Enderton (set). Isomorphism from ℕ to peano systems.

Bookshelf/Enderton/Set.tex

@@ -6378,7 +6378,7 @@
 
   \end{proof}
 
-\subsection{\pending{Theorem 4H}}%
+\subsection{\verified{Theorem 4H}}%
 \hyperlabel{sub:theorem-4h}
 
   \begin{theorem}[4H]
@@ -6389,6 +6389,9 @@
       $$h(\sigma(n)) = S(h(n))$$ and the zero element $$h(0) = e.$$
   \end{theorem}
 
+  \code{Common/Set/Peano}
+    {Peano.nat\_isomorphism}
+
   \begin{proof}
 
     Let $\langle N, S, e \rangle$ be a \nameref{ref:peano-system}.

Bookshelf/Enderton/Set/Chapter_4.lean

@@ -1,5 +1,6 @@
 import Common.Logic.Basic
 import Common.Set.Basic
+import Common.Set.Peano
 import Mathlib.Data.Set.Basic
 import Mathlib.SetTheory.Ordinal.Basic
 

Common/Set/Peano.lean

@@ -1,5 +1,6 @@
 import Mathlib.Data.Rel
 import Mathlib.Data.Set.Basic
+import Mathlib.Tactic.LibrarySearch
 
 /-! # Common.Set.Peano
 
@@ -18,13 +19,16 @@ met:
 3. Every subset `A` of `N` containing `e` and closed under `S` is `N` itself.
 -/
 class System (N : Set α) (S : α → α) (e : α) where
+  maps_to : Set.MapsTo S N N
+  mem : e ∈ N
   zero_range : e ∉ Set.range S
   injective : Function.Injective S
   induction : ∀ A, A ⊆ N ∧ e ∈ A ∧ (∀ a ∈ A, S a ∈ A) → A = N
 
 instance : System (N := @Set.univ ℕ) (S := Nat.succ) (e := 0) where
-  zero_range := by
-    simp
+  maps_to := by simp
+  mem := by simp
+  zero_range := by simp
   injective := by
     intro x₁ x₂ h
     injection h
@@ -39,4 +43,105 @@ instance : System (N := @Set.univ ℕ) (S := Nat.succ) (e := 0) where
       refine h.right.right n (ih ?_)
       simp
 
+/--
+The unique (by virtue of the Recursion Theorem for `ω`) function that maps the
+natural numbers to an arbitrary Peano system.
+-/
+def of_nat {e : α} [h : System N S e] : Nat → α
+  | 0 => e
+  | n + 1 => S (@of_nat _ _ _ _ h n)
+
+/--
+The `of_nat` function maps all natural numbers to the set `N` defined in the
+provided Peano system.
+-/
+theorem nat_maps_to [h : System N S e]
+  : Set.MapsTo (@of_nat _ _ _ _ h) Set.univ N := by
+  unfold Set.MapsTo
+  intro x hx
+  unfold of_nat
+  induction x with
+  | zero => exact h.mem
+  | succ x ih =>
+    simp only
+    by_cases hx' : x = 0
+    · rw [hx']
+      unfold of_nat
+      exact h.maps_to h.mem
+    · have ⟨p, hp⟩ := Nat.exists_eq_succ_of_ne_zero hx'
+      rw [hp] at ih ⊢
+      simp only [Set.mem_univ, forall_true_left] at ih
+      exact h.maps_to ih
+
+/--
+Let `⟨N, S, e⟩` be a Peano system. Then `⟨ω, σ, 0⟩` is isomorphic to
+`⟨N, S, e⟩`, i.e., there is a function `h` mapping `ω` one-to-one onto `N` in a
+way that preserves the successor operation
+```
+h(σ(n)) = S(h(n))
+```
+and the zero element
+```
+h(0) = e.
+```
+-/
+theorem nat_isomorphism [h : System N S e]
+  : let n2p := @of_nat _ _ _ _ h
+    Set.BijOn n2p Set.univ N ∧
+      (∀ n : ℕ, n2p n.succ = S (n2p n)) ∧ n2p 0 = e := by
+  let n2p := @of_nat _ _ _ _ h
+  refine ⟨⟨nat_maps_to, ?_, ?_⟩, ?_, rfl⟩
+  · -- Prove `of_nat` is one-to-one.
+    suffices ∀ n : ℕ, ∀ m, n2p m = n2p n → m = n by
+      unfold Set.InjOn
+      intro m _ n _
+      exact this n m
+    intro n
+    induction n with
+    | zero =>
+      intro m hm
+      by_contra nm
+      have ⟨p, hp⟩ := Nat.exists_eq_succ_of_ne_zero nm
+      rw [hp] at hm
+      unfold of_nat at hm
+      exact absurd ⟨n2p p, hm⟩ h.zero_range
+    | succ n ih =>
+      intro m hm
+      by_cases nm : m = 0
+      · rw [nm] at hm
+        unfold of_nat at hm
+        exact absurd ⟨n2p n, hm.symm⟩ h.zero_range
+      · have ⟨p, hp⟩ := Nat.exists_eq_succ_of_ne_zero nm
+        rw [hp] at hm
+        unfold of_nat at hm
+        have := h.injective hm
+        rwa [← ih p this]
+  · -- Prove `of_nat` is onto.
+    suffices e ∈ Set.range n2p ∧ (∀ y, y ∈ Set.range n2p → S y ∈ Set.range n2p) by
+      unfold Set.SurjOn
+      simp only [Set.image_univ]
+      have hN : Set.range n2p = N := by
+        refine h.induction (Set.range n2p) ⟨?_, this⟩
+        show ∀ t, t ∈ Set.range n2p → t ∈ N
+        intro t ht
+        unfold Set.range at ht
+        simp only [Set.mem_setOf_eq] at ht
+        have ⟨y, hy⟩ := ht
+        rw [← hy]
+        exact nat_maps_to (by simp)
+      exact Eq.subset (id (Eq.symm hN))
+    refine ⟨?_, ?_⟩
+    · unfold Set.range
+      simp only [Set.mem_setOf_eq]
+      exact ⟨0, rfl⟩
+    · intro y hy
+      have ⟨n, hn⟩ := hy
+      refine ⟨n + 1, ?_⟩
+      unfold of_nat
+      simp only [Nat.add_eq, Nat.add_zero]
+      dsimp only at hn
+      rw [hn]
+  · intro _
+    conv => left; unfold of_nat
+
 end Peano