prune import lists

3 files changed, 10 insertions, 153 deletions

diff --git a/Projections.thy b/Projections.thy
index 9eeb57f..a28c66b 100644
--- a/Projections.thy
+++ b/Projections.thy
@@ -7,7 +7,7 @@ Projection functions for the dependent sum type.
 ********)
 
 theory Projections
-imports HoTT_Methods Prod Sum
+imports Prod Sum
 
 begin
 
diff --git a/Type_Families.thy b/Type_Families.thy
index 6c784e5..b9e1049 100644
--- a/Type_Families.thy
+++ b/Type_Families.thy
@@ -7,7 +7,7 @@ Various results viewing type families as fibrations: transport, path lifting, de
 ********)
 
 theory Type_Families
-imports HoTT_Methods Sum Projections Eq
+imports Eq Projections
 
 begin
 
diff --git a/Univalence.thy b/Univalence.thy
index 322dbbd..dd8a13c 100644
--- a/Univalence.thy
+++ b/Univalence.thy
@@ -5,158 +5,14 @@ Feb 2019
 ********)
 
 theory Univalence
-imports HoTT_Methods Prod Sum Eq Type_Families
+imports Equivalence
 
 begin
 
 
-section \<open>Homotopy\<close>
-
-definition homotopic :: "[t, t \<Rightarrow> t, t, t] \<Rightarrow> t"  ("(2homotopic[_, _] _ _)" [0, 0, 1000, 1000])
-where "homotopic[A, B] f g \<equiv> \<Prod>x: A. f`x =[B x] g`x"
-
-declare homotopic_def [comp]
-
-syntax "_homotopic" :: "[t, idt, t, t, t] \<Rightarrow> t"  ("(1_ ~[_: _. _]/ _)" [101, 0, 0, 0, 101] 100)
-translations "f ~[x: A. B] g" \<rightleftharpoons> "(CONST homotopic) A (\<lambda>x. B) f g"
-
-(*
-syntax "_homotopic'" :: "[t, t] \<Rightarrow> t"  ("(2_ ~ _)" [1000, 1000])
-
-ML \<open>val pretty_homotopic = Attrib.setup_config_bool @{binding "pretty_homotopic"} (K true)\<close>
-
-print_translation \<open>
-let fun homotopic_tr' ctxt [A, B, f, g] =
-  if Config.get ctxt pretty_homotopic
-  then Syntax.const @{syntax_const "_homotopic'"} $ f $ g
-  else @{const homotopic} $ A $ B $ f $ g
-in
-  [(@{const_syntax homotopic}, homotopic_tr')]
-end
-\<close>
-*)
-
-lemma homotopic_type:
-  assumes [intro]: "A: U i" "B: A \<leadsto> U i" "f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x"
-  shows "f ~[x: A. B x] g: U i"
-by derive
-
-declare homotopic_type [intro]
-
-schematic_goal fun_eq_imp_homotopic:
-  assumes [intro]:
-    "p: f =[\<Prod>x: A. B x] g"
-    "f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x"
-    "A: U i" "B: A \<leadsto> U i"
-  shows "?prf: f ~[x: A. B x] g"
-proof (path_ind' f g p)
-  show "\<And>f. f : \<Prod>(x: A). B x \<Longrightarrow> \<lambda>x: A. refl(f`x): f ~[x: A. B x] f" by derive
-qed routine
-
-definition happly :: "[t, t \<Rightarrow> t, t, t, t] \<Rightarrow> t"
-where "happly A B f g p \<equiv> indEq (\<lambda>f g. & f ~[x: A. B x] g) (\<lambda>f. \<lambda>(x: A). refl(f`x)) f g p"
-
-syntax "_happly" :: "[idt, t, t, t, t, t] \<Rightarrow> t"
-  ("(2happly[_: _. _] _ _ _)" [0, 0, 0, 1000, 1000, 1000])
-translations "happly[x: A. B] f g p"  \<rightleftharpoons> "(CONST happly) A (\<lambda>x. B) f g p"
-
-corollary happly_type:
-  assumes [intro]:
-    "p: f =[\<Prod>x: A. B x] g"
-    "f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x"
-    "A: U i" "B: A \<leadsto> U i"
-  shows "happly[x: A. B x] f g p: f ~[x: A. B x] g"
-unfolding happly_def by (derive lems: fun_eq_imp_homotopic)
-
-
-section \<open>Equivalence\<close>
-
-text \<open>For now, we define equivalence in terms of bi-invertibility.\<close>
-
-definition biinv :: "[t, t, t] \<Rightarrow> t"  ("(2biinv[_, _]/ _)")
-where "biinv[A, B] f \<equiv>
-  (\<Sum>g: B \<rightarrow> A. g o[A] f ~[x:A. A] id A) \<times> (\<Sum>g: B \<rightarrow> A. f o[B] g ~[x: B. B] id B)"
-
-text \<open>
-The meanings of the syntax defined above are:
-\<^item> @{term "f ~[x: A. B x] g"} expresses that @{term f} and @{term g} are homotopic functions of type @{term "\<Prod>x:A. B x"}.
