clean up Eckmann-Hilton and move to Identity

3 files changed, 206 insertions, 209 deletions

diff --git a/hott/Eckmann_Hilton.thy b/hott/Eckmann_Hilton.thy
deleted file mode 100644
index 8320256..0000000
--- a/hott/Eckmann_Hilton.thy
+++ /dev/null
@@ -1,188 +0,0 @@
-theory Eckmann_Hilton
-imports Identity
-
-begin
-
-section \<open>Whiskering and horizontal composition\<close>
-
-Lemma (derive) right_whisker:
-  assumes "A: U i" "a: A" "b: A" "c: A"
-  shows "\<lbrakk>p: a = b; q: a = b; r: b = c; \<alpha>: p =\<^bsub>a = b\<^esub> q\<rbrakk> \<Longrightarrow> p \<bullet> r = q \<bullet> r"
-  apply (eq r)
-  focus prems vars x s t \<gamma>
-    proof -
-      have "t \<bullet> refl x = t" by (rule pathcomp_refl)
-      also have ".. = s" by (rule \<open>\<gamma>: t = s\<close>)
-      also have ".. = s \<bullet> refl x" by (rule pathcomp_refl[symmetric])
-      finally show "t \<bullet> refl x = s \<bullet> refl x" by this
-    qed
-  done
-
-Lemma (derive) left_whisker:
-  assumes "A: U i" "a: A" "b: A" "c: A"
-  shows "\<lbrakk>p: b = c; q: b = c; r: a = b; \<alpha>: p =\<^bsub>b = c\<^esub> q\<rbrakk> \<Longrightarrow> r \<bullet> p = r \<bullet> q"
-  apply (eq r)
-  focus prems prms vars x s t \<gamma>
-    proof -
-      have "refl x \<bullet> t = t" by (rule refl_pathcomp)
-      also have ".. = s" by (rule \<open>\<gamma>:_ t = s\<close>)
-      also have ".. = refl x \<bullet> s" by (rule refl_pathcomp[symmetric])
-      finally show "refl x \<bullet> t = refl x \<bullet> s" by this
-    qed
-  done
-
-definition right_whisker_i (infix "\<bullet>\<^sub>r\<^bsub>_\<^esub>" 121)
-  where [implicit]: "\<alpha> \<bullet>\<^sub>r\<^bsub>a\<^esub> r \<equiv> right_whisker ? a ? ? ? ? r \<alpha>"
-
-definition left_whisker_i (infix "\<bullet>\<^sub>l\<^bsub>_\<^esub>" 121)
-  where [implicit]: "r \<bullet>\<^sub>l\<^bsub>c\<^esub> \<alpha> \<equiv> left_whisker ? ? ? c ? ? r \<alpha>"
-
-translations
-  "\<alpha> \<bullet>\<^sub>r\<^bsub>a\<^esub> r" \<leftharpoondown> "CONST right_whisker A a b c p q r \<alpha>"
-  "r \<bullet>\<^sub>l\<^bsub>c\<^esub> \<alpha>" \<leftharpoondown> "CONST left_whisker A a b c p q r \<alpha>"
-
-Lemma whisker_refl [comps]:
-  assumes "A: U i" "a: A" "b: A"
-  shows "\<lbrakk>p: a = b; q: a = b; \<alpha>: p =\<^bsub>a = b\<^esub> q\<rbrakk> \<Longrightarrow>
-    \<alpha> \<bullet>\<^sub>r\<^bsub>a\<^esub> (refl b) \<equiv> ru p \<bullet> \<alpha> \<bullet> (ru q)\<inverse>"
-  unfolding right_whisker_def by reduce
-
-Lemma refl_whisker [comps]:
-  assumes "A: U i" "a: A" "b: A"
