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Proof step 9 from proof lls-unfold-tail (previous step)

((lls(r0, r2, (a0 + x0) + a ') * P))(H), ∀ r0, r2, a, P, H. ((lls(r0, r2, x0 + a ') * P))(H) → (∃ r. ((lls(r0, r, x0) * r => a × r2 * P))(H)) ⊦ ∃ r. ((lls(r0, r, a0 + x0) * r => a × r2 * P))(H)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
(((r0 => a0 × r * lls(r, r2, x0 + a ')) * P))(H), ∀ r0, r2, a, P, H. ((lls(r0, r2, x0 + a ') * P))(H) → (∃ r. ((lls(r0, r, x0) * r => a × r2 * P))(H)) ⊦ ∃ r1, r. (((r0 => a0 × r * lls(r, r1, x0)) * r1 => a × r2 * P))(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: ⊦ P0 * P1 = P1 * P0; from specification heap-sep%sep-ass: ⊦ (P0 * P1) * P2 = P0 * P1 * P2; from specification heap-sep% ⊦ ((_∃ rP * P))(H) ↔ (∃ r. ((rP(r) * P))(H)); (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ a + x = a ' + x; (s)