[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 17 from proof exin2pot (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 18:
⊦ 0 < i → j ^ i = j ^ i -1 * j
Proof step 19:
0 < i + 1, 0 ≤ i, 2 ^ i ≤ i0, i0 < 2 ^ (i + 1)-1 * 2, 0 < i0 * 2 + j1, 2 ^ i < 2 ^ (i + 1)-1 * 2, 0 ≤ j1, j1 < 2, 0 ≤ i0, 0 < 2 ^ (i + 1)-1 * 2, ∀ j0. i→n(j0) < i→n(i0 * 2 + j1) → 0 < j0 → (∃ i. 0 ≤ i ∧ 2 ^ i ≤ j0 ∧ j0 < 2 ^ (i + 1)) ⊦ i0 * 2 + j1 = 1, i0 * 2 + j1 = 0, 0 ≤ i + 1 ∧ 2 ^ (i + 1)-1 * 2 ≤ i0 * 2 + j1 ∧ i0 * 2 + j1 < 2 ^ (2 + i)

Rule argument

spec: int-pot inst: name: Pot-pos seq: ⊦ 0 < i → j ^ i = j ^ i -1 * j subst: [i, j] → [i + 1, 2] flags: rewrite, current

Additional information

Used simplifier rules: numls-02: ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0; from specification int-abs%mone-06: ⊦ ~1 < i ↔ 0 ≤ i; from specification int-abs%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%lels-04: ⊦ i ≤ j → (¬ ¬ j < i ↔ false); from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: i + 1, 2