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Proof step 50 from proof uint2bits-restsgn (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 51:
⊦ (i * 2) % 2 = 0
Proof step 52:
0 < 2, i0 + 2 ^ n→i(# bits0) * 2 = 2 * 2 ^ i + 1, i0 ≤ 2 * 2 ^ i, i0 < 2 ^ i * 2, 2 ^ i < 2 * i0, i0 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ i0, 0 < i0, 1 < i, 1 < i0, 0 < i, 0 ≤ i, 0 ≤ 2 ^ i, ∀ j0, i. i→n(j0) < i→n(2 * i0 + 0) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ sgn(true ' + (0 = 1)') = (true ' + bits0) + (0 = 1)', (2 * 2) * 2 ^ i = 1 + 2 * i0, false ∈ bits0

Rule argument

seq: ⊦ (i * 2) % 2 = 0 subst: [i] → [2 * 2 ^ i - i0] rotate: F spec: int-div inst: name: mod-two-02

Additional information

Used simplifier rules: ⊦ i * j = j * i; Used terms: 2 * 2 ^ i - i0