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Proof step 20 from proof bits2int-le8 (previous step)

# sgn(bits) = 2 + i→n(i), bits2int(bits) < i, bits2int(bits) < 2 ^ i, 0 < 2 ^ i, bits2int(bits) + 1 < 2 ^ i, bits2int(bits) < 2 ^ i * 2, bits2int(bits) + 1 < 2 ^ i * 2, 0 ≤ bits2int(bits) + 2 ^ i * 2, 0 ≤ i, bits2int(bits) + 2 ^ i < 0, # bits ≤ 8, bits2int(bits) < ~128, bits2int(bits) + 2 ^ i < i, bits2int(bits) < ~1, bits2int(bits) < 0 ⊦ ~(2 ^ (i + 1)) < ~(2 ^ 7), bits = [], i = ~1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 21:
# sgn(bits) = 2 + i→n(i), bits2int(bits) < 64, 2 ^ i ≤ 64, bits2int(bits) < 0, bits2int(bits) < ~1, bits2int(bits) + 2 ^ i < i, bits2int(bits) < ~128, # bits ≤ 8, bits2int(bits) + 2 ^ i < 0, 0 ≤ i, 0 ≤ bits2int(bits) + 2 ^ i * 2, bits2int(bits) + 1 < 2 ^ i * 2, bits2int(bits) < 2 ^ i * 2, bits2int(bits) + 1 < 2 ^ i, 0 < 2 ^ i, bits2int(bits) < 2 ^ i, bits2int(bits) < i, bits2int(bits) < 63 ⊦ i = ~1, bits = []

Rule argument

Additional information

Used simplifier rules: ⊦ i + (* j) < (* j0) ↔ i < j0 - j;% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0);% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0;% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;