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Proof step 4 from proof valid-indices-nonrec (previous step)

∀ ar. nx0 < # ar ↔ (∀ n. n ∈ nx0 → n < # ar) ⊦ m + nx0 < # ar ↔ (∀ n. n ∈ m + nx0 → n < # ar)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
∀ ar. nx0 < # ar ↔ (∀ n. n ∈ nx0 → n < # ar) ⊦ m ' + nx0 < # ar ↔ (∀ n. n = m ∨ n ∈ nx0 → n < # ar)

Rule argument

Additional information

Used simplifier rules: a: ⊦ (nx + nx0) + nx1 = nx + nx0 + nx1; from specification list[nat-list-set]% ⊦ n ∈ n0 ' + nx ↔ n = n0 ∨ n ∈ nx; (s)% ⊦ n + nx = n ' + nx; (s)