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docs(README): fix typos and update text
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favonia committed Jun 1, 2024
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Expand Up @@ -12,27 +12,25 @@ This is a formalization of the displacement algebras, their properties, and part

🚧 The links are currently broken.

| Displacements | Paper Section | Agda Module | OCaml Module(s) |
| :------------------------------------ | :-------------- | :----------------------------------------------------------------------------- | :------------------------------------------------------------------------------------------------------------------------------------------------------------------ |
| Natural numbers | 3.3.1 | [Nat](src/Mugen/Algebra/Displacement/Instances/Nat.agda) | [Nat](https://redprl.org/mugen/mugen/Mugen/Shift/Nat) and [Nat](https://redprl.org/mugen/mugen/Mugen/ShiftWithJoin/Nat) |
| Integers | 3.3.1 | [Int](src/Mugen/Algebra/Displacement/Instances/Int.agda) | [Int](https://redprl.org/mugen/mugen/Mugen/Shift/Int) and [Int](https://redprl.org/mugen/mugen/Mugen/ShiftWithJoin/Int) |
| Non-positive integers | 3.3.1 | [NonPositive](src/Mugen/Algebra/Displacement/Instances/NonPositive.agda) | [NonPositive](https://redprl.org/mugen/mugen/Mugen/Shift/NonPositive) and [NonPositive](https://redprl.org/mugen/mugen/Mugen/ShiftWithJoin/NonPositive) |
| Constant displacements | 3.3.2 | [Constant](src/Mugen/Algebra/Displacement/Instances/Constant.agda) | [Constant](https://redprl.org/mugen/mugen/Mugen/Shift/Constant) |
| Binary products | 3.3.3 | [Product](src/Mugen/Algebra/Displacement/Instances/Product.agda) | [Product](https://redprl.org/mugen/mugen/Mugen/Shift/Product) and [Product](https://redprl.org/mugen/mugen/Mugen/ShiftWithJoin/Product) |
| Lexicographic binary products | 3.3.4 | [Lexicographic](src/Mugen/Algebra/Displacement/Instances/Lexicographic.agda) | [Lexicographic](https://redprl.org/mugen/mugen/Mugen/Shift/Lexicographic) and [Lexicographic](https://redprl.org/mugen/mugen/Mugen/ShiftWithJoin/Lexicographic) |
| Indexed products | 3.3.5 | [IndexedProduct](src/Mugen/Algebra/Displacement/Instances/IndexedProduct.agda) | _(not implementable)_ |
| Nearly constant infinite products | 3.3.5 | [NearlyConstant](src/Mugen/Algebra/Displacement/Instances/NearlyConstant.agda) | [NearlyConstant](https://redprl.org/mugen/mugen/Mugen/Shift/NearlyConstant) and [NearlyConstant](https://redprl.org/mugen/mugen/Mugen/ShiftWithJoin/NearlyConstant) |
| Infinite products with finite support | 3.3.5 | [FiniteSupport](src/Mugen/Algebra/Displacement/Instances/FiniteSupport.agda) | [FiniteSupport](https://redprl.org/mugen/mugen/Mugen/Shift/FiniteSupport) and [FiniteSupport](https://redprl.org/mugen/mugen/Mugen/ShiftWithJoin/FiniteSupport) |
| Prefix order | 3.3.6 | [Prefix](src/Mugen/Algebra/Displacement/Instances/Prefix.agda) | [Prefix](https://redprl.org/mugen/mugen/Mugen/Shift/Prefix) |
| Fractal displacements | 3.3.7 | [Fractal](src/Mugen/Algebra/Displacement/Instances/Fractal.agda) | [Fractal](https://redprl.org/mugen/mugen/Mugen/Shift/Fractal) |
| Opposite displacements | 3.3.8 | [Opposite](src/Mugen/Algebra/Displacement/Instances/Opposite.agda) | [Opposite](https://redprl.org/mugen/mugen/Mugen/Shift/Opposite) |
| Weird frarctal displacements | 3.3.9 | [WeirdFractal](src/Mugen/Algebra/Displacement/Instances/WeirdFractal.agda) | [Fractal](https://redprl.org/mugen/mugen/Mugen/Shift/Fractal) |
| Endomorphisms | 3.4 (Lemma 3.7) | [Endomorphism](src/Mugen/Algebra/Displacement/Instances/Endomorphism.agda) | _(not implementable)_ |
