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Proof step 59 from proof uint2bits-restsgn (previous step)

2 ^ n→i(# bits0) * 2 ^ (i - n→i(# bits0)) = 2 ^ i, (2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0) < 2 * ((2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0) * 2 + 1), (2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0) + 1 < 2 * ((2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0) * 2 + 1), n→i(# bits0) ≤ i, 0 ≤ 2 ^ i, 0 ≤ i, 0 < i, 0 < (2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0), 1 < i, 0 ≤ (2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0), 0 ≤ 2 ^ n→i(# bits0) * (2 ^ (i - n→i(# bits0)) - 1), 0 < 2 ^ i, 1 < 2 ^ i, (2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0) * 2 + 1 < 2 ^ i, 2 ^ i < 2 * ((2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0) * 2 + 1), 2 ^ n→i(# bits0) * (2 ^ (i - n→i(# bits0)) - 1) < 2 ^ i, (2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0) < 2 ^ i, ∀ j0, i1. i→n(j0) < i→n(2 * ((2 ^ (i - n→i(# bits0)) - 1) * 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 * 2 ^ i = 1 + 2 * ((2 ^ (i - n→i(# bits0)) - 1) * 2 ^ n→i(# bits0) * 2 + 1), bits0 = [], false ∈ bits0

User applied rule insert elim lemma

And obtained the following new goals

Proof step 60:
i = i0 + n→i(# bits0), 2 ^ n→i(# bits0) * 2 ^ i0 = 2 ^ i, (2 ^ i0 - 1) * 2 ^ n→i(# bits0) < 2 * ((2 ^ i0 - 1) * 2 ^ n→i(# bits0) * 2 + 1), (2 ^ i0 - 1) * 2 ^ n→i(# bits0) + 1 < 2 * ((2 ^ i0 - 1) * 2 ^ n→i(# bits0) * 2 + 1), n→i(# bits0) ≤ i, 0 ≤ 2 ^ i, 0 ≤ i, 0 < i, 0 < (2 ^ i0 - 1) * 2 ^ n→i(# bits0), 1 < i, 0 ≤ (2 ^ i0 - 1) * 2 ^ n→i(# bits0), 0 ≤ 2 ^ n→i(# bits0) * (2 ^ i0 - 1), 0 < 2 ^ i, 1 < 2 ^ i, (2 ^ i0 - 1) * 2 ^ n→i(# bits0) * 2 + 1 < 2 ^ i, 2 ^ i < 2 * ((2 ^ i0 - 1) * 2 ^ n→i(# bits0) * 2 + 1), 2 ^ n→i(# bits0) * (2 ^ i0 - 1) < 2 ^ i, (2 ^ i0 - 1) * 2 ^ n→i(# bits0) < 2 ^ i, ∀ j0, i1. i→n(j0) < i→n(2 * ((2 ^ i0 - 1) * 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 * 2 ^ i = 1 + 2 * ((2 ^ i0 - 1) * 2 ^ n→i(# bits0) * 2 + 1), bits0 = [], false ∈ bits0

Rule argument

seq: ⊦ i0 = i - j ↔ i = i0 + j subst: [i, j, i0] → [i, n→i(# bits0), 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, n→i(# bits0), i0