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Proof step 9 from proof noeth-modlep (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ (i - j) * j0 = i * j0 - j * j0
Proof step 11:
~1 * i0 - j0 * i0 ≤ (~1 - j0) * i + j0, i0 < i, 0 ≤ j0, 0 < j2, (~1 * i0 - j0 * i0) + i0 ≤ i + (~1 - j0) * i + j0, j0 < j2, ∀ j, i, i0, j0. abs(j) < j2 → abs(j) * i ≤ abs(j) * i0 + j0 ∧ 0 ≤ j0 ∧ j0 < abs(j) → i ≤ i0 ⊦ ~1 = j0

Rule argument

spec: inst: name: dist-sub-split seq: ⊦ (i - j) * j0 = i * j0 - j * j0 subst: [j0, j, i] → [i0, j0, ~1] flags: rewrite, current

Additional information

Used terms: i0, j0, ~1