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Proof step 17 from proof reachable-addnew-rev (previous step)

path(r ' + x1 + r1 ', H[r1, ce]), r ∈ H ⊦ dups(x1), r1 ∈ x1, r1 ∈ H, r ∈ x1, reachable(r, (r ' + x1).last, H) ∧ H[(r ' + x1).last].nxt = r1 ∧ H[(r ' + x1).last].nxt ≠ null

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 18:
⊦ path(r ' + x + r0 ', H) ↔ path(r ' + x, H) ∧ H[(r ' + x).last].nxt = r0 ∧ r0 ∈ H ∧ r0 ≠ null
Proof step 19:
H[r1, ce][(r ' + x1).last].nxt = r1, path(r ' + x1, H[r1, ce]), r ∈ H, r1 ∈ H[r1, ce] ⊦ r1 = null, dups(x1), r ∈ x1, r1 ∈ H, r1 ∈ x1, reachable(r, (r ' + x1).last, H) ∧ H[(r ' + x1).last].nxt = r1 ∧ H[(r ' + x1).last].nxt ≠ null

Rule argument

spec: <toplevel> name: path-split-last seq: ⊦ path(r ' + x + r0 ', H) ↔ path(r ' + x, H) ∧ H[(r ' + x).last].nxt = r0 ∧ r0 ∈ H ∧ r0 ≠ null subst: [H, r0, x, r] → [H[r1, ce], r1, x1, r] flags: <all paths>

Additional information

Used terms: H[r1, ce], r1, x1, r