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Proof step 10 from proof int2int-lastn-negneg (previous step)

int2bits(i) = lastn(# int2bits(i), int2bits(i)), i < 0, 2 ^ n→i(# int2bits(i)) + i < 0 ⊦ i + 2 ^ n→i(# int2bits(i)) = 0, i = i % 2 ^ n→i(# int2bits(i))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ i < ~1 ∧ 0 ≤ j → (2 ^ j + i < 0 ↔ ¬ # int2bits(i) ≤ i→n(j) + 1)

Rule argument

name: int2bits-lelen-02 rotate: F subst: [i, j] → [i, n→i(# int2bits(i))]

Additional information

Used simplifier rules: int2bits-neg-num: i < 0 ⊦ int2bits((* i)) = sgn(bitsinvert(uint2bits(abs(i))) +1);%uint2bits-one: ⊦ uint2bits(1) = false + true;% ⊦ m + n = n + m;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ ¬ m + n < m;% ⊦ ¬ m ≤ n ↔ n < m;% ⊦ i→n(n→i(n)) = n;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < 0 → (~1 ≤ i ↔ i = ~1);% ⊦ 0 ≤ n→i(n);% ⊦ i < j ∧ i = i0 → i0 < j; (forward)% ⊦ 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;% ⊦ lastn(1, boolvar ') = boolvar ';% ⊦ boolvar ' = boolvar0 ' ↔ boolvar ↔ boolvar0; Used terms: i, n→i(# int2bits(i))