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Proof step 4 from proof inapp (previous step)

(a0 + x0) + a ' + y = z + a ' + y0, ∀ a, y, z, y0. x0 + a ' + y = z + a ' + y0 → a ∈ x0 ∨ a ∈ y ∨ y = y0 ∧ x0 = z ⊦ a ∈ y, a ∈ a0 + x0, y = y0 ∧ a0 + x0 = z

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
(a0 ' + x0) + a ' + y = z + a ' + y0, a0 ∈ z, ∀ a, y, z, y0. x0 + a ' + y = z + a ' + y0 → a ∈ x0 ∨ a ∈ y ∨ y = y0 ∧ x0 = z ⊦ a = a0, a ∈ x0, a ∈ y, y = y0 ∧ a0 ' + x0 = z

Rule argument

Additional information

Used simplifier rules: c: ⊦ a + x = a ' + x;%a: ⊦ (x + y) + z = x + y + z;%in-06: ⊦ a ∈ b ' + x ↔ ¬(a ≠ b ∧ ¬ a ∈ x);%in-14: ⊦ a ' + x = y → a ∈ y;%in-append: ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y;%inapp-02: ⊦ ¬ a ∈ y → (y + c ' + z = a ' + z0 ↔ y = [] ∧ c = a ∧ z = z0);