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| (* Title: Modification | |
| Author: Eugene W. Stark <stark@cs.stonybrook.edu>, 2020 | |
| Maintainer: Eugene W. Stark <stark@cs.stonybrook.edu> | |
| *) | |
| section "Modifications" | |
| text \<open> | |
| Modifications are morphisms of pseudonatural transformations. | |
| In this section, we give a definition of ``modification'', and we prove that | |
| the mappings \<open>\<eta>\<close> and \<open>\<epsilon>\<close>, which were chosen in the course of showing that a | |
| pseudonatural equivalence \<open>\<tau>\<close> has a converse pseudonatural equivalence \<open>\<tau>'\<close>, | |
| are invertible modifications that relate the composites \<open>\<tau>'\<tau>\<close> and \<open>\<tau>\<tau>'\<close> | |
| to the identity pseudonatural transformations on \<open>F\<close> and \<open>G\<close>. | |
| This means that \<open>\<tau>\<close> and \<open>\<tau>'\<close> are quasi-inverses in a suitable bicategory | |
| of pseudofunctors, pseudonatural transformations, and modifications, though | |
| we do not go so far as to give a formal construction of such a bicategory. | |
| \<close> | |
| theory Modification | |
| imports PseudonaturalTransformation | |
| begin | |
| locale modification = (* 12 sec *) | |
| C: bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C + | |
| D: bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D + | |
| \<tau>: pseudonatural_transformation | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1 + | |
| \<tau>': pseudonatural_transformation | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0' \<tau>\<^sub>1' | |
| for V\<^sub>C :: "'c comp" (infixr "\<cdot>\<^sub>C" 55) | |
| and H\<^sub>C :: "'c comp" (infixr "\<star>\<^sub>C" 53) | |
| and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c" ("\<a>\<^sub>C[_, _, _]") | |
| and \<i>\<^sub>C :: "'c \<Rightarrow> 'c" ("\<i>\<^sub>C[_]") | |
| and src\<^sub>C :: "'c \<Rightarrow> 'c" | |
| and trg\<^sub>C :: "'c \<Rightarrow> 'c" | |
| and V\<^sub>D :: "'d comp" (infixr "\<cdot>\<^sub>D" 55) | |
| and H\<^sub>D :: "'d comp" (infixr "\<star>\<^sub>D" 53) | |
| and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd" ("\<a>\<^sub>D[_, _, _]") | |
| and \<i>\<^sub>D :: "'d \<Rightarrow> 'd" ("\<i>\<^sub>D[_]") | |
| and src\<^sub>D :: "'d \<Rightarrow> 'd" | |
| and trg\<^sub>D :: "'d \<Rightarrow> 'd" | |
| and F :: "'c \<Rightarrow> 'd" | |
| and \<Phi>\<^sub>F :: "'c * 'c \<Rightarrow> 'd" | |
| and G :: "'c \<Rightarrow> 'd" | |
| and \<Phi>\<^sub>G :: "'c * 'c \<Rightarrow> 'd" | |
| and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd" | |
| and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" | |
| and \<tau>\<^sub>0' :: "'c \<Rightarrow> 'd" | |
| and \<tau>\<^sub>1' :: "'c \<Rightarrow> 'd" | |
| and \<Gamma> :: "'c \<Rightarrow> 'd" + | |
| assumes \<Gamma>_in_hom: "C.obj a \<Longrightarrow> \<guillemotleft>\<Gamma> a : \<tau>\<^sub>0 a \<Rightarrow>\<^sub>D \<tau>\<^sub>0' a\<guillemotright>" | |
| and naturality: "\<lbrakk> \<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>; C.ide f \<rbrakk> \<Longrightarrow> \<tau>\<^sub>1' f \<cdot>\<^sub>D (G f \<star>\<^sub>D \<Gamma> a) = (\<Gamma> b \<star>\<^sub>D F f) \<cdot>\<^sub>D \<tau>\<^sub>1 f" | |
| locale invertible_modification = (* 13 sec *) | |
| modification + | |
| assumes components_are_iso: "C.obj a \<Longrightarrow> D.iso (\<Gamma> a)" | |
| locale identity_modification = (* 12 sec *) | |
| C: bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C + | |
| D: bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D + | |
| \<tau>: pseudonatural_transformation | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1 | |
| for V\<^sub>C :: "'c comp" (infixr "\<cdot>\<^sub>C" 55) | |
| and H\<^sub>C :: "'c comp" (infixr "\<star>\<^sub>C" 53) | |
| and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c" ("\<a>\<^sub>C[_, _, _]") | |
| and \<i>\<^sub>C :: "'c \<Rightarrow> 'c" ("\<i>\<^sub>C[_]") | |
| and src\<^sub>C :: "'c \<Rightarrow> 'c" | |
| and trg\<^sub>C :: "'c \<Rightarrow> 'c" | |
| and V\<^sub>D :: "'d comp" (infixr "\<cdot>\<^sub>D" 55) | |
| and H\<^sub>D :: "'d comp" (infixr "\<star>\<^sub>D" 53) | |
| and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd" ("\<a>\<^sub>D[_, _, _]") | |
| and \<i>\<^sub>D :: "'d \<Rightarrow> 'd" ("\<i>\<^sub>D[_]") | |
| and src\<^sub>D :: "'d \<Rightarrow> 'd" | |
| and trg\<^sub>D :: "'d \<Rightarrow> 'd" | |
| and F :: "'c \<Rightarrow> 'd" | |
| and \<Phi>\<^sub>F :: "'c * 'c \<Rightarrow> 'd" | |
| and G :: "'c \<Rightarrow> 'd" | |
| and \<Phi>\<^sub>G :: "'c * 'c \<Rightarrow> 'd" | |
| and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd" | |
| and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" | |
| begin | |
| abbreviation map | |
| where "map \<equiv> \<tau>\<^sub>0" | |
| sublocale invertible_modification | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1 \<tau>\<^sub>0 \<tau>\<^sub>1 map | |
| using D.comp_arr_dom D.comp_cod_arr | |
| by unfold_locales auto | |
| end | |
| locale composite_modification = (* 13 sec *) | |
| C: bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C + | |
| D: bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D + | |
| \<rho>: pseudonatural_transformation | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<rho>\<^sub>0 \<rho>\<^sub>1 + | |
| \<sigma>: pseudonatural_transformation | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<sigma>\<^sub>0 \<sigma>\<^sub>1 + | |
| \<tau>: pseudonatural_transformation | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1 + | |
| \<Gamma>: modification | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<rho>\<^sub>0 \<rho>\<^sub>1 \<sigma>\<^sub>0 \<sigma>\<^sub>1 \<Gamma> + | |
| \<Delta>: modification | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<sigma>\<^sub>0 \<sigma>\<^sub>1 \<tau>\<^sub>0 \<tau>\<^sub>1 \<Delta> | |
| for V\<^sub>C :: "'c comp" (infixr "\<cdot>\<^sub>C" 55) | |
| and H\<^sub>C :: "'c comp" (infixr "\<star>\<^sub>C" 53) | |
| and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c" ("\<a>\<^sub>C[_, _, _]") | |
| and \<i>\<^sub>C :: "'c \<Rightarrow> 'c" ("\<i>\<^sub>C[_]") | |
| and src\<^sub>C :: "'c \<Rightarrow> 'c" | |
| and trg\<^sub>C :: "'c \<Rightarrow> 'c" | |
| and V\<^sub>D :: "'d comp" (infixr "\<cdot>\<^sub>D" 55) | |
| and H\<^sub>D :: "'d comp" (infixr "\<star>\<^sub>D" 53) | |
| and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd" ("\<a>\<^sub>D[_, _, _]") | |
| and \<i>\<^sub>D :: "'d \<Rightarrow> 'd" ("\<i>\<^sub>D[_]") | |
| and src\<^sub>D :: "'d \<Rightarrow> 'd" | |
| and trg\<^sub>D :: "'d \<Rightarrow> 'd" | |
| and F :: "'c \<Rightarrow> 'd" | |
| and \<Phi>\<^sub>F :: "'c * 'c \<Rightarrow> 'd" | |
| and G :: "'c \<Rightarrow> 'd" | |
| and \<Phi>\<^sub>G :: "'c * 'c \<Rightarrow> 'd" | |
| and \<rho>\<^sub>0 :: "'c \<Rightarrow> 'd" | |
| and \<rho>\<^sub>1 :: "'c \<Rightarrow> 'd" | |
| and \<sigma>\<^sub>0 :: "'c \<Rightarrow> 'd" | |
| and \<sigma>\<^sub>1 :: "'c \<Rightarrow> 'd" | |
| and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd" | |
| and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" | |
| and \<Gamma> :: "'c \<Rightarrow> 'd" | |
| and \<Delta> :: "'c \<Rightarrow> 'd" | |
| begin | |
| abbreviation map | |
| where "map a \<equiv> \<Delta> a \<cdot>\<^sub>D \<Gamma> a" | |
| sublocale modification | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<rho>\<^sub>0 \<rho>\<^sub>1 \<tau>\<^sub>0 \<tau>\<^sub>1 map | |
