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Proof step 60 from proof acyclic-rem-not-reachable (previous step)

x1 + H[r0].nxt ' = x2 + r0 ' ∨ (∃ r, y0. x1 + H[r0].nxt ' = (x2 + r0 ') + r ' + y0) ∨ (∃ r, x0. x2 + r0 ' = (x1 + H[r0].nxt ') + r ' + x0), ce.nxt = H[H[r0].nxt].nxt, r0 ∈ H, path(r ' + x1 + H[r0].nxt ', H), path(r ' + x2 + r0 ', H), H[r0].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = null, r = H[r0].nxt, r0 = r, r0 = H[r0].nxt, r0 ∈ x1, r ∈ x1, H[r0].nxt ∈ x1, dups(x1), r ∈ x2, r0 ∈ x2, dups(x2)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 61:
ce.nxt = H[H[r0].nxt].nxt, r1 ' + x = x0 + r0 ', r0 ∈ H, path(r ' + x1 + H[r0].nxt ', H), path(r ' + (x1 + H[r0].nxt ' + x0) + r0 ', H), H[r0].nxt ∈ H, r1 ∈ x0 ∨ r1 = r0, ∀ r1. ¬ reachable(H[r1].nxt, r1, H), r0 = r1 ∨ r0 ∈ x ⊦ H[r0].nxt = null, r = H[r0].nxt, r0 = r, r0 = H[r0].nxt, r0 ∈ x1, r ∈ x1, H[r0].nxt ∈ x1, r ∈ x0, r0 ∈ x0, H[r0].nxt ∈ x0, dups(x1 + x0), dups(x1), dups(x0)

Rule argument

Additional information

Used simplifier rules: a: ⊦ ¬ dups(x0 + x1) → ¬ dups(x0) ∧ ¬ dups(x1); (forward) from specification list-dup[reflist]%f-01: ⊦ dups(x0 + r ' + x1) ↔ r ∈ x0 ∨ r ∈ x1 ∨ dups(x0 + x1); from specification list-dup[reflist]%e-02: ⊦ x0 + r ' ≠ []; from specification list[reflist]%app-04: ⊦ x0 = x0 + x1 ↔ x1 = []; from specification list[reflist]%app-01: ⊦ x0 + x1 = x0 + x2 ↔ x1 = x2; from specification list[reflist]%app-03: ⊦ x0 + x1 = x2 + x1 ↔ x0 = x2; from specification list[reflist]%in-17: ⊦ r ∈ r0 ' ↔ r = r0; from specification list[reflist]%in-14: ⊦ r ' + x0 = x1 → r ∈ x1; (forward) from specification list[reflist]%in-01: ⊦ r ≠ r0 → (r ∈ r0 ' + x0 ↔ r ∈ x0); from specification list[reflist]%in-append: ⊦ r ∈ x0 + x1 ↔ ¬ ¬(r ∈ x0 ∨ r ∈ x1); from specification list[reflist]%in-rec: ⊦ r ∈ r0 ' + x0 ↔ ¬ ¬(r = r0 ∨ r ∈ x0); from specification list[reflist]%in-15: ⊦ x0 + r ' = x1 → r ∈ x1; (forward) from specification list[reflist]%inex: ⊦ x0 + r ' = x1 + r0 ' + x2 ∧ r ≠ r0 → (∃ x3. x2 = x3 + r '); (forward) from specification list[reflist]%in-16: ⊦ x0 + r ' = x1 + r0 ' + x2 ∧ r ≠ r0 → r ∈ x2; (forward) from specification list[reflist]%app-06: ⊦ x0 + r ' = x1 + r0 ' ↔ x0 = x1 ∧ r = r0; from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%neq-03: ⊦ ¬ r ∈ x ∧ r ≠ r0 → x + r0 ' ≠ x0 + r ' + x1; from specification list[reflist]