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Proof step 4 from proof lagrange-primedivisors (previous step)

(∀ j. j ∈ ints0 → (∃ i, i0, i1, i2. j = i ^2 + i0 ^2 + i1 ^2 + i2 ^2)) → (∃ i, i0, i1, i2. Π ints0 = i ^2 + i0 ^2 + i1 ^2 + i2 ^2), ∀ j. j ∈ i + ints0 → (∃ i, i0, i1, i2. j = i ^2 + i0 ^2 + i1 ^2 + i2 ^2) ⊦ ∃ i3, i0, i1, i2. Π(i + ints0) = i3 ^2 + i0 ^2 + i1 ^2 + i2 ^2

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
(j ∈ ints0 → (∃ i, i0, i1, i2. j = i ^2 + i0 ^2 + i1 ^2 + i2 ^2)) → Π ints0 = i0 ^2 + i1 ^2 + i2 ^2 + i3 ^2, ∀ j0. j0 = i ∨ j0 ∈ ints0 → (∃ i, i0, i1, i2. j0 = i ^2 + i0 ^2 + i1 ^2 + i2 ^2) ⊦ ∃ i4, i5, i6, i7. i * Π ints0 = i4 ^2 + i5 ^2 + i6 ^2 + i7 ^2

Rule argument

Additional information

Used simplifier rules: prod-rec: ⊦ Π(i ' + ints) = i * Π ints; from specification intlist%a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%in-17: ⊦ i ∈ i0 ' ↔ i = i0; from specification list[intlista]%in-append: ⊦ i ∈ ints + ints0 ↔ ¬ ¬(i ∈ ints ∨ i ∈ ints0); from specification list[intlista]%c: ⊦ i + ints = i ' + ints; from specification list[intlista]