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Proof step 30 from proof cast-mod-pos-high (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 31:
0 ≤ i, 2 ^ i ≤ j, j < 2 ^ (i + 1) ⊦ # int2bits(j) = i→n(i + 2)
Proof step 32:
0 ≤ i - 1 ∧ 2 ^ (i - 1) ≤ j ∧ j < 2 ^ ((i - 1) + 1), i→n((i - 1) + 2) = i→n(i) + 1, 0 < i, 0 ≤ i, 2 ^ (i - 1) ≤ j % 2 ^ i, 0 ≤ j, 2 ^ (i - 1) ≤ j, 0 ≤ 2 ^ (i - 1) + j, j < 2 ^ i, 0 < j ⊦ j = j % 2 ^ i

Rule argument

seq: 0 ≤ i, 2 ^ i ≤ j, j < 2 ^ (i + 1) ⊦ # int2bits(j) = i→n(i + 2) subst: [i, j] → [i - 1, j] rotate: F spec: int2bits inst: name: int2bits-len-pos

Additional information

Used simplifier rules: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i ≤ j → (j ≤ i ↔ i = j);% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ i + (j - i) = j;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false);% ⊦ i % 1 = 0;% ⊦ 0 ≤ i → (i < 1 ↔ i = 0); Used terms: i - 1, j