Proof step 1 from proof tr-null (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  tr(null, t) ≠ (emp ∧ ⌜t = leaf⌝ ) ⊦

Rule argument

Additional information

Proof step 2 from proof tr-null (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  tr(null, t) ≠ (emp ∧ ⌜t = leaf⌝ ) ⊦

Rule argument

Additional information

Proof step 3 from proof tr-null (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 4:

  t = branch(t0, a, t1), tr(null, t) ≠ (emp ∧ ⌜t = leaf⌝ ) ⊦

Rule argument

Additional information

Used simplifier rules: tr-base: ⊦ tr(r, leaf) = (emp ∧ ⌜r = null⌝ ); Tree[A] freely generated by leaf, branch; ⊦ ⌜true⌝ = true; (s) ⊦ (true ∧ hP) = hP; (s) and-ass: ⊦ ((hP0 ∧ hP1) ∧ hP2) = (hP0 ∧ hP1 ∧ hP2); from specification map-sep and-com: ⊦ (hP0 ∧ hP1) = (hP1 ∧ hP0); from specification map-sep

Proof step 4 from proof tr-null (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  tr(null, branch(t0, a, t1)) ≠ false ⊦

Rule argument

Additional information

Used simplifier rules: and-com: ⊦ (hP0 ∧ hP1) = (hP1 ∧ hP0); from specification map-sep and-ass: ⊦ ((hP0 ∧ hP1) ∧ hP2) = (hP0 ∧ hP1 ∧ hP2); from specification map-sep ⊦ (false ∧ hP) = false; (s) ⊦ ⌜false⌝ = false; (s) ⊦ leaf ≠ branch(t, val, t0); (s)

Proof step 5 from proof tr-null (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 6:

  ⊦ tr(r, branch(t0, a, t1)) = _∃(λ r0, r1. r => mknode(r0, a, r1) * tr(r0, t0) * tr(r1, t1))

• Proof step 7:

  _∃(λ r0, r1. null => mknode(r0, a, r1) * tr(r0, t0) * tr(r1, t1)) ≠ false ⊦

Rule argument

Additional information

Used terms: null, t0, a, t1

Proof step 6 from proof tr-null (previous step) (back to tree)

Lemma tr-rec

Proof step 7 from proof tr-null (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

  _∃(λ r0, r1. false) ≠ false ⊦

Rule argument

Additional information

Used simplifier rules: sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification map-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification map-sep ⊦ false * hP = false; (s) ⊦ ⌜false⌝ = false; (s) ⊦ null => o = ⌜false⌝ ; (s)

Proof step 8 from proof tr-null (previous step) (back to tree)

User applied rule expand left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ (_∃ bcP)(h) ↔ (∃ b, c. bcP(b, c)(h)); (s) ⊦ ¬ false(h); (s)