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

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ∀ n. ii(n) ≤ ii(n + 1) ⊦ ∀ n. ∃ m. ii(m) ≥ n, ∃ n. ∀ m. m ≥ n → ii(m) = ii(n)

Rule argument

Additional information

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  ∀ n. ii(n) ≤ ii(n + 1) ⊦ ∃ n0. ∀ m. n0 ≤ m → ii(m) = ii(n0), ∃ m. n ≤ ii(m)

Rule argument

Additional information

Used simplifier rules: ⊦ n ≥ m ↔ m ≤ n; (s)

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

User applied rule cut formula

And obtained the following new goals

• Proof step 4:

  ∀ n. ii(n) ≤ ii(n + 1), ∃ m, n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n) ⊦ ∃ m. n ≤ ii(m), ∃ n0. ∀ m. n0 ≤ m → ii(m) = ii(n0)

• Proof step 18:

  ∀ n. ii(n) ≤ ii(n + 1) ⊦ ∃ m. n ≤ ii(m), ∃ n0. ∀ m. n0 ≤ m → ii(m) = ii(n0), ∃ m, n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)

Rule argument

Additional information

Used formulas: ∃ m, n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  ∀ m. ii(m) ≤ ii(m0), ∀ n. ii(n) ≤ ii(n + 1) ⊦ ∃ n0. ∀ m. n0 ≤ m → ii(m) = ii(n0), ∃ m. n ≤ ii(m)

Rule argument

Additional information

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

User applied rule exists right

And obtained the following new goals

• Proof step 6:

  ∀ n. ii(n) ≤ ii(n + 1), ∀ m. ii(m) ≤ ii(m0) ⊦ ∀ m. m0 ≤ m → ii(m) = ii(m0), ∃ m. n ≤ ii(m)

Rule argument

Additional information

Used terms: m0

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  ii(m) ≠ ii(m0), m0 ≤ m, ∀ m. ii(m) ≤ ii(m0), ∀ n. ii(n) ≤ ii(n + 1) ⊦ ∃ m. n ≤ ii(m)

Rule argument

Additional information

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

User applied rule weakening

And obtained the following new goals

• Proof step 8:

  ii(m) ≠ ii(m0), m0 ≤ m, ∀ n. ii(n) ≤ ii(n + 1), ∀ m. ii(m) ≤ ii(m0) ⊦

Rule argument

Additional information

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

User applied rule structural induction

And obtained the following new goals

• Proof step 9:

  ii(m1 +1) ≠ ii(m0), m0 ≤ m1 +1, ∀ m. ii(m) ≤ ii(m0), ∀ n. ii(n) ≤ ii(n + 1), ∀ ii, m0. ¬(ii(m1) ≠ ii(m0) ∧ m0 ≤ m1 ∧ (∀ n. ii(n) ≤ ii(n + 1)) ∧ (∀ m. ii(m) ≤ ii(m0))) ⊦

Rule argument

Additional information

Used simplifier rules: nat freely generated by 0, +1;; ⊦ m ≤ 0 ↔ m = 0; (s)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  ii(m1 + 1) ≠ ii(m0), m0 ≤ m1 + 1, ∀ n. ii(n) ≤ ii(n + 1), ∀ m. ii(m) ≤ ii(m0), ∀ ii0, m. ii0(m1) = ii0(m) ∨ ¬ m ≤ m1 ∨ ¬(∀ n. ii0(n) ≤ ii0(n + 1)) ∨ ¬(∀ m1. ii0(m1) ≤ ii0(m)) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)

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

User applied rule all left

And obtained the following new goals

• Proof step 11:

  ii(m1 + 1) ≠ ii(m0), m0 ≤ m1 + 1, ∀ m. ii(m) ≤ ii(m0), ∀ n. ii(n) ≤ ii(n + 1), ii(m1) = ii(m0) ∨ ¬ m0 ≤ m1 ∨ ¬(∀ n. ii(n) ≤ ii(n + 1)) ∨ ¬(∀ m1. ii(m1) ≤ ii(m0)) ⊦

