Proof step 1 from proof writersok-rm-writer (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  noPending(t, bq, pq), oneMember(t, l, writers), writersOk(bq, pq, writers), ¬ writersOk(bq, pq, writers(l := writers(l) -- t)) ⊦

Rule argument

Additional information

Proof step 2 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  bq(t) = [], oneMember(t, l, writers), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []), ∀ l. pq(l) | t = [] ⊦ (t0 ∈ (l = l0 ⊃ writers(l0) -- t;writers(l0)) → l1 ≠ l0 → ¬ t0 ∈ (l = l1 ⊃ writers(l1) -- t;writers(l1))) ∧ (¬ t0 ∈ (l = l0 ⊃ writers(l0) -- t;writers(l0)) → pq(l0) |W | t0 = [] ∧ bq(t0) |WL l0 = [])

Rule argument

Additional information

Used simplifier rules: noPending: ⊦ noPending(t, bq, p) ↔ (∀ l. p(l) | t = [] ∧ bq(t) = []); 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 = []));

Proof step 3 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 4:

  bq(t) = [], l1 ≠ l0, oneMember(t, l, writers), t0 ∈ (l = l1 ⊃ writers(l1) -- t;writers(l1)), t0 ∈ (l = l0 ⊃ writers(l0) -- t;writers(l0)), ∀ l. pq(l) | t = [], ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []) ⊦

• Proof step 20:

  bq(t) = [], oneMember(t, l, writers), ¬ t0 ∈ (l = l0 ⊃ writers(l0) -- t;writers(l0)), ∀ l. pq(l) | t = [], ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []) ⊦ pq(l0) |W | t0 = [] ∧ bq(t0) |WL l0 = []

Rule argument

Additional information

Proof step 4 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  bq(t) = [], # (l = l1 ⊃ writers(l1) -- t;writers(l1)) ≠ 0, # (l = l0 ⊃ writers(l0) -- t;writers(l0)) ≠ 0, (l = l1 ⊃ writers(l1) -- t;writers(l1)) ≠ ∅, (l = l0 ⊃ writers(l0) -- t;writers(l0)) ≠ ∅, l1 ≠ l0, oneMember(t, l, writers), t0 ∈ (l = l1 ⊃ writers(l1) -- t;writers(l1)), t0 ∈ (l = l0 ⊃ writers(l0) -- t;writers(l0)), ∀ l. pq(l) | t = [], ∀ 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-rm-writer (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 6:

  bq(t) = [], l = l1, #(writers(l1) -- t) ≠ 0, # (l = l0 ⊃ writers(l0) -- t;writers(l0)) ≠ 0, (l = l1 ⊃ writers(l1) -- t;writers(l1)) ≠ ∅, (l = l0 ⊃ writers(l0) -- t;writers(l0)) ≠ ∅, l1 ≠ l0, oneMember(t, l, writers), t0 ∈ (l = l0 ⊃ writers(l0) -- t;writers(l0)), t0 ∈ (l = l1 ⊃ writers(l1) -- t;writers(l1)), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []), ∀ l. pq(l) | t = [] ⊦

• Proof step 10:

  bq(t) = [], # writers(l1) ≠ 0, # (l = l0 ⊃ writers(l0) -- t;writers(l0)) ≠ 0, (l = l1 ⊃ writers(l1) -- t;writers(l1)) ≠ ∅, (l = l0 ⊃ writers(l0) -- t;writers(l0)) ≠ ∅, l ≠ l1, l1 ≠ l0, oneMember(t, l, writers), t0 ∈ (l = l0 ⊃ writers(l0) -- t;writers(l0)), t0 ∈ (l = l1 ⊃ writers(l1) -- t;writers(l1)), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []), ∀ l. pq(l) | t = [] ⊦

Rule argument

Additional information

Proof step 6 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  bq(t) = [], # writers(l1) ≠ 0, # writers(l0) ≠ 0, #(writers(l1) -- t) ≠ 0, l1 ≠ l0, t0 ≠ t, writers(l0) ≠ ∅, writers(l1) ≠ ∅ ++ t, writers(l1) ≠ ∅, oneMember(t, l1, writers), t0 ∈ writers(l0), t0 ∈ writers(l1), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []), ∀ l. pq(l) | t = [] ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ s -- a = ∅ ↔ s = ∅ ∨ s = ∅ ++ a; (s) size-zero: ⊦ # s ≠ 0 → s ≠ ∅; (forward) from specification set-basic size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic ne-01: ⊦ a ∈ s → s ≠ ∅; (forward) from specification set-basic ⊦ a ∈ s -- a0 ↔ a ≠ a0 ∧ a ∈ s; (s)

