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Proof step 4 from proof get-in (previous step)

m < #(a + x0), ∀ m. m < # x0 → x0 = [] ∨ x0[m] ∈ x0 ⊦ a + x0 = [], (a + x0)[m] ∈ a + x0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
m ≤ # x0, ∀ m. m < # x0 → x0 = [] ∨ x0[m] ∈ x0 ⊦ (a ' + x0)[m] = a, (a ' + x0)[m] ∈ x0

Rule argument

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ a ∈ b ' + x ↔ ¬(a ≠ b ∧ ¬ a ∈ x); (ss)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ # a ' = 0 +1; (s)% ⊦ #(x + y) = # x + # y; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ a ' + x ≠ []; (s)% ⊦ a + x = a ' + x; (s)