Rule argument

Additional information

Used terms: ii, m0

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

  ii(m1 + 1) ≠ ii(m0), m0 ≤ m1 + 1, ∀ n. ii(n) ≤ ii(n + 1), ∀ m. ii(m) ≤ ii(m0), ii(m1) = ii(m0) ∨ ¬ m0 ≤ m1 ∨ ¬ ii(m) ≤ ii(m0) ⊦

Rule argument

Additional information

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

User applied rule all left

And obtained the following new goals

• Proof step 13:

  ii(m1 + 1) ≠ ii(m0), m0 ≤ m1 + 1, ii(m1) ≤ ii(m1 + 1), ∀ m. ii(m) ≤ ii(m0), ∀ n. ii(n) ≤ ii(n + 1), ii(m1) = ii(m0) ∨ ¬ m0 ≤ m1 ∨ ¬ ii(m) ≤ ii(m0) ⊦

Rule argument

Additional information

Used terms: m1

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

User applied rule case distinction

And obtained the following new goals

• Proof step 14:

  ii(m1) = ii(m0), ii(m1 + 1) ≠ ii(m0), ii(m1) ≤ ii(m1 + 1), m0 ≤ m1 + 1, ∀ n. ii(n) ≤ ii(n + 1), ∀ m. ii(m) ≤ ii(m0) ⊦

• Proof step 16:

  ii(m1 + 1) ≠ ii(m0), ii(m1) ≤ ii(m1 + 1), m0 ≤ m1 + 1, ¬ ii(m) ≤ ii(m0), ∀ n. ii(n) ≤ ii(n + 1), ∀ m. ii(m) ≤ ii(m0) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ m < n → (n ≤ m + 1 ↔ n = m + 1); (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 15:

  ii(m1) = ii(m0), ii(m1 + 1) ≠ ii(m0), ii(m0) < ii(m1 + 1), m0 ≤ m1 + 1, ∀ m. ii(m) ≤ ii(m0), ∀ n. ii(n) ≤ ii(n + 1) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)

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

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: m1 + 1

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 17:

  ii(m1 + 1) ≠ ii(m0), ii(m0) < ii(m), ii(m1) ≤ ii(m1 + 1), m0 ≤ m1 + 1, ∀ n. ii(n) ≤ ii(n + 1), ∀ m. ii(m) ≤ ii(m0) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m ≤ n ↔ ¬ n < m; (ss)

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

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: m

Proof step 18 from proof weakmon-lem (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 19:

  ∀ n. ii(n) ≤ ii(n + 1) ⊦ ∃ n0. ∀ m. n0 ≤ m → ii(m) = ii(n0), ∃ m. n ≤ ii(m), ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)

Rule argument

Additional information

Proof step 19 from proof weakmon-lem (previous step) (back to tree)

User applied rule weakening

And obtained the following new goals

• Proof step 20:

  ∀ n. ii(n) ≤ ii(n + 1) ⊦ ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n), ∃ m. n ≤ ii(m)

Rule argument

Additional information

Proof step 20 from proof weakmon-lem (previous step) (back to tree)

User applied rule induction

And obtained the following new goals

• Proof step 21:

  ∀ n. ii(n) ≤ ii(n + 1), ∀ n0, ii. n0 < n → (∀ n. ii(n) ≤ ii(n + 1)) → (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. n0 ≤ ii(m)) ⊦ ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n), ∃ m. n ≤ ii(m)

Rule argument

Additional information

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

Proof step 21 from proof weakmon-lem (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 22:

  ∀ n. ii(n) ≤ ii(n + 1), ∀ n0, ii. n0 < n → (∀ n. ii(n) ≤ ii(n + 1)) → (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. n0 ≤ ii(m)) ⊦ ∃ m. n ≤ ii(m), ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)

Rule argument

Additional information

Proof step 22 from proof weakmon-lem (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 23:

  n ≠ 0, ∀ n. ii(n) ≤ ii(n + 1), ∀ n0, ii. n0 < n → (∀ n. ii(n) ≤ ii(n + 1)) → (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. n0 ≤ ii(m)) ⊦ ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n), ∃ m. n ≤ ii(m)

Rule argument

Additional information

Used simplifier rules: ⊦ 0 ≤ m; (s) Used formulas: n = 0

Proof step 23 from proof weakmon-lem (previous step) (back to tree)

