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Proof step 65 from proof rem-stays-reachable-aux2 (previous step)

x2 = (x + H[(r ' + x2).last].nxt ') + r0 ' + y, ((r ' + x2).last ' + x0).last = (r ' + x).last, (r ' + x1).last = (r ' + x).last, path(r ' + x1, H), path(r ' + x + H[(r ' + x2).last].nxt ', H), path(r ' + x2, H), H[(r ' + x2).last].nxt ∈ H, (r ' + x2).last ∈ H, path((r ' + x2).last ' + x0, H), (r ' + x).last ∈ H ⊦ H[(r ' + x2).last].nxt = null, (r ' + x2).last = r, (r ' + x2).last ∈ x, (r ' + 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 66:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 67:
x2 ≠ [], x2 = (x + H[x2.last].nxt ') + r0 ' + y, (x2.last ' + x0).last = (r ' + x).last, (r ' + x1).last = (r ' + x).last, path(r ' + x1, H), path(r ' + x + H[x2.last].nxt ', 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 = 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] → [x2, 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]%app-02-ant: ⊦ ¬ x0 + x1 ≠ [] ↔ x0 = [] ∧ x1 = []; from specification list[reflist]%e-01: ⊦ r ' + x0 ≠ []; from specification list[reflist] Used terms: x2, r '