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Proof step 34 from proof int2int-lastn-negpos (previous step)

0 < bits2int(lastn(m, int2bits(i))) ∧ 0 ≤ n→i(m), # 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), ¬ # int2bits(bits2int(lastn(m, int2bits(i)))) ≤ i→n(n→i(m)) + 1, 0 < bits2int(lastn(m, int2bits(i))), i + 1 < bits2int(lastn(m, int2bits(i))), bits2int(bits) + 1 < bits2int(lastn(m, int2bits(i))) ⊦ 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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 35:
bits2int(bits) * 2 ^ n→i(m) + bits2int(lastn(m, int2bits(i))) = i, int2bits(i) = bits + lastn(m, int2bits(i)), # bits = # int2bits(i) - m, 1 + m < # sgn(lastn(m, int2bits(i))), bits2int(bits) + 1 < bits2int(lastn(m, int2bits(i))), i + 1 < bits2int(lastn(m, int2bits(i))), 0 < bits2int(lastn(m, int2bits(i))), bits2int(bits) + 1 < 2 ^ n→i(m), bits2int(bits) < bits2int(lastn(m, int2bits(i))), i < 2 ^ n→i(m), i + 1 < 2 ^ n→i(m), i < bits2int(lastn(m, int2bits(i))), i < 0, 0 < 2 ^ n→i(m), 0 ≤ bits2int(lastn(m, int2bits(i))), m ≤ # int2bits(i), bits2int(bits) < 0, bits2int(bits) < 2 ^ n→i(m) ⊦ bits2int(lastn(m, int2bits(i))) % 2 ^ n→i(m) = 0, bits = [], m = 0, i % 2 ^ n→i(m) = 0, bits2int(lastn(m, int2bits(i))) = bits2int(lastn(m, int2bits(i))) % 2 ^ n→i(m)

Rule argument

Additional information

Used simplifier rules: bits2bits-def: bits ≠ [] ⊦ int2bits(bits2int(bits)) = sgn(bits);% ⊦ sgn(bits) = [] ↔ bits = [];% ⊦ # bits = 0 ↔ bits = [];% ⊦ i * j = j * i;% ⊦ m < n → n ≠ 0; (forward)% ⊦ ¬ m ≤ n ↔ n < m;% ⊦ i→n(n→i(n)) = n;% ⊦ lastn(n, bits) = [] ↔ n = 0;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ 0 ≤ n→i(n);