| using \<Delta>.\<Gamma>_in_hom \<Gamma>.\<Gamma>_in_hom \<Delta>.naturality \<Gamma>.naturality | |
| apply unfold_locales | |
| apply auto[1] | |
| proof - | |
| fix f a b | |
| assume f: "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" and ide_f: "C.ide f" | |
| have "\<tau>\<^sub>1 f \<cdot>\<^sub>D (G f \<star>\<^sub>D map a) = (\<tau>\<^sub>1 f \<cdot>\<^sub>D (G f \<star>\<^sub>D \<Delta> a)) \<cdot>\<^sub>D (G f \<star>\<^sub>D \<Gamma> a)" | |
| proof - | |
| have "D.seq (\<Delta> a) (\<Gamma> a)" | |
| using f \<Delta>.\<Gamma>_in_hom \<Gamma>.\<Gamma>_in_hom by blast | |
| thus ?thesis | |
| using f ide_f D.whisker_left [of "G f" "\<Delta> a" "\<Gamma> a"] D.comp_assoc | |
| by simp | |
| qed | |
| also have "... = (\<Delta> b \<star>\<^sub>D F f) \<cdot>\<^sub>D \<sigma>\<^sub>1 f \<cdot>\<^sub>D (G f \<star>\<^sub>D \<Gamma> a)" | |
| using f ide_f \<Delta>.naturality [of f a b] D.comp_assoc by simp | |
| also have "... = ((\<Delta> b \<star>\<^sub>D F f) \<cdot>\<^sub>D (\<Gamma> b \<star>\<^sub>D F f)) \<cdot>\<^sub>D \<rho>\<^sub>1 f" | |
| using f ide_f \<Gamma>.naturality [of f a b] D.comp_assoc by simp | |
| also have "... = (map b \<star>\<^sub>D F f) \<cdot>\<^sub>D \<rho>\<^sub>1 f" | |
| proof - | |
| have "D.seq (\<Delta> b) (\<Gamma> b)" | |
| using f \<Delta>.\<Gamma>_in_hom \<Gamma>.\<Gamma>_in_hom by blast | |
| thus ?thesis | |
| using f ide_f D.whisker_right [of "F f" "\<Delta> b" "\<Gamma> b"] by simp | |
| qed | |
| finally show "\<tau>\<^sub>1 f \<cdot>\<^sub>D (G f \<star>\<^sub>D map a) = (map b \<star>\<^sub>D F f) \<cdot>\<^sub>D \<rho>\<^sub>1 f" by simp | |
| qed | |
| end | |
| context converse_pseudonatural_equivalence | |
| begin | |
| interpretation EV: self_evaluation_map V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D .. | |
| notation EV.eval ("\<lbrace>_\<rbrace>") | |
| interpretation \<iota>\<^sub>F: identity_pseudonatural_transformation | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F | |
| .. | |
| interpretation \<iota>\<^sub>G: identity_pseudonatural_transformation | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G | |
| .. | |
| interpretation \<tau>'\<tau>: composite_pseudonatural_equivalence | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G F \<Phi>\<^sub>F | |
| \<tau>\<^sub>0 \<tau>\<^sub>1 map\<^sub>0 map\<^sub>1 | |
| .. | |
| interpretation \<tau>\<tau>': composite_pseudonatural_equivalence | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G F \<Phi>\<^sub>F G \<Phi>\<^sub>G | |
| map\<^sub>0 map\<^sub>1 \<tau>\<^sub>0 \<tau>\<^sub>1 | |
| .. | |
| interpretation \<eta>: invertible_modification | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F F \<Phi>\<^sub>F | |
| \<iota>\<^sub>F.map\<^sub>0 \<iota>\<^sub>F.map\<^sub>1 \<tau>'\<tau>.map\<^sub>0 \<tau>'\<tau>.map\<^sub>1 \<eta> | |
| proof | |
| show "\<And>a. C.obj a \<Longrightarrow> \<guillemotleft>\<eta> a : F.map\<^sub>0 a \<Rightarrow>\<^sub>D \<tau>'\<tau>.map\<^sub>0 a\<guillemotright>" | |
| using unit_in_hom \<tau>'\<tau>.map\<^sub>0_def by simp | |
| show "\<And>a. C.obj a \<Longrightarrow> D.iso (\<eta> a)" | |
| by simp | |
| show "\<And>f a b. \<lbrakk> \<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>; C.ide f\<rbrakk> | |
| \<Longrightarrow> \<tau>'\<tau>.map\<^sub>1 f \<cdot>\<^sub>D (F f \<star>\<^sub>D \<eta> a) = (\<eta> b \<star>\<^sub>D F f) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<cdot>\<^sub>D \<r>\<^sub>D[F f]" | |
| proof - | |
| fix f a b | |
| assume f: "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" and ide_f: "C.ide f" | |
| have a: "C.obj a" and b: "C.obj b" | |
| using f by auto | |
| have "\<tau>'\<tau>.map\<^sub>1 f \<cdot>\<^sub>D (F f \<star>\<^sub>D \<eta> a) = | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) | |
| \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[F f, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a]) \<cdot>\<^sub>D | |
| (F f \<star>\<^sub>D \<eta> a)" | |
| unfolding \<tau>'\<tau>.map\<^sub>1_def map\<^sub>1_def | |
| using a b f D.comp_assoc by auto | |
| also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (((\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[F f, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a]) \<cdot>\<^sub>D | |
| (F f \<star>\<^sub>D \<eta> a)" | |
| proof - | |
| have "(\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) | |
| \<star>\<^sub>D \<tau>\<^sub>0 a | |
| = ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a)" | |
| proof - | |
| have "D.arr ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a))" | |
| using a b f ide_f \<tau>.iso_map\<^sub>1_ide | |
| by (metis (no_types, lifting) C.in_hhomE map\<^sub>1_def map\<^sub>1_simps(1)) | |
| thus ?thesis | |
| using a b f D.whisker_right [of "\<tau>\<^sub>0 a"] by fastforce | |
| qed | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (F f \<star>\<^sub>D \<eta> a)" | |
| proof - | |
| have "((\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F f, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] = | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a)" | |
| using a b f ide_f D.assoc'_naturality [of "\<l>\<^sub>D\<^sup>-\<^sup>1[F f]" "\<tau>\<^sub>0' a" "\<tau>\<^sub>0 a"] by auto | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a]) \<cdot>\<^sub>D | |
| ((F\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<eta> a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| proof - | |
| have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<eta> a) = \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<eta> a" | |
| using a b f ide_f D.comp_arr_dom D.comp_cod_arr | |
| D.interchange [of "\<l>\<^sub>D\<^sup>-\<^sup>1[F f]" "F f" "\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a" "\<eta> a"] | |
| by simp | |
| also have "... = ((F\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<eta> a) \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| using a b f ide_f D.comp_arr_dom D.comp_cod_arr | |
| D.interchange [of "F\<^sub>0 b \<star>\<^sub>D F f" "\<l>\<^sub>D\<^sup>-\<^sup>1[F f]" "\<eta> a" "F\<^sub>0 a"] | |
| by auto | |
| finally have "(\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<eta> a) = | |
| ((F\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<eta> a) \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| by blast | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| (((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((F\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<eta> a)) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| proof - | |
| have "(((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] = | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a)" | |
| using a b f ide_f D.assoc'_naturality [of "\<eta> b \<star>\<^sub>D F f" "\<tau>\<^sub>0' a" "\<tau>\<^sub>0 a"] by auto | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a)) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| (((\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f) \<star>\<^sub>D \<eta> a) \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D F\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| proof - | |
| have "((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D ((F\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<eta> a) | |
| = (\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<eta> a" | |
| using a b f ide_f D.comp_arr_dom D.comp_cod_arr | |
