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Proof step 6 from proof valid-indices-nonrec (previous step)

m ' + nx0 < # ar, ∀ ar. nx0 < # ar ↔ (∀ n. n ∈ nx0 → n < # ar) ⊦ m ' + nx0 < # ar ↔ (∀ n. n = m ∨ n ∈ nx0 → n < # ar)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
m < # ar, nx0 < # ar, m < n, n ∈ nx0, # ar ≤ n, ∀ ar. nx0 < # ar ↔ (∀ n. n ∈ nx0 → n < # ar) ⊦ n = m

Rule argument

Additional information

Used simplifier rules: ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ m < n → n ≠ 0; (s)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m < n ↔ ¬ n ≤ m; (ss)%a: ⊦ (nx + nx0) + nx1 = nx + nx0 + nx1; from specification list[nat-list-set]% ⊦ m ' + nx < n ↔ m < n ∧ nx < n; (ws)