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Proof step 69 from proof refine-add (previous step)

((ls(rxs, xs) * ls(rys, ys) * del-list(to_delete, n1 + 2)))(H), ((del-list(to_delete, n1 + 2) * ls(rys, ys) * ls(rxs, xs)))(H), ((del-list(to_delete0, n1) * ls(rys, ys) * ls(rxs, xs)))(H1), # xs < h0, # xs + 1 < h0, n1 + # xs < h0, n1 + 2 ≤ # xs + # ys, (n1 + 2) + # xs ≤ h0, 2 ≤ # xs + # ys, n1 ≤ (# xs + 1) + # ys, ys = [] ∨ # ys = h0 ⊦ # xs + # ys = 0, # xs + 1 = h0, r = null, ((ls(r, a ' + xs) * ls(rys, ys) * del-list(to_delete0, n1)))((H1 ++ r)[r, a × rxs]), r ∈ H1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 70:
⊦ ls(r, a ' + x) = _∃(λ r0. r => a × r0 * ls(r0, x))
Proof step 71:
((del-list(to_delete0, n1) * ls(rys, ys) * ls(rxs, xs)))(H1), ((del-list(to_delete, n1 + 2) * ls(rys, ys) * ls(rxs, xs)))(H), ((ls(rxs, xs) * ls(rys, ys) * del-list(to_delete, n1 + 2)))(H), n1 + # xs < h0, # xs + 1 < h0, # xs < h0, n1 ≤ (# xs + 1) + # ys, 2 ≤ # xs + # ys, (n1 + 2) + # xs ≤ h0, n1 + 2 ≤ # xs + # ys, ys = [] ∨ # ys = h0 ⊦ # xs + 1 = h0, # xs + # ys = 0, r = null, ((_∃(λ r0. r => a × r0 * ls(r0, xs)) * ls(rys, ys) * del-list(to_delete0, n1)))((H1 ++ r)[r, a × rxs]), r ∈ H1

Rule argument

spec: linked-list name: ls-rec seq: ⊦ ls(r, a ' + x) = _∃(λ r0. r => a × r0 * ls(r0, x)) subst: [x, a, r] → [xs, a, r] flags: <all paths>

Additional information

Used terms: xs, a, r