## [aa4262]: thys / Category2 / NatTrans.thy Maximize Restore History

### NatTrans.thy    640 lines (578 with data), 38.0 kB

```(*
Author: Alexander Katovsky
*)

theory NatTrans imports Functors begin

record ('o1, 'o2, 'm1, 'm2, 'a, 'b) NatTrans =
NTDom :: "('o1, 'o2, 'm1, 'm2, 'a, 'b) Functor"
NTCod :: "('o1, 'o2, 'm1, 'm2, 'a, 'b) Functor"
NatTransMap :: "'o1 \<Rightarrow> 'm2"

abbreviation
NatTransApp :: "('o1, 'o2, 'm1, 'm2, 'a, 'b) NatTrans \<Rightarrow> 'o1 \<Rightarrow> 'm2" (infixr "\$\$" 70) where
"NatTransApp \<eta> X \<equiv> (NatTransMap \<eta>) X"

definition  "NTCatDom \<eta> \<equiv> CatDom (NTDom \<eta>)"
definition  "NTCatCod \<eta> \<equiv> CatCod (NTCod \<eta>)"

locale NatTransExt =
fixes \<eta> :: "('o1, 'o2, 'm1, 'm2, 'a, 'b) NatTrans" (structure)
assumes  NTExt : "NatTransMap \<eta> \<in> extensional (Obj (NTCatDom \<eta>))"

locale NatTransP =
fixes \<eta> :: "('o1, 'o2, 'm1, 'm2, 'a, 'b) NatTrans" (structure)
assumes NatTransFtor:   "Functor (NTDom \<eta>)"
and     NatTransFtor2:  "Functor (NTCod \<eta>)"
and     NatTransFtorDom:   "NTCatDom \<eta> = CatDom (NTCod \<eta>)"
and     NatTransFtorCod:   "NTCatCod \<eta> = CatCod (NTDom \<eta>)"
and    NatTransMapsTo:  "X \<in> obj\<^bsub>NTCatDom \<eta>\<^esub> \<Longrightarrow>
(\<eta> \$\$ X) maps\<^bsub>NTCatCod \<eta>\<^esub> ((NTDom \<eta>) @@ X) to ((NTCod \<eta>) @@ X)"
and    NatTrans:  "f maps\<^bsub>NTCatDom \<eta>\<^esub> X to Y \<Longrightarrow>
((NTDom \<eta>) ## f) ;;\<^bsub>NTCatCod \<eta>\<^esub> (\<eta> \$\$ Y) = (\<eta> \$\$ X) ;;\<^bsub>NTCatCod \<eta>\<^esub> ((NTCod \<eta>) ## f)"

locale NatTrans = NatTransP + NatTransExt

lemma [simp]: "NatTrans \<eta> \<Longrightarrow> NatTransP \<eta>"

definition MakeNT :: "('o1, 'o2, 'm1, 'm2, 'a, 'b) NatTrans \<Rightarrow> ('o1, 'o2, 'm1, 'm2, 'a, 'b) NatTrans" where
"MakeNT \<eta> \<equiv> \<lparr>
NTDom = NTDom \<eta> ,
NTCod = NTCod \<eta> ,
NatTransMap = restrict (NatTransMap \<eta>) (Obj (NTCatDom \<eta>))
\<rparr>"

definition
nt_abbrev ("NT _ : _ \<Longrightarrow> _" [81]) where
"NT f : F \<Longrightarrow> G \<equiv> (NatTrans f) \<and> (NTDom f = F) \<and> (NTCod f = G)"

lemma nt_abbrevE[elim]: "\<lbrakk>NT f : F \<Longrightarrow> G ; \<lbrakk>(NatTrans f) ; (NTDom f = F) ; (NTCod f = G)\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"

lemma MakeNT: "NatTransP \<eta> \<Longrightarrow> NatTrans (MakeNT \<eta>)"
by(auto simp add: NatTransP_def NatTrans_def MakeNT_def NTCatDom_def NTCatCod_def Category.MapsToObj
NatTransExt_def)

lemma MakeNT_comp: "X \<in> Obj (NTCatDom f) \<Longrightarrow> (MakeNT f) \$\$ X = f \$\$ X"

lemma MakeNT_dom: "NTCatDom f = NTCatDom (MakeNT f)"

lemma MakeNT_cod: "NTCatCod f = NTCatCod (MakeNT f)"

lemma MakeNTApp: "X \<in> Obj (NTCatDom (MakeNT f)) \<Longrightarrow> f \$\$ X = (MakeNT f) \$\$ X"

lemma NatTransMapsTo:
assumes "NT \<eta> : F \<Longrightarrow> G" and "X \<in> Obj (CatDom F)"
shows "\<eta> \$\$ X maps\<^bsub>CatCod G \<^esub>(F @@ X) to (G @@ X)"
proof-
have NTP: "NatTransP \<eta>" using assms by auto
have NTC: "NTCatCod \<eta> = CatCod G" using assms by (auto simp add: NTCatCod_def)
have NTD: "NTCatDom \<eta> = CatDom F" using assms by (auto simp add: NTCatDom_def)
hence Obj: "X \<in> Obj (NTCatDom \<eta>)" using assms by simp
have DF: "NTDom \<eta> = F" and CG: "NTCod \<eta> = G" using assms by auto
have NTmapsTo: "\<eta> \$\$ X maps\<^bsub>NTCatCod \<eta> \<^esub>((NTDom \<eta>) @@ X) to ((NTCod \<eta>) @@ X)"
using NTP Obj by (simp add: NatTransP.NatTransMapsTo)
thus ?thesis using NTC NTD DF CG by simp
qed

definition
NTCompDefined :: "('o1, 'o2, 'm1, 'm2, 'a, 'b) NatTrans
\<Rightarrow> ('o1, 'o2, 'm1, 'm2, 'a, 'b) NatTrans \<Rightarrow> bool" (infixl "\<approx>>\<bullet>" 65) where
"NTCompDefined \<eta>1 \<eta>2 \<equiv> NatTrans \<eta>1 \<and> NatTrans \<eta>2 \<and> NTCatDom \<eta>2 = NTCatDom \<eta>1 \<and>
NTCatCod \<eta>2 = NTCatCod \<eta>1 \<and> NTCod \<eta>1 = NTDom \<eta>2"

lemma NTCompDefinedE[elim]: "\<lbrakk>\<eta>1 \<approx>>\<bullet> \<eta>2 ; \<lbrakk>NatTrans \<eta>1 ; NatTrans \<eta>2 ; NTCatDom \<eta>2 = NTCatDom \<eta>1 ;
NTCatCod \<eta>2 = NTCatCod \<eta>1 ; NTCod \<eta>1 = NTDom \<eta>2\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"

lemma NTCompDefinedI: "\<lbrakk>NatTrans \<eta>1 ; NatTrans \<eta>2 ; NTCatDom \<eta>2 = NTCatDom \<eta>1 ;
NTCatCod \<eta>2 = NTCatCod \<eta>1 ; NTCod \<eta>1 = NTDom \<eta>2\<rbrakk> \<Longrightarrow> \<eta>1 \<approx>>\<bullet> \<eta>2"

lemma NatTransExt0:
assumes "NTDom \<eta>1 = NTDom \<eta>2" and "NTCod \<eta>1 = NTCod \<eta>2"
and     "\<And>X . X \<in> Obj (NTCatDom \<eta>1) \<Longrightarrow> \<eta>1 \$\$ X = \<eta>2 \$\$ X"
and     "NatTransMap \<eta>1 \<in> extensional (Obj (NTCatDom \<eta>1))"
and     "NatTransMap \<eta>2 \<in> extensional (Obj (NTCatDom \<eta>2))"
