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Proof step 46 from proof rem-stays-reachable-aux (previous step)

ce.nxt = H[H[r0].nxt].nxt, path(r ' + x1 + r0 ', H), path(r ' + x2 + H[r0].nxt ', H[r0, ce]), H[r0].nxt ∈ H[r0, ce], r0 ∈ H, H[r0].nxt ∈ H ⊦ H[r0].nxt = null, H[r0].nxt = ce.nxt, r = H[r0].nxt, r = r0, dups(x2), dups(x1), H[r0].nxt ∈ x2, r0 ∈ x1, r ∈ x2, r ∈ x1, r0 ∈ x2, ∃ x. path(r ' + x + H[r0].nxt ', H) ∧ ¬ r0 ∈ x, ∃ x0. path(H[r0].nxt ' + x0 + H[r0].nxt ', H) ∧ ¬ r0 ∈ x0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 47:
⊦ ¬ r0 ∈ x ∧ r0 ∈ H → (path(x + r1 ', H[r0, ce]) ↔ path(x + r1 ', H))
Proof step 48:
ce.nxt = H[H[r0].nxt].nxt, path((r ' + x2) + H[r0].nxt ', H), path(r ' + x1 + r0 ', H), H[r0].nxt ∈ H, H[r0].nxt ∈ H[r0, ce], r0 ∈ H ⊦ H[r0].nxt = ce.nxt, H[r0].nxt = null, r = r0, r = H[r0].nxt, dups(x1), dups(x2), r0 ∈ x2, r ∈ x1, r ∈ x2, r0 ∈ x1, H[r0].nxt ∈ x2, r0 ∈ r ' + x2, ∃ x0. path(H[r0].nxt ' + x0 + H[r0].nxt ', H) ∧ ¬ r0 ∈ x0, ∃ x. path(r ' + x + H[r0].nxt ', H) ∧ ¬ r0 ∈ x

Rule argument

spec: <toplevel> name: path-modify-strong seq: ⊦ ¬ r0 ∈ x ∧ r0 ∈ H → (path(x + r1 ', H[r0, ce]) ↔ path(x + r1 ', H)) subst: [ce, r0, H, r1, x] → [ce, r0, H, H[r0].nxt, r ' + x2] flags: <all paths>

Additional information

Used simplifier rules: ⊦ r ≠ r0 → (r ∈ r0 ' + x0 ↔ r ∈ x0); (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist] Used terms: ce, r0, H, H[r0].nxt, r ' + x2