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Proof step 20 from proof acyclic-rem-not-reachable (previous step)

x = r2 + x1, path(r0 ' + x + H[r0].nxt ', H[r0, ce]), 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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 21:
ce.nxt = H[H[r0].nxt].nxt, path(r ' + x0 + r0 ' + (ce.nxt ' + x1) + H[r0].nxt ', H[r0, ce]), r0 ∈ H, path(ce.nxt ' + x1 + H[r0].nxt ', H[r0, ce]), H[r0].nxt ∈ H, reachable(r, r0, H), ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = ce.nxt, r = ce.nxt, r0 = ce.nxt, r0 = null, H[r0].nxt = r0, r = r0, r = H[r0].nxt, H[r0].nxt = null, dups(x1), dups(x0), dups(x0 + x1), ce.nxt ∈ x1, ce.nxt ∈ x0, r0 ∈ x0, H[r0].nxt ∈ x1, H[r0].nxt ∈ x0, r ∈ x1, r ∈ x0, r0 ∈ x1

Rule argument

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, H) ↔ ¬ ¬(r ∈ H ∧ r ≠ null ∧ H[r].nxt = r0 ∧ path(r0 ' + x, H));%a: ⊦ ¬ dups(x0 + x1) → ¬ dups(x0) ∧ ¬ dups(x1); (forward) from specification list-dup[reflist]%c-02: ⊦ ¬ r ∈ x0 → (dups(r ' + x0) ↔ dups(x0)); from specification list-dup[reflist]%f-01: ⊦ dups(x0 + r ' + x1) ↔ r ∈ x0 ∨ r ∈ x1 ∨ dups(x0 + x1); from specification list-dup[reflist]%in-06: ⊦ ¬ r ∈ r0 ' + x0 ↔ r ≠ r0 ∧ ¬ r ∈ x0; from specification list[reflist]%At-same: ⊦ H[r, ce][r] = ce; from specification store[linheap]%insin: ⊦ r ∈ H[r, ce]; from specification store[linheap]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%c: ⊦ r + x0 = r ' + x0; from specification list[reflist]