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diff --git a/spartan/theories/Equivalence.thy b/spartan/theories/Equivalence.thy
deleted file mode 100644
index 2975738..0000000
--- a/spartan/theories/Equivalence.thy
+++ /dev/null
@@ -1,416 +0,0 @@
-theory Equivalence
-imports Identity
-
-begin
-
-section \<open>Homotopy\<close>
-
-definition "homotopy A B f g \<equiv> \<Prod>x: A. f `x =\<^bsub>B x\<^esub> g `x"
-
-definition homotopy_i (infix "~" 100)
-  where [implicit]: "f ~ g \<equiv> homotopy ? ? f g"
-
-translations "f ~ g" \<leftharpoondown> "CONST homotopy A B f g"
-
-Lemma homotopy_type [typechk]:
-  assumes
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> B x: U i"
-    "f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x"
-  shows "f ~ g: U i"
-  unfolding homotopy_def by typechk
-
-Lemma (derive) homotopy_refl [refl]:
-  assumes
-    "A: U i"
-    "f: \<Prod>x: A. B x"
-  shows "f ~ f"
-  unfolding homotopy_def by intros
-
-Lemma (derive) hsym:
-  assumes
-    "f: \<Prod>x: A. B x"
-    "g: \<Prod>x: A. B x"
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> B x: U i"
-  shows "H: f ~ g \<Longrightarrow> g ~ f"
-  unfolding homotopy_def
-  apply intros
-    apply (rule pathinv)
-      \<guillemotright> by (elim H)
-      \<guillemotright> by typechk
-  done
-
-Lemma (derive) htrans:
-  assumes
-    "f: \<Prod>x: A. B x"
-    "g: \<Prod>x: A. B x"
-    "h: \<Prod>x: A. B x"
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> B x: U i"
-  shows "\<lbrakk>H1: f ~ g; H2: g ~ h\<rbrakk> \<Longrightarrow> f ~ h"
-  unfolding homotopy_def
-  apply intro
-    \<guillemotright> vars x
-      apply (rule pathcomp[where ?y="g x"])
-        \<^item> by (elim H1)
-        \<^item> by (elim H2)
-      done
-    \<guillemotright> by typechk
-  done
-
-text \<open>For calculations:\<close>
-
-lemmas
-  homotopy_sym [sym] = hsym[rotated 4] and
-  homotopy_trans [trans] = htrans[rotated 5]
-
-Lemma (derive) commute_homotopy:
-  assumes
-    "A: U i" "B: U i"
-    "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
-    "f: A \<rightarrow> B" "g: A \<rightarrow> B"
-    "H: homotopy A (\<lambda>_. B) f g"
-  shows "(H x) \<bullet> g[p] = f[p] \<bullet> (H y)"
-  \<comment> \<open>Need this assumption unfolded for typechecking:\<close>
-  supply assms(8)[unfolded homotopy_def]
-  apply (eq p)
-    focus vars x
-      apply reduce
-        \<comment> \<open>Here it would really be nice to have automation for transport and
-          propositional equality.\<close>
-        apply (rule use_transport[where ?y="H x \<bullet> refl (g x)"])
-          \<guillemotright> by (rule pathcomp_right_refl)
-          \<guillemotright> by (rule pathinv) (rule pathcomp_left_refl)
-          \<guillemotright> by typechk
-    done
-  done
-
-Corollary (derive) commute_homotopy':
-  assumes
-    "A: U i"
-    "x: A"
-    "f: A \<rightarrow> A"
-    "H: homotopy A (\<lambda>_. A) f (id A)"
-  shows "H (f x) = f [H x]"
-oops
-
-Lemma homotopy_id_left [typechk]:
-  assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
-  shows "homotopy_refl A f: (id B) \<circ> f ~ f"
-  unfolding homotopy_refl_def homotopy_def by reduce
-
-Lemma homotopy_id_right [typechk]:
-  assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
-  shows "homotopy_refl A f: f \<circ> (id A) ~ f"
