Reorganized HoTT_Base, updated theories

16 files changed, 635 insertions, 207 deletions

diff --git a/Coprod.thy b/Coprod.thy
index 4607d1d..97e0566 100644
--- a/Coprod.thy
+++ b/Coprod.thy
@@ -12,45 +12,45 @@ begin
 section \<open>Constants and type rules\<close>
 
 axiomatization
-  Coprod :: "[Term, Term] \<Rightarrow> Term"  (infixr "+" 50) and
-  inl :: "Term \<Rightarrow> Term" and
-  inr :: "Term \<Rightarrow> Term" and
-  indCoprod :: "[Term \<Rightarrow> Term, Term \<Rightarrow> Term, Term] \<Rightarrow> Term"  ("(1ind\<^sub>+)")
+  Coprod :: "[t, t] \<Rightarrow> t"  (infixr "+" 50) and
+  inl :: "t \<Rightarrow> t" and
+  inr :: "t \<Rightarrow> t" and
+  indCoprod :: "[t \<Rightarrow> t, t \<Rightarrow> t, t] \<Rightarrow> t"  ("(1ind\<^sub>+)")
 where
-  Coprod_form: "\<lbrakk>A: U(i); B: U(i)\<rbrakk> \<Longrightarrow> A + B: U(i)"
+  Coprod_form: "\<lbrakk>A: U i; B: U i\<rbrakk> \<Longrightarrow> A + B: U i"
 and
-  Coprod_intro_inl: "\<lbrakk>a: A; B: U(i)\<rbrakk> \<Longrightarrow> inl(a): A + B"
+  Coprod_intro_inl: "\<lbrakk>a: A; B: U i\<rbrakk> \<Longrightarrow> inl a: A + B"
 and
-  Coprod_intro_inr: "\<lbrakk>b: B; A: U(i)\<rbrakk> \<Longrightarrow> inr(b): A + B"
+  Coprod_intro_inr: "\<lbrakk>b: B; A: U i\<rbrakk> \<Longrightarrow> inr b: A + B"
 and
   Coprod_elim: "\<lbrakk>
-    C: A + B \<longrightarrow> U(i);
-    \<And>x. x: A \<Longrightarrow> c(x): C(inl(x));
-    \<And>y. y: B \<Longrightarrow> d(y): C(inr(y));
+    C: A + B \<longrightarrow> U i;
+    \<And>x. x: A \<Longrightarrow> c x: C (inl x);
+    \<And>y. y: B \<Longrightarrow> d y: C (inr y);
     u: A + B
-    \<rbrakk> \<Longrightarrow> ind\<^sub>+(c)(d)(u) : C(u)"
+    \<rbrakk> \<Longrightarrow> ind\<^sub>+ c d u: C u"
 and
   Coprod_comp_inl: "\<lbrakk>
-    C: A + B \<longrightarrow> U(i);
-    \<And>x. x: A \<Longrightarrow> c(x): C(inl(x));
-    \<And>y. y: B \<Longrightarrow> d(y): C(inr(y));
+    C: A + B \<longrightarrow> U i;
+    \<And>x. x: A \<Longrightarrow> c x: C (inl x);
+    \<And>y. y: B \<Longrightarrow> d y: C (inr y);
     a: A
-    \<rbrakk> \<Longrightarrow> ind\<^sub>+(c)(d)(inl(a)) \<equiv> c(a)"
+    \<rbrakk> \<Longrightarrow> ind\<^sub>+ c d (inl a) \<equiv> c a"
 and
   Coprod_comp_inr: "\<lbrakk>
-    C: A + B \<longrightarrow> U(i);
-    \<And>x. x: A \<Longrightarrow> c(x): C(inl(x));
-    \<And>y. y: B \<Longrightarrow> d(y): C(inr(y));
+    C: A + B \<longrightarrow> U i;
+    \<And>x. x: A \<Longrightarrow> c x: C (inl x);
+    \<And>y. y: B \<Longrightarrow> d y: C (inr y);
     b: B
-    \<rbrakk> \<Longrightarrow> ind\<^sub>+(c)(d)(inr(b)) \<equiv> d(b)"
+    \<rbrakk> \<Longrightarrow> ind\<^sub>+ c d (inr b) \<equiv> d b"
 
 
 text "Admissible formation inference rules:"
 
 axiomatization where
-  Coprod_wellform1: "A + B: U(i) \<Longrightarrow> A: U(i)"
+  Coprod_wellform1: "A + B: U i \<Longrightarrow> A: U i"
 and
-  Coprod_wellform2: "A + B: U(i) \<Longrightarrow> B: U(i)"
+  Coprod_wellform2: "A + B: U i \<Longrightarrow> B: U i"
 
 
 text "Rule attribute declarations:"
diff --git a/Empty.thy b/Empty.thy
index 1b339ba..107f2d7 100644
--- a/Empty.thy
+++ b/Empty.thy
@@ -14,12 +14,12 @@ section \<open>Constants and type rules\<close>
 section \<open>Empty type\<close>
 
 axiomatization
-  Empty :: Term  ("\<zero>") and
-  indEmpty :: "Term \<Rightarrow> Term"  ("(1ind\<^sub>\<zero>)")
+  Empty :: t  ("\<zero>") and
+  indEmpty :: "t \<Rightarrow> t"  ("(1ind\<^sub>\<zero>)")
 where
-  Empty_form: "\<zero> : U(O)"
+  Empty_form: "\<zero>: U O"
 and
-  Empty_elim: "\<lbrakk>C: \<zero> \<longrightarrow> U(i); z: \<zero>\<rbrakk> \<Longrightarrow> ind\<^sub>\<zero>(z): C(z)"
+  Empty_elim: "\<lbrakk>C: \<zero> \<longrightarrow> U i; z: \<zero>\<rbrakk> \<Longrightarrow> ind\<^sub>\<zero> z: C z"
 
 
 text "Rule attribute declarations:"
diff --git a/Equal.thy b/Equal.thy
index 772072a..7b6acb5 100644
--- a/Equal.thy
+++ b/Equal.thy
@@ -12,13 +12,13 @@ begin
 section \<open>Constants and syntax\<close>
 
 axiomatization
-  Equal :: "[Term, Term, Term] \<Rightarrow> Term" and
-  refl :: "Term \<Rightarrow> Term" and
-  indEqual :: "[Term \<Rightarrow> Term, Term] \<Rightarrow> Term"  ("(1ind\<^sub>=)")
+  Equal :: "[t, t, t] \<Rightarrow> t" and
+  refl :: "t \<Rightarrow> t" and
+  indEqual :: "[t \<Rightarrow> t, t] \<Rightarrow> t"  ("(1ind\<^sub>=)")
 
