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Proof step 97 from proof uint2bits-restsgn (previous step)

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 * 2 ^ i - i0) - 1) / 2).rest + ((2 * (2 * 2 ^ i - i0) - 1) % 2 = 1)') = uint2bits((2 * (2 * 2 ^ i - i0) - 1) / 2).rest + ((2 * (2 * 2 ^ i - i0) - 1) % 2 = 1)'

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 98:
⊦ 0 < i → (2 * i - 1) / 2 = i - 1
Proof step 99:
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)'

Rule argument

seq: ⊦ 0 < i → (2 * i - 1) / 2 = i - 1 subst: [i] → [2 * 2 ^ i - i0] rotate: F spec: int-div inst: name: two-mult-01

Additional information

Used simplifier rules: ⊦ i < j - i0 ↔ i + i0 < j;% ⊦ i + 0 = i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ 0 < i → (i * j < i ↔ j < 1);% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i; Used terms: 2 * 2 ^ i - i0