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Proof step 28 from proof int2int-lastn-negpos (previous step)

bits2int(bits) * 2 ^ n→i(m) + bits2int(lastn(m, int2bits(i))) < 0 ∧ 0 < 2 ^ n→i(m) ∧ 0 ≤ bits2int(lastn(m, int2bits(i))) ∧ bits2int(lastn(m, int2bits(i))) % 2 ^ n→i(m) ≠ 0, bits2int(bits) * 2 ^ n→i(m) + bits2int(lastn(m, int2bits(i))) = i, int2bits(i) = bits + lastn(m, int2bits(i)), # bits = # int2bits(i) - m, m ≤ # int2bits(i), i < 0, bits.first ⊦ bits = [], m = 0, i % 2 ^ n→i(m) = 0, bits2int(lastn(m, int2bits(i))) = (bits2int(lastn(m, int2bits(i))) % 2 ^ n→i(m) - 2 ^ n→i(m)) + 2 ^ n→i(m), lastn(m, int2bits(i)).first

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 29:
# bits = # int2bits(i) - m, int2bits(i) = bits + lastn(m, int2bits(i)), bits2int(bits) * 2 ^ n→i(m) + bits2int(lastn(m, int2bits(i))) = i, bits2int(bits) < 2 ^ n→i(m), bits2int(bits) < 0, m ≤ # int2bits(i), 0 ≤ bits2int(lastn(m, int2bits(i))), 0 < 2 ^ n→i(m), i < 0, i < bits2int(lastn(m, int2bits(i))), i + 1 < 2 ^ n→i(m), i < 2 ^ n→i(m), bits2int(bits) < bits2int(lastn(m, int2bits(i))), bits2int(bits) + 1 < 2 ^ n→i(m) ⊦ bits2int(lastn(m, int2bits(i))) = bits2int(lastn(m, int2bits(i))) % 2 ^ n→i(m), i % 2 ^ n→i(m) = 0, m = 0, bits = [], bits2int(lastn(m, int2bits(i))) % 2 ^ n→i(m) = 0

Rule argument

Additional information

Used simplifier rules: bits2int-first: bits ≠ [] ⊦ 0 ≤ bits2int(bits) → ¬ bits.first;%bits2int-first-01: bits ≠ [] ⊦ bits2int(bits) < 0 → bits.first;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 0 < i → (i * j < 0 ↔ j < 0);% ⊦ i + j < j ↔ i < 0;% ⊦ i < j ∧ i = i0 → i0 < j; (forward)% ⊦ i +1 = i + 1;% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ i * j = j * i;% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ lastn(n, bits) = [] ↔ n = 0;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + (j - i) = j;