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Proof step 24 from proof cyclic-floyd-def (previous step)

m * n = n1 + m, 2 * m = m0 + 1, reachable-n(H, n, r2, r2), reachable-n(H, n + m, r, r2), reachable-n(H, n1 + m, r, r1), reachable-n(H, m, r, r2), ∀ n0. reachable-n(H, m + n0 * n, r, r2) ⊦ n1 ≤ n, reachable-n(H, n1, r2, r1), n = 0, m = 0, reachable-n(H, 2 * (n1 + m), r, r1)

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 21:
reachable-n(H, n, r, r0), m < n ⊦ ∃ r1. reachable-n(H, m, r, r1) ∧ reachable-n(H, n - m, r1, r0)
Proof step 23:
∃ r2. reachable-n(H, n + m, r, r2) ∧ reachable-n(H, n1 + m - n + m, r2, r1), m * n = n1 + m, 2 * m = m0 + 1, reachable-n(H, n, r2, r2), reachable-n(H, n + m, r, r2), reachable-n(H, n1 + m, r, r1), reachable-n(H, m, r, r2), ∀ n0. reachable-n(H, m + n0 * n, r, r2) ⊦ n1 ≤ n, reachable-n(H, n1, r2, r1), n = 0, m = 0, reachable-n(H, 2 * (n1 + m), r, r1)

Rule argument

[x0lemmaarg: spec-name: path-reachable instance-name: lemma-name: reach-n-split antecedent: reachable-n(H, n, r, r0), m < n succedent: ∃ r1. reachable-n(H, m, r, r1) ∧ reachable-n(H, n - m, r1, r0) substlist: [H, n, r, r0, m] → [H, n1 + m, r, r1, n + m] add preconditions?: F is rewrite lemma?: F current spec?: F rotate lemma?: F replace all?: F paths: () dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ m + n < m + n0 ↔ n < n0], [CSIMPSEQ : ⊦ ¬ m < n ↔ n ≤ m], [CSIMPSEQ : ⊦ m0 * m * n = (m0 * m) * n], [CSIMPSEQ : ⊦ m * n = n * m], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m])], [EXTRATERMS : (H, n1 + m, r, r1, n + m)]]