From e49f5a5717b4d3459f2522a6343d64492a8db225 Mon Sep 17 00:00:00 2001 From: jlh Date: Fri, 30 Jul 2021 12:22:22 +0100 Subject: [PATCH] Updated Trinitarianism/Quest2 --- README.md | 8 ++-- Trinitarianism/.DS_Store | Bin 6148 -> 0 bytes Trinitarianism/Preambles/P2.agda | 1 + Trinitarianism/Quest0.md | 3 +- Trinitarianism/Quest1.agda | 3 ++ Trinitarianism/Quest1.md | 31 ++++++++++++++-- Trinitarianism/Quest1Solutions.agda | 4 +- Trinitarianism/Quest2.md | 26 +++++++++++-- Trinitarianism/Quest2Solutions.agda | 35 ++++++++++++++++++ Trinitarianism/README.md | 6 +-- .../agda/Trinitarianism/Preambles/P2.agdai | Bin 23615 -> 25324 bytes .../agda/Trinitarianism/Quest1Solutions.agdai | Bin 29333 -> 30124 bytes .../agda/Trinitarianism/Quest2Solutions.agdai | Bin 29110 -> 40283 bytes 13 files changed, 101 insertions(+), 16 deletions(-) delete mode 100644 Trinitarianism/.DS_Store diff --git a/README.md b/README.md index 8465b57..7cc6e4d 100644 --- a/README.md +++ b/README.md @@ -45,15 +45,15 @@ Some text should be highlighted, and any `?` should turn into `{ }`. ## How the game works -Our Game is under development. Please contact the devs. -Currently you can try the _quests_ in the `Trinitarianism` folder. +Our Game is currently under development. +As of now you can try the _quests_ in the `Trinitarianism` folder. Each quest consists of three files, for example : -- `Trinitarianism/Quest0.md` is the guide for the quest +- `Trinitarianism/Quest0.md` is the guide for the quest. In there, you will find details of the tasks you must finish in order to complete the quest. For now, it is recommended that you view these online within github. -- `Trinitarianism/Quest0.agda` is the actual file in which +- `Trinitarianism/Quest0.agda` is the agda file in which you do the quest. - `Trinitarianism/Quest0Solutions.agda` contains solutions to the tasks in the quest. diff --git a/Trinitarianism/.DS_Store b/Trinitarianism/.DS_Store deleted file mode 100644 index 5008ddfcf53c02e82d7eee2e57c38e5672ef89f6..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 6148 zcmeH~Jr2S!425mzP>H1@V-^m;4Wg<&0T*E43hX&L&p$$qDprKhvt+--jT7}7np#A3 zem<@ulZcFPQ@L2!n>{z**++&mCkOWA81W14cNZlEfg7;MkzE(HCqgga^y>{tEnwC%0;vJ&^%eQ zLs35+`xjp>T0 @@ -68,3 +65,26 @@ someone else's definition of addition, it might not work anymore. > But `_+_` and `_+'_` compute the same values. > Are `_+_` and `_+'_` 'the same'? What is 'the same'? + +As the final task of the Quest, +try to express and prove in agda the statement +> For any natural number it is even or is is not even. +We will make a summary of what is needed: +- a definition of the type `A ⊕ B` (input `\oplus`), + which has three interpretations + - the proposition '`A` or `B`' + - the construction with two ways of making recipes + `left : A → A ⊕ B` + and `right : B → A ⊕ B`. + - the coproduct of two objects `A` and `B`. + The type needs to take in parameters `A : Type` and `B : Type` + ```agda + data _⊕_ (A : Type) (B : Type) : Type where + ??? + ``` +- a definition of negation. One can motivate it by the following + - Define `A ↔ B : Type` for two types `A : Type` and `B : Type`. + - Show that for any `A : Type` we have `(A ↔ ⊥) ↔ (A → ⊥)` + - Define `¬ : Type → Type` to be `λ A → (A → ⊥)`. +- a formulation and proof of the statement above + diff --git a/Trinitarianism/Quest2Solutions.agda b/Trinitarianism/Quest2Solutions.agda index 56f3250..be2ecfc 100644 --- a/Trinitarianism/Quest2Solutions.agda +++ b/Trinitarianism/Quest2Solutions.agda @@ -21,3 +21,38 @@ Sum'OfEven x (zero , hy) = x .snd Sum'OfEven x (suc (suc y) , hy) = Sum'OfEven x (y , hy) -} + +data _⊕_ (A : Type) (B : Type) : Type where + left : A → A ⊕ B + right : B → A ⊕ B + +_↔_ : Type → Type → Type +_↔_ A B = (A → B) × (B → A) + +¬Motivation : (A : Type) → ((A ↔ ⊥) ↔ (A → ⊥)) +¬Motivation A = + -- forward direction + ( + -- suppose we have a proof `hiff : A ↔ ⊥` + λ hiff → + -- give the forward map only + fst hiff + ) , + -- backward direction; assume a proof hto : A → ⊥ + λ hto → + -- we need to show A → ⊥ which we have already + hto + , + -- we need to show ⊥ → A, which is the principle of explosion + λ () + +¬ : Type → Type +¬ A = A → ⊥ + +isEvenDecidable : (n : ℕ) → isEven n ⊕ ¬ (isEven n) +-- zero is even; go left +isEvenDecidable zero = left tt +-- one is not even; go right +isEvenDecidable (suc zero) = right (λ ()) +-- inductive step +isEvenDecidable (suc (suc n)) = isEvenDecidable n diff --git a/Trinitarianism/README.md b/Trinitarianism/README.md index 85e12c8..5713794 100644 --- a/Trinitarianism/README.md +++ b/Trinitarianism/README.md @@ -41,9 +41,9 @@ In category theory, types are objects and terms are generalised elements. - existence / Σ type / total space of bundles - for all / Π type / space of sections of bundles -## Something doesn't feel the Same +## What is 'the Same'? There will be one thing missing from this 'place to do maths' and that is a notion of _equality_. -This is where HoTT deviates from its predecessors, -and is the theme of the next arc. +How HoTT treats equality is where it deviates from its predecessors. +This is the theme of the next arc. diff --git a/_build/2.6.2/agda/Trinitarianism/Preambles/P2.agdai b/_build/2.6.2/agda/Trinitarianism/Preambles/P2.agdai index 37e8e374eeb6267f9c781875a0e2c0937c61b28d..6f40a83272c5576d424bb04c9353123c755000ae 100644 GIT binary patch literal 25324 zcmXuL2|QKX`#-++KC|%{GGyAJ5QS1^b|OcVgU{!tkTIO1E%|!tdUq z?r9`6TuFmN=X2BPN`;%0hI3J5_&=NPzx{fZ*V=18>lxn9u%5NF|9|ts56b7SK6=hx z;QQU|9|V8YRNtzZWpe2s5tG5Z%c*}6SIGJL;{E8Tm-ibjVMypM58~H%GnZ|S7HmB1 zzWd~g=ncKOo3b?&Aup;HWd2q1^51d)L(A8<&D*+S{^gkECzoBn(U(w|P^tg##k7PM z3^Y~3ClM=}@?kJ#$VAvvfS3+g59RX^aU=C43>A|<5cUMyL!OJ8Pf*(?W-z#srV>o$ zIEAjWoXE9CBznb|Dsj-~*K1I0>n99m6<>x94N)i_hu14fR1K?CnGn=!OkLP!%CG+n z)9hd!SoWR 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