 syntax
-  "_EQUAL" :: "[Term, Term, Term] \<Rightarrow> Term"        ("(3_ =\<^sub>_/ _)" [101, 0, 101] 100)
-  "_EQUAL_ASCII" :: "[Term, Term, Term] \<Rightarrow> Term"  ("(3_ =[_]/ _)" [101, 0, 101] 100)
+  "_EQUAL" :: "[t, t, t] \<Rightarrow> t"        ("(3_ =\<^sub>_/ _)" [101, 0, 101] 100)
+  "_EQUAL_ASCII" :: "[t, t, t] \<Rightarrow> t"  ("(3_ =[_]/ _)" [101, 0, 101] 100)
 translations
   "a =[A] b" \<rightleftharpoons> "CONST Equal A a b"
   "a =\<^sub>A b" \<rightharpoonup> "CONST Equal A a b"
@@ -27,33 +27,33 @@ translations
 section \<open>Type rules\<close>
 
 axiomatization where
-  Equal_form: "\<lbrakk>A: U(i); a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U(i)"
+  Equal_form: "\<lbrakk>A: U i; a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U i"
 and
-  Equal_intro: "a : A \<Longrightarrow> refl(a): a =\<^sub>A a"
+  Equal_intro: "a : A \<Longrightarrow> (refl a): a =\<^sub>A a"
 and
   Equal_elim: "\<lbrakk>
     x: A;
     y: A;
     p: x =\<^sub>A y;
-    \<And>x. x: A \<Longrightarrow> f(x) : C(x)(x)(refl x);
-    \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C(x)(y): x =\<^sub>A y \<longrightarrow> U(i)
-    \<rbrakk> \<Longrightarrow> ind\<^sub>=(f)(p) : C(x)(y)(p)"
+    \<And>x. x: A \<Longrightarrow> f x: C x x (refl x);
+    \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U i
+    \<rbrakk> \<Longrightarrow> ind\<^sub>= f p : C x y p"
 and
   Equal_comp: "\<lbrakk>
     a: A;
-    \<And>x. x: A \<Longrightarrow> f(x) : C(x)(x)(refl x);
-    \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C(x)(y): x =\<^sub>A y \<longrightarrow> U(i)
-    \<rbrakk> \<Longrightarrow> ind\<^sub>=(f)(refl(a)) \<equiv> f(a)"
+    \<And>x. x: A \<Longrightarrow> f x: C x x (refl x);
+    \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U i
+    \<rbrakk> \<Longrightarrow> ind\<^sub>= f (refl a) \<equiv> f a"
 
 
 text "Admissible inference rules for equality type formation:"
 
 axiomatization where
-  Equal_wellform1: "a =\<^sub>A b: U(i) \<Longrightarrow> A: U(i)"
+  Equal_wellform1: "a =\<^sub>A b: U i \<Longrightarrow> A: U i"
 and
-  Equal_wellform2: "a =\<^sub>A b: U(i) \<Longrightarrow> a: A"
+  Equal_wellform2: "a =\<^sub>A b: U i \<Longrightarrow> a: A"
 and
-  Equal_wellform3: "a =\<^sub>A b: U(i) \<Longrightarrow> b: A"
+  Equal_wellform3: "a =\<^sub>A b: U i \<Longrightarrow> b: A"
 
 
 text "Rule attribute declarations:"
diff --git a/EqualProps.thy b/EqualProps.thy
index be8e53e..19c788c 100644
--- a/EqualProps.thy
+++ b/EqualProps.thy
@@ -14,7 +14,7 @@ begin
 
 section \<open>Symmetry / Path inverse\<close>
 
-definition inv :: "Term \<Rightarrow> Term"  ("_\<inverse>" [1000] 1000) where "p\<inverse> \<equiv> ind\<^sub>= (\<lambda>x. refl(x)) p"
+definition inv :: "t \<Rightarrow> t"  ("_\<inverse>" [1000] 1000) where "p\<inverse> \<equiv> ind\<^sub>= (\<lambda>x. (refl x)) p"
 
 text "
   In the proof below we begin by using path induction on \<open>p\<close> with the application of \<open>rule Equal_elim\<close>, telling Isabelle the specific substitutions to use.
@@ -22,7 +22,7 @@ text "
 "
 
 lemma inv_type:
-  assumes "A : U(i)" and "x : A" and "y : A" and "p: x =\<^sub>A y" shows "p\<inverse>: y =\<^sub>A x"
+  assumes "A : U i" and "x : A" and "y : A" and "p: x =\<^sub>A y" shows "p\<inverse>: y =\<^sub>A x"
 unfolding inv_def
 by (rule Equal_elim[where ?x=x and ?y=y]) (routine lems: assms)
   \<comment> \<open>The type doesn't depend on \<open>p\<close> so we don't need to specify \<open>?p\<close> in the \<open>where\<close> clause above.\<close>
@@ -33,7 +33,7 @@ text "
 "
 
 lemma inv_comp:
-  assumes "A : U(i)" and "a : A" shows "(refl(a))\<inverse> \<equiv> refl(a)"
+  assumes "A : U i " and "a : A" shows "(refl a)\<inverse> \<equiv> refl a"
 unfolding inv_def
 proof compute
   show "\<And>x. x: A \<Longrightarrow> refl x: x =\<^sub>A x" ..
@@ -46,14 +46,14 @@ text "
   Raw composition function, of type \<open>\<Prod>x:A. \<Prod>y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)\<close> polymorphic over the type \<open>A\<close>.
 "
 
-definition rpathcomp :: Term where "rpathcomp \<equiv> \<^bold>\<lambda>_ _ p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>_ q. ind\<^sub>= (\<lambda>x. refl(x)) q) p"
+definition rpathcomp :: t where "rpathcomp \<equiv> \<^bold>\<lambda>_ _ p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>_ q. ind\<^sub>= (\<lambda>x. (refl x)) q) p"
 
 text "
   More complicated proofs---the nested path inductions require more explicit step-by-step rule applications:
 "
 
 lemma rpathcomp_type:
-  assumes "A: U(i)"
+  assumes "A: U i"
   shows "rpathcomp: \<Prod>x:A. \<Prod>y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)"
 unfolding rpathcomp_def
 proof
@@ -81,7 +81,7 @@ proof
 qed fact
 
 corollary
-  assumes "A: U(i)" "x: A" "y: A" "z: A" "p: x =\<^sub>A y" "q: y =\<^sub>A z"
+  assumes "A: U i" "x: A" "y: A" "z: A" "p: x =\<^sub>A y" "q: y =\<^sub>A z"
   shows "rpathcomp`x`y`p`z`q: x =\<^sub>A z"
   by (routine lems: assms rpathcomp_type)
 
@@ -90,11 +90,11 @@ text "
 "
 
 lemma rpathcomp_comp:
-  assumes "A: U(i)" and "a: A"
-  shows "rpathcomp`a`a`refl(a)`a`refl(a) \<equiv> refl(a)"
+  assumes "A: U i" and "a: A"
+  shows "rpathcomp`a`a`(refl a)`a`(refl a) \<equiv> refl a"
 unfolding rpathcomp_def
 proof compute
-  { fix x assume 1: "x: A"
+  fix x assume 1: "x: A"
   show "\<^bold>\<lambda>y p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p: \<Prod>y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)"
   proof
     fix y assume 2: "y: A"
@@ -114,11 +114,11 @@ proof compute
         qed (rule assms)
       qed (routine lems: assms 1 2 3)
     qed (routine lems: assms 1 2)
-  qed (rule assms) }
+  qed (rule assms)
 
