[Merged by Bors] - feat: Port SetTheory.Ordinal.CantorNormalForm

Mathlib.lean

@@ -1033,6 +1033,7 @@ import Mathlib.SetTheory.Cardinal.SchroederBernstein
 import Mathlib.SetTheory.Lists
 import Mathlib.SetTheory.Ordinal.Arithmetic
 import Mathlib.SetTheory.Ordinal.Basic
+import Mathlib.SetTheory.Ordinal.CantorNormalForm
 import Mathlib.SetTheory.Ordinal.Exponential
 import Mathlib.SetTheory.Ordinal.FixedPoint
 import Mathlib.SetTheory.Ordinal.NaturalOps

Mathlib/SetTheory/Ordinal/CantorNormalForm.lean

@@ -0,0 +1,190 @@
+/-
+Copyright (c) 2018 Mario Carneiro. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Mario Carneiro
+
+! This file was ported from Lean 3 source module set_theory.ordinal.cantor_normal_form
+! leanprover-community/mathlib commit f1e061e3caef3022f0daa99d670ecf2c30e0b5c6
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.SetTheory.Ordinal.Arithmetic
+import Mathlib.SetTheory.Ordinal.Exponential
+
+/-!
+# Cantor Normal Form
+
+The Cantor normal form of an ordinal is generally defined as its base `ω` expansion, with its
+non-zero exponents in decreasing order. Here, we more generally define a base `b` expansion
+`Ordinal.CNF` in this manner, which is well-behaved for any `b ≥ 2`.
+
+# Implementation notes
+
+We implement `Ordinal.CNF` as an association list, where keys are exponents and values are
+coefficients. This is because this structure intrinsically reflects two key properties of the Cantor
+normal form:
+
+- It is ordered.
+- It has finitely many entries.
+
+# Todo
+
+- Add API for the coefficients of the Cantor normal form.
+- Prove the basic results relating the CNF to the arithmetic operations on ordinals.
+-/
+
+
+noncomputable section
+
+universe u
+
+open Order
+
+namespace Ordinal
+
+/-- Inducts on the base `b` expansion of an ordinal. -/
+@[elab_as_elim]
+noncomputable def CNFRec (b : Ordinal) {C : Ordinal → Sort _} (H0 : C 0)
+    (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o := fun o ↦ by
+    by_cases (o = 0)
+    · rw [h]; exact H0
+    · exact H o h (CNFRec _ H0 H (o % b ^ log b o))
+    termination_by CNFRec b H0 H o => o
+    decreasing_by exact mod_opow_log_lt_self b h
+set_option linter.uppercaseLean3 false in
+#align ordinal.CNF_rec Ordinal.CNFRec
+
+@[simp]
+theorem CNFRec_zero {C : Ordinal → Sort _} (b : Ordinal) (H0 : C 0)
+    (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNFRec b C H0 H 0 = H0 := by
+  rw [CNFRec, dif_pos rfl]
+  rfl
+set_option linter.uppercaseLean3 false in
+#align ordinal.CNF_rec_zero Ordinal.CNFRec_zero
+
+theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort _} (ho : o ≠ 0) (H0 : C 0)
+    (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :
+    @CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _) := by rw [CNFRec, dif_neg ho]
+set_option linter.uppercaseLean3 false in
+#align ordinal.CNF_rec_pos Ordinal.CNFRec_pos
+
+-- Porting note: unknown attribute @[pp_nodot]
+/-- The Cantor normal form of an ordinal `o` is the list of coefficients and exponents in the
+base-`b` expansion of `o`.
+
+We special-case `CNF 0 o = CNF 1 o = [(0, o)]` for `o ≠ 0`.
+
+`CNF b (b ^ u₁ * v₁ + b ^ u₂ * v₂) = [(u₁, v₁), (u₂, v₂)]` -/
+def CNF (b o : Ordinal) : List (Ordinal × Ordinal) :=
+  CNFRec b [] (fun o _ho IH ↦ (log b o, o / b ^ log b o)::IH) o
+set_option linter.uppercaseLean3 false in
+#align ordinal.CNF Ordinal.CNF
+
+@[simp]
+theorem CNF_zero (b : Ordinal) : CNF b 0 = [] :=
+  CNFRec_zero b _ _
+set_option linter.uppercaseLean3 false in
+#align ordinal.CNF_zero Ordinal.CNF_zero
+
+/-- Recursive definition for the Cantor normal form. -/
+theorem CNF_ne_zero {b o : Ordinal} (ho : o ≠ 0) :
+    CNF b o = (log b o, o / b ^ log b o)::CNF b (o % b ^ log b o) :=
+  CNFRec_pos b ho _ _
+set_option linter.uppercaseLean3 false in
+#align ordinal.CNF_ne_zero Ordinal.CNF_ne_zero
+
+theorem zero_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩] := by simp [CNF_ne_zero ho]