-  shows "\<lbrakk>p: a = b; q: a = b; \<alpha>: p = q\<rbrakk> \<Longrightarrow>
-    (refl a) \<bullet>\<^sub>l\<^bsub>b\<^esub> \<alpha> \<equiv> (lu p) \<bullet> \<alpha> \<bullet> (lu q)\<inverse>"
-  unfolding left_whisker_def by reduce
-
-text \<open>Define the conditions under which horizontal composition is well-defined:\<close>
-
-locale horiz_pathcomposable =
-fixes
-  i A a b c p q r s
-assumes assums:
-  "A: U i" "a: A" "b: A" "c: A"
-  "p: a =\<^bsub>A\<^esub> b" "q: a =\<^bsub>A\<^esub> b"
-  "r: b =\<^bsub>A\<^esub> c" "s: b =\<^bsub>A\<^esub> c"
-begin
-
-  Lemma (derive) horiz_pathcomp:
-    notes assums
-    shows "\<lbrakk>\<alpha>: p = q; \<beta>: r = s\<rbrakk> \<Longrightarrow> ?prf \<alpha> \<beta>: p \<bullet> r = q \<bullet> s"
-  proof (rule pathcomp)
-    show "\<alpha>: p = q \<Longrightarrow> p \<bullet> r = q \<bullet> r" by (rule right_whisker)
-    show "\<beta>: r = s \<Longrightarrow> .. = q \<bullet> s" by (rule left_whisker)
-  qed typechk
-
-  text \<open>A second horizontal composition:\<close>
-
-  Lemma (derive) horiz_pathcomp':
-    notes assums
-    shows "\<lbrakk>\<alpha>: p = q; \<beta>: r = s\<rbrakk> \<Longrightarrow> ?prf \<alpha> \<beta>: p \<bullet> r = q \<bullet> s"
-  proof (rule pathcomp)
-    show "\<beta>: r = s \<Longrightarrow> p \<bullet> r = p \<bullet> s" by (rule left_whisker)
-    show "\<alpha>: p = q \<Longrightarrow> .. = q \<bullet> s" by (rule right_whisker)
-  qed typechk
-
-  abbreviation horiz_pathcomp_abbr :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> (infix "\<star>" 121)
-    where "\<alpha> \<star> \<beta> \<equiv> horiz_pathcomp \<alpha> \<beta>"
-
-  abbreviation horiz_pathcomp'_abbr (infix "\<star>''" 121)
-    where "\<alpha> \<star>' \<beta> \<equiv> horiz_pathcomp' \<alpha> \<beta>"
-
-  Lemma (derive) horiz_pathcomp_eq_horiz_pathcomp':
-    notes assums
-    shows "\<lbrakk>\<alpha>: p = q; \<beta>: r = s\<rbrakk> \<Longrightarrow> \<alpha> \<star> \<beta> = \<alpha> \<star>' \<beta>"
-    unfolding horiz_pathcomp_def horiz_pathcomp'_def
-    apply (eq \<alpha>, eq \<beta>)
-      focus vars p apply (eq p)
-        focus vars _ q by (eq q) (reduce; refl)
-      done
-    done
-
-end
-
-
-section \<open>Loop space\<close>
-
-definition \<Omega> where "\<Omega> A a \<equiv> a =\<^bsub>A\<^esub> a"
-definition \<Omega>2 where "\<Omega>2 A a \<equiv> \<Omega> (\<Omega> A a) (refl a)"
-
-Lemma \<Omega>2_alt_def:
-  "\<Omega>2 A a \<equiv> refl a = refl a"
-  unfolding \<Omega>2_def \<Omega>_def .
