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Proof step 42 from proof rem-stays-reachable-aux2 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 43:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 44:
¬ x1 ≠ [], (H[r].nxt ' + x0).last = r, (r ' + x1).last = r, H[r].nxt = H[H[r].nxt].nxt, H[r].nxt ∈ H, path(H[r].nxt ' + x0, H), r ∈ H, path(r ' + x1, H) ⊦ r = null, H[r].nxt = null, H[r].nxt = r, H[r].nxt ∈ x0, dups(x0), r ∈ x1, dups(x1)

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 simplifier rules: lin-01: ⊦ x0.last ∈ x0 ↔ x0 ≠ []; from specification list-last[reflist] Used terms: x1, r '