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Proof step 25 from proof int2bits-len-neg (previous step)

0 ≤ i, 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 + 1), 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 26:
⊦ 0 ≤ i → (2 ^ i ≤ 2 ^ j ↔ i ≤ j)
Proof step 27:
0 ≤ n→i(# bits), 0 ≤ i, 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, n→i(# bits) ≤ i + 1, 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: ⊦ 0 ≤ i → (2 ^ i ≤ 2 ^ j ↔ i ≤ j) subst: [j, i] → [i + 1, n→i(# bits)] rotate: F spec: int-pot inst: name: two-le

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j;% ⊦ 0 < i → (0 < i ^ j ↔ i = 1 ∨ 0 ≤ j); Used terms: i + 1, n→i(# bits)