| D.interchange [of "\<eta> b \<star>\<^sub>D F f" "F\<^sub>0 b \<star>\<^sub>D F f" "\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a" "\<eta> a"] | |
| by auto | |
| also have "... = (((\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f) \<star>\<^sub>D \<eta> a) \<cdot>\<^sub>D ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D F\<^sub>0 a)" | |
| using a b f ide_f D.comp_arr_dom D.comp_cod_arr | |
| D.interchange [of "(\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f" "\<eta> b \<star>\<^sub>D F f" "\<eta> a" "F\<^sub>0 a"] | |
| by auto | |
| finally have "((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D ((F\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<eta> a) = | |
| (((\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f) \<star>\<^sub>D \<eta> a) \<cdot>\<^sub>D ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D F\<^sub>0 a)" | |
| by blast | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a]) \<cdot>\<^sub>D | |
| (((\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f) \<star>\<^sub>D \<eta> a) \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D F\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| proof - | |
| have "(((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) = | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a" | |
| using a b f ide_f D.whisker_right \<tau>.iso_map\<^sub>1_ide by auto | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 a), \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| (((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (((\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f) \<star>\<^sub>D \<eta> a)) \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D F\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| proof - | |
| have "(((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] = | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 a), \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a)" | |
| using a b f ide_f \<tau>.iso_map\<^sub>1_ide | |
| D.assoc'_naturality | |
| [of "(\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f]" "\<tau>\<^sub>0' a" "\<tau>\<^sub>0 a"] | |
| by auto | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 a), \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<eta> a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D F\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D F\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| proof - | |
| have "((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (((\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f) \<star>\<^sub>D \<eta> a) = | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f]) \<cdot>\<^sub>D ((\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f) \<star>\<^sub>D | |
| (\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D \<eta> a" | |
| using a b f ide_f \<tau>.iso_map\<^sub>1_ide | |
| D.interchange [of "(\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f]" | |
| "(\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f" "\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a" "\<eta> a"] | |
| by auto | |
| also have "... = (\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<eta> a" | |
| using a b f ide_f \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr D.comp_assoc | |
| by auto | |
| also have "... = (\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] | |
| \<star>\<^sub>D \<eta> a \<cdot>\<^sub>D F\<^sub>0 a" | |
| using a b f ide_f \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr by auto | |
| also have "... = ((\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<eta> a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| proof - | |
| have "D.seq (\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a) ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f])" | |
| using a b f ide_f \<tau>.iso_map\<^sub>1_ide | |
| by (intro D.seqI D.hseqI') auto | |
| thus ?thesis | |
| using a b f ide_f \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr | |
| D.interchange [of "(\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a)" | |
| "(\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f]" | |
| "\<eta> a" "F\<^sub>0 a"] | |
| by simp | |
| qed | |
| finally have "((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (((\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f) \<star>\<^sub>D \<eta> a) = | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<eta> a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| by blast | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, G\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D G f, G\<^sub>0 a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<epsilon> a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 a), \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a]) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<eta> a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a, F\<^sub>0 a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<star>\<^sub>D F\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D F\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D F\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| proof - | |
| have "(\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<star>\<^sub>D \<tau>\<^sub>0 a | |
| = (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, G\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D G f, G\<^sub>0 a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<epsilon> a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a, \<tau>\<^sub>0 a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a)" | |
| proof - | |
| have "(\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<star>\<^sub>D \<tau>\<^sub>0 a | |
| = (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, G\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<epsilon> a) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a)" | |
| using a b f ide_f D.hcomp_reassoc D.whisker_right [of "\<tau>\<^sub>0 a"] by auto | |
| also have "... = (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, G\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D G f, G\<^sub>0 a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<epsilon> a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a, \<tau>\<^sub>0 a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a)" | |
| using a b f ide_f D.hcomp_reassoc(1) [of "\<tau>\<^sub>0' b \<star>\<^sub>D G f" "\<epsilon> a" "\<tau>\<^sub>0 a"] | |
| by auto | |
| finally show ?thesis by blast | |
| qed | |
| moreover have "(\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<eta> a | |
| = (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<eta> a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a, F\<^sub>0 a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<star>\<^sub>D F\<^sub>0 a)" | |
| proof - | |
| have "(\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<eta> a = | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<cdot>\<^sub>D ((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<star>\<^sub>D | |
| (\<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D \<eta> a \<cdot>\<^sub>D F\<^sub>0 a" | |
| using a b f ide_f D.comp_arr_dom D.comp_cod_arr | |
| D.hcomp_reassoc(2) [of "\<tau>\<^sub>0' b" "G f" "\<tau>\<^sub>0 a"] | |
| by auto | |
| also have "... = (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<eta> a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<star>\<^sub>D F\<^sub>0 a)" | |
| using a b f ide_f D.inv_inv D.iso_inv_iso D.interchange by auto | |
