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Proof step 47 from proof reachable-addnew-rev (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 48:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 49:
¬ x1 ≠ [], path(r ' + x1 + H[(r ' + x1).last].nxt ' + x0, H[H[(r ' + x1).last].nxt, ce]), r ∈ H, (H[(r ' + x1).last].nxt ' + x0).last ∈ H, path(r ' + x1, H) ⊦ reachable(ce.nxt, (H[(r ' + x1).last].nxt ' + x0).last, H), H[(r ' + x1).last].nxt = null, (r ' + x1).last = H[(r ' + x1).last].nxt, H[(r ' + x1).last].nxt ∈ H, r ∈ x1, r ∈ x0, H[(r ' + x1).last].nxt ∈ x1, H[(r ' + x1).last].nxt ∈ x0, dups(x1 + x0), reachable(r, (H[(r ' + x1).last].nxt ' + x0).last, H), dups(x1), dups(x0)
Proof step 60:
x1 ≠ [], path(r ' + x1 + H[x1.last].nxt ' + x0, H[H[x1.last].nxt, ce]), r ∈ H, (H[x1.last].nxt ' + x0).last ∈ H, path(r ' + x1, H) ⊦ reachable(ce.nxt, (H[x1.last].nxt ' + x0).last, H), H[x1.last].nxt = null, x1.last = H[x1.last].nxt, H[x1.last].nxt ∈ H, r ∈ x1, r ∈ x0, H[x1.last].nxt ∈ x1, H[x1.last].nxt ∈ x0, dups(x1 + x0), reachable(r, (H[x1.last].nxt ' + x0).last, H), dups(x1), dups(x0)

Rule argument

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

Additional information

Used terms: x1, r '