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Proof step 51 from proof uint2bits-add-low (previous step)

i = i2 + 2, j0 < 2 ^ i, 1 < i1 * 2 + i, 1 < i1 * 2 + 2 ^ (i - 1), 2 < 2 ^ i, 0 < i1 * 2 + j0, 1 < (i1 * 2 + j0) + j0, 0 ≤ 2 ^ (i - 1), i1 * 2 + j0 < 2 ^ i, 2 < (i1 * 2 + j0) + i, 0 < 2 ^ (i - 1), 0 < i, i1 * 2 + j0 ≤ 2 ^ (i - 1) + 1, 2 < (i1 * 2 + j0) + 2 ^ (i - 1), i1 * 2 + j0 ≤ 2 ^ i, 2 < (i1 * 2 + j0) + 2 ^ i, 0 ≤ 2 ^ i, i1 * 2 + j0 ≤ 2 ^ i + 1, 2 ^ i2 < i1, 0 ≤ j0, j0 < 2, 0 ≤ i1, 1 < i1 * 2 + j0, 0 < (i1 * 2 + j0) + j0, 0 ≤ (i1 * 2 + j0) + j0, 0 ≤ (i1 * 2 + j0) + i, 0 ≤ (i1 * 2 + j0) + 2 ^ (i - 1), 0 ≤ (i1 * 2 + j0) + 2 ^ i, 0 ≤ i1 * 2 + i, 0 < i1 * 2 + i, 0 < i1 * 2 + 2 ^ (i - 1), 0 < 2 ^ i, j0 + 1 < 2 ^ i, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ abs(j) ≤ 2 ^ (i0 - 1) → uint2bits(j + 2 ^ i0) = false ' + fillsgn(i→n(i0), int2bits(j)) ⊦ i = 1, 1 = i1 * 2 + j0, uint2bits(2 ^ (i - 1) - i1) + (~1 = (1 - i1 * 2 + j0) % 2)' = false ' + fillsgn(i→n(i), int2bits(1 - i1 * 2 + j0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 52:
j0 < 2 ^ (i2 + 2), 0 ≤ (i1 * 2 + j0) + 2 ^ (i2 + 2), 0 ≤ 2 ^ (1 + i2) + i1 * 2 + j0, 0 ≤ 2 * (i1 + 1) + j0 + i2, 0 < 2 * (i1 + 1) + j0 + i2, 0 ≤ 2 ^ (i2 + 2), 0 ≤ 2 * (i1 + 1) + i2, 0 ≤ (i1 * 2 + j0) + j0, 0 < (i1 * 2 + j0) + j0, 1 < i1 * 2 + j0, 0 ≤ i1, j0 < 2, 0 ≤ j0, 2 ^ i2 < i1, i1 * 2 + j0 ≤ 2 ^ (i2 + 2) + 1, 2 < (i1 * 2 + j0) + 2 ^ (i2 + 2), i1 * 2 + j0 ≤ 2 ^ (i2 + 2), 2 < 2 ^ (1 + i2) + i1 * 2 + j0, i1 * 2 + j0 ≤ 1 + 2 ^ (1 + i2), ~2 < i2, 2 < 2 * (i1 + 1) + j0 + i2, i1 * 2 + j0 < 2 ^ (i2 + 2), 0 ≤ 2 ^ (1 + i2), 1 < (i1 * 2 + j0) + j0, 0 < i1 * 2 + j0, j0 + 1 < 2 ^ (i2 + 2), 0 < 2 ^ (i2 + 2), 0 < 2 ^ (1 + i2), 2 < 2 ^ (i2 + 2), 1 < 2 ^ (1 + i2) + i1 * 2, 1 < 2 * (i1 + 1) + i2, 0 < 2 ^ (1 + i2) + i1 * 2, 0 < 2 * (i1 + 1) + i2, ∀ i0, j. i→n(i0) < i→n(i2 + 2) → true → j < 0 ∧ 0 < i0 ∧ abs(j) ≤ 2 ^ (i0 - 1) → uint2bits(j + 2 ^ i0) = false ' + fillsgn(i→n(i0), int2bits(j)) ⊦ uint2bits(2 ^ (1 + i2) - i1) + (~1 = (1 - i1 * 2 + j0) % 2)' = false ' + fillsgn(i→n(i2 + 2), int2bits(1 - i1 * 2 + j0)), 1 = i1 * 2 + j0, ~1 = i2

Rule argument

Additional information

Used simplifier rules: i0 ≠ 0, i0 ≤ i ⊦ (* i) < j → (* i0) < j;% ⊦ 0 ≤ j ∧ 0 ≤ j0 ∧ 0 ≤ i0 → 0 ≤ i0 * j + j0;% ⊦ 0 < i ∧ 0 < j → 0 < i + j;% ⊦ i < i + j ↔ 0 < j;% ⊦ i + j < i0 ∧ ¬ j < j0 → i + j0 < i0;% ⊦ i < j + i0 ∧ j < j0 → i < j0 + i0;% ⊦ i + j < i + i0 ↔ j < i0;% ⊦ i +1 = i + 1;%i0 ≠ 0, i0 < i ⊦ (* i) < j → (* i0) < j;% ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward)% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0;% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j + i = i * (j + 1);% ⊦ (* i) + i0 - (* j) = (i - j) + i0;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ (* j) = i0 + (* j0) ↔ j - j0 = i0;