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Proof step 7 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) ⊦ reachable((r ' + x).last, r1, H), reachable(r1, (r ' + x).last, H)

User applied rule cut formula

And obtained the following new goals

Proof step 8:
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)

Rule argument

fma: x = []

Additional information

Used simplifier rules: path-one: ⊦ path(r ', H) ↔ ¬ ¬(r ∈ H ∧ r ≠ null);%app: ⊦ x0 + [] = x0; from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%l: ⊦ r '.last = r; from specification list-last[reflist] Used formulas: x = []