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Proof step 78 from proof int2bits-add-low (previous step)

0 < i, j < 0, j0 < 0, 0 < 2 ^ (i - 1) + j0, n→i(# int2bits(~(j * 2 ^ i + j0))) ≤ 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 79:
0 ≤ i ⊦ n→i(n) ≤ i ↔ n ≤ i→n(i)
Proof step 80:
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))

Rule argument

seq: 0 ≤ i ⊦ n→i(n) ≤ (* i) ↔ n ≤ i→n(i) subst: [i, n] → [i, # int2bits(~(j * 2 ^ i + j0))] rotate: F spec: intnat inst: name: num-n2i-ls-04

Additional information

Used simplifier rules: ⊦ i + j = j + i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i; Used terms: i, # int2bits(~(j * 2 ^ i + j0))