42 lines
1.1 KiB
Plaintext
42 lines
1.1 KiB
Plaintext
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import Mathlib.Data.Rel
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import Mathlib.Data.Set.Basic
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/-! # Common.Set.Peano
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Data types and theorems used to define Peano systems.
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-/
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namespace Peano
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/--
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A `Peano system` is a triple `⟨N, S, e⟩` consisting of a set `N`, a function
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`S : N → N`, and a member `e ∈ N` such that the following three conditions are
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met:
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1. `e ∉ ran S`.
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2. `S` is one-to-one.
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3. Every subset `A` of `N` containing `e` and closed under `S` is `N` itself.
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-/
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class System (N : Set α) (S : α → α) (e : α) where
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zero_range : e ∉ Set.range S
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injective : Function.Injective S
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induction : ∀ A, A ⊆ N ∧ e ∈ A ∧ (∀ a ∈ A, S a ∈ A) → A = N
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instance : System (N := @Set.univ ℕ) (S := Nat.succ) (e := 0) where
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zero_range := by
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simp
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injective := by
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intro x₁ x₂ h
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injection h
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induction := by
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intro A h
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suffices Set.univ ⊆ A from Set.Subset.antisymm h.left this
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show ∀ n, n ∈ Set.univ → n ∈ A
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intro n hn
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induction n with
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| zero => exact h.right.left
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| succ n ih =>
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refine h.right.right n (ih ?_)
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simp
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end Peano
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