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Proof step 7 from proof int2bits-rec-2pot-neg-03 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ 1 < j ∧ i < 0 ∧ i % j ≠ 0 → (i - (j - 1)) / j = i / j - 1

Rule argument

seq: ⊦ 1 < j ∧ i < 0 ∧ i % j ≠ 0 → (i - (j - 1)) / j = i / j - 1 subst: [j, i] → [2 ^ i, j] rotate: F spec: int-div inst: name: div-add-succ-neg

Additional information

Used simplifier rules: int2bits-inj: ⊦ int2bits(i) = int2bits(j) ↔ i = j;% ⊦ m + n = n + m;% ⊦ 0 < i → i→n(i + 1) = i→n(i) + 1;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i - (j - j0) = i + j0 - j;% ⊦ bits + bits0 = bits1 + bits0 ↔ bits = bits1;% ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ i ≤ ~ j ↔ i + j ≤ 0;% ⊦ 1 < 2 ^ i ↔ 0 < i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false);% ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward)% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward)% ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward)% ⊦ i + j ≤ i ↔ j ≤ 0;% ⊦ 2 ^ i ≤ 0 ↔ i < 0;% ⊦ i < j → ¬ j < i; Used terms: 2 ^ i, j