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Proof step 156 from proof uint2bits-restsgn (previous step)

~1 ≤ i0, false ∈ uint2bits(2 * 2 ^ i - i0 + 1).rest, i0 < 2 * 2 ^ i, 0 < i0, 0 ≤ i0, 0 ≤ i, i0 * 2 + 1 < 4 * 2 ^ i, 0 < i, 2 ^ i ≤ i0 * 2, i0 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ 2 ^ i, ∀ 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 ⊦ i0 = 1, sgn(uint2bits(2 * 2 ^ i - i0 + 1).rest) = uint2bits(2 * 2 ^ i - i0 + 1).rest, 4 * 2 ^ i = 2 * (i0 + 1), i = 1, ¬ i0 < 2 ^ (i - 1) ∧ ¬ 2 * 2 ^ (i - 1) ≤ i0 + 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 157:
⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j
Proof step 158:
0 ≤ i - 1, ~1 ≤ i0, false ∈ uint2bits(2 * 2 ^ i - i0 + 1).rest, i0 < 2 * 2 ^ i, 0 < i0, 0 ≤ i0, 0 ≤ i, i0 * 2 + 1 < 4 * 2 ^ i, 0 < i, 2 ^ i ≤ i0 * 2, i0 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ 2 ^ i, ∀ 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 ⊦ i0 = 1, sgn(uint2bits(2 * 2 ^ i - i0 + 1).rest) = uint2bits(2 * 2 ^ i - i0 + 1).rest, 4 * 2 ^ i = 2 * (i0 + 1), i = 1, ¬ i0 < 2 ^ (i - 1) ∧ ¬ 2 ^ ((i - 1) + 1) ≤ i0 + 1

Rule argument

seq: ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j subst: [i, j] → [i - 1, 2] rotate: T spec: int-pot inst: name: succ

Additional information

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