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Proof step 207 from proof rem-stays-reachable-aux2 (previous step)

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

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, r, x1]

Additional information

Used simplifier rules: ⊦ x0 + r ' ≠ []; (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]% ⊦ r ∈ x0 + x1 ↔ r ∈ x0 ∨ r ∈ x1; (s)% ⊦ r ∈ r0 ' ↔ r = r0; (s)% ⊦ dups(x0 + r ') ↔ r ∈ x0 ∨ dups(x0); (ws)%in-15: ⊦ x0 + r ' = x1 → r ∈ x1; (forward) from specification list[reflist]% ⊦ ¬ r ∈ x ∧ r ≠ r0 → x + r0 ' ≠ x0 + r ' + x1; (s)% ⊦ ¬ r ∈ x0 → (x0 + r ' = x1 + r ' + x2 ↔ x1 = x0 ∧ x2 = []); (ws)% ⊦ r ' + x0 ≠ []; (s)%x0 ≠ [] ⊦ r = x0.last ∧ x1 = x0.butlast ↔ x0 = x1 + r '; (elim) Used terms: x2, r, x1