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Proof step 21 from proof rem-stays-reachable-aux1 (previous step)

x = [], path((r0 ' + x) + H[r0].nxt ', H), r0 ∈ H, H[r0].nxt ∈ H, path(H[H[r0].nxt].nxt ' + x0 + (r0 ' + x) + H[r0].nxt ', H) ⊦ H[r0].nxt = H[H[r0].nxt].nxt, H[r0].nxt = null, H[H[r0].nxt].nxt = r0, H[r0].nxt = r0, reachable(H[H[r0].nxt].nxt, r0, H), H[H[r0].nxt].nxt ∈ x0, H[H[r0].nxt].nxt ∈ x, H[r0].nxt ∈ x0, H[r0].nxt ∈ x, r0 ∈ x0, r0 ∈ x, dups(x0 + x), dups(x0), dups(x), ∃ x. path(H[r0].nxt ' + x + H[r0].nxt ', H) ∧ ¬ r0 ∈ x

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 22:
r0 ∈ H, H[r0].nxt ∈ H, path(H[H[r0].nxt].nxt ' + x0 + r0 ' + H[r0].nxt ', H) ⊦ H[r0].nxt = H[H[r0].nxt].nxt, H[r0].nxt = null, H[H[r0].nxt].nxt = r0, H[r0].nxt = r0, r0 = null, reachable(H[H[r0].nxt].nxt, r0, H), H[H[r0].nxt].nxt ∈ x0, H[r0].nxt ∈ x0, r0 ∈ x0, dups(x0), ∃ x. path(H[r0].nxt ' + x + H[r0].nxt ', H) ∧ ¬ r0 ∈ x

Rule argument

Additional information

Used simplifier rules: path-two: ⊦ path(r ' + r0 ', H) ↔ ¬ ¬(r ∈ H ∧ r0 ∈ H ∧ r ≠ null ∧ r0 ≠ null ∧ H[r].nxt = r0);%e-01: ⊦ ¬ dups([]); from specification list-dup[reflist]%app: ⊦ x0 + [] = x0; from specification list[reflist]%in-empty: ⊦ ¬ r ∈ []; from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%Nil: ⊦ [] + x0 = x0; from specification list[reflist]