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Proof step 9 from proof i2np (previous step)

0 ≤ j, 0 ≤ i0, ∀ j0. 0 ≤ j0 → i→n(i0) * i→n(j0) = i→n(i0 * j0) ⊦ i→n(j) * i→n(i0) + i→n(j) = i→n(i0 * j + j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
0 ≤ i, 0 ≤ j ⊦ i→n(i) + i→n(j) = i→n(i + j)
Proof step 11:
0 ≤ i0, 0 ≤ i0 * j, 0 ≤ j, ∀ j0. 0 ≤ j0 → i→n(i0) * i→n(j0) = i→n(i0 * j0) ⊦ i→n(j) * i→n(i0) + i→n(j) = i→n(i0 * j) + i→n(j)

Rule argument

spec: <toplevel> name: add seq: 0 ≤ i, 0 ≤ j ⊦ i→n(i) + i→n(j) = i→n(i + j) subst: [j, i] → [j, i0 * j] flags: <rotated> <all paths>

Additional information

Used simplifier rules: ⊦ i * j + i = i * (j + 1); (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ 0 ≤ i → (0 ≤ i * j ↔ i = 0 ∨ 0 ≤ j); (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s) Used terms: j, i0 * j