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Proof step 99 from proof uint2bits-restsgn (previous step)

0 < 2 * 2 ^ i - i0, 0 < 2 ^ i, 1 < 2 ^ i, i0 < 2 ^ i, 2 ^ i ≤ i0 * 2, 0 < i, 0 ≤ 2 ^ i, i0 * 2 + 1 < 4 * 2 ^ i, 0 ≤ i, 0 ≤ i0, 0 < i0, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 1) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ 4 * 2 ^ i = 2 * (i0 + 1), sgn(uint2bits(2 * 2 ^ i - i0 - 1).rest + ((2 * (2 * 2 ^ i - i0) - 1) % 2 = 1)') = uint2bits(2 * 2 ^ i - i0 - 1).rest + ((2 * (2 * 2 ^ i - i0) - 1) % 2 = 1)'

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 100:
⊦ 0 < i → (i * 2 - 1) % 2 = 1
Proof step 101:
0 < 2 * 2 ^ i - i0, 0 < 2 * 2 ^ i - i0, 0 < 2 ^ i, 1 < 2 ^ i, i0 < 2 ^ i, 2 ^ i ≤ i0 * 2, 0 < i, 0 ≤ 2 ^ i, i0 * 2 + 1 < 4 * 2 ^ i, 0 ≤ i, 0 ≤ i0, 0 < i0, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 1) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ 4 * 2 ^ i = 2 * (i0 + 1), sgn(uint2bits(2 * 2 ^ i - i0 - 1).rest + (1 = 1)') = uint2bits(2 * 2 ^ i - i0 - 1).rest + (1 = 1)'

Rule argument

seq: ⊦ 0 < i → (i * 2 - 1) % 2 = 1 subst: [i] → [2 * 2 ^ i - i0] rotate: F spec: int-div inst: name: mod-two-03

Additional information

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