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Proof step 1 from proof int2bits-rec-2pot-pos (conclusion)

⊦ 2 ^ i ≤ j ∧ 0 < i → int2bits(j) = int2bits(j / 2 ^ i) + fillsgn(i→n(i + 1), int2bits(j % 2 ^ i)).rest

User applied rule insert elim lemma

And obtained the following new goals

Proof step 2:
j = i0 * 2 ^ i + j0 ∧ 0 ≤ j0 ∧ j0 < 2 ^ i ∧ 0 ≤ i0 ∧ 0 < 2 ^ i ∧ 0 ≤ j ⊦ 2 ^ i ≤ j ∧ 0 < i → int2bits(j) = int2bits(i0) + fillsgn(i→n(i + 1), int2bits(j0)).rest

Rule argument

seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0) subst: [i, j, i0, j0] → [j, 2 ^ i, i0, j0] rotate: T spec: int-div inst: name: div-elim-pospos

Additional information

Used simplifier rules: ⊦ 0 < i → i→n(i + 1) = i→n(i) + 1;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ 0 < 2 ^ i ↔ 0 ≤ i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i;% ⊦ j < i ∧ i ≤ 0 → j < 0; (forward)% ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward)% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ ¬ 2 ^ i < 0;%⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0); (elim) Used terms: j, 2 ^ i, i0, j0