| also have "... = (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<eta> a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a, F\<^sub>0 a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<star>\<^sub>D F\<^sub>0 a)" | |
| using a b f ide_f D.hcomp_reassoc(1) [of "\<tau>\<^sub>0' b \<star>\<^sub>D G f" "\<tau>\<^sub>0 a" "\<eta> a"] by auto | |
| finally show ?thesis by blast | |
| qed | |
| ultimately show ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, G\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D G f, G\<^sub>0 a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| (((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<epsilon> a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a]) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<eta> a)) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a, F\<^sub>0 a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<star>\<^sub>D F\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D F\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D F\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| proof - | |
| have "\<a>\<^sub>D[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 a), \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a] = | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a]" | |
| proof - | |
| have "\<a>\<^sub>D[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D (G f \<star>\<^sub>D \<tau>\<^sub>0 a), \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a \<star>\<^sub>D \<tau>\<^sub>0 a] | |
| = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> | |
| (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle>, \<^bold>\<langle>G f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>) \<^bold>\<cdot> | |
| ((\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[ \<^bold>\<langle>G f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>) \<^bold>\<cdot> | |
| (\<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle>, \<^bold>\<langle>G f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>) \<^bold>\<cdot> | |
| \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>G f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>), \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> | |
| (\<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle>, \<^bold>\<langle>G f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>) \<^bold>\<cdot> | |
| \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>\<^bold>]\<rbrace>" | |
| using a b f ide_f D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char | |
| D.VV.ide_char D.VV.arr_char | |
| by auto | |
| also have "... = \<lbrace>(\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>\<^bold>]\<rbrace>" | |
| using a b f ide_f by (intro EV.eval_eqI, auto) | |
| also have "... = (\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a]" | |
| using a b f ide_f D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char | |
| D.VV.ide_char D.VV.arr_char | |
| by auto | |
| finally show ?thesis by blast | |
| qed | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, G\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D G f, G\<^sub>0 a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 a]) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a, F\<^sub>0 a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<star>\<^sub>D F\<^sub>0 a)) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D F\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D F\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| proof - | |
| interpret adjoint_equivalence_in_bicategory | |
| V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>\<tau>\<^sub>0 a\<close> \<open>\<tau>\<^sub>0' a\<close> \<open>\<eta> a\<close> \<open>\<epsilon> a\<close> | |
| using a chosen_adjoint_equivalence by simp | |
| have "((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<epsilon> a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a]) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<eta> a) | |
| = (\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D (\<epsilon> a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 a \<star>\<^sub>D \<eta> a)" | |
| using a b f ide_f D.whisker_left by auto | |
| also have "... = (\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 a]" | |
| using triangle_left by simp | |
| finally have "((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<epsilon> a \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a, \<tau>\<^sub>0' a, \<tau>\<^sub>0 a]) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<eta> a) | |
| = (\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 a]" | |
| by blast | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D | |
| (\<r>\<^sub>D[\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D F\<^sub>0 a)) \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D F\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| proof - | |
| have "\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, G\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D G f, G\<^sub>0 a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 a]) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a, F\<^sub>0 a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<star>\<^sub>D F\<^sub>0 a) | |
| = \<r>\<^sub>D[\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a]" | |
| proof - | |
| have "\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, G f, G\<^sub>0 a] \<star>\<^sub>D \<tau>\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D G f, G\<^sub>0 a, \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D G f) \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 a]) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b \<star>\<^sub>D G f, \<tau>\<^sub>0 a, F\<^sub>0 a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, G f, \<tau>\<^sub>0 a] \<star>\<^sub>D F\<^sub>0 a) | |
| = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle>, \<^bold>\<langle>G f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> | |
| ((\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<r>\<^bold>[\<^bold>\<langle>G f\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>) \<^bold>\<cdot> | |
| (\<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle>, \<^bold>\<langle>G f\<^bold>\<rangle>, \<^bold>\<langle>G\<^sub>0 a\<^bold>\<rangle>\<^sub>0\<^bold>] \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>) \<^bold>\<cdot> | |
| (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G f\<^bold>\<rangle>, \<^bold>\<langle>G\<^sub>0 a\<^bold>\<rangle>\<^sub>0, \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> | |
| ((\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<r>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> | |
| \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>, \<^bold>\<langle>F\<^sub>0 a\<^bold>\<rangle>\<^sub>0\<^bold>]) \<^bold>\<cdot> | |
| (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle>, \<^bold>\<langle>G f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>F\<^sub>0 a\<^bold>\<rangle>\<^sub>0)\<rbrace>" | |
| using a b f ide_f D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char | |
| D.VV.ide_char D.VV.arr_char D.\<ll>_ide_simp D.\<rr>_ide_simp | |
| by auto | |
| also have "... = \<lbrace>\<^bold>\<r>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>G f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 a\<^bold>\<rangle>\<^bold>]\<rbrace>" | |
| using a b f ide_f by (intro EV.eval_eqI, auto) | |
| also have "... = \<r>\<^sub>D[\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a]" | |
