From 4a0fb237cac8796e45d7b217db0d808452454820 Mon Sep 17 00:00:00 2001
From: Akshay Nair <phenax5@gmail.com>
Date: Sat, 16 Dec 2023 17:31:12 +0530
Subject: feat: lambda calculus

---
 src/equation.ts        |   9 ++--
 src/lambda-calculus.ts | 114 +++++++++++++++++++++++++++++++++++++++++++++++++
 src/utils/theorem.ts   |   7 +--
 3 files changed, 122 insertions(+), 8 deletions(-)
 create mode 100644 src/lambda-calculus.ts

(limited to 'src')

diff --git a/src/equation.ts b/src/equation.ts
index 0ec25d6..cc668b1 100644
--- a/src/equation.ts
+++ b/src/equation.ts
@@ -1,4 +1,4 @@
-import { Op, Rewrite, ChainRewrites, Sym, Equation, VerifyEquation, assert, ApplyRewrite, Evaluate, OpToStr } from './utils/theorem';
+import { Op, Rewrite, ChainRewrites, Sym, Equation, VerifyEquation, assert, ApplyRewrite, Evaluate, OpToStr, SubstEq } from './utils/theorem';
 
 export namespace equation {
   type F<X extends Op> = { op: 'F', a: X };
@@ -9,7 +9,6 @@ export namespace equation {
 
   export type Congruence<A extends Op, B extends Op> = Rewrite<Equation<A, B>, Equation<F<A>, F<B>>>;
 
