diff options
Diffstat (limited to 'Common/Finite.lean')
| -rw-r--r-- | Common/Finite.lean | 81 |
1 files changed, 72 insertions, 9 deletions
diff --git a/Common/Finite.lean b/Common/Finite.lean index 25041a8..95f90a0 100644 --- a/Common/Finite.lean +++ b/Common/Finite.lean @@ -1,4 +1,5 @@ import Std.Data.HashSet +import Common.Nat class Finite (α : Type u) where cardinality : Nat @@ -23,6 +24,68 @@ theorem Finite.injective {α : Type u} [Finite α] {a b : α} : enumerate a = en have h₆ := Function.comp_apply.subst (h₄ b).symm h₅.substr $ h₆.substr h₃ +def Finite.tuple_enumerate {α : Type u} [Finite α] {β : Type v} [Finite β] (x : α × β) : Fin ((Finite.cardinality α) * (Finite.cardinality β)) := + let (a, b) := x + let idxa := (Finite.enumerate a) + let idxb := (Finite.enumerate b) + let idx := idxa.val + (Finite.cardinality α) * idxb.val + have h : idx < (Finite.cardinality α) * (Finite.cardinality β) := + Nat.two_d_coordinate_to_index_lt_size idxa.isLt idxb.isLt + ⟨idx,h⟩ + +def Finite.tuple_nth {α : Type u} [Finite α] {β : Type v} [Finite β] (idx : Fin ((Finite.cardinality α) * (Finite.cardinality β))) := + let idxav := idx % (Finite.cardinality α) + let idxbv := idx / (Finite.cardinality α) + have h₁ : Finite.cardinality α > 0 := + if h : 0 = Finite.cardinality α then + have : cardinality α * cardinality β = 0 := h.subst (motive := λx↦x*cardinality β = 0) $ Nat.zero_mul (cardinality β) + (Fin.cast this idx).elim0 + else + Nat.pos_of_ne_zero (Ne.symm h) + let idxa : Fin (Finite.cardinality α) := ⟨idxav, Nat.mod_lt _ h₁⟩ + let idxb : Fin (Finite.cardinality β):= ⟨idxbv, Nat.div_lt_of_lt_mul idx.isLt⟩ + (Finite.nth idxa, Finite.nth idxb) + +theorem Finite.tuple_nth_inverse_enumerate {α : Type u} [Finite α] {β : Type v} [Finite β] : Finite.tuple_nth (α := α) (β := β) ∘ Finite.tuple_enumerate (α := α) (β := β) = id := by + unfold Finite.tuple_enumerate Finite.tuple_nth + funext + simp + congr + case h.e_fst x => + simp[Nat.mod_eq_of_lt] + rw[←Function.comp_apply (f := Finite.nth) (x := x.fst), Finite.nth_inverse_enumerate] + rfl + case h.e_snd x => + have h₁ : (↑(Finite.enumerate x.fst) + (Finite.cardinality α) * ↑(Finite.enumerate x.snd)) / Finite.cardinality α = ↑(Finite.enumerate x.snd) := by + rw[Nat.add_mul_div_left] + simp[Nat.div_eq_of_lt] + exact Nat.zero_lt_of_lt (Finite.enumerate x.fst).isLt + simp[h₁] + rw[←Function.comp_apply (f := Finite.nth) (x := x.snd), Finite.nth_inverse_enumerate] + rfl + +theorem Finite.tuple_enumerate_inerse_nth {α : Type u} [Finite α] {β : Type v} [Finite β] : Finite.tuple_enumerate (α := α) (β := β) ∘ Finite.tuple_nth (α := α) (β := β) = id := by + funext + unfold Finite.tuple_enumerate Finite.tuple_nth + simp + rename_i x + rw[Fin.eq_mk_iff_val_eq] + simp + rw[←Function.comp_apply (f := Finite.enumerate), Finite.enumerate_inverse_nth] + rw[←Function.comp_apply (f := Finite.enumerate), Finite.enumerate_inverse_nth] + simp[Nat.mod_add_div] + +instance {α : Type u} [Finite α] {β : Type v} [Finite β] : Finite (Prod α β) where + cardinality := (Finite.cardinality α) * (Finite.cardinality β) + enumerate := Finite.tuple_enumerate + nth := Finite.tuple_nth + enumerate_inverse_nth := Finite.tuple_enumerate_inerse_nth + nth_inverse_enumerate := Finite.tuple_nth_inverse_enumerate + +theorem Finite.forall_nth {α : Type u} [Finite α] (p : α → Prop) (h₁ : ∀(o : Fin (Finite.cardinality α)), p (Finite.nth o)) : ∀(a : α), p a := λa↦ + have : p ((nth ∘ enumerate) a) := Function.comp_apply.substr $ h₁ (Finite.enumerate a) + Finite.nth_inverse_enumerate.subst (motive := λx ↦ p (x a)) this + def Finite.set (α : Type u) [Finite α] [BEq α] [Hashable α] : Std.HashSet α := match h: (Finite.cardinality α) with | 0 => Std.HashSet.empty @@ -65,15 +128,15 @@ protected theorem Finite.set_worker_contains (α : Type u) [Finite α] [BEq α] exact hi (oldSet.insert (nth ⟨↑(enumerate a) + (p + 1), h₁⟩)) this theorem Finite.set_contains (α : Type u) [Finite α] [BEq α] [Hashable α] [LawfulBEq α] : ∀ (a : α), (Finite.set α).contains a := λa ↦ by - unfold set - split - case h_1 h => exact (Fin.cast h $ Finite.enumerate a).elim0 - case h_2 l h => - let o := l - enumerate a - have h₁ : (Finite.enumerate a).val + o = l := by omega - have h₂ := Finite.set_worker_contains _ a Std.HashSet.empty o (by omega) - simp[h₁] at h₂ - assumption + unfold set + split + case h_1 h => exact (Fin.cast h $ Finite.enumerate a).elim0 + case h_2 l h => + let o := l - enumerate a + have h₁ : (Finite.enumerate a).val + o = l := by omega + have h₂ := Finite.set_worker_contains _ a Std.HashSet.empty o (by omega) + simp[h₁] at h₂ + assumption protected theorem Finite.set_worker_size (α : Type u) [Finite α] [BEq α] [Hashable α] [LawfulBEq α] : ∀(set : Std.HashSet α) (n : Fin (Finite.cardinality α)) (_ : ∀(x : Fin (Finite.cardinality α)) (_ : x ≤ n), |