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Proof step 132 from proof uint2bits-add-high (previous step)

2 ^ (i - 1) < i1, 0 ≤ i1 * 2 + 2 ^ i, 0 < i1 * 2 + 2 ^ i, 2 ^ (i - 1) ≤ 2 ^ i, 0 ≤ i1 * 2 + i, 0 < i1 * 2 + i, 0 ≤ i1 + 2 ^ i, 0 ≤ i1 + i, 2 ^ (i - 1) + 2 < i1 * 2 + 2 ^ i, 2 < i1 * 2 + 2 ^ i, 0 < 2 ^ i, 1 < 2 ^ i, 2 ^ (i - 1) + 1 < 3 * i1, 2 ^ (i - 1) ≤ i1 * 2, 2 ^ (i - 1) ≤ i1 * 2 + 2 ^ i, 1 < i1 * 2 + i, 2 ^ (i - 1) + 1 < i1 * 2 + i, 0 < i, 0 < i + i1 * 2, 0 < 2 ^ i + i1 * 2, 0 ≤ 2 ^ i, i1 * 2 ≤ 2 ^ i, i1 * 2 < 2 ^ i, 2 ^ (i - 1) < i1 * 2, 2 ^ (i - 1) < 3 * i1, i1 < 2 ^ i, 1 < i1 + 2 ^ i, i1 + 1 < 2 ^ i, 1 < i1 + i, 0 < i1, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ 2 ^ i = 1 + i1 * 2, uint2bits(2 ^ (i - 1) - (i1 * 2 + 1) / 2) + false ' = sgn(int2bits(~(i1 * 2)).rest), i = 1, i1 = 0

User applied rule insert elim lemma

And obtained the following new goals

Proof step 133:
i = i2 + 1, 2 ^ i2 < i1, 0 ≤ i1 * 2 + 2 ^ i, 0 < i1 * 2 + 2 ^ i, 2 ^ i2 ≤ 2 ^ i, 0 ≤ i1 * 2 + i, 0 < i1 * 2 + i, 0 ≤ i1 + 2 ^ i, 0 ≤ i1 + i, 2 ^ i2 + 2 < i1 * 2 + 2 ^ i, 2 < i1 * 2 + 2 ^ i, 0 < 2 ^ i, 1 < 2 ^ i, 2 ^ i2 + 1 < 3 * i1, 2 ^ i2 ≤ i1 * 2, 2 ^ i2 ≤ i1 * 2 + 2 ^ i, 1 < i1 * 2 + i, 2 ^ i2 + 1 < i1 * 2 + i, 0 < i, 0 < i + i1 * 2, 0 < 2 ^ i + i1 * 2, 0 ≤ 2 ^ i, i1 * 2 ≤ 2 ^ i, i1 * 2 < 2 ^ i, 2 ^ i2 < i1 * 2, 2 ^ i2 < 3 * i1, i1 < 2 ^ i, 1 < i1 + 2 ^ i, i1 + 1 < 2 ^ i, 1 < i1 + i, 0 < i1, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ 2 ^ i = 1 + i1 * 2, uint2bits(2 ^ i2 - (i1 * 2 + 1) / 2) + false ' = sgn(int2bits(~(i1 * 2)).rest), i = 1, i1 = 0

Rule argument

seq: ⊦ i0 = i - j ↔ i = i0 + j subst: [i, j, i0] → [i, 1, i2] rotate: T spec: int-abs inst: name: elim-sub

Additional information

Used simplifier rules: ⊦ i0 = i - j ↔ i = i0 + j; (elim) Used terms: i, 1, i2