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Proof step 55 from proof rem-stays-reachable-aux2 (previous step)

(r ' + x1).last = (r0 ' + x0).last, r0 = H[r0].nxt, path(r ' + (x2 + (r0 ' + x0).last ') + r0 ', H), path(r0 ' + x0, H), path(r ' + x1, H), (r0 ' + x0).last ∈ H, r0 ∈ H ⊦ r = (r0 ' + x0).last, r0 = null, r0 = r, r0 = (r0 ' + x0).last, dups(x1), dups(x0), dups(x2), r ∈ x2, r ∈ x1, r0 ∈ x0, r0 ∈ x2, (r0 ' + x0).last ∈ x2

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 56:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 57:
x1.last = (r0 ' + x0).last, r0 = H[r0].nxt, path(r ' + x1, H), path(r0 ' + x0, H), path(r ' + (x2 + (r0 ' + x0).last ') + r0 ', H), r0 ∈ H, (r0 ' + x0).last ∈ H ⊦ r = (r0 ' + x0).last, r0 = (r0 ' + x0).last, r0 = r, r0 = null, x1 = [], dups(x2), dups(x0), dups(x1), (r0 ' + x0).last ∈ x2, r0 ∈ x2, r0 ∈ x0, r ∈ x1, r ∈ x2

Rule argument

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