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Proof step 83 from proof int2bits-add-low (previous step)

0 ≤ 2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0, j * 2 ^ i + j0 < 0, 0 < i, j < 0, j0 < 0, 0 < 2 ^ (i - 1) + j0, 0 ≤ i, 0 < 2 ^ (i - 1) + j * 2 ^ i + j0 ⊦ sgn(uint2bits(2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 84:
⊦ 0 < i → j ^ i = j ^ i -1 * j
Proof step 85:
0 < i, 0 ≤ 2 ^ n→i(# int2bits(~(j * 2 ^ i -1 * 2 + j0))) + j * 2 ^ i -1 * 2 + j0, j * 2 ^ i -1 * 2 + j0 < 0, 0 < i, j < 0, j0 < 0, 0 < 2 ^ (i - 1) + j0, 0 ≤ i, 0 < 2 ^ (i - 1) + j * 2 ^ i -1 * 2 + j0 ⊦ sgn(uint2bits(2 ^ n→i(# int2bits(~(j * 2 ^ i -1 * 2 + j0))) + j * 2 ^ i -1 * 2 + j0).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0))

Rule argument

seq: ⊦ 0 < i → j ^ i = j ^ i -1 * j subst: [i, j] → [i, 2] rotate: F spec: int-pot inst: name: Pot-pos

Additional information

Used terms: i, 2