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Proof step 30 from proof acyclic-rem-not-reachable (previous step)

ce.nxt = H[H[r0].nxt].nxt, path(r ' + x1 + H[r0].nxt ', H[r0, ce]), reachable(r, r0, H), H[r0].nxt ∈ H[r0, ce], r0 ∈ H, H[r0].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = null, r = H[r0].nxt, dups(x1), H[r0].nxt ∈ x1, r ∈ x1, r0 ∈ x1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 31:
⊦ ¬ r0 ∈ x → (path(x, H[r0, ce]) ↔ path(x, H))
Proof step 32:
ce.nxt = H[H[r0].nxt].nxt, path(r ' + x1 + H[r0].nxt ', H[r0, ce]), reachable(r, r0, H), r0 ∈ r ' + x1 + H[r0].nxt ', H[r0].nxt ∈ H, r0 ∈ H, H[r0].nxt ∈ H[r0, ce], ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = null, r = H[r0].nxt, dups(x1), r0 ∈ x1, r ∈ x1, H[r0].nxt ∈ x1
Proof step 50:
ce.nxt = H[H[r0].nxt].nxt, path(r ' + x1 + H[r0].nxt ', H), reachable(r, r0, H), H[r0].nxt ∈ H, r0 ∈ H, H[r0].nxt ∈ H[r0, ce], ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = null, r = H[r0].nxt, dups(x1), r0 ∈ x1, r ∈ x1, H[r0].nxt ∈ x1, r0 ∈ r ' + x1 + H[r0].nxt '

Rule argument

spec: <toplevel> name: path-modify seq: ⊦ ¬ r0 ∈ x → (path(x, H[r0, ce]) ↔ path(x, H)) subst: [ce, r0, H, x] → [ce, r0, H, r ' + x1 + H[r0].nxt '] flags: <all paths>

Additional information

Used terms: ce, r0, H, r ' + x1 + H[r0].nxt '