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Proof step 21 from proof acyclic-rem-not-reachable (previous step)

ce.nxt = H[H[r0].nxt].nxt, x = r2 + x1, path(r0 ' + x + H[r0].nxt ', H[r0, ce]), path(r ' + (x0 + r0 ' + x) + H[r0].nxt ', H[r0, ce]), reachable(r, r0, H), r0 ∈ H, H[r0].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = null, H[r0].nxt = r0, r = H[r0].nxt, r = r0, dups(x0 + x), dups(x0), dups(x), r0 ∈ x, H[r0].nxt ∈ x0, H[r0].nxt ∈ x, r ∈ x0, r ∈ x, r0 ∈ x0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 22:
ce.nxt = H[H[r0].nxt].nxt, path(ce.nxt ' + x1 + H[r0].nxt ', H[r0, ce]), path(r ' + x0 + r0 ' + (ce.nxt ' + x1) + H[r0].nxt ', H[r0, ce]), reachable(r, r0, H), r0 ∈ H, H[r0].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = ce.nxt, H[r0].nxt = null, H[r0].nxt = r0, r = ce.nxt, r = H[r0].nxt, r = r0, r0 = ce.nxt, r0 = null, dups(x1), dups(x0), dups(x0 + x1), r0 ∈ x1, H[r0].nxt ∈ x0, H[r0].nxt ∈ x1, r ∈ x0, r ∈ x1, r0 ∈ x0, ce.nxt ∈ x1, ce.nxt ∈ x0

Rule argument

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, H) ↔ r ∈ H ∧ r ≠ null ∧ H[r].nxt = r0 ∧ path(r0 ' + x, H);%a: ⊦ ¬ dups(x0 + x1) → ¬ dups(x0) ∧ ¬ dups(x1); (forward) from specification list-dup[reflist]% ⊦ ¬ r ∈ x0 → (dups(r ' + x0) ↔ dups(x0)); (s)% ⊦ dups(x0 + r ' + x1) ↔ r ∈ x0 ∨ r ∈ x1 ∨ dups(x0 + x1); (ws)% ⊦ r ∈ r0 ' + x0 ↔ ¬(r ≠ r0 ∧ ¬ r ∈ x0); (ss)% ⊦ H[r, ce][r] = ce; (s)% ⊦ r ∈ H[r, ce]; (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]% ⊦ r + x0 = r ' + x0; (s)