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Proof step 22 from proof nummult-ls-03 (previous step)

¬ i0 ≤ i3, i2 * (i3 + 1) + j0 ≤ 0, abs(j0) < i2, i3 * i2 + j0 < 0, 0 < i2, j0 < i2, j0 ≤ 0 ⊦ i2 * (i3 + 1) + j0 ≤ i2 * i0

User applied rule insert spec-lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: modlsp-01 seq: j * i < j * i0 + j0, abs(j0) < j ⊦ i ≤ i0 subst: [j, i, i0, j0] → [i2, i0, i3 + 1, j0] flags: []

Additional information

Used simplifier rules: lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%les: ⊦ i < j → (j ≤ i + 1 ↔ j = i + 1); from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%zero-ls: ⊦ 0 < i → (i * j < 0 ↔ j < 0); from specification int-mult%numls-03: ⊦ i + (* j) < (* j0) ↔ i < j0 - j; from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%ls-02: ⊦ i < i + j ↔ 0 < j; from specification int-basic2%ls-dist: ⊦ i * (j + i0) < i * (j0 + i0) + j1 ↔ i * j < i * j0 + j1; from specification int-mult%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs Used terms: i2, i0, i3 + 1, j0