 = i * (j + j0); (s)