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Proof step 57 from proof acyclic-rem-not-reachable (previous step)

ce.nxt = H[H[x.last].nxt].nxt, path(r ' + x, H), path(r ' + x1 + H[x.last].nxt ', H), x.last ∈ H, H[x.last].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ x.last = r, x.last = H[x.last].nxt, H[x.last].nxt = null, r = H[x.last].nxt, x = [], dups(x1), dups(x), H[x.last].nxt ∈ x1, r ∈ x1, x.last ∈ x1, r ∈ x

User applied rule insert elim lemma

And obtained the following new goals

Proof step 58:
ce.nxt = H[H[r0].nxt].nxt, x = x2 + r0 ', path(r ' + x1 + H[r0].nxt ', H), path(r ' + x, H), H[r0].nxt ∈ H, r0 ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = null, r = H[r0].nxt, r0 = H[r0].nxt, r0 = r, x = [], dups(x), dups(x1), r ∈ x, r0 ∈ x1, r ∈ x1, H[r0].nxt ∈ x1

Rule argument

spec: list-last, inst: reflist name: last-elim seq: x0 ≠ [] ⊦ r = x0.last ∧ x1 = x0.butlast ↔ x0 = x1 + r ' subst: [x0, r, x1] → [x, r0, x1]

Additional information

Used simplifier rules: x0 ≠ [] ⊦ r = x0.last ∧ x1 = x0.butlast ↔ x0 = x1 + r '; (elim) Used terms: x, r0, x1