Proof step 1 from proof writersOk-persist (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  pq(l) = fow ' + fowl, writersOk(bq, pq, writers), ¬ writersOk(bq, pq(l := fowl), writers) ⊦

Rule argument

Additional information

Proof step 2 from proof writersOk-persist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  pq(l) = fow ' + fowl, fow ∈ pq(l), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []) ⊦ (t ∈ writers(l0) → l1 ≠ l0 → ¬ t ∈ writers(l1)) ∧ (¬ t ∈ writers(l0) → (l = l0 ⊃ fowl;pq(l0)) |W | t = [] ∧ bq(t) |WL l0 = [])

Rule argument

Additional information

Used simplifier rules: writersOk: ⊦ writersOk(bq, p, writers) ↔ (∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = [] ∧ bq(t) |WL l = [])); a: ⊦ (x + y) + z = x + y + z; from specification list in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list

Proof step 3 from proof writersOk-persist (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 4:

  pq(l) = fow ' + fowl, l1 ≠ l0, t ∈ writers(l1), t ∈ writers(l0), fow ∈ pq(l), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []) ⊦

• Proof step 8:

  pq(l) = fow ' + fowl, fow ∈ pq(l), ¬ t ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []) ⊦ (l = l0 ⊃ fowl;pq(l0)) |W | t = [] ∧ bq(t) |WL l0 = []

Rule argument

Additional information

Proof step 4 from proof writersOk-persist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  pq(l) = fow ' + fowl, # writers(l1) ≠ 0, # writers(l0) ≠ 0, l1 ≠ l0, writers(l1) ≠ ∅, writers(l0) ≠ ∅, t ∈ writers(l1), t ∈ writers(l0), fow ∈ pq(l), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []) ⊦

Rule argument

Additional information

Used simplifier rules: size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic ne-01: ⊦ a ∈ s → s ≠ ∅; (forward) from specification set-basic

Proof step 5 from proof writersOk-persist (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 6:

  pq(l) = fow ' + fowl, # writers(l1) ≠ 0, # writers(l0) ≠ 0, l1 ≠ l0, writers(l1) ≠ ∅, writers(l0) ≠ ∅, fow ∈ pq(l), t ∈ writers(l0), t ∈ writers(l1), t ∈ writers(l0) → (∀ l1. l1 ≠ l0 → ¬ t ∈ writers(l1)), ¬ t ∈ writers(l0) → pq(l0) |W | t = [] ∧ bq(t) |WL l0 = [], ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []) ⊦

Rule argument

Additional information

Used terms: l0, t

Proof step 6 from proof writersOk-persist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  pq(l) = fow ' + fowl, # writers(l1) ≠ 0, # writers(l0) ≠ 0, l1 ≠ l0, writers(l1) ≠ ∅, writers(l0) ≠ ∅, t ∈ writers(l1), t ∈ writers(l0), fow ∈ pq(l), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []), ∀ l1. l1 ≠ l0 → ¬ t ∈ writers(l1) ⊦

Rule argument

Additional information

Proof step 7 from proof writersOk-persist (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l1

Proof step 8 from proof writersOk-persist (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 9:

  pq(l) = fow ' + fowl, (l = l0 ⊃ fowl;pq(l0)) |W | t ≠ [], fow ∈ pq(l), ¬ t ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []) ⊦

• Proof step 17:

  pq(l) = fow ' + fowl, bq(t) |WL l0 ≠ [], fow ∈ pq(l), ¬ t ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []) ⊦

Rule argument

Additional information

Proof step 9 from proof writersOk-persist (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 10:

  l = l0, pq(l) = fow ' + fowl, fowl |W | t ≠ [], fow ∈ pq(l), ¬ t ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []) ⊦

• Proof step 16:

  pq(l) = fow ' + fowl, l ≠ l0, pq(l0) |W | t ≠ [], fow ∈ pq(l), ¬ t ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []) ⊦

Rule argument

Additional information

Proof step 10 from proof writersOk-persist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

  pq(l0) = fow ' + fowl, fowl |W | t ≠ [], fow ∈ pq(l0), ¬ t ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []) ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list

Proof step 11 from proof writersOk-persist (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 12:

  pq(l0) = fow ' + fowl, fowl |W | t ≠ [], fow ∈ pq(l0), ¬ t ∈ writers(l0), t ∈ writers(l0) → (∀ l1. l1 ≠ l0 → ¬ t ∈ writers(l1)), ¬ t ∈ writers(l0) → pq(l0) |W | t = [] ∧ bq(t) |WL l0 = [], ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []) ⊦

Rule argument

Additional information

Used terms: l0, t

Proof step 12 from proof writersOk-persist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 13:

  pq(l0) = fow ' + fowl, pq(l0) |W | t = [], bq(t) |WL l0 = [], fowl |W | t ≠ [], fow ∈ pq(l0), ¬ t ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []) ⊦

Rule argument

Additional information

Proof step 13 from proof writersOk-persist (previous step) (back to tree)

User applied rule subst equation

And obtained the following new goals

• Proof step 14:

  pq(l0) = fow ' + fowl, (fow ' + fowl)|W | t = [], bq(t) |WL l0 = [], fowl |W | t ≠ [], fow ∈ fow ' + fowl, ¬ t ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []) ⊦

Rule argument

Additional information

Proof step 14 from proof writersOk-persist (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 15:

  ⊦ (fowl + fowl0)|W = fowl |W + fowl0 |W

Rule argument

Additional information

Used simplifier rules: ⊦ fowl + fowl0 | t = (fowl | t) + (fowl0 | t); (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: fow ', fowl

Proof step 15 from proof writersOk-persist (previous step) (back to tree)

A spec-lemma

Proof step 16 from proof writersOk-persist (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l0, t

Proof step 17 from proof writersOk-persist (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l0, t