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Proof step 5 from proof int2bits-mod2pot (previous step)

0 < j, 0 ≤ j0, j0 < 2 ^ i, 0 ≤ i, 0 < 2 ^ i, 0 ≤ j ⊦ uint2bits(j * 2 ^ i + j0) = int2bits(j) + fillsgn(i→n(i) + 1, int2bits(j0)).rest

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
0 ≤ i, 2 ^ i ≤ j ⊦ uint2bits(j) = uint2bits(j / 2 ^ i) + fillsgn(i→n(i + 1), uint2bits(j % 2 ^ i)).rest
Proof step 7:
0 ≤ i ∧ 2 ^ i ≤ j * 2 ^ i + j0, 0 < j, 0 ≤ j0, j0 < 2 ^ i, 0 ≤ i, 0 < 2 ^ i, 0 ≤ j ⊦ uint2bits((j * 2 ^ i + j0) / 2 ^ i) + fillsgn(i→n(i + 1), uint2bits((j * 2 ^ i + j0) % 2 ^ i)).rest = int2bits(j) + fillsgn(i→n(i) + 1, int2bits(j0)).rest

Rule argument

name: uint2bits-2potrec rotate: F subst: [i, j] → [i, j * 2 ^ i + j0]

Additional information

Used simplifier rules: ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ 0 ≤ j1 ∧ j1 < j → (i0 * j + j1 < j ↔ i0 ≤ 0);% ⊦ i ≤ j → (j ≤ i ↔ i = j);% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i < j ∧ i = i0 → i0 < j; (forward)% ⊦ i < j ∧ j = i0 → i < i0; (forward) Used terms: i, j * 2 ^ i + j0