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Proof step 38 from proof rem-stays-reachable-aux2 (previous step)

(r ' + x1).last = (r ' + x).last, (x2.last ' + x0).last = (r ' + x).last, x2 = x + H[x2.last].nxt ', path(r ' + x1, H), path(x2.last ' + x0, H), path(r ' + x2, H), H[x2.last].nxt ∈ H, x2.last ∈ H, (r ' + x).last ∈ H, H[x2.last].nxt ∈ x2 ⊦ x2.last = r, H[x2.last].nxt = null, dups(x0), dups(x1), dups(x2), x2.last ∈ x, x2.last ∈ x0, r ∈ x1, r ∈ x2

User applied rule insert elim lemma

And obtained the following new goals

Proof step 39:
(r0 ' + x0).last = (r ' + x).last, (r ' + x1).last = (r ' + x).last, x2 = x + H[r0].nxt ', x2 = x3 + r0 ', path(r ' + x2, H), path(r0 ' + x0, H), path(r ' + x1, H), H[r0].nxt ∈ x2, (r ' + x).last ∈ H, r0 ∈ H, H[r0].nxt ∈ H ⊦ H[r0].nxt = null, r0 = r, x2 = [], dups(x2), dups(x1), dups(x0), r ∈ x2, r ∈ x1, r0 ∈ x0, r0 ∈ x

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] → [x2, r0, x3]

Additional information

Used simplifier rules: ⊦ x0 + x1 = [] ↔ x0 = [] ∧ x1 = []; (as)% ⊦ r ' ≠ []; (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%x0 ≠ [] ⊦ r = x0.last ∧ x1 = x0.butlast ↔ x0 = x1 + r '; (elim) Used terms: x2, r0, x3