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Proof step 56 from proof rem-stays-reachable-aux1 (previous step)

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

User applied rule insert elim lemma

And obtained the following new goals

Proof step 57:
x = x1 + r ', path(H[H[r].nxt].nxt ' + x0 + H[r].nxt ' + r1 ' + x, H), path(H[H[r].nxt].nxt ' + x0 + H[r].nxt ', H), r ∈ H, H[r].nxt ∈ H ⊦ H[H[r].nxt].nxt = r1, H[r].nxt = r1, H[r].nxt = H[H[r].nxt].nxt, H[r].nxt = null, r = r1, r = H[H[r].nxt].nxt, r = H[r].nxt, x = [], dups(x1), dups(x0), dups(x0 + x1), r1 ∈ x0, r1 ∈ x1, r ∈ x1, H[H[r].nxt].nxt ∈ x0, H[H[r].nxt].nxt ∈ x1, r ∈ x0, H[r].nxt ∈ x0, H[r].nxt ∈ x1

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, r, x1]

Additional information

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