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Proof step 53 from proof acyclic-rem-not-reachable (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 54:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 55:
x ≠ [], ce.nxt = H[H[x.last].nxt].nxt, x.last ∈ H, path(r ' + x1 + H[x.last].nxt ', H), path(r ' + x, H), H[x.last].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[x.last].nxt = null, r = H[x.last].nxt, x.last = r, x.last = H[x.last].nxt, x.last ∈ x1, r ∈ x1, H[x.last].nxt ∈ x1, dups(x1), r ∈ x, dups(x)

Rule argument

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

Additional information

Used simplifier rules: app: ⊦ x0 + [] = x0; from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%l: ⊦ r '.last = r; from specification list-last[reflist] Used terms: x, r '