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Proof step 9 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(r1, (r ' + x0).last, H) ∨ reachable((r ' + x0).last, r1, H) ⊦ x = [], reachable(r1, (r ' + x).last, H), reachable((r ' + x).last, r1, H)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
path(r ' + x, H), reachable(r, r1, H), 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(r1, (r ' + x0).last, H) ∨ reachable((r ' + x0).last, r1, 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]% ⊦ x0 ≠ [] → (r ' + x0).last = x0.last; (s)