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Proof step 84 from proof uint2bits-add-high (previous step)

uint2bits(2 ^ (i - 1) - 2 ^ i0) = false ', 0 ≤ ~(2 ^ i0) * 2 + 2 ^ i, 0 ≤ 2 ^ i, (2 ^ (i - 1) + ~(2 ^ i0) * 2) + 1 < 2 ^ i, ~(2 ^ i0) * 2 < 2 ^ i, ~(2 ^ i0) * 2 + 1 < 2 ^ i, ~(2 ^ i0) * 2 < i, 2 ^ (i - 1) + ~(2 ^ i0) * 2 < 0, 0 ≤ i, (2 ^ (i - 1) + ~(2 ^ i0) * 2) + 1 < i, 2 ^ (i - 1) + ~(2 ^ i0) * 2 < i, ~(2 ^ i0) * 2 + 1 < i, 2 ^ (i - 1) + ~(2 ^ i0) * 2 < 2 ^ i, 0 < ~(2 ^ i0) * 2 + 2 ^ i, i ≤ i0, 1 < 2 ^ i, 0 < 2 ^ i, 1 < i, 0 < i, 0 < i0, ~(2 ^ i0) * 2 < i0, 2 ^ (i - 1) + ~(2 ^ i0) * 2 < i0, ~(2 ^ i0) * 2 + 1 < i0, (2 ^ (i - 1) + ~(2 ^ i0) * 2) + 1 < i0, abs(~(2 ^ i0) * 2 - 1) / 2 < 2 ^ i, 2 ^ (i - 2) < abs(~(2 ^ i0) * 2 - 1) / 2, abs(~(2 ^ i0) * 2 - 1) / 2 ≤ 2 ^ (i - 1), ~(2 ^ i0) < 0, 0 ≤ 2 ^ i0 * 2 + 2 ^ i, ∀ 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) ⊦ i = 1, i0 = 0, 2 ^ i = 1 + 2 ^ i0 * 2

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 85:
⊦ j < 0 → abs(j) = 0 - j
Proof step 86:
¬ ~(2 ^ i0) * 2 - 1 < 0, uint2bits(2 ^ (i - 1) - 2 ^ i0) = false ', 0 ≤ ~(2 ^ i0) * 2 + 2 ^ i, 0 ≤ 2 ^ i, (2 ^ (i - 1) + ~(2 ^ i0) * 2) + 1 < 2 ^ i, ~(2 ^ i0) * 2 < 2 ^ i, ~(2 ^ i0) * 2 + 1 < 2 ^ i, ~(2 ^ i0) * 2 < i, 2 ^ (i - 1) + ~(2 ^ i0) * 2 < 0, 0 ≤ i, (2 ^ (i - 1) + ~(2 ^ i0) * 2) + 1 < i, 2 ^ (i - 1) + ~(2 ^ i0) * 2 < i, ~(2 ^ i0) * 2 + 1 < i, 2 ^ (i - 1) + ~(2 ^ i0) * 2 < 2 ^ i, 0 < ~(2 ^ i0) * 2 + 2 ^ i, i ≤ i0, 1 < 2 ^ i, 0 < 2 ^ i, 1 < i, 0 < i, 0 < i0, ~(2 ^ i0) * 2 < i0, 2 ^ (i - 1) + ~(2 ^ i0) * 2 < i0, ~(2 ^ i0) * 2 + 1 < i0, (2 ^ (i - 1) + ~(2 ^ i0) * 2) + 1 < i0, abs(~(2 ^ i0) * 2 - 1) / 2 < 2 ^ i, 2 ^ (i - 2) < abs(~(2 ^ i0) * 2 - 1) / 2, abs(~(2 ^ i0) * 2 - 1) / 2 ≤ 2 ^ (i - 1), ~(2 ^ i0) < 0, 0 ≤ 2 ^ i0 * 2 + 2 ^ i, ∀ 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) ⊦ i = 1, i0 = 0, 2 ^ i = 1 + 2 ^ i0 * 2

Rule argument

seq: ⊦ j < 0 → abs(j) = 0 - j subst: [j] → [~(2 ^ i0) * 2 - 1] rotate: F spec: int-abs inst: name: Abs-neg

Additional information

Used simplifier rules: uint2bits-zero-01: 0 ≤ i ⊦ uint2bits(i) = false ' ↔ i = 0;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 0 ≤ j ∧ 0 ≤ j0 ∧ 0 ≤ i0 → 0 ≤ i0 * j + j0;% ⊦ ~ i < 0 ↔ 0 < i;% ⊦ 0 ≤ i → (2 * i + 1) / 2 = i;% ⊦ ~ ~ i = i;% ⊦ 0 - i = ~ i;% ⊦ ~ i - j = ~(i + j);% ⊦ ~ i * j = ~(i * j);% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ i + (j - i0) = i + j - i0;% ⊦ i + ~ j = i - j;% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ i + 0 = i;% ⊦ i < j - i0 ↔ i + i0 < j;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j);% ⊦ ~1 < i ↔ 0 ≤ i;% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0;% ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ i ≠ j → (i ≤ j ↔ i < j);%0 ≤ j ⊦ 2 ^ j < 2 ^ j0 ↔ j < j0; Used terms: ~(2 ^ i0) * 2 - 1