-  show "(\<^bold>\<lambda>y p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p)`a`refl(a)`a`refl(a) \<equiv> refl(a)"
+  next show "(\<^bold>\<lambda>y p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p)`a`(refl a)`a`(refl a) \<equiv> refl a"
   proof compute
-    { fix y assume 1: "y: A"
+    fix y assume 1: "y: A"
     show "\<^bold>\<lambda>p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p: a =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> a =\<^sub>A z)"
     proof
       fix p assume 2: "p: a =\<^sub>A y"
@@ -135,11 +135,11 @@ proof compute
           qed (routine lems: assms 3 4)
         qed fact
       qed (routine lems: assms 1 2)
-    qed (routine lems: assms 1) }
+    qed (routine lems: assms 1)
     
-    show "(\<^bold>\<lambda>p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z. \<^bold>\<lambda>q. ind\<^sub>= refl q) p)`refl(a)`a`refl(a) \<equiv> refl(a)"
+    next show "(\<^bold>\<lambda>p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z. \<^bold>\<lambda>q. ind\<^sub>= refl q) p)`(refl a)`a`(refl a) \<equiv> refl a"
     proof compute
-      { fix p assume 1: "p: a =\<^sub>A a"
+      fix p assume 1: "p: a =\<^sub>A a"
       show "ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p: \<Prod>z:A. a =\<^sub>A z \<rightarrow> a =\<^sub>A z"
       proof (rule Equal_elim[where ?x=a and ?y=a])
         fix u assume 2: "u: A"
@@ -152,11 +152,11 @@ proof compute
             by (rule Equal_elim[where ?x=u and ?y=z]) (routine lems: assms 2 3)
           qed (routine lems: assms 2 3)
         qed fact
-      qed (routine lems: assms 1) }
+      qed (routine lems: assms 1)
       
-      show "(ind\<^sub>=(\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q)(refl(a)))`a`refl(a) \<equiv> refl(a)"
+      next show "(ind\<^sub>=(\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q)(refl a))`a`(refl a) \<equiv> refl a"
       proof compute
-        { fix u assume 1: "u: A"
+        fix u assume 1: "u: A"
         show "\<^bold>\<lambda>z q. ind\<^sub>= refl q: \<Prod>z:A. u =\<^sub>A z \<rightarrow> u =\<^sub>A z"
         proof
           fix z assume 2: "z: A"
@@ -165,22 +165,22 @@ proof compute
             show "\<And>q. q: u =\<^sub>A z \<Longrightarrow> ind\<^sub>= refl q: u =\<^sub>A z"
             by (rule Equal_elim[where ?x=u and ?y=z]) (routine lems: assms 1 2)
           qed (routine lems: assms 1 2)
-        qed fact }
+        qed fact
         
-        show "(\<^bold>\<lambda>z q. ind\<^sub>= refl q)`a`refl(a) \<equiv> refl(a)"
+        next show "(\<^bold>\<lambda>z q. ind\<^sub>= refl q)`a`(refl a) \<equiv> refl a"
         proof compute
-          { fix a assume 1: "a: A"
+          fix a assume 1: "a: A"
           show "\<^bold>\<lambda>q. ind\<^sub>= refl q: a =\<^sub>A a \<rightarrow> a =\<^sub>A a"
           proof
             show "\<And>q. q: a =\<^sub>A a \<Longrightarrow> ind\<^sub>= refl q: a =\<^sub>A a"
             by (rule Equal_elim[where ?x=a and ?y=a]) (routine lems: assms 1)
-          qed (routine lems: assms 1) }
+          qed (routine lems: assms 1)
           
-          show "(\<^bold>\<lambda>q. ind\<^sub>= refl q)`refl(a) \<equiv> refl(a)"
+          next show "(\<^bold>\<lambda>q. ind\<^sub>= refl q)`(refl a) \<equiv> refl a"
           proof compute
             show "\<And>p. p: a =\<^sub>A a \<Longrightarrow> ind\<^sub>= refl p: a =\<^sub>A a"
             by (rule Equal_elim[where ?x=a and ?y=a]) (routine lems: assms)
-            show "ind\<^sub>= refl (refl(a)) \<equiv> refl(a)"
+            show "ind\<^sub>= refl (refl a) \<equiv> refl a"
             proof compute
               show "\<And>x. x: A \<Longrightarrow> refl(x): x =\<^sub>A x" ..
             qed (routine lems: assms)
@@ -194,23 +194,25 @@ qed fact
 
 text "The raw object lambda term is cumbersome to use, so we define a simpler constant instead."
 
-axiomatization pathcomp :: "[Term, Term] \<Rightarrow> Term"  (infixl "\<bullet>" 120) where
+axiomatization pathcomp :: "[t, t] \<Rightarrow> t"  (infixl "\<bullet>" 120) where
   pathcomp_def: "\<lbrakk>
-    A: U(i);
+    A: U i;
     x: A; y: A; z: A;
     p: x =\<^sub>A y; q: y =\<^sub>A z
   \<rbrakk> \<Longrightarrow> p \<bullet> q \<equiv> rpathcomp`x`y`p`z`q"
 
 
 lemma pathcomp_type:
-  assumes "A: U(i)" "x: A" "y: A" "z: A" "p: x =\<^sub>A y" "q: y =\<^sub>A z"
+  assumes "A: U i" "x: A" "y: A" "z: A" "p: x =\<^sub>A y" "q: y =\<^sub>A z"
   shows "p \<bullet> q: x =\<^sub>A z"
+
 proof (subst pathcomp_def)
-  show "A: U(i)" "x: A" "y: A" "z: A" "p: x =\<^sub>A y" "q: y =\<^sub>A z" by fact+
+  show "A: U i" "x: A" "y: A" "z: A" "p: x =\<^sub>A y" "q: y =\<^sub>A z" by fact+
 qed (routine lems: assms rpathcomp_type)
 
+
 lemma pathcomp_comp:
-  assumes "A : U(i)" and "a : A" shows "refl(a) \<bullet> refl(a) \<equiv> refl(a)"
+  assumes "A : U i" and "a : A" shows "(refl a) \<bullet> (refl a) \<equiv> refl a"
 by (subst pathcomp_def) (routine lems: assms rpathcomp_comp)
 
 
@@ -281,4 +283,79 @@ proof (rule Equal_elim[where ?x=x and ?y=y and ?p=p])
   proof (compute lems: whg1b)
   
