[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 24 from proof shiftright-neg-div (previous step)

2 * 0 ≤ 2 * 2 ^ (abs(j) - 2) + 2 * (i - 1) / 2, 2 * 2 ^ (abs(j) - 2) + i < 0, 1 < abs(j), 0 ≤ 2 ^ (abs(j) - 2) + (i - 1) / 2, i < 0, 2 ^ (abs(j) - 2) + i < (i - 1) / 2, ∀ i, j0. i→n(j0) < i→n(j) → i + 2 ^ (abs(j0) - 1) < 0 → j0 = 0 ∨ i >> abs(j0) = (i + 1 - 2 ^ abs(j0)) / 2 ^ abs(j0) ⊦ (i - 1) / 2 >> (abs(j) - 1) = (i + 1 - 2 ^ abs(j)) / 2 ^ abs(j), j = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 25:
j ≠ 0 ⊦ i % j = 0 → i / j * j = i
Proof step 26:
¬(2 ≠ 0 ∧ (i - 1) % 2 = 0), 2 * 0 ≤ 2 * 2 ^ (abs(j) - 2) + 2 * (i - 1) / 2, 2 * 2 ^ (abs(j) - 2) + i < 0, 1 < abs(j), 0 ≤ 2 ^ (abs(j) - 2) + (i - 1) / 2, i < 0, 2 ^ (abs(j) - 2) + i < (i - 1) / 2, ∀ i, j0. i→n(j0) < i→n(j) → i + 2 ^ (abs(j0) - 1) < 0 → j0 = 0 ∨ i >> abs(j0) = (i + 1 - 2 ^ abs(j0)) / 2 ^ abs(j0) ⊦ (i - 1) / 2 >> (abs(j) - 1) = (i + 1 - 2 ^ abs(j)) / 2 ^ abs(j), j = 0

Rule argument

seq: j ≠ 0 ⊦ i % j = 0 → i / j * j = i subst: [j, i] → [2, i - 1] rotate: F spec: int-div inst: name: divmult

Additional information

Used simplifier rules: ⊦ i + (j - i0) = i + j - i0;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ i < j → ¬ j < i;% ⊦ i * j = j * i; Used terms: 2, i - 1