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Proof step 6 from proof prime-smallest (previous step)

∀ m. 2 ≤ m ∧ m * m ≤ n → n % m ≠ 0 ⊦ n = 1, n = 0, prime(n)

User applied rule insert lemma

And obtained the following new goals

Proof step 7:
¬ prime(n), 2 ≤ n ⊦ ∃ m. m | n ∧ prime(m) ∧ m * m ≤ n
Proof step 8:
∃ m. m | n ∧ prime(m) ∧ m * m ≤ n, ∀ m. 2 ≤ m ∧ m * m ≤ n → n % m ≠ 0 ⊦ n = 1, n = 0, prime(n)

Rule argument

spec: nat-prime inst: name: not-prime seq: ¬ prime(n), 2 ≤ n ⊦ ∃ m. m | n ∧ prime(m) ∧ m * m ≤ n subst: [n] → [n] flags: current, []

Additional information

Used simplifier rules: lels-01: ⊦ ¬ m ≤ n ↔ n < m; from specification nat%twols-02: ⊦ m ≠ 0 → (m < 2 ↔ m = 1); from specification nat-basic1 Used terms: n