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Proof step 174 from proof rem-stays-reachable-aux2 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 175:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 176:
x0 ≠ [], H[x0.last].nxt = H[r1].nxt, x0.last ∈ H, H[r1].nxt ∈ H, path(H[H[r1].nxt].nxt ' + x3 + r1 ', H), path(r1 ' + x0, H), r1 ∈ H ⊦ x0.last = null, H[r1].nxt = r1, r1 = x0.last, r1 = H[H[r1].nxt].nxt, H[r1].nxt = H[H[r1].nxt].nxt, x0.last = H[H[r1].nxt].nxt, H[r1].nxt = null, dups(x3), r1 ∈ x3, H[r1].nxt ∈ x3, x0.last ∈ x3, dups(x0), r1 ∈ x0, H[H[r1].nxt].nxt ∈ x3

Rule argument

spec: list-last inst: reflist name: lapp seq: ⊦ x0 ≠ [] → (x1 + x0).last = x0.last subst: [x0, x1] → [x0, r1 '] 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: x0, r1 '