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Proof step 8 from proof linearity (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
r1 ∈ H, path(r ' + x, H), reachable(r, r1, H), x.last ∈ H, ∀ x0, r, H, r1. # x0 < # x → path(r ' + x0, H) ∧ reachable(r, r1, H) ∧ (r ' + x0).last ∈ H ∧ r1 ∈ H → reachable((r ' + x0).last, r1, H) ∨ reachable(r1, (r ' + x0).last, H) ⊦ x = [], reachable(x.last, r1, H), reachable(r1, x.last, H)

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%l-02: ⊦ x0 ≠ [] → (r ' + x0).last = x0.last; from specification list-last[reflist]