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Proof step 15 from proof acyclic-rem-not-reachable (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 16:
ce.nxt = H[H[r0].nxt].nxt, r0 ∈ H, path(r ' + (x0 + r0 ' + x) + H[r0].nxt ', H[r0, ce]), reachable(r, r0, H), H[r0].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = null, r = H[r0].nxt, r = r0, H[r0].nxt = r0, r0 ∈ x, r ∈ x0, r ∈ x, H[r0].nxt ∈ x0, H[r0].nxt ∈ x, r0 ∈ x0, dups(x0 + x), dups(x0), dups(x)

Rule argument

Additional information

Used simplifier rules: a: ⊦ ¬ dups(x0 + x1) → ¬ dups(x0) ∧ ¬ dups(x1); (forward) from specification list-dup[reflist]%f-01: ⊦ dups(x0 + r ' + x1) ↔ r ∈ x0 ∨ r ∈ x1 ∨ dups(x0 + x1); from specification list-dup[reflist]%in-neq: ⊦ r ≠ r0 → (r ∈ H[r0, ce] ↔ r ∈ H); from specification store[linheap]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%in-rec: ⊦ r ∈ r0 ' + x0 ↔ ¬ ¬(r = r0 ∨ r ∈ x0); from specification list[reflist]%in-append: ⊦ r ∈ x0 + x1 ↔ ¬ ¬(r ∈ x0 ∨ r ∈ x1); from specification list[reflist]