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Proof step 21 from proof int2bits-len-neg (previous step)

0 ≤ 2 ^ n→i(# bits) + 2 ^ i, 2 ^ i + 1 < 2 ^ n→i(# bits) + n→i(# bits), 2 ^ i < 2 ^ n→i(# bits) + 2 ^ n→i(# bits), 2 ^ i + 1 < 2 ^ n→i(# bits) + 2 ^ n→i(# bits), 1 < 2 ^ n→i(# bits), 2 ^ i < 2 ^ n→i(# bits) + i, 2 ^ n→i(# bits) ≤ 2 ^ i * 2, 0 ≤ i, 0 < i + 2 ^ n→i(# bits), 1 < 2 ^ n→i(# bits) + 2 ^ i, i < n→i(# bits), 0 ≤ 2 ^ n→i(# bits) + 2 ^ n→i(# bits), 0 ≤ 2 ^ n→i(# bits) + n→i(# bits), 1 < 2 ^ n→i(# bits) + 2 ^ n→i(# bits), 1 < 2 ^ n→i(# bits) + i, 0 ≤ 2 ^ n→i(# bits), 0 < 2 ^ n→i(# bits), 0 < 2 ^ i, 0 < 2 ^ n→i(# bits) + i, 0 ≤ 2 ^ n→i(# bits) + i ⊦ # bits = 1 + i→n(i), bits = [], true ∈ bits

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 22:
⊦ i * j = j * i
Proof step 23:
0 ≤ 2 ^ n→i(# bits) + 2 ^ i, 2 ^ i + 1 < 2 ^ n→i(# bits) + n→i(# bits), 2 ^ i < 2 ^ n→i(# bits) + 2 ^ n→i(# bits), 2 ^ i + 1 < 2 ^ n→i(# bits) + 2 ^ n→i(# bits), 1 < 2 ^ n→i(# bits), 2 ^ i < 2 ^ n→i(# bits) + i, 2 ^ n→i(# bits) ≤ 2 * 2 ^ i, 0 ≤ i, 0 < i + 2 ^ n→i(# bits), 1 < 2 ^ n→i(# bits) + 2 ^ i, i < n→i(# bits), 0 ≤ 2 ^ n→i(# bits) + 2 ^ n→i(# bits), 0 ≤ 2 ^ n→i(# bits) + n→i(# bits), 1 < 2 ^ n→i(# bits) + 2 ^ n→i(# bits), 1 < 2 ^ n→i(# bits) + i, 0 ≤ 2 ^ n→i(# bits), 0 < 2 ^ n→i(# bits), 0 < 2 ^ i, 0 < 2 ^ n→i(# bits) + i, 0 ≤ 2 ^ n→i(# bits) + i ⊦ # bits = 1 + i→n(i), bits = [], true ∈ bits

Rule argument

seq: ⊦ i * j = j * i subst: [j, i] → [2, 2 ^ i] rotate: F spec: int-mult inst: name: commutativity

Additional information

Used terms: 2, 2 ^ i