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Proof step 14 from proof root-elim (previous step)

i < (sqrt(i) + 1) * (sqrt(i) + 1), sqrt(i) * sqrt(i) ≤ i, 0 ≤ i, 0 < (sqrt(i) + 1) * (sqrt(i) + 1), sqrt(i) < abs(sqrt(i) + 1) ⊦ (sqrt(i) + 1) * (sqrt(i) + 1) = sqrt(i) * sqrt(i) + 2 * sqrt(i) + 1, ∃ j0. i = sqrt(i) * sqrt(i) + j0 ∧ j0 ≤ 2 * sqrt(i) ∧ 0 ≤ sqrt(i) ∧ 0 ≤ j0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: dist-lsplit seq: ⊦ (i + i0) * j = i * j + i0 * j subst: [j, i0, i] → [sqrt(i) + 1, 1, sqrt(i)] flags: rewrite

Additional information

Used simplifier rules: one: ⊦ 1 * i = i; from specification int-mult%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%numeq-02: ⊦ (* i2) * i + j = i + j0 ↔ (i2 - 1) * i + j = j0; from specification int-mult%add-in: ⊦ i + j = i + i0 ↔ j = i0; from specification int-basic2%eq-dist: ⊦ (i + j) * i0 = j0 + j * i0 ↔ i * i0 = j0; from specification int-mult%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: sqrt(i) + 1, 1, sqrt(i)