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Proof step 12 from proof nummult-le-02 (previous step)

0 < i, j0 < i, i2 * i + j0 < i * i0, 0 ≤ i + i2 * i + j0, 0 ≤ j0, ~1 ≤ i2 * i + j0, 0 ≤ i0 ⊦ i2 < i0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-abs name: le-def seq: ⊦ i ≤ j ↔ ¬ j < i subst: [j, i] → [i2 * i + j0, ~1] flags: <all paths>

Additional information

Used simplifier rules: ⊦ i < j ↔ ¬ j ≤ i; (ss)%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs% ⊦ j0 < j ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 ≤ j0 → (i * j + j0 < i0 * j ↔ i < i0); (s)% ⊦ i ≤ j → ¬ j < i; (s)%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 Used terms: i2 * i + j0, ~1