shows   "\<eta>1 = \<eta>2"
proof-
have "NatTransMap \<eta>1 = NatTransMap \<eta>2"
proof(rule extensionalityI [of "NatTransMap \<eta>1" "Obj (NTCatDom \<eta>1)"])
show "NatTransMap \<eta>1 \<in> extensional (Obj (NTCatDom \<eta>1))" using assms by simp
have "NTCatDom \<eta>1 = NTCatDom \<eta>2" using assms by (simp add: NTCatDom_def)
moreover have "NatTransMap \<eta>2 \<in> extensional (Obj (NTCatDom \<eta>2))" using assms by simp
ultimately show "NatTransMap \<eta>2 \<in> extensional (Obj (NTCatDom \<eta>1))" by simp
{fix X assume "X \<in> Obj (NTCatDom \<eta>1)" thus "\<eta>1 \$\$ X = \<eta>2 \$\$ X" using assms by simp}
qed
thus ?thesis using assms by (simp)
qed

lemma NatTransExt':
assumes "NTDom \<eta>1' = NTDom \<eta>2'" and "NTCod \<eta>1' = NTCod \<eta>2'"
and     "\<And>X . X \<in> Obj (NTCatDom \<eta>1') \<Longrightarrow> \<eta>1' \$\$ X = \<eta>2' \$\$ X"
shows   "MakeNT \<eta>1' = MakeNT \<eta>2'"
proof(rule NatTransExt0)
show "NatTransMap (MakeNT \<eta>1') \<in> extensional (Obj (NTCatDom (MakeNT \<eta>1')))" and
"NatTransMap (MakeNT \<eta>2') \<in> extensional (Obj (NTCatDom (MakeNT \<eta>2')))" using assms
by(simp add: MakeNT_def NTCatDom_def NTCatCod_def NatTransExt_def)+
show "NTDom (MakeNT \<eta>1') = NTDom (MakeNT \<eta>2')" and
"NTCod (MakeNT \<eta>1') = NTCod (MakeNT \<eta>2')" using assms by (simp add: MakeNT_def)+
{
fix X assume 1: "X \<in> Obj (NTCatDom (MakeNT \<eta>1'))"
show "(MakeNT \<eta>1') \$\$ X = (MakeNT \<eta>2') \$\$ X"
proof-
have "NTCatDom (MakeNT \<eta>1') = NTCatDom (MakeNT \<eta>2')" using assms by(simp add: NTCatDom_def MakeNT_def)
hence 2: "X \<in> Obj (NTCatDom (MakeNT \<eta>2'))" using 1 by simp
have "(NTCatDom \<eta>1') = (NTCatDom (MakeNT \<eta>1'))" by (rule MakeNT_dom)
hence "X \<in> Obj (NTCatDom \<eta>1')" using 1 assms by simp
hence "\<eta>1' \$\$ X = \<eta>2' \$\$ X" using assms by simp
moreover have "\<eta>1' \$\$ X = (MakeNT \<eta>1') \$\$ X" using 1 assms by (simp add: MakeNTApp)
moreover have "\<eta>2' \$\$ X = (MakeNT \<eta>2') \$\$ X" using 2 assms by (simp add: MakeNTApp)
ultimately have "(MakeNT \<eta>1') \$\$ X = (MakeNT \<eta>2') \$\$ X" by simp
thus ?thesis using assms by simp
qed
}
qed

lemma NatTransExt:
assumes "NatTrans \<eta>1" and "NatTrans \<eta>2" and "NTDom \<eta>1 = NTDom \<eta>2" and "NTCod \<eta>1 = NTCod \<eta>2"
and     "\<And>X . X \<in> Obj (NTCatDom \<eta>1) \<Longrightarrow> \<eta>1 \$\$ X = \<eta>2 \$\$ X"
shows   "\<eta>1 = \<eta>2"
proof-
have "NatTransMap \<eta>1 \<in> extensional (Obj (NTCatDom \<eta>1))" and
"NatTransMap \<eta>2 \<in> extensional (Obj (NTCatDom \<eta>2))" using assms
by(simp only: NatTransExt_def NatTrans_def)+
thus ?thesis using assms by (simp add: NatTransExt0)
qed

definition
IdNatTrans' :: "('o1, 'o2, 'm1, 'm2, 'a1, 'a2) Functor \<Rightarrow> ('o1, 'o2, 'm1, 'm2, 'a1, 'a2) NatTrans" where
"IdNatTrans' F \<equiv> \<lparr>
NTDom = F ,
NTCod = F ,
NatTransMap = \<lambda> X . id\<^bsub>CatCod F\<^esub> (F @@ X)
\<rparr>"

definition "IdNatTrans F \<equiv> MakeNT(IdNatTrans' F)"

lemma IdNatTrans_map: "X \<in> obj\<^bsub>CatDom F\<^esub> \<Longrightarrow> (IdNatTrans F) \$\$ X = id\<^bsub>CatCod F\<^esub> (F @@ X)"
by(auto simp add: IdNatTrans_def IdNatTrans'_def MakeNT_comp MakeNT_def NTCatDom_def)

lemmas IdNatTrans_defs = IdNatTrans_def IdNatTrans'_def MakeNT_def IdNatTrans_map NTCatCod_def NTCatDom_def

lemma IdNatTransNatTrans': "Functor F \<Longrightarrow> NatTransP(IdNatTrans' F)"
proof(auto simp add:NatTransP_def IdNatTrans'_def NTCatDom_def NTCatCod_def Category.Simps
PreFunctor.FunctorId2 functor_simps Functor.FunctorMapsTo)
{
fix f X Y
assume a: "Functor F" and b: "f maps\<^bsub>CatDom F\<^esub> X to Y"
show "(F ## f) ;;\<^bsub>CatCod F\<^esub> (id\<^bsub>CatCod F\<^esub> (F @@ Y)) = (id\<^bsub>CatCod F\<^esub> (F @@ X)) ;;\<^bsub>CatCod F\<^esub> (F ## f)"
proof-
have 1: "Category (CatCod F)" using a by simp
have "F ## f maps\<^bsub>CatCod F\<^esub> (F @@ X) to (F @@ Y)" using a b by (auto simp add: Functor.FunctorMapsTo)
hence 2: "F ## f \<in> mor\<^bsub>CatCod F\<^esub>" and 3: "cod\<^bsub>CatCod F\<^esub> (F ## f) = (F @@ Y)"
and 4: "dom\<^bsub>CatCod F\<^esub> (F ## f) = (F @@ X)" by auto
have "(F ## f) ;;\<^bsub>CatCod F\<^esub> (id\<^bsub>CatCod F\<^esub> (F @@ Y)) = (F ## f) ;;\<^bsub>CatCod F\<^esub> (id\<^bsub>CatCod F\<^esub> (cod\<^bsub>CatCod F\<^esub> (F ## f)))"
using 3 by simp
also have "... = F ## f" using 1 2 by (auto simp add: Category.Cidr)
also have "... = (id\<^bsub>CatCod F\<^esub> (dom\<^bsub>CatCod F\<^esub> (F ## f))) ;;\<^bsub>CatCod F\<^esub> (F ## f)"
using 1 2 by (auto simp add: Category.Cidl)
also have "... = (id\<^bsub>CatCod F\<^esub> (F @@ X)) ;;\<^bsub>CatCod F\<^esub> (F ## f)" using 4 by simp
finally show ?thesis .