Proof step 7 from proof writersok-rm-writer (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 8:

  bq(t) = [], # writers(l1) ≠ 0, # writers(l0) ≠ 0, #(writers(l1) -- t) ≠ 0, l1 ≠ l0, t0 ≠ t, writers(l0) ≠ ∅, writers(l1) ≠ ∅ ++ t, writers(l1) ≠ ∅, oneMember(t, l1, writers), t0 ∈ writers(l1), t0 ∈ writers(l0), t0 ∈ writers(l1) → (∀ l0. l0 ≠ l1 → ¬ t0 ∈ writers(l0)), ¬ t0 ∈ writers(l1) → pq(l1) |W | t0 = [] ∧ bq(t0) |WL l1 = [], ∀ l. pq(l) | t = [], ∀ 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: l1, t0

Proof step 8 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 9:

  bq(t) = [], # writers(l1) ≠ 0, # writers(l0) ≠ 0, #(writers(l1) -- t) ≠ 0, l1 ≠ l0, t0 ≠ t, writers(l0) ≠ ∅, writers(l1) ≠ ∅ ++ t, writers(l1) ≠ ∅, oneMember(t, l1, writers), t0 ∈ writers(l0), t0 ∈ writers(l1), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []), ∀ l. pq(l) | t = [], ∀ l0. l0 ≠ l1 → ¬ t0 ∈ writers(l0) ⊦

Rule argument

Additional information

Proof step 9 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l0

Proof step 10 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

  bq(t) = [], # writers(l1) ≠ 0, # (l = l0 ⊃ writers(l0) -- t;writers(l0)) ≠ 0, (l = l0 ⊃ writers(l0) -- t;writers(l0)) ≠ ∅, l ≠ l1, l1 ≠ l0, writers(l1) ≠ ∅, oneMember(t, l, writers), t0 ∈ (l = l0 ⊃ writers(l0) -- t;writers(l0)), t0 ∈ writers(l1), ∀ l. pq(l) | t = [], ∀ 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 11 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 12:

  bq(t) = [], l = l0, #(writers(l0) -- t) ≠ 0, # writers(l1) ≠ 0, (l = l0 ⊃ writers(l0) -- t;writers(l0)) ≠ ∅, l ≠ l1, l1 ≠ l0, writers(l1) ≠ ∅, oneMember(t, l, writers), t0 ∈ writers(l1), t0 ∈ (l = l0 ⊃ writers(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 = []), ∀ l. pq(l) | t = [] ⊦

• Proof step 16:

  bq(t) = [], # writers(l0) ≠ 0, # writers(l1) ≠ 0, (l = l0 ⊃ writers(l0) -- t;writers(l0)) ≠ ∅, l ≠ l0, l ≠ l1, l1 ≠ l0, writers(l1) ≠ ∅, oneMember(t, l, writers), t0 ∈ writers(l1), t0 ∈ (l = l0 ⊃ writers(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 = []), ∀ l. pq(l) | t = [] ⊦

Rule argument

Additional information

Proof step 12 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 13:

  bq(t) = [], # writers(l0) ≠ 0, # writers(l1) ≠ 0, #(writers(l0) -- t) ≠ 0, l1 ≠ l0, t0 ≠ t, writers(l0) ≠ ∅ ++ t, writers(l0) ≠ ∅, writers(l1) ≠ ∅, oneMember(t, l0, writers), t0 ∈ writers(l0), t0 ∈ writers(l1), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []), ∀ l. pq(l) | t = [] ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ s -- a = ∅ ↔ s = ∅ ∨ s = ∅ ++ a; (s) size-zero: ⊦ # s ≠ 0 → s ≠ ∅; (forward) from specification set-basic size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic ne-01: ⊦ a ∈ s → s ≠ ∅; (forward) from specification set-basic ⊦ a ∈ s -- a0 ↔ a ≠ a0 ∧ a ∈ s; (s)

Proof step 13 from proof writersok-rm-writer (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 14:

  bq(t) = [], # writers(l0) ≠ 0, # writers(l1) ≠ 0, #(writers(l0) -- t) ≠ 0, l1 ≠ l0, t0 ≠ t, writers(l0) ≠ ∅ ++ t, writers(l0) ≠ ∅, writers(l1) ≠ ∅, oneMember(t, l0, writers), t0 ∈ writers(l1), t0 ∈ writers(l0), t0 ∈ writers(l1) → (∀ l0. l0 ≠ l1 → ¬ t0 ∈ writers(l0)), ¬ t0 ∈ writers(l1) → pq(l1) |W | t0 = [] ∧ bq(t0) |WL l1 = [], ∀ l. pq(l) | t = [], ∀ 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: l1, t0

Proof step 14 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 15:

  bq(t) = [], # writers(l0) ≠ 0, # writers(l1) ≠ 0, #(writers(l0) -- t) ≠ 0, l1 ≠ l0, t0 ≠ t, writers(l0) ≠ ∅ ++ t, writers(l0) ≠ ∅, writers(l1) ≠ ∅, oneMember(t, l0, writers), t0 ∈ writers(l0), t0 ∈ writers(l1), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []), ∀ l. pq(l) | t = [], ∀ l0. l0 ≠ l1 → ¬ t0 ∈ writers(l0) ⊦

