clean up Eckmann-Hilton and move to Identity

1 files changed, 0 insertions, 188 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