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Proof step 118 from proof rem-stays-reachable-aux2 (previous step)

(r0 ' + x0).last = (r ' + x1).last, path(r ' + x1 + H[r0].nxt ' + r0 ', H), path(r ' + x1 + H[r0].nxt ', H), path(r0 ' + x0, H), path(r ' + x1, H), (r ' + x1).last ∈ H, r0 ∈ H, H[r0].nxt ∈ H ⊦ H[r0].nxt = null, H[r0].nxt = r0, r = H[r0].nxt, r0 = r, dups(x1), dups(x0), H[r0].nxt ∈ x1, r0 ∈ x1, r ∈ x1, r0 ∈ x0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 119:
⊦ path(r ' + x + r0 ', H) ↔ path(r ' + x, H) ∧ H[(r ' + x).last].nxt = r0 ∧ r0 ∈ H ∧ r0 ≠ null
Proof step 120:
(r0 ' + x0).last = (r ' + x1).last, H[(r ' + x1 + H[r0].nxt ').last].nxt = r0, path(r ' + x1, H), path(r0 ' + x0, H), path(r ' + x1 + H[r0].nxt ', H), H[r0].nxt ∈ H, (r ' + x1).last ∈ H, r0 ∈ H ⊦ H[r0].nxt = r0, H[r0].nxt = null, r = H[r0].nxt, r0 = r, r0 = null, dups(x0), dups(x1), r0 ∈ x0, r ∈ x1, r0 ∈ x1, H[r0].nxt ∈ x1

Rule argument

spec: <toplevel> name: path-split-last seq: ⊦ path(r ' + x + r0 ', H) ↔ path(r ' + x, H) ∧ H[(r ' + x).last].nxt = r0 ∧ r0 ∈ H ∧ r0 ≠ null subst: [H, r0, x, r] → [H, r0, x1 + H[r0].nxt ', r] flags: <all paths>

Additional information

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