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Proof step 3 from proof del1 (previous step)

∀ y, a. filter(x0, y -1l a) = filter(x0, y) ∨ a ∈ x0 ⊦ filter(a0 + x0, y -1l a) = filter(a0 + x0, y), a ∈ a0 + x0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 4:
∀ y, a. filter(x0, y -1l a) = filter(x0, y) ∨ a ∈ x0 ⊦ filter(a0 ' + x0, y -1l a) = filter(a0 ' + x0, y), a = a0, a ∈ x0

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list%in-06: ⊦ ¬ a ∈ b ' + x ↔ a ≠ b ∧ ¬ a ∈ x; from specification list%c: ⊦ a + x = a ' + x; from specification list