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Proof step 29 from proof uint2bits-add-low (previous step)

¬(j < 0 ∧ j ≠ ~1), uint2bits(2 ^ (i - 1) + (j - 1) / 2) = false ' + fillsgn(i→n(i) - 1, int2bits((j - 1) / 2)), 0 < j + 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ 2 ^ i, 0 ≤ j + 2 ^ i, j ≤ 2 ^ (i - 1), j ≤ 2 ^ i, 0 ≤ 2 ^ (i - 1), 1 < j + 2 ^ i, j < 2 ^ (i - 1), j + 1 < i, 0 < 2 ^ (i - 1), 0 < i, 0 ≤ j + 2 ^ (i - 1), j < i, j + 1 < 2 ^ (i - 1), j < 0, abs(j - 1) / 2 ≤ 2 ^ (i - 2), ∀ 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, j = 0, fillsgn(i→n(i) - 1, int2bits((j - 1) / 2)) + (~1 = j % 2)' = fillsgn(i→n(i), int2bits(j))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 30:
mklist(true, i→n(i) - 1) = fillsgn(i→n(i) - 1, true '), 0 < 2 ^ (i - 2), 0 < 2 ^ (i - 1), 1 < i, 0 ≤ 2 ^ (i - 1), 0 < 2 ^ i, 0 ≤ 2 ^ i, 1 < 2 ^ i, 0 < i, 0 ≤ 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, fillsgn(i→n(i) - 1, true ') + true ' = mklist(true, i→n(i))

Rule argument

Additional information

Used simplifier rules: uint2bits-one: ⊦ uint2bits(1) = false + true;%int2bits-neg-num: i < 0 ⊦ int2bits((* i)) = sgn(bitsinvert(uint2bits(abs(i))) +1);%uint2bits-2pot-02: 0 < i ⊦ uint2bits(2 ^ i - 1) = false ' + mklist(true, i→n(i));% ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward)%i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j;% ⊦ 0 ≤ i → (2 < 2 ^ i ↔ 1 < i);% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ ~1 < i ↔ 0 ≤ i;% ⊦ bits + bits0 = bits + bits1 ↔ bits0 = bits1;% ⊦ n -1 = n - 1;% ⊦ 0 < i ∧ i ≠ 1 → i→n(i - 1) = i→n(i) -1;% ⊦ i < j - i0 ↔ i + i0 < j;% ⊦ boolvar + boolvar0 = boolvar ' + boolvar0 ';% ⊦ bitsinvert(false ' + bits) = true + bitsinvert(bits);% ⊦ bitsinvert(true ') = false ';% ⊦ boolvar + bits = boolvar ' + bits;% ⊦ (bits + false ')+1 = bits + true;% ⊦ bits + boolvar = bits + boolvar ';% ⊦ sgn(true ' + true ') = true ';% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ 0 < i → fillsgn(i→n(i), boolvar ') = mklist(boolvar, i→n(i));% ⊦ ¬ i < j → (i ≠ j ↔ j < i);% ⊦ i < 1 ↔ i ≤ 0;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false);% ⊦ i + ~1 = i - 1;% ⊦ i < j ∧ i = i0 → i0 < j; (forward)% ⊦ i < j → ¬ j < i;% ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;