Proof step 1 from proof reorder-firstPersistW-eq (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  fowl0 = W(v, t) ' + fowl1, fowl[firstPersistW(fowl)] ≠ W(v, t), reorder(l, fowl, fowl0, rfo) ⊦

Rule argument

Additional information

Proof step 2 from proof reorder-firstPersistW-eq (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  fowl[firstPersistW(fowl)] ≠ W(v, t), reorder(l, fowl, W(v, t) ' + fowl1, rfo) ⊦

Rule argument

Additional information

Proof step 3 from proof reorder-firstPersistW-eq (previous step) (back to tree)

User applied rule structural induction

And obtained the following new goals

• Proof step 4:

  (fow + fowl0)[firstPersistW(fow + fowl0)] ≠ W(v, t), reorder(l, fow + fowl0, W(v, t) ' + fowl1, rfo), ∀ v, t, l, fowl1, rfo. ¬(fowl0[firstPersistW(fowl0)] ≠ W(v, t) ∧ reorder(l, fowl0, W(v, t) ' + fowl1, rfo)) ⊦

Rule argument

Additional information

Used simplifier rules: reorder-base: ⊦ reorder(l, [], fowl0, rfo) ↔ fowl0 = [] ∧ (∀ t. propFOs(t, l, rfo(t)) = []); List('a) freely generated by [], +;; ⊦ firstPersistW([]) = 0; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 4 from proof reorder-firstPersistW-eq (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  (fow ' + fowl0)[firstPersistW(fow ' + fowl0)] ≠ W(v, t), reorder(l, fow ' + fowl0, W(v, t) ' + fowl1, rfo), ∀ v0, t0, l0, fowl, rfo0. fowl0[firstPersistW(fowl0)] = W(v0, t0) ∨ ¬ reorder(l0, fowl0, W(v0, t0) ' + fowl, rfo0) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s)

Proof step 5 from proof reorder-firstPersistW-eq (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 6:

  fow = FO(t1), (fow ' + fowl0)[firstPersistW(fow ' + fowl0)] ≠ W(v, t), reorder(l, fow ' + fowl0, W(v, t) ' + fowl1, rfo), ∀ v0, t0, l0, fowl, rfo0. fowl0[firstPersistW(fowl0)] = W(v0, t0) ∨ ¬ reorder(l0, fowl0, W(v0, t0) ' + fowl, rfo0) ⊦

Rule argument

Additional information

Used simplifier rules: reorder-rec-W: ⊦ reorder(l, W(v, t) ' + fowl0, fowl1, rfo) ↔ fowl1 ≠ [] ∧ fowl1.head = W(v, t) ∧ reorder(l, fowl0, fowl1.tail, rfo); FOW freely generated by FO, W;; ⊦ firstPersistW(W(v, t) ' + fowl) = 0; (s) ⊦ (a ' + x)[0] = a; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) ⊦ (a ' + x).head = a; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 6 from proof reorder-firstPersistW-eq (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  fowl0[firstPersistW(fowl0)] ≠ W(v, t), ∀ v0, t0, l0, fowl, rfo0. fowl0[firstPersistW(fowl0)] = W(v0, t0) ∨ ¬ reorder(l0, fowl0, W(v0, t0) ' + fowl, rfo0), W(v, t) ' + fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl0, fowl2 + fowl3, rfo) ∨ t1 ≠ t ∧ fowl1 | t1 = [] ∧ FO(l) ∈ rfo(t1) ∧ reorder(l, fowl0, W(v, t) ' + fowl1, rfo(t1 := rfo(t1) -1l FO(l))) ⊦

Rule argument

Additional information

Used simplifier rules: reorder-rec-FO: ⊦ reorder(l, FO(t) ' + fowl, fowl0, rfo) ↔ (∃ fowl1, fowl2. fowl0 = fowl1 + FO(t) ' + fowl2 ∧ fowl1 | t = [] ∧ reorder(l, fowl, fowl1 + fowl2, rfo)) ∨ fowl0 | t = [] ∧ FO(l) ∈ rfo(t) ∧ reorder(l, fowl, fowl0, rfo(t := rfo(t) -1l FO(l))); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ W(v, tid).tid = tid; (s) ⊦ fow ' | t = (t = fow.tid ⊃ fow ';[]); (s) ⊦ fowl + fowl0 | t = (fowl | t) + (fowl0 | t); (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + x)[n + 1] = x[n]; (s) ⊦ firstPersistW(FO(t) ' + fowl) = firstPersistW(fowl) + 1; (s)