Rule argument

Additional information

Proof step 15 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l0

Proof step 16 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 17:

  bq(t) = [], # writers(l0) ≠ 0, # writers(l1) ≠ 0, l ≠ l0, l ≠ l1, l1 ≠ l0, writers(l1) ≠ ∅, writers(l0) ≠ ∅, oneMember(t, l, writers), t0 ∈ writers(l1), t0 ∈ writers(l0), ∀ l. pq(l) | t = [], ∀ 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 17 from proof writersok-rm-writer (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 18:

  bq(t) = [], # writers(l0) ≠ 0, # writers(l1) ≠ 0, l ≠ l0, l ≠ l1, l1 ≠ l0, writers(l1) ≠ ∅, writers(l0) ≠ ∅, oneMember(t, l, writers), t0 ∈ writers(l0), t0 ∈ writers(l1), t0 ∈ writers(l0) → (∀ l1. l1 ≠ l0 → ¬ t0 ∈ writers(l1)), ¬ t0 ∈ writers(l0) → pq(l0) |W | t0 = [] ∧ bq(t0) |WL l0 = [], ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []), ∀ l. pq(l) | t = [] ⊦

Rule argument

Additional information

Used terms: l0, t0

Proof step 18 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 19:

  bq(t) = [], # writers(l0) ≠ 0, # writers(l1) ≠ 0, l ≠ l0, l ≠ l1, l1 ≠ l0, writers(l1) ≠ ∅, writers(l0) ≠ ∅, oneMember(t, l, writers), t0 ∈ writers(l1), t0 ∈ writers(l0), ∀ l. pq(l) | t = [], ∀ 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 → ¬ t0 ∈ writers(l1) ⊦

Rule argument

Additional information

Proof step 19 from proof writersok-rm-writer (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 20 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 21:

  bq(t) = [], l = l0, oneMember(t, l, writers), ¬ t0 ∈ writers(l0) -- t, ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []), ∀ l. pq(l) | t = [] ⊦ pq(l0) |W | t0 = [] ∧ bq(t0) |WL l0 = []

• Proof step 27:

  bq(t) = [], l ≠ l0, oneMember(t, l, writers), ¬ t0 ∈ 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. pq(l) | t = [] ⊦ pq(l0) |W | t0 = [] ∧ bq(t0) |WL l0 = []

Rule argument

Additional information

Proof step 21 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 22:

  bq(t) = [], oneMember(t, l0, writers), ∀ l. pq(l) | t = [], ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []) ⊦ pq(l0) |W | t0 = [] ∧ bq(t0) |WL l0 = [], t0 ≠ t ∧ t0 ∈ writers(l0)

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ s -- a0 ↔ a ≠ a0 ∧ a ∈ s; (s)

Proof step 22 from proof writersok-rm-writer (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 23:

  bq(t) = [], oneMember(t, l0, writers), t0 ∈ writers(l0) → (∀ l1. l1 ≠ l0 → ¬ t0 ∈ writers(l1)), ¬ t0 ∈ writers(l0) → pq(l0) |W | t0 = [] ∧ bq(t0) |WL l0 = [], ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []), ∀ l. pq(l) | t = [] ⊦ t0 ≠ t ∧ t0 ∈ writers(l0), pq(l0) |W | t0 = [] ∧ bq(t0) |WL l0 = []

Rule argument

Additional information

Used terms: l0, t0

Proof step 23 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 24:

  bq(t) = [], # writers(l0) ≠ 0, writers(l0) ≠ ∅, pq(l0) |W | t ≠ [], oneMember(t, l0, writers), ∀ l. pq(l) | t = [], ∀ 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: restrict-l-empty: ⊦ [] |WL l = []; size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic ⊦ oneMember(t, l, tsf) → (t ∈ tsf(l0) ↔ l = l0); (s) ne-01: ⊦ a ∈ s → s ≠ ∅; (forward) from specification set-basic

Proof step 24 from proof writersok-rm-writer (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 25:

  bq(t) = [], pq(l0) | t = [], # writers(l0) ≠ 0, writers(l0) ≠ ∅, pq(l0) |W | t ≠ [], oneMember(t, l0, writers), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → pq(l) |W | t = [] ∧ bq(t) |WL l = []), ∀ l. pq(l) | t = [] ⊦

Rule argument

Additional information

Used terms: l0

Proof step 25 from proof writersok-rm-writer (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 26:

  ⊦ fowl |W | t = (fowl | t)|W

Rule argument

Additional information

Used simplifier rules: ⊦ [] |W = []; (s) Used terms: pq(l0), t

Proof step 26 from proof writersok-rm-writer (previous step) (back to tree)

A spec-lemma

Proof step 27 from proof writersok-rm-writer (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l0, t0