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Proof step 45 from proof int2bits-add-high (previous step)

1 = 2 ^ i0 + (1 - 2 ^ i0), 1 - 2 ^ i0 < 0, j0 < 0, 0 < i, 2 ^ (i - 1) + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, (1 - 2 ^ i0) * 2 ^ i + j0 < 0, 0 < 2 ^ n→i(# int2bits(~(1 - 2 ^ i0))) + (1 - 2 ^ i0), 0 ≤ (2 ^ n→i(# int2bits(~(1 - 2 ^ i0))) + (1 - 2 ^ i0)) * 2 ^ i + j0, 0 ≤ j0 + 2 ^ i, 0 < 2 ^ n→i(# int2bits(1 - (1 - 2 ^ i0))) + (1 - 2 ^ i0) ⊦ sgn(sgn(uint2bits(2 ^ n→i(# int2bits(~(1 - 2 ^ i0))) + (1 - 2 ^ i0) - 1) + fillsgn(i→n(i), int2bits(j0).rest)).rest) = sgn(uint2bits(2 ^ n→i(# int2bits(1 - (1 - 2 ^ i0))) + (1 - 2 ^ i0) - 1).rest) + fillsgn(i→n(i), int2bits(j0).rest)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 46:
j0 < 0, 0 < i, 2 ^ (i - 1) + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, 2 ^ i0 ≤ 2 ^ (1 + i0), 0 ≤ j0 + 2 ^ i, 2 ^ i0 ≤ 2 ^ (2 + i0), 0 < i0, (1 - 2 ^ i0) * 2 ^ i + j0 < 0, 0 ≤ 2 ^ i * (2 ^ (1 + i0) + 1 - 2 ^ i0) + j0 ⊦ true ' + mklist(false, i→n(i0)) = sgn(uint2bits(2 ^ (2 + i0) - 2 ^ i0).rest)

Rule argument

Additional information

Used simplifier rules: int2bits-2pot-02: 0 ≤ i ⊦ int2bits(2 ^ i - 1) = false ' + mklist(true, i→n(i));%int2bits-2pot: 0 ≤ i ⊦ int2bits(2 ^ i) = false ' + true ' + mklist(false, i→n(i));%uint2bits-2pot: 0 ≤ i ⊦ uint2bits(2 ^ i) = false ' + true ' + mklist(false, i→n(i));% ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward)% ⊦ j < i ∧ i ≤ 0 → j < 0; (forward)% ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward)% ⊦ 0 < i → n→i(i→n(i)) = i;% ⊦ i < j - i0 ↔ i + i0 < j;% ⊦ i < j + 1 ↔ i ≤ j;% ⊦ bits + bits0 = bits1 + bits0 ↔ bits = bits1;%⊦ 0 < i → sgn(true ' + mklist(false, i→n(i)) + bits) = true ' + mklist(false, i→n(i)) + bits;% ⊦ (boolvar ' + bits).rest = bits;% ⊦ sgn(false ' + true ' + bits) = false ' + true ' + bits;% ⊦ i < j → ¬ j < i;% ⊦ 0 < i → 2 ^ (i + 1) - 2 ^ i = 2 ^ i;% ⊦ 0 < j → abs(j) = j;% ⊦ 1 < 2 ^ i ↔ 0 < i;% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ i + 0 = i;% ⊦ (* i) + i0 - (* j) = (i - j) + i0;% ⊦ i - (j - j0) = i + j0 - j;% ⊦ i + j - i0 + j = i - i0;% ⊦ i - j - i0 = i - j + i0;% ⊦ i + (j - i0) = i + j - i0;% ⊦ m + n = n + m;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ n→i(i→n(i)) = abs(i);% ⊦ n→i(m + n) = n→i(m) + n→i(n);% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ # mklist(boolvar, n) = n;% ⊦ # boolvar ' = 0 +1;% ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < 0 → 2 ^ i = 0;% ⊦ ~(i - j) = j - i;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + (j - i) = j;