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Proof step 166 from proof rem-stays-reachable-aux2 (previous step)

path(r ' + x + H[r1].nxt ', H), H[(r ' + x).last].nxt = H[r1].nxt, (r1 ' + x0).last = (r ' + x).last, path(r1 ' + x0, H), r1 ∈ H, path(r ' + x + H[r1].nxt ' + (r0 ' + x3) + r1 ', H), path(r ' + x, H), (r ' + x).last ∈ H, H[r1].nxt ∈ H ⊦ H[r1].nxt = null, H[r1].nxt = r1, r = H[r1].nxt, r1 = r, r1 = r0, H[r1].nxt = r0, r = r0, r1 ∈ x3, H[r1].nxt ∈ x3, H[r1].nxt ∈ x, r ∈ x3, r ∈ x, dups(x0), r1 ∈ x0, r1 ∈ x, r0 ∈ x3, r0 ∈ x, dups(x + x3), dups(x), dups(x3)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 167:
⊦ path(r ' + x + r0 ', H) ↔ path(r ' + x, H) ∧ H[(r ' + x).last].nxt = r0 ∧ r0 ∈ H ∧ r0 ≠ null
Proof step 168:
path(r ' + x, H) ∧ H[(r ' + x).last].nxt = H[r1].nxt ∧ H[r1].nxt ∈ H ∧ H[r1].nxt ≠ null, H[(r ' + x).last].nxt = H[r1].nxt, (r1 ' + x0).last = (r ' + x).last, path(r1 ' + x0, H), r1 ∈ H, path(r ' + x + H[r1].nxt ' + (r0 ' + x3) + r1 ', H), path(r ' + x, H), (r ' + x).last ∈ H, H[r1].nxt ∈ H ⊦ H[r1].nxt = null, H[r1].nxt = r1, r = H[r1].nxt, r1 = r, r1 = r0, H[r1].nxt = r0, r = r0, r1 ∈ x3, H[r1].nxt ∈ x3, H[r1].nxt ∈ x, r ∈ x3, r ∈ x, dups(x0), r1 ∈ x0, r1 ∈ x, r0 ∈ x3, r0 ∈ x, dups(x + x3), dups(x), dups(x3)

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, H[r1].nxt, x, r] flags: rewrite, current

Additional information

Used terms: H, H[r1].nxt, x, r