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Proof step 3 from proof product-del1-02 (previous step)

∀ i. i ∈ ints0 → i * Π(ints0 -1l i) = Π ints0, i ∈ i0 + ints0 ⊦ i * Π((i0 + ints0) -1l i) = Π(i0 + ints0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 4:
i ∈ ints0, ∀ i. i ∈ ints0 → i * Π(ints0 -1l i) = Π ints0 ⊦ i = i0, i0 = 0, i * Π(ints0 -1l i) = Π ints0

Rule argument

Additional information

Used simplifier rules: product-app: ⊦ Π(ints + ints0) = Π ints * Π ints0;%product-one: ⊦ Π i ' = i;%eq-nonzero: ⊦ i ≠ 0 → (i * j = i * j0 ↔ j = j0); from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%neq-not-zero: ⊦ i * j ≠ i * j0 → i ≠ 0; (forward) from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%del1-n: ⊦ i ≠ i0 → (i0 ' + ints) -1l i = i0 ' + (ints -1l i); from specification list-del[intlista]%del1-y: ⊦ (i ' + ints) -1l i = ints; from specification list-del[intlista]%in-17: ⊦ i ∈ i0 ' ↔ i = i0; from specification list[intlista]%in-append: ⊦ i ∈ ints + ints0 ↔ ¬ ¬(i ∈ ints ∨ i ∈ ints0); from specification list[intlista]%a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%c: ⊦ i + ints = i ' + ints; from specification list[intlista]