| using a b f ide_f D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char | |
| D.VV.ide_char D.VV.arr_char D.\<ll>_ide_simp D.\<rr>_ide_simp | |
| by auto | |
| finally show ?thesis by blast | |
| qed | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)))) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D \<r>\<^sub>D[(\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f] \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D F\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| proof - | |
| have "\<r>\<^sub>D[\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D F\<^sub>0 a) | |
| = (\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D | |
| \<r>\<^sub>D[(\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f]" | |
| using a b f ide_f D.comp_assoc \<tau>.iso_map\<^sub>1_ide | |
| D.runit_naturality [of "(\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f]"] | |
| by auto | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = ((\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f]) \<cdot>\<^sub>D | |
| \<r>\<^sub>D[(\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f]) \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D F\<^sub>0 a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| proof - | |
| have "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D ((\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D (\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f))) | |
| = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f]" | |
| using a b f ide_f D.comp_arr_inv' D.comp_arr_dom \<tau>.iso_map\<^sub>1_ide | |
| D.whisker_left [of "\<tau>\<^sub>0' b" "\<tau>\<^sub>1 f" "D.inv (\<tau>\<^sub>1 f)"] | |
| by auto | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (\<r>\<^sub>D[(\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f] \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D F\<^sub>0 a)) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| proof - | |
| have "(\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f]) \<cdot>\<^sub>D \<r>\<^sub>D[(\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f] | |
| = \<r>\<^sub>D[(\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f]" | |
| using a b f ide_f D.comp_inv_arr' D.comp_cod_arr by auto | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (\<eta> b \<star>\<^sub>D F f) \<cdot>\<^sub>D \<r>\<^sub>D[F\<^sub>0 b \<star>\<^sub>D F f] \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D F\<^sub>0 a)" | |
| proof - | |
| have "\<r>\<^sub>D[(\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f] \<cdot>\<^sub>D ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D F\<^sub>0 a) | |
| = (\<eta> b \<star>\<^sub>D F f) \<cdot>\<^sub>D \<r>\<^sub>D[F\<^sub>0 b \<star>\<^sub>D F f]" | |
| using a b f ide_f D.runit_naturality [of "\<eta> b \<star>\<^sub>D F f"] by auto | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (\<eta> b \<star>\<^sub>D F f) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<cdot>\<^sub>D \<r>\<^sub>D[F f]" | |
| using a b f ide_f D.runit_naturality [of "\<l>\<^sub>D\<^sup>-\<^sup>1[F f]"] by auto | |
| finally show "\<tau>'\<tau>.map\<^sub>1 f \<cdot>\<^sub>D (F f \<star>\<^sub>D \<eta> a) = (\<eta> b \<star>\<^sub>D F f) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<cdot>\<^sub>D \<r>\<^sub>D[F f]" | |
| by blast | |
| qed | |
| qed | |
| lemma unit_is_invertible_modification: | |
| shows "invertible_modification | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F F \<Phi>\<^sub>F | |
| \<iota>\<^sub>F.map\<^sub>0 \<iota>\<^sub>F.map\<^sub>1 \<tau>'\<tau>.map\<^sub>0 \<tau>'\<tau>.map\<^sub>1 \<eta>" | |
| .. | |
| interpretation \<epsilon>: invertible_modification | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G G \<Phi>\<^sub>G | |
| \<tau>\<tau>'.map\<^sub>0 \<tau>\<tau>'.map\<^sub>1 \<iota>\<^sub>G.map\<^sub>0 \<iota>\<^sub>G.map\<^sub>1 \<epsilon> | |
| proof | |
| show "\<And>a. C.obj a \<Longrightarrow> \<guillemotleft>\<epsilon> a : \<tau>\<tau>'.map\<^sub>0 a \<Rightarrow>\<^sub>D G\<^sub>0 a\<guillemotright>" | |
| using counit_in_hom \<tau>\<tau>'.map\<^sub>0_def by simp | |
| show "\<And>a. C.obj a \<Longrightarrow> D.iso (\<epsilon> a)" | |
| by simp | |
| show "\<And>f a b. \<lbrakk>\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>; C.ide f\<rbrakk> | |
| \<Longrightarrow> (\<l>\<^sub>D\<^sup>-\<^sup>1[G f] \<cdot>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D (G f \<star>\<^sub>D \<epsilon> a) = (\<epsilon> b \<star>\<^sub>D G f) \<cdot>\<^sub>D \<tau>\<tau>'.map\<^sub>1 f" | |
| proof - | |
| fix f a b | |
| assume f: "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" and ide_f: "C.ide f" | |
| have a: "C.obj a" and b: "C.obj b" | |
| using f by auto | |
| have "(\<epsilon> b \<star>\<^sub>D G f) \<cdot>\<^sub>D \<tau>\<tau>'.map\<^sub>1 f | |
| = (\<epsilon> b \<star>\<^sub>D G f) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, G f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| unfolding \<tau>\<tau>'.map\<^sub>1_def map\<^sub>1_def | |
| using a b f D.comp_assoc by auto | |
| also have "... = (\<epsilon> b \<star>\<^sub>D G f) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, G f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D (\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D (\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| proof - | |
| have "\<tau>\<^sub>0 b \<star>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) | |
| = (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D (\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D (\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a)" | |
| proof - | |
| have "D.arr ((\<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| ((\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a))" | |
| using a b f ide_f \<tau>.iso_map\<^sub>1_ide | |
| by (intro D.seqI) auto | |
| thus ?thesis | |
| using a b f D.whisker_left [of "\<tau>\<^sub>0 b"] by fastforce | |
| qed | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (\<epsilon> b \<star>\<^sub>D G f) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, G f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D (\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| proof - | |
| have "(\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D (\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a) | |
| = \<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a)" | |
| using a b f ide_f D.whisker_left [of "\<tau>\<^sub>0 b"] \<tau>.iso_map\<^sub>1_ide by auto | |
| also have "... = \<tau>\<^sub>0 b \<star>\<^sub>D (\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a]" | |
| using a b f ide_f \<tau>.iso_map\<^sub>1_ide | |
| D.assoc_naturality [of "\<tau>\<^sub>0' b" "D.inv (\<tau>\<^sub>1 f)" "\<tau>\<^sub>0' a"] | |
| by auto | |
| also have "... = (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a])" | |
| using a b f ide_f D.whisker_left [of "\<tau>\<^sub>0 b"] \<tau>.iso_map\<^sub>1_ide by auto | |
| finally have "(\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D (\<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f)) \<star>\<^sub>D \<tau>\<^sub>0' a) | |
| = (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a])" | |
| by blast | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (\<epsilon> b \<star>\<^sub>D G f) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, G f] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f])) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D (\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| proof - | |
