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Proof step 10 from proof int2bits-len-eq (previous step)

j = 2 ^ n→i(# int2bits(j)), j < 0, ~1 ≤ i, j + 2 ^ i < 0, 0 ≤ 2 ^ (i + 1) + j ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

name: int2bits-len-neg rotate: F subst: [i, j] → [i, 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;% ⊦ ¬ 2 ^ i < 0;% ⊦ 0 ≤ i → abs(i) = i;% ⊦ 0 ≤ i ∧ 0 ≤ j → abs(i + j) = abs(i) + abs(j);% ⊦ n→i(i→n(i)) = abs(i);% ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ ~ i ≤ j ↔ 0 ≤ i + j;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i ≤ j → ¬ j < i;% ⊦ i < ~ j ↔ i + j < 0;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false);% ⊦ i < 0 → (~1 ≤ i ↔ i = ~1);% ⊦ i < 0 → 2 ^ i = 0;% ⊦ i + 0 = i;% ⊦ i < 0 ∧ 0 ≤ j → i < j;% ⊦ j < i ∧ i ≤ 0 → j < 0; (forward)% ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0;% ⊦ 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 ' = 0 +1; Used terms: i, j