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Proof step 56 from proof cast-mod-neg-high (previous step)

j + 2 ^ i ≤ 0, j + 2 ^ i < i, 2 ^ (i - 1) + j < 0, j < 2 ^ (i - 1), 0 < i, j < 0, 0 ≤ 2 ^ (i - 1) + j % 2 ^ i, j < j % 2 ^ i, i→n(i) ≤ (# fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) - 1) + # int2bits(j / 2 ^ i - 1) ⊦ j % 2 ^ i = 0, i = 1, j = ~1, bits2int(lastn(i→n(i), int2bits(j / 2 ^ i - 1) + fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest)) = j % 2 ^ i, fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) = [], i→n(i) = # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) - 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 57:
⊦ # bits ≤ n → # fillsgn(n, bits) = n
Proof step 58:
¬ # int2bits(j % 2 ^ i) ≤ i→n(i) + 1, j + 2 ^ i ≤ 0, j + 2 ^ i < i, 2 ^ (i - 1) + j < 0, j < 2 ^ (i - 1), 0 < i, j < 0, 0 ≤ 2 ^ (i - 1) + j % 2 ^ i, j < j % 2 ^ i, i→n(i) ≤ (# fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) - 1) + # int2bits(j / 2 ^ i - 1) ⊦ j % 2 ^ i = 0, i = 1, j = ~1, bits2int(lastn(i→n(i), int2bits(j / 2 ^ i - 1) + fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest)) = j % 2 ^ i, fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) = [], i→n(i) = # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) - 1

Rule argument

seq: ⊦ # bits ≤ n → # fillsgn(n, bits) = n subst: [bits, n] → [int2bits(j % 2 ^ i), i→n(i) + 1] rotate: F spec: bitopsaux inst: name: fill-length

Additional information

Used simplifier rules: ⊦ m + n = n + m;% ⊦ n ≤ m → m + n0 - n = (m - n) + n0;% ⊦ 0 + m = m;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ i→n(i) = i→n(j) ↔ abs(i) = abs(j); Used terms: int2bits(j % 2 ^ i), i→n(i) + 1