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Proof step 1 from proof int2bits-lelen-04 (conclusion)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

name: int2bits-lelen-03 rotate: F subst: [m, i] → [i→n(j) + 1, 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;% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ (* i) ≤ j ∧ i ≠ 0 → i - 1 < j; (forward)% ⊦ i < (* j) ∧ j < 0 → i < 0; (forward)% ⊦ i < j ∧ j = i0 → i < i0; (forward)% ⊦ i ≤ j → (j ≤ i ↔ i = j);% ⊦ i < j → ¬ j < i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i + 0 = i;% ⊦ (* i) + i0 - (* j) = (i - j) + i0;% ⊦ 0 ≤ i → n→i(i→n(i)) = i;% ⊦ n→i(m + n) = n→i(m) + n→i(n);% ⊦ ~ i = j ↔ i + j = 0;% ⊦ i + j = j + i;% ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < 0 → (~1 ≤ i ↔ i = ~1);% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ i < j ∧ i = i0 → i0 < j; (forward)% ⊦ i + ~1 = i - 1;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ 2 ^ i < 1 ↔ i < 0;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ 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;% ⊦ ¬ m ≤ n ↔ n < m;% ⊦ ¬ m + n < m;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m; Used terms: i→n(j) + 1, i