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Proof step 57 from proof uint2bits-add-high (previous step)

uint2bits((j - 1) / 2 + 2 ^ (i - 1)) = sgn(int2bits((j - 1) / 2).rest), 2 ^ (i - 2) < 2 ^ i, 0 < 2 ^ i, 1 < 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 0 < j + 2 ^ i, 2 ^ (i - 1) + j < 2 ^ i, j + 1 < i, 2 ^ (i - 1) + j < i, (2 ^ (i - 1) + j) + 1 < i, 2 ^ (i - 1) < 2 ^ i, 0 < i, 2 ^ (i - 1) + j < 0, j < i, j + 1 < 2 ^ i, j < 2 ^ i, (2 ^ (i - 1) + j) + 1 < 2 ^ i, 0 ≤ 2 ^ i, 0 ≤ j + 2 ^ i, j ≤ 2 ^ i, j < 0, abs(j - 1) / 2 ≤ 2 ^ (i - 1), 2 ^ (i - 2) < abs(j - 1) / 2, abs(j - 1) / 2 < 2 ^ i, 2 ^ (i - 2) < 2 ^ (i - 1), ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ j + 2 ^ i = 1, sgn(int2bits((j - 1) / 2).rest) + (~1 = j % 2)' = sgn(int2bits(j).rest), i = 1, j = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 58:
⊦ j < 0 ∧ j ≠ ~1 → int2bits(j) = int2bits((j - 1) / 2) + (~1 = j % 2)'
Proof step 59:
j < 0 ∧ j ≠ ~1, uint2bits((j - 1) / 2 + 2 ^ (i - 1)) = sgn(int2bits((j - 1) / 2).rest), 2 ^ (i - 2) < 2 ^ i, 0 < 2 ^ i, 1 < 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 0 < j + 2 ^ i, 2 ^ (i - 1) + j < 2 ^ i, j + 1 < i, 2 ^ (i - 1) + j < i, (2 ^ (i - 1) + j) + 1 < i, 2 ^ (i - 1) < 2 ^ i, 0 < i, 2 ^ (i - 1) + j < 0, j < i, j + 1 < 2 ^ i, j < 2 ^ i, (2 ^ (i - 1) + j) + 1 < 2 ^ i, 0 ≤ 2 ^ i, 0 ≤ j + 2 ^ i, j ≤ 2 ^ i, j < 0, abs(j - 1) / 2 ≤ 2 ^ (i - 1), 2 ^ (i - 2) < abs(j - 1) / 2, abs(j - 1) / 2 < 2 ^ i, 2 ^ (i - 2) < 2 ^ (i - 1), ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ j + 2 ^ i = 1, sgn(int2bits((j - 1) / 2).rest) + (~1 = j % 2)' = sgn((int2bits((j - 1) / 2) + (~1 = j % 2)').rest), i = 1, j = 0

Rule argument

name: int2bits-rec-neg rotate: F subst: [j] → [j]

Additional information

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