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Proof step 43 from proof intshiftrsign32 (previous step)

n0 + i→n(band(j, 31)) = n1 + # int2bits(i), i→n(band(j, 31)) < # int2bits(i), n0 ≤ n1, band(j, 31) ≤ 32, band(j, 31) < 32, i→n(band(j, 31)) < n1 + # int2bits(i) ⊦ bits2int(sublist(0, # int2bits(i), mklist(¬ 0 ≤ i, i→n(band(j, 31))) + int2bits(i))) = bits2int((¬ 0 ≤ i)'), band(j, 31) = 0, n0 = 0

User applied rule cut formula

And obtained the following new goals

Proof step 44:
n0 + i→n(band(j, 31)) = n1 + # int2bits(i), i→n(band(j, 31)) < # int2bits(i), n0 ≤ n1, band(j, 31) ≤ 32, band(j, 31) < 32, i→n(band(j, 31)) < n1 + # int2bits(i) ⊦ n0 + # int2bits(i) ≤ n0 + i→n(band(j, 31)), bits2int(sublist(0, # int2bits(i), mklist(¬ 0 ≤ i, i→n(band(j, 31))) + int2bits(i))) = bits2int((¬ 0 ≤ i)'), band(j, 31) = 0, n0 = 0

Rule argument

fma: n0 + # int2bits(i) ≤ n0 + i→n(band(j, 31))

Additional information

Used simplifier rules: band-commutativity: ⊦ band(i, j) = band(j, i);%lep-02: ⊦ m + n ≤ m + n0 ↔ n ≤ n0; from specification nat%ls-le-simp: ⊦ m < n → (n ≤ m ↔ false); from specification nat%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2 Used formulas: n0 + # int2bits(i) ≤ n0 + i→n(band(j, 31))