Proof step 1 from proof relyp-no-pending-w-stable (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  noPendingW(t, p0), relyp(t, p0, p1), ¬ noPendingW(t, p1) ⊦

Rule argument

Additional information

Proof step 2 from proof relyp-no-pending-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  p1(l).filter(.W?).filter(λ fow. fow.tid = t) ≠ [], relyp(t, p0, p1), ∀ l. p0(l).filter(.W?).filter(λ fow. fow.tid = t) = [] ⊦

Rule argument

Additional information

Used simplifier rules: noPendingW: ⊦ noPendingW(t, p) ↔ (∀ 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 relyp-no-pending-w-stable (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 4:

  p0(l).filter(.W?).filter(λ fow. fow.tid = t) = [], p1(l).filter(.W?).filter(λ fow. fow.tid = t) ≠ [], relyp(t, p0, p1) ⊦

Rule argument

Additional information

Used terms: l

Proof step 4 from proof relyp-no-pending-w-stable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 5:

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

• Proof step 6:

  p0(l).filter(.W?).filter(λ fow. fow.tid = t) = [], p1(l).filter(.W?).filter(λ fow. fow.tid = t) ≠ [], ∀ l. relyp(t, p0(l), p1(l)) ⊦

Rule argument

Additional information

Used terms: t, p0, p1

Proof step 5 from proof relyp-no-pending-w-stable (previous step) (back to tree)

Lemma allrelyp-def

Proof step 6 from proof relyp-no-pending-w-stable (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 7:

  p0(l).filter(.W?).filter(λ fow. fow.tid = t) = [], p1(l).filter(.W?).filter(λ fow. fow.tid = t) ≠ [], relyp(t, p0(l), p1(l)) ⊦

Rule argument

Additional information

Used terms: l

Proof step 7 from proof relyp-no-pending-w-stable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 8:

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

• Proof step 9:

  p0(l).filter(.W?).filter(λ fow. fow.tid = t) = [], p1(l).filter(.W?).filter(λ fow. fow.tid = t) ≠ [], ∃ fowl1, fowl2. p0(l) ⊒ fowl1 ∧ p1(l) = fowl1 + fowl2 ∧ fowl2 | t = [] ⊦

Rule argument

Additional information

Used terms: t, p0(l), p1(l)

Proof step 8 from proof relyp-no-pending-w-stable (previous step) (back to tree)

Lemma relyp-def

Proof step 9 from proof relyp-no-pending-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  fowl2.filter(λ fow. fow.tid = t) = [], p0(l).filter(.W?).filter(λ fow. fow.tid = t) = [], p1(l) = fowl1 + fowl2, p0(l) ⊒ fowl1 ⊦ fowl1.filter(.W?).filter(λ fow. fow.tid = t) = [] ∧ fowl2.filter(.W?).filter(λ fow. fow.tid = t) = []

Rule argument

Additional information

Used simplifier rules: filtertid-def: ⊦ fowl | t = fowl.filter(λ fow. fow.tid = t); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ (x + y).filter(p) = x.filter(p) + y.filter(p); (s)

Proof step 10 from proof relyp-no-pending-w-stable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 11:

  ⊦ x ⊒ y ↔ (∃ z. z + y = x)

• Proof step 12:

  p0(l).filter(.W?).filter(λ fow. fow.tid = t) = [], fowl2.filter(λ fow. fow.tid = t) = [], p1(l) = fowl1 + fowl2, ∃ z. z + fowl1 = p0(l) ⊦ fowl1.filter(.W?).filter(λ fow. fow.tid = t) = [] ∧ fowl2.filter(.W?).filter(λ fow. fow.tid = t) = []

Rule argument

Additional information

Used terms: p0(l), fowl1

Proof step 11 from proof relyp-no-pending-w-stable (previous step) (back to tree)

A spec-lemma

Proof step 12 from proof relyp-no-pending-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 13:

  fowl + fowl1 = p0(l), fowl2.filter(λ fow. fow.tid = t) = [], fowl.filter(.W?).filter(λ fow. fow.tid = t) = [], fowl1.filter(.W?).filter(λ fow. fow.tid = t) = [], p1(l) = fowl1 + fowl2, fowl2.filter(.W?).filter(λ fow. fow.tid = t) ≠ [] ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ (x + y).filter(p) = x.filter(p) + y.filter(p); (s)

Proof step 13 from proof relyp-no-pending-w-stable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 14:

  ⊦ x = [] ↔ ¬(∃ a. a ∈ x)

• Proof step 15:

  fowl + fowl1 = p0(l), fowl1.filter(.W?).filter(λ fow. fow.tid = t) = [], fowl.filter(.W?).filter(λ fow. fow.tid = t) = [], fowl2.filter(λ fow. fow.tid = t) = [], p1(l) = fowl1 + fowl2, ∃ a. a ∈ fowl2.filter(.W?).filter(λ fow. fow.tid = t) ⊦

Rule argument

Additional information

Used terms: fowl2.filter(.W?).filter(λ fow. fow.tid = t)

Proof step 14 from proof relyp-no-pending-w-stable (previous step) (back to tree)

A spec-lemma

Proof step 15 from proof relyp-no-pending-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 16:

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

Rule argument

Additional information

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

Proof step 16 from proof relyp-no-pending-w-stable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 17:

  ⊦ x = [] ↔ ¬(∃ a. a ∈ x)

• Proof step 18:

  fowl + fowl1 = p0(l), fowl.filter(.W?).filter(λ fow0. fow0.tid = fow.tid) = [], fowl1.filter(.W?).filter(λ fow0. fow0.tid = fow.tid) = [], p1(l) = fowl1 + fowl2, fow.W?, fow ∈ fowl2, ∀ 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 17 from proof relyp-no-pending-w-stable (previous step) (back to tree)

A spec-lemma

Proof step 18 from proof relyp-no-pending-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 19:

  fowl + fowl1 = p0(l), fowl1.filter(.W?).filter(λ fow0. fow0.tid = fow.tid) = [], fowl.filter(.W?).filter(λ fow0. fow0.tid = fow.tid) = [], p1(l) = fowl1 + fowl2, fow.W?, fow ∈ fowl2, ∀ 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 19 from proof relyp-no-pending-w-stable (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: fow