Proof step 1 from proof twopot-mone-div (conclusion) (back to tree)

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

System applied rule normalize

And obtained the following new goals

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

Rule argument

Additional information

Proof step 2 from proof twopot-mone-div (previous step) (back to tree)

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

User applied rule induction

And obtained the following new goals

Proof step 3:
(2 ^ i - 1) / 2 ^ j ≠ 2 ^ i / 2 ^ j - 1, j ≤ i, 0 ≤ j, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Rule argument

indvar: i→n(i) wfpred: <

Additional information

Used simplifier rules: <; (wellfounded) Used terms: i→n(i)

Proof step 3 from proof twopot-mone-div (previous step) (back to tree)

(2 ^ i - 1) / 2 ^ j ≠ 2 ^ i / 2 ^ j - 1, j ≤ i, 0 ≤ j, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 4:
2 ^ i / 2 ^ j ≠ (2 ^ i - 1) / 2 ^ j + 1, j ≤ i, 0 ≤ j, 0 ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Rule argument

Additional information

Used simplifier rules: forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs ⊦ i - j = i0 ↔ i = i0 + j; (s)

Proof step 4 from proof twopot-mone-div (previous step) (back to tree)

2 ^ i / 2 ^ j ≠ (2 ^ i - 1) / 2 ^ j + 1, j ≤ i, 0 ≤ j, 0 ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ 0 < i → j ^ i = j ^ i -1 * j
Proof step 6:
(2 ^ i -1 * 2) / 2 ^ j ≠ (2 ^ i -1 * 2 - 1) / 2 ^ j + 1, 0 < i, 0 ≤ i, 0 ≤ j, j ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Rule argument

spec: <toplevel>, name:Pot-pos, seq: ⊦ 0 < i → j ^ i = j ^ i -1 * j, subst:[i, j] → [i, 2], <all paths>

Additional information

Used simplifier rules: two-zero: ⊦ 0 ≤ j → 2 ^ j ≠ 0; eq-one: ⊦ 2 ^ i = 1 ↔ i = 0; ⊦ i < j ↔ ¬ j ≤ i; (ss) j ≠ 0 ⊦ 0 / j = 0; (s) j ≠ 0 ⊦ 1 / j = 1 ↔ j = 1; (s) ⊦ i ≤ j → (j ≤ i ↔ i = j); (s) Used terms: i, 2

Proof step 5 from proof twopot-mone-div (previous step) (back to tree)

⊦ 0 < i → j ^ i = j ^ i -1 * j

Lemma Pot-pos

Proof step 6 from proof twopot-mone-div (previous step) (back to tree)

(2 ^ i -1 * 2) / 2 ^ j ≠ (2 ^ i -1 * 2 - 1) / 2 ^ j + 1, 0 < i, 0 ≤ i, 0 ≤ j, j ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
(2 ^ (i - 1) * 2) / 2 ^ j ≠ (2 ^ (i - 1) * 2 - 1) / 2 ^ j + 1, 0 < i, j ≤ i, 0 ≤ j, 0 ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Rule argument

Additional information

Used simplifier rules: forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs ⊦ i -1 = i - 1; (s)

Proof step 7 from proof twopot-mone-div (previous step) (back to tree)

(2 ^ (i - 1) * 2) / 2 ^ j ≠ (2 ^ (i - 1) * 2 - 1) / 2 ^ j + 1, 0 < i, j ≤ i, 0 ≤ j, 0 ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ 0 < i → j ^ i = j ^ i -1 * j
Proof step 9:
(2 ^ (i - 1) * 2) / (2 ^ j -1 * 2) ≠ (2 ^ (i - 1) * 2 - 1) / (2 ^ j -1 * 2) + 1, 0 < i, 0 < j, 0 ≤ i, 0 ≤ j, j ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Rule argument

spec: <toplevel>, name:Pot-pos, seq: ⊦ 0 < i → j ^ i = j ^ i -1 * j, subst:[i, j] → [j, 2], <all paths>

Additional information

Used simplifier rules: ⊦ i < j ↔ ¬ j ≤ i; (ss) ⊦ i ≤ j → (j ≤ i ↔ i = j); (s) ⊦ i / 1 = i; (s) ⊦ i + (j - i) = j; (s) add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2 add-c: ⊦ i + j = j + i; from specification int-basic2 ⊦ 0 < i → (i * j = i * j0 ↔ j = j0); (s) associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: j, 2

Proof step 8 from proof twopot-mone-div (previous step) (back to tree)

⊦ 0 < i → j ^ i = j ^ i -1 * j

Lemma Pot-pos

Proof step 9 from proof twopot-mone-div (previous step) (back to tree)

