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Proof step 4 from proof uint2bits-len-01 (previous step)

0 < 2 ^ i, 2 * 2 ^ i ≤ j, j < 3 * 2 ^ i, 0 ≤ i, ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ i ∧ j0 < 3 * 2 ^ i ∧ 2 * 2 ^ i ≤ j0 ∧ 0 < 2 ^ i → # uint2bits(j0) = i→n(i + 3) ⊦ # uint2bits(j) = i→n(i + 3)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ 0 < i ∧ i ≠ 1 → uint2bits(i) = uint2bits(i / 2) + (i % 2 = 1)
Proof step 6:
0 < j ∧ j ≠ 1, 0 < 2 ^ i, 2 * 2 ^ i ≤ j, j < 3 * 2 ^ i, 0 ≤ i, ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ i ∧ j0 < 3 * 2 ^ i ∧ 2 * 2 ^ i ≤ j0 ∧ 0 < 2 ^ i → # uint2bits(j0) = i→n(i + 3) ⊦ #(uint2bits(j / 2) + (j % 2 = 1)) = i→n(i + 3)

Rule argument

name: uint2bits-rec-01 rotate: F subst: [i] → [j]

Additional information

Used simplifier rules: uint2bits-one: ⊦ uint2bits(1) = false + true;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ boolvar + boolvar0 = boolvar ' + boolvar0 ';% ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ # boolvar ' = 0 +1;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0);% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ 1 < j → (i * j ≤ 1 ↔ i ≤ 0);% ⊦ 0 ≤ i → ¬ 2 ^ i ≤ 0;% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ 0 < i → (i * j ≤ 0 ↔ j ≤ 0); Used terms: j