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Proof step 73 from proof reachable-addnew-rev (previous step)

path(ce.nxt ' + x1, H), path(r ' + x + r0 ' + H[r0].nxt ' + ce.nxt ' + x1, H[H[r0].nxt, ce]), path(r ' + x + r0 ', H), ce.nxt ∈ H, r0 ∈ H, (ce.nxt ' + x1).last ∈ H, r ∈ H ⊦ H[r0].nxt = ce.nxt, H[r0].nxt = null, r = ce.nxt, r = r0, r0 = ce.nxt, r0 = null, r0 = H[r0].nxt, dups(x1), dups(x), dups(x + x1), reachable(ce.nxt, (ce.nxt ' + x1).last, H), reachable(r, (ce.nxt ' + x1).last, H), H[r0].nxt ∈ H, H[r0].nxt ∈ x1, ce.nxt ∈ x1, ce.nxt ∈ x, r0 ∈ x1, r0 ∈ x, H[r0].nxt ∈ x, r ∈ x1, r ∈ x

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 74:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 75:
path(r ' + x + r0 ', H), path(r ' + x + r0 ' + H[r0].nxt ' + ce.nxt ' + x1, H[H[r0].nxt, ce]), path(ce.nxt ' + x1, H), r ∈ H, (ce.nxt ' + x1).last ∈ H, r0 ∈ H, ce.nxt ∈ H ⊦ H[r0].nxt = null, H[r0].nxt = ce.nxt, r = r0, r = ce.nxt, r0 = H[r0].nxt, r0 = null, r0 = ce.nxt, dups(x + x1), dups(x), dups(x1), reachable(r, (ce.nxt ' + x1).last, H), r ∈ x, r ∈ x1, H[r0].nxt ∈ x, r0 ∈ x, r0 ∈ x1, ce.nxt ∈ x, ce.nxt ∈ x1, H[r0].nxt ∈ x1, H[r0].nxt ∈ H, ∃ x. path(ce.nxt ' + x, H) ∧ (ce.nxt ' + x).last = (ce.nxt ' + x1).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, (ce.nxt ' + x1).last, ce.nxt] flags: <all paths>

Additional information

Used terms: H, (ce.nxt ' + x1).last, ce.nxt