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Proof step 97 from proof reachable-addnew-rev (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 98:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 99:
path(ce.nxt ' + x, H[H[r0].nxt, ce]), reachable(r, r0, H), reachable(r, r0, H[H[r0].nxt, ce]), reachable(ce.nxt, (ce.nxt ' + x).last, H), (ce.nxt ' + x).last ∈ H, (ce.nxt ' + x).last ∈ H[H[r0].nxt, ce], r0 ∈ H[H[r0].nxt, ce], r0 ∈ H, r ∈ H ⊦ H[r0].nxt = null, reachable(r, (ce.nxt ' + x).last, H), reachable(r, (ce.nxt ' + x).last, H[H[r0].nxt, ce]), H[r0].nxt ∈ H, ∃ x0. path(r0 ' + x0, H[H[r0].nxt, ce]) ∧ (r0 ' + x0).last = (ce.nxt ' + 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[H[r0].nxt, ce], (ce.nxt ' + x).last, r0] flags: <all paths>

Additional information

Used terms: H[H[r0].nxt, ce], (ce.nxt ' + x).last, r0