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, p), oneMember(t, l, writers), writersOk(p, writers), ¬ writersOk(p, 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:

  ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []), ∀ l0. t ∈ writers(l0) ↔ l0 = l, ∀ l. p(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)) → p(l0) |W | t0 = [])

Rule argument

Additional information

Used simplifier rules: oneWriter: ⊦ oneMember(t, l, tsf) ↔ (∀ l0. t ∈ tsf(l0) ↔ l0 = l); noPending: ⊦ noPending(t, p) ↔ (∀ l. p(l) | t = []); writersOk: ⊦ writersOk(p, writers) ↔ (∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []));

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:

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

• Proof step 20:

  p(l0) |W | t0 ≠ [], ¬ t0 ∈ (l = l0 ⊃ writers(l0) -- t;writers(l0)), ∀ l. p(l) | t = [], ∀ l0. t ∈ writers(l0) ↔ l0 = l, ∀ 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-rm-writer (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  # (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, t0 ∈ (l = l1 ⊃ writers(l1) -- t;writers(l1)), t0 ∈ (l = l0 ⊃ writers(l0) -- t;writers(l0)), ∀ l. p(l) | t = [], ∀ l0. t ∈ writers(l0) ↔ l0 = 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-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:

  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, 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) → p(l) |W | t = []), ∀ l0. t ∈ writers(l0) ↔ l0 = l, ∀ l. p(l) | t = [] ⊦

• Proof step 10:

  # 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, 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) → p(l) |W | t = []), ∀ l0. t ∈ writers(l0) ↔ l0 = l, ∀ l. p(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:

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

  # writers(l1) ≠ 0, # writers(l0) ≠ 0, #(writers(l1) -- t) ≠ 0, l1 ≠ l0, t0 ≠ t, writers(l0) ≠ ∅, writers(l1) ≠ ∅ ++ t, writers(l1) ≠ ∅, t0 ∈ writers(l1), t0 ∈ writers(l0), t0 ∈ writers(l1) → (∀ l0. l0 ≠ l1 → ¬ t0 ∈ writers(l0)), ¬ t0 ∈ writers(l1) → p(l1) |W | t0 = [], ∀ l. p(l) | t = [], ∀ l0. t ∈ writers(l0) ↔ l0 = l1, ∀ 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, 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:

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

  # writers(l1) ≠ 0, # (l = l0 ⊃ writers(l0) -- t;writers(l0)) ≠ 0, (l = l0 ⊃ writers(l0) -- t;writers(l0)) ≠ ∅, l ≠ l1, l1 ≠ l0, writers(l1) ≠ ∅, t0 ∈ (l = l0 ⊃ writers(l0) -- t;writers(l0)), t0 ∈ writers(l1), ∀ l. p(l) | t = [], ∀ l0. t ∈ writers(l0) ↔ l0 = 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 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:

  l = l0, #(writers(l0) -- t) ≠ 0, # writers(l1) ≠ 0, (l = l0 ⊃ writers(l0) -- t;writers(l0)) ≠ ∅, l ≠ l1, l1 ≠ l0, writers(l1) ≠ ∅, 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) → p(l) |W | t = []), ∀ l0. t ∈ writers(l0) ↔ l0 = l, ∀ l. p(l) | t = [] ⊦

• Proof step 16:

  # writers(l0) ≠ 0, # writers(l1) ≠ 0, (l = l0 ⊃ writers(l0) -- t;writers(l0)) ≠ ∅, l ≠ l0, l ≠ l1, l1 ≠ l0, writers(l1) ≠ ∅, 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) → p(l) |W | t = []), ∀ l0. t ∈ writers(l0) ↔ l0 = l, ∀ l. p(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:

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

  # writers(l0) ≠ 0, # writers(l1) ≠ 0, #(writers(l0) -- t) ≠ 0, l1 ≠ l0, t0 ≠ t, writers(l0) ≠ ∅ ++ t, writers(l0) ≠ ∅, writers(l1) ≠ ∅, t0 ∈ writers(l1), t0 ∈ writers(l0), t0 ∈ writers(l1) → (∀ l0. l0 ≠ l1 → ¬ t0 ∈ writers(l0)), ¬ t0 ∈ writers(l1) → p(l1) |W | t0 = [], ∀ l. p(l) | t = [], ∀ l1. t ∈ writers(l1) ↔ l1 = 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: 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:

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

  # writers(l0) ≠ 0, # writers(l1) ≠ 0, l ≠ l0, l ≠ l1, l1 ≠ l0, writers(l1) ≠ ∅, writers(l0) ≠ ∅, t0 ∈ writers(l1), t0 ∈ writers(l0), ∀ l. p(l) | t = [], ∀ l0. t ∈ writers(l0) ↔ l0 = 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 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:

