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Proof step 56 from proof uint2bits-restsgn (previous step)

0 ≤ n→i(# bits0), 2 * 2 ^ i = 2 ^ (n→i(# bits0) + 1) + (i0 - 1), i0 + 2 ^ (n→i(# bits0) + 1) = 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 ⊦ bits0 = [], 4 * 2 ^ i = 1 + 2 * i0, false ∈ bits0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

seq: ⊦ 0 ≤ j ∧ 0 ≤ i ∧ 0 ≤ i0 → ( 2 * 2 ^ i = 2 ^ i0 + j ↔ j = 2 ^ i0 * (2 ^ (i + 1 - i0) - 1) ∧ i0 ≤ i + 1) subst: [j, i0, i] → [i0 - 1, n→i(# bits0) + 1, i] rotate: F spec: int-pot inst: name: add-eq2pot-01

Additional information

Used simplifier rules: ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i + (j - i0) = i + j - i0;% ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ 0 ≤ n→i(n);% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i ≤ j → (j ≤ i ↔ i = j);% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i;% ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0;%i < 0 ⊦ (* i) ≤ n→i(n);% ⊦ i * j = j * i;% ⊦ i < j ∧ i = i0 → i0 < j; (forward)% ⊦ i < j ∧ j = i0 → i < i0; (forward) Used terms: i0 - 1, n→i(# bits0) + 1, i