[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 15 from proof exin2pot (previous step)

0 ≤ i, 2 ^ i ≤ j / 2, j / 2 < 2 ^ (i + 1), 0 < j, 2 ^ i < 2 ^ (i + 1), ∀ j0. i→n(j0) < i→n(j) → 0 < j0 → (∃ i. 0 ≤ i ∧ 2 ^ i ≤ j0 ∧ j0 < 2 ^ (i + 1)) ⊦ j = 1, j = 0, 0 ≤ i + 1 ∧ 2 ^ (i + 1) ≤ j ∧ j < 2 ^ (2 + i)

User applied rule insert elim lemma

And obtained the following new goals

Proof step 16:
j = i0 * 2 + j1 ∧ 0 ≤ j1 ∧ j1 < 2 ∧ 0 < 2 ∧ 0 ≤ j, 0 ≤ i, 2 ^ i ≤ i0, i0 < 2 ^ (i + 1), 0 < j, 2 ^ i < 2 ^ (i + 1), ∀ j0. i→n(j0) < i→n(j) → 0 < j0 → (∃ i. 0 ≤ i ∧ 2 ^ i ≤ j0 ∧ j0 < 2 ^ (i + 1)) ⊦ j = 1, j = 0, 0 ≤ i + 1 ∧ 2 ^ (i + 1) ≤ j ∧ j < 2 ^ (2 + i)

Rule argument

spec: int-div inst: name: div-elim-pos seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j) subst: [i, j, i0, j0] → [j, 2, i0, j1] flags:

Additional information

Used simplifier rules: lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1%⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j); (elim) Used terms: j, 2, i0, j1