| have "(\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a) | |
| = \<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)" | |
| using a b f ide_f D.whisker_left \<tau>.iso_map\<^sub>1_ide by auto | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (\<epsilon> b \<star>\<^sub>D G f) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, G f \<star>\<^sub>D G\<^sub>0 a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a)) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D (\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| proof - | |
| have "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, G f] \<cdot>\<^sub>D (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D \<r>\<^sub>D[G f]) | |
| = ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, G f \<star>\<^sub>D G\<^sub>0 a]" | |
| using a b f ide_f D.assoc'_naturality [of "\<tau>\<^sub>0 b" "\<tau>\<^sub>0' b" "\<r>\<^sub>D[G f]"] | |
| by auto | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (\<epsilon> b \<star>\<^sub>D G f) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a))) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D (\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| proof - | |
| have "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, G f \<star>\<^sub>D G\<^sub>0 a] \<cdot>\<^sub>D (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) | |
| = ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a]" | |
| using a b f ide_f D.assoc'_naturality [of "\<tau>\<^sub>0 b" "\<tau>\<^sub>0' b" "G f \<star>\<^sub>D \<epsilon> a"] | |
| by auto | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = ((\<epsilon> b \<star>\<^sub>D G f) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<r>\<^sub>D[G f])) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, (\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D (\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| proof - | |
| have "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) | |
| = ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, (\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a]" | |
| using a b f ide_f \<tau>.iso_map\<^sub>1_ide | |
| D.assoc'_naturality | |
| [of "\<tau>\<^sub>0 b" "\<tau>\<^sub>0' b" "\<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)"] | |
| by auto | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (G\<^sub>0 b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| ((\<epsilon> b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a)) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, (\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D (\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| proof - | |
| have "(\<epsilon> b \<star>\<^sub>D G f) \<cdot>\<^sub>D ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<r>\<^sub>D[G f]) = \<epsilon> b \<star>\<^sub>D \<r>\<^sub>D[G f]" | |
| using a b f ide_f D.comp_arr_dom D.comp_cod_arr | |
| D.interchange [of "\<epsilon> b" "\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b" "G f" "\<r>\<^sub>D[G f]"] | |
| by simp | |
| also have "... = (G\<^sub>0 b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D (\<epsilon> b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 a)" | |
| using a b f ide_f D.comp_arr_dom D.comp_cod_arr | |
| D.interchange [of "G\<^sub>0 b" "\<epsilon> b" "\<r>\<^sub>D[G f]" "G f \<star>\<^sub>D G\<^sub>0 a"] | |
| by auto | |
| finally have "(\<epsilon> b \<star>\<^sub>D G f) \<cdot>\<^sub>D ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<r>\<^sub>D[G f]) = | |
| (G\<^sub>0 b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D (\<epsilon> b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 a)" | |
| by blast | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (G\<^sub>0 b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (G\<^sub>0 b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| ((\<epsilon> b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a))) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, (\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D (\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| proof - | |
| have "(\<epsilon> b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 a) \<cdot>\<^sub>D ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) | |
| = \<epsilon> b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a" | |
| using a b f ide_f D.comp_arr_dom D.comp_cod_arr | |
| D.interchange [of "\<epsilon> b" "\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b" "G f \<star>\<^sub>D G\<^sub>0 a" "G f \<star>\<^sub>D \<epsilon> a"] | |
| by auto | |
| also have "... = (G\<^sub>0 b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D (\<epsilon> b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a)" | |
| using a b f ide_f D.comp_arr_dom D.comp_cod_arr | |
| D.interchange [of "G\<^sub>0 b" "\<epsilon> b" "G f \<star>\<^sub>D \<epsilon> a" "G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a"] | |
| by auto | |
| finally have "(\<epsilon> b \<star>\<^sub>D G f \<star>\<^sub>D G\<^sub>0 a) \<cdot>\<^sub>D ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) = | |
| (G\<^sub>0 b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D (\<epsilon> b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a)" | |
| by blast | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (G\<^sub>0 b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (G\<^sub>0 b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| (G\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| (\<epsilon> b \<star>\<^sub>D (\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, (\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D (\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| proof - | |
| have 1: "D.seq (G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a) | |
| (\<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a))" | |
| using a b f ide_f \<tau>.iso_map\<^sub>1_ide | |
| by (intro D.seqI D.hseqI') auto | |
| have 2: "D.seq (\<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a)" | |
| using a b f ide_f \<tau>.iso_map\<^sub>1_ide | |
| by (intro D.seqI D.hseqI') auto | |
| have "(\<epsilon> b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) = | |
| \<epsilon> b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)" | |
| using 1 a b f ide_f D.comp_arr_dom D.comp_cod_arr \<tau>.iso_map\<^sub>1_ide | |
| D.interchange [of "\<epsilon> b" "\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b" "G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a" | |
| "\<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)"] | |
| by auto | |
| also have "... = G\<^sub>0 b \<cdot>\<^sub>D \<epsilon> b \<star>\<^sub>D | |
| (\<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a)" | |
| using a b f ide_f \<tau>.iso_map\<^sub>1_ide D.comp_arr_dom D.comp_cod_arr by auto | |
| also have "... = (G\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| (\<epsilon> b \<star>\<^sub>D (\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a)" | |
| using 2 a b f ide_f \<tau>.iso_map\<^sub>1_ide D.comp_assoc | |
| D.interchange [of "G\<^sub>0 b" "\<epsilon> b" | |
| "\<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)" | |
| "(\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a"] | |
| by simp | |
| finally have "(\<epsilon> b \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) | |
| = (G\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| (\<epsilon> b \<star>\<^sub>D (\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a)" | |
