[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 57 from proof div-div-rec (previous step)

2 ^ i * (j1 + 2 * (i1 + 1)) + j0 < i, 2 ^ i * (j1 + 2 * (i1 + 1)) + j0 ≤ 0, 0 ≤ i, (2 * i1 + j1) + 1 < i, 2 ^ i * ((2 * i1 + j1) + 1) + j0 < 0, 2 * i1 + j1 < 0, j0 < i, j0 ≤ 0, abs(j1) < 2, i1 < i, i1 ≤ 0, j1 < i, j1 ≤ 0, abs(j0) < 2 ^ i, 0 < i ⊦ i1 = ((2 ^ i * 2) * i1 + 2 ^ i * j1 + j0) / (2 ^ i * 2)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 58:
⊦ 0 < j ∧ i ≤ 0 ∧ i0 ≤ 0 ∧ abs(i0) < j → (j * i + i0) / j = i
Proof step 59:
¬(0 < 2 ^ i * 2 ∧ i1 ≤ 0 ∧ 2 ^ i * j1 + j0 ≤ 0 ∧ abs(2 ^ i * j1 + j0) < 2 ^ i * 2), 2 ^ i * (j1 + 2 * (i1 + 1)) + j0 < i, 2 ^ i * (j1 + 2 * (i1 + 1)) + j0 ≤ 0, 0 ≤ i, (2 * i1 + j1) + 1 < i, 2 ^ i * ((2 * i1 + j1) + 1) + j0 < 0, 2 * i1 + j1 < 0, j0 < i, j0 ≤ 0, abs(j1) < 2, i1 < i, i1 ≤ 0, j1 < i, j1 ≤ 0, abs(j0) < 2 ^ i, 0 < i ⊦ i1 = ((2 ^ i * 2) * i1 + 2 ^ i * j1 + j0) / (2 ^ i * 2)

Rule argument

spec: int-div inst: name: div-mult-neg seq: ⊦ 0 < j ∧ i ≤ 0 ∧ i0 ≤ 0 ∧ abs(i0) < j → (j * i + i0) / j = i subst: [i0, i, j] → [2 ^ i * j1 + j0, i1, 2 ^ i * 2] flags: rewrite

Additional information

Used terms: 2 ^ i * j1 + j0, i1, 2 ^ i * 2