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

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  dlls(null, r0, r1, x) ≠ (emp ∧ ⌜r1 = null⌝ ∧ ⌜x = []⌝ ) ⊦

Rule argument

Additional information

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

User applied rule constructor cut

And obtained the following new goals

• Proof step 3:

  x = a + x0, dlls(null, r0, r1, x) ≠ (emp ∧ ⌜r1 = null⌝ ∧ ⌜x = []⌝ ) ⊦

Rule argument

Additional information

Used simplifier rules: dlls-base: ⊦ dlls(r, r0, r1, []) = (emp ∧ ⌜r = r1⌝ ); List[A] freely generated by [], +; ⊦ ⌜true⌝ = true; (s) ⊦ (true ∧ hP) = hP; (s) ⊦ hP0 = hP1 → ((hP ∧ hP0) = (hP ∧ hP1) ↔ true); (ss) and-ass: ⊦ ((hP0 ∧ hP1) ∧ hP2) = (hP0 ∧ hP1 ∧ hP2); from specification map-sep and-com: ⊦ (hP0 ∧ hP1) = (hP1 ∧ hP0); from specification map-sep

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 4:

  _∃(λ r2. false) ≠ false ⊦

Rule argument

Additional information

Used simplifier rules: dlls-rec: ⊦ dlls(r, r0, r1, a ' + x) = _∃(λ r2. r => mknode(r0, a, r2) * dlls(r2, r, r1, x)); and-ass: ⊦ ((hP0 ∧ hP1) ∧ hP2) = (hP0 ∧ hP1 ∧ hP2); from specification map-sep and-com: ⊦ (hP0 ∧ hP1) = (hP1 ∧ hP0); from specification map-sep ⊦ (false ∧ hP) = false; (s) ⊦ ⌜false⌝ = false; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a + x = a ' + x; (s) sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification map-sep sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification map-sep ⊦ false * hP = false; (s) ⊦ null => o = ⌜false⌝ ; (s)

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

User applied rule expand left

And closed the goal!

Rule argument

Additional information

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