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Proof step 15 from proof linearity (previous step)

reachable((r ' + x1).last, r1, H) ∨ reachable(r1, (r ' + x1).last, H), path(r ' + x1, H), H[(r ' + x1).last].nxt ∈ H, reachable(r, r1, H), path(r ' + x1 + H[(r ' + x1).last].nxt ', H), r1 ∈ H, ∀ x0, r2, H0, r1. # x0 < #(x1 + H[(r ' + x1).last].nxt ') → path(r2 ' + x0, H0) ∧ reachable(r2, r1, H0) ∧ (r2 ' + x0).last ∈ H0 ∧ r1 ∈ H0 → reachable((r2 ' + x0).last, r1, H0) ∨ reachable(r1, (r2 ' + x0).last, H0) ⊦ H[(r ' + x1).last].nxt = null, reachable(r1, H[(r ' + x1).last].nxt, H), reachable(H[(r ' + x1).last].nxt, r1, H)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 16:
r1 ∈ H, path(r ' + x1 + H[(r ' + x1).last].nxt ', H), reachable(r, r1, H), H[(r ' + x1).last].nxt ∈ H, path(r ' + x1, H), reachable((r ' + x1).last, r1, H), ∀ x0, r2, H0, r1. # x0 < #(x1 + H[(r ' + x1).last].nxt ') → path(r2 ' + x0, H0) ∧ reachable(r2, r1, H0) ∧ (r2 ' + x0).last ∈ H0 ∧ r1 ∈ H0 → reachable((r2 ' + x0).last, r1, H0) ∨ reachable(r1, (r2 ' + x0).last, H0) ⊦ H[(r ' + x1).last].nxt = null, reachable(H[(r ' + x1).last].nxt, r1, H), reachable(r1, H[(r ' + x1).last].nxt, H), reachable(r1, (r ' + x1).last, H)

Rule argument

Additional information

Used simplifier rules: reach-reachnext: ⊦ reachable(r, r1, H) → (¬ reachable(r, H[r1].nxt, H) ↔ H[r1].nxt = null ∨ ¬ H[r1].nxt ∈ H);