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Proof step 72 from proof int2bits-add-low (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

name: int2bits-sgn-02 rotate: F subst: [bits, j] → [fillsgn(i→n(i), int2bits(j0)), j - 1]

Additional information

Used simplifier rules: int2bits-zero: ⊦ int2bits(0) = false ';%int2bits-neg-num: i < 0 ⊦ int2bits((* i)) = sgn(bitsinvert(uint2bits(abs(i))) +1);%uint2bits-one: ⊦ uint2bits(1) = false + true;%int2bits-pos-num: 0 ≤ i ⊦ int2bits((* i)) = uint2bits(i);% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ 1 * i = i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ # boolvar ' = 0 +1;% ⊦ i < j → ¬ j < i;% ⊦ 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 ';% ⊦ i * 0 = 0;% ⊦ i + 0 = i;% ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ i < 0 ∧ 0 ≤ j → i < j; Used terms: fillsgn(i→n(i), int2bits(j0)), j - 1