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Proof step 118 from proof uint2bits-restsgn (previous step)

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

User applied rule insert elim lemma

And obtained the following new goals

Proof step 119:
i + 1 = i0 + n→i(# bits), 2 ^ i0 ≤ (2 ^ i0 - 1) * 2, 0 ≤ 2 ^ i0, 0 ≤ i, (2 ^ n→i(# bits) * (2 ^ i0 - 1)) * 2 + 1 < 4 * 2 ^ i, 0 ≤ 2 ^ i, 0 < i, 2 ^ i ≤ (2 ^ n→i(# bits) * (2 ^ i0 - 1)) * 2, 2 ^ n→i(# bits) * (2 ^ i0 - 1) < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 < 2 ^ n→i(# bits), n→i(# bits) ≤ i + 1, 2 ^ n→i(# bits) * (2 ^ i0 - 1) < 2 * 2 ^ i, 1 < 2 ^ i0, 0 < 2 ^ i0, ∀ j0, i1. i→n(j0) < i→n((2 ^ n→i(# bits) * (2 ^ i0 - 1)) * 2 + 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 ⊦ sgn(bits + true ') = bits + true ', 4 * 2 ^ i = 2 * (2 ^ n→i(# bits) * (2 ^ i0 - 1) + 1), false ∈ bits

Rule argument

seq: ⊦ i0 = i - j ↔ i = i0 + j subst: [i, j, i0] → [i + 1, n→i(# bits), i0] rotate: T spec: int-abs inst: name: elim-sub

Additional information

Used simplifier rules: ⊦ i0 = i - j ↔ i = i0 + j; (elim) Used terms: i + 1, n→i(# bits), i0