| Displacements | Paper Section | Agda Module | OCaml Module(s) |
| :------------------------------------ | :------------------ | :----------------------------------------------------------------------------- | :------------------------------------------------------------------------------------------------------------------------------------------------------------------ |
| Natural numbers | 3.3.1 | [Nat](src/Mugen/Algebra/Displacement/Instances/Nat.agda) | [Nat](https://redprl.org/mugen/mugen/Mugen/Shift/Nat) and [Nat](https://redprl.org/mugen/mugen/Mugen/ShiftWithJoin/Nat) |
| Integers | 3.3.1 | [Int](src/Mugen/Algebra/Displacement/Instances/Int.agda) | [Int](https://redprl.org/mugen/mugen/Mugen/Shift/Int) and [Int](https://redprl.org/mugen/mugen/Mugen/ShiftWithJoin/Int) |
| Non-positive integers | 3.3.1 | [NonPositive](src/Mugen/Algebra/Displacement/Instances/NonPositive.agda) | [NonPositive](https://redprl.org/mugen/mugen/Mugen/Shift/NonPositive) and [NonPositive](https://redprl.org/mugen/mugen/Mugen/ShiftWithJoin/NonPositive) |
| Constant displacements | 3.3.2 | [Constant](src/Mugen/Algebra/Displacement/Instances/Constant.agda) | [Constant](https://redprl.org/mugen/mugen/Mugen/Shift/Constant) |
| Binary products | 3.3.3 | [Product](src/Mugen/Algebra/Displacement/Instances/Product.agda) | [Product](https://redprl.org/mugen/mugen/Mugen/Shift/Product) and [Product](https://redprl.org/mugen/mugen/Mugen/ShiftWithJoin/Product) |
| Lexicographic binary products | 3.3.4 | [Lexicographic](src/Mugen/Algebra/Displacement/Instances/Lexicographic.agda) | [Lexicographic](https://redprl.org/mugen/mugen/Mugen/Shift/Lexicographic) and [Lexicographic](https://redprl.org/mugen/mugen/Mugen/ShiftWithJoin/Lexicographic) |
| Indexed products | 3.3.5 | [IndexedProduct](src/Mugen/Algebra/Displacement/Instances/IndexedProduct.agda) | _(not implementable)_ |
| Nearly constant infinite products | 3.3.5 | [NearlyConstant](src/Mugen/Algebra/Displacement/Instances/NearlyConstant.agda) | [NearlyConstant](https://redprl.org/mugen/mugen/Mugen/Shift/NearlyConstant) and [NearlyConstant](https://redprl.org/mugen/mugen/Mugen/ShiftWithJoin/NearlyConstant) |
| Infinite products with finite support | 3.3.5 | [FiniteSupport](src/Mugen/Algebra/Displacement/Instances/FiniteSupport.agda) | [FiniteSupport](https://redprl.org/mugen/mugen/Mugen/Shift/FiniteSupport) and [FiniteSupport](https://redprl.org/mugen/mugen/Mugen/ShiftWithJoin/FiniteSupport) |
| Prefix order | 3.3.6 | [Prefix](src/Mugen/Algebra/Displacement/Instances/Prefix.agda) | [Prefix](https://redprl.org/mugen/mugen/Mugen/Shift/Prefix) |
| Fractal displacements | 3.3.7 | [Fractal](src/Mugen/Algebra/Displacement/Instances/Fractal.agda) | [Fractal](https://redprl.org/mugen/mugen/Mugen/Shift/Fractal) |
| Opposite displacements | 3.3.8 | [Opposite](src/Mugen/Algebra/Displacement/Instances/Opposite.agda) | [Opposite](https://redprl.org/mugen/mugen/Mugen/Shift/Opposite) |
| Weird fractal displacements | 3.3.9 (not in POPL) | [WeirdFractal](src/Mugen/Algebra/Displacement/Instances/WeirdFractal.agda) | [Fractal](https://redprl.org/mugen/mugen/Mugen/Shift/Fractal) |
| Endomorphisms | 3.4 (Lemma 3.7) | [Endomorphism](src/Mugen/Algebra/Displacement/Instances/Endomorphism.agda) | _(not implementable)_ |

### Other Theorems

🚧 The links are currently broken.

| Theorems | Paper Section | Agda Module |
| :------------------------------------ | :----------------- | :----------------------------------------------------------------------------------------------- |
| Validity of McBride monads | 3.1 | [McBride](./src/Mugen/Cat/HierarchyTheory/McBride.agda) |
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