Proof step 1 from proof log2ub-lem (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  n /↑ 2^ log2↑(n) ≠ 1, n ≠ 0 ⊦

Rule argument

Additional information

Proof step 2 from proof log2ub-lem (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 3:

  ⊦ log2↑(n) = (∃ m. n = 2^ m ⊃ log2 n;log2(2 * n + 1))

• Proof step 4:

  n ≠ 0 ⊦ n /↑ 2^ (∃ m. n = 2^ m ⊃ log2 n;log2(2 * n + 1)) = 1

Rule argument

Additional information

Used terms: n

Proof step 3 from proof log2ub-lem (previous step) (back to tree)

Lemma log2up-def

Proof step 4 from proof log2ub-lem (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  n ≠ 0 ⊦ n /↑ 2^ (∃ m. n = 2^ m ⊃ log2 n;log2 n + 1) = 1

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic mult-c: ⊦ m * n = n * m; from specification nat-mult mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult ⊦ n ≠ 0 → log2(2 * n + 1) = log2 n + 1; (s)

Proof step 5 from proof log2ub-lem (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 6:

  n /↑ 2^ log2 n ≠ 1, n ≠ 0, ∃ m. n = 2^ m ⊦

• Proof step 9:

  n /↑ 2^(log2 n + 1) ≠ 1, n ≠ 0, ∀ m. n ≠ 2^ m ⊦

Rule argument

Additional information

Proof step 6 from proof log2ub-lem (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  2^ m /↑ 2^ m ≠ 1 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ 2^ n ≠ 0; (s) ⊦ log2 2^ n = n; (s)

Proof step 7 from proof log2ub-lem (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 8:

  ⊦ m /↑ n = (∃ n0. m = n0 * n ⊃ m / n;m / n + 1)

Rule argument

Additional information

Used simplifier rules: ⊦ m * n = m ↔ m = 0 ∨ n = 1; (s) mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult mult-c: ⊦ m * n = n * m; from specification nat-mult ⊦ 2^ n ≠ 0; (s) n ≠ 0 ⊦ n / n = 1; (s) Used terms: 2^ m, 2^ m

Proof step 8 from proof log2ub-lem (previous step) (back to tree)

Lemma divup-def

Proof step 9 from proof log2ub-lem (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  n /↑ (2 * 2^ log2 n) ≠ 1, n ≠ 0, ∀ m. n ≠ 2^ m ⊦

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ 2^(n + 1) = 2 * 2^ n; (s)

Proof step 10 from proof log2ub-lem (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 11:

  n ≠ 0 ⊦ ∃ m. 2^ m ≤ n ∧ n < 2^(m + 1)

• Proof step 12:

  n /↑ (2 * 2^ log2 n) ≠ 1, n ≠ 0, ∀ m. n ≠ 2^ m, ∃ m. 2^ m ≤ n ∧ n < 2^(m + 1) ⊦

Rule argument

Additional information

Used terms: n

Proof step 11 from proof log2ub-lem (previous step) (back to tree)

A spec-lemma

Proof step 12 from proof log2ub-lem (previous step) (back to tree)

User applied rule exists left

And obtained the following new goals

• Proof step 13:

  n /↑ (2 * 2^ log2 n) ≠ 1, n ≠ 0, n < 2^(m + 1), 2^ m ≤ n, ∀ m. n ≠ 2^ m ⊦

Rule argument

Additional information

Proof step 13 from proof log2ub-lem (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 14:

  ⊦ 2^ m ≤ n ∧ n < 2^ m +1 → log2 n = m

• Proof step 15:

  n /↑ (2 * 2^ m) ≠ 1, n ≠ 0, n < 2^(m + 1), n < 2^ m +1, 2^ m ≤ n, ∀ m. n ≠ 2^ m ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) ⊦ 2^(n + 1) = 2 * 2^ n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: m, n

Proof step 14 from proof log2ub-lem (previous step) (back to tree)

A spec-lemma

Proof step 15 from proof log2ub-lem (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 16:

  n /↑ (2 * 2^ m) ≠ 1, n ≠ 0, 2^ m < 2 * 2^ m, n < 2 * 2^ m, 2^ m ≤ n, ∀ m. n ≠ 2^ m ⊦

Rule argument

Additional information

Used simplifier rules: mult-c: ⊦ m * n = n * m; from specification nat-mult mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult ⊦ 2^ n ≠ 0; (s) ⊦ 2 * m = 0 ↔ m = 0; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ 2^(n + 1) = 2 * 2^ n; (s) ⊦ m +1 = m + 1; (s)

Proof step 16 from proof log2ub-lem (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 17:

  ⊦ m /↑ n = (∃ n0. m = n0 * n ⊃ m / n;m / n + 1)

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → m / n = 0; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic ⊦ 2 * m = 0 ↔ m = 0; (s) ⊦ m0 * m = 0 ↔ m0 = 0 ∨ m = 0; (ws) ⊦ 2^ n ≠ 0; (s) ⊦ 2 * m < 2 * n ↔ m < n; (s) ⊦ m0 * m < m ↔ m0 = 0 ∧ m ≠ 0; (ws) mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult mult-c: ⊦ m * n = n * m; from specification nat-mult Used terms: n, 2 * 2^ m

Proof step 17 from proof log2ub-lem (previous step) (back to tree)

Lemma divup-def