-
-
-section \<open>Eckmann-Hilton\<close>
-
-context
-fixes A a
-assumes "A: U i" "a: A"
-begin
-
-  interpretation \<Omega>2:
-    horiz_pathcomposable i A a a a "refl a" "refl a" "refl a" "refl a"
-    by unfold_locales typechk+
-
-  abbreviation horiz_pathcomp (infix "\<star>" 121)
-    where "\<alpha> \<star> \<beta> \<equiv> \<Omega>2.horiz_pathcomp \<alpha> \<beta>"
-
-  abbreviation horiz_pathcomp' (infix "\<star>''" 121)
-    where "\<alpha> \<star>' \<beta> \<equiv> \<Omega>2.horiz_pathcomp' \<alpha> \<beta>"
-
-  Lemma (derive) pathcomp_eq_horiz_pathcomp:
-    assumes "\<alpha>: \<Omega>2 A a" "\<beta>: \<Omega>2 A a"
-    shows "\<alpha> \<bullet> \<beta> = \<alpha> \<star> \<beta>"
-    unfolding \<Omega>2.horiz_pathcomp_def
-    using assms[unfolded \<Omega>2_alt_def]
-    proof (reduce, rule pathinv)
-      \<comment> \<open>Propositional equality rewriting needs to be improved\<close>
-      have "refl (refl a) \<bullet> \<alpha> \<bullet> refl (refl a) = refl (refl a) \<bullet> \<alpha>" by (rule pathcomp_refl)
-      also have ".. = \<alpha>" by (rule refl_pathcomp)
-      finally have eq\<alpha>: "refl (refl a) \<bullet> \<alpha> \<bullet> refl (refl a) = \<alpha>" by this
-
-      have "refl (refl a) \<bullet> \<beta> \<bullet> refl (refl a) = refl (refl a) \<bullet> \<beta>" by (rule pathcomp_refl)
-      also have ".. = \<beta>" by (rule refl_pathcomp)
-      finally have eq\<beta>: "refl (refl a) \<bullet> \<beta> \<bullet> refl (refl a) = \<beta>" by this
-
-      have "refl (refl a) \<bullet> \<alpha> \<bullet> refl (refl a) \<bullet> (refl (refl a) \<bullet> \<beta> \<bullet> refl (refl a))
-        = \<alpha> \<bullet> (refl (refl a) \<bullet> \<beta> \<bullet> refl (refl a))" by (rule right_whisker) (rule eq\<alpha>)
-      also have ".. = \<alpha> \<bullet> \<beta>" by (rule left_whisker) (rule eq\<beta>)
-      finally show "refl (refl a) \<bullet> \<alpha> \<bullet> refl (refl a) \<bullet> (refl (refl a) \<bullet> \<beta> \<bullet> refl (refl a))
-        = \<alpha> \<bullet> \<beta>" by this
-    qed
-
-  Lemma (derive) pathcomp_eq_horiz_pathcomp':
-    assumes "\<alpha>: \<Omega>2 A a" "\<beta>: \<Omega>2 A a"
-    shows "\<alpha> \<star>' \<beta> = \<beta> \<bullet> \<alpha>"
-    unfolding \<Omega>2.horiz_pathcomp'_def
-    using assms[unfolded \<Omega>2_alt_def]
-    proof reduce
-      have "refl (refl a) \<bullet> \<beta> \<bullet> refl (refl a) = refl (refl a) \<bullet> \<beta>" by (rule pathcomp_refl)
-      also have ".. = \<beta>" by (rule refl_pathcomp)
-      finally have eq\<beta>: "refl (refl a) \<bullet> \<beta> \<bullet> refl (refl a) = \<beta>" by this
-
-      have "refl (refl a) \<bullet> \<alpha> \<bullet> refl (refl a) = refl (refl a) \<bullet> \<alpha>" by (rule pathcomp_refl)
-      also have ".. = \<alpha>" by (rule refl_pathcomp)
-      finally have eq\<alpha>: "refl (refl a) \<bullet> \<alpha> \<bullet> refl (refl a) = \<alpha>" by this
-
-      have "refl (refl a) \<bullet> \<beta> \<bullet> refl (refl a) \<bullet> (refl (refl a) \<bullet> \<alpha> \<bullet> refl (refl a))
