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Proof step 19 from proof cyclic-alt-def (previous step)

⊦ (∃ m, n, r0. m < n ∧ reachable-n(H, m, r, r0) ∧ reachable-n(H, n, r, r0)) ↔ (∃ m, r0. reachable-n(H, m, r, r0) ∧ (∃ n. n ≠ 0 ∧ (∀ n0. reachable-n(H, m + n0 * n, r, r0))))

User applied rule case distinction

And obtained the following new goals

Proof step 14:
∃ m, n, r0. m < n ∧ reachable-n(H, m, r, r0) ∧ reachable-n(H, n, r, r0) ⊦ ∃ m, r0. reachable-n(H, m, r, r0) ∧ (∃ n. n ≠ 0 ∧ (∀ n0. reachable-n(H, m + n0 * n, r, r0)))
Proof step 18:
∃ m, r0. reachable-n(H, m, r, r0) ∧ (∃ n. n ≠ 0 ∧ (∀ n0. reachable-n(H, m + n0 * n, r, r0))) ⊦ ∃ m, n, r0. m < n ∧ reachable-n(H, m, r, r0) ∧ reachable-n(H, n, r, r0)

Rule argument

[CASEDARG : pos: right 1, ()]

Additional information

[[CONCRETESIMPRULES : ()]]