Proof step 1 from proof relyp-no-event-added-by-other (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  fow.tid = t, relyp(t, p0, p1), fow ∈ p1(l), ¬ fow ∈ p0(l) ⊦

Rule argument

Additional information

Proof step 2 from proof relyp-no-event-added-by-other (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  relyp(fow.tid, p0, p1), fow ∈ p1(l), ¬ fow ∈ p0(l) ⊦

Rule argument

Additional information

Proof step 3 from proof relyp-no-event-added-by-other (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 4:

  ⊦ relyp(t, p, p0) ↔ (∀ l. relyp(t, p(l), p0(l)))

• Proof step 5:

  fow ∈ p1(l), ¬ fow ∈ p0(l), ∀ l. relyp(fow.tid, p0(l), p1(l)) ⊦

Rule argument

Additional information

Used terms: fow.tid, p0, p1

Proof step 4 from proof relyp-no-event-added-by-other (previous step) (back to tree)

Lemma allrelyp-def

Proof step 5 from proof relyp-no-event-added-by-other (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 6:

  relyp(fow.tid, p0(l), p1(l)), fow ∈ p1(l), ¬ fow ∈ p0(l) ⊦

Rule argument

Additional information

Used terms: l

Proof step 6 from proof relyp-no-event-added-by-other (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 7:

  ⊦ relyp(t, fowl, fowl0) ↔ (∃ fowl1, fowl2. fowl ⊒ fowl1 ∧ fowl0 = fowl1 + fowl2 ∧ fowl2 | t = [])

• Proof step 8:

  fow ∈ p1(l), ¬ fow ∈ p0(l), ∃ fowl1, fowl2. p0(l) ⊒ fowl1 ∧ p1(l) = fowl1 + fowl2 ∧ fowl2 | fow.tid = [] ⊦

Rule argument

Additional information

Used terms: fow.tid, p0(l), p1(l)

Proof step 7 from proof relyp-no-event-added-by-other (previous step) (back to tree)

Lemma relyp-def

Proof step 8 from proof relyp-no-event-added-by-other (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 9:

  fowl2.filter(λ fow0. fow0.tid = fow.tid) = [], p1(l) = fowl1 + fowl2, fow ∈ fowl2, p0(l) ⊒ fowl1, ¬ fow ∈ p0(l), ¬ fow ∈ fowl1 ⊦

Rule argument

Additional information

Used simplifier rules: filtertid-def: ⊦ fowl | t = fowl.filter(λ fow. fow.tid = t); ⊦ a ∈ x ∧ ¬ a ∈ y → ¬ y ⊒ x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s)

Proof step 9 from proof relyp-no-event-added-by-other (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 10:

  ⊦ x = [] ↔ ¬(∃ a. a ∈ x)

• Proof step 11:

  p1(l) = fowl1 + fowl2, fow ∈ fowl2, p0(l) ⊒ fowl1, ¬ fow ∈ p0(l), ¬ fow ∈ fowl1, ∀ a. ¬ a ∈ fowl2.filter(λ fow0. fow0.tid = fow.tid) ⊦

Rule argument

Additional information

Used terms: fowl2.filter(λ fow0. fow0.tid = fow.tid)

Proof step 10 from proof relyp-no-event-added-by-other (previous step) (back to tree)

A spec-lemma

Proof step 11 from proof relyp-no-event-added-by-other (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

  p1(l) = fowl1 + fowl2, fow ∈ fowl2, p0(l) ⊒ fowl1, ¬ fow ∈ p0(l), ¬ fow ∈ fowl1, ∀ a. ¬ a ∈ fowl2 ∨ a.tid ≠ fow.tid ⊦

Rule argument

Additional information

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

Proof step 12 from proof relyp-no-event-added-by-other (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: fow