 
+section \<open>Higher groupoid structure of types\<close>
+
+lemma
+  assumes "A: U i" "x: A" "y: A" "p: x =\<^sub>A y"
+  shows
+    "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p =[x =\<^sub>A y] p \<bullet> (refl y)" and
+    "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p =[x =\<^sub>A y] (refl x) \<bullet> p"
+
+proof -
+  show "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p =[x =[A] y] p \<bullet> (refl y)"
+  by (rule Equal_elim[where ?p=p and ?x=x and ?y=y]) (derive lems: assms)+
+
+  show "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p =[x =[A] y] (refl x) \<bullet> p"
+  by (rule Equal_elim[where ?p=p and ?x=x and ?y=y]) (derive lems: assms)+
+qed
+
+
+lemma
+  assumes "A: U i" "x: A" "y: A" "p: x =\<^sub>A y"
+  shows
+    "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p\<inverse> \<bullet> p =[y =\<^sub>A y] (refl y)" and
+    "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p \<bullet> p\<inverse> =[x =\<^sub>A x] (refl x)"
+
+proof -
+  show "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p\<inverse> \<bullet> p =[y =\<^sub>A y] (refl y)"
+  by (rule Equal_elim[where ?p=p and ?x=x and ?y=y]) (derive lems: assms)+
+
+  show "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p \<bullet> p\<inverse> =[x =\<^sub>A x] (refl x)"
+  by (rule Equal_elim[where ?p=p and ?x=x and ?y=y]) (derive lems: assms)+
+qed
+
+
+lemma
+  assumes "A: U i" "x: A" "y: A" "p: x =\<^sub>A y"
+  shows "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p\<inverse>\<inverse> =[x =\<^sub>A y] p"
+by (rule Equal_elim[where ?p=p and ?x=x and ?y=y]) (derive lems: assms)
+
+text "Next we construct a proof term of associativity of path composition."
+
+schematic_goal
+  assumes
+    "A: U i"
+    "x: A" "y: A" "z: A" "w: A"
+    "p: x =\<^sub>A y" "q: y =\<^sub>A z" "r: z =\<^sub>A w"
+  shows
+    "?a: p \<bullet> (q \<bullet> r) =[x =\<^sub>A z] (p \<bullet> q) \<bullet> r"
+
+apply (rule Equal_elim[where ?p=p and ?x=x and ?y=y])
+apply (rule assms)+
+\<comment> \<open>Continue by substituting \<open>refl x \<bullet> q = q\<close> etc.\<close>
+sorry
+
+
+section \<open>Transport\<close>
+
+definition transport :: "t \<Rightarrow> t" where
+  "transport p \<equiv> ind\<^sub>= (\<lambda>x. (\<^bold>\<lambda>x. x)) p"
+
+text "Note that \<open>transport\<close> is a polymorphic function in our formulation."
+
+lemma transport_type:
+  assumes
+    "A: U i" "P: A \<longrightarrow> U i"
+    "x: A" "y: A"
+    "p: x =\<^sub>A y"
+  shows "transport p: P x \<rightarrow> P y"
+unfolding transport_def
+by (rule Equal_elim[where ?p=p and ?x=x and ?y=y]) (routine lems: assms)
+
+lemma transport_comp:
+  assumes "A: U i" and "x: A"
+  shows "transport (refl x) \<equiv> id"
+unfolding transport_def by (derive lems: assms)
+
+
 end
diff --git a/HoTT_Base.thy b/HoTT_Base.thy
index 8ea767f..07fbfc4 100644
--- a/HoTT_Base.thy
+++ b/HoTT_Base.thy
@@ -1,84 +1,77 @@
 (*  Title:  HoTT/HoTT_Base.thy
     Author: Joshua Chen
 
-Basic setup and definitions of a homotopy type theory object logic with a cumulative universe hierarchy à la Russell.
+Basic setup of a homotopy type theory object logic with a Russell-style cumulative universe hierarchy.
 *)
 
 theory HoTT_Base
-imports Pure
+imports
+  Pure
+  "HOL-Eisbach.Eisbach"
 
 begin
 
 
-section \<open>Foundational definitions\<close>
+section \<open>Basic setup\<close>
 
-text "Meta syntactic type for object-logic types and terms."
-
-typedecl t
+typedecl t  \<comment> \<open>Type of object types and terms\<close>
+typedecl ord  \<comment> \<open>Type of meta-level numerals\<close>
 
+axiomatization
+  O :: ord and
+  Suc :: "ord \<Rightarrow> ord" and
+  lt :: "[ord, ord] \<Rightarrow> prop"  (infix "<" 999)
+where
+  lt_Suc [intro]: "n < (Suc n)" and
+  lt_trans [intro]: "\<lbrakk>m1 < m2; m2 < m3\<rbrakk> \<Longrightarrow> m1 < m3" and
+  Suc_monotone [simp]: "m < n \<Longrightarrow> (Suc m) < (Suc n)"
 
-section \<open>Judgments\<close>
+method proveSuc = (rule lt_Suc | (rule lt_trans, (rule lt_Suc)+)+)
 
-text "
-  Formalize the typing judgment \<open>a: A\<close>.
-  For judgmental/definitional equality we use the existing Pure equality \<open>\<equiv>\<close> and hence do not need to define a separate judgment for it.
-"
+text \<open>Method @{method proveSuc} proves statements of the form \<open>n < (Suc (... (Suc n) ...))\<close>.\<close>
 
-judgment hastype :: "[t, t] \<Rightarrow> prop"  ("(3_:/ _)")
 
+section \<open>Judgment\<close>
 
-section \<open>Universe hierarchy\<close>
+judgment hastype :: "[t, t] \<Rightarrow> prop"  ("(3_:/ _)")
 
-text "Meta-numerals to index the universes."
 
-typedecl ord
+section \<open>Universes\<close>
 
 axiomatization
-  O :: ord and
-  S :: "ord \<Rightarrow> ord"  ("S_") and
-  lt :: "[ord, ord] \<Rightarrow> prop"  (infix "<-" 999)
+  U :: "ord \<Rightarrow> t"
 where
-  Ord_min: "\<And>n. O <- S(n)"
-and
-  Ord_monotone: "\<And>m n. m <- n \<Longrightarrow> S(m) <- S(n)"
+  U_hierarchy: "i < j \<Longrightarrow> U i: U j" and                               
+  U_cumulative: "\<lbrakk>A: U i; i < j\<rbrakk> \<Longrightarrow> A: U j"
 
-lemmas Ord_rules [intro] = Ord_min Ord_monotone
-  \<comment> \<open>Enables \<open>standard\<close> to automatically solve inequalities.\<close>
+text \<open>
+Using method @{method rule} with @{thm U_cumulative} is unsafe, if applied blindly it will typically lead to non-termination.
+One should instead use @{method elim}, or instantiate @{thm U_cumulative} suitably.
+\<close>
 
