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Proof step 12 from proof cyclic-floyd-def (previous step)

reachable-n(H, 2 + m1, r, H[r] .nxt), reachable-n(H, m1 + 1, H[r] .nxt, H[r] .nxt), H[r] .nxt ∈ H, r ∈ H, reachable-n(H, 1 + m1, r, r1), ∀ n. reachable-n(H, 1 + n * m1 + n, r, H[r] .nxt) ⊦ r = null, H[r] .nxt = null, m1 = 0, reachable-n(H, 2 + 2 * m1, r, r1), reachable-n(H, m1, H[r] .nxt, r1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ reachable-n(H, n + 1, r, r0) ↔ (∃ r1. reachable-1(H, r, r1) ∧ reachable-n(H, n, r1, r0))

Rule argument

[x0lemmaarg: spec-name: path-reachable instance-name: lemma-name: reachable-n-rec antecedent: succedent: reachable-n(H, n + 1, r, r0) ↔ (∃ r1. reachable-1(H, r, r1) ∧ reachable-n(H, n, r1, r0)) substlist: [r0, r, n, H] → [r1, r, m1, H] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((0, 0, 4)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ reachable-1(H, r, r0) ↔ r ≠ null ∧ r0 ≠ null ∧ r ∈ H ∧ r0 ∈ H ∧ H[r] .nxt = r0], [CSIMPSEQ : ⊦ m + n = n + m])], [EXTRATERMS : (r1, r, m1, H)]]