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Proof step 11 from proof linearity (previous step)

r1 ∈ H, path(r ' + x1 + r0 ', H), reachable(r, r1, H), r0 ∈ H, ∀ x0, r, H, r1. # x0 < #(x1 + r0 ') → 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(r0, r1, H), reachable(r1, r0, H)

User applied rule insert lemma

And obtained the following new goals

Proof step 12:
⊦ path(r ' + x + r0 ', H) ↔ path(r ' + x, H) ∧ H[(r ' + x).last].nxt = r0 ∧ r0 ∈ H ∧ r0 ≠ null
Proof step 13:
path(r ' + x1 + r0 ', H) ↔ path(r ' + x1, H) ∧ H[(r ' + x1).last].nxt = r0 ∧ r0 ∈ H ∧ r0 ≠ null, r1 ∈ H, path(r ' + x1 + r0 ', H), reachable(r, r1, H), r0 ∈ H, ∀ x0, r, H, r1. # x0 < #(x1 + r0 ') → 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(r0, r1, H), reachable(r1, r0, H)

Rule argument

spec: inst: name: path-split-last seq: ⊦ path(r ' + x + r0 ', H) ↔ path(r ' + x, H) ∧ H[(r ' + x).last].nxt = r0 ∧ r0 ∈ H ∧ r0 ≠ null subst: [r, x, r0, H] → [r, x1, r0, H] flags: current, []

Additional information

Used terms: r, x1, r0, H