Proof step 1 from proof rowperms-clean0unchanged (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  #c mx = #c mx0, #r mx = #r mx0, #c mx ≠ 0, ¬(#clean0rows(mx0, i) = #clean0rows(mx, i) ↔ true), ∀ r. r < #r mx → perm(mx0[r]r, mx[r]r) ⊦

Rule argument

Additional information

Proof step 2 from proof rowperms-clean0unchanged (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  #c mx = #c mx0, #r mx = #r mx0, #c mx0 ≠ 0, #clean0rows(mx0, i) ≠ #clean0rows(mx, i), ∀ r. r < #r mx0 → perm(mx0[r]r, mx[r]r) ⊦

Rule argument

Additional information

Proof step 3 from proof rowperms-clean0unchanged (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 4:

  #c mx = #c mx0, #r mx = #r mx0, #c mx0 ≠ 0, #clean0rows(mx0, i) ≠ #clean0rows(mx, i), #clean0rows(mx0, i) < #clean0rows(mx, i), ∀ r. r < #r mx0 → perm(mx0[r]r, mx[r]r) ⊦

• Proof step 16:

  #c mx = #c mx0, #r mx = #r mx0, #c mx0 ≠ 0, #clean0rows(mx0, i) ≠ #clean0rows(mx, i), ¬ #clean0rows(mx0, i) < #clean0rows(mx, i), ∀ r. r < #r mx0 → perm(mx0[r]r, mx[r]r) ⊦

Rule argument

Additional information

Used formulas: #clean0rows(mx0, i) < #clean0rows(mx, i)

Proof step 4 from proof rowperms-clean0unchanged (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 5:

  ⊦ ¬ #r mx < #clean0rows(mx, i)

• Proof step 6:

  #c mx = #c mx0, #r mx = #r mx0, #c mx0 ≠ 0, #clean0rows(mx0, i) ≠ #clean0rows(mx, i), #clean0rows(mx0, i) < #clean0rows(mx, i), ¬ #r mx < #clean0rows(mx, i), ∀ r. r < #r mx0 → perm(mx0[r]r, mx[r]r) ⊦

Rule argument

Additional information

Used terms: i, mx

Proof step 5 from proof rowperms-clean0unchanged (previous step) (back to tree)

Lemma clean0rows-bounded

Proof step 6 from proof rowperms-clean0unchanged (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  #c mx = #c mx0, #r mx = #r mx0, #c mx0 ≠ 0, #clean0rows(mx0, i) ≠ #clean0rows(mx, i), #clean0rows(mx0, i) < #clean0rows(mx, i), #clean0rows(mx0, i) < #r mx0, #clean0rows(mx, i) ≤ #r mx0, ∀ r. r < #r mx0 → perm(mx0[r]r, mx[r]r) ⊦

Rule argument

Additional information

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

Proof step 7 from proof rowperms-clean0unchanged (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 8:

  ⊦ #clean0rows(mx, i) < #r mx → count(mx[#clean0rows(mx, i)]r, λ j. ¬ j < i) ≠ 0

• Proof step 9:

  #c mx = #c mx0, #r mx = #r mx0, #c mx0 ≠ 0, #clean0rows(mx0, i) ≠ #clean0rows(mx, i), #clean0rows(mx0, i) < #r mx0, #clean0rows(mx0, i) < #clean0rows(mx, i), #clean0rows(mx, i) ≤ #r mx0, #clean0rows(mx0, i) < #r mx0 → count(mx0[#clean0rows(mx0, i)]r, λ j. ¬ j < i) ≠ 0, ∀ r. r < #r mx0 → perm(mx0[r]r, mx[r]r) ⊦

Rule argument

Additional information

Used terms: mx0, i

Proof step 8 from proof rowperms-clean0unchanged (previous step) (back to tree)

Lemma clean0rows-notallls

Proof step 9 from proof rowperms-clean0unchanged (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  #c mx = #c mx0, #r mx = #r mx0, #c mx0 ≠ 0, #clean0rows(mx0, i) ≠ #clean0rows(mx, i), count(mx0[#clean0rows(mx0, i)]r, λ j. ¬ j < i) ≠ 0, #clean0rows(mx0, i) < #clean0rows(mx, i), #clean0rows(mx0, i) < #r mx0, #clean0rows(mx, i) ≤ #r mx0, ∀ r. r < #r mx0 → perm(mx0[r]r, mx[r]r) ⊦

Rule argument

Additional information

Proof step 10 from proof rowperms-clean0unchanged (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 11:

  ⊦ r < #clean0rows(mx, i) → count(mx[r]r, λ j. ¬ j < i) = 0

• Proof step 12:

  #c mx = #c mx0, #r mx = #r mx0, #c mx0 ≠ 0, #clean0rows(mx0, i) ≠ #clean0rows(mx, i), count(mx0[#clean0rows(mx0, i)]r, λ j. ¬ j < i) ≠ 0, #clean0rows(mx0, i) < #r mx0, #clean0rows(mx0, i) < #clean0rows(mx, i), #clean0rows(mx, i) ≤ #r mx0, #clean0rows(mx0, i) < #clean0rows(mx, i) → count(mx[#clean0rows(mx0, i)]r, λ j. ¬ j < i) = 0, ∀ r. r < #r mx0 → perm(mx0[r]r, mx[r]r) ⊦

Rule argument

Additional information

Used terms: #clean0rows(mx0, i), mx, i

Proof step 11 from proof rowperms-clean0unchanged (previous step) (back to tree)

