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Proof step 12 from proof int2bits-len-add (previous step)

0 ≤ i, j < 0, j0 < 0, 0 < 2 ^ i + j0, j * 2 ^ i + j0 ≤ 0 ⊦ # uint2bits(~ j * 2 ^ i + ~ j0) = # int2bits(~ j) + i→n(i)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 13:
⊦ 0 < j ∧ 0 ≤ j0 ∧ j0 < 2 ^ i ∧ 0 ≤ i → # uint2bits(j * 2 ^ i + j0) = # uint2bits(j) + i→n(i)
Proof step 14:
0 < ~ j ∧ 0 ≤ ~ j0 ∧ ~ j0 < 2 ^ i ∧ 0 ≤ i, 0 ≤ i, j < 0, j0 < 0, 0 < 2 ^ i + j0, j * 2 ^ i + j0 ≤ 0 ⊦ # uint2bits(~ j) + i→n(i) = # int2bits(~ j) + i→n(i)

Rule argument

name: uint2bits-len-add rotate: F subst: [j0, i, j] → [~ j0, i, ~ j]

Additional information

Used simplifier rules: ⊦ ~ i * j = ~(i * j);% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ ~ i + ~ j = ~(i + j);% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ 0 < ~ i ↔ i < 0;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false);% ⊦ i ≤ ~ j ↔ i + j ≤ 0;% ⊦ i + 0 = i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i;% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false); Used terms: ~ j0, i, ~ j