-text "Define the universe types."
-
-axiomatization
-  U :: "Ord \<Rightarrow> Term"
-where
-  U_hierarchy: "\<And>i j. i <- j \<Longrightarrow> U(i): U(j)"
-and
-  U_cumulative: "\<And>A i j. \<lbrakk>A: U(i); i <- j\<rbrakk> \<Longrightarrow> A: U(j)"
-    \<comment> \<open>WARNING: \<open>rule Universe_cumulative\<close> can result in an infinite rewrite loop!\<close>
+method cumulativity = (elim U_cumulative, proveSuc)  \<comment> \<open>Proves \<open>A: U i \<Longrightarrow> A: U (Suc (... (Suc i) ...))\<close>.\<close>
+method hierarchy = (rule U_hierarchy, proveSuc)  \<comment> \<open>Proves \<open>U i: U (Suc (... (Suc i) ...)).\<close>\<close>
 
 
 section \<open>Type families\<close>
 
-text "
-  The following abbreviation constrains the output type of a meta lambda expression when given input of certain type.
-"
-
-abbreviation (input) constrained :: "[Term \<Rightarrow> Term, Term, Term] \<Rightarrow> prop"  ("(1_: _ \<longrightarrow> _)")
-  where "f: A \<longrightarrow> B \<equiv> (\<And>x. x : A \<Longrightarrow> f(x): B)"
+abbreviation (input) constrained :: "[t \<Rightarrow> t, t, t] \<Rightarrow> prop"  ("(1_:/ _ \<longrightarrow> _)")
+  where "f: A \<longrightarrow> B \<equiv> (\<And>x. x : A \<Longrightarrow> f x: B)"
 
-text "
-  The above is used to define type families, which are constrained meta-lambdas \<open>P: A \<longrightarrow> B\<close> where \<open>A\<close> and \<open>B\<close> are small types.
-"
+text \<open>
+The abbreviation @{abbrev constrained} is used to define type families, which are constrained expressions @{term "P: A \<longrightarrow> B"} where @{term "A::t"} and @{term "B::t"} are small types.
+\<close>
 
-type_synonym Typefam = "Term \<Rightarrow> Term"
+type_synonym tf = "t \<Rightarrow> t"  \<comment> \<open>Type of type families.\<close>
 
 
 section \<open>Named theorems\<close>
 
-text "
-  Named theorems to be used by proof methods later (see HoTT_Methods.thy).
-  
-  \<open>wellform\<close> declares necessary wellformedness conditions for type and inhabitation judgments, while \<open>comp\<close> declares computation rules, which are usually passed to invocations of the method \<open>subst\<close> to perform equational rewriting.
-"
+text \<open>
+Declare named theorems to be used by proof methods defined in @{file HoTT_Methods.thy}.
+\<open>wellform\<close> declares necessary well-formedness conditions for type and inhabitation judgments.
+\<open>comp\<close> declares computation rules, which are usually passed to invocations of the method \<open>subst\<close> to perform equational rewriting.
+\<close>
 
 named_theorems wellform
 named_theorems comp
diff --git a/HoTT_Methods.thy b/HoTT_Methods.thy
index 32e412b..abb6dda 100644
--- a/HoTT_Methods.thy
+++ b/HoTT_Methods.thy
@@ -14,12 +14,12 @@ begin
 
 section \<open>Deriving typing judgments\<close>
 
+method routine uses lems = (assumption | rule lems | standard)+
+
 text "
-  \<open>routine\<close> proves routine type judgments \<open>a : A\<close> using the rules declared [intro] in the respective theory files, as well as additional provided lemmas.
+  @{method routine} proves routine type judgments \<open>a : A\<close> using the rules declared [intro] in the respective theory files, as well as additional provided lemmas.
 "
 
-method routine uses lems = (assumption | rule lems | standard)+
-
 text "
   \<open>wellformed'\<close> finds a proof of any valid typing judgment derivable from the judgment passed as \<open>jdmt\<close>.
   If no judgment is passed, it will try to resolve with the theorems declared \<open>wellform\<close>.
diff --git a/LICENSE b/LICENSE
new file mode 100644
index 0000000..65c5ca8
--- /dev/null
+++ b/LICENSE
@@ -0,0 +1,165 @@
+                   GNU LESSER GENERAL PUBLIC LICENSE
+                       Version 3, 29 June 2007
+
+ Copyright (C) 2007 Free Software Foundation, Inc. <http://fsf.org/>
+ Everyone is permitted to copy and distribute verbatim copies
+ of this license document, but changing it is not allowed.
+
+
+  This version of the GNU Lesser General Public License incorporates
+the terms and conditions of version 3 of the GNU General Public
+License, supplemented by the additional permissions listed below.
+
+  0. Additional Definitions.
+
+  As used herein, "this License" refers to version 3 of the GNU Lesser
+General Public License, and the "GNU GPL" refers to version 3 of the GNU
+General Public License.
+
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+other than an Application or a Combined Work as defined below.
+
+  An "Application" is any work that makes use of an interface provided
+by the Library, but which is not otherwise based on the Library.
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diff --git a/Nat.thy b/Nat.thy
index 45c3a2e..a82e7cb 100644
--- a/Nat.thy
+++ b/Nat.thy
@@ -12,36 +12,36 @@ begin
 section \<open>Constants and type rules\<close>
 
 axiomatization
-  Nat :: Term   ("\<nat>") and
-  zero :: Term  ("0") and
-  succ :: "Term \<Rightarrow> Term" and
-  indNat :: "[[Term, Term] \<Rightarrow> Term, Term, Term] \<Rightarrow> Term"  ("(1ind\<^sub>\<nat>)")
+  Nat :: t   ("\<nat>") and
+  zero :: t  ("0") and
+  succ :: "t \<Rightarrow> t" and
+  indNat :: "[[t, t] \<Rightarrow> t, t, t] \<Rightarrow> t"  ("(1ind\<^sub>\<nat>)")
 where
-  Nat_form: "\<nat>: U(O)"
+  Nat_form: "\<nat>: U O"
 and
   Nat_intro_0: "0: \<nat>"
 and
-  Nat_intro_succ: "n: \<nat> \<Longrightarrow> succ(n): \<nat>"
+  Nat_intro_succ: "n: \<nat> \<Longrightarrow> succ n: \<nat>"
 and
   Nat_elim: "\<lbrakk>
-    C: \<nat> \<longrightarrow> U(i);
-    \<And>n c. \<lbrakk>n: \<nat>; c: C(n)\<rbrakk> \<Longrightarrow> f(n)(c): C(succ n);
-    a: C(0);
+    C: \<nat> \<longrightarrow> U i;
+    \<And>n c. \<lbrakk>n: \<nat>; c: C n\<rbrakk> \<Longrightarrow> f n c: C (succ n);
+    a: C 0;
     n: \<nat>
-    \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat>(f)(a)(n): C(n)"
+    \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat> f a n: C n"
 and
   Nat_comp_0: "\<lbrakk>
-    C: \<nat> \<longrightarrow> U(i);
-    \<And>n c. \<lbrakk>n: \<nat>; c: C(n)\<rbrakk> \<Longrightarrow> f(n)(c): C(succ n);
-    a: C(0)
-    \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat>(f)(a)(0) \<equiv> a"
+    C: \<nat> \<longrightarrow> U i;
+    \<And>n c. \<lbrakk>n: \<nat>; c: C(n)\<rbrakk> \<Longrightarrow> f n c: C (succ n);
+    a: C 0
+    \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat> f a 0 \<equiv> a"
 and
   Nat_comp_succ: "\<lbrakk>
-    C: \<nat> \<longrightarrow> U(i);
-    \<And>n c. \<lbrakk>n: \<nat>; c: C(n)\<rbrakk> \<Longrightarrow> f(n)(c): C(succ n);
-    a: C(0);
+    C: \<nat> \<longrightarrow> U i;
+    \<And>n c. \<lbrakk>n: \<nat>; c: C n\<rbrakk> \<Longrightarrow> f n c: C (succ n);
+    a: C 0;
     n: \<nat>
-    \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat>(f)(a)(succ n) \<equiv> f(n)(ind\<^sub>\<nat> f a n)"
+    \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat> f a (succ n) \<equiv> f n (ind\<^sub>\<nat> f a n)"
 
