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Proof step 11 from proof div-add1-negpos-neg-01 (previous step)

i1 * j + i2 * j + j0 = i0 * j + j1, (i0 * j + j1) + 1 < j0, abs(j1) < j, i1 * j < j0, i1 * j + i2 * j < j, i1 * j + i2 * j < j0, (i1 * j + i2 * j) + 1 < j0, 2 + i1 < j, i1 < j0, 0 ≤ i2, 0 < j, 1 < j, j + i1 * j ≤ 0, i1 < 0, i1 < j, j0 < j, 0 < j0, i1 * j + i2 * j < ~1, 0 ≤ i2 * j + j0, j + i1 * j < j0, i1 + 1 < j0, i1 * j + i2 * j < 0, 2 + i1 * j + i2 * j < j, j1 ≤ 0, j1 < j, j1 < j0, i0 ≤ 0, i0 < j, i0 < j0, i0 * j + j1 < 0, i0 * j + j1 < j, i0 * j + j1 < j0, 2 + i0 * j + j1 < j, j1 + 1 < j, i0 + 1 < j ⊦ j0 = 0, j = 1, 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] → [i2 + i1, j, j0 - j1, i0, 0] 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%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%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%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%eq-dist: ⊦ (i + j) * i0 = j0 + j * i0 ↔ i * i0 = 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%succ-01: ⊦ i +1 = i + 1; from specification int-basic2%lst-04: ⊦ i + j < i0 ∧ j0 < j → i + j0 < i0; from specification int-basic2%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%zero-ls: ⊦ 0 < i → (i * j < 0 ↔ j < 0); from specification int-mult%ls-add-02: ⊦ i * (j + j1) + i0 < i * j + j0 ↔ i * j1 + i0 < j0; from specification int-mult%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%modzero-02: ⊦ i < 0 ∧ j0 < j ∧ 0 < j → i * j + j0 < 0; from specification int-mult%zero-gels-03: ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%lep-01: ⊦ i + j ≤ i ↔ j ≤ 0; from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used terms: i2 + i1, j, j0 - j1, i0, 0