Lemma clean0rows-allls

Proof step 12 from proof rowperms-clean0unchanged (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 13:

  #c mx = #c mx0, #r mx = #r mx0, count(mx[#clean0rows(mx0, i)]r, λ j. ¬ j < i) = 0, #c mx0 ≠ 0, #clean0rows(mx0, i) ≠ #clean0rows(mx, i), count(mx0[#clean0rows(mx0, i)]r, λ j. ¬ j < i) ≠ 0, #clean0rows(mx0, i) < #clean0rows(mx, i), #clean0rows(mx0, i) < #r mx0, #clean0rows(mx, i) ≤ #r mx0, ∀ r. r < #r mx0 → perm(mx0[r]r, mx[r]r) ⊦

Rule argument

Additional information

Proof step 13 from proof rowperms-clean0unchanged (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 14:

  ⊦ perm(x, y) → count(x, p) = count(y, p)

• Proof step 15:

  #c mx = #c mx0, #r mx = #r mx0, count(mx[#clean0rows(mx0, i)]r, λ j. ¬ j < i) = 0, #c mx0 ≠ 0, #clean0rows(mx0, i) ≠ #clean0rows(mx, i), count(mx0[#clean0rows(mx0, i)]r, λ j. ¬ j < i) ≠ 0, #clean0rows(mx0, i) < #r mx0, #clean0rows(mx0, i) < #clean0rows(mx, i), #clean0rows(mx, i) ≤ #r mx0, ¬ perm(mx0[#clean0rows(mx0, i)]r, mx[#clean0rows(mx0, i)]r), ∀ r. r < #r mx0 → perm(mx0[r]r, mx[r]r) ⊦

Rule argument

Additional information

Used terms: mx0[#clean0rows(mx0, i)]r, mx[#clean0rows(mx0, i)]r, λ j. ¬ j < i

Proof step 14 from proof rowperms-clean0unchanged (previous step) (back to tree)

A spec-lemma

Proof step 15 from proof rowperms-clean0unchanged (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: #clean0rows(mx0, i)

Proof step 16 from proof rowperms-clean0unchanged (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 17:

  #c mx = #c mx0, #r mx = #r mx0, #c mx0 ≠ 0, #clean0rows(mx0, i) ≠ #clean0rows(mx, i), #clean0rows(mx, i) < #clean0rows(mx0, i), ∀ r. r < #r mx0 → perm(mx0[r]r, mx[r]r) ⊦

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 ↔ ¬ n < m); (ss)

Proof step 17 from proof rowperms-clean0unchanged (previous step) (back to tree)

User applied rule apply proof lemma

And obtained the following new goals

• Proof step 18:

  #c mx = #c mx0, #r mx = #r mx0, #c mx0 ≠ 0, #clean0rows(mx0, i) ≠ #clean0rows(mx, i), #clean0rows(mx0, i) < #clean0rows(mx, i), ∀ r. r < #r mx0 → perm(mx0[r]r, mx[r]r) ⊦

• Proof step 19:

  #c mx = #c mx0, #r mx = #r mx0, #c mx0 ≠ 0, #clean0rows(mx0, i) ≠ #clean0rows(mx, i), #clean0rows(mx, i) < #clean0rows(mx0, i), ∀ r. r < #r mx0 → perm(mx0[r]r, mx[r]r) ⊦ #c mx0 = #c mx ∧ #r mx0 = #r mx ∧ #c mx ≠ 0 ∧ #clean0rows(mx, i) ≠ #clean0rows(mx0, i) ∧ (∀ r. r < #r mx → perm(mx[r]r, mx0[r]r))

Rule argument

Additional information

Proof step 18 from proof rowperms-clean0unchanged (previous step) (back to tree)

Proof step 4 used as proof-lemma

Proof step 19 from proof rowperms-clean0unchanged (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 20:

  #c mx = #c mx0, #r mx = #r mx0, #c mx0 ≠ 0, #clean0rows(mx0, i) ≠ #clean0rows(mx, i), r < #r mx0, #clean0rows(mx, i) < #clean0rows(mx0, i), ¬ perm(mx[r]r, mx0[r]r), ∀ r. r < #r mx0 → perm(mx0[r]r, mx[r]r) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 20 from proof rowperms-clean0unchanged (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 21:

  #c mx = #c mx0, #r mx = #r mx0, #c mx0 ≠ 0, #clean0rows(mx0, i) ≠ #clean0rows(mx, i), #clean0rows(mx, i) < #clean0rows(mx0, i), r < #r mx0, ¬ perm(mx[r]r, mx0[r]r), r < #r mx0 → perm(mx0[r]r, mx[r]r) ⊦

Rule argument

Additional information

Used terms: r

Proof step 21 from proof rowperms-clean0unchanged (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 22:

  #c mx = #c mx0, #r mx = #r mx0, #c mx0 ≠ 0, #clean0rows(mx0, i) ≠ #clean0rows(mx, i), r < #r mx0, #clean0rows(mx, i) < #clean0rows(mx0, i), perm(mx0[r]r, mx[r]r), ¬ perm(mx[r]r, mx0[r]r) ⊦

Rule argument

Additional information

Proof step 22 from proof rowperms-clean0unchanged (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 23:

  ⊦ perm(x, y) ↔ perm(y, x)

Rule argument

Additional information

Used terms: mx[r]r, mx0[r]r

Proof step 23 from proof rowperms-clean0unchanged (previous step) (back to tree)

A spec-lemma