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https://github.com/choukh/agda-veblen

veblen function in agda
https://github.com/choukh/agda-veblen

agda formalization googology mathematical-logic mathematics ordinal ordinal-numbers

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veblen function in agda

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README 

        
# Veblen Function in Agda



## Features



- --without-K and --safe

- Ready for googology function such as fast growing hierarchy

- Literate agda script (but in Chinese) and [html5 rendering](https://choukh.github.io/agda-veblen/NonWellFormed.html)



## Contents



### [Ordinal.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Ordinal.lagda.md)



- Inductive definition of Brouwer ordinal

- Inductive definition of non-strict order `_≤_`

- Equality `_≈_` and strict order `_<_` are defined by `_≤_`

- Partial ordering of `_≤_` and strict ordering of `_<_` is proved

- No total ordering, but that's fine



### [WellFormed.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/WellFormed.lagda.md)



- Well formed (WF) ordinals are those constructed hereditarily by strictly increasing sequence

- WF of finite ordinals and `ω` is proved



### [Function.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Function.lagda.md)



- Common properties of ordinal functions are defined

- Definition of normal function is adapted for constructive setup



### [Recursion.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Recursion.lagda.md)



- General form of ordianl recursive function is defined and its properties are proved



### [Arithmetic.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Arithmetic.lagda.md)



- `_+_`, `_*_` and `_^_` are defined by recursion and their WF preserving properties are proved

- Association law, distribution law, etc



### [Tetration.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Tetration.lagda.md)



- We show that tetration is no-go since `α ^^ suc ω ≈ α ^^ ω` and, moreover, `α ^^ β ≈ α ^^ ω` forall `β ≥ ω`



### [Fixpoint.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Fixpoint.lagda.md)



- Infinite iteration `_⋱_` is defined

- If `F` is normal then `F ⋱ α` is a fixed point not less than `α`

- Recursion of `F ⋱_ ∘ suc` is the fixed point yielding function of `F`, written `F ′`

- We proved that higher order function `_′` is normal-preserving and wf-preserving-preserving



### [Fixpoint.Lower.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Fixpoint/Lower.lagda.md)



- Trivial examples of fixed point



### [Epsilon.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Epsilon.lagda.md)



- ε function is defined as `(ω ^_) ′`

- We have `ε (suc α) ≡ ω ^^[ suc (ε α) ] ω` forall `α`

- ζ is defined as `ε ′` and η as `ζ ′`

- `ε`, `ζ`, `η`, ... are all normal and WF preserving



### [Epsilon.Alternative.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Epsilon/Alternative.lagda.md)



- We proved `ε (suc α) ≈ ε α ^^ ω` forall WF `α`



### [VeblenFunction.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/VeblenFunction.lagda.md)



- Veblen function `φ α β` is defined

- We show that `φ` is normal, monotonic, congruence and wf-preserving

- Γ₀ is defined as `(λ α → φ α zero) ⋱ zero`



There is also a well formed version of most of the above in [src/WellFormed](https://github.com/choukh/agda-veblen/blob/main/src/WellFormed). From Γ₀ on, there will be only the well formed version.



## License



[AGPL-3.0](https://github.com/choukh/agda-veblen/blob/main/LICENSE)