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Proof step 31 from proof prime-def-prime (previous step)

0 < i0, 1 < i0 * i1, 1 < i0, 1 < i1, 0 < i1, i1 ≤ i0, ∀ j. prime(j) ∧ j * j ≤ i0 * i1 → ¬ j | i0 * i1, ∀ i2. i→n(i2) < i→n(i0 * i1) → 0 < i2 ∧ 1 < i2 ∧ (∀ j. prime(j) ∧ j * j ≤ i2 → ¬ j | i2) → prime(i2) ⊦ prime(i1)

User applied rule apply induction

And obtained the following new goals

Rule argument

subst: [i2] → [i1]

Additional information

Used simplifier rules: ⊦ i→n(i) < i→n(j) ↔ abs(i) < abs(j); (s)% ⊦ 1 < i → abs(i) = i; (s)% ⊦ abs(i * j) = abs(i) * abs(j); (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ 0 < i → (i * j ≤ i ↔ j ≤ 1); (s)% ⊦ 0 < i → (i ≤ 1 ↔ i = 1); (s)% ⊦ i = 1 → i * j = j; (s)%i0 ≠ 0, i0 ≤ i ⊦ (* i) < j → (* i0) < j; (s) Used terms: i1