The ordinal induction principle for the natural numbers

Content created by Fredrik Bakke, Jonathan Prieto-Cubides and Egbert Rijke.

Created on 2022-02-08.
Last modified on 2023-04-08.

module elementary-number-theory.ordinal-induction-natural-numbers where
open import elementary-number-theory.natural-numbers
open import elementary-number-theory.strict-inequality-natural-numbers

open import foundation.empty-types
open import foundation.universe-levels


The ordinal induction principle of the natural numbers is the well-founded induction principle of ℕ.

To Do

The computation rule should still be proven.

□-<-ℕ :
  {l : Level}  (  UU l)    UU l
□-<-ℕ P n = (m : )  (le-ℕ m n)  P m

reflect-□-<-ℕ :
  {l : Level} (P :   UU l) 
  (( n : )  □-<-ℕ P n)  (n : )  P n
reflect-□-<-ℕ P f n = f (succ-ℕ n) n (succ-le-ℕ n)

zero-ordinal-ind-ℕ :
  { l : Level} (P :   UU l)  □-<-ℕ P zero-ℕ
zero-ordinal-ind-ℕ P m t = ex-falso (contradiction-le-zero-ℕ m t)

succ-ordinal-ind-ℕ :
  {l : Level} (P :   UU l)  ((n : )  (□-<-ℕ P n)  P n) 
  (k : )  □-<-ℕ P k  □-<-ℕ P (succ-ℕ k)
succ-ordinal-ind-ℕ P f k g m t =
  f m  m' t'  g m' (transitive-le-ℕ' m' m k t' t))

induction-ordinal-ind-ℕ :
  { l : Level} (P :   UU l) 
  ( qS : (k : )  □-<-ℕ P k  □-<-ℕ P (succ-ℕ k))
  ( n : )  □-<-ℕ P n
induction-ordinal-ind-ℕ P qS zero-ℕ = zero-ordinal-ind-ℕ P
induction-ordinal-ind-ℕ P qS (succ-ℕ n) =
  qS n (induction-ordinal-ind-ℕ P qS n)

ordinal-ind-ℕ :
  { l : Level} (P :   UU l) 
  ( (n : )  (□-<-ℕ P n)  P n) 
  ( n : )  P n
ordinal-ind-ℕ P f =
  reflect-□-<-ℕ P
    ( induction-ordinal-ind-ℕ P (succ-ordinal-ind-ℕ P f))

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