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Proof step 36 from proof rem-stays-reachable-aux1 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 37:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 38:
x ≠ [], x.last = r0, path(H[H[r0].nxt].nxt ' + x, H), path(H[H[r0].nxt].nxt ' + x0 + H[r0].nxt ', H), H[r0].nxt ∈ H, r0 ∈ H ⊦ r0 = H[H[r0].nxt].nxt, H[r0].nxt = null, H[r0].nxt = H[H[r0].nxt].nxt, r0 ∈ x0, H[H[r0].nxt].nxt ∈ x, dups(x)

Rule argument

spec: list-last inst: reflist name: lapp seq: ⊦ x0 ≠ [] → (x1 + x0).last = x0.last subst: [x0, x1] → [x, H[H[r0].nxt].nxt '] flags: rewrite

Additional information

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