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Proof step 9 from proof zero-gr-add-02 (previous step)

j < 0, 0 < i1 * j + i0, 0 < i1, 0 < i1 * (j + 1), i1 * j + i0 ≤ i1 * (j + 1), 0 ≤ i0, i0 ≤ i1 ⊦ i1 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: zero-gr-02 seq: ⊦ 0 ≤ i → (0 < i * j ↔ 0 < j ∧ i ≠ 0) subst: [j, i] → [j + 1, i1] flags: <all paths>

Additional information

Used simplifier rules: dist-lsplit: ⊦ (i + i0) * j = i * j + i0 * j;%one: ⊦ 1 * i = i;%associativity: ⊦ i * j * j0 = (i * j) * j0;%commutativity: ⊦ i * j = j * i;%dist-lsplit: ⊦ (i + i0) * j = i * j + i0 * j;%one: ⊦ 1 * i = i;%associativity: ⊦ i * j * j0 = (i * j) * j0;%commutativity: ⊦ i * j = j * i;% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i + j ≤ i0 + j ↔ i ≤ i0; (s)% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0; (s)% ⊦ ~1 < i ↔ 0 ≤ i; (s)% ⊦ i < j → (j ≤ i ↔ false); (as) Used terms: j + 1, i1