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Proof step 18 from proof acyclic-rem-not-reachable (previous step)

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

User applied rule constructor cut

And obtained the following new goals

Proof step 19:
ce.nxt = H[H[r0].nxt].nxt, x = [], 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
Proof step 21:
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

Rule argument

var: x

Additional information

Used simplifier rules: reflist freely generated by [], +;;