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Proof step 139 from proof uint2bits-add-high (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 140:
⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j
Proof step 141:
0 ≤ i2, 0 < 2 ^ (i2 - 1), 1 < 2 ^ i2 * 2, 0 ≤ i1 + 2 ^ i2 * 2, 2 ^ i2 ≤ 2 ^ i2 * 2, 0 ≤ 2 ^ i2 * 2, 0 < 2 ^ i2 * 2, 0 ≤ i1 * 2 + 2 ^ i2 * 2, 0 < i1 * 2 + 2 ^ i2 * 2, 0 < i1, i1 + 1 < 2 ^ i2 * 2, 1 < i1 + 2 ^ i2 * 2, i1 < 2 ^ i2 * 2, 2 ^ i2 < 3 * i1, 2 ^ i2 < i1 * 2, i1 * 2 < 2 ^ i2 * 2, i1 * 2 ≤ 2 ^ i2 * 2, 0 < 2 ^ i2 * 2 + i1 * 2, 0 < i2, 2 ^ i2 < i1 * 2 + i2, 0 < i1 * 2 + i2, 2 ^ i2 + 1 < 2 ^ i2 * 2, 2 ^ i2 ≤ i1 * 2 + 2 ^ i2 * 2, 2 ^ i2 ≤ i1 * 2, 2 ^ i2 + 1 < 3 * i1, 2 < i1 * 2 + 2 ^ i2 * 2, 2 ^ i2 + 2 < i1 * 2 + 2 ^ i2 * 2, ~1 ≤ i1 + i2, 0 ≤ i1 * 2 + i2, i1 ≤ 2 ^ (i2 - 1), ∀ i0, j. i→n(i0) < i→n(i2 + 1) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ 0 = i2, uint2bits(2 ^ i2 - (i1 * 2 + 1) / 2) + false ' = sgn(int2bits(~(i1 * 2)).rest), 2 ^ i2 * 2 = 1 + i1 * 2, i1 = 0

Rule argument

seq: ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j subst: [i, j] → [i2, 2] rotate: F spec: int-pot inst: name: succ

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i; Used terms: i2, 2