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Proof step 63 from proof acyclic-rem-not-reachable (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 64:
⊦ path(r ' + x + r0 ', H) ↔ path(r ' + x, H) ∧ H[(r ' + x).last].nxt = r0 ∧ r0 ∈ H ∧ r0 ≠ null
Proof step 65:
ce.nxt = H[H[r0].nxt].nxt, path(r ' + x1 + H[r0].nxt ', H) ∧ H[(r ' + x1 + H[r0].nxt ').last].nxt = r0 ∧ r0 ∈ H ∧ r0 ≠ null, path(r ' + x1 + H[r0].nxt ', H), r0 ∈ H, H[r0].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ r0 = H[r0].nxt, r0 = r, r = H[r0].nxt, H[r0].nxt = null, H[r0].nxt ∈ x1, r ∈ x1, r0 ∈ x1, dups(x1)

Rule argument

spec: inst: 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: rewrite, current

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