Proof step 1 from proof exists-max0column-ind (conclusion) (back to tree)

User applied rule structural induction

And obtained the following new goals

• Proof step 2:

  n0 +1 < #c mx, ∀ mx, i. n0 < #c mx → (∃ c. c ≤ n0 ∧ (∀ c0. c0 ≤ n0 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i))) ⊦ ∃ c. c ≤ n0 +1 ∧ (∀ c0. c0 ≤ n0 +1 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i))

Rule argument

Additional information

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

Proof step 2 from proof exists-max0column-ind (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  n0 + 1 < #c mx, ∀ mx, i. n0 < #c mx → (∃ c. c ≤ n0 ∧ (∀ c0. c0 ≤ n0 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i))) ⊦ ∃ c. c ≤ n0 + 1 ∧ (∀ c0. c0 ≤ n0 + 1 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i))

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m +1 = m + 1; (s)

Proof step 3 from proof exists-max0column-ind (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 4:

  n0 + 1 < #c mx, n0 < #c mx → (∃ c. c ≤ n0 ∧ (∀ c0. c0 ≤ n0 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i))) ⊦ ∃ c. c ≤ n0 + 1 ∧ (∀ c0. c0 ≤ n0 + 1 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i))

Rule argument

Additional information

Used terms: mx, i

Proof step 4 from proof exists-max0column-ind (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  n0 + 1 < #c mx, n0 < #c mx, c < #c mx, c ≤ n0, ∀ c0. c0 ≤ n0 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i) ⊦ ∃ c. c ≤ n0 + 1 ∧ (∀ c0. c0 ≤ n0 + 1 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i))

Rule argument

Additional information

Used simplifier rules: fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ¬ m < n → ¬ m + m0 < n; (s)

Proof step 5 from proof exists-max0column-ind (previous step) (back to tree)

User applied rule exists right

And obtained the following new goals

• Proof step 6:

  c < #c mx, n0 < #c mx, n0 + 1 < #c mx, c ≤ n0, ∀ c0. c0 ≤ n0 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i) ⊦ ∃ c. c ≤ n0 + 1 ∧ (∀ c0. c0 ≤ n0 + 1 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i)), c ≤ n0 + 1 ∧ (∀ c0. c0 ≤ n0 + 1 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i))

Rule argument

Additional information

Used terms: c

Proof step 6 from proof exists-max0column-ind (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  c0 < #c mx, count(mx[c]c, λ j. j < i) < count(mx[c0]c, λ j. j < i), n0 + 1 < #c mx, n0 < #c mx, c < #c mx, c ≤ n0 + 1, c0 ≤ n0 + 1, c ≤ n0, ∀ c0. c0 ≤ n0 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i) ⊦ ∃ c. c ≤ n0 + 1 ∧ (∀ c0. c0 ≤ n0 + 1 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i))

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s) ⊦ m < n → n ≠ 0; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic le-one: ⊦ m ≤ n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ ¬ m + n < m; (s) ⊦ m +1 = m + 1; (s) trans-nat: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic

Proof step 7 from proof exists-max0column-ind (previous step) (back to tree)

User applied rule exists right

And obtained the following new goals

• Proof step 8:

  c < #c mx, n0 < #c mx, n0 + 1 < #c mx, count(mx[c]c, λ j. j < i) < count(mx[c0]c, λ j. j < i), c0 < #c mx, c ≤ n0, c0 ≤ n0 + 1, c ≤ n0 + 1, ∀ c0. c0 ≤ n0 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i) ⊦ n0 + 1 ≤ n0 + 1 ∧ (∀ c0. c0 ≤ n0 + 1 → count(mx[c0]c, λ j. j < i) ≤ count(mx[n0 + 1]c, λ j. j < i))

Rule argument

Additional information

Used terms: n0 + 1

Proof step 8 from proof exists-max0column-ind (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 9:

  c1 < #c mx, count(mx[n0 + 1]c, λ j. j < i) < count(mx[c1]c, λ j. j < i), c < #c mx, n0 < #c mx, n0 + 1 < #c mx, count(mx[c]c, λ j. j < i) < count(mx[c0]c, λ j. j < i), c0 < #c mx, c1 ≤ n0 + 1, c ≤ n0, c0 ≤ n0 + 1, c ≤ n0 + 1, ∀ c0. c0 ≤ n0 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s) ⊦ m < n → n ≠ 0; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic le-one: ⊦ m ≤ n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)

Proof step 9 from proof exists-max0column-ind (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 10:

  c0 < #c mx, count(mx[c]c, λ j. j < i) < count(mx[c0]c, λ j. j < i), n0 + 1 < #c mx, n0 < #c mx, c < #c mx, count(mx[n0 + 1]c, λ j. j < i) < count(mx[c1]c, λ j. j < i), c1 < #c mx, c ≤ n0 + 1, c0 ≤ n0 + 1, c ≤ n0, c1 ≤ n0 + 1, c0 ≤ n0 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i), ∀ c0. c0 ≤ n0 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i) ⊦

Rule argument

Additional information

Used terms: c0

Proof step 10 from proof exists-max0column-ind (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

  count(mx[c]c, λ j. j < i) < count(mx[c1]c, λ j. j < i), c1 < #c mx, count(mx[n0 + 1]c, λ j. j < i) < count(mx[c1]c, λ j. j < i), c < #c mx, count(mx[c]c, λ j. j < i) < count(mx[n0 + 1]c, λ j. j < i), n0 + 1 < #c mx, n0 < #c mx, count(mx[c]c, λ j. j < i) + 1 < count(mx[c1]c, λ j. j < i), c1 ≤ n0 + 1, c ≤ n0, c ≤ n0 + 1, ∀ c0. c0 ≤ n0 → count(mx[c0]c, λ j. j < i) ≤ count(mx[c]c, λ j. j < i) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) ⊦ m < n → n ≠ 0; (s) trans-nat: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic 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 ⊦ ¬ n < n; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m ≤ n → (n < m ↔ false); (s) ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s) ⊦ m + 1 ≠ 0; (s) le-one: ⊦ m ≤ n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat ⊦ m < n → (n ≤ m + 1 ↔ n = m + 1); (s) ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ m < n → (n ≤ m ↔ false); (s)

Proof step 11 from proof exists-max0column-ind (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) ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ m < n → (n ≤ m + 1 ↔ n = m + 1); (s) eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic le-one: ⊦ m ≤ n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat ⊦ m + 1 ≠ 0; (s) ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s) ⊦ m < n → m ≤ n; (s) ⊦ m ≤ n → (n < m ↔ false); (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ¬ n < n; (s) Used terms: c1