-        = \<beta> \<bullet> (refl (refl a) \<bullet> \<alpha> \<bullet> refl (refl a))" by (rule right_whisker) (rule eq\<beta>)
-      also have ".. = \<beta> \<bullet> \<alpha>" by (rule left_whisker) (rule eq\<alpha>)
-      finally show "refl (refl a) \<bullet> \<beta> \<bullet> refl (refl a) \<bullet> (refl (refl a) \<bullet> \<alpha> \<bullet> refl (refl a))
-        = \<beta> \<bullet> \<alpha>" by this
-    qed
-
-  Lemma (derive) eckmann_hilton:
-    assumes "\<alpha>: \<Omega>2 A a" "\<beta>: \<Omega>2 A a"
-    shows "\<alpha> \<bullet> \<beta> = \<beta> \<bullet> \<alpha>"
-    using assms[unfolded \<Omega>2_alt_def]
-    proof -
-      have "\<alpha> \<bullet> \<beta> = \<alpha> \<star> \<beta>" by (rule pathcomp_eq_horiz_pathcomp)
-      also have ".. = \<alpha> \<star>' \<beta>" by (rule \<Omega>2.horiz_pathcomp_eq_horiz_pathcomp'[simplified comps])
-      also have ".. = \<beta> \<bullet> \<alpha>" by (rule pathcomp_eq_horiz_pathcomp')
-      finally show "\<alpha> \<bullet> \<beta> = \<beta> \<bullet> \<alpha>" by this (reduce add: \<Omega>2.horiz_pathcomp_def \<Omega>2.horiz_pathcomp'_def)
-    qed
-
-end
-
-
-end
diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy
index 9e7b83a..9c86a95 100644
--- a/hott/Equivalence.thy
+++ b/hott/Equivalence.thy
@@ -138,6 +138,11 @@ Lemma homotopy_funcomp_right:
       apply (rule ap, assumption)
   done
 
+method id_htpy = (rule homotopy_id_left)
+method htpy_id = (rule homotopy_id_right)
+method htpy_o = (rule homotopy_funcomp_left)
+method o_htpy = (rule homotopy_funcomp_right)
+
 
 section \<open>Quasi-inverse and bi-invertibility\<close>
 
@@ -187,15 +192,20 @@ Lemma (derive) funcomp_qinv:
     "f: A \<rightarrow> B" "g: B \<rightarrow> C"
   shows "qinv f \<rightarrow> qinv g \<rightarrow> qinv (g \<circ> f)"
   apply (intros, unfold qinv_def, elims)
-  focus
-    prems prms
-    vars _ _ finv _ ginv _ HfA HfB HgB HgC
-
-    apply intro
-    apply (rule funcompI[where ?f=ginv and ?g=finv])
-
-    text \<open>Now a whole bunch of rewrites and we're done.\<close>
-oops
+  focus prems vars _ _ finv _ ginv
+    apply (intro, rule funcompI[where ?f=ginv and ?g=finv])
+    proof (reduce, intro)
+      have "finv \<circ> ginv \<circ> g \<circ> f ~ finv \<circ> (ginv \<circ> g) \<circ> f" by reduce refl
+      also have ".. ~ finv \<circ> id B \<circ> f" by (o_htpy, htpy_o) fact
+      also have ".. ~ id A" by reduce fact
+      finally show "finv \<circ> ginv \<circ> g \<circ> f ~ id A" by this
+
+      have "g \<circ> f \<circ> finv \<circ> ginv ~ g \<circ> (f \<circ> finv) \<circ> ginv" by reduce refl
+      also have ".. ~ g \<circ> id B \<circ> ginv" by (o_htpy, htpy_o) fact
+      also have ".. ~ id C" by reduce fact
+      finally show "g \<circ> f \<circ> finv \<circ> ginv ~ id C" by this
+    qed
+  done
 
 subsection \<open>Bi-invertible maps\<close>
 
@@ -246,10 +256,10 @@ Lemma (derive) biinv_imp_qinv:
   \<close>
   unfolding qinv_def
   apply intro
-    \<guillemotright> by (rule \<open>g: _\<close>)