-  unfolding homotopy_refl_def homotopy_def by reduce
-
-Lemma homotopy_funcomp_left:
-  assumes
-    "H: homotopy B C g g'"
-    "f: A \<rightarrow> B"
-    "g: \<Prod>x: B. C x"
-    "g': \<Prod>x: B. C x"
-    "A: U i" "B: U i"
-    "\<And>x. x: B \<Longrightarrow> C x: U i"
-  shows "homotopy A {} (g \<circ>\<^bsub>A\<^esub> f) (g' \<circ>\<^bsub>A\<^esub> f)"
-  unfolding homotopy_def
-  apply (intro; reduce)
-    apply (insert \<open>H: _\<close>[unfolded homotopy_def])
-      apply (elim H)
-  done
-
-Lemma homotopy_funcomp_right:
-  assumes
-    "H: homotopy A (\<lambda>_. B) f f'"
-    "f: A \<rightarrow> B"
-    "f': A \<rightarrow> B"
-    "g: B \<rightarrow> C"
-    "A: U i" "B: U i" "C: U i"
-  shows "homotopy A {} (g \<circ>\<^bsub>A\<^esub> f) (g \<circ>\<^bsub>A\<^esub> f')"
-  unfolding homotopy_def
-  apply (intro; reduce)
-    apply (insert \<open>H: _\<close>[unfolded homotopy_def])
-      apply (dest PiE, assumption)
-      apply (rule ap, assumption)
-  done
-
-
-section \<open>Quasi-inverse and bi-invertibility\<close>
-
-subsection \<open>Quasi-inverses\<close>
-
-definition "qinv A B f \<equiv> \<Sum>g: B \<rightarrow> A.
-  homotopy A (\<lambda>_. A) (g \<circ>\<^bsub>A\<^esub> f) (id A) \<times> homotopy B (\<lambda>_. B) (f \<circ>\<^bsub>B\<^esub> g) (id B)"
-
-lemma qinv_type [typechk]:
-  assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
-  shows "qinv A B f: U i"
-  unfolding qinv_def by typechk
-
-definition qinv_i ("qinv")
-  where [implicit]: "qinv f \<equiv> Equivalence.qinv ? ? f"
-
-translations "qinv f" \<leftharpoondown> "CONST Equivalence.qinv A B f"
-
-Lemma (derive) id_qinv:
-  assumes "A: U i"
-  shows "qinv (id A)"
-  unfolding qinv_def
-  apply intro defer
-    apply intro defer
-      apply (rule homotopy_id_right)
-      apply (rule homotopy_id_left)
-  done
-
-Lemma (derive) quasiinv_qinv:
-  assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
-  shows "prf: qinv f \<Longrightarrow> qinv (fst prf)"
-  unfolding qinv_def
-  apply intro
-    \<guillemotright> by (rule \<open>f:_\<close>)
-    \<guillemotright> apply (elim "prf")
-        focus vars g HH
-          apply intro
-            \<^item> by reduce (snd HH)
-            \<^item> by reduce (fst HH)
-        done
-      done
-  done
-
-Lemma (derive) funcomp_qinv:
-  assumes
-    "A: U i" "B: U i" "C: U i"
-    "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
-
-subsection \<open>Bi-invertible maps\<close>
-
-definition "biinv A B f \<equiv>
-  (\<Sum>g: B \<rightarrow> A. homotopy A (\<lambda>_. A) (g \<circ>\<^bsub>A\<^esub> f) (id A))
-    \<times> (\<Sum>g: B \<rightarrow> A. homotopy B (\<lambda>_. B) (f \<circ>\<^bsub>B\<^esub> g) (id B))"
-
-lemma biinv_type [typechk]:
-  assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
-  shows "biinv A B f: U i"
-  unfolding biinv_def by typechk
-
-definition biinv_i ("biinv")
-  where [implicit]: "biinv f \<equiv> Equivalence.biinv ? ? f"
-
-translations "biinv f" \<leftharpoondown> "CONST Equivalence.biinv A B f"
-
-Lemma (derive) qinv_imp_biinv:
-  assumes
-    "A: U i" "B: U i"
-    "f: A \<rightarrow> B"
-  shows "?prf: qinv f \<rightarrow> biinv f"
-  apply intros
-  unfolding qinv_def biinv_def
-  by (rule Sig_dist_exp)
-
-text \<open>
-  Show that bi-invertible maps are quasi-inverses, as a demonstration of how to
-  work in this system.
