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Proof step 13 from proof modnotreachable-reachable-rev (previous step)

path(H[r].nxt ' + x0, H[r0, ce]), reachable(H[r].nxt, (H[r].nxt ' + x0).last, H), r ∈ H, r0 ∈ H, (H[r].nxt ' + x0).last ∈ H[r0, ce], (H[r].nxt ' + x0).last ∈ H ⊦ r = null, r = r0, reachable(r, r0, H), reachable(r, (H[r].nxt ' + x0).last, H), reachable(H[r].nxt, r0, H)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 15:
path(H[r].nxt ' + x0, H[r0, ce]), (H[r].nxt ' + x0).last ∈ H, (H[r].nxt ' + x0).last ∈ H[r0, ce], r0 ∈ H, r ∈ H, ∃ x. path(H[r].nxt ' + x, H) ∧ (H[r].nxt ' + x).last = (H[r].nxt ' + x0).last ⊦ r = r0, r = null, reachable(H[r].nxt, r0, H), reachable(r, (H[r].nxt ' + x0).last, H), reachable(r, r0, 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, (H[r].nxt ' + x0).last, H[r].nxt] flags: <all paths>

Additional information

Used terms: H, (H[r].nxt ' + x0).last, H[r].nxt