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Proof step 66 from proof uint2bits-restsgn (previous step)

0 ≤ i0 ∧ 0 ≤ n→i(# bits0), 2 ^ i0 = i + 1, 2 ^ n→i(# bits0) * (i + 1) = (i + 1) * 2 ^ n→i(# bits0), i * 2 ^ n→i(# bits0) < 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), 0 ≤ (i + 1) * 2 ^ n→i(# bits0), 0 < i * 2 ^ n→i(# bits0), 1 < i0 + n→i(# bits0), 0 ≤ i * 2 ^ n→i(# bits0), 0 ≤ 2 ^ n→i(# bits0) * i, i * 2 ^ n→i(# bits0) * 2 + 1 < (i + 1) * 2 ^ n→i(# bits0), (i + 1) * 2 ^ n→i(# bits0) < 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), 2 ^ n→i(# bits0) * i < (i + 1) * 2 ^ n→i(# bits0), i * 2 ^ n→i(# bits0) < (i + 1) * 2 ^ n→i(# bits0), i * 2 ^ n→i(# bits0) + 1 < 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), 0 ≤ i0, 1 < (i + 1) * 2 ^ n→i(# bits0), 0 < i0 + n→i(# bits0), 0 ≤ i0 + n→i(# bits0), 0 < (i + 1) * 2 ^ n→i(# bits0), ∀ j0, i1. i→n(j0) < i→n(2 * (i * 2 ^ n→i(# bits0) * 2 + 1) + 0) → 0 < i1 ∧ 0 ≤ j0 ∧ 2 ^ i1 < j0 ∧ j0 < 2 * 2 ^ i1 → sgn(uint2bits(2 ^ (i1 + 2) - j0).rest) = uint2bits(2 ^ (i1 + 2) - j0).rest ⊦ 4 * (i + 1) * 2 ^ n→i(# bits0) = 1 + 2 * (i * 2 ^ n→i(# bits0) * 2 + 1), bits0 = [], false ∈ bits0

User applied rule insert spec-lemma

And obtained the following new goals

Rule argument

seq: j * i + j0 < j * i0, abs(j0) < j ⊦ i ≤ i0 subst: [j, i, j0, i0] → [2 ^ n→i(# bits0), 2 * i, 1, i + 1] rotate: F spec: int-mult inst: name: modlsp-02

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ 0 ≤ n→i(n);% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i + j < i0 ∧ ¬ j < j0 → i + j0 < i0;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ 2 ^ i ≤ 1 ↔ i < 1;% ⊦ n→i(n) < 1 ↔ n = 0;% ⊦ # bits = 0 ↔ bits = []; Used terms: 2 ^ n→i(# bits0), 2 * i, 1, i + 1