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Proof step 37 from proof rem-stays-reachable-aux2 (previous step)

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

User applied rule insert elim lemma

And obtained the following new goals

Proof step 38:
x2 = x3 + r0 ' ∧ x2 ≠ [], x2 = x + H[r0].nxt ', (r0 ' + x0).last = (r ' + x).last, (r ' + x1).last = (r ' + x).last, path(r ' + x1, H), path(r ' + x2, H), H[r0].nxt ∈ H, r0 ∈ H, path(r0 ' + x0, H), (r ' + x).last ∈ H, H[r0].nxt ∈ x2 ⊦ H[r0].nxt = null, r0 = r, r0 ∈ x, r0 ∈ x0, dups(x0), r ∈ x1, dups(x1), r ∈ x2, dups(x2)

Rule argument

spec: list-last inst: reflist name: last-elim seq: x0 ≠ [] ⊦ r = x0.last ∧ x1 = x0.butlast ↔ x0 = x1 + r ' subst: [x0, r, x1] → [x2, r0, x3] flags: []

Additional information

Used simplifier rules: app-02-ant: ⊦ ¬ x0 + x1 ≠ [] ↔ x0 = [] ∧ x1 = []; from specification list[reflist]%e: ⊦ r ' ≠ []; from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%x0 ≠ [] ⊦ r = x0.last ∧ x1 = x0.butlast ↔ x0 = x1 + r '; (elim) Used terms: x2, r0, x3