Proof step 1 from proof div2-2pot-neg (conclusion) (back to tree)

⊦ i < 0 ∧ 0 < j → (i * 2 ^ j) / 2 = i * 2 ^ (j - 1)

System applied rule normalize

And obtained the following new goals

Proof step 2:
(i * 2 ^ j) / 2 ≠ i * 2 ^ (j - 1), i < 0, 0 < j ⊦

Rule argument

Additional information

Proof step 2 from proof div2-2pot-neg (previous step) (back to tree)

(i * 2 ^ j) / 2 ≠ i * 2 ^ (j - 1), i < 0, 0 < j ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
(i * 2 ^ j) / 2 ≠ i * 2 ^ (j - 1), 0 < j, i < 0 ⊦

Rule argument

Additional information

Used simplifier rules: forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs

Proof step 3 from proof div2-2pot-neg (previous step) (back to tree)

(i * 2 ^ j) / 2 ≠ i * 2 ^ (j - 1), 0 < j, i < 0 ⊦

User applied rule cut formula

And obtained the following new goals

Proof step 4:
i = ~ abs(i), (i * 2 ^ j) / 2 ≠ i * 2 ^ (j - 1), i < 0, 0 < j ⊦

Rule argument

fma: i = ~ abs(i)

Additional information

Used simplifier rules: ⊦ i < 0 → abs(i) = ~ i; (s) ⊦ ~ ~ i = i; (s) Used formulas: i = ~ abs(i)

Proof step 4 from proof div2-2pot-neg (previous step) (back to tree)

i = ~ abs(i), (i * 2 ^ j) / 2 ≠ i * 2 ^ (j - 1), i < 0, 0 < j ⊦

User applied rule subst equation

And obtained the following new goals

Proof step 5:
(~ abs(i) * 2 ^ j) / 2 ≠ ~ abs(i) * 2 ^ (j - 1), 0 < j, ~ abs(i) < 0 ⊦

Rule argument

pos: left 1, flags: discard

Additional information

Proof step 5 from proof div2-2pot-neg (previous step) (back to tree)

(~ abs(i) * 2 ^ j) / 2 ≠ ~ abs(i) * 2 ^ (j - 1), 0 < j, ~ abs(i) < 0 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ ~ i * j = ~(i * j)
Proof step 7:
~(abs(i) * 2 ^ j) / 2 ≠ ~ abs(i) * 2 ^ (j - 1), ~ abs(i) < 0, 0 < j ⊦

Rule argument

spec: int-mult, name:minus-left, seq: ⊦ ~ i * j = ~(i * j), subst:[j, i] → [2 ^ j, abs(i)], <all paths>

Additional information

Used terms: 2 ^ j, abs(i)

Proof step 6 from proof div2-2pot-neg (previous step) (back to tree)

⊦ ~ i * j = ~(i * j)

A spec-lemma

Proof step 7 from proof div2-2pot-neg (previous step) (back to tree)

~(abs(i) * 2 ^ j) / 2 ≠ ~ abs(i) * 2 ^ (j - 1), ~ abs(i) < 0, 0 < j ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ j ≠ 0 → ~ i / j = ~(i / j)
Proof step 9:
2 ≠ 0, ~((abs(i) * 2 ^ j) / 2) ≠ ~ abs(i) * 2 ^ (j - 1), 0 < j, ~ abs(i) < 0 ⊦

Rule argument

spec: int-div, name:Div-left, seq: ⊦ j ≠ 0 → ~ i / j = ~(i / j), subst:[j, i] → [2, abs(i) * 2 ^ j], <all paths>

Additional information

Used terms: 2, abs(i) * 2 ^ j

Proof step 8 from proof div2-2pot-neg (previous step) (back to tree)

⊦ j ≠ 0 → ~ i / j = ~(i / j)

A spec-lemma

Proof step 9 from proof div2-2pot-neg (previous step) (back to tree)

2 ≠ 0, ~((abs(i) * 2 ^ j) / 2) ≠ ~ abs(i) * 2 ^ (j - 1), 0 < j, ~ abs(i) < 0 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ 0 ≤ i ∧ 0 < j → (i * 2 ^ j) / 2 = i * 2 ^ (j - 1)

Rule argument

spec: <toplevel>, name:div2-2pot-02, seq: ⊦ 0 ≤ i ∧ 0 < j → (i * 2 ^ j) / 2 = i * 2 ^ (j - 1), subst:[j, i] → [j, abs(i)], <all paths>

Additional information

Used simplifier rules: ⊦ ~ i * j = ~(i * j); (s) associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult commutativity: ⊦ i * j = j * i; from specification int-mult ⊦ ~ i = ~ j ↔ i = j; (s) ⊦ ~ i < 0 ↔ 0 < i; (s) ⊦ 0 < abs(i) ↔ i ≠ 0; (s) ⊦ 0 ≤ abs(i); (s) Used terms: j, abs(i)

Proof step 10 from proof div2-2pot-neg (previous step) (back to tree)

⊦ 0 ≤ i ∧ 0 < j → (i * 2 ^ j) / 2 = i * 2 ^ (j - 1)

Lemma div2-2pot-02

Proof Tree for div2-2pot-neg (View conclusion)

The following image shows the proof tree of sequent calculus. The proved goal is at the bottom. You can click on the nodes to view additional rules. See here for details on colors in a proof tree.