User applied rule exists right

And obtained the following new goals

• Proof step 24:

  n ≠ 0, ∀ n. ii(n) ≤ ii(n + 1), ∀ n0, ii. n0 < n → (∀ n. ii(n) ≤ ii(n + 1)) → (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. n0 ≤ ii(m)) ⊦ ∃ m. n ≤ ii(m), ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n), (∀ m. ii(m) ≤ n - 1) ∧ (∃ m. ii(m) = n - 1)

Rule argument

Additional information

Used terms: n - 1

Proof step 24 from proof weakmon-lem (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 25:

  n ≠ 0, ∀ n. ii(n) ≤ ii(n + 1), ∀ n0, ii. n0 < n → (∀ n. ii(n) ≤ ii(n + 1)) → (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. n0 ≤ ii(m)) ⊦ ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n), ∃ m. n ≤ ii(m), ii(m) < n ∧ (∃ m. ii(m) = n - 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 ⊦ m + 1 ≤ n ↔ m < n; (s) n0 ≤ n ⊦ m ≤ n - n0 ↔ m + n0 ≤ n; (s) ⊦ m < 1 ↔ m = 0; (s) ⊦ m ≤ n ↔ ¬ n < m; (ss)

Proof step 25 from proof weakmon-lem (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 26:

  n = m0 + 1, n ≠ 0, ∀ n. ii(n) ≤ ii(n + 1), ∀ n0, ii. n0 < n → (∀ n. ii(n) ≤ ii(n + 1)) → (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. n0 ≤ ii(m)) ⊦ ∃ m. n ≤ ii(m), ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n), ii(m) < n ∧ (∃ m. ii(m) = m0)

Rule argument

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: n, m0

Proof step 26 from proof weakmon-lem (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 27:

  ∀ n. ii(n) ≤ ii(n + 1), ∀ n0, ii. n0 < m0 + 1 → (∀ n. ii(n) ≤ ii(n + 1)) → (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. n0 ≤ ii(m)) ⊦ ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n), ∃ m1. m0 < ii(m1), ¬ m0 < ii(m) ∧ (∃ m. ii(m) = m0)

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 ⊦ m + 1 ≤ n ↔ m < n; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m + 1 ≠ 0; (s)

Proof step 27 from proof weakmon-lem (previous step) (back to tree)

User applied rule exists right

And obtained the following new goals

• Proof step 28:

  ¬ m0 < ii(m), ∀ n. ii(n) ≤ ii(n + 1), ∀ n0, ii. n0 < m0 + 1 → (∀ n. ii(n) ≤ ii(n + 1)) → (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. n0 ≤ ii(m)) ⊦ ∃ m1. m0 < ii(m1), ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n), ¬ m0 < ii(m) ∧ (∃ m. ii(m) = m0)

Rule argument

Additional information

Used terms: m

Proof step 28 from proof weakmon-lem (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 29:

  ii(m) ≤ m0, ∀ n. ii(n) ≤ ii(n + 1), ∀ n0, ii. n0 < m0 + 1 → (∀ n. ii(n) ≤ ii(n + 1)) → (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. n0 ≤ ii(m)) ⊦ ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n), ∃ m1. m0 < ii(m1), ∃ m. ii(m) = m0

Rule argument

Additional information

Used simplifier rules: ⊦ m ≤ n → (n < m ↔ false); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 29 from proof weakmon-lem (previous step) (back to tree)

User applied rule apply induction

And obtained the following new goals

• Proof step 30:

  ii(m) ≤ m0, ∀ n. ii(n) ≤ ii(n + 1), ∀ n0, ii. n0 < m0 + 1 → (∀ n. ii(n) ≤ ii(n + 1)) → (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. n0 ≤ ii(m)) ⊦ ∀ n. ii(n) ≤ ii(n + 1), ∃ m. ii(m) = m0, ∃ m1. m0 < ii(m1), ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)

• Proof step 32:

  ii(m) ≤ m0, ∀ n. ii(n) ≤ ii(n + 1), (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. m0 ≤ ii(m)), ∀ n0, ii. n0 < m0 + 1 → (∀ n. ii(n) ≤ ii(n + 1)) → (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. n0 ≤ ii(m)) ⊦ ∃ m. ii(m) = m0, ∃ m1. m0 < ii(m1), ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)