 
 text "Rule attribute declarations:"
diff --git a/Prod.thy b/Prod.thy
index 3b74531..a3cc347 100644
--- a/Prod.thy
+++ b/Prod.thy
@@ -12,14 +12,14 @@ begin
 section \<open>Constants and syntax\<close>
 
 axiomatization
-  Prod :: "[Term, Typefam] \<Rightarrow> Term" and
-  lambda :: "(Term \<Rightarrow> Term) \<Rightarrow> Term"  (binder "\<^bold>\<lambda>" 30) and
-  appl :: "[Term, Term] \<Rightarrow> Term"  (infixl "`" 60)
+  Prod :: "[t, tf] \<Rightarrow> t" and
+  lambda :: "(t \<Rightarrow> t) \<Rightarrow> t"  (binder "\<^bold>\<lambda>" 30) and
+  appl :: "[t, t] \<Rightarrow> t"  (infixl "`" 60)
     \<comment> \<open>Application binds tighter than abstraction.\<close>
 
 syntax
-  "_PROD" :: "[idt, Term, Term] \<Rightarrow> Term"          ("(3\<Prod>_:_./ _)" 30)
-  "_PROD_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term"    ("(3PROD _:_./ _)" 30)
+  "_PROD" :: "[idt, t, t] \<Rightarrow> t"          ("(3\<Prod>_:_./ _)" 30)
+  "_PROD_ASCII" :: "[idt, t, t] \<Rightarrow> t"    ("(3PROD _:_./ _)" 30)
 
 text "The translations below bind the variable \<open>x\<close> in the expressions \<open>B\<close> and \<open>b\<close>."
 
@@ -29,24 +29,24 @@ translations
 
 text "Nondependent functions are a special case."
 
-abbreviation Function :: "[Term, Term] \<Rightarrow> Term"  (infixr "\<rightarrow>" 40)
+abbreviation Function :: "[t, t] \<Rightarrow> t"  (infixr "\<rightarrow>" 40)
   where "A \<rightarrow> B \<equiv> \<Prod>_: A. B"
 
 
 section \<open>Type rules\<close>
 
 axiomatization where
-  Prod_form: "\<lbrakk>A: U(i); B: A \<longrightarrow> U(i)\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x): U(i)"
+  Prod_form: "\<lbrakk>A: U i; B: A \<longrightarrow> U i\<rbrakk> \<Longrightarrow> \<Prod>x:A. B x: U i"
 and
-  Prod_intro: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b(x): B(x); A: U(i)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b(x): \<Prod>x:A. B(x)"
+  Prod_intro: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; A: U i\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b x: \<Prod>x:A. B x"
 and
-  Prod_elim: "\<lbrakk>f: \<Prod>x:A. B(x); a: A\<rbrakk> \<Longrightarrow> f`a: B(a)"
+  Prod_elim: "\<lbrakk>f: \<Prod>x:A. B x; a: A\<rbrakk> \<Longrightarrow> f`a: B a"
 and
-  Prod_appl: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b(x): B(x); a: A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. b(x))`a \<equiv> b(a)"
+  Prod_appl: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; a: A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. b x)`a \<equiv> b a"
 and
-  Prod_uniq: "f : \<Prod>x:A. B(x) \<Longrightarrow> \<^bold>\<lambda>x. (f`x) \<equiv> f"
+  Prod_uniq: "f : \<Prod>x:A. B x \<Longrightarrow> \<^bold>\<lambda>x. (f`x) \<equiv> f"
 and
-  Prod_eq: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b(x) \<equiv> b'(x); A: U(i)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b(x) \<equiv> \<^bold>\<lambda>x. b'(x)"
+  Prod_eq: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x \<equiv> c x; A: U i\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b x \<equiv> \<^bold>\<lambda>x. c x"
 
 text "
   The Pure rules for \<open>\<equiv>\<close> only let us judge strict syntactic equality of object lambda expressions; Prod_eq is the actual definitional equality rule.
@@ -55,15 +55,15 @@ text "
 "
 
 text "
-  In addition to the usual type rules, it is a meta-theorem that whenever \<open>\<Prod>x:A. B x: U(i)\<close> is derivable from some set of premises \<Gamma>, then so are \<open>A: U(i)\<close> and \<open>B: A \<longrightarrow> U(i)\<close>.
+  In addition to the usual type rules, it is a meta-theorem that whenever \<open>\<Prod>x:A. B x: U i\<close> is derivable from some set of premises \<Gamma>, then so are \<open>A: U i\<close> and \<open>B: A \<longrightarrow> U i\<close>.
   
   That is to say, the following inference rules are admissible, and it simplifies proofs greatly to axiomatize them directly.
 "
 
 axiomatization where
-  Prod_wellform1: "(\<Prod>x:A. B(x): U(i)) \<Longrightarrow> A: U(i)"
+  Prod_wellform1: "(\<Prod>x:A. B x: U i) \<Longrightarrow> A: U i"
 and
-  Prod_wellform2: "(\<Prod>x:A. B(x): U(i)) \<Longrightarrow> B: A \<longrightarrow> U(i)"
+  Prod_wellform2: "(\<Prod>x:A. B x: U i) \<Longrightarrow> B: A \<longrightarrow> U i"
 
 
 text "Rule attribute declarations---set up various methods to use the type rules."
@@ -75,19 +75,15 @@ lemmas Prod_routine [intro] = Prod_form Prod_intro Prod_elim
 
 section \<open>Function composition\<close>
 
-definition compose :: "[Term, Term] \<Rightarrow> Term"  (infixr "o" 70) where "g o f \<equiv> \<^bold>\<lambda>x. g`(f`x)"
+definition compose :: "[t, t] \<Rightarrow> t"  (infixr "o" 110) where "g o f \<equiv> \<^bold>\<lambda>x. g`(f`x)"
 
