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

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  (dlls(hd, lst0, prv, hd0, x) * dlls(hd0, lst, lst0, nxt, y) * hP)(h), ¬ (dlls(hd, lst, prv, nxt, x + y) * hP)(h) ⊦

Rule argument

Additional information

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

User applied rule structural induction

And obtained the following new goals

• Proof step 3:

  (dlls(hd, lst0, prv, hd0, a + x0) * dlls(hd0, lst, lst0, nxt, y) * hP)(h), ¬ (dlls(hd, lst, prv, nxt, (a + x0) + y) * hP)(h), ∀ hd, lst0, prv, hd0, lst, nxt, y, hP, h. ¬((dlls(hd, lst0, prv, hd0, x0) * dlls(hd0, lst, lst0, nxt, y) * hP)(h) ∧ ¬ (dlls(hd, lst, prv, nxt, x0 + y) * hP)(h)) ⊦

Rule argument

Additional information

Used simplifier rules: dlls-last-base: ⊦ dlls(hd, lst, prv, nxt, []) = (emp ∧ ⌜hd = nxt⌝ ∧ ⌜lst = prv⌝ ); List[A] freely generated by [], +; ⊦ emp * hP = hP; (s) ⊦ (⌜bv⌝ ∧ hP0) * hP1 = (⌜bv⌝ ∧ hP0 * hP1); (s) sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification map-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification map-sep ⊦ (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 ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 4:

  ((hd => mknode(prv, a, nxt0) * dlls(nxt0, lst0, hd, hd0, x0)) * dlls(hd0, lst, lst0, nxt, y) * hP)(h), ∀ hd1, lst1, prv0, hd2, lst2, nxt1, y0, hP0, h0. ¬ (dlls(hd1, lst1, prv0, hd2, x0) * dlls(hd2, lst2, lst1, nxt1, y0) * hP0)(h0) ∨ (dlls(hd1, lst2, prv0, nxt1, x0 + y0) * hP0)(h0) ⊦ ∃ nxt0. ((hd => mknode(prv, a, nxt0) * dlls(nxt0, lst, hd, nxt, x0 + y)) * hP)(h)

Rule argument

Additional information

Used simplifier rules: dlls-last-rec: ⊦ dlls(hd, lst, prv, nxt, a ' + x) = _∃(λ nxt0. hd => mknode(prv, a, nxt0) * dlls(nxt0, lst, hd, nxt, x)); sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification map-sep sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification map-sep ⊦ (_∃ bP * hP)(h) ↔ (∃ b. (bP(b) * hP)(h)); (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a + x = a ' + x; (s)

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

User applied rule all left

And obtained the following new goals

• Proof step 5:

  ((hd => mknode(prv, a, nxt0) * dlls(nxt0, lst0, hd, hd0, x0)) * dlls(hd0, lst, lst0, nxt, y) * hP)(h), ∀ hd1, lst1, prv0, hd2, lst2, nxt1, y0, hP0, h0. ¬ (dlls(hd1, lst1, prv0, hd2, x0) * dlls(hd2, lst2, lst1, nxt1, y0) * hP0)(h0) ∨ (dlls(hd1, lst2, prv0, nxt1, x0 + y0) * hP0)(h0), ¬ (dlls(nxt0, lst0, hd, hd0, x0) * dlls(hd0, lst, lst0, nxt, y) * hP * hd => mknode(prv, a, nxt0))(h) ∨ (dlls(nxt0, lst, hd, nxt, x0 + y) * hP * hd => mknode(prv, a, nxt0))(h) ⊦ ∃ nxt0. ((hd => mknode(prv, a, nxt0) * dlls(nxt0, lst, hd, nxt, x0 + y)) * hP)(h)

Rule argument

Additional information

Used terms: nxt0, lst0, hd, hd0, lst, nxt, y, hP * hd => mknode(prv, a, nxt0), h

Proof step 5 from proof dlls-last-consecutive (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

  (dlls(nxt0, lst, hd, nxt, x0 + y) * hP * hd => mknode(prv, a, nxt0))(h), ((hd => mknode(prv, a, nxt0) * dlls(nxt0, lst0, hd, hd0, x0)) * dlls(hd0, lst, lst0, nxt, y) * hP)(h), ∀ hd1, lst1, prv0, hd2, lst2, nxt1, y0, hP0, h0. ¬ (dlls(hd1, lst1, prv0, hd2, x0) * dlls(hd2, lst2, lst1, nxt1, y0) * hP0)(h0) ∨ (dlls(hd1, lst2, prv0, nxt1, x0 + y0) * hP0)(h0) ⊦ ∃ nxt0. ((hd => mknode(prv, a, nxt0) * dlls(nxt0, lst, hd, nxt, x0 + y)) * hP)(h)

Rule argument

Additional information

Used simplifier rules: sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification map-sep sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification map-sep eq-objs: ⊦ (r => o0 * hP0)(h) ∧ (r => o1 * hP1)(h) → o0 = o1; (forward) from specification refmap-maplet

Proof step 6 from proof dlls-last-consecutive (previous step) (back to tree)

Heuristic quantifier closing applied rule exists right

And closed the goal!

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 Used terms: nxt0