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Proof step 25 from proof modmod-add-case (previous step)

j + i0 ≤ 0, 0 ≤ i + i0 % j, i + i0 < j, i + i0 < 0, i0 + 1 < j, 0 < j, 0 ≤ i, i0 < 0, i0 < j, (i + i0) + 1 < j ⊦ j = 1, (i + i0 % j) % j = (i + i0) % j, (i + i0 % j) % j = (i + i0) % j + j, (i + i0) % j = (i + i0 % j) % j + j

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: modmod-add2-posneg-neg-01 seq: ⊦ 0 ≤ i + i0 % j ∧ i + i0 < 0 ∧ 0 ≤ i ∧ i0 < 0 ∧ (i + i0 % j) % j ≠ 0 ∧ 0 < j → (i + i0 % j) % j = (i + i0) % j + j subst: [j, i0, i] → [j, i0, i] flags: rewrite, current

Additional information

Used simplifier rules: modmod-add2-posneg-neg-02: ⊦ 0 ≤ i + i0 % j ∧ i + i0 < 0 ∧ 0 ≤ i ∧ i0 < 0 ∧ (i + i0 % j) % j = 0 ∧ 0 < j → (i + i0 % j) % j = (i + i0) % j;%modmod-add-zero: 0 < j ⊦ (i + i0 % j) % j = 0 → (i + i0) % j = 0;%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%lels-03: ⊦ i < j → (¬ ¬ j ≤ i ↔ false); from specification int-abs Used terms: j, i0, i