(2 ^ (i - 1) * 2) / (2 ^ j -1 * 2) ≠ (2 ^ (i - 1) * 2 - 1) / (2 ^ j -1 * 2) + 1, 0 < i, 0 < j, 0 ≤ i, 0 ≤ j, j ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ i * j = j * i
Proof step 11:
(2 ^ (i - 1) * 2) / (2 * 2 ^ j -1) ≠ (2 ^ (i - 1) * 2 - 1) / (2 * 2 ^ j -1) + 1, 0 < j, 0 < i, j ≤ i, 0 ≤ j, 0 ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Rule argument

spec: int-mult, name:commutativity, seq: ⊦ i * j = j * i, subst:[j, i] → [2, 2 ^ j -1], <all paths>

Additional information

Used terms: 2, 2 ^ j -1

Proof step 10 from proof twopot-mone-div (previous step) (back to tree)

⊦ i * j = j * i

A spec-lemma

Proof step 11 from proof twopot-mone-div (previous step) (back to tree)

(2 ^ (i - 1) * 2) / (2 * 2 ^ j -1) ≠ (2 ^ (i - 1) * 2 - 1) / (2 * 2 ^ j -1) + 1, 0 < j, 0 < i, j ≤ i, 0 ≤ j, 0 ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

User applied rule apply lemma

And obtained the following new goals

Proof step 12:
j ≠ 0 ⊦ (2 * i) / (2 * j) = i / j
Proof step 13:
2 ^ j -1 = 0, (2 ^ (i - 1) * 2) / (2 * 2 ^ j -1) ≠ (2 ^ (i - 1) * 2 - 1) / (2 * 2 ^ j -1) + 1, 0 < i, 0 < j, 0 ≤ i, 0 ≤ j, j ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦
Proof step 15:
(2 * 2 ^ (i - 1)) / (2 * 2 ^ j -1) = 2 ^ (i - 1) / 2 ^ j -1, (2 ^ (i - 1) * 2) / (2 * 2 ^ j -1) ≠ (2 ^ (i - 1) * 2 - 1) / (2 * 2 ^ j -1) + 1, 0 < i, 0 < j, 0 ≤ i, 0 ≤ j, j ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Rule argument

spec: int-div, name:div-two-01, seq:j ≠ 0 ⊦ (2 * i) / (2 * j) = i / j, subst:[j, i] → [2 ^ j -1, 2 ^ (i - 1)]

Additional information

Used terms: 2 ^ j -1, 2 ^ (i - 1)

Proof step 12 from proof twopot-mone-div (previous step) (back to tree)

j ≠ 0 ⊦ (2 * i) / (2 * j) = i / j

A spec-lemma

Proof step 13 from proof twopot-mone-div (previous step) (back to tree)

2 ^ j -1 = 0, (2 ^ (i - 1) * 2) / (2 * 2 ^ j -1) ≠ (2 ^ (i - 1) * 2 - 1) / (2 * 2 ^ j -1) + 1, 0 < i, 0 < j, 0 ≤ i, 0 ≤ j, j ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 14:
2 ^ (j - 1) = 0, (2 ^ (i - 1) * 2) / 0 ≠ (2 ^ (i - 1) * 2 - 1) / 0 + 1, 0 < i, 0 < j, 0 ≤ i, 0 ≤ j, j ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Rule argument

Additional information

Used simplifier rules: forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs ⊦ i -1 = i - 1; (s)

Proof step 14 from proof twopot-mone-div (previous step) (back to tree)

2 ^ (j - 1) = 0, (2 ^ (i - 1) * 2) / 0 ≠ (2 ^ (i - 1) * 2 - 1) / 0 + 1, 0 < i, 0 < j, 0 ≤ i, 0 ≤ j, j ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Heuristic elimination applied rule apply elim lemma

And closed the goal!

Rule argument

spec: int-abs, name:elim-sub, seq: ⊦ i0 = i - j ↔ i = i0 + j, subst:[i, j, i0] → [j, 1, i1]

Additional information

Used simplifier rules: two-zero: ⊦ 0 ≤ j → 2 ^ j ≠ 0; ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0; (s) ⊦ ~1 < i ↔ 0 ≤ i; (s) add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2 add-c: ⊦ i + j = j + i; from specification int-basic2 ⊦ i0 = i - j ↔ i = i0 + j; (elim) Used terms: j, 1, i1

Proof step 15 from proof twopot-mone-div (previous step) (back to tree)

(2 * 2 ^ (i - 1)) / (2 * 2 ^ j -1) = 2 ^ (i - 1) / 2 ^ j -1, (2 ^ (i - 1) * 2) / (2 * 2 ^ j -1) ≠ (2 ^ (i - 1) * 2 - 1) / (2 * 2 ^ j -1) + 1, 0 < i, 0 < j, 0 ≤ i, 0 ≤ j, j ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 16:
(2 * 2 ^ (i - 1)) / (2 * 2 ^ (j - 1)) = 2 ^ (i - 1) / 2 ^ (j - 1), 2 ^ (i - 1) / 2 ^ (j - 1) ≠ (2 ^ (i - 1) * 2 - 1) / (2 * 2 ^ (j - 1)) + 1, 0 < i, 0 < j, j ≤ i, 0 ≤ j, 0 ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Rule argument

