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Proof step 15 from proof nummult-le-01 (previous step)

i1 < i, i1 < ~1, i0 < i, i0 + 1 < i, i * (i1 + 1) + j0 < i, j0 < i, 0 < i, i * (i1 + 1) + j0 ≤ 0, abs(j0) < i, j0 ≤ 0, i * i0 < i * (i1 + 1) + j0, i1 < i0, i * i0 + 1 < i, i * i0 < i, i0 < 0, i1 < 0 ⊦ i = i * (i1 + 1) + j0

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] → [i, i0, i1 + 1, j0] flags:

Additional information

Used simplifier rules: 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%lf-03: ⊦ i < j ∧ i = i0 → i0 < j; (forward) from specification int-basic1%lf-02: ⊦ i < j ∧ j = i0 → i < i0; (forward) from specification int-basic1%numls-03: ⊦ i + (* j) < (* j0) ↔ i < j0 - j; from specification int-abs%succ-02: ⊦ i < j + 1 ↔ i ≤ j; from specification int-abs%ler: ⊦ i ≤ i; from specification int-abs%lst-03: ⊦ i + j < i0 ∧ ¬ j < j0 → i + j0 < i0; from specification int-basic2%ls-02: ⊦ i < i + j ↔ 0 < j; from specification int-basic2%lels-04: ⊦ i ≤ j → (¬ ¬ j < i ↔ false); from specification int-abs Used terms: i, i0, i1 + 1, j0