qed
}
qed

lemma IdNatTransNatTrans: "Functor F \<Longrightarrow> NatTrans (IdNatTrans F)"
by (simp add: IdNatTransNatTrans' IdNatTrans_def MakeNT)

definition
NatTransComp' :: "('o1, 'o2, 'm1, 'm2, 'a, 'b) NatTrans \<Rightarrow>
('o1, 'o2, 'm1, 'm2, 'a, 'b) NatTrans \<Rightarrow>
('o1, 'o2, 'm1, 'm2, 'a, 'b) NatTrans" (infixl "\<bullet>1" 75) where
"NatTransComp' \<eta>1 \<eta>2 = \<lparr>
NTDom = NTDom \<eta>1 ,
NTCod = NTCod \<eta>2 ,
NatTransMap = \<lambda> X . (\<eta>1 \$\$ X) ;;\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>2 \$\$ X)
\<rparr>"

definition NatTransComp (infixl "\<bullet>" 75) where "\<eta>1 \<bullet> \<eta>2 \<equiv> MakeNT(\<eta>1 \<bullet>1 \<eta>2)"

lemma NatTransComp_Comp1: "\<lbrakk>x \<in> Obj (NTCatDom f) ; f \<approx>>\<bullet> g\<rbrakk> \<Longrightarrow> (f \<bullet> g) \$\$ x = (f \$\$ x) ;;\<^bsub>NTCatCod g\<^esub> (g \$\$ x)"
by(auto simp add: NatTransComp_def NatTransComp'_def MakeNT_def NTCatCod_def NTCatDom_def)

lemma NatTransComp_Comp2: "\<lbrakk>x \<in> Obj (NTCatDom f) ; f \<approx>>\<bullet> g\<rbrakk> \<Longrightarrow> (f \<bullet> g) \$\$ x = (f \$\$ x) ;;\<^bsub>NTCatCod f\<^esub> (g \$\$ x)"
by(auto simp add: NatTransComp_def NatTransComp'_def MakeNT_def NTCatCod_def NTCatDom_def)

lemmas NatTransComp_defs = NatTransComp_def NatTransComp'_def MakeNT_def
NatTransComp_Comp1  NTCatCod_def NTCatDom_def

lemma [simp]: "\<eta>1 \<approx>>\<bullet> \<eta>2 \<Longrightarrow> NatTrans \<eta>1" by auto
lemma [simp]: "\<eta>1 \<approx>>\<bullet> \<eta>2 \<Longrightarrow> NatTrans \<eta>2" by auto
lemma NTCatDom:        "\<eta>1 \<approx>>\<bullet> \<eta>2 \<Longrightarrow> NTCatDom \<eta>1 = NTCatDom \<eta>2" by auto
lemma NTCatCod:        "\<eta>1 \<approx>>\<bullet> \<eta>2 \<Longrightarrow> NTCatCod \<eta>1 = NTCatCod \<eta>2" by auto
lemma [simp]: "\<eta>1 \<approx>>\<bullet> \<eta>2 \<Longrightarrow> NTCatDom (\<eta>1 \<bullet>1 \<eta>2) = NTCatDom \<eta>1" by (auto simp add: NatTransComp'_def NTCatDom_def)
lemma [simp]: "\<eta>1 \<approx>>\<bullet> \<eta>2 \<Longrightarrow> NTCatCod (\<eta>1 \<bullet>1 \<eta>2) = NTCatCod \<eta>1" by (auto simp add: NatTransComp'_def NTCatCod_def)
lemma [simp]: "\<eta>1 \<approx>>\<bullet> \<eta>2 \<Longrightarrow> NTCatDom (\<eta>1 \<bullet> \<eta>2) = NTCatDom \<eta>1" by (auto simp add: NatTransComp_defs)
lemma [simp]: "\<eta>1 \<approx>>\<bullet> \<eta>2 \<Longrightarrow> NTCatCod (\<eta>1 \<bullet> \<eta>2) = NTCatCod \<eta>1" by (auto simp add: NatTransComp_defs)
lemma [simp]: "NatTrans \<eta> \<Longrightarrow> Category(NTCatDom \<eta>)" by (simp add:  NatTransP.NatTransFtor NTCatDom_def)
lemma [simp]: "NatTrans \<eta> \<Longrightarrow> Category(NTCatCod \<eta>)" by (simp add:  NatTransP.NatTransFtor2 NTCatCod_def)
lemma DDDC: assumes "NatTrans f" shows "CatDom (NTDom f) = CatDom (NTCod f)"
proof-
have "CatDom (NTDom f) = NTCatDom f" by (simp add: NTCatDom_def)
thus ?thesis using assms by (simp add: NatTransP.NatTransFtorDom)
qed
lemma CCCD: assumes "NatTrans f" shows "CatCod (NTCod f) = CatCod (NTDom f)"
proof-
have "CatCod (NTCod f) = NTCatCod f" by (simp add: NTCatCod_def)
thus ?thesis using assms by (simp add: NatTransP.NatTransFtorCod)
qed

lemma IdNatTransCompDefDom: "NatTrans f \<Longrightarrow> (IdNatTrans (NTDom f)) \<approx>>\<bullet> f"
apply(rule NTCompDefinedI)
done

lemma IdNatTransCompDefCod: "NatTrans f \<Longrightarrow> f \<approx>>\<bullet> (IdNatTrans (NTCod f))"
apply(rule NTCompDefinedI)
done

lemma NatTransCompDefCod:
assumes "NatTrans \<eta>" and "f maps\<^bsub>NTCatDom \<eta>\<^esub> X to Y"
shows "(\<eta> \$\$ X) \<approx>>\<^bsub>NTCatCod \<eta>\<^esub> (NTCod \<eta> ## f)"
proof(rule CompDefinedI)
have b: "X \<in> obj\<^bsub>NTCatDom \<eta>\<^esub>" and c: "Y \<in> obj\<^bsub>NTCatDom \<eta>\<^esub>" using assms by (auto simp add: Category.MapsToObj)
have d: "(\<eta> \$\$ X) maps\<^bsub>NTCatCod \<eta>\<^esub> ((NTDom \<eta>) @@ X) to ((NTCod \<eta>) @@ X)" using assms b
thus "\<eta> \$\$ X \<in> mor\<^bsub>NTCatCod \<eta>\<^esub>" by auto
have "f maps\<^bsub>CatDom (NTCod \<eta>)\<^esub> X to Y" using assms by (simp add: NatTransP.NatTransFtorDom)
hence e: "NTCod \<eta> ## f maps\<^bsub>CatCod (NTCod \<eta>)\<^esub> (NTCod \<eta> @@ X) to (NTCod \<eta> @@ Y)" using assms
thus "NTCod \<eta> ## f \<in> mor\<^bsub>NTCatCod \<eta>\<^esub>" by (auto simp add: NTCatCod_def)
