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Proof step 203 from proof rem-stays-reachable-aux2 (previous step)

x + H[x2.last].nxt ' = x2 + r1 ' + x3, (x2.last ' + x0).last = (r ' + x).last, (r ' + x1).last = (r ' + x).last, path(r ' + x1, H), path(r ' + x2 + r1 ' + x3, H), path(r ' + x2, H), H[x2.last].nxt ∈ H, x2.last ∈ H, path(x2.last ' + x0, H), (r ' + x).last ∈ H, H[x2.last].nxt ∈ x2 ∨ H[x2.last].nxt = r1 ∨ H[x2.last].nxt ∈ x3 ⊦ x2 = [], H[x2.last].nxt = null, x2.last = r, x2.last ∈ x, x2.last ∈ x0, dups(x0), r ∈ x1, dups(x1), r ∈ x2, dups(x2)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 204:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 205:
¬ x1 ≠ [], x + H[x2.last].nxt ' = x2 + r1 ' + x3, (x2.last ' + x0).last = (r ' + x).last, (r ' + x1).last = (r ' + x).last, path(r ' + x1, H), path(r ' + x2 + r1 ' + x3, H), path(r ' + x2, H), H[x2.last].nxt ∈ H, x2.last ∈ H, path(x2.last ' + x0, H), (r ' + x).last ∈ H, H[x2.last].nxt ∈ x2 ∨ H[x2.last].nxt = r1 ∨ H[x2.last].nxt ∈ x3 ⊦ x2 = [], H[x2.last].nxt = null, x2.last = r, x2.last ∈ x, x2.last ∈ x0, dups(x0), r ∈ x1, dups(x1), r ∈ x2, dups(x2)
Proof step 207:
x1 ≠ [], x + H[x2.last].nxt ' = x2 + r1 ' + x3, (x2.last ' + x0).last = (r ' + x).last, x1.last = (r ' + x).last, path(r ' + x1, H), path(r ' + x2 + r1 ' + x3, H), path(r ' + x2, H), H[x2.last].nxt ∈ H, x2.last ∈ H, path(x2.last ' + x0, H), (r ' + x).last ∈ H, H[x2.last].nxt ∈ x2 ∨ H[x2.last].nxt = r1 ∨ H[x2.last].nxt ∈ x3 ⊦ x2 = [], H[x2.last].nxt = null, x2.last = r, x2.last ∈ x, x2.last ∈ x0, dups(x0), r ∈ x1, dups(x1), r ∈ x2, dups(x2)

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 '