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Proof step 210 from proof rem-stays-reachable-aux2 (previous step)

x + H[x2.last].nxt ' = x2 + r1 ' + x3, (x2.last ' + x0).last = (r ' + x).last, path(r ' + x4 + (r ' + x).last ', 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, r = (r ' + x).last, r ∈ x2, dups(x2), x2.last ∈ x, x2.last ∈ x0, dups(x0), r ∈ x4, (r ' + x).last ∈ x4, dups(x4)

User applied rule insert elim lemma

And closed the goal!

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, x1] flags: []

Additional information

Used simplifier rules: e-02: ⊦ x0 + r ' ≠ []; from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%in-append: ⊦ r ∈ x0 + x1 ↔ ¬ ¬(r ∈ x0 ∨ r ∈ x1); from specification list[reflist]%in-17: ⊦ r ∈ r0 ' ↔ r = r0; from specification list[reflist]%f: ⊦ dups(x0 + r ') ↔ r ∈ x0 ∨ dups(x0); from specification list-dup[reflist]%in-15: ⊦ x0 + r ' = x1 → r ∈ x1; (forward) from specification list[reflist]%in-rec: ⊦ r ∈ r0 ' + x0 ↔ ¬ ¬(r = r0 ∨ r ∈ x0); from specification list[reflist]%neq-03: ⊦ ¬ r ∈ x ∧ r ≠ r0 → x + r0 ' ≠ x0 + r ' + x1; from specification list[reflist]%inapp-14: ⊦ ¬ r ∈ x0 → (x0 + r ' = x1 + r ' + x2 ↔ x1 = x0 ∧ x2 = []); from specification list[reflist]%e-01: ⊦ r ' + x0 ≠ []; from specification list[reflist]%x0 ≠ [] ⊦ r = x0.last ∧ x1 = x0.butlast ↔ x0 = x1 + r '; (elim) Used terms: x2, r0, x1