-\<close>
-
-Lemma (derive) biinv_imp_qinv:
-  assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
-  shows "biinv f \<rightarrow> qinv f"
-
-  text \<open>Split the hypothesis \<^term>\<open>biinv f\<close> into its components:\<close>
-  apply intro
-  unfolding biinv_def
-  apply elims
-
-  text \<open>Name the components:\<close>
-  focus prems vars _ _ _ g H1 h H2
-  thm \<open>g:_\<close> \<open>h:_\<close> \<open>H1:_\<close> \<open>H2:_\<close>
-
-  text \<open>
-    \<^term>\<open>g\<close> is a quasi-inverse to \<^term>\<open>f\<close>, so the proof will be a triple
-    \<^term>\<open><g, <?H1, ?H2>>\<close>.
-  \<close>
-  unfolding qinv_def
-  apply intro
-    \<guillemotright> by (rule \<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>)
-
-      text \<open>
-        It remains to prove \<^prop>\<open>?H2: f \<circ> g ~ id B\<close>. First show that \<open>g ~ h\<close>,
-        then the result follows from @{thm \<open>H2: f \<circ> h ~ id B\<close>}. Here a proof
-        block is used to calculate "forward".
-      \<close>
-      proof -
-        have "g ~ g \<circ> (id B)" by reduce refl
-        also have "g \<circ> (id B) ~ g \<circ> f \<circ> h"
-          by (rule homotopy_funcomp_right) (rule \<open>H2:_\<close>[symmetric])
-        also have "g \<circ> f \<circ> h ~ (id A) \<circ> h"
-          by (subst funcomp_assoc[symmetric])
-             (rule homotopy_funcomp_left, rule \<open>H1:_\<close>)
-        also have "(id A) \<circ> h ~ 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)
-      qed
-    done
-  done
-
-Lemma (derive) id_biinv:
-  "A: U i \<Longrightarrow> biinv (id A)"
-    by (rule qinv_imp_biinv) (rule id_qinv)
-
-Lemma (derive) funcomp_biinv:
-  assumes
-    "A: U i" "B: U i" "C: U i"
-    "f: A \<rightarrow> B" "g: B \<rightarrow> C"
-  shows "biinv f \<rightarrow> biinv g \<rightarrow> biinv (g \<circ> f)"
-  apply intros
-  focus vars biinv\<^sub>f biinv\<^sub>g
-
-  text \<open>Follows from \<open>funcomp_qinv\<close>.\<close>
-oops
-
-
-section \<open>Equivalence\<close>
-
-text \<open>
-  Following the HoTT book, we first define equivalence in terms of
-  bi-invertibility.
-\<close>
-
-definition equivalence (infix "\<simeq>" 110)
-  where "A \<simeq> B \<equiv> \<Sum>f: A \<rightarrow> B. Equivalence.biinv A B f"
-
-lemma equivalence_type [typechk]:
-  assumes "A: U i" "B: U i"
-  shows "A \<simeq> B: U i"
-  unfolding equivalence_def by typechk
-
-Lemma (derive) equivalence_refl:
-  assumes "A: U i"
-  shows "A \<simeq> A"
-  unfolding equivalence_def
-  apply intro defer
-    apply (rule qinv_imp_biinv) defer
-      apply (rule id_qinv)
-  done
-
-text \<open>
-  The following could perhaps be easier with transport (but then I think we need
-  univalence?)...
