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Proof step 115 from proof uint2bits-restsgn (previous step)

2 * 2 ^ i = 2 ^ n→i(# bits) + i0, i0 < 2 * 2 ^ i, 0 < i0, 0 ≤ i0, 0 ≤ i, i0 * 2 + 1 < 4 * 2 ^ i, 0 ≤ 2 ^ i, 0 < i, 2 ^ i ≤ i0 * 2, i0 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 < 2 ^ n→i(# bits), ∀ 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 ⊦ sgn(bits + true ') = bits + true ', 4 * 2 ^ i = 2 * (i0 + 1), false ∈ bits

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 116:
⊦ 0 ≤ j ∧ 0 ≤ i ∧ 0 ≤ i0 → (2 * 2 ^ i = 2 ^ i0 + j ↔ j = 2 ^ i0 * (2 ^ (i + 1 - i0) - 1) ∧ i0 ≤ i + 1)
Proof step 117:
0 ≤ i0 ∧ 0 ≤ i ∧ 0 ≤ n→i(# bits), i0 = 2 ^ n→i(# bits) * (2 ^ (i + 1 - n→i(# bits)) - 1) ∧ n→i(# bits) ≤ i + 1, i0 < 2 * 2 ^ i, 0 < i0, 0 ≤ i0, 0 ≤ i, i0 * 2 + 1 < 4 * 2 ^ i, 0 ≤ 2 ^ i, 0 < i, 2 ^ i ≤ i0 * 2, i0 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 < 2 ^ n→i(# bits), ∀ 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 ⊦ sgn(bits + true ') = bits + true ', 4 * 2 ^ i = 2 * (i0 + 1), false ∈ bits

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, n→i(# bits), i] rotate: F spec: int-pot inst: name: add-eq2pot-01

Additional information

Used simplifier rules: ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i;% ⊦ 0 ≤ n→i(n); Used terms: i0, n→i(# bits), i