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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 2:

  x ≠ [] ⊦ ((lls(r0, r2, x) * hP))(h) ↔ (∃ a, y, r1. ((r0 => a × r1 * lls(r1, r2, y) * hP))(h) ∧ x = a ' + y)

Rule argument

Additional information

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

User applied rule constructor cut

And obtained the following new goals

• Proof step 3:

  x = a0 + x0, x ≠ [] ⊦ ((lls(r0, r2, x) * hP))(h) ↔ (∃ a, y, r1. ((r0 => a × r1 * lls(r1, r2, y) * hP))(h) ∧ x = a ' + y)

Rule argument

Additional information

Used simplifier rules: list freely generated by [], +;;

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 4:

  ⊦ (∃ r. (((r0 => a0 × r * lls(r, r2, x0)) * hP))(h)) ↔ (∃ a, y, r1. ((r0 => a × r1 * lls(r1, r2, y) * hP))(h) ∧ a0 ' + x0 = a ' + y)

Rule argument

Additional information

Used simplifier rules: lls-rec: ⊦ lls(r, r1, a ' + x) = _∃(λ r0. r => a × r0 * lls(r0, r1, x)); ⊦ a + x = a ' + x; (s) 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 ≠ []; (s)

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

User applied rule case distinction

And obtained the following new goals

• Proof step 5:

  ∃ r. (((r0 => a0 × r * lls(r, r2, x0)) * hP))(h) ⊦ ∃ a, y, r1. ((r0 => a × r1 * lls(r1, r2, y) * hP))(h) ∧ a0 ' + x0 = a ' + y

• Proof step 7:

  ∃ a, y, r1. ((r0 => a × r1 * lls(r1, r2, y) * hP))(h) ∧ a0 ' + x0 = a ' + y ⊦ ∃ r. (((r0 => a0 × r * lls(r, r2, x0)) * hP))(h)

Rule argument

Additional information

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

  (((r0 => a0 × r * lls(r, r2, x0)) * hP))(h) ⊦ ∃ a, y, r1. ((r0 => a × r1 * lls(r1, r2, y) * hP))(h) ∧ a0 ' + x0 = a ' + y

Rule argument

Additional information

Proof step 6 from proof lls-unfold-head (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: a0, x0, r

Proof step 7 from proof lls-unfold-head (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

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

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ' + x = b ' + y ↔ a = b ∧ x = y; (ws)

Proof step 8 from proof lls-unfold-head (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: r1