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Proof step 32 from proof cyclic-brent-def (previous step)

2 ^ m0 < 2 ^ m0 * 2, n0 < 2 ^ m0 * 2, 2 ^ m0 ≤ n0, cyclic(r, H), reachable-n(H, m, r, r0), reachable-n(H, n0, r0, r0), reachable-n(H, n0 + m, r, r0) ⊦ n0 = 0, ∃ n. ¬ 2 ^ n ≤ n0 ∧ ¬ 2 ^ n ≤ m, ∃ n, m, r0. m ≠ 0 ∧ m ≤ 2 ^ n ∧ reachable-n(H, 2 ^ n - 1, r, r0) ∧ reachable-n(H, m, r0, r0)

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 25:
n ≠ 0 ⊦ ∃ m. 2^ m ≤ n ∧ n < 2^ (m + 1)
Proof step 27:
2 ^ m0 < 2 ^ m0 * 2, n0 < 2 ^ m0 * 2, 2 ^ m0 ≤ n0, cyclic(r, H), reachable-n(H, m, r, r0), reachable-n(H, n0, r0, r0), reachable-n(H, n0 + m, r, r0) ⊦ m ≠ 0, n0 = 0, ∃ n. ¬ 2 ^ n ≤ n0 ∧ ¬ 2 ^ n ≤ m, ∃ n, m, r0. m ≠ 0 ∧ m ≤ 2 ^ n ∧ reachable-n(H, 2 ^ n - 1, r, r0) ∧ reachable-n(H, m, r0, r0)
Proof step 31:
∃ m0. 2^ m0 ≤ m ∧ m < 2^ (m0 + 1), 2 ^ m0 < 2 ^ m0 * 2, n0 < 2 ^ m0 * 2, 2 ^ m0 ≤ n0, cyclic(r, H), reachable-n(H, m, r, r0), reachable-n(H, n0, r0, r0), reachable-n(H, n0 + m, r, r0) ⊦ n0 = 0, ∃ n. ¬ 2 ^ n ≤ n0 ∧ ¬ 2 ^ n ≤ m, ∃ n, m, r0. m ≠ 0 ∧ m ≤ 2 ^ n ∧ reachable-n(H, 2 ^ n - 1, r, r0) ∧ reachable-n(H, m, r0, r0)

Rule argument

[x0lemmaarg: spec-name: nat-pot instance-name: lemma-name: potex antecedent: n ≠ 0 succedent: ∃ m. 2^ m ≤ n ∧ n < 2^ (m + 1) substlist: [n] → [m] add preconditions?: F is rewrite lemma?: F current spec?: F rotate lemma?: F replace all?: F paths: () dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (m)]]