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Proof step 11 from proof uint2bits-restsgn (previous step)

2 ^ i < 2 ^ (i + 2), j < 2 ^ (i + 2), 0 < 2 ^ i, 1 < 2 ^ i, 0 < j, 0 ≤ j, 1 < j, 0 ≤ 2 ^ i, 2 < j, 0 < i, 2 ^ i < j, j < 2 * 2 ^ i, 1 < 2 ^ (i + 2), 2 < 2 ^ (i + 2), 3 < 2 ^ (i + 2), 2 ^ i + 1 < 2 ^ (i + 2), 0 < 2 ^ (i + 2), 0 ≤ 2 ^ (i + 2), ∀ j0, i. i→n(j0) < i→n(j) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ 2 ^ (i + 2) = 1 + j, sgn((uint2bits((2 ^ (i + 2) - j) / 2) + ((2 ^ (i + 2) - j) % 2 = 1)').rest) = (uint2bits((2 ^ (i + 2) - j) / 2) + ((2 ^ (i + 2) - j) % 2 = 1)').rest

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

seq: ⊦ bits ≠ [] → (bits + bits0).rest = bits.rest + bits0 subst: [bits0, bits] → [((2 ^ (i + 2) - j) % 2 = 1)', uint2bits((2 ^ (i + 2) - j) / 2)] rotate: F spec: list inst: bitlista name: rest-01

Additional information

Used simplifier rules: uint2bits-empty: 0 ≤ i ⊦ uint2bits(i) ≠ [];% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j < j ↔ i < 0;% ⊦ i +1 = i + 1;% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i - j ≤ i0 ↔ i ≤ j + i0;% ⊦ i + ~1 = i - 1;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ 0 ≤ i / 2 ↔ ~1 ≤ i; Used terms: ((2 ^ (i + 2) - j) % 2 = 1)', uint2bits((2 ^ (i + 2) - j) / 2)