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

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  p(l) = fow ' + fowl, writersOk(p, writers), ¬ writersOk(p(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:

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

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 = [])); 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:

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

• Proof step 8:

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

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:

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

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:

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

Rule argument

Additional information

Used terms: l1, 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:

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

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: l0

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:

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

• Proof step 33:

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

Rule argument

Additional information

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

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

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

User applied rule all left

And obtained the following new goals

• Proof step 11:

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

Rule argument

Additional information

Used terms: l0, t

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

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

Rule argument

Additional information

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 13:

  ⊦ x = [] ↔ ¬(∃ a. a ∈ x)

• Proof step 14:

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

Rule argument

Additional information

Used terms: p(l0) |W | t

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

A spec-lemma

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:

  ⊦ x = [] ↔ ¬(∃ a. a ∈ x)

• Proof step 16:

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

Rule argument

Additional information

Used terms: fowl |W | t

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 simplifier applied rule simplifier

And obtained the following new goals

• Proof step 17:

  p(l0) = fow ' + fowl, fow0 ∈ fowl |W | t, fow ∈ p(l0), ¬ t ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []), ∀ a. ¬ a ∈ p(l0) |W | t ⊦

Rule argument

Additional information

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

User applied rule all left

And obtained the following new goals

• Proof step 18:

  p(l0) = fow ' + fowl, fow ∈ p(l0), fow0 ∈ fowl |W | t, ¬ fow0 ∈ p(l0) |W | t, ¬ t ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []) ⊦

Rule argument

Additional information

Used terms: fow0

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 19:

  ⊦ fowl |W = fowl.filter(λ fow. fow.W?)

• Proof step 20:

  p(l0) = fow ' + fowl, fow0 ∈ fowl |W | t, fow ∈ p(l0), ¬ fow0 ∈ p(l0).filter(λ fow. fow.W?) | t, ¬ t ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []) ⊦

Rule argument

Additional information

Used terms: p(l0)

Proof step 19 from proof writersok-persist (previous step) (back to tree)

Lemma filterwrites-def

Proof step 20 from proof writersok-persist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 21:

  p(l0) = fow ' + fowl, fow ∈ p(l0), fow0 ∈ fowl |W | t, ¬ t ∈ writers(l0), ¬ fow0 ∈ p(l0).filter(.W?) | t, ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []) ⊦

Rule argument

Additional information

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 22:

  ⊦ fowl | t = fowl.filter(λ fow. fow.tid = t)

• Proof step 23:

  p(l0) = fow ' + fowl, fow0 ∈ fowl |W | t, fow ∈ p(l0), ¬ t ∈ writers(l0), ¬ fow0 ∈ p(l0).filter(.W?).filter(λ fow. fow.tid = t), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []) ⊦

Rule argument

Additional information

Used terms: p(l0).filter(.W?), t

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

Lemma filtertid-def

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 24:

  p(l0) = fow ' + fowl, fow ∈ p(l0), fow0 ∈ fowl |W | t, ¬ t ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []) ⊦ fow0 ∈ p(l0) ∧ fow0.W? ∧ fow0.tid = t

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ x.filter(p) ↔ a ∈ x ∧ p(a); (s)

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 25:

  ⊦ fowl | t = fowl.filter(λ fow. fow.tid = t)

• Proof step 26:

  p(l0) = fow ' + fowl, fow0 ∈ fowl |W.filter(λ fow. fow.tid = t), fow ∈ p(l0), ¬ t ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []) ⊦ fow0 ∈ p(l0) ∧ fow0.W? ∧ fow0.tid = t

Rule argument

Additional information

Used terms: fowl |W, t

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

Lemma filtertid-def

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 27:

  p(l0) = fow ' + fowl, fow0 ∈ fowl |W, fow ∈ p(l0), ¬ fow0.tid ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []) ⊦ fow0 ∈ p(l0) ∧ fow0.W?

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ x.filter(p) ↔ a ∈ x ∧ p(a); (s)

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 28:

  ⊦ fowl |W = fowl.filter(λ fow. fow.W?)

• Proof step 29:

  p(l0) = fow ' + fowl, fow ∈ p(l0), fow0 ∈ fowl.filter(λ fow. fow.W?), ¬ fow0.tid ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []) ⊦ fow0 ∈ p(l0) ∧ fow0.W?

Rule argument

Additional information

Used terms: fowl

Proof step 28 from proof writersok-persist (previous step) (back to tree)

Lemma filterwrites-def

Proof step 29 from proof writersok-persist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 30:

  p(l0) = fow ' + fowl, fow0.W?, fow ∈ p(l0), fow0 ∈ fowl, ¬ fow0.tid ∈ writers(l0), ¬ fow0 ∈ p(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ x.filter(p) ↔ a ∈ x ∧ p(a); (s)

Proof step 30 from proof writersok-persist (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 31:

  fow0 = W(v, t0), p(l0) = fow ' + fowl, fow0.W?, fow0 ∈ fowl, fow ∈ p(l0), ¬ t0 ∈ writers(l0), ¬ fow0 ∈ p(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ fow.W? → (val = fow.val ∧ tid = fow.tid ↔ fow = W(val, tid)); (elim) Used terms: fow0, v, t0

Proof step 31 from proof writersok-persist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 32:

  p(l0) = fow ' + fowl, fow ∈ p(l0), W(v, t0) ∈ fowl, ¬ t0 ∈ writers(l0), ¬ W(v, t0) ∈ p(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ W(v, tid).W?; (s)

Proof step 32 from proof writersok-persist (previous step) (back to tree)

User applied rule subst equation

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ∈ y → a ∈ x + y; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 33 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