have "cod\<^bsub>NTCatCod \<eta>\<^esub> (\<eta> \$\$ X) = (NTCod \<eta> @@ X)" using d by auto
also have "... = dom\<^bsub>CatCod (NTCod \<eta>)\<^esub> (NTCod \<eta> ## f)" using e by auto
finally show "cod\<^bsub>NTCatCod \<eta>\<^esub> (\<eta> \$\$ X) = dom\<^bsub>NTCatCod \<eta>\<^esub> (NTCod \<eta> ## f)" by (auto simp add: NTCatCod_def)
qed

lemma NatTransCompDefDom:
assumes "NatTrans \<eta>" and "f maps\<^bsub>NTCatDom \<eta>\<^esub> X to Y"
shows "(NTDom \<eta> ## f)  \<approx>>\<^bsub>NTCatCod \<eta>\<^esub> (\<eta> \$\$ Y)"
proof(rule CompDefinedI)
have b: "X \<in> obj\<^bsub>NTCatDom \<eta>\<^esub>" and c: "Y \<in> obj\<^bsub>NTCatDom \<eta>\<^esub>" using assms by (auto simp add: Category.MapsToObj)
have d: "(\<eta> \$\$ Y) maps\<^bsub>NTCatCod \<eta>\<^esub> ((NTDom \<eta>) @@ Y) to ((NTCod \<eta>) @@ Y)" using assms c
thus "\<eta> \$\$ Y \<in> mor\<^bsub>NTCatCod \<eta>\<^esub>" by auto
have "f maps\<^bsub>CatDom (NTDom \<eta>)\<^esub> X to Y" using assms by (simp add: NTCatDom_def)
hence e: "NTDom \<eta> ## f maps\<^bsub>CatCod (NTDom \<eta>)\<^esub> (NTDom \<eta> @@ X) to (NTDom \<eta> @@ Y)" using assms
thus "NTDom \<eta> ## f \<in> mor\<^bsub>NTCatCod \<eta>\<^esub>" using assms by (auto simp add: NatTransP.NatTransFtorCod)
have "dom\<^bsub>NTCatCod \<eta>\<^esub> (\<eta> \$\$ Y) = (NTDom \<eta> @@ Y)" using d by auto
also have "... = cod\<^bsub>CatCod (NTDom \<eta>)\<^esub> (NTDom \<eta> ## f)" using e by auto
finally show "cod\<^bsub>NTCatCod \<eta>\<^esub> (NTDom \<eta> ## f) = dom\<^bsub>NTCatCod \<eta>\<^esub> (\<eta> \$\$ Y)"
using assms by (auto simp add: NatTransP.NatTransFtorCod)
qed

lemma NatTransCompCompDef:
assumes "\<eta>1 \<approx>>\<bullet> \<eta>2" and "X \<in> obj\<^bsub>NTCatDom \<eta>1\<^esub>"
shows "(\<eta>1 \$\$ X) \<approx>>\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>2 \$\$ X)"
proof(rule CompDefinedI)
have 1: "(\<eta>1 \$\$ X) maps\<^bsub>NTCatCod \<eta>1\<^esub> ((NTDom \<eta>1) @@ X) to ((NTCod \<eta>1) @@ X)" using assms
have "NTCatCod \<eta>1 = NTCatCod \<eta>2" using assms by auto
hence 2: "(\<eta>2 \$\$ X) maps\<^bsub>NTCatCod \<eta>1\<^esub> ((NTDom \<eta>2) @@ X) to ((NTCod \<eta>2) @@ X)" using assms
show "\<eta>1 \$\$ X \<in> mor\<^bsub>NTCatCod \<eta>1\<^esub>"
and "\<eta>2 \$\$ X \<in> mor\<^bsub>NTCatCod \<eta>1\<^esub>"  using 1 2 by auto
have "cod\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>1 \$\$ X) = ((NTCod \<eta>1) @@ X)" using 1 by auto
also have "... = ((NTDom \<eta>2) @@ X)" using assms by auto
finally show "cod\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>1 \$\$ X) = dom\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>2 \$\$ X)" using 2 by auto
qed

lemma NatTransCompNatTrans':
assumes "\<eta>1 \<approx>>\<bullet> \<eta>2"
shows   "NatTransP (\<eta>1 \<bullet>1 \<eta>2)"
show "Functor (NTDom (\<eta>1 \<bullet>1 \<eta>2))" and "Functor (NTCod (\<eta>1 \<bullet>1 \<eta>2))" using assms
by (auto simp add: NatTransComp'_def NatTransP.NatTransFtor NatTransP.NatTransFtor2)
show "NTCatDom (\<eta>1 \<bullet>1 \<eta>2) = CatDom (NTCod (\<eta>1 \<bullet>1 \<eta>2))" and
"NTCatCod (\<eta>1 \<bullet>1 \<eta>2) = CatCod (NTDom (\<eta>1 \<bullet>1 \<eta>2))"
proof (auto simp add: NatTransComp'_def NTCatCod_def NTCatDom_def)
have "CatDom (NTDom \<eta>1) = NTCatDom \<eta>1" by (simp add: NTCatDom_def)
thus "CatDom (NTDom \<eta>1) = CatDom (NTCod \<eta>2)" using assms by (auto simp add: NatTransP.NatTransFtorDom)
have "CatCod (NTCod \<eta>2) = NTCatCod \<eta>2" by (simp add: NTCatCod_def)
thus "CatCod (NTCod \<eta>2) = CatCod (NTDom \<eta>1)" using assms by (auto simp add: NatTransP.NatTransFtorCod)
qed
{
fix X assume aa: "X \<in> obj\<^bsub>NTCatDom (\<eta>1 \<bullet>1 \<eta>2)\<^esub>"
show "(\<eta>1 \<bullet>1 \<eta>2) \$\$ X maps\<^bsub>NTCatCod (\<eta>1 \<bullet>1 \<eta>2)\<^esub> NTDom (\<eta>1 \<bullet>1 \<eta>2) @@ X to NTCod (\<eta>1 \<bullet>1 \<eta>2) @@ X"
proof-
have "X \<in> obj\<^bsub>NTCatDom \<eta>1\<^esub>" and "NatTrans \<eta>1" using assms aa by simp+
hence "(\<eta>1 \$\$ X) maps\<^bsub>NTCatCod \<eta>1\<^esub> ((NTDom \<eta>1) @@ X) to ((NTCod \<eta>1) @@ X)"
moreover have "(\<eta>2 \$\$ X) maps\<^bsub>NTCatCod \<eta>1\<^esub> ((NTCod \<eta>1) @@ X) to ((NTCod \<eta>2) @@ X)"
proof-
