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Proof step 49 from proof uint2bits-add-low (previous step)

i1 * 2 + j0 < 2 ^ i, 1 - i1 * 2 + j0 < j0, 1 - i1 * 2 + j0 ≤ 2 ^ (i - 1), 1 - i1 * 2 + j0 ≤ 2 ^ i, 0 ≤ 2 ^ (i - 1), 1 < (1 - i1 * 2 + j0) + 2 ^ i, 1 - i1 * 2 + j0 < 2 ^ (i - 1), (1 - i1 * 2 + j0) + 1 < i, 0 < 2 ^ (i - 1), 0 < i, 0 ≤ (1 - i1 * 2 + j0) + 2 ^ (i - 1), 1 - i1 * 2 + j0 < i, (1 - i1 * 2 + j0) + 1 < 2 ^ (i - 1), 0 < (1 - i1 * 2 + j0) + 2 ^ i, 1 - i1 * 2 + j0 < 2 ^ i, (1 - i1 * 2 + j0) + 1 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ 2 ^ i, 0 ≤ (1 - i1 * 2 + j0) + 2 ^ i, 2 ^ (i - 2) < i1, 0 ≤ j0, j0 < 2, 1 - i1 * 2 + j0 ≤ 1, 1 - i1 * 2 + j0 < 0, ∀ 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)) ⊦ uint2bits(2 ^ (i - 1) + (1 - i1 * 2 + j0 - 1) / 2) + (~1 = (1 - i1 * 2 + j0) % 2)' = false ' + fillsgn(i→n(i), int2bits(1 - i1 * 2 + j0)), i = 1, 1 - i1 * 2 + j0 = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 50:
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 ^ (i - 2) < 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))

Rule argument

Additional information

Used simplifier rules: ⊦ i < j + i0 ∧ j < j0 → i < j0 + i0;% ⊦ 0 ≤ j ∧ 0 ≤ j0 ∧ 0 ≤ i0 → 0 ≤ i0 * j + j0;% ⊦ 0 < i ∧ 0 < j → 0 < i + j;% ⊦ i < i + j ↔ 0 < j;% ⊦ i +1 = i + 1;% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward)% ⊦ 0 ≤ i0 ∧ i0 < j ∧ 0 ≤ i → (i * j + i0) / j = i;% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 0 ≤ j1 ∧ j1 < j → (0 ≤ i0 * j + j1 ↔ 0 ≤ i0);% ⊦ ~1 < i ↔ 0 ≤ i;% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0;% ⊦ i < j + 1 ↔ i ≤ j;%i0 ≠ 0, i0 < i ⊦ (* i) < j → (* i0) < j;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ i + j < i + i0 ↔ j < i0;% ⊦ i < j - i0 ↔ i + i0 < j;% ⊦ i + (j - i0) = i + j - i0;% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ i - j ≤ i0 ↔ i ≤ j + i0;% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ i + 0 = i;% ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ i + ~ j = i - j;%j ≠ 0 ⊦ ~ i / j = ~(i / j);% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i - j + i = ~ j;% ⊦ i - j - i0 = i - j + i0;