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Proof step 5 from proof bits2int-le8-01 (previous step)

bits2int(bits) < 2 ^ i * 2, 2 ^ i ≤ bits2int(bits), # bits ≤ 8, 127 < bits2int(bits), 64 < 2 ^ i, 63 < 2 ^ i, 64 < bits2int(bits), 63 < bits2int(bits), 0 < 2 ^ i, 0 ≤ 2 ^ i, 32 < 2 ^ i, 31 < 2 ^ i, 0 < bits2int(bits), 32 < bits2int(bits), 31 < bits2int(bits), 16 < 2 ^ i, 15 < 2 ^ i, 16 < bits2int(bits), 15 < bits2int(bits), 8 < 2 ^ i, 7 < 2 ^ i, 8 < bits2int(bits), 7 < bits2int(bits), 4 < 2 ^ i, 3 < 2 ^ i, 4 < bits2int(bits), 3 < bits2int(bits), 1 < i, 0 < i, 0 ≤ i, 1 < bits2int(bits), 1 < 2 ^ i ⊦ 0 = bits2int(bits), bits = []

User applied rule insert lemma

And obtained the following new goals

Proof step 6:
0 ≤ i, 2 ^ i ≤ j, j < 2 ^ (i + 1) ⊦ # int2bits(j) = i→n(i + 2)
Proof step 7:
# int2bits(bits2int(bits)) = i→n(i + 2), bits2int(bits) < 2 ^ i * 2, 2 ^ i ≤ bits2int(bits), # bits ≤ 8, 127 < bits2int(bits), 64 < 2 ^ i, 63 < 2 ^ i, 64 < bits2int(bits), 63 < bits2int(bits), 0 < 2 ^ i, 0 ≤ 2 ^ i, 32 < 2 ^ i, 31 < 2 ^ i, 0 < bits2int(bits), 32 < bits2int(bits), 31 < bits2int(bits), 16 < 2 ^ i, 15 < 2 ^ i, 16 < bits2int(bits), 15 < bits2int(bits), 8 < 2 ^ i, 7 < 2 ^ i, 8 < bits2int(bits), 7 < bits2int(bits), 4 < 2 ^ i, 3 < 2 ^ i, 4 < bits2int(bits), 3 < bits2int(bits), 1 < i, 0 < i, 0 ≤ i, 1 < bits2int(bits), 1 < 2 ^ i ⊦ 0 = bits2int(bits), bits = []

Rule argument

name: int2bits-len-pos subst: [i, j] → [i, bits2int(bits)]

Additional information

Used simplifier rules: ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i * j = j * i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false); Used terms: i, bits2int(bits)