have "X \<in> obj\<^bsub>NTCatDom \<eta>2\<^esub>" and "NatTrans \<eta>2" using assms aa by auto
hence "(\<eta>2 \$\$ X) maps\<^bsub>NTCatCod \<eta>2\<^esub> ((NTDom \<eta>2) @@ X) to ((NTCod \<eta>2) @@ X)"
thus ?thesis using assms by auto
qed
ultimately have "(\<eta>1 \$\$ X) ;;\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>2 \$\$ X) maps\<^bsub>NTCatCod \<eta>1\<^esub> ((NTDom \<eta>1) @@ X) to ((NTCod \<eta>2) @@ X)"
using assms by (simp add: Category.Ccompt)
thus ?thesis using assms by (auto simp add: NatTransComp'_def NTCatCod_def)
qed
}
{
fix f X Y assume a: "f maps\<^bsub>(NTCatDom (\<eta>1 \<bullet>1 \<eta>2))\<^esub> X to Y"
show "(NTDom (\<eta>1 \<bullet>1 \<eta>2) ## f) ;;\<^bsub>NTCatCod (\<eta>1 \<bullet>1 \<eta>2)\<^esub> (\<eta>1 \<bullet>1 \<eta>2 \$\$ Y) =
((\<eta>1 \<bullet>1 \<eta>2) \$\$ X) ;;\<^bsub>NTCatCod (\<eta>1 \<bullet>1 \<eta>2)\<^esub> (NTCod (\<eta>1 \<bullet>1 \<eta>2) ## f)"
proof-
have b: "X \<in> obj\<^bsub>NTCatDom \<eta>1\<^esub>" and c: "Y \<in> obj\<^bsub>NTCatDom \<eta>1\<^esub>" using assms a by (auto simp add: Category.MapsToObj)
have "((NTDom \<eta>1) ## f) ;;\<^bsub>NTCatCod \<eta>1\<^esub> ((\<eta>1 \$\$ Y) ;;\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>2 \$\$ Y)) =
(((NTDom \<eta>1) ## f) ;;\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>1 \$\$ Y)) ;;\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>2 \$\$ Y)"
proof-
have "((NTDom \<eta>1) ## f) \<approx>>\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>1 \$\$ Y)" using assms a by (auto simp add: NatTransCompDefDom)
moreover have "(\<eta>1 \$\$ Y) \<approx>>\<^bsub>NTCatCod \<eta>1\<^esub>  (\<eta>2 \$\$ Y)" using assms by (simp add: NatTransCompCompDef c)
ultimately show ?thesis using assms by (simp add: Category.Cassoc)
qed
also have "... = ((\<eta>1 \$\$ X) ;;\<^bsub>NTCatCod \<eta>1\<^esub> ((NTDom \<eta>2) ## f)) ;;\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>2 \$\$ Y)"
using assms a by (auto simp add: NatTransP.NatTrans)
also have "... = (\<eta>1 \$\$ X) ;;\<^bsub>NTCatCod \<eta>1\<^esub> (((NTDom \<eta>2) ## f) ;;\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>2 \$\$ Y))"
proof-
have "(\<eta>1 \$\$ X) \<approx>>\<^bsub>NTCatCod \<eta>1\<^esub> ((NTCod \<eta>1) ## f)" using assms a by (simp add: NatTransCompDefCod)
moreover have "((NTDom \<eta>2) ## f) \<approx>>\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>2 \$\$ Y)" using assms a
by (simp add: NatTransCompDefDom NTCatDom NTCatCod)
ultimately show ?thesis using assms by (auto simp add: Category.Cassoc)
qed
also have "... = (\<eta>1 \$\$ X) ;;\<^bsub>NTCatCod \<eta>1\<^esub> ((\<eta>2 \$\$ X) ;;\<^bsub>NTCatCod \<eta>1\<^esub> ((NTCod \<eta>2) ## f))"
using assms a by (simp add: NatTransP.NatTrans NTCatDom NTCatCod)
also have "... = (\<eta>1 \$\$ X) ;;\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>2 \$\$ X) ;;\<^bsub>NTCatCod \<eta>1\<^esub> ((NTCod \<eta>2) ## f)"
proof-
have "(\<eta>1 \$\$ X) \<approx>>\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>2 \$\$ X)" using assms by (simp add: NatTransCompCompDef b)
moreover have "(\<eta>2 \$\$ X) \<approx>>\<^bsub>NTCatCod \<eta>1\<^esub> ((NTCod \<eta>2) ## f)" using assms a
by (simp add: NatTransCompDefCod NTCatCod NTCatDom)
ultimately show ?thesis using assms by (simp add: Category.Cassoc)
qed
finally show ?thesis using assms by (auto simp add: NatTransComp'_def NTCatCod_def)
qed
}
qed

lemma NatTransCompNatTrans: "\<eta>1 \<approx>>\<bullet> \<eta>2 \<Longrightarrow> NatTrans (\<eta>1 \<bullet> \<eta>2)"
by (simp add: NatTransCompNatTrans' NatTransComp_def MakeNT)

definition
CatExp' :: "('o1,'m1,'a) Category_scheme \<Rightarrow> ('o2,'m2,'b) Category_scheme \<Rightarrow>
(('o1, 'o2, 'm1, 'm2, 'a, 'b) Functor,
('o1, 'o2, 'm1, 'm2, 'a, 'b) NatTrans) Category"  where
"CatExp' A B \<equiv> \<lparr>
Category.Obj = {F . Ftor F : A \<longrightarrow> B} ,
Category.Mor = {\<eta> . NatTrans \<eta> \<and> NTCatDom \<eta> = A \<and> NTCatCod \<eta> = B} ,
Category.Dom = NTDom ,
Category.Cod = NTCod ,
Category.Id  = IdNatTrans ,
Category.Comp = \<lambda>f g. (f \<bullet> g)
\<rparr>"

definition "CatExp A B \<equiv> MakeCat(CatExp' A B)"

lemma IdNatTransMapL:
assumes NT: "NatTrans f"
shows "IdNatTrans (NTDom f) \<bullet> f = f"
proof(rule NatTransExt)
show "NatTrans f" using assms .
