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Proof step 11 from proof modeq-neg-02 (previous step)

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

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 12:
i * j + j0 = i0 * j + j1, 0 ≤ j0, j0 < j, 0 ≤ j1, j1 < j ⊦ i = i0 ∧ j0 = j1
Proof step 13:
j * (i1 + i0) + j1 = i2 * j + j0, abs(j0) < j, 0 ≤ i1 * j + j1, 0 < j, 1 < j, 0 < j1, i0 < 0, j1 < j, j0 ≤ 0, j0 < j, j0 < j1, i2 * j + j0 < 0, j0 ≤ j1, j0 + j < j1 ⊦ (i1 + i0 + 1) * j + (j1 - j0 - j) = i2 * j + 0 ∧ 0 ≤ j1 - j0 - j ∧ j1 - j0 - j < j ∧ 0 ≤ 0, j1 = j0 + j, j1 = 0

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 + 1, j, j1 - j0 - j, i2, 0] flags: []

Additional information

Used simplifier rules: sub-sub: ⊦ i - j - i0 = i - j + i0; from specification int-abs%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%eq-succ-01: ⊦ (i + i0) * j + j0 = (i + i1) * j + j1 ↔ i0 * j + j0 = i1 * j + j1; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: i1 + i0 + 1, j, j1 - j0 - j, i2, 0