| import Mathlib | |
| namespace OAI | |
| namespace LargeIndependentSets.IndexedBlocks | |
| lemma range_blocks {α : Type*} (f : ℕ → ℕ → α) (k : ℕ) (hk : 0<k) (n : ℕ) : | |
| (List.range (k*n)).map (fun i => f (i/k) (i%k)) = | |
| (List.range n).flatMap (fun j => (List.range k).map (f j)) := by | |
| induction n with | |
| | zero => simp | |
| | succ n ih => | |
| rw [Nat.mul_succ,List.range_add,List.map_append,ih,List.range_succ,List.flatMap_append] | |
| simp only [List.flatMap_cons,List.flatMap_nil,List.append_nil,List.map_map] | |
| congr 1 | |
| apply List.map_congr_left | |
| intro j hj | |
| have hj' := List.mem_range.mp hj | |
| have hd : (k*n+j)/k=n := by | |
| rw [Nat.add_comm,Nat.add_mul_div_left _ _ hk,Nat.div_eq_of_lt hj',Nat.zero_add] | |
| simp [hd,Nat.mod_eq_of_lt hj'] | |
| end LargeIndependentSets.IndexedBlocks | |
| end OAI | |