Additional information

Used simplifier rules: forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs commutativity: ⊦ i * j = j * i; from specification int-mult ⊦ i -1 = i - 1; (s)

Proof step 16 from proof twopot-mone-div (previous step) (back to tree)

(2 * 2 ^ (i - 1)) / (2 * 2 ^ (j - 1)) = 2 ^ (i - 1) / 2 ^ (j - 1), 2 ^ (i - 1) / 2 ^ (j - 1) ≠ (2 ^ (i - 1) * 2 - 1) / (2 * 2 ^ (j - 1)) + 1, 0 < i, 0 < j, j ≤ i, 0 ≤ j, 0 ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ 0 < i ∧ 0 < j0 → (2 * i - 1) / (2 * j0) = (i - 1) / j0
Proof step 18:
(2 * 2 ^ (i - 1)) / (2 * 2 ^ (j - 1)) = 2 ^ (i - 1) / 2 ^ (j - 1), 2 ^ (i - 1) / 2 ^ (j - 1) ≠ (2 ^ (i - 1) - 1) / 2 ^ (j - 1) + 1, 0 < j, 0 < i, 0 < 2 ^ (i - 1), 0 < 2 ^ (j - 1), 0 ≤ i, 0 ≤ j, j ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Rule argument

spec: int-div, name:two-mult, seq: ⊦ 0 < i ∧ 0 < j0 → (2 * i - 1) / (2 * j0) = (i - 1) / j0, subst:[j0, i] → [2 ^ (j - 1), 2 ^ (i - 1)], <all paths>

Additional information

Used simplifier rules: pot2-pos: ⊦ 0 < 2 ^ i ↔ 0 ≤ i; ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; (s) ⊦ 1 ≤ i ↔ 0 < i; (s) commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: 2 ^ (j - 1), 2 ^ (i - 1)

Proof step 17 from proof twopot-mone-div (previous step) (back to tree)

⊦ 0 < i ∧ 0 < j0 → (2 * i - 1) / (2 * j0) = (i - 1) / j0

A spec-lemma

Proof step 18 from proof twopot-mone-div (previous step) (back to tree)

(2 * 2 ^ (i - 1)) / (2 * 2 ^ (j - 1)) = 2 ^ (i - 1) / 2 ^ (j - 1), 2 ^ (i - 1) / 2 ^ (j - 1) ≠ (2 ^ (i - 1) - 1) / 2 ^ (j - 1) + 1, 0 < j, 0 < i, 0 < 2 ^ (i - 1), 0 < 2 ^ (j - 1), 0 ≤ i, 0 ≤ j, j ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 19:
(2 * 2 ^ (i - 1)) / (2 * 2 ^ (j - 1)) = 2 ^ (i - 1) / 2 ^ (j - 1), 2 ^ (i - 1) / 2 ^ (j - 1) ≠ (2 ^ (i - 1) - 1) / 2 ^ (j - 1) + 1, 0 < j, 0 < i, j ≤ i, 0 ≤ j, 0 ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

Rule argument

Additional information

Used simplifier rules: pot2-pos: ⊦ 0 < 2 ^ i ↔ 0 ≤ i; ⊦ 1 ≤ i ↔ 0 < i; (s) ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; (s)

Proof step 19 from proof twopot-mone-div (previous step) (back to tree)

(2 * 2 ^ (i - 1)) / (2 * 2 ^ (j - 1)) = 2 ^ (i - 1) / 2 ^ (j - 1), 2 ^ (i - 1) / 2 ^ (j - 1) ≠ (2 ^ (i - 1) - 1) / 2 ^ (j - 1) + 1, 0 < j, 0 < i, j ≤ i, 0 ≤ j, 0 ≤ i, ∀ i0, j. i→n(i0) < i→n(i) → (2 ^ i0 - 1) / 2 ^ j ≠ 2 ^ i0 / 2 ^ j - 1 ∧ j ≤ i0 ∧ 0 ≤ j → false ⊦

User applied rule apply induction

And closed the goal!

Rule argument

subst: [i0, j] → [i - 1, j - 1]

Additional information

Used simplifier rules: ⊦ i - j = i0 ↔ i = i0 + j; (s) ⊦ i - j ≤ i0 ↔ i ≤ j + i0; (s) ⊦ i + (j - i) = j; (s) ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; (s) add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2 add-c: ⊦ i + j = j + i; from specification int-basic2 ⊦ 1 ≤ i ↔ 0 < i; (s) ⊦ i→n(i - 1) < i→n(i) ↔ 0 < i; (s) j ≠ 0, i0 ≠ 0 ⊦ (i * i0) / (j * i0) = i / j; (s) associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: i - 1, j - 1

Proof Tree for twopot-mone-div (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.