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Proof step 12 from proof div-add1-negpos-neg-01 (previous step)

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

User applied rule insert spec-lemma

And obtained the following new goals

Rule argument

spec: int-mult 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]

Additional information

Used simplifier rules: ⊦ i - j = i0 ↔ i = i0 + j; (s)% ⊦ i + 0 = i; (s)%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% ⊦ i + j = i + i0 ↔ j = i0; (s)% ⊦ i = i + j ↔ j = 0; (s)%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% ⊦ i + j < i + i0 ↔ j < i0; (s)% ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); (s)% ⊦ i + j < j ↔ i < 0; (s)% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ i + j ≤ i ↔ j ≤ 0; (s)% ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ 0 < j → abs(j) = j; (s)% ⊦ i ≤ j → ¬ j < i; (s) Used terms: i2 + i1, j, j0 - j1, i0, 0