| by blast | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (G\<^sub>0 b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (G\<^sub>0 b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| (G\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| (G\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[G\<^sub>0 b, \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| ((\<epsilon> b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, (\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<eta> b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| proof - | |
| have "\<epsilon> b \<star>\<^sub>D (\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a | |
| = (G\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[G\<^sub>0 b, \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| ((\<epsilon> b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a])" | |
| proof - | |
| have "\<epsilon> b \<star>\<^sub>D (\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a | |
| = (G\<^sub>0 b \<cdot>\<^sub>D \<epsilon> b \<cdot>\<^sub>D (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b)) \<star>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (\<tau>\<^sub>0 b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]" | |
| using a b f ide_f D.comp_arr_dom D.comp_cod_arr | |
| D.hcomp_reassoc(1) [of "\<tau>\<^sub>0 b" "F f" "\<tau>\<^sub>0' a"] | |
| by auto | |
| also have "... = (G\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<epsilon> b \<star>\<^sub>D \<tau>\<^sub>0 b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a])" | |
| using a b f ide_f D.comp_arr_dom D.comp_cod_arr D.assoc'_is_natural_1 | |
| D.interchange [of "G\<^sub>0 b" "\<epsilon> b \<cdot>\<^sub>D (\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b)" "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]" | |
| "(\<tau>\<^sub>0 b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]"] | |
| D.interchange [of "\<epsilon> b" "\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b" "\<tau>\<^sub>0 b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a" | |
| "\<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]"] | |
| by fastforce | |
| also have "... = (G\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[G\<^sub>0 b, \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| ((\<epsilon> b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a])" | |
| using a b f ide_f D.hcomp_reassoc(2) [of "\<epsilon> b" "\<tau>\<^sub>0 b" "F f \<star>\<^sub>D \<tau>\<^sub>0' a"] | |
| by auto | |
| finally show ?thesis by blast | |
| qed | |
| moreover have "\<tau>\<^sub>0 b \<star>\<^sub>D (\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a | |
| = (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<eta> b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 b, F f, \<tau>\<^sub>0' a])" | |
| proof - | |
| have "\<tau>\<^sub>0 b \<star>\<^sub>D (\<eta> b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a = | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<eta> b \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 b, F f, \<tau>\<^sub>0' a])" | |
| using a b f ide_f D.hcomp_reassoc(1) [of "\<eta> b" "F f" "\<tau>\<^sub>0' a"] | |
| D.whisker_left [of "\<tau>\<^sub>0 b"] | |
| by auto | |
| also have "... = (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<eta> b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 b, F f, \<tau>\<^sub>0' a])" | |
| using a b f ide_f D.hcomp_reassoc(2) [of "\<tau>\<^sub>0 b" "\<eta> b" "F f \<star>\<^sub>D \<tau>\<^sub>0' a"] | |
| by auto | |
| finally show ?thesis by blast | |
| qed | |
| ultimately show ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (G\<^sub>0 b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (G\<^sub>0 b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| (G\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| (G\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[G\<^sub>0 b, \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (((\<epsilon> b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, \<tau>\<^sub>0 b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<eta> b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| proof - | |
| have "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, (\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a] | |
| = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, \<tau>\<^sub>0 b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a" | |
| proof - | |
| have "\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<tau>\<^sub>0' b) \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, (\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b \<star>\<^sub>D F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0' b, \<tau>\<^sub>0 b, F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b \<star>\<^sub>D \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a] | |
| = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> | |
| ((\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> | |
| \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle>, (\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> | |
| (\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> | |
| (\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>) \<^bold>\<cdot> | |
| (\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> | |
| \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>\<^bold>]\<rbrace>" | |
| using a b f ide_f D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char | |
| D.VV.ide_char D.VV.arr_char | |
| by auto | |
| also have "... = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' b\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>\<rbrace>" | |
| using a b f ide_f by (intro EV.eval_eqI, auto) | |
| also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, \<tau>\<^sub>0 b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a" | |
| using a b f ide_f D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char | |
| D.VV.ide_char D.VV.arr_char | |
| by auto | |
| finally show ?thesis by blast | |
| qed | |
| thus ?thesis | |
| using D.comp_assoc by auto | |
| qed | |
| also have "... = (G\<^sub>0 b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (G\<^sub>0 b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| (G\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| ((G\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[G\<^sub>0 b, \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| proof - | |
| have "((\<epsilon> b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, \<tau>\<^sub>0 b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<eta> b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) | |
| = \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a" | |
| proof - | |
| interpret adjoint_equivalence_in_bicategory | |
| V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<open>\<tau>\<^sub>0 b\<close> \<open>\<tau>\<^sub>0' b\<close> \<open>\<eta> b\<close> \<open>\<epsilon> b\<close> | |
| using b chosen_adjoint_equivalence by simp | |
| have "((\<epsilon> b \<star>\<^sub>D \<tau>\<^sub>0 b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, \<tau>\<^sub>0 b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| ((\<tau>\<^sub>0 b \<star>\<^sub>D \<eta> b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) | |
| = (\<epsilon> b \<star>\<^sub>D \<tau>\<^sub>0 b) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, \<tau>\<^sub>0' b, \<tau>\<^sub>0 b] \<cdot>\<^sub>D (\<tau>\<^sub>0 b \<star>\<^sub>D \<eta> b) \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a" | |
