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Proof step 8 from proof lookup-append (previous step)

a ∈ ax0, ∀ a, ay. a ∈ ax0 → (ax0 + ay)[a] = ax0[a] ⊦ ((p ' + ax0) + ay)[a] = (p ' + ax0)[a], a ∈ p '

User applied rule structural induction

And obtained the following new goals

Proof step 9:
⊦ a0 ∈ ax0 ∧ (∀ a, ay. a ∈ ax0 → (ax0 + ay)[a] = ax0[a]) → (((a × d)' + ax0) + ay)[a0] = ((a × d)' + ax0)[a0] ∨ a0 ∈ (a × d)'

Rule argument

indvar: p

Additional information

Used simplifier rules: pair freely generated by ×;;