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Proof step 80 from proof int2bits-add-low (previous step)

0 ≤ i, 0 < i, j < 0, j0 < 0, 0 < 2 ^ (i - 1) + j0, # int2bits(~(j * 2 ^ i + j0)) ≤ i→n(i), 0 ≤ 2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0, j * 2 ^ i + j0 < 0 ⊦ 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 81:
⊦ 0 < i → (# int2bits(i) ≤ m ↔ i < 2 ^ (n→i(m) - 1))
Proof step 82:
0 < ~(j * 2 ^ i + j0), 0 ≤ i, 0 < i, j < 0, j0 < 0, 0 < 2 ^ (i - 1) + j0, ~(j * 2 ^ i + j0) < 2 ^ (n→i(i→n(i)) - 1), 0 ≤ 2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0, j * 2 ^ i + j0 < 0 ⊦ sgn(uint2bits(2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0))

Rule argument

name: int2bits-lelen rotate: F subst: [m, i] → [i→n(i), ~(j * 2 ^ i + j0)]

Additional information

Used simplifier rules: ⊦ i + j = j + i;% ⊦ 0 < ~ i ↔ i < 0;% ⊦ i * j = j * i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false); Used terms: i→n(i), ~(j * 2 ^ i + j0)