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Proof step 37 from proof uint2bits-add-high (previous step)

0 = j + 2 ^ i, (2 ^ (i - 1) + j) + 1 < 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, j < i, 2 ^ (i - 1) < 2 ^ i, (2 ^ (i - 1) + j) + 1 < i, j + 1 < i, 2 ^ (i - 1) + j < 2 ^ i, 0 < 2 ^ i, 2 ^ (i - 1) + j < 0, 0 < i, j < 0, ∀ 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) ⊦ false ' = sgn(int2bits(j).rest)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 38:
⊦ j = i0 + j0 ↔ j - j0 = i0
Proof step 39:
0 - 2 ^ i = j, (2 ^ (i - 1) + j) + 1 < 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, j < i, 2 ^ (i - 1) < 2 ^ i, (2 ^ (i - 1) + j) + 1 < i, j + 1 < i, 2 ^ (i - 1) + j < 2 ^ i, 0 < 2 ^ i, 2 ^ (i - 1) + j < 0, 0 < i, j < 0, ∀ 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) ⊦ false ' = sgn(int2bits(j).rest)

Rule argument

seq: ⊦ (* j) = i0 + (* j0) ↔ j - j0 = i0 subst: [j0, i0, j] → [2 ^ i, j, 0] rotate: F spec: int-abs inst: name: numadd-01

Additional information

Used terms: 2 ^ i, j, 0