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Proof step 57 from proof reachable-addnew-rev (previous step)

path(ce.nxt ' + x, H[H[r].nxt, ce]), r ∈ H, (ce.nxt ' + x).last ∈ H ⊦ H[r].nxt = null, H[r].nxt = ce.nxt, r = H[r].nxt, r = null, r = ce.nxt, dups(x), path(ce.nxt ' + x, H), reachable(r, (ce.nxt ' + x).last, H), H[r].nxt ∈ H, r ∈ x, H[r].nxt ∈ x, ce.nxt ∈ x

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 58:
⊦ ¬ r2 ∈ H ∧ ¬ r2 ∈ x ∧ r ∈ H → (path(r ' + x, H[r2, ce]) ↔ path(r ' + x, H))
Proof step 59:
path(ce.nxt ' + x, H[H[r].nxt, ce]), (ce.nxt ' + x).last ∈ H, r ∈ H ⊦ H[r].nxt = ce.nxt, H[r].nxt = null, r = ce.nxt, r = null, r = H[r].nxt, dups(x), path(ce.nxt ' + x, H), reachable(r, (ce.nxt ' + x).last, H), ce.nxt ∈ x, H[r].nxt ∈ x, r ∈ x, H[r].nxt ∈ H, ¬ H[r].nxt ∈ H ∧ ¬ H[r].nxt ∈ x ∧ ce.nxt ∈ H

Rule argument

spec: <toplevel> name: path-new-irrelevant-strong seq: ⊦ ¬ r2 ∈ H ∧ ¬ r2 ∈ x ∧ r ∈ H → (path(r ' + x, H[r2, ce]) ↔ path(r ' + x, H)) subst: [ce, r2, H, x, r] → [ce, H[r].nxt, H, x, ce.nxt] flags: <all paths>

Additional information

Used terms: ce, H[r].nxt, H, x, ce.nxt