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Proof step 139 from proof uint2bits-restsgn (previous step)

n→i(# bits) + i0 = i + 1, (2 ^ i0 - 1) * 2 ^ n→i(# bits) < 2 ^ i * 2, n→i(# bits) ≤ i + 1, 0 < 2 ^ n→i(# bits), 0 < 2 ^ i, 1 < 2 ^ i, 2 ^ i0 ≤ 2 ^ (i0 - 1), 2 ^ i ≤ 2 ^ (n→i(# bits) + 1) * (2 ^ i0 - 1), 0 < i, 0 ≤ 2 ^ i, 2 ^ (n→i(# bits) + 1) * (2 ^ i0 - 1) + 1 < 4 * 2 ^ i, 0 ≤ i, 0 < 2 ^ (n→i(# bits) + 1), 0 ≤ 2 ^ i0, 1 < 2 ^ (i0 - 1), 0 < 2 ^ (i0 - 1), 1 < 2 ^ (n→i(# bits) + 1) * (2 ^ i0 - 1), 0 ≤ 2 ^ (i0 - 1), 0 < 2 ^ i0, 1 < 2 ^ i0, ∀ j0, i1. i→n(j0) < i→n(2 ^ (n→i(# bits) + 1) * (2 ^ i0 - 1) + 1) → 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 * 2 ^ i = 2 * ((2 ^ i0 - 1) * 2 ^ n→i(# bits) + 1), sgn(bits + true ') = bits + true ', false ∈ bits

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

seq: ⊦ 1 < i ∧ 0 ≤ j → (i ^ j ≤ i ^ j0 ↔ j ≤ j0) subst: [j0, j, i] → [i0 - 1, i0, 2] rotate: F spec: int-pot inst: name: le-pos

Additional information

Used simplifier rules: ⊦ i + j ≤ i ↔ j ≤ 0;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < 0 → 2 ^ i = 0;% ⊦ i * ~1 = ~ i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ i + ~ j = i - j;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ i ≤ ~ j ↔ i + j ≤ 0;% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ i < ~ j ↔ i + j < 0;% ⊦ i + (* j) < (* j0) ↔ i < j0 - j; Used terms: i0 - 1, i0, 2