Added Agda code for some HITs

Agda-HITs/Expressions/Expressions.agda

@@ -0,0 +1,59 @@
+{-# OPTIONS --without-K --rewriting #-}
+
+open import HoTT
+
+module Expressions where
+
+private
+  data Exp' : Set where
+    value : Nat -> Exp'
+    addition : Exp' -> Exp' -> Exp'
+
+Exp : Set
+Exp = Exp'
+
+val : Nat -> Exp
+val = value
+
+plus : Exp -> Exp -> Exp
+plus = addition
+
+postulate 
+  add : (n m : Nat) -> plus (val n) (val m) == val (n + m)
+  trunc : is-set Exp
+
+Exp-ind                : (C : Exp -> Set) 
+                       -> (vC : (n : Nat) -> C (val n))
+                       -> (pC : (e₁ e₂ : Exp) -> C e₁ -> C e₂ -> C(plus e₁ e₂))
+                       -> (addC : (n m : Nat) -> PathOver C (add n m) (pC (val n) (val m) (vC n) (vC m)) (vC (n + m)))
+                       -> (t : (e : Exp) -> is-set (C e))
+                       -> (x : Exp) -> C x
+Exp-ind C vC pC addC t (value n)  = vC n
+Exp-ind C vC pC addC t (addition e₁ e₂)  = pC e₁ e₂ (Exp-ind C vC pC addC t e₁) (Exp-ind C vC pC addC t e₂)
+
+postulate
+  Exp-ind-βadd : (C : Exp -> Set) 
+                 -> (vC : (n : Nat) -> C (val n))
+                 -> (pC : (e₁ e₂ : Exp) -> C e₁ -> C e₂ -> C(plus e₁ e₂))
+                 -> (addC : (n m : Nat) -> PathOver C (add n m) (pC (val n) (val m) (vC n) (vC m)) (vC (n + m)))
+                 -> (t : (e : Exp) -> is-set (C e))
+                 -> (n m : Nat)
+                 -> apd (Exp-ind C vC pC addC t) (add n m) == addC n m
+
+Exp-rec                : {C : Set} 
+                       -> (vC : Nat -> C)
+                       -> (pC : C -> C -> C)
+                       -> (addC : (n m : Nat) -> pC (vC n) (vC m) == vC (n + m))
+                       -> (t : is-set C)
+                       -> Exp -> C
+Exp-rec vC pC addC t (value n)  = vC n
+Exp-rec vC pC addC t (addition e₁ e₂)  = pC (Exp-rec vC pC addC t e₁) (Exp-rec vC pC addC t e₂)
+
+postulate
+  Exp-rec-βadd : {C : Set} 
+                 -> (vC : Nat -> C)
+                 -> (pC : C -> C -> C)
+                 -> (addC : (n m : Nat) -> pC (vC n) (vC m) == vC (n + m))
+                 -> (t : is-set C)
+                 -> (n m : Nat)
+                 -> ap (Exp-rec vC pC addC t) (add n m) == addC n m

Agda-HITs/Expressions/Thms.agda

@@ -0,0 +1,16 @@
+{-# OPTIONS --without-K --rewriting #-}
+
+open import HoTT
+open import Expressions
+
+module Thms where
+
+value : (e : Exp) -> Σ Nat (\n -> e == val n)
+value = Exp-ind
+  (\e ->  Σ Nat (\n -> e == val n))
+  (\n -> n , idp)
+  (\e₁ e₂ v₁ v₂  -> fst v₁ + fst v₂ ,
+    (ap (\e -> plus e e₂) (snd v₁) ∙ ap (plus (val (fst v₁))) (snd v₂)) ∙ add (fst v₁) (fst v₂)
+  )
+  (\n m -> from-transp! (\e -> Σ Nat (\n -> e == val n)) (add n m) (pair= {!!} {!!}))
+  (\e -> {!!})