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Proof step 3 from proof reachable-trans (previous step)

reachable(r0, r1, H), reachable(r, r0, H), r1 ∈ H, r0 ∈ H ⊦ reachable(r, r1, H)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 5:
reachable(r0, r1, H), r0 ∈ H, r1 ∈ H, ∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0 ⊦ reachable(r, r1, H)

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, r0, r] flags: <all paths>

Additional information

Used terms: H, r0, r