-\<^item> @{term "biinv[A, B] f"} expresses that the function @{term f} of type @{term "A \<rightarrow> B"} is bi-invertible.
-\<close>
-
-lemma biinv_type:
-  assumes [intro]: "A: U i" "B: U i" "f: A \<rightarrow> B"
-  shows "biinv[A, B] f: U i"
-unfolding biinv_def by derive
-
-declare biinv_type [intro]
-
-schematic_goal id_is_biinv:
-  assumes [intro]: "A: U i"
-  shows "?prf: biinv[A, A] (id A)"
-unfolding biinv_def proof (rule Sum_routine, compute)
-  show "<id A, \<lambda>x: A. refl x>: \<Sum>(g: A \<rightarrow> A). (g o[A] id A) ~[x: A. A] (id A)" by derive
-  show "<id A, \<lambda>x: A. refl x>: \<Sum>(g: A \<rightarrow> A). (id A o[A] g) ~[x: A. A] (id A)" by derive
-qed routine
-
-definition equivalence :: "[t, t] \<Rightarrow> t"  (infix "\<simeq>" 100)
-where "A \<simeq> B \<equiv> \<Sum>f: A \<rightarrow> B. biinv[A, B] f"
-
-schematic_goal equivalence_symmetric:
-  assumes [intro]: "A: U i"
-  shows "?prf: A \<simeq> A"
-unfolding equivalence_def proof (rule Sum_routine)
-  show "\<And>f. f : A \<rightarrow> A \<Longrightarrow> biinv[A, A] f : U i" unfolding biinv_def by derive
-  show "id A: A \<rightarrow> A" by routine
-qed (routine add: id_is_biinv)
-
-
-section \<open>Transport, homotopy, and bi-invertibility\<close>
-
-schematic_goal transport_invl_hom:
-  assumes [intro]:
-    "P: A \<leadsto> U j" "A: U i"
-    "x: A" "y: A" "p: x =[A] y"
-  shows "?prf:
-    (transport[A, P, y, x]`(inv[A, x, y]`p)) o[P x] (transport[A, P, x, y]`p) ~[w: P x. P x] id P x"
-by (rule happly_type[OF transport_invl], derive)
-
-schematic_goal transport_invr_hom:
-  assumes [intro]:
-    "A: U i" "P: A \<leadsto> U j" 
-    "y: A" "x: A" "p: x =[A] y"
-  shows "?prf:
-    (transport[A, P, x, y]`p) o[P y] (transport[A, P, y, x]`(inv[A, x, y]`p)) ~[w: P y. P y] id P y"
-by (rule happly_type[OF transport_invr], derive)
-
-declare
-  transport_invl_hom [intro]
-  transport_invr_hom [intro]
-
-text \<open>
-The following result states that the transport of an equality @{term p} is bi-invertible, with inverse given by the transport of the inverse @{text "~p"}.
-It is used in the derivation of @{text idtoeqv} in the next section.
-\<close>
-
-schematic_goal transport_biinv:
-  assumes [intro]: "p: A =[U i] B" "A: U i" "B: U i"
-  shows "?prf: biinv[A, B] (transport[U i, Id, A, B]`p)"
-unfolding biinv_def
-apply (rule Sum_routine)
-prefer 2
-  apply (rule Sum_routine)
-  prefer 3 apply (rule transport_invl_hom)
-prefer 9
-  apply (rule Sum_routine)
-  prefer 3 apply (rule transport_invr_hom)
-\<comment> \<open>The remaining subgoals are now handled easily\<close>
-by derive
-
-
-section \<open>Univalence\<close>
-
 schematic_goal type_eq_imp_equiv:
   assumes [intro]: "A: U i" "B: U i"
-  shows "?prf: A =[U i] B \<rightarrow> A \<simeq> B"
+  shows "?prf: A =[U i] B \<rightarrow> A \<cong> B"
 unfolding equivalence_def
 apply (rule Prod_routine, rule Sum_routine)
 prefer 3 apply (derive lems: transport_biinv)
@@ -195,15 +51,16 @@ idtoeqv[i, A, B] \<equiv>
 
 corollary idtoeqv_type:
   assumes [intro]: "A: U i" "B: U i" "p: A =[U i] B"
-  shows "idtoeqv[i, A, B]: A =[U i] B \<rightarrow> A \<simeq> B"
+  shows "idtoeqv[i, A, B]: A =[U i] B \<rightarrow> A \<cong> B"
 unfolding idtoeqv_def by (routine add: type_eq_imp_equiv)
 
 declare idtoeqv_type [intro]
 
-(* I'll formalize a more explicit constructor for the inverse equivalence of idtoeqv later;
-   the following is a placeholder for now.
-*)
-axiomatization ua :: "[ord, t, t] \<Rightarrow> t" where univalence: "ua i A B: biinv[A, B] idtoeqv[i, A, B]"
+
+text \<open>For now, we use bi-invertibility as our definition of equivalence.\<close>
+
+axiomatization univalance :: "[ord, t, t] \<Rightarrow> t"
+where univalence: "univalence i A B: biinv[A, B] idtoeqv[i, A, B]"
 
 
 end