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Proof step 16 from proof acyclic-rem-not-reachable (previous step)

ce.nxt = H[H[r0].nxt].nxt, path(r ' + x1 + H[r0].nxt ', H[r0, ce]), reachable(r, r0, H), H[r0].nxt ∈ H, r0 ∈ H, H[r0].nxt ∈ H[r0, ce], ∀ r1. ¬ reachable(H[r1].nxt, r1, H), ∃ x, x2. x1 = x + r0 ' + x2 ∧ ¬ r0 ∈ x2 ⊦ H[r0].nxt = null, r = H[r0].nxt, dups(x1), r ∈ x1, H[r0].nxt ∈ x1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 17:
ce.nxt = H[H[r0].nxt].nxt, 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

Rule argument

Additional information

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