[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 27 from proof cast-add-2pot-h (previous step)

0 ≤ 2 ^ i + j, 2 ^ (i - 1) ≤ (2 ^ i + j) % 2 ^ i, 0 < 2 ^ i + j, 0 < i, j < 0, 0 < 2 ^ i + j, 0 ≤ j + 2 ^ (i - 1) ⊦ (2 ^ i + j) % 2 ^ i = 2 ^ i + j

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 28:
⊦ i % j = 0 ∧ 0 ≤ i + i0 ∧ 0 ≤ i ∧ i0 < 0 ∧ i0 % j ≠ 0 ∧ 0 < j → (i + i0) % j = i0 % j + j
Proof step 29:
¬(2 ^ i % 2 ^ i = 0 ∧ 0 ≤ 2 ^ i + j ∧ 0 ≤ 2 ^ i ∧ j < 0 ∧ j % 2 ^ i ≠ 0 ∧ 0 < 2 ^ i), 0 ≤ 2 ^ i + j, 2 ^ (i - 1) ≤ (2 ^ i + j) % 2 ^ i, 0 < 2 ^ i + j, 0 < i, j < 0, 0 < 2 ^ i + j, 0 ≤ j + 2 ^ (i - 1) ⊦ (2 ^ i + j) % 2 ^ i = 2 ^ i + j
Proof step 32:
2 ^ i % 2 ^ i = 0 ∧ 0 ≤ 2 ^ i + j ∧ 0 ≤ 2 ^ i ∧ j < 0 ∧ j % 2 ^ i ≠ 0 ∧ 0 < 2 ^ i, 0 ≤ 2 ^ i + j, 2 ^ (i - 1) ≤ j % 2 ^ i + 2 ^ i, 0 < 2 ^ i + j, 0 < i, j < 0, 0 < 2 ^ i + j, 0 ≤ j + 2 ^ (i - 1) ⊦ j % 2 ^ i + 2 ^ i = 2 ^ i + j

Rule argument

seq: ⊦ i % j = 0 ∧ 0 ≤ i + i0 ∧ 0 ≤ i ∧ i0 < 0 ∧ i0 % j ≠ 0 ∧ 0 < j → (i + i0) % j = i0 % j + j subst: [j, i0, i] → [2 ^ i, j, 2 ^ i] rotate: F spec: int-div inst: name: mod-add1-posneg-pos

Additional information

Used terms: 2 ^ i, j, 2 ^ i