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Proof step 11 from proof mod-add-neg (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: modeq-neg seq: i0 * j + j0 < 0, 0 < j, j0 < 0 ⊦ (i0 * j + j0) % j = j0 % j subst: [j0, j, i0] → [j0, j, i0 + i1] flags: <all paths>

Additional information

Used simplifier rules: mod-mult-04: j ≠ 0 ⊦ (i * j) % j = 0;%mod-zero-05: ⊦ abs(i) < j → i % j = i;% ⊦ i < j ↔ ¬ j ≤ i; (ss)%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs% ⊦ i + j < i + i0 ↔ j < i0; (s)% ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); (s)% ⊦ i < i + j ↔ 0 < j; (s)% ⊦ i < j → ¬ j < i; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i ≤ j → (j ≤ i ↔ i = j); (s)%lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1%trans: ⊦ i < j ∧ j < j0 → i < j0; (forward) from specification int-basic1%lf-02: ⊦ i < j ∧ j = i0 → i < i0; (forward) from specification int-basic1%lf-03: ⊦ i < j ∧ i = i0 → i0 < j; (forward) from specification int-basic1% ⊦ i + 0 = i; (s)% ⊦ i * 0 = 0; (s)%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: j0, j, i0 + i1