Proof step 1 from proof dll-prev-one (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  dll(r, r0, a ') ≠ r => mknode(r0, a, null) ⊦

Rule argument

Additional information

Proof step 2 from proof dll-prev-one (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 3:

  ⊦ dll(r, r0, a ' + x) = _∃(λ r1. r => mknode(r0, a, r1) * dll(r1, r, x))

• Proof step 4:

  dll(r, r0, a ' + []) = _∃(λ r1. r => mknode(r0, a, r1) * dll(r1, r, [])), dll(r, r0, a ') ≠ r => mknode(r0, a, null) ⊦

Rule argument

Additional information

Used terms: [], r0, a, r

Proof step 3 from proof dll-prev-one (previous step) (back to tree)

Lemma dll-rec

Proof step 4 from proof dll-prev-one (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  dll(r, r0, a ') = _∃(λ r1. ⌜r1 = null⌝ ∧ r => mknode(r0, a, r1)), dll(r, r0, a ') ≠ r => mknode(r0, a, null) ⊦

Rule argument

Additional information

Used simplifier rules: dll-base: ⊦ dll(r, r0, []) = (emp ∧ ⌜r = null⌝ ); sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification map-sep sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification map-sep and-com: ⊦ (hP0 ∧ hP1) = (hP1 ∧ hP0); from specification map-sep and-ass: ⊦ ((hP0 ∧ hP1) ∧ hP2) = (hP0 ∧ hP1 ∧ hP2); from specification map-sep ⊦ emp * hP = hP; (s) ⊦ (⌜bv⌝ ∧ hP0) * hP1 = (⌜bv⌝ ∧ hP0 * hP1); (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + [] = x; (s)

Proof step 5 from proof dll-prev-one (previous step) (back to tree)

User applied rule subst equation

And obtained the following new goals

• Proof step 6:

  _∃(λ r1. ⌜r1 = null⌝ ∧ r => mknode(r0, a, r1)) ≠ r => mknode(r0, a, null) ⊦

Rule argument

Additional information

Proof step 6 from proof dll-prev-one (previous step) (back to tree)

User applied rule expand left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜false⌝ = false; (s) ⊦ ¬ false(h); (s) ⊦ (_∃ bP)(h) ↔ (∃ b. bP(b)(h)); (s) ⊦ (hP0 ∧ hP1)(h) ↔ hP0(h) ∧ hP1(h); (s) ⊦ ⌜bv⌝ (h) ↔ bv; (s) and-ass: ⊦ ((hP0 ∧ hP1) ∧ hP2) = (hP0 ∧ hP1 ∧ hP2); from specification map-sep and-com: ⊦ (hP0 ∧ hP1) = (hP1 ∧ hP0); from specification map-sep