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Proof step 2 from proof prime-product-between (previous step)

primes(ints), j + 2 ≤ min(ints), max(ints) ≤ j * 2 + 1, 4 ^ j < Π ints, 0 < j ⊦ dups(ints)

User applied rule insert lemma

And obtained the following new goals

Proof step 3:
∀ i. i ∈ ints → prime(i) ∧ j + 2 ≤ i ∧ i ≤ j * 2 + 1, ¬ dups(ints), 0 < j ⊦ Π ints ≤ binom_coef(2 * j + 1, j)
Proof step 4:
Π ints ≤ binom_coef(2 * j + 1, j), primes(ints), j + 2 ≤ min(ints), max(ints) ≤ j * 2 + 1, 4 ^ j < Π ints, 0 < j ⊦ dups(ints)

Rule argument

spec: inst: name: coef-product-le seq: ∀ i. i ∈ ints → prime(i) ∧ j + 2 ≤ i ∧ i ≤ j * 2 + 1, ¬ dups(ints), 0 < j ⊦ Π ints ≤ binom_coef(2 * j + 1, j) subst: [ints, j] → [ints, j] flags: current, []

Additional information

Used simplifier rules: primes-in: ⊦ primes(ints) ∧ i ∈ ints → (¬ prime(i) ↔ false); from specification intlist%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%min-in-not-ls: ⊦ i ∈ ints → ¬ i < min(ints); from specification intlist%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%max-in-not-ls: ⊦ i ∈ ints → ¬ max(ints) < i; from specification intlist Used terms: ints, j