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Proof step 163 from proof rem-stays-reachable-aux2 (previous step)

x1 + r2 ' + x2 = x + H[r1].nxt ' + r0 ' + x3, (r1 ' + x0).last = (r ' + x).last, (r ' + x1).last = (r ' + x).last, H[(r ' + x).last].nxt = H[r1].nxt, path(r1 ' + x0, H), path(r ' + x, H), path(r ' + x1 + r2 ' + x2 + r1 ', H), path(r ' + x1, H), path(r ' + x1 + r2 ', H), r1 ∈ H, (r ' + x).last ∈ H, H[r1].nxt ∈ H ⊦ H[r1].nxt = null, r = r2, r1 = r, r2 = r1, dups(x2), dups(x1), dups(x1 + x2), dups(x0), r1 ∈ x2, r1 ∈ x1, r2 ∈ x2, r2 ∈ x1, r1 ∈ x, r1 ∈ x0, r ∈ x1, r ∈ x2

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 164:
⊦ path(r ' + x + r0 ', H) ↔ path(r ' + x, H) ∧ H[(r ' + x).last].nxt = r0 ∧ r0 ∈ H ∧ r0 ≠ null
Proof step 165:
x1 + r2 ' + x2 = x + H[r1].nxt ' + r0 ' + x3, (r ' + x1).last = (r ' + x).last, (r1 ' + x0).last = (r ' + x).last, H[(r ' + x1).last].nxt = r2, H[(r ' + x).last].nxt = H[r1].nxt, path(r ' + x1, H), path(r ' + x1 + r2 ' + x2 + r1 ', H), path(r ' + x, H), path(r1 ' + x0, H), H[r1].nxt ∈ H, (r ' + x).last ∈ H, r1 ∈ H, r2 ∈ H ⊦ H[r1].nxt = null, r = r2, r1 = r, r2 = r1, r2 = null, dups(x0), dups(x1 + x2), dups(x1), dups(x2), r ∈ x2, r ∈ x1, r1 ∈ x0, r1 ∈ x, r2 ∈ x1, r2 ∈ x2, r1 ∈ x1, r1 ∈ x2

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, r2, x1, r] flags: <all paths>

Additional information

Used terms: H, r2, x1, r