{"id":20458282,"url":"https://github.com/choukh/agda-veblen","last_synced_at":"2026-03-10T05:04:19.433Z","repository":{"id":63089318,"uuid":"551457750","full_name":"choukh/agda-veblen","owner":"choukh","description":"veblen function in 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Veblen Function in Agda\n\n## Features\n\n- --without-K and --safe\n- Ready for googology function such as fast growing hierarchy\n- Literate agda script (but in Chinese) and [html5 rendering](https://choukh.github.io/agda-veblen/NonWellFormed.html)\n\n## Contents\n\n### [Ordinal.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Ordinal.lagda.md)\n\n- Inductive definition of Brouwer ordinal\n- Inductive definition of non-strict order `_≤_`\n- Equality `_≈_` and strict order `_\u003c_` are defined by `_≤_`\n- Partial ordering of `_≤_` and strict ordering of `_\u003c_` is proved\n- No total ordering, but that's fine\n\n### [WellFormed.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/WellFormed.lagda.md)\n\n- Well formed (WF) ordinals are those constructed hereditarily by strictly increasing sequence\n- WF of finite ordinals and `ω` is proved\n\n### [Function.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Function.lagda.md)\n\n- Common properties of ordinal functions are defined\n- Definition of normal function is adapted for constructive setup\n\n### [Recursion.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Recursion.lagda.md)\n\n- General form of ordianl recursive function is defined and its properties are proved\n\n### [Arithmetic.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Arithmetic.lagda.md)\n\n- `_+_`, `_*_` and `_^_` are defined by recursion and their WF preserving properties are proved\n- Association law, distribution law, etc\n\n### [Tetration.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Tetration.lagda.md)\n\n- We show that tetration is no-go since `α ^^ suc ω ≈ α ^^ ω` and, moreover, `α ^^ β ≈ α ^^ ω` forall `β ≥ ω`\n\n### [Fixpoint.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Fixpoint.lagda.md)\n\n- Infinite iteration `_⋱_` is defined\n- If `F` is normal then `F ⋱ α` is a fixed point not less than `α`\n- Recursion of `F ⋱_ ∘ suc` is the fixed point yielding function of `F`, written `F ′`\n- We proved that higher order function `_′` is normal-preserving and wf-preserving-preserving\n\n### [Fixpoint.Lower.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Fixpoint/Lower.lagda.md)\n\n- Trivial examples of fixed point\n\n### [Epsilon.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Epsilon.lagda.md)\n\n- ε function is defined as `(ω ^_) ′`\n- We have `ε (suc α) ≡ ω ^^[ suc (ε α) ] ω` forall `α`\n- ζ is defined as `ε ′` and η as `ζ ′`\n- `ε`, `ζ`, `η`, ... are all normal and WF preserving\n\n### [Epsilon.Alternative.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/Epsilon/Alternative.lagda.md)\n\n- We proved `ε (suc α) ≈ ε α ^^ ω` forall WF `α`\n\n### [VeblenFunction.lagda.md](https://github.com/choukh/agda-veblen/blob/main/src/NonWellFormed/VeblenFunction.lagda.md)\n\n- Veblen function `φ α β` is defined\n- We show that `φ` is normal, monotonic, congruence and wf-preserving\n- Γ₀ is defined as `(λ α → φ α zero) ⋱ zero`\n\nThere 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.\n\n## License\n\n[AGPL-3.0](https://github.com/choukh/agda-veblen/blob/main/LICENSE)\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fchoukh%2Fagda-veblen","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fchoukh%2Fagda-veblen","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fchoukh%2Fagda-veblen/lists"}