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Proof step 44 from proof uint2bits-restsgn (previous step)

i0 + 2 ^ n→i(# bits0) * 2 = 2 * 2 ^ i + 1, i0 ≤ 2 * 2 ^ i, i0 * 2 < (2 * 2) * 2 ^ i, 2 ^ i < i0 * 2, 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(i0 * 2 + 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) % 2 = 1)') = (true ' + bits0) + (((2 * 2) * 2 ^ i - i0 * 2) % 2 = 1)', (2 * 2) * 2 ^ i = 1 + i0 * 2, false ∈ bits0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 45:
⊦ 0 < j → (i * j < i0 * j ↔ i < i0)
Proof step 46:
0 < 2, i0 + 2 ^ n→i(# bits0) * 2 = 2 * 2 ^ i + 1, i0 ≤ 2 * 2 ^ i, i0 < 2 ^ i * 2, 2 ^ i < i0 * 2, 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(i0 * 2 + 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) % 2 = 1)') = (true ' + bits0) + (((2 * 2) * 2 ^ i - i0 * 2) % 2 = 1)', (2 * 2) * 2 ^ i = 1 + i0 * 2, false ∈ bits0

Rule argument

seq: ⊦ 0 < j → (i * j < i0 * j ↔ i < i0) subst: [i0, j, i] → [2 ^ i * 2, 2, i0] rotate: F spec: int-mult inst: name: ls-pos-01

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0; Used terms: 2 ^ i * 2, 2, i0