+    \<guillemotright> by (fact \<open>g: _\<close>)
     \<guillemotright> apply intro
       text \<open>The first part \<^prop>\<open>?H1: g \<circ> f ~ id A\<close> is given by \<^term>\<open>H1\<close>.\<close>
-      apply (rule \<open>H1: _\<close>)
+      apply (fact \<open>H1: _\<close>)
 
       text \<open>
         It remains to prove \<^prop>\<open>?H2: f \<circ> g ~ id B\<close>. First show that \<open>g ~ h\<close>,
@@ -258,19 +268,13 @@ Lemma (derive) biinv_imp_qinv:
       \<close>
       proof -
         have "g ~ g \<circ> (id B)" by reduce refl
-        also have ".. ~ g \<circ> f \<circ> h"
-          by (rule homotopy_funcomp_right) (rule \<open>H2:_\<close>[symmetric])
-        also have ".. ~ (id A) \<circ> h"
-          by (subst funcomp_assoc[symmetric])
-             (rule homotopy_funcomp_left, rule \<open>H1:_\<close>)
+        also have ".. ~ g \<circ> f \<circ> h" by o_htpy (rule \<open>H2:_\<close>[symmetric])
+        also have ".. ~ (id A) \<circ> h" by (subst funcomp_assoc[symmetric]) (htpy_o, fact)
         also have ".. ~ h" by reduce refl
         finally have "g ~ h" by this
-
         then have "f \<circ> g ~ f \<circ> h" by (rule homotopy_funcomp_right)
-
-        with \<open>H2:_\<close>
-        show "f \<circ> g ~ id B"
-          by (rule homotopy_trans) (assumption+, typechk)
+        also note \<open>H2:_\<close>
+        finally show "f \<circ> g ~ id B" by this
       qed
     done
   done
diff --git a/hott/Identity.thy b/hott/Identity.thy
index 64aea5a..571617a 100644
--- a/hott/Identity.thy
+++ b/hott/Identity.thy
@@ -460,4 +460,185 @@ Lemma (derive) apd_ap:
   by (eq p) (reduce; intro)
 
 
+section \<open>Whiskering\<close>
+
+Lemma (derive) right_whisker:
+  assumes "A: U i" "a: A" "b: A" "c: A"
+  shows "\<lbrakk>p: a = b; q: a = b; r: b = c; \<alpha>: p =\<^bsub>a = b\<^esub> q\<rbrakk> \<Longrightarrow> p \<bullet> r = q \<bullet> r"
+  apply (eq r)
+  focus prems vars x s t
+    proof -
+      have "t \<bullet> refl x = t" by (rule pathcomp_refl)
+      also have ".. = s" by fact
+      also have ".. = s \<bullet> refl x" by (rule pathcomp_refl[symmetric])
+      finally show "t \<bullet> refl x = s \<bullet> refl x" by this
+    qed
+  done
+
+Lemma (derive) left_whisker:
+  assumes "A: U i" "a: A" "b: A" "c: A"
+  shows "\<lbrakk>p: b = c; q: b = c; r: a = b; \<alpha>: p =\<^bsub>b = c\<^esub> q\<rbrakk> \<Longrightarrow> r \<bullet> p = r \<bullet> q"
+  apply (eq r)
+  focus prems prms vars x s t
+    proof -
+      have "refl x \<bullet> t = t" by (rule refl_pathcomp)
+      also have ".. = s" by fact
+      also have ".. = refl x \<bullet> s" by (rule refl_pathcomp[symmetric])
+      finally show "refl x \<bullet> t = refl x \<bullet> s" by this
+    qed
+  done
+
+definition right_whisker_i (infix "\<bullet>\<^sub>r\<^bsub>_\<^esub>" 121)
+  where [implicit]: "\<alpha> \<bullet>\<^sub>r\<^bsub>a\<^esub> r \<equiv> right_whisker ? a ? ? ? ? r \<alpha>"
+
+definition left_whisker_i (infix "\<bullet>\<^sub>l\<^bsub>_\<^esub>" 121)
