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Proof step 38 from proof prime-product (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-basic1 inst: name: ls-rec seq: ⊦ i ≠ j -1 → (i < j ↔ i < j -1) subst: [j, i] → [j + 2, min(ints0)] flags: rewrite

Additional information

Used simplifier rules: sub-pred: ⊦ i -1 = i - 1; from specification int-abs%numsub: ⊦ (* i) + i0 - (* j) = (i - j) + i0; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%succ-02: ⊦ i < j + 1 ↔ i ≤ j; from specification int-abs%prod-empty: ⊦ Π [] = 1; from specification intlist%one: ⊦ 1 * i = i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%leone: ⊦ 1 ≤ i ↔ 0 < i; from specification int-abs%Nil: ⊦ [] + ints = ints; from specification list[intlista]%a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%e-01: ⊦ ¬ dups([]); from specification list-dup[intlista]%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%irreflexivity: ⊦ ¬ i < i; from specification oelem[intlista] Used terms: j + 2, min(ints0)