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Proof step 4 from proof append-mod (previous step)

n0 < # x0 ⊦ x0 = [], (x + x0)[n0 + # x, a] = x + x1 ↔ x0[n0, a] = x1, # x + # x0 = 0

Heuristic structural induction applied rule structural induction

And obtained the following new goals

Proof step 5:
∀ n0, x0, a, x1. n0 < # x0 → x0 = [] ∨ ((x2 + x0)[n0 + # x2, a] = x2 + x1 ↔ x0[n0, a] = x1) ∨ # x2 + # x0 = 0, n0 < # x0 ⊦ x0 = [], ((a0 + x2) + x0)[n0 + #(a0 + x2), a] = (a0 + x2) + x1 ↔ x0[n0, a] = x1, #(a0 + x2) + # x0 = 0

Rule argument

term: 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%add-base: ⊦ n + 0 = n; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2