+set_option linter.uppercaseLean3 false in
+#align ordinal.zero_CNF Ordinal.zero_CNF
+
+theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩] := by simp [CNF_ne_zero ho]
+set_option linter.uppercaseLean3 false in
+#align ordinal.one_CNF Ordinal.one_CNF
+
+theorem CNF_of_le_one {b o : Ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩] := by
+  rcases le_one_iff.1 hb with (rfl | rfl)
+  · exact zero_CNF ho
+  · exact one_CNF ho
+set_option linter.uppercaseLean3 false in
+#align ordinal.CNF_of_le_one Ordinal.CNF_of_le_one
+
+theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩] := by
+  simp only [CNF_ne_zero ho, log_eq_zero hb, opow_zero, div_one, mod_one, CNF_zero]
+set_option linter.uppercaseLean3 false in
+#align ordinal.CNF_of_lt Ordinal.CNF_of_lt
+
+/-- Evaluating the Cantor normal form of an ordinal returns the ordinal. -/
+theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o :=
+  CNFRec b (by rw [CNF_zero]; rfl)
+    (fun o ho IH ↦ by rw [CNF_ne_zero ho, List.foldr_cons, IH, div_add_mod]) o
+set_option linter.uppercaseLean3 false in
+#align ordinal.CNF_foldr Ordinal.CNF_foldr
+
+/-- Every exponent in the Cantor normal form `CNF b o` is less or equal to `log b o`. -/
+theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → x.1 ≤ log b o :=
+  by
+  refine' CNFRec b _ (fun o ho H ↦ _) o
+  · rw [CNF_zero]
+    intro contra; contradiction
+  · rw [CNF_ne_zero ho, List.mem_cons]
+    rintro (rfl | h)
+    · exact le_rfl
+    · exact (H h).trans (log_mono_right _ (mod_opow_log_lt_self b ho).le)
+set_option linter.uppercaseLean3 false in
+#align ordinal.CNF_fst_le_log Ordinal.CNF_fst_le_log
+
+/-- Every exponent in the Cantor normal form `CNF b o` is less or equal to `o`. -/
+theorem CNF_fst_le {b o : Ordinal.{u}} {x : Ordinal × Ordinal} (h : x ∈ CNF b o) : x.1 ≤ o :=
+  (CNF_fst_le_log h).trans <| log_le_self _ _
+set_option linter.uppercaseLean3 false in
+#align ordinal.CNF_fst_le Ordinal.CNF_fst_le
+
+/-- Every coefficient in a Cantor normal form is positive. -/
+theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2 := by
+  refine' CNFRec b _ (fun o ho IH ↦ _) o
+  · simp only [CNF_zero, List.not_mem_nil, IsEmpty.forall_iff]
+  · rcases eq_zero_or_pos b with (rfl | hb)
+    · rw [zero_CNF ho, List.mem_singleton]
+      rintro rfl
+      exact Ordinal.pos_iff_ne_zero.2 ho
+    · rw [CNF_ne_zero ho]
+      intro h
+      cases' (List.mem_cons.mp h) with h h
+      · rw [h]; rw [div_pos]
+        · exact opow_log_le_self _ ho
+        · exact (opow_pos _ hb).ne'
+      · exact IH h
+set_option linter.uppercaseLean3 false in
+#align ordinal.CNF_lt_snd Ordinal.CNF_lt_snd
+
+/-- Every coefficient in the Cantor normal form `CNF b o` is less than `b`. -/
+theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :
+    x ∈ CNF b o → x.2 < b := by
+  refine' CNFRec b _ (fun o ho IH ↦ _) o
+  · simp only [CNF_zero, List.not_mem_nil, IsEmpty.forall_iff]
+  · rw [CNF_ne_zero ho]
+    intro h
+    cases' (List.mem_cons.mp h) with h h
+    · rw [h]; simpa only using div_opow_log_lt o hb
+    · exact IH h
+set_option linter.uppercaseLean3 false in
+#align ordinal.CNF_snd_lt Ordinal.CNF_snd_lt
+
+/-- The exponents of the Cantor normal form are decreasing. -/
+theorem CNF_sorted (b o : Ordinal) : ((CNF b o).map Prod.fst).Sorted (· > ·) := by
+  refine' CNFRec b _ (fun o ho IH ↦ _) o
+  · simp only [CNF_zero]
+  · cases' le_or_lt b 1 with hb hb
+    · simp only [CNF_of_le_one hb ho, List.map]
+    · cases' lt_or_le o b with hob hbo
+      · simp only [CNF_of_lt ho hob, List.map]
+      · rw [CNF_ne_zero ho, List.map_cons, List.sorted_cons]
+        refine' ⟨fun a H ↦ _, IH⟩
+        rw [List.mem_map'] at H
+        rcases H with ⟨⟨a, a'⟩, H, rfl⟩
+        exact (CNF_fst_le_log H).trans_lt (log_mod_opow_log_lt_log_self hb ho hbo)
+set_option linter.uppercaseLean3 false in
+#align ordinal.CNF_sorted Ordinal.CNF_sorted
+
+end Ordinal
+