show "NatTrans (IdNatTrans (NTDom f) \<bullet> f)" using NT
by (simp add: NatTransP.NatTransFtor IdNatTransNatTrans IdNatTransCompDefDom NatTransCompNatTrans)
show "NTDom (IdNatTrans (NTDom f) \<bullet> f) = NTDom f" and
"NTCod (IdNatTrans (NTDom f) \<bullet> f) = NTCod f" by (simp add: IdNatTrans_defs NatTransComp_defs)+
{
fix x assume aa: "x \<in> Obj (NTCatDom (IdNatTrans (NTDom f) \<bullet> f))"
show "(IdNatTrans (NTDom f) \<bullet> f) \$\$ x = f \$\$ x"
proof-
have XObj: "x \<in> Obj(NTCatDom f)" using aa by (simp add: IdNatTrans_defs NatTransComp_defs)
have fMap: "f \$\$ x maps\<^bsub>NTCatCod f\<^esub> NTDom f @@ x to NTCod f @@ x" using NT XObj
have "(IdNatTrans (NTDom f) \<bullet> f) \$\$ x = (IdNatTrans (NTDom f) \$\$ x) ;;\<^bsub>NTCatCod f \<^esub>(f \$\$ x)"
proof(rule NatTransComp_Comp1)
show "x \<in> obj\<^bsub>NTCatDom (IdNatTrans (NTDom f))\<^esub>" using XObj by (simp add: IdNatTrans_defs)
show "IdNatTrans (NTDom f) \<approx>>\<bullet> f" using NT by (simp add: IdNatTransCompDefDom)
qed
also have "... = id\<^bsub>NTCatCod f\<^esub> (dom\<^bsub>NTCatCod f\<^esub> (f \$\$ x)) ;;\<^bsub>NTCatCod f \<^esub>(f \$\$ x)"
using XObj NT fMap by (auto simp add: IdNatTrans_map NTCatDom_def CCCD NTCatCod_def)
also have "... = f \$\$ x"
proof-
have "f \$\$ x \<in> mor\<^bsub>NTCatCod f\<^esub>" using fMap by (auto)
thus ?thesis using NT by (simp add: Category.Cidl)
qed
finally show ?thesis .
qed
}
qed

lemma IdNatTransMapR:
assumes NT: "NatTrans f"
shows "f \<bullet> IdNatTrans (NTCod f) = f"
proof(rule NatTransExt)
show "NatTrans f" using assms .
show "NatTrans (f \<bullet> IdNatTrans (NTCod f))" using NT
by (simp add: NatTransP.NatTransFtor IdNatTransNatTrans IdNatTransCompDefCod NatTransCompNatTrans)
show "NTDom (f \<bullet> IdNatTrans (NTCod f)) = NTDom f" and
"NTCod (f \<bullet> IdNatTrans (NTCod f)) = NTCod f" by (simp add: IdNatTrans_defs NatTransComp_defs)+
{
fix x assume aa: "x \<in> Obj (NTCatDom (f \<bullet> IdNatTrans (NTCod f)))"
show "(f \<bullet> IdNatTrans (NTCod f)) \$\$ x = f \$\$ x"
proof-
have XObj: "x \<in> Obj(NTCatDom f)" using aa by (simp add:  NatTransComp_defs)
have fMap: "f \$\$ x maps\<^bsub>NTCatCod f\<^esub> NTDom f @@ x to NTCod f @@ x" using NT XObj
have "(f \<bullet> IdNatTrans (NTCod f)) \$\$ x = (f \$\$ x) ;;\<^bsub>NTCatCod f\<^esub> (IdNatTrans (NTCod f) \$\$ x)"
using XObj NT by (auto simp add: NatTransComp_Comp2 IdNatTransCompDefCod)
also have "... = (f \$\$ x) ;;\<^bsub>NTCatCod f\<^esub> (id\<^bsub>NTCatCod f\<^esub> (cod\<^bsub>NTCatCod f\<^esub> (f \$\$ x)))"
proof-
have "x \<in> obj\<^bsub>CatDom (NTCod f)\<^esub>" using XObj NT by (simp add: IdNatTrans_defs DDDC)
moreover have "(cod\<^bsub>NTCatCod f\<^esub> (f \$\$ x)) = (NTCod f) @@ x" using fMap by auto
ultimately have "(IdNatTrans (NTCod f) \$\$ x) = (id\<^bsub>NTCatCod f\<^esub> (cod\<^bsub>NTCatCod f\<^esub> (f \$\$ x)))"
thus ?thesis by simp
qed
also have "... = f \$\$ x"
proof-
have "f \$\$ x \<in> mor\<^bsub>NTCatCod f\<^esub>" using fMap by (auto)
thus ?thesis using NT by (simp add: Category.Cidr)
qed
finally show ?thesis .