+  where [implicit]: "r \<bullet>\<^sub>l\<^bsub>c\<^esub> \<alpha> \<equiv> left_whisker ? ? ? c ? ? r \<alpha>"
+
+translations
+  "\<alpha> \<bullet>\<^sub>r\<^bsub>a\<^esub> r" \<leftharpoondown> "CONST right_whisker A a b c p q r \<alpha>"
+  "r \<bullet>\<^sub>l\<^bsub>c\<^esub> \<alpha>" \<leftharpoondown> "CONST left_whisker A a b c p q r \<alpha>"
+
+Lemma whisker_refl [comps]:
+  assumes "A: U i" "a: A" "b: A"
+  shows "\<lbrakk>p: a = b; q: a = b; \<alpha>: p =\<^bsub>a = b\<^esub> q\<rbrakk> \<Longrightarrow>
+    \<alpha> \<bullet>\<^sub>r\<^bsub>a\<^esub> (refl b) \<equiv> ru p \<bullet> \<alpha> \<bullet> (ru q)\<inverse>"
+  unfolding right_whisker_def by reduce
+
+Lemma refl_whisker [comps]:
+  assumes "A: U i" "a: A" "b: A"
+  shows "\<lbrakk>p: a = b; q: a = b; \<alpha>: p = q\<rbrakk> \<Longrightarrow>
+    (refl a) \<bullet>\<^sub>l\<^bsub>b\<^esub> \<alpha> \<equiv> (lu p) \<bullet> \<alpha> \<bullet> (lu q)\<inverse>"
+  unfolding left_whisker_def by reduce
+
+method left_whisker = (rule left_whisker)
+method right_whisker = (rule right_whisker)
+
+
+section \<open>Horizontal path-composition\<close>
+
+text \<open>Conditions under which horizontal path-composition is defined.\<close>
+locale horiz_pathcomposable =
+fixes
+  i A a b c p q r s
+assumes assums:
+  "A: U i" "a: A" "b: A" "c: A"
+  "p: a =\<^bsub>A\<^esub> b" "q: a =\<^bsub>A\<^esub> b"
+  "r: b =\<^bsub>A\<^esub> c" "s: b =\<^bsub>A\<^esub> c"
+begin
+
+Lemma (derive) horiz_pathcomp:
+  notes assums
+  shows "\<lbrakk>\<alpha>: p = q; \<beta>: r = s\<rbrakk> \<Longrightarrow> ?prf \<alpha> \<beta>: p \<bullet> r = q \<bullet> s"
+proof (rule pathcomp)
+  show "\<alpha>: p = q \<Longrightarrow> p \<bullet> r = q \<bullet> r" by right_whisker
+  show "\<beta>: r = s \<Longrightarrow> .. = q \<bullet> s" by left_whisker
+qed typechk
+
+text \<open>A second horizontal composition:\<close>
+
+Lemma (derive) horiz_pathcomp':
+  notes assums
+  shows "\<lbrakk>\<alpha>: p = q; \<beta>: r = s\<rbrakk> \<Longrightarrow> ?prf \<alpha> \<beta>: p \<bullet> r = q \<bullet> s"
+proof (rule pathcomp)
+  show "\<beta>: r = s \<Longrightarrow> p \<bullet> r = p \<bullet> s" by left_whisker
+  show "\<alpha>: p = q \<Longrightarrow> .. = q \<bullet> s" by right_whisker
+qed typechk
+
+notation horiz_pathcomp (infix "\<star>" 121)
+notation horiz_pathcomp' (infix "\<star>''" 121)
+
+Lemma (derive) horiz_pathcomp_eq_horiz_pathcomp':
+  notes assums
+  shows "\<lbrakk>\<alpha>: p = q; \<beta>: r = s\<rbrakk> \<Longrightarrow> \<alpha> \<star> \<beta> = \<alpha> \<star>' \<beta>"
+  unfolding horiz_pathcomp_def horiz_pathcomp'_def
+  apply (eq \<alpha>, eq \<beta>)
+    focus vars p apply (eq p)
+      focus vars _ q by (eq q) (reduce; refl)
+    done
+  done
+
+end
+
+
+section \<open>Loop space\<close>
+
+definition \<Omega> where "\<Omega> A a \<equiv> a =\<^bsub>A\<^esub> a"
+definition \<Omega>2 where "\<Omega>2 A a \<equiv> \<Omega> (\<Omega> A a) (refl a)"
+
+Lemma \<Omega>2_alt_def:
+  "\<Omega>2 A a \<equiv> refl a = refl a"
+  unfolding \<Omega>2_def \<Omega>_def .
