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Proof step 5 from proof uint2bits-mod2pot-01 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
0 ≤ i0 * j + j0, 0 ≤ j0, j0 < j ⊦ (i0 * j + j0) / j = i0
Proof step 7:
0 ≤ j * 2 ^ i + j0 ∧ 0 ≤ j0 ∧ j0 < 2 ^ i, 2 ^ i ≤ j * 2 ^ i + j0, 0 ≤ j0, 0 < j, 0 ≤ i, j0 < 2 ^ i ⊦ uint2bits(j) + fillsgn(i→n(i) + 1, uint2bits((j * 2 ^ i + j0) % 2 ^ i)).rest = uint2bits(j) + fillsgn(i→n(i) + 1, uint2bits(j0)).rest

Rule argument

seq: 0 ≤ i0 * j + j0, 0 ≤ j0, j0 < j ⊦ (i0 * j + j0) / j = i0 subst: [j0, j, i0] → [j0, 2 ^ i, j] rotate: F spec: int-div inst: name: diveq-pos

Additional information

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