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Proof step 25 from proof rem-stays-reachable-aux (previous step)

ce.nxt = H[H[x.last].nxt].nxt, (r ' + x0).last = H[x.last].nxt, path(r ' + x0, H[x.last, ce]), path(r ' + x, H), H[x.last].nxt ∈ H, x.last ∈ H, H[x.last].nxt ∈ H[x.last, ce] ⊦ H[x.last].nxt = ce.nxt, H[x.last].nxt = null, x = [], r ∈ x, dups(x), r ∈ x0, dups(x0), H[x.last].nxt ≠ x.last ∧ r = H[x.last].nxt, ∃ x0. path(r ' + x0 + H[x.last].nxt ', H) ∧ x.last ≠ r ∧ ¬ x.last ∈ x0, ∃ x0. path(H[x.last].nxt ' + x0 + H[x.last].nxt ', H) ∧ ¬ x.last ∈ x0

User applied rule insert elim lemma

And obtained the following new goals

Proof step 26:
x = x1 + r0 ', ce.nxt = H[H[r0].nxt].nxt, (r ' + x0).last = H[r0].nxt, path(r ' + x0, H[r0, ce]), path(r ' + x, H), H[r0].nxt ∈ H, r0 ∈ H, H[r0].nxt ∈ H[r0, ce] ⊦ H[r0].nxt = ce.nxt, H[r0].nxt = null, x = [], r ∈ x, dups(x), r ∈ x0, dups(x0), H[r0].nxt ≠ r0 ∧ r = H[r0].nxt, ∃ x0. path(r ' + x0 + H[r0].nxt ', H) ∧ r0 ≠ r ∧ ¬ r0 ∈ x0, ∃ x0. path(H[r0].nxt ' + x0 + H[r0].nxt ', H) ∧ ¬ r0 ∈ x0

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] → [x, r0, x1] flags: []

Additional information

Used simplifier rules: x0 ≠ [] ⊦ r = x0.last ∧ x1 = x0.butlast ↔ x0 = x1 + r '; (elim) Used terms: x, r0, x1