- mid T m$ xs m$ mid f
-== mid T m$ ((mid id) m$ xs) m$ mid f, by 1
-== mid (○) m$ mid T m$ mid id m$ xs m$ mid f, by 3
-== mid ($id) m$ (mid (○) m$ mid T) m$ xs m$ mid f, by 4
-== mid (○) m$ mid ($id) m$ mid (○) m$ mid T m$ xs m$ mid f, by 3
-== mid ((○) ($id)) m$ mid (○) m$ mid T m$ xs m$ mid f, by 2
-== mid ((○) ($id) (○)) m$ mid T m$ xs m$ mid f, by 2
-== mid id m$ mid T m$ xs m$ mid f, by definitions of ○ and $
-== mid T m$ xs m$ mid f, by 1
-== mid ($f) m$ (mid T m$ xs), by 4
-== mid (○) m$ mid ($f) m$ mid T m$ xs, by 3
-== mid ((○) ($f)) m$ mid T m$ xs, by 2
-== mid ((○) ($f) T) m$ xs, by 2
-== mid f m$ xs, by definitions of ○ and $ and T == flip ($)
+ ⇧ T m$ xs m$ ⇧ f
+== ⇧ T m$ ((⇧ id) m$ xs) m$ ⇧ f, by 1
+== ⇧ (○) m$ ⇧ T m$ ⇧ id m$ xs m$ ⇧ f, by 3
+== ⇧ ($id) m$ (⇧ (○) m$ ⇧ T) m$ xs m$ ⇧ f, by 4
+== ⇧ (○) m$ ⇧ ($id) m$ ⇧ (○) m$ ⇧ T m$ xs m$ ⇧ f, by 3
+== ⇧ ((○) ($id)) m$ ⇧ (○) m$ ⇧ T m$ xs m$ ⇧ f, by 2
+== ⇧ ((○) ($id) (○)) m$ ⇧ T m$ xs m$ ⇧ f, by 2
+== ⇧ id m$ ⇧ T m$ xs m$ ⇧ f, by definitions of ○ and $
+== ⇧ T m$ xs m$ ⇧ f, by 1
+== ⇧ ($f) m$ (⇧ T m$ xs), by 4
+== ⇧ (○) m$ ⇧ ($f) m$ ⇧ T m$ xs, by 3
+== ⇧ ((○) ($f)) m$ ⇧ T m$ xs, by 2
+== ⇧ ((○) ($f) T) m$ xs, by 2
+== ⇧ f m$ xs, by definitions of ○ and $ and T == flip ($)