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

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ((lls(r0, r1, x) * lls(r1, r2, y) * hP))(h), ¬ ((lls(r0, r2, x + y) * hP))(h) ⊦

Rule argument

Additional information

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  ((lls(r0, r1, x) * lls(r1, r2, y) * hP))(h), ¬ ((lls(r0, r2, x + y) * hP))(h) ⊦

Rule argument

Additional information

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

User applied rule structural induction

And obtained the following new goals

• Proof step 4:

  ((lls(r0, r1, a + x0) * lls(r1, r2, y) * hP))(h), ¬ ((lls(r0, r2, (a + x0) + y) * hP))(h), ∀ r0, r1, r2, y, hP, h. ¬(((lls(r0, r1, x0) * lls(r1, r2, y) * hP))(h) ∧ ¬ ((lls(r0, r2, x0 + y) * hP))(h)) ⊦

Rule argument

Additional information

Used simplifier rules: lls-base: ⊦ lls(r0, r1, []) = (emp ∧ ⌜r0 = r1⌝ ); list freely generated by [], +;; ⊦ emp * hP = hP; (s) ⊦ (⌜bv⌝ ∧ hP0) * hP1 = (⌜bv⌝ ∧ hP0 * hP1); (s) sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep ⊦ ((hP0 ∧ hP1))(h) ↔ hP0(h) ∧ hP1(h); (s) ⊦ ⌜bv⌝ (h) ↔ bv; (s) and-ass: ⊦ ((hP0 ∧ hP1) ∧ hP2) = (hP0 ∧ hP1 ∧ hP2); from specification heap-sep and-com: ⊦ (hP0 ∧ hP1) = (hP1 ∧ hP0); from specification heap-sep ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  (((r0 => a × r * lls(r, r1, x0)) * lls(r1, r2, y) * hP))(h), ∀ r3, r4, r5, y0, hP0, h0. ¬ ((lls(r3, r4, x0) * lls(r4, r5, y0) * hP0))(h0) ∨ ((lls(r3, r5, x0 + y0) * hP0))(h0) ⊦ ∃ r. (((r0 => a × r * lls(r, r2, x0 + y)) * hP))(h)

Rule argument

Additional information

Used simplifier rules: lls-rec: ⊦ lls(r, r1, a ' + x) = _∃(λ r0. r => a × r0 * lls(r0, r1, x)); sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep ⊦ ((_∃ rP * hP))(h) ↔ (∃ r. ((rP(r) * hP))(h)); (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a + x = a ' + x; (s)

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

User applied rule all left

And obtained the following new goals

• Proof step 6:

  (((r0 => a × r * lls(r, r1, x0)) * lls(r1, r2, y) * hP))(h), ¬ ((lls(r, r1, x0) * lls(r1, r2, y) * hP * r0 => a × r))(h) ∨ ((lls(r, r2, x0 + y) * hP * r0 => a × r))(h) ⊦ ∃ r. (((r0 => a × r * lls(r, r2, x0 + y)) * hP))(h)

Rule argument

Additional information

Used terms: r, r1, r2, y, hP * r0 => a × r, h

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  (((r0 => a × r * lls(r, r1, x0)) * lls(r1, r2, y) * hP))(h), ((lls(r, r2, x0 + y) * hP * r0 => a × r))(h) ⊦ ∃ r. (((r0 => a × r * lls(r, r2, x0 + y)) * hP))(h)

Rule argument

Additional information

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

Proof step 7 from proof lls-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 heap-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep Used terms: r