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

x = [], 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 19:
ce.nxt = H[ce.nxt].nxt, ce.nxt = H[r0].nxt, reachable(r, r0, H), ce.nxt ∈ H, r0 ∈ H, path(r ' + x0 + r0 ' + ce.nxt ', H[r0, ce]), ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ ce.nxt = null, r = ce.nxt, r = r0, ce.nxt = r0, r0 = null, r ∈ x0, ce.nxt ∈ x0, r0 ∈ x0, dups(x0)

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]%At-same: ⊦ H[r, ce][r] = ce; from specification store[linheap]%in-neq: ⊦ r ≠ r0 → (r ∈ H[r0, ce] ↔ r ∈ H); from specification store[linheap]%insin: ⊦ r ∈ H[r, ce]; from specification store[linheap]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%Nil: ⊦ [] + x0 = x0; from specification list[reflist]