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Proof step 9 from proof linearity (previous step)

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)

User applied rule insert elim lemma

And obtained the following new goals

Proof step 10:
x = x1 + r0 ', r1 ∈ H, path(r ' + x, H), reachable(r, r1, H), r0 ∈ 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(r0, r1, H), reachable(r1, r0, H)

Rule argument

spec: list-last inst: reflist name: last-elim seq: x0 ≠ [] ⊦ r = x0.last ∧ x1 = x0.butlast ↔ x0 = x1 + r ' subst: [x0, r, x1] → [x, r0, x1] flags: []

Additional information

Used simplifier rules: x0 ≠ [] ⊦ r = x0.last ∧ x1 = x0.butlast ↔ x0 = x1 + r '; (elim) Used terms: x, r0, x1