qed
}
qed

lemma NatTransCompDefined:
assumes "f \<approx>>\<bullet> g" and "g \<approx>>\<bullet> h"
shows "(f \<bullet> g) \<approx>>\<bullet> h" and "f \<approx>>\<bullet> (g \<bullet> h)"
proof-
show "(f \<bullet> g) \<approx>>\<bullet> h"
proof(rule NTCompDefinedI)
show "NatTrans (f \<bullet> g)" and "NatTrans h" using assms by (auto simp add: NatTransCompNatTrans)
have "NTCatDom f = NTCatDom h" using assms by (simp add: NTCatDom)
thus "NTCatDom h = NTCatDom (f \<bullet> g)" by (simp add: NatTransComp_defs)
have "NTCatCod h = NTCatCod g" using assms by (simp add: NTCatCod)
thus "NTCatCod h = NTCatCod (f \<bullet> g)" by ( simp add: NatTransComp_defs)
show "NTCod (f \<bullet> g) = NTDom h" using assms by (auto simp add: NatTransComp_defs)
qed
show "f \<approx>>\<bullet> (g \<bullet> h)"
proof(rule NTCompDefinedI)
show "NatTrans f" and "NatTrans (g \<bullet> h)" using assms by (auto simp add: NatTransCompNatTrans)
have "NTCatDom f = NTCatDom g" using assms by (simp add: NTCatDom)
thus "NTCatDom (g \<bullet> h) = NTCatDom f" by (simp add: NatTransComp_defs)
have "NTCatCod h = NTCatCod f" using assms by (simp add: NTCatCod)
thus "NTCatCod (g \<bullet> h) = NTCatCod f" by ( simp add: NatTransComp_defs)
show "NTCod f = NTDom (g \<bullet> h)" using assms by (auto simp add: NatTransComp_defs)
qed
qed

lemma NatTransCompAssoc:
assumes "f \<approx>>\<bullet> g" and "g \<approx>>\<bullet> h"
shows "(f \<bullet> g) \<bullet> h = f \<bullet> (g \<bullet> h)"
proof(rule NatTransExt)
show "NatTrans ((f \<bullet> g) \<bullet> h)" using assms by (simp add: NatTransCompNatTrans NatTransCompDefined)
show "NatTrans (f \<bullet> (g \<bullet> h))" using assms by (simp add: NatTransCompNatTrans NatTransCompDefined)
show "NTDom (f \<bullet> g \<bullet> h) = NTDom (f \<bullet> (g \<bullet> h))" and "NTCod (f \<bullet> g \<bullet> h) = NTCod (f \<bullet> (g \<bullet> h))"
{
fix x assume aa: "x \<in> obj\<^bsub>NTCatDom (f \<bullet> g \<bullet> h)\<^esub>" show "((f \<bullet> g) \<bullet> h) \$\$ x = (f \<bullet> (g \<bullet> h)) \$\$ x"
proof-
have ntd1: "NTCatDom (f \<bullet> g) = NTCatDom (f \<bullet> g \<bullet> h)" and ntd2: "NTCatDom f = NTCatDom (f \<bullet> g \<bullet> h)"
using assms by (simp add: NatTransCompDefined)+
have obj1: "x \<in> Obj (NTCatDom f)" using aa ntd2 by simp
have  1: "(f \<bullet> g) \$\$ x = (f \$\$ x) ;;\<^bsub>NTCatCod h\<^esub> (g \$\$ x)" and
2: "(g \<bullet> h) \$\$ x = (g \$\$ x) ;;\<^bsub>NTCatCod h\<^esub> (h \$\$ x)" using obj1
using assms by (auto simp add: NatTransComp_Comp1)
have "((f \<bullet> g) \<bullet> h) \$\$ x = ((f \<bullet> g) \$\$ x) ;;\<^bsub>NTCatCod h\<^esub> (h \$\$ x)"
proof(rule NatTransComp_Comp1)
show "x \<in> obj\<^bsub>NTCatDom (f \<bullet> g)\<^esub>" using aa ntd1 by simp
show "f \<bullet> g \<approx>>\<bullet> h" using assms by (simp add: NatTransCompDefined)
qed
also have "... = ((f \$\$ x) ;;\<^bsub>NTCatCod h\<^esub> (g \$\$ x)) ;;\<^bsub>NTCatCod h\<^esub> (h \$\$ x)" using 1 by simp
also have "... = (f \$\$ x) ;;\<^bsub>NTCatCod h\<^esub> ((g \$\$ x) ;;\<^bsub>NTCatCod h\<^esub> (h \$\$ x))"
proof-
have 1: "NTCatCod h = NTCatCod f" and 2: "NTCatCod h = NTCatCod g" using assms by (simp add: NTCatCod)+
hence "(f \$\$ x) \<approx>>\<^bsub>NTCatCod h\<^esub> (g \$\$ x)" using obj1 assms by (simp add: NatTransCompCompDef)
moreover have "(g \$\$ x) \<approx>>\<^bsub>NTCatCod h\<^esub> (h \$\$ x)" using obj1 assms 2 by (simp add: NatTransCompCompDef NTCatDom)
moreover have "Category (NTCatCod h)" using assms by auto
ultimately show ?thesis by (simp add: Category.Cassoc)
qed
also have "... = (f \$\$ x) ;;\<^bsub>NTCatCod h\<^esub> ((g \<bullet> h) \$\$ x)" using 2 by simp
also have "... = (f \<bullet> (g \<bullet> h)) \$\$ x"
proof-
have "NTCatCod f = NTCatCod h" using assms by (simp add: NTCatCod)
moreover have "(f \<bullet> (g \<bullet> h)) \$\$ x = (f \$\$ x) ;;\<^bsub>NTCatCod f\<^esub> ((g \<bullet> h) \$\$ x)"
proof(rule NatTransComp_Comp2)
show "x \<in> obj\<^bsub>NTCatDom f\<^esub>" using obj1 assms by (simp add: NTCatDom)
show "f \<approx>>\<bullet> g \<bullet> h" using assms by (simp add: NatTransCompDefined)
qed
ultimately show ?thesis by simp
qed
finally show ?thesis .
