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Proof step 8 from proof modeq-neg-02 (previous step)

j * (i1 + i0) + j1 = i2 * j + j0, i2 * j + j0 < 0, j0 < j1, j0 < j, j0 ≤ 0, j1 < j, i0 < 0, 0 < j1, 1 < j, 0 < j, 0 ≤ i1 * j + j1, abs(j0) < j ⊦ j1 = 0, j1 = j0 + j

User applied rule insert spec-lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: modeqp seq: i * j + j0 = i0 * j + j1, 0 ≤ j0, j0 < j, 0 ≤ j1, j1 < j ⊦ i = i0 ∧ j0 = j1 subst: [i, j, j0, i0, j1] → [i1 + i0, j, j1 - j0, i2, 0] flags: []

Additional information

Used simplifier rules: sub-eq: ⊦ i - j = i0 ↔ i = i0 + j; from specification int-abs%add-zero: ⊦ i + 0 = i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%commutativity: ⊦ i * j = j * i; from specification int-mult%add-in: ⊦ i + j = i + i0 ↔ j = i0; from specification int-basic2%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%eq: ⊦ i = i + j ↔ j = 0; from specification int-basic2%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%ls-pos: ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); from specification int-mult%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%abs-16: ⊦ 0 < j → abs(j) = j; from specification int-abs%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs Used terms: i1 + i0, j, j1 - j0, i2, 0