--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/agda-term.agda.replaced	Tue Sep 08 18:38:08 2020 +0900
@@ -0,0 +1,28 @@
+module agda-term where
+
+open import Data.Nat.Base
+open import Relation.Binary.PropositionalEquality
+
++zero : {y : @$\mathbb{N}$@} @$\rightarrow$@ y + zero @$\equiv$@ y
++zero {zero} = refl
++zero {suc y} = cong (@$\lambda$@ yy @$\rightarrow$@ suc yy) (+zero {y})
+
++-suc : {x y : @$\mathbb{N}$@} @$\rightarrow$@ x + suc y @$\equiv$@ suc (x + y)
++-suc {zero} {y} = refl
++-suc {suc x} {y} = cong suc (+-suc {x} {y})
+
++-comm : (x y : @$\mathbb{N}$@) @$\rightarrow$@ x + y @$\equiv$@ y + x
++-comm zero y rewrite (+zero {y}) = refl
++-comm (suc x) y = let open @$\equiv$@-Reasoning in
+  begin
+  suc (x + y) @$\equiv$@@$\langle$@@$\rangle$@
+  suc (x + y) @$\equiv$@@$\langle$@ cong suc (+-comm x y) @$\rangle$@
+  suc (y + x) @$\equiv$@@$\langle$@ sym (+-suc {y} {x}) @$\rangle$@
+  y + suc x @$\blacksquare$@
+
++-come : (x y : @$\mathbb{N}$@) @$\rightarrow$@ x + y @$\equiv$@ y + x
++-come zero y rewrite (+zero {y}) = refl
++-come (suc x) y
+  rewrite (cong suc (+-come x y)) | sym (+-suc {y} {x}) = refl
+
+