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Proof step 57 from proof acyclic-rem-not-reachable (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 58:
ce.nxt = H[H[r0].nxt].nxt, r0 ∈ H, path(r ' + x1 + H[r0].nxt ', H), path(r ' + x2 + r0 ', H), H[r0].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = null, r = H[r0].nxt, r0 = r, r0 = H[r0].nxt, r0 ∈ x1, r ∈ x1, H[r0].nxt ∈ x1, dups(x1), r ∈ x2, r0 ∈ x2, dups(x2)

Rule argument

Additional information

Used simplifier rules: f: ⊦ dups(x0 + r ') ↔ r ∈ x0 ∨ dups(x0); from specification list-dup[reflist]%in-17: ⊦ r ∈ r0 ' ↔ r = r0; from specification list[reflist]%in-append: ⊦ r ∈ x0 + x1 ↔ ¬ ¬(r ∈ x0 ∨ r ∈ x1); from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%e-02: ⊦ x0 + r ' ≠ []; from specification list[reflist]