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Proof step 11 from proof int2bits-rec-2pot-neg-02 (previous step)

0 ≤ i, j < 0, j + 2 ^ i < i, j + 2 ^ i ≤ 0, 0 < i ⊦ j % 2 ^ i = 0, sgn(bitsinvert(uint2bits(~(j / 2 ^ i))) +1 + fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest) = int2bits(j / 2 ^ i - 1) + fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, true ∈ fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 12:
⊦ n ≤ # bits → fillsgn(n, bits) = bits
Proof step 13:
¬ i→n(i) + 1 ≤ # uint2bits(~(j % 2 ^ i)), 0 ≤ i, j < 0, j + 2 ^ i < i, j + 2 ^ i ≤ 0, 0 < i ⊦ j % 2 ^ i = 0, sgn(bitsinvert(uint2bits(~(j / 2 ^ i))) +1 + fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest) = int2bits(j / 2 ^ i - 1) + fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, true ∈ fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest
Proof step 28:
i→n(i) + 1 ≤ # uint2bits(~(j % 2 ^ i)), 0 ≤ i, j < 0, j + 2 ^ i < i, j + 2 ^ i ≤ 0, 0 < i ⊦ j % 2 ^ i = 0, sgn(bitsinvert(uint2bits(~(j / 2 ^ i))) +1 + uint2bits(~(j % 2 ^ i)).rest) = int2bits(j / 2 ^ i - 1) + fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, true ∈ uint2bits(~(j % 2 ^ i)).rest

Rule argument

seq: ⊦ n ≤ # bits → fillsgn(n, bits) = bits subst: [bits, n] → [uint2bits(~(j % 2 ^ i)), i→n(i) + 1] rotate: F spec: bitopsaux inst: name: fillsgn-longer

Additional information

Used simplifier rules: ⊦ m + n = n + m; Used terms: uint2bits(~(j % 2 ^ i)), i→n(i) + 1