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Proof step 10 from proof bits2int-le8 (previous step)

# sgn(bits) = i→n(i + 2), 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 ⊦ bits = [], i = ~1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
0 ≤ i, 0 ≤ j ⊦ i→n(i + j) = i→n(i) + i→n(j)
Proof step 12:
0 ≤ i ∧ 0 ≤ 2, # sgn(bits) = i→n(i) + i→n(2), 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 ⊦ bits = [], i = ~1

Rule argument

seq: 0 ≤ i, 0 ≤ j ⊦ i→n(i + j) = i→n(i) + i→n(j) subst: [j, i] → [2, i] rotate: F spec: intnat inst: name: add-01

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false); Used terms: 2, i