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Proof step 11 from proof sum-all-elems-le (previous step)

(1 + n→i(# ints0)) * i0 < i + i0 * n→i(# ints0), Σ ints0 ≤ i0 * n→i(# ints0), i0 * (1 + n→i(# ints0)) < i + Σ ints0, i ≤ i0, ∀ i1. i1 ∈ ints0 → i1 ≤ i0 ⊦

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] → [i0, n→i(# ints0), 1] 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%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: i0, n→i(# ints0), 1