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Proof step 28 from proof rem-stays-reachable-aux (previous step)

(r ' + x0).last = H[r0].nxt, ce.nxt = H[H[r0].nxt].nxt, path(r ' + x0, H[r0, ce]), path(r ' + x1 + r0 ', H), H[r0].nxt ∈ H, r0 ∈ H, H[r0].nxt ∈ H[r0, ce] ⊦ H[r0].nxt = ce.nxt, H[r0].nxt = null, r = r0, r = H[r0].nxt, dups(x0), dups(x1), r ∈ x1, r ∈ x0, r0 ∈ x1, ∃ x0. path(H[r0].nxt ' + x0 + H[r0].nxt ', H) ∧ ¬ r0 ∈ x0, ∃ x. path(r ' + x + H[r0].nxt ', H) ∧ ¬ r0 ∈ x

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 29:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 30:
x0.last = H[r0].nxt, ce.nxt = H[H[r0].nxt].nxt, path(r ' + x1 + r0 ', H), path(r ' + x0, H[r0, ce]), H[r0].nxt ∈ H[r0, ce], r0 ∈ H, H[r0].nxt ∈ H ⊦ H[r0].nxt = null, H[r0].nxt = ce.nxt, r = H[r0].nxt, r = r0, x0 = [], dups(x1), dups(x0), r0 ∈ x1, r ∈ x0, r ∈ x1, ∃ x. path(r ' + x + H[r0].nxt ', H) ∧ ¬ r0 ∈ x, ∃ x0. path(H[r0].nxt ' + x0 + H[r0].nxt ', H) ∧ ¬ r0 ∈ x0

Rule argument

spec: list-last, inst: reflist name: lapp seq: ⊦ x0 ≠ [] → (x1 + x0).last = x0.last subst: [x0, x1] → [x0, r '] 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: x0, r '