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Proof step 5 from proof reachable-modbehind (previous step)

path(r ' + x, H), reachable((r ' + x).last, r1, H), (r ' + x).last ∈ H, r1 ∈ H ⊦ reachable(r1, (r ' + x).last, H), reachable(r, (r ' + x).last, H[r1, ce])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 7:
path(r ' + x, H), reachable((r ' + x).last, r1, H), (r ' + x).last ∈ H, r1 ∈ H ⊦ reachable(r1, (r ' + x).last, H), ∃ x0. path(r ' + x0, H[r1, ce]) ∧ (r ' + x0).last = (r ' + x).last

Rule argument

spec: inst: name: reachable-def seq: ⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0) subst: [H, r0, r] → [H[r1, ce], (r ' + x).last, r] flags: rewrite, current

Additional information

Used terms: H[r1, ce], (r ' + x).last, r