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Proof step 77 from proof reachable-addnew-rev (previous step)

path(r ' + x, H), r ∈ H, (r ' + x).last ∈ H ⊦ dups(x), reachable(r, (r ' + x).last, H), r2 ∈ x, r2 ∈ H, r ∈ x, ∃ r0. reachable(r, r0, H) ∧ H[r0].nxt = r2 ∧ H[r0].nxt ≠ null ∧ reachable(ce.nxt, (r ' + x).last, H)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 78:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 79:
path(r ' + x, H), (r ' + x).last ∈ H, r ∈ H ⊦ dups(x), r ∈ x, r2 ∈ H, r2 ∈ x, ∃ r0. reachable(r, r0, H) ∧ H[r0].nxt = r2 ∧ H[r0].nxt ≠ null ∧ reachable(ce.nxt, (r ' + x).last, H), ∃ x0. path(r ' + x0, H) ∧ (r ' + x0).last = (r ' + x).last

Rule argument

spec: <toplevel> name: reachable-def seq: ⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0) subst: [H, r0, r] → [H, (r ' + x).last, r] flags: <all paths>

Additional information

Used terms: H, (r ' + x).last, r