-syntax "_COMPOSE" :: "[Term, Term] \<Rightarrow> Term"  (infixr "\<circ>" 70)
+syntax "_COMPOSE" :: "[t, t] \<Rightarrow> t"  (infixr "\<circ>" 110)
 translations "g \<circ> f" \<rightleftharpoons> "g o f"
 
 
-section \<open>Atomization\<close>
+section \<open>Polymorphic identity function\<close>
 
-text "
-  Universal statements can be internalized within the theory; the following rule is admissible.
-" (* PROOF NEEDED *)
-
-axiomatization where Prod_atomize: "(\<^bold>\<lambda>x. b(x): \<Prod>x:A. B(x)) \<Longrightarrow> (\<And>x. x: A \<Longrightarrow> b(x): B(x))"
+abbreviation id :: t where "id \<equiv> \<^bold>\<lambda>x. x"
 
 
 end
diff --git a/ProdProps.thy b/ProdProps.thy
index 1af6ad3..a68f79b 100644
--- a/ProdProps.thy
+++ b/ProdProps.thy
@@ -14,11 +14,11 @@ begin
 section \<open>Composition\<close>
 
 text "
-  The proof of associativity needs some guidance; it involves telling Isabelle to use the correct rule for \<Pi>-type definitional equality, and the correct substitutions in the subgoals thereafter.
+  The proof of associativity needs some guidance; it involves telling Isabelle to use the correct rule for \<Prod>-type definitional equality, and the correct substitutions in the subgoals thereafter.
 "
 
 lemma compose_assoc:
-  assumes "A: U(i)" and "f: A \<rightarrow> B" "g: B \<rightarrow> C" "h: \<Prod>x:C. D(x)"
+  assumes "A: U i" and "f: A \<rightarrow> B" "g: B \<rightarrow> C" "h: \<Prod>x:C. D x"
   shows "(h \<circ> g) \<circ> f \<equiv> h \<circ> (g \<circ> f)"
 proof (subst (0 1 2 3) compose_def)
   show "\<^bold>\<lambda>x. (\<^bold>\<lambda>y. h`(g`y))`(f`x) \<equiv> \<^bold>\<lambda>x. h`((\<^bold>\<lambda>y. g`(f`y))`x)"
@@ -31,22 +31,27 @@ proof (subst (0 1 2 3) compose_def)
       proof compute
         show "\<And>x. x: A \<Longrightarrow> g`(f`x): C" by (routine lems: assms)
       qed
-      show "\<And>x. x: B \<Longrightarrow> h`(g`x): D(g`x)" by (routine lems: assms)
+      show "\<And>x. x: B \<Longrightarrow> h`(g`x): D (g`x)" by (routine lems: assms)
     qed (routine lems: assms)
   qed fact
 qed
 
 
 lemma compose_comp:
-  assumes "A: U(i)" and "\<And>x. x: A \<Longrightarrow> b(x): B" and "\<And>x. x: B \<Longrightarrow> c(x): C(x)"
-  shows "(\<^bold>\<lambda>x. c(x)) \<circ> (\<^bold>\<lambda>x. b(x)) \<equiv> \<^bold>\<lambda>x. c(b(x))"
+  assumes "A: U i" and "\<And>x. x: A \<Longrightarrow> b x: B" and "\<And>x. x: B \<Longrightarrow> c x: C x"
+  shows "(\<^bold>\<lambda>x. c x) \<circ> (\<^bold>\<lambda>x. b x) \<equiv> \<^bold>\<lambda>x. c (b x)"
 proof (subst compose_def, subst Prod_eq)
-  show "\<And>a. a: A \<Longrightarrow> (\<^bold>\<lambda>x. c(x))`((\<^bold>\<lambda>x. b(x))`a) \<equiv> (\<^bold>\<lambda>x. c (b x))`a"
+  show "\<And>a. a: A \<Longrightarrow> (\<^bold>\<lambda>x. c x)`((\<^bold>\<lambda>x. b x)`a) \<equiv> (\<^bold>\<lambda>x. c (b x))`a"
   proof compute
-    show "\<And>a. a: A \<Longrightarrow> c((\<^bold>\<lambda>x. b(x))`a) \<equiv> (\<^bold>\<lambda>x. c(b(x)))`a"
+    show "\<And>a. a: A \<Longrightarrow> c ((\<^bold>\<lambda>x. b x)`a) \<equiv> (\<^bold>\<lambda>x. c (b x))`a"
     by (derive lems: assms)
   qed (routine lems: assms)
 qed (derive lems: assms)
 
 
+text "Set up the \<open>compute\<close> method to automatically simplify function compositions."
+
+lemmas compose_comp [comp]
+
+
 end
diff --git a/Proj.thy b/Proj.thy
index a1c4c8f..74c561c 100644
--- a/Proj.thy
+++ b/Proj.thy
@@ -12,44 +12,44 @@ theory Proj
 begin
 
 
-definition fst :: "Term \<Rightarrow> Term" where "fst(p) \<equiv> ind\<^sub>\<Sum> (\<lambda>x y. x) p"
-definition snd :: "Term \<Rightarrow> Term" where "snd(p) \<equiv> ind\<^sub>\<Sum> (\<lambda>x y. y) p"
+definition fst :: "Term \<Rightarrow> Term" where "fst p \<equiv> ind\<^sub>\<Sum> (\<lambda>x y. x) p"
+definition snd :: "Term \<Rightarrow> Term" where "snd p \<equiv> ind\<^sub>\<Sum> (\<lambda>x y. y) p"
 
 text "Typing judgments and computation rules for the dependent and non-dependent projection functions."
 
 lemma fst_type:
-  assumes "\<Sum>x:A. B(x): U(i)" and "p: \<Sum>x:A. B(x)" shows "fst(p): A"
+  assumes "\<Sum>x:A. B x: U i" and "p: \<Sum>x:A. B x" shows "fst p: A"
 unfolding fst_def by (derive lems: assms)
 
 lemma fst_comp:
-  assumes "A: U(i)" and "B: A \<longrightarrow> U(i)" and "a: A" and "b: B(a)" shows "fst(<a,b>) \<equiv> a"
+  assumes "A: U i" and "B: A \<longrightarrow> U i" and "a: A" and "b: B a" shows "fst <a,b> \<equiv> a"
 unfolding fst_def
 proof compute
-  show "a: A" and "b: B(a)" by fact+
+  show "a: A" and "b: B a" by fact+
 qed (routine lems: assms)+
 
 lemma snd_type:
-  assumes "\<Sum>x:A. B(x): U(i)" and "p: \<Sum>x:A. B(x)" shows "snd(p): B(fst p)"
+  assumes "\<Sum>x:A. B x: U i" and "p: \<Sum>x:A. B x" shows "snd p: B (fst p)"
 unfolding snd_def
 proof
-  show "\<And>p. p: \<Sum>x:A. B(x) \<Longrightarrow> B(fst p): U(i)" by (derive lems: assms fst_type)
+  show "\<And>p. p: \<Sum>x:A. B x \<Longrightarrow> B (fst p): U i" by (derive lems: assms fst_type)
   