+
+
+section \<open>Eckmann-Hilton\<close>
+
+context fixes i A a assumes "A: U i" "a: A"
+begin
+
+interpretation \<Omega>2:
+  horiz_pathcomposable i A a a a "refl a" "refl a" "refl a" "refl a"
+  by unfold_locales typechk+
+
+notation \<Omega>2.horiz_pathcomp (infix "\<star>" 121)
+notation \<Omega>2.horiz_pathcomp' (infix "\<star>''" 121)
+
+Lemma (derive) pathcomp_eq_horiz_pathcomp:
+  assumes "\<alpha>: \<Omega>2 A a" "\<beta>: \<Omega>2 A a"
+  shows "\<alpha> \<bullet> \<beta> = \<alpha> \<star> \<beta>"
+  unfolding \<Omega>2.horiz_pathcomp_def
+  using assms[unfolded \<Omega>2_alt_def]
+  proof (reduce, rule pathinv) 
+    \<comment> \<open>Propositional equality rewriting needs to be improved\<close>
+    have "{} = refl (refl a) \<bullet> \<alpha>" by (rule pathcomp_refl)
+    also have ".. = \<alpha>" by (rule refl_pathcomp)
+    finally have eq\<alpha>: "{} = \<alpha>" by this
+
+    have "{} = refl (refl a) \<bullet> \<beta>" by (rule pathcomp_refl)
+    also have ".. = \<beta>" by (rule refl_pathcomp)
+    finally have eq\<beta>: "{} = \<beta>" by this
+
+    have "refl (refl a) \<bullet> \<alpha> \<bullet> refl (refl a) \<bullet> (refl (refl a) \<bullet> \<beta> \<bullet> refl (refl a))
+      = \<alpha> \<bullet> {}" by right_whisker (rule eq\<alpha>)
+    also have ".. = \<alpha> \<bullet> \<beta>" by left_whisker (rule eq\<beta>)
+    finally show "{} = \<alpha> \<bullet> \<beta>" by this
+  qed
+
+Lemma (derive) pathcomp_eq_horiz_pathcomp':
+  assumes "\<alpha>: \<Omega>2 A a" "\<beta>: \<Omega>2 A a"
+  shows "\<alpha> \<star>' \<beta> = \<beta> \<bullet> \<alpha>"
+  unfolding \<Omega>2.horiz_pathcomp'_def
+  using assms[unfolded \<Omega>2_alt_def]
+  proof reduce
+    have "{} = refl (refl a) \<bullet> \<beta>" by (rule pathcomp_refl)
+    also have ".. = \<beta>" by (rule refl_pathcomp)
+    finally have eq\<beta>: "{} = \<beta>" by this
+
+    have "{} = refl (refl a) \<bullet> \<alpha>" by (rule pathcomp_refl)
+    also have ".. = \<alpha>" by (rule refl_pathcomp)
+    finally have eq\<alpha>: "{} = \<alpha>" by this
+
+    have "refl (refl a) \<bullet> \<beta> \<bullet> refl (refl a) \<bullet> (refl (refl a) \<bullet> \<alpha> \<bullet> refl (refl a))
+      = \<beta> \<bullet> {}" by right_whisker (rule eq\<beta>)
+    also have ".. = \<beta> \<bullet> \<alpha>" by left_whisker (rule eq\<alpha>)
+    finally show "{} = \<beta> \<bullet> \<alpha>" by this
+  qed
+
+Lemma (derive) eckmann_hilton:
+  assumes "\<alpha>: \<Omega>2 A a" "\<beta>: \<Omega>2 A a"
+  shows "\<alpha> \<bullet> \<beta> = \<beta> \<bullet> \<alpha>"
+  using assms[unfolded \<Omega>2_alt_def]
+  proof -
+    have "\<alpha> \<bullet> \<beta> = \<alpha> \<star> \<beta>"
+      by (rule pathcomp_eq_horiz_pathcomp)
+    also have ".. = \<alpha> \<star>' \<beta>"
+      by (rule \<Omega>2.horiz_pathcomp_eq_horiz_pathcomp'[simplified comps])
+    also have ".. = \<beta> \<bullet> \<alpha>"
+      by (rule pathcomp_eq_horiz_pathcomp')
+    finally show "\<alpha> \<bullet> \<beta> = \<beta> \<bullet> \<alpha>"
+      by this (reduce add: \<Omega>2.horiz_pathcomp_def \<Omega>2.horiz_pathcomp'_def)
+  qed
+
+end
+
+
 end