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Proof step 15 from proof one-neg (previous step)

1 = i0 * j + j0, i0 < 0, j0 < j, i0 + 1 < j, i0 < j, i0 < j0, 0 < j, 0 ≤ j0 ⊦ j = ~1, j = 0

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] → [0, j, 1, i0, j0] flags: []

Additional information

Used simplifier rules: Mult-zero: ⊦ i * 0 = 0; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%bet: ⊦ 0 < i → (i ≤ 1 ↔ i = 1); from specification int-abs%one-01: ⊦ i = 1 → i * j = j; from specification int-mult%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%lsone: ⊦ i < 1 ↔ i ≤ 0; from specification int-abs%int-add-pos-01: ⊦ 0 ≤ j ∧ j < j1 ∧ j + i = j1 → ¬ i ≤ 0; from specification int-abs Used terms: 0, j, 1, i0, j0