| using a b f ide_f D.whisker_right by auto | |
| also have "... = \<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a" | |
| using triangle_left by simp | |
| finally show ?thesis by blast | |
| qed | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (G\<^sub>0 b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (G\<^sub>0 b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| ((G\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| \<l>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| proof - | |
| have "(G\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[G\<^sub>0 b, \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a] | |
| = \<l>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a]" | |
| proof - | |
| have "(G\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<a>\<^sub>D[G\<^sub>0 b, \<tau>\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (\<l>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b] \<cdot>\<^sub>D \<r>\<^sub>D[\<tau>\<^sub>0 b] \<star>\<^sub>D F f \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 b, F\<^sub>0 b, F f \<star>\<^sub>D \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[F\<^sub>0 b, F f, \<tau>\<^sub>0' a]) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>0 b \<star>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| \<a>\<^sub>D[\<tau>\<^sub>0 b, F f, \<tau>\<^sub>0' a] | |
| = \<lbrace>(\<^bold>\<langle>G\<^sub>0 b\<^bold>\<rangle>\<^sub>0 \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> | |
| (\<^bold>\<a>\<^bold>[\<^bold>\<langle>G\<^sub>0 b\<^bold>\<rangle>\<^sub>0, \<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> | |
| (\<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> \<^bold>\<r>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>) \<^bold>\<cdot> | |
| \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle>, \<^bold>\<langle>F\<^sub>0 b\<^bold>\<rangle>\<^sub>0, \<^bold>\<langle>F f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> | |
| (\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>F\<^sub>0 b\<^bold>\<rangle>\<^sub>0, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> | |
| (\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>F f\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>) \<^bold>\<cdot> | |
| \<^bold>\<a>\<^bold>[\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle>, \<^bold>\<langle>F f\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>\<^bold>]\<rbrace>" | |
| using a b f ide_f D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char | |
| D.VV.ide_char D.VV.arr_char D.\<ll>_ide_simp D.\<rr>_ide_simp | |
| by auto | |
| also have "... = \<lbrace>\<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[(\<^bold>\<langle>\<tau>\<^sub>0 b\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>F f\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<^sub>0' a\<^bold>\<rangle>\<^bold>]\<rbrace>" | |
| using a b f ide_f by (intro EV.eval_eqI, auto) | |
| also have "... = \<l>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a]" | |
| using a b f ide_f D.\<alpha>_def D.\<alpha>'.map_ide_simp D.VVV.ide_char D.VVV.arr_char | |
| D.VV.ide_char D.VV.arr_char D.\<ll>_ide_simp | |
| by auto | |
| finally show ?thesis by blast | |
| qed | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (G\<^sub>0 b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (G\<^sub>0 b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| \<l>\<^sub>D\<^sup>-\<^sup>1[G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| ((D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D | |
| (\<tau>\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| proof - | |
| have "(G\<^sub>0 b \<star>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D | |
| \<l>\<^sub>D\<^sup>-\<^sup>1[(\<tau>\<^sub>0 b \<star>\<^sub>D F f) \<star>\<^sub>D \<tau>\<^sub>0' a] | |
| = \<l>\<^sub>D\<^sup>-\<^sup>1[G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)" | |
| using a b f ide_f \<tau>.iso_map\<^sub>1_ide | |
| D.lunit'_naturality [of "\<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D (D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a)"] | |
| by auto | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (G\<^sub>0 b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (G\<^sub>0 b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| \<l>\<^sub>D\<^sup>-\<^sup>1[G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D | |
| \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| proof - | |
| have "((D.inv (\<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D (\<tau>\<^sub>1 f \<star>\<^sub>D \<tau>\<^sub>0' a)) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] = | |
| ((D.inv (\<tau>\<^sub>1 f) \<cdot>\<^sub>D \<tau>\<^sub>1 f) \<star>\<^sub>D \<tau>\<^sub>0' a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| using a b f ide_f \<tau>.iso_map\<^sub>1_ide D.whisker_right [of "\<tau>\<^sub>0' a"] by simp | |
| also have "... = \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a]" | |
| using a b f ide_f \<tau>.iso_map\<^sub>1_ide D.comp_inv_arr' D.comp_cod_arr by auto | |
| finally show ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (G\<^sub>0 b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D | |
| (G\<^sub>0 b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D | |
| \<l>\<^sub>D\<^sup>-\<^sup>1[G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a]" | |
| proof - | |
| have "\<l>\<^sub>D\<^sup>-\<^sup>1[G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a] \<cdot>\<^sub>D \<a>\<^sub>D[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 a, \<tau>\<^sub>0' a] = | |
| \<l>\<^sub>D\<^sup>-\<^sup>1[G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a]" | |
| using a b f ide_f D.comp_arr_inv' D.comp_arr_dom by auto | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = ((G\<^sub>0 b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[G f \<star>\<^sub>D G\<^sub>0 a]) \<cdot>\<^sub>D (G f \<star>\<^sub>D \<epsilon> a)" | |
| proof - | |
| have "(G\<^sub>0 b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D (G\<^sub>0 b \<star>\<^sub>D G f \<star>\<^sub>D \<epsilon> a) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[G f \<star>\<^sub>D \<tau>\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0' a] = | |
| (G\<^sub>0 b \<star>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[G f \<star>\<^sub>D G\<^sub>0 a] \<cdot>\<^sub>D (G f \<star>\<^sub>D \<epsilon> a)" | |
| using a b f ide_f D.lunit'_naturality [of "G f \<star>\<^sub>D \<epsilon> a"] by auto | |
| thus ?thesis | |
| using D.comp_assoc by simp | |
| qed | |
| also have "... = (\<l>\<^sub>D\<^sup>-\<^sup>1[G f] \<cdot>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D (G f \<star>\<^sub>D \<epsilon> a)" | |
| using a b f ide_f D.lunit'_naturality [of "\<r>\<^sub>D[G f]"] by auto | |
| finally show "(\<l>\<^sub>D\<^sup>-\<^sup>1[G f] \<cdot>\<^sub>D \<r>\<^sub>D[G f]) \<cdot>\<^sub>D (G f \<star>\<^sub>D \<epsilon> a) = (\<epsilon> b \<star>\<^sub>D G f) \<cdot>\<^sub>D \<tau>\<tau>'.map\<^sub>1 f" | |
| by simp | |
| qed | |
| qed | |
| lemma counit_is_invertible_modification: | |
| shows "invertible_modification | |
| V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G G \<Phi>\<^sub>G | |
| \<tau>\<tau>'.map\<^sub>0 \<tau>\<tau>'.map\<^sub>1 \<iota>\<^sub>G.map\<^sub>0 \<iota>\<^sub>G.map\<^sub>1 \<epsilon>" | |
| .. | |
| end | |
| end | |