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Proof step 54 from proof uint2bits-add-low (previous step)

0 ≤ i2, j0 < 2 ^ (i2 + 2), 0 ≤ (i1 * 2 + j0) + 2 ^ (i2 + 2), 0 ≤ 2 ^ i2 * 2 + 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 ^ i2 * 2 + i1 * 2 + j0, i1 * 2 + j0 ≤ 1 + 2 ^ i2 * 2, ~2 < i2, 2 < 2 * (i1 + 1) + j0 + i2, i1 * 2 + j0 < 2 ^ (i2 + 2), 0 ≤ 2 ^ i2 * 2, 1 < (i1 * 2 + j0) + j0, 0 < i1 * 2 + j0, j0 + 1 < 2 ^ (i2 + 2), 0 < 2 ^ (i2 + 2), 0 < 2 ^ i2 * 2, 2 < 2 ^ (i2 + 2), 1 < 2 ^ i2 * 2 + i1 * 2, 1 < 2 * (i1 + 1) + i2, 0 < 2 ^ i2 * 2 + 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 ^ i2 * 2 - i1) + (~1 = (1 - i1 * 2 + j0) % 2)' = false ' + fillsgn(i→n(i2 + 2), int2bits(1 - i1 * 2 + j0)), 1 = i1 * 2 + j0, ~1 = i2

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 55:
0 < 2 * (i1 + 2 ^ i2) + j0, 1 < 2 * (i1 + 1) + i2, 2 < 2 ^ (i2 + 2), 0 < 2 ^ i2, 0 < 2 ^ (i2 + 2), j0 + 1 < 2 ^ (i2 + 2), 0 < i1 * 2 + j0, 1 < (i1 * 2 + j0) + j0, 0 ≤ 2 ^ i2, i1 * 2 + j0 < 2 ^ (i2 + 2), 2 < 2 * (i1 + 1) + j0 + i2, i1 * 2 + j0 ≤ 1 + 2 ^ i2 * 2, 2 < 2 * (i1 + 2 ^ i2) + j0, i1 * 2 + j0 ≤ 2 ^ (i2 + 2), 2 < (i1 * 2 + j0) + 2 ^ (i2 + 2), i1 * 2 + j0 ≤ 2 ^ (i2 + 2) + 1, 2 ^ i2 < i1, 0 ≤ j0, j0 < 2, 1 < i1 * 2 + j0, 0 < (i1 * 2 + j0) + j0, 0 ≤ (i1 * 2 + j0) + j0, 0 ≤ 2 * (i1 + 1) + i2, 0 ≤ 2 ^ (i2 + 2), 0 < 2 * (i1 + 1) + j0 + i2, 0 ≤ 2 * (i1 + 1) + j0 + i2, 0 ≤ (i1 * 2 + j0) + 2 ^ (i2 + 2), j0 < 2 ^ (i2 + 2), 0 ≤ i2, 0 < 2 * (i1 + 1) + i2, 1 < i1, 0 ≤ i1 + 2 ^ i2, 0 < i1, 0 ≤ i1, ∀ 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)) ⊦ ~1 = i2, 1 = i1 * 2 + j0, uint2bits(2 ^ i2 * 2 - i1) + (~1 = (1 - i1 * 2 + j0) % 2)' = false ' + fillsgn(i→n(i2 + 2), int2bits(1 - i1 * 2 + j0))

Rule argument

Additional information

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