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Proof step 30 from proof cast-mod-pos-low (previous step)

j % 2 ^ i < i, j % 2 ^ i ≤ 0, 2 ^ i ≤ j, 0 ≤ 2 ^ (i - 1) + j, j % 2 ^ i < j, 2 ^ (i - 1) ≤ j, 0 ≤ j, 0 < i, j % 2 ^ i < 2 ^ (i - 1), i→n(i) ≤ (# fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) - 1) + # int2bits(j / 2 ^ i), i→n(i) + 1 < # int2bits(j % 2 ^ i) ⊦ bits2int(lastn(i→n(i), int2bits(j / 2 ^ i) + 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 31:
⊦ n ≤ # bits → fillsgn(n, bits) = bits
Proof step 32:
i→n(i) + 1 ≤ # int2bits(j % 2 ^ i), j % 2 ^ i < i, j % 2 ^ i ≤ 0, 2 ^ i ≤ j, 0 ≤ 2 ^ (i - 1) + j, j % 2 ^ i < j, 2 ^ (i - 1) ≤ j, 0 ≤ j, 0 < i, j % 2 ^ i < 2 ^ (i - 1), i→n(i) ≤ (# int2bits(j % 2 ^ i) - 1) + # int2bits(j / 2 ^ i), i→n(i) + 1 < # int2bits(j % 2 ^ i) ⊦ bits2int(lastn(i→n(i), int2bits(j / 2 ^ i) + int2bits(j % 2 ^ i).rest)) = j % 2 ^ i, int2bits(j % 2 ^ i) = [], i→n(i) = # int2bits(j % 2 ^ i) - 1

Rule argument

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

Additional information

Used simplifier rules: ⊦ m + 1 ≤ n ↔ m < n;% ⊦ ¬ m < n ↔ n ≤ m;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward)% ⊦ m < n ∧ n ≤ n0 → m < n0; (forward)% ⊦ ¬ m + n < m; Used terms: int2bits(j % 2 ^ i), i→n(i) + 1