From 4749b3ca572618fa1a7cdb17e2abba9b498279ca Mon Sep 17 00:00:00 2001 From: Andreas Grois Date: Sun, 5 Jan 2025 23:55:54 +0100 Subject: Move AStar into a separte library and use it for Day17-1 --- Common/Finite.lean | 186 ----------------------------------------------------- 1 file changed, 186 deletions(-) delete mode 100644 Common/Finite.lean (limited to 'Common/Finite.lean') diff --git a/Common/Finite.lean b/Common/Finite.lean deleted file mode 100644 index 95f90a0..0000000 --- a/Common/Finite.lean +++ /dev/null @@ -1,186 +0,0 @@ -import Std.Data.HashSet -import Common.Nat - -class Finite (α : Type u) where - cardinality : Nat - enumerate : α → Fin cardinality - nth : Fin cardinality → α - nth_inverse_enumerate : nth ∘ enumerate = id - enumerate_inverse_nth : enumerate ∘ nth = id - -theorem Finite.surjective {α : Type u} [Finite α] {a b : Fin (Finite.cardinality α)} : nth a = nth b → a = b := λh₃ ↦ - have h₁ := Finite.enumerate_inverse_nth (α := α) - have h₄ := congrArg enumerate h₃ - have h₂ : ∀(x : Fin (Finite.cardinality α)), (enumerate ∘ nth) x = x := λ_↦h₁.substr rfl - have h₅ := Function.comp_apply.subst (h₂ a).symm - have h₆ := Function.comp_apply.subst (h₂ b).symm - h₅.substr $ h₆.substr h₄ - -theorem Finite.injective {α : Type u} [Finite α] {a b : α} : enumerate a = enumerate b → a = b := λh₁↦ - have h₂ := Finite.nth_inverse_enumerate (α := α) - have h₃ := congrArg nth h₁ - have h₄ : ∀(x : α), (nth ∘ enumerate) x = x := λ_↦h₂.substr rfl - have h₅ := Function.comp_apply.subst (h₄ a).symm - 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 - | l+1 => set_worker Std.HashSet.empty ⟨l,h.substr (p := λx ↦ l < x) $ Nat.lt.base l⟩ -where set_worker (set : Std.HashSet α) (n : Fin (Finite.cardinality α)) : Std.HashSet α := - let e := Finite.nth n - let set := set.insert e - match n with - | ⟨0,_⟩ => set - | ⟨m+1,h₁⟩ => set_worker set ⟨m, Nat.lt_of_succ_lt h₁⟩ - -protected theorem Finite.set_worker_contains_self' (α : Type u) [Finite α] [BEq α] [Hashable α] [LawfulBEq α] (a : α) (oldSet : Std.HashSet α) (h₁ : oldSet.contains a) (n : Fin (Finite.cardinality α)) : (Finite.set.set_worker α oldSet n).contains a := by - cases n - case mk n h₂ => - induction n generalizing oldSet - case zero => unfold set.set_worker; simp[h₁] - case succ m hm => - unfold set.set_worker - exact hm (oldSet.insert (nth ⟨m + 1, h₂⟩)) (by simp[h₁]) (Nat.lt_of_succ_lt h₂) - -protected theorem Finite.set_worker_contains_self (α : Type u) [Finite α] [BEq α] [Hashable α] [LawfulBEq α] : ∀ (a : α) (set : Std.HashSet α), (Finite.set.set_worker α set (Finite.enumerate a)).contains a := by - intros a oldSet - unfold set.set_worker - rw[←Function.comp_apply (f := nth), Finite.nth_inverse_enumerate, id_def] - split - case h_1 => apply Std.HashSet.contains_insert_self - case h_2 => - apply Finite.set_worker_contains_self' - exact Std.HashSet.contains_insert_self - -protected theorem Finite.set_worker_contains (α : Type u) [Finite α] [BEq α] [Hashable α] [LawfulBEq α] : ∀ (a : α) (set : Std.HashSet α) (o : Nat) (h₁ : Finite.enumerate a + o < Finite.cardinality α), (Finite.set.set_worker α set ⟨Finite.enumerate a + o, h₁⟩).contains a := by - intros a oldSet offset h₁ - induction offset generalizing oldSet - case zero => - exact Finite.set_worker_contains_self _ _ _ - case succ p hi => - unfold set.set_worker - simp - have : ↑(enumerate a) + p < cardinality α := Nat.lt_of_succ_lt $ (Nat.add_assoc (enumerate a) p 1).substr h₁ - 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 - -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), - ¬set.contains (Finite.nth x)), (Finite.set.set_worker α set n).size = set.size + n + 1 -:= by - intros set n h₂ - simp at h₂ - unfold Finite.set.set_worker - cases n - case mk n h₁ => - split - case h_1 m isLt he => - simp at he - simp[Std.HashSet.size_insert, Std.HashSet.mem_iff_contains, h₂, he] - case h_2 m isLt he => - simp - have h₄ : m < n := have : n = m.succ := Fin.val_eq_of_eq he; this.substr (Nat.lt_succ_self m) - have h₅ : ∀ (x : Fin (cardinality α)), x ≤ ⟨m, Nat.lt_of_succ_lt isLt⟩ → ¬(set.insert (nth ⟨n, h₁⟩)).contains (nth x) = true := by - simp - intros x hx - constructor - case right => exact h₂ x (Nat.le_trans hx (Nat.le_of_lt h₄)) - case left => - have h₅ : x ≠ ⟨n, h₁⟩ := Fin.ne_of_val_ne $ Nat.ne_of_lt $ Nat.lt_of_le_of_lt hx h₄ - have h₆ := Finite.surjective (α := α) (a := x) (b := ⟨n,h₁⟩) - exact Ne.symm (h₅ ∘ h₆) - have h₃ := Finite.set_worker_size α (set.insert (nth ⟨n, h₁⟩)) ⟨m, Nat.lt_of_succ_lt isLt⟩ (h₅) - rw[h₃] - simp at he - simp[he, Std.HashSet.size_insert] - split - case isFalse => rw[Nat.add_assoc, Nat.add_comm 1 m] - case isTrue hx => - subst n - have h₂ := h₂ ⟨m+1,h₁⟩ (Fin.le_refl _) - have hx := Std.HashSet.mem_iff_contains.mp hx - exact absurd hx (Bool.eq_false_iff.mp h₂) -termination_by _ n => n.val - -theorem Finite.set_size_eq_cardinality (α : Type u) [Finite α] [BEq α] [Hashable α] [LawfulBEq α] : (Finite.set α).size = Finite.cardinality α := by - unfold set - split - case h_1 h => exact Std.HashSet.size_empty.substr h.symm - case h_2 l h => - rewrite(occs := .pos [3])[h] - have := Finite.set_worker_size α Std.HashSet.empty ⟨l,h.substr $ Nat.lt_succ_self l⟩ (λx _↦Bool.eq_false_iff.mp (Std.HashSet.contains_empty (a:=Finite.nth x))) - simp only [this, Std.HashSet.size_empty, Nat.zero_add] -- cgit v1.2.3