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Proof step 11 from proof div-add1-posneg-pos-01 (previous step)

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

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] → [i0 + i1, j, 0, i2, j1 - j0] flags: []

Additional information

Used simplifier rules: sub-eq: ⊦ i - j = i0 ↔ i = i0 + j; from specification int-abs%add-zero-rev: ⊦ 0 + i = 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%add-in: ⊦ i + j = i + i0 ↔ j = i0; from specification int-basic2%eq: ⊦ i = i + j ↔ j = 0; from specification int-basic2%ls-add-03: ⊦ i * j + i0 < i * (j + j1) + j0 ↔ i0 < i * j1 + j0; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%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%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%eq-dist: ⊦ (i + j) * i0 = j0 + j * i0 ↔ i * i0 = j0; from specification int-mult%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%ls-02: ⊦ i < i + j ↔ 0 < j; from specification int-basic2%zero-gr: ⊦ 0 < i → (0 < i * j ↔ 0 < j); from specification int-mult%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs Used terms: i0 + i1, j, 0, i2, j1 - j0