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Proof step 61 from proof cyclic-floyd-def (previous step)

reachable-n(H, m, r, r0), reachable-n(H, m + m * n, r, r0) ⊦ ∃ r1. reachable-n(H, m * n, r, r1), n = 0, ∃ n, r0. n ≠ 0 ∧ reachable-n(H, n, r, r0) ∧ reachable-n(H, 2 * n, r, r0)

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 58:
reachable-n(H, n, r, r0), m < n ⊦ ∃ r1. reachable-n(H, m, r, r1) ∧ reachable-n(H, n - m, r1, r0)
Proof step 60:
∃ r1. reachable-n(H, m * n, r, r1) ∧ reachable-n(H, m + m * n - m * n, r1, r0), reachable-n(H, m, r, r0), reachable-n(H, m + m * n, r, r0) ⊦ ∃ r1. reachable-n(H, m * n, r, r1), n = 0, ∃ n, r0. n ≠ 0 ∧ reachable-n(H, n, r, r0) ∧ reachable-n(H, 2 * n, r, r0)

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, m + m * n, r, r0, m * n] add preconditions?: F is rewrite lemma?: F current spec?: F rotate lemma?: F replace all?: F paths: () dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ m * n < m1 * n + n ↔ ¬ ¬ (m < m1 + 1 ∧ n ≠ 0)], [CSIMPSEQ : ⊦ m < n + 1 ↔ ¬ n < m], [CSIMPSEQ : ⊦ ¬ n < n], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ m0 * m * n = (m0 * m) * n], [CSIMPSEQ : ⊦ m * n = n * m], [CSIMPSEQ : ⊦ m * 0 = 0], [CSIMPSEQ : ⊦ reachable-n(H, 0, r, r0) ↔ r = r0])], [EXTRATERMS : (H, m + m * n, r, r0, m * n)]]