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Proof step 74 from proof refine-add (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 75:
⊦ ls(r, a ' + x) = _∃(λ r0. r => a × r0 * ls(r0, x))
Proof step 76:
((ls(rxs, xs) * ls(rys, ys) * del-list(to_delete, n)))(H), ((del-list(to_delete, n) * ls(rys, ys) * ls(rxs, xs)))(H), ((ls(rys, ys) * ls(rxs, xs)))(H1), # xs < h0, # xs + 1 < h0, n < 2, n ≤ # xs + # ys, n + # xs ≤ h0, ys = [] ∨ # ys = h0 ⊦ # xs + 1 = h0, r = null, ((ls(rys, ys) * _∃(λ r0. r => a × r0 * ls(r0, xs))))((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