-  export type SubstituteEq<O extends Equation<Op, Op>> = Rewrite<O['a'], O['b']>
   export interface SubstituteEq_Proof<A extends Op, B extends Op> {
     type: 'rewrite',
     known: {
@@ -18,7 +17,7 @@ export namespace equation {
     left: Equation<F<A>, F<B>>; // f(a) = f(b)
     right: ChainRewrites<[
       Sym<Congruence<A, B>>,                   // a = b
-      SubstituteEq<this['known']['a = b']>,    // b = b
+      SubstEq<this['known']['a = b']>,         // b = b
       Reflexivity<B>,
     ], this['left']>;
   };
@@ -36,8 +35,8 @@ export namespace equation {
     },
     left: Equation<Equation<A, B>, Equation<B, C>>; // (a = b) = (b = c)
     right: ChainRewrites<[
-      SubstituteEq<this['known']['a = b']>,                                 // (b = b) = (b = c)
-      SubstituteEq<ApplyRewrite<this['known']['b = c'], Symmetry<B, C>>>,   // (b = b) = (c = b)
+      SubstEq<this['known']['a = b']>,                                 // (b = b) = (b = c)
+      SubstEq<ApplyRewrite<this['known']['b = c'], Symmetry<B, C>>>,   // (b = b) = (c = b)
       Reflexivity<B>,                                                       // true = (b = b)
       Reflexivity<B>,                                                       // true = true
       Reflexivity<'true'>,
diff --git a/src/lambda-calculus.ts b/src/lambda-calculus.ts
new file mode 100644
index 0000000..bbd2d1e
--- /dev/null
+++ b/src/lambda-calculus.ts
@@ -0,0 +1,114 @@
+import { Op, Rewrite, ChainRewrites, Sym, Equation, Refl, VerifyEquation, assert, Subst, ApplyRewrite, Evaluate, OpToStr, SubstEq } from './utils/theorem';
+
+export type Abs<x extends Op, o extends Op> = { op: '->', a: x, b: o }; // \x. o
+export type Ap<F extends Op, x extends Op> = { op: '$', a: F, b: x }; // F(x)
+export type Comp<F extends Op, G extends Op> = { op: '.', a: F, b: G }; // F . G
+export type Id<x extends Op = 'x'> = Abs<x, x> // \x. x
+export type Const<x extends Op = 'x', y extends Op = 'y'> = Abs<x, Abs<y, x>> // \x. \y. x
+export type Inv<F extends Op> = { op: 'inv', a: F }
+
+export namespace composition {
+  export type Abstraction<x extends Op, y extends Op, v extends Op> = Rewrite<Ap<Abs<x, y>, v>, y>;
+
+  export type Composition<F extends Op, G extends Op, x extends Op> = Rewrite<Ap<Comp<F, G>, x>, Ap<F, Ap<G, x>>>;
+
+  // Identity
+  export type Identity<x extends Op> = Rewrite<Ap<Id, x>, x>;
+  export namespace proof {
+    export interface Identity_Proof<x extends Op> {
+      type: 'rewrite';
+      left: Equation<Ap<Id<x>, x>, x>;  // id(x) = x
+      right: ChainRewrites<[
+        Abstraction<x, x, x>,  // x = x
+        Refl,
+      ], this['left']>;
+    }
+
+    export interface Identity_Composition_Proof<F extends Op, x extends Op> {
+      type: 'rewrite';
+      left: Equation<Ap<Comp<F, Id>, x>, Ap<F, x>>;  // (f . id)(x) = f(x)
+      right: ChainRewrites<[
+        Composition<F, Id, x>,    // f(id(x)) = f(x)
+        Identity<x>,              // f(x) = f(x)
+        Refl,
+      ], this['left']>;
+    }
+
+    export type identity = [
+      assert<VerifyEquation<Identity_Proof<'x'>>>,
+      assert<VerifyEquation<Identity_Composition_Proof<'F', 'x'>>>,
+    ]
+  }
+
+  // Associativity
+  export type Associativity<F extends Op, G extends Op, H extends Op> = Rewrite<
+    Comp<Comp<F, G>, H>,
+    Comp<F, Comp<G, H>>
+  >
+  export namespace proof {
+    export interface Assoc_Proof<F extends Op, G extends Op, H extends Op, x extends Op> {
+      type: 'rewrite';
+      left: Equation<Ap<Comp<Comp<F, G>, H>, x>, Ap<Comp<F, Comp<G, H>>, x>>;  // ((f . g) . h)(x) = (f . (g . h))(x)
+      right: ChainRewrites<[
+        Composition<Comp<F, G>, H, x>,       // (f . g)(h(x)) = (f . (g . h))(x)
+        Composition<F, G, Ap<H, x>>,         // f(g(h(x))) = (f . (g . h))(x)
+        Composition<F, Comp<G, H>, x>,       // f(g(h(x))) = f((g . h)(x))
+        Composition<G, H, x>,                // f(g(h(x))) = f(g(h(x)))
+        Refl,
+      ], this['left']>;
+    }
+    export type associativity = [
+      assert<VerifyEquation<Assoc_Proof<'F', 'G', 'H', 'x'>>>,
+    ]
+  }
+