qed
}
qed

lemma CatExpCatAx:
assumes "Category A" and "Category B"
shows "Category_axioms (CatExp' A B)"
{
fix f assume "f \<in> mor\<^bsub>CatExp' A B\<^esub>"
thus "dom\<^bsub>CatExp' A B\<^esub> f \<in> obj\<^bsub>CatExp' A B\<^esub>" and
"cod\<^bsub>CatExp' A B\<^esub> f \<in> obj\<^bsub>CatExp' A B\<^esub>"
NatTransP.NatTransFtor2 NatTransP.NatTransFtorDom NatTransP.NatTransFtorCod DDDC CCCD functor_abbrev_def)
}
{
fix F assume a: "F \<in> obj\<^bsub>CatExp' A B\<^esub>"
show "id\<^bsub>CatExp' A B\<^esub> F maps\<^bsub>CatExp' A B\<^esub> F to F"
proof(rule MapsToI)
have "Ftor F : A \<longrightarrow> B" using a by (simp add: CatExp'_def)
thus "id\<^bsub>CatExp' A B\<^esub> F \<in> mor\<^bsub>CatExp' A B\<^esub>"
apply(simp add: CatExp'_def NTCatDom_def NTCatCod_def IdNatTransNatTrans functor_abbrev_def)
done
show "dom\<^bsub>CatExp' A B\<^esub> (id\<^bsub>CatExp' A B\<^esub> F) = F" by (simp add: CatExp'_def IdNatTrans_defs)
show "cod\<^bsub>CatExp' A B\<^esub> (id\<^bsub>CatExp' A B\<^esub> F) = F" by (simp add: CatExp'_def IdNatTrans_defs)
qed
}
{
fix f assume a: "f \<in> mor\<^bsub>CatExp' A B\<^esub>"
show "(id\<^bsub>CatExp' A B\<^esub> (dom\<^bsub>CatExp' A B\<^esub> f)) ;;\<^bsub>CatExp' A B\<^esub> f = f" and
"f ;;\<^bsub>CatExp' A B\<^esub> (id\<^bsub>CatExp' A B\<^esub> (cod\<^bsub>CatExp' A B\<^esub> f)) = f"
have NT: "NatTrans f" using a by (simp add: CatExp'_def)
show "IdNatTrans (NTDom f) \<bullet> f = f" using NT by (simp add:IdNatTransMapL)
show "f \<bullet> IdNatTrans (NTCod f) = f" using NT by (simp add:IdNatTransMapR)
qed
}
{
fix f g h assume aa: "f \<approx>>\<^bsub>CatExp' A B\<^esub> g" and bb: "g \<approx>>\<^bsub>CatExp' A B\<^esub> h"
{
fix f g assume "f \<approx>>\<^bsub>CatExp' A B\<^esub> g" hence "f \<approx>>\<bullet> g"
apply(simp only: NTCompDefined_def)
}
hence "f \<approx>>\<bullet> g" and "g \<approx>>\<bullet> h" using aa bb by auto
thus "f ;;\<^bsub>CatExp' A B\<^esub> g ;;\<^bsub>CatExp' A B\<^esub> h = f ;;\<^bsub>CatExp' A B\<^esub> (g ;;\<^bsub>CatExp' A B\<^esub> h)"
}
{
fix f g X Y Z assume a: "f maps\<^bsub>CatExp' A B\<^esub> X to Y" and b: "g maps\<^bsub>CatExp' A B\<^esub> Y to Z"
show "f ;;\<^bsub>CatExp' A B\<^esub> g maps\<^bsub>CatExp' A B\<^esub> X to Z"
proof(rule MapsToI, auto simp add: CatExp'_def)
have nt1: "NatTrans f" and cd1: "NTCatDom f = A"
and cc1: "NTCatCod f = B" and d1: "NTDom f = X" and c1: "NTCod f = Y"
using a by (auto simp add: CatExp'_def)
moreover have nt2: "NatTrans g" and cd2: "NTCatDom g = A"
and cc2: "NTCatCod g = B" and d2: "NTDom g = Y" and c2: "NTCod g = Z"
using b by (auto simp add: CatExp'_def)
ultimately have Comp: "f \<approx>>\<bullet> g" by(auto intro: NTCompDefinedI)
thus "NatTrans (f \<bullet> g)" by (simp add: NatTransCompNatTrans)
show "NTCatDom (f \<bullet> g) = A" using Comp cd1 by (simp add: NTCatDom)
show "NTCatCod (f \<bullet> g) = B" using Comp cc2 by (simp add: NTCatCod)
show "NTDom (f \<bullet> g) = X" using d1 by (simp add: NatTransComp_defs)
show "NTCod (f \<bullet> g) = Z" using c2 by (simp add: NatTransComp_defs)
qed
}
qed

lemma CatExpCat: "\<lbrakk>Category A ; Category B\<rbrakk> \<Longrightarrow> Category (CatExp A B)"

lemmas CatExp_defs = CatExp_def CatExp'_def MakeCat_def

lemma CatExpDom: "f \<in> Mor (CatExp A B) \<Longrightarrow> dom\<^bsub>CatExp A B\<^esub> f = NTDom f"

lemma CatExpCod: "f \<in> Mor (CatExp A B) \<Longrightarrow> cod\<^bsub>CatExp A B\<^esub> f = NTCod f"

lemma CatExpId: "X \<in> Obj (CatExp A B) \<Longrightarrow> Id (CatExp A B) X = IdNatTrans X"

lemma CatExpNatTransCompDef: assumes "f \<approx>>\<^bsub>CatExp A B\<^esub> g" shows "f \<approx>>\<bullet> g"
proof-
have 1: "f \<approx>>\<^bsub>CatExp' A B\<^esub> g" using assms by (simp add: CatExp_def MakeCatCompDef)
show "f \<approx>>\<bullet> g"
proof(rule NTCompDefinedI)
show "NatTrans f" using 1 by (auto simp add: CatExp'_def)
show "NatTrans g" using 1 by (auto simp add: CatExp'_def)
show "NTCatDom g = NTCatDom f" using 1 by (auto simp add: CatExp'_def)
show "NTCatCod g = NTCatCod f" using 1 by (auto simp add: CatExp'_def)
show "NTCod f = NTDom g" using 1 by (auto simp add: CatExp'_def)
qed
qed

lemma CatExpDist:
assumes "X \<in> Obj A" and "f \<approx>>\<^bsub>CatExp A B\<^esub> g"
shows "(f ;;\<^bsub>CatExp A B\<^esub> g) \$\$ X = (f \$\$ X) ;;\<^bsub>B\<^esub> (g \$\$ X)"
proof-
have "f \<in> Mor (CatExp' A B)" using assms by (auto simp add: CatExp_def MakeCatMor)
hence 1: "NTCatDom f = A" and 2: "NTCatCod f = B" by (simp add: CatExp'_def)+
hence 4: "X \<in> Obj (NTCatDom f)" using assms by simp
have 3: "f \<approx>>\<bullet> g" using assms(2) by (simp add: CatExpNatTransCompDef)
have "(f ;;\<^bsub>CatExp A B\<^esub> g) \$\$ X = (f ;;\<^bsub>CatExp' A B\<^esub> g) \$\$ X" using assms(2) by (simp add: CatExp_def MakeCatComp2)
also have "... = (f \<bullet> g) \$\$ X" by (simp add: CatExp'_def)
also have "... = (f \$\$ X) ;;\<^bsub>B\<^esub> (g \$\$ X)" using 4 2 3 by (simp add: NatTransComp_Comp2[of X f g])
finally show ?thesis .
qed

lemma CatExpMorNT: "f \<in> Mor (CatExp A B) \<Longrightarrow> NatTrans f"