Rule argument

Additional information

Used simplifier rules: ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ¬ n < n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: m0, ii

Proof step 30 from proof weakmon-lem (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 31:

  ii(n + 1) < ii(n), ii(m) ≤ m0, ∀ n. ii(n) ≤ ii(n + 1), ∀ n0, ii. n0 < m0 + 1 → (∀ n. ii(n) ≤ ii(n + 1)) → (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. n0 ≤ ii(m)) ⊦ ∃ m. ii(m) = m0, ∃ m1. m0 < ii(m1), ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m ≤ n ↔ ¬ n < m; (ss)

Proof step 31 from proof weakmon-lem (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ m ≤ n → (n < m ↔ false); (s) Used terms: n

Proof step 32 from proof weakmon-lem (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 33:

  ii(m) ≤ m0, m0 ≤ ii(m1), ii(m) ≤ ii(m1), ∀ n. ii(n) ≤ ii(n + 1), ∀ n0, ii. n0 < m0 + 1 → (∀ n. ii(n) ≤ ii(n + 1)) → (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. n0 ≤ ii(m)) ⊦ ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n), ∃ m1. m0 < ii(m1), ∃ m. ii(m) = m0

Rule argument

Additional information

Used simplifier rules: let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat

Proof step 33 from proof weakmon-lem (previous step) (back to tree)

User applied rule exists right

And obtained the following new goals

• Proof step 34:

  ii(m) ≠ m0, ii(m) ≤ ii(m1), m0 ≤ ii(m1), ii(m) ≤ m0, ∀ n. ii(n) ≤ ii(n + 1), ∀ n0, ii. n0 < m0 + 1 → (∀ n. ii(n) ≤ ii(n + 1)) → (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. n0 ≤ ii(m)) ⊦ ∃ m. ii(m) = m0, ∃ m1. m0 < ii(m1), ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)

Rule argument

Additional information

Used terms: m

Proof step 34 from proof weakmon-lem (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 35:

  ii(m) ≠ m0, ii(m) < m0, ii(m) < ii(m1), m0 ≤ ii(m1), ∀ n. ii(n) ≤ ii(n + 1), ∀ n0, ii. n0 < m0 + 1 → (∀ n. ii(n) ≤ ii(n + 1)) → (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. n0 ≤ ii(m)) ⊦ ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n), ∃ m1. m0 < ii(m1), ∃ m. ii(m) = m0

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m < n → m ≤ n; (s) fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)

Proof step 35 from proof weakmon-lem (previous step) (back to tree)

User applied rule exists right

And obtained the following new goals

• Proof step 36:

  ii(m) ≠ m0, ii(m) < ii(m1), ii(m) < m0, m0 ≤ ii(m1), ¬ m0 < ii(m1), ∀ n. ii(n) ≤ ii(n + 1), ∀ n0, ii. n0 < m0 + 1 → (∀ n. ii(n) ≤ ii(n + 1)) → (∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)) ∨ (∃ m. n0 ≤ ii(m)) ⊦ ∃ m. ii(m) = m0, ∃ m1. m0 < ii(m1), ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)

Rule argument

Additional information

Used terms: m1

Proof step 36 from proof weakmon-lem (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 37:

  ii(m) ≠ ii(m1), ii(m) < ii(m1), ∀ n. ii(n) ≤ ii(n + 1), ∀ n0, ii0. n0 < ii(m1) + 1 → (∀ n. ii0(n) ≤ ii0(n + 1)) → (∃ n. (∀ m. ii0(m) ≤ n) ∧ (∃ m. ii0(m) = n)) ∨ (∃ m. n0 ≤ ii0(m)) ⊦ ∃ m. ii(m) = ii(m1), ∃ m. ii(m1) < ii(m), ∃ n. (∀ m. ii(m) ≤ n) ∧ (∃ m. ii(m) = n)

Rule argument

Additional information

Used simplifier rules: ⊦ m ≤ m; (s) ⊦ m ≤ n → (n ≤ m ↔ m = n); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 37 from proof weakmon-lem (previous step) (back to tree)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

Additional information

Used terms: m1