Proof step 7 from proof reorder-firstPersistW-eq (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 8:

  W(v, t) ' + fowl1 = fowl2 + FO(t1) ' + fowl3, fowl2 | t1 = [], fowl0[firstPersistW(fowl0)] ≠ W(v, t), reorder(l, fowl0, fowl2 + fowl3, rfo), ∀ v0, t0, l0, fowl, rfo0. fowl0[firstPersistW(fowl0)] = W(v0, t0) ∨ ¬ reorder(l0, fowl0, W(v0, t0) ' + fowl, rfo0) ⊦

• Proof step 13:

  fowl1 | t1 = [], fowl0[firstPersistW(fowl0)] ≠ W(v, t), t1 ≠ t, reorder(l, fowl0, W(v, t) ' + fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ∀ v0, t0, l0, fowl, rfo0. fowl0[firstPersistW(fowl0)] = W(v0, t0) ∨ ¬ reorder(l0, fowl0, W(v0, t0) ' + fowl, rfo0) ⊦

Rule argument

Additional information

Proof step 8 from proof reorder-firstPersistW-eq (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 9:

  W(v, t) ' + fowl1 = fowl2 + FO(t1) ' + fowl3, fowl2 | t1 = [], fowl0[firstPersistW(fowl0)] ≠ W(v, t), reorder(l, fowl0, fowl2 + fowl3, rfo), W(v, t) ∈ fowl2, ∀ v0, t0, l0, fowl, rfo0. fowl0[firstPersistW(fowl0)] = W(v0, t0) ∨ ¬ reorder(l0, fowl0, W(v0, t0) ' + fowl, rfo0) ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ FO(tid) ≠ W(v, tid0); (s) ⊦ ¬ a ∈ y → (y + a0 ' + z = a ' + z0 ↔ y = [] ∧ a0 = a ∧ z = z0); (ws) ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list

Proof step 9 from proof reorder-firstPersistW-eq (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 10:

  ⊦ a ∈ x ↔ (∃ y, z. x = y + a ' + z ∧ ¬ a ∈ y)

• Proof step 11:

  W(v, t) ' + fowl1 = fowl2 + FO(t1) ' + fowl3, fowl2 | t1 = [], fowl0[firstPersistW(fowl0)] ≠ W(v, t), reorder(l, fowl0, fowl2 + fowl3, rfo), ∀ v0, t0, l0, fowl, rfo0. fowl0[firstPersistW(fowl0)] = W(v0, t0) ∨ ¬ reorder(l0, fowl0, W(v0, t0) ' + fowl, rfo0), ∃ y, z. fowl2 = y + W(v, t) ' + z ∧ ¬ W(v, t) ∈ y ⊦

Rule argument

Additional information

Used terms: W(v, t), fowl2

Proof step 10 from proof reorder-firstPersistW-eq (previous step) (back to tree)

A spec-lemma

Proof step 11 from proof reorder-firstPersistW-eq (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

  fowl4 | t1 = [], fowl0[firstPersistW(fowl0)] ≠ W(v, t), t1 ≠ t, reorder(l, fowl0, W(v, t) ' + fowl4 + fowl3, rfo), ∀ v0, t0, l0, fowl, rfo0. fowl0[firstPersistW(fowl0)] = W(v0, t0) ∨ ¬ reorder(l0, fowl0, W(v0, t0) ' + fowl, rfo0) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ a ∈ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ [] + x = x; (s) ⊦ [] | t = []; (s) ⊦ ¬ a ∈ y → (y + a0 ' + z = a ' + z0 ↔ y = [] ∧ a0 = a ∧ z = z0); (ws) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ W(v, tid).tid = tid; (s) ⊦ fow ' | t = (t = fow.tid ⊃ fow ';[]); (s) ⊦ fowl + fowl0 | t = (fowl | t) + (fowl0 | t); (s)

Proof step 12 from proof reorder-firstPersistW-eq (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: v, t, l, fowl4 + fowl3, rfo

Proof step 13 from proof reorder-firstPersistW-eq (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: v, t, l, fowl1, rfo(t1 := rfo(t1) -1l FO(l))