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Proof step 23 from proof prime-product (previous step)

Π ints = Π ints1 * Π ints0, perm(ints, ints1 + ints0), primes(ints), 4 ^ (2 * j + 1) < Π ints, max(ints) ≤ 2 * j + 1, 0 ≤ j, 0 < j, primes(ints0), primes(ints1), i→n(2 * j + 1) ≤ i→n(j) + 1, ints1 = [] ∨ max(ints1) ≤ j + 1, ints0 = [] ∨ j + 1 < min(ints0), j + 2 ≤ min(ints0) ∧ max(ints0) ≤ j * 2 + 1 → Π ints0 ≤ 4 ^ j, ∀ i0, ints. i→n(i0) < i→n(j + j + 1) → primes(ints) ∧ max(ints) ≤ i0 ∧ ¬ dups(ints) → Π ints ≤ 4 ^ i0 ⊦ ints = [], dups(ints1), dups(ints0), dups(ints)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: intnat inst: name: num-i2n-le seq: 0 ≤ i ⊦ i→n(i) ≤ (* m) ↔ i ≤ n→i(m) subst: [m, i] → [i→n(j) + 1, 2 * j + 1] flags: rewrite

Additional information

Used simplifier rules: zero-ge-add: ⊦ 0 ≤ j1 ∧ j1 < j → (0 ≤ i0 * j + j1 ↔ 0 ≤ i0); from specification int-mult%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%n2i-add: ⊦ n→i(m + n) = n→i(m) + n→i(n); from specification intnat%i2i-pos: ⊦ 0 < i → n→i(i→n(i)) = i; from specification intnat%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2%le-add: ⊦ i + j ≤ i0 + j ↔ i ≤ i0; from specification int-abs%le-same: ⊦ 0 < i → (i * j ≤ i ↔ j ≤ 1); from specification int-mult%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used terms: i→n(j) + 1, 2 * j + 1