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Proof step 9 from proof modmod-add-if (previous step)

i0 < 0, 0 < j, i + i0 < 0, 0 ≤ i ⊦ (i + i0 % j) % j = ((i + i0 % j) % j ≠ 0 ⊃ (i + i0) % j + j;0), i + i0 % j < 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
i + i0 < 0, 0 < j, i0 < 0, j + i0 ≤ 0, 0 ≤ i, 0 ≤ i + i0 % j ⊦ (i + i0 % j) % j = ((i + i0 % j) % j ≠ 0 ⊃ (i + i0) % j + j;0), j = 1

Rule argument

Additional information

Used simplifier rules: mod-ls-same-02: j ≠ 0 ⊦ i < i % j ↔ i < 0 ∧ (abs(j) = 1 ∨ abs(j) ≤ abs(i));%mod-one: ⊦ i % 1 = 0;% ⊦ i + j < j ↔ i < 0; (s)%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%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs% ⊦ i ≤ ~ j ↔ i + j ≤ 0; (s)% ⊦ i < 0 → abs(i) = ~ i; (s)% ⊦ 0 < j → abs(j) = j; (s)%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 + i0 ↔ j < i0; (s)% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs% ⊦ i < j ↔ ¬ j ≤ i; (ss)