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Proof step 16 from proof rem-stays-reachable-aux (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 18:
x1.last = H[r].nxt, ce.nxt = H[H[r].nxt].nxt, path(ce.nxt ' + x1, H[r, ce]), H[r].nxt ∈ H, r ∈ H, H[r].nxt ∈ H[r, ce] ⊦ H[r].nxt = ce.nxt, H[r].nxt = null, r = null, r = ce.nxt, x1 = [], dups(x1), r ∈ x1, ce.nxt ∈ x1, ∃ x. path(H[r].nxt ' + x + H[r].nxt ', H) ∧ ¬ r ∈ x

Rule argument

spec: list-last, inst: reflist name: lapp seq: ⊦ x0 ≠ [] → (x1 + x0).last = x0.last subst: [x0, x1] → [x1, ce.nxt '] flags: <all paths>

Additional information

Used simplifier rules: ⊦ x0 + [] = x0; (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]% ⊦ r '.last = r; (s) Used terms: x1, ce.nxt '