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Proof step 1 from proof put-app-left (conclusion)

⊦ n < # x → (x + y)[n, a] = x[n, a] + y

User applied rule structural induction

And obtained the following new goals

Proof step 2:
∀ n, y, a. n < # x0 → (x0 + y)[n, a] = x0[n, a] + y ⊦ n < #(a0 + x0) → ((a0 + x0) + y)[n, a] = (a0 + x0)[n, a] + y

Rule argument

indvar: x

Additional information

Used simplifier rules: list freely generated by [], +;;%Nil: ⊦ [] + x = x; from specification list%a: ⊦ (x + y) + z = x + y + z; from specification list%size-01: ⊦ # [] = 0; from specification list-data%zero-01: ⊦ ¬ m < 0; from specification nat-basic1