-\<close>
-
-Lemma (derive) equivalence_symmetric:
-  assumes "A: U i" "B: U i"
-  shows "A \<simeq> B \<rightarrow> B \<simeq> A"
-  apply intros
-  unfolding equivalence_def
-  apply elim
-  \<guillemotright> vars _ f "prf"
-    apply (dest (4) biinv_imp_qinv)
-    apply intro
-      \<^item> unfolding qinv_def by (rule fst[of "(biinv_imp_qinv A B f) prf"])
-      \<^item> by (rule qinv_imp_biinv) (rule quasiinv_qinv)
-    done
-  done
-
-Lemma (derive) equivalence_transitive:
-  assumes "A: U i" "B: U i" "C: U i"
-  shows "A \<simeq> B \<rightarrow> B \<simeq> C \<rightarrow> A \<simeq> C"
-  apply intros
-  unfolding equivalence_def
-
-  text \<open>Use \<open>funcomp_biinv\<close>.\<close>
-oops
-
-text \<open>
-  Equal types are equivalent. We give two proofs: the first by induction, and
-  the second by following the HoTT book and showing that transport is an
-  equivalence.
-\<close>
-
-Lemma
-  assumes
-    "A: U i" "B: U i" "p: A =\<^bsub>U i\<^esub> B"
-  shows "A \<simeq> B"
-  by (eq p) (rule equivalence_refl)
-
-text \<open>
-  The following proof is wordy because (1) the typechecker doesn't rewrite, and
-  (2) we don't yet have universe automation.
-\<close>
-
-Lemma (derive) id_imp_equiv:
-  assumes
-    "A: U i" "B: U i" "p: A =\<^bsub>U i\<^esub> B"
-  shows "<trans (id (U i)) p, ?isequiv>: A \<simeq> B"
-  unfolding equivalence_def
-  apply intros defer
-
-  \<comment> \<open>Switch off auto-typechecking, which messes with universe levels\<close>
-  supply [[auto_typechk=false]]
-
-    \<guillemotright> apply (eq p, typechk)
-      \<^item> prems vars A B
-        apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric])
-        apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric])
-        apply (rule transport, rule U_in_U)
-        apply (rule lift_universe_codomain, rule U_in_U)
-        apply (typechk, rule U_in_U)
-        done
-      \<^item> prems vars A
-        apply (subst transport_comp)
-          \<^enum> by (rule U_in_U)
-          \<^enum> by reduce (rule lift_universe)
-          \<^enum> by reduce (rule id_biinv)
-        done
-      done
-
-    \<guillemotright> \<comment> \<open>Similar proof as in the first subgoal above\<close>
-      apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric])
-      apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric])
-      apply (rule transport, rule U_in_U)
-      apply (rule lift_universe_codomain, rule U_in_U)
-      apply (typechk, rule U_in_U)
-      done
-  done
-
-(*Uncomment this to see all implicits from here on.
-no_translations
-  "f x" \<leftharpoondown> "f `x"
-  "x = y" \<leftharpoondown> "x =\<^bsub>A\<^esub> y"
-  "g \<circ> f" \<leftharpoondown> "g \<circ>\<^bsub>A\<^esub> f"
-  "p\<inverse>" \<leftharpoondown> "CONST pathinv A x y p"
-  "p \<bullet> q" \<leftharpoondown> "CONST pathcomp A x y z p q"
-  "fst" \<leftharpoondown> "CONST Spartan.fst A B"
-  "snd" \<leftharpoondown> "CONST Spartan.snd A B"
-  "f[p]" \<leftharpoondown> "CONST ap A B x y f p"
-  "trans P p" \<leftharpoondown> "CONST transport A P x y p"
-  "trans_const B p" \<leftharpoondown> "CONST transport_const A B x y p"
-  "lift P p u" \<leftharpoondown> "CONST pathlift A P x y p u"
-  "apd f p" \<leftharpoondown> "CONST Identity.apd A P f x y p"
-  "f ~ g" \<leftharpoondown> "CONST homotopy A B f g"
-  "qinv f" \<leftharpoondown> "CONST Equivalence.qinv A B f"
-  "biinv f" \<leftharpoondown> "CONST Equivalence.biinv A B f"
-*)
-
-
-end