Proof step 1 from proof writersok-add-fo (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  writersOk(p, writers), ¬ writersOk(p(l := p(l) + FO(t) '), writers) ⊦

Rule argument

Additional information

Proof step 2 from proof writersok-add-fo (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  ∀ l0, t0. (t0 ∈ writers(l0) → (∀ l1. l1 ≠ l0 → ¬ t0 ∈ writers(l1))) ∧ (¬ t0 ∈ writers(l0) → p(l0).filter(.W?).filter(λ fow. fow.tid = t0) = []) ⊦ (t0 ∈ writers(l0) → l1 ≠ l0 → ¬ t0 ∈ writers(l1)) ∧ (¬ t0 ∈ writers(l0) → (l = l0 ⊃ p(l0) + FO(t) ';p(l0)).filter(.W?).filter(λ fow. fow.tid = t0) = [])

Rule argument

Additional information

Used simplifier rules: writersOk: ⊦ writersOk(p, writers) ↔ (∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = [])); filterwrites-def: ⊦ fowl |W = fowl.filter(λ fow. fow.W?); filtertid-def: ⊦ fowl | t = fowl.filter(λ fow. fow.tid = t);

Proof step 3 from proof writersok-add-fo (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 4:

  l1 ≠ l0, t0 ∈ writers(l1), t0 ∈ writers(l0), ∀ l0, t0. (t0 ∈ writers(l0) → (∀ l1. l1 ≠ l0 → ¬ t0 ∈ writers(l1))) ∧ (¬ t0 ∈ writers(l0) → p(l0).filter(.W?).filter(λ fow. fow.tid = t0) = []) ⊦

• Proof step 8:

  (l = l0 ⊃ p(l0) + FO(t) ';p(l0)).filter(.W?).filter(λ fow. fow.tid = t0) ≠ [], ¬ t0 ∈ writers(l0), ∀ l0, t0. (t0 ∈ writers(l0) → (∀ l1. l1 ≠ l0 → ¬ t0 ∈ writers(l1))) ∧ (¬ t0 ∈ writers(l0) → p(l0).filter(.W?).filter(λ fow. fow.tid = t0) = []) ⊦

Rule argument

Additional information

Proof step 4 from proof writersok-add-fo (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  # writers(l1) ≠ 0, # writers(l0) ≠ 0, l1 ≠ l0, writers(l1) ≠ ∅, writers(l0) ≠ ∅, t0 ∈ writers(l1), t0 ∈ writers(l0), ∀ l0, t0. (t0 ∈ writers(l0) → (∀ l1. l1 ≠ l0 → ¬ t0 ∈ writers(l1))) ∧ (¬ t0 ∈ writers(l0) → p(l0).filter(.W?).filter(λ fow. fow.tid = t0) = []) ⊦

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-add-fo (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 6:

  # writers(l1) ≠ 0, # writers(l0) ≠ 0, l1 ≠ l0, writers(l1) ≠ ∅, writers(l0) ≠ ∅, t0 ∈ writers(l0), t0 ∈ writers(l1), t0 ∈ writers(l0) → (∀ l1. l1 ≠ l0 → ¬ t0 ∈ writers(l1)), ¬ t0 ∈ writers(l0) → p(l0).filter(.W?).filter(λ fow. fow.tid = t0) = [], ∀ l0, t0. (t0 ∈ writers(l0) → (∀ l1. l1 ≠ l0 → ¬ t0 ∈ writers(l1))) ∧ (¬ t0 ∈ writers(l0) → p(l0).filter(.W?).filter(λ fow. fow.tid = t0) = []) ⊦

Rule argument

Additional information

Used terms: l0, t0

Proof step 6 from proof writersok-add-fo (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  # writers(l1) ≠ 0, # writers(l0) ≠ 0, l1 ≠ l0, writers(l1) ≠ ∅, writers(l0) ≠ ∅, t0 ∈ writers(l1), t0 ∈ writers(l0), ∀ l0, t0. (t0 ∈ writers(l0) → (∀ l1. l1 ≠ l0 → ¬ t0 ∈ writers(l1))) ∧ (¬ t0 ∈ writers(l0) → p(l0).filter(.W?).filter(λ fow. fow.tid = t0) = []), ∀ l1. l1 ≠ l0 → ¬ t0 ∈ writers(l1) ⊦

Rule argument

Additional information

Proof step 7 from proof writersok-add-fo (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-add-fo (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 9:

  l = l0, (p(l0) + FO(t) ').filter(.W?).filter(λ fow. fow.tid = t0) ≠ [], ¬ t0 ∈ writers(l0), ∀ l0, t0. (t0 ∈ writers(l0) → (∀ l1. l1 ≠ l0 → ¬ t0 ∈ writers(l1))) ∧ (¬ t0 ∈ writers(l0) → p(l0).filter(.W?).filter(λ fow. fow.tid = t0) = []) ⊦

• Proof step 11:

  p(l0).filter(.W?).filter(λ fow. fow.tid = t0) ≠ [], l ≠ l0, ¬ t0 ∈ writers(l0), ∀ l0, t0. (t0 ∈ writers(l0) → (∀ l1. l1 ≠ l0 → ¬ t0 ∈ writers(l1))) ∧ (¬ t0 ∈ writers(l0) → p(l0).filter(.W?).filter(λ fow. fow.tid = t0) = []) ⊦

Rule argument

Additional information

Proof step 9 from proof writersok-add-fo (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  p(l0).filter(.W?).filter(λ fow. fow.tid = t0) ≠ [], ¬ t0 ∈ writers(l0), ∀ l0, t0. (t0 ∈ writers(l0) → (∀ l1. l1 ≠ l0 → ¬ t0 ∈ writers(l1))) ∧ (¬ t0 ∈ writers(l0) → p(l0).filter(.W?).filter(λ fow. fow.tid = t0) = []) ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + [] = x; (s) ⊦ ¬ FO(tid).W?; (s) ⊦ a '.filter(p) = (p(a) ⊃ a ';[]); (s) ⊦ (x + y).filter(p) = x.filter(p) + y.filter(p); (s)

Proof step 10 from proof writersok-add-fo (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l0, t0

Proof step 11 from proof writersok-add-fo (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l0, t0