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Proof step 16 from proof bits2int-le8 (previous step)

2 ^ 7 < 2 ^ (i + 1), # sgn(bits) = 2 + i→n(i), 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 17:
⊦ 1 < i ∧ 0 ≤ j → (i ^ j < i ^ j0 ↔ j < j0)
Proof step 18:
1 < 2 ∧ 0 ≤ 7, 7 < i + 1, # sgn(bits) = 2 + i→n(i), 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: ⊦ 1 < i ∧ 0 ≤ j → (i ^ j < i ^ j0 ↔ j < j0) subst: [j0, j, i] → [i + 1, 7, 2] rotate: F spec: int-pot inst: name: ls-pos

Additional information

Used simplifier rules: ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0);% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i; Used terms: i + 1, 7, 2