+  // Identity
+  export type Transitivity<x extends Op, y extends Op, z extends Op> = Rewrite<Comp<Abs<y, z>, Abs<x, y>>, Abs<x, z>>;
+  export namespace proof {
+    export interface Transitivity_Proof<x extends Op, y extends Op, z extends Op> {
+      type: 'rewrite';
+      left: Equation<Ap<Comp<Abs<y, z>, Abs<x, y>>, x>, Ap<Abs<x, z>, x>>;  // ((\y -> z) . (\x -> y))(x) = (\x -> z)(x)
+      right: ChainRewrites<[
+        Composition<Abs<y, z>, Abs<x, y>, x>,        // (\y -> z) $ (\x -> y)(x) = (\x -> z)(x)
+        Abstraction<x, y, x>,                        // (\y -> z) $ y = (\x -> z)(x)
+        Abstraction<y, z, y>,                        // z = (\x -> z)(x)
+        Abstraction<x, z, x>,                        // z = z
+        Refl,
+      ], this['left']>;
+    }
+    export type transitivity = [
+      // OpToStr<Evaluate<Transitivity_Proof<'x', 'y', 'z'>>>,
+      assert<VerifyEquation<Transitivity_Proof<'x', 'y', 'z'>>>,
+    ]
+  }
+
+  // Inverse functions
+  export type Inverse<x extends Op, y extends Op> = Rewrite<Inv<Abs<x, y>>, Abs<y, x>>;
+  export namespace proof {
+    export interface Composition_Inverse_Proof<F extends Op, G extends Op> {
+      type: 'rewrite',
+      known: {
+        'f': Equation<F, Abs<'y', 'z'>>,
+        'g': Equation<G, Abs<'x', 'y'>>,
+      },
+      left: Equation<Inv<Comp<F, G>>, Comp<Inv<G>, Inv<F>>>, // inv(f . g) = inv(g) . inv(f)
+      right: ChainRewrites<[
+        SubstEq<this['known']['f']>,
+        SubstEq<this['known']['f']>,         // inv((\y -> z) . g) = inv(g) . inv(\y -> z)
+        SubstEq<this['known']['g']>,
+        SubstEq<this['known']['g']>,         // inv((\y -> z) . (\x -> y)) = inv(\x -> y) . inv(\y -> z)
+        Inverse<'x', 'y'>,
+        Inverse<'y', 'z'>,                   // inv((\x -> y) . (\y -> z)) = (\z -> y) . (\y -> x)
+        Transitivity<'x', 'y', 'z'>,         // inv(\x -> z) = (\z -> y) . (\y -> x)
+        Transitivity<'z', 'y', 'x'>,         // inv(\x -> z) = \z -> x
+        Inverse<'x', 'z'>,                   // \z -> x = \z -> x
+        Refl
+      ], this['left']>,
+    }
+    export type inverse = [
+      // OpToStr<Evaluate<Composition_Inverse_Proof<'F', 'G'>>>,
+      assert<VerifyEquation<Composition_Inverse_Proof<'F', 'G'>>>,
+    ]
+  }
+}
+
diff --git a/src/utils/theorem.ts b/src/utils/theorem.ts
index a4dd2e9..be19b94 100644
--- a/src/utils/theorem.ts
+++ b/src/utils/theorem.ts
@@ -6,9 +6,9 @@ export type OpToStr<O extends Op> =
   O extends string
   ? O
   : O extends { op: string, a: infer a extends string, b: infer b extends string } ? `(${a} ${O['op']} ${b})`
-  : O extends { op: string, a: infer a extends DyadicOp, b: infer b extends string } ? `(${OpToStr<a>} ${O['op']} ${b})`
-  : O extends { op: string, a: infer a extends string, b: infer b extends DyadicOp } ? `(${a} ${O['op']} ${OpToStr<b>})`
-  : O extends { op: string, a: infer a extends DyadicOp, b: infer b extends DyadicOp } ? `(${OpToStr<a>} ${O['op']} ${OpToStr<b>})`
+  : O extends { op: string, a: infer a extends DyadicOp | MonadicOp, b: infer b extends string } ? `(${OpToStr<a>} ${O['op']} ${b})`
+  : O extends { op: string, a: infer a extends string, b: infer b extends DyadicOp | MonadicOp } ? `(${a} ${O['op']} ${OpToStr<b>})`
+  : O extends { op: string, a: infer a extends DyadicOp | MonadicOp, b: infer b extends DyadicOp | MonadicOp } ? `(${OpToStr<a>} ${O['op']} ${OpToStr<b>})`
   : O extends { op: string, a: infer a extends string } ? `${O['op']}(${a})`
   : O extends { op: string, a: infer a extends DyadicOp } ? `${O['op']}(${OpToStr<a>})`
   : never;
@@ -30,6 +30,7 @@ export interface Rewrite<left extends Op, right extends Op, Typ extends RwType =
 
 export type Sym<R extends RewriteBase> = Rewrite<R['right'], R['left'], R['type']>;
 export type Subst<A extends Op, B extends Op> = Rewrite<A, B>
+export type SubstEq<O extends Equation<Op, Op>> = Rewrite<O['a'], O['b']>
 
 export type ApplyRewrite<O extends Op, R extends RewriteBase> =
   R['type'] extends 'imperative' ? (R & { left: O })['right']
-- 
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