  # writers(l0) ≠ 0, # writers(l1) ≠ 0, l ≠ l0, l ≠ l1, 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) |W | t0 = [], ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []), ∀ l0. t ∈ writers(l0) ↔ l0 = l, ∀ l. p(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:

  # writers(l0) ≠ 0, # writers(l1) ≠ 0, l ≠ l0, l ≠ l1, l1 ≠ l0, writers(l1) ≠ ∅, writers(l0) ≠ ∅, t0 ∈ writers(l1), t0 ∈ writers(l0), ∀ l. p(l) | t = [], ∀ l0. t ∈ writers(l0) ↔ l0 = l, ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []), ∀ 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:

  l = l0, p(l0) |W | t0 ≠ [], ¬ t0 ∈ writers(l0) -- t, ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []), ∀ l0. t ∈ writers(l0) ↔ l0 = l, ∀ l. p(l) | t = [] ⊦

• Proof step 40:

  l ≠ l0, p(l0) |W | t0 ≠ [], ¬ t0 ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []), ∀ l0. t ∈ writers(l0) ↔ l0 = l, ∀ l. p(l) | t = [] ⊦

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:

  p(l0) |W | t0 ≠ [], ∀ l. p(l) | t = [], ∀ l1. t ∈ writers(l1) ↔ l1 = l0, ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []) ⊦ 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 case distinction

And obtained the following new goals

• Proof step 23:

  t0 = t, p(l0) |W | t0 ≠ [], ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []), ∀ l1. t ∈ writers(l1) ↔ l1 = l0, ∀ l. p(l) | t = [] ⊦

• Proof step 39:

  p(l0) |W | t0 ≠ [], ¬ t0 ∈ writers(l0), ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []), ∀ l1. t ∈ writers(l1) ↔ l1 = l0, ∀ l. p(l) | t = [] ⊦

Rule argument

Additional information

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:

  p(l0) |W | t ≠ [], ∀ l. p(l) | t = [], ∀ l1. t ∈ writers(l1) ↔ l1 = 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 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:

  p(l0) | t = [], p(l0) |W | t ≠ [], ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []), ∀ l1. t ∈ writers(l1) ↔ l1 = l0 ⊦

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:

  ⊦ x = [] ↔ ¬(∃ a. a ∈ x)

• Proof step 27:

  p(l0) | t = [], ∀ l1. t ∈ writers(l1) ↔ l1 = 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 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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 28:

  p(l0) | t = [], fow ∈ p(l0) |W | t, ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []), ∀ l1. t ∈ writers(l1) ↔ l1 = l0 ⊦

Rule argument

Additional information

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 29:

  ⊦ x = [] ↔ ¬(∃ a. a ∈ x)

• Proof step 30:

  fow ∈ p(l0) |W | t, ∀ l1. t ∈ writers(l1) ↔ l1 = l0, ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []), ∀ a. ¬ a ∈ p(l0) | t ⊦

Rule argument

Additional information

Used terms: p(l0) | t

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

A spec-lemma

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

User applied rule all left

And obtained the following new goals

• Proof step 31:

  fow ∈ p(l0) |W | t, ¬ fow ∈ p(l0) | t, ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []), ∀ l1. t ∈ writers(l1) ↔ l1 = l0 ⊦

Rule argument

Additional information

Used terms: fow

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 32:

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

• Proof step 33:

  fow ∈ p(l0) |W | t, ¬ fow ∈ p(l0).filter(λ fow. fow.tid = t), ∀ l1. t ∈ writers(l1) ↔ l1 = 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), t

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

Lemma filtertid-def

Proof step 33 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 34:

  fow ∈ p(l0) |W | t, ∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = []), ∀ l1. t ∈ writers(l1) ↔ l1 = l0 ⊦ fow ∈ p(l0) ∧ fow.tid = t

Rule argument

Additional information

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

Proof step 34 from proof writersok-rm-writer (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 35:

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

• Proof step 36:

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

Rule argument

Additional information

Used terms: p(l0) |W, t

Proof step 35 from proof writersok-rm-writer (previous step) (back to tree)

Lemma filtertid-def

Proof step 36 from proof writersok-rm-writer (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 37:

  fow ∈ p(l0) |W, ¬ fow ∈ p(l0), ∀ l1. fow.tid ∈ writers(l1) ↔ l1 = 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 37 from proof writersok-rm-writer (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 38:

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

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ x.filter(p) ↔ a ∈ x ∧ p(a); (s) Used terms: p(l0)

Proof step 38 from proof writersok-rm-writer (previous step) (back to tree)

Lemma filterwrites-def

Proof step 39 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, t0

Proof step 40 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, t0