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Proof step 7 from proof bits2int-le8 (previous step)

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 lemma

And obtained the following new goals

Proof step 8:
0 ≤ i, ~(2 ^ (i + 1)) ≤ j, j < ~(2 ^ i) ⊦ # int2bits(j) = i→n(i + 2)
Proof step 9:
# int2bits(bits2int(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

Rule argument

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

Additional information

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