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Proof step 20 from proof noeth-modlep (previous step)

i * j0 < (i0 + 1) * j0, i0 < i, 0 < j2, j0 < j2, j0 * i + i + i0 - i0 + j0 * i0 ≤ j0 + i, i + j0 * i ≤ j0 * i0 + i0 + j0, 0 ≤ j0, ∀ j0, j, i, i0. j0 < abs(j) ∧ abs(j) < j2 ∧ abs(j) * i ≤ abs(j) * i0 + j0 ∧ 0 ≤ j0 → i ≤ i0 ⊦ ~1 = j0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: ls-pos-01 seq: ⊦ 0 < j → (i * j < i0 * j ↔ i < i0) subst: [i0, j, i] → [i0 + 1, j0, i] 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;%Mult-zero: ⊦ i * 0 = 0;% ⊦ i < j + 1 ↔ i ≤ j; (s)% ⊦ i < j → (j ≤ i ↔ false); (as)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ i ≤ j → (j ≤ i ↔ i = j); (s) Used terms: i0 + 1, j0, i