   fix x y
-  assume asm: "x: A" "y: B(x)"
-  show "y: B(fst <x,y>)"
+  assume asm: "x: A" "y: B x"
+  show "y: B (fst <x,y>)"
   proof (subst fst_comp)
-    show "A: U(i)" by (wellformed lems: assms(1))
-    show "\<And>x. x: A \<Longrightarrow> B(x): U(i)" by (wellformed lems: assms(1))
+    show "A: U i" by (wellformed lems: assms(1))
+    show "\<And>x. x: A \<Longrightarrow> B x: U i" by (wellformed lems: assms(1))
   qed fact+
 qed fact
 
 lemma snd_comp:
-  assumes "A: U(i)" and "B: A \<longrightarrow> U(i)" and "a: A" and "b: B(a)" shows "snd(<a,b>) \<equiv> b"
+  assumes "A: U i" and "B: A \<longrightarrow> U i" and "a: A" and "b: B a" shows "snd <a,b> \<equiv> b"
 unfolding snd_def
 proof compute
-  show "\<And>x y. y: B(x) \<Longrightarrow> y: B(x)" .
+  show "\<And>x y. y: B x \<Longrightarrow> y: B x" .
   show "a: A" by fact
-  show "b: B(a)" by fact
+  show "b: B a" by fact
 qed (routine lems: assms)
 
 
diff --git a/README.md b/README.md
index 4747be7..d557f3e 100644
--- a/README.md
+++ b/README.md
@@ -4,15 +4,19 @@ An experimental implementation of [homotopy type theory](https://en.wikipedia.or
 
 ### Installation & Usage
 
-Clone the contents of this repository into `<Isabelle root directory>/src/HoTT`.
+Clone or copy the contents of this repository into `<Isabelle root directory>/src/Isabelle-HoTT`.
 
 To use, set Isabelle's prover to Pure in the Theories panel, and import `HoTT`.
 
+### Some comments on the implementation
+
+The implementation in the `master` branch is polymorphic without type annotations, and as such has some differences with the standard theory as presented in the [Homotopy Type Theory book](https://homotopytypetheory.org/book/).
+
 ### Collaboration
 
-I've been flying solo on this library as part of my Masters project, and there are very many improvements and developments that have yet to be implemented, so ***collaborators are welcome!***
+I've been flying solo on this library as part of my Masters project, and there are very many improvements and developments that have yet to be implemented, so **collaborators are welcome!**
 
-If you're interested in working together on any part of this do drop me a line at `joshua DOT chen AT uni-bonn DOT de`.
+If you're interested in working together on any part of this library do drop me a line at `joshua DOT chen AT uni-bonn DOT de`.
 
 ### License
 
diff --git a/Sum.thy b/Sum.thy
index 8d0b0e6..92375b9 100644
--- a/Sum.thy
+++ b/Sum.thy
@@ -12,13 +12,13 @@ begin
 section \<open>Constants and syntax\<close>
 
 axiomatization
-  Sum :: "[Term, Typefam] \<Rightarrow> Term" and
-  pair :: "[Term, Term] \<Rightarrow> Term"  ("(1<_,/ _>)") and
-  indSum :: "[[Term, Term] \<Rightarrow> Term, Term] \<Rightarrow> Term"  ("(1ind\<^sub>\<Sum>)")
+  Sum :: "[t, Typefam] \<Rightarrow> t" and
+  pair :: "[t, t] \<Rightarrow> t"  ("(1<_,/ _>)") and
+  indSum :: "[[t, t] \<Rightarrow> t, t] \<Rightarrow> t"  ("(1ind\<^sub>\<Sum>)")
 
 syntax
-  "_SUM" :: "[idt, Term, Term] \<Rightarrow> Term"        ("(3\<Sum>_:_./ _)" 20)
-  "_SUM_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term"  ("(3SUM _:_./ _)" 20)
+  "_SUM" :: "[idt, t, t] \<Rightarrow> t"        ("(3\<Sum>_:_./ _)" 20)
+  "_SUM_ASCII" :: "[idt, t, t] \<Rightarrow> t"  ("(3SUM _:_./ _)" 20)
 
 translations
   "\<Sum>x:A. B" \<rightleftharpoons> "CONST Sum A (\<lambda>x. B)"
@@ -26,37 +26,37 @@ translations
 
 text "Nondependent pair."
 
-abbreviation Pair :: "[Term, Term] \<Rightarrow> Term"  (infixr "\<times>" 50)
+abbreviation Pair :: "[t, t] \<Rightarrow> t"  (infixr "\<times>" 50)
   where "A \<times> B \<equiv> \<Sum>_:A. B"
 
 
 section \<open>Type rules\<close>
 
 axiomatization where
-  Sum_form: "\<lbrakk>A: U(i); B: A \<longrightarrow> U(i)\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x): U(i)"
+  Sum_form: "\<lbrakk>A: U i; B: A \<longrightarrow> U i\<rbrakk> \<Longrightarrow> \<Sum>x:A. B x: U i"
 and
-  Sum_intro: "\<lbrakk>B: A \<longrightarrow> U(i); a: A; b: B(a)\<rbrakk> \<Longrightarrow> <a,b>: \<Sum>x:A. B(x)"
+  Sum_intro: "\<lbrakk>B: A \<longrightarrow> U i; a: A; b: B a\<rbrakk> \<Longrightarrow> <a,b>: \<Sum>x:A. B x"
 and
   Sum_elim: "\<lbrakk>
-    p: \<Sum>x:A. B(x);
-    \<And>x y. \<lbrakk>x: A; y: B(x)\<rbrakk> \<Longrightarrow> f(x)(y): C(<x,y>);
-    C: \<Sum>x:A. B(x) \<longrightarrow> U(i)
-    \<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum>(f)(p): C(p)"  (* What does writing \<lambda>x y. f(x, y) change? *)
+    p: \<Sum>x:A. B x;
+    \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x,y>;
+    C: \<Sum>x:A. B x \<longrightarrow> U i
+    \<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum> f p: C p"  (* What does writing \<lambda>x y. f(x, y) change? *)
 and
   Sum_comp: "\<lbrakk>
     a: A;
-    b: B(a);
-    \<And>x y. \<lbrakk>x: A; y: B(x)\<rbrakk> \<Longrightarrow> f(x)(y): C(<x,y>);
-    B: A \<longrightarrow> U(i);
-    C: \<Sum>x:A. B(x) \<longrightarrow> U(i)
-    \<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum>(f)(<a,b>) \<equiv> f(a)(b)"
+    b: B a;
+    \<And>x y. \<lbrakk>x: A; y: B(x)\<rbrakk> \<Longrightarrow> f x y: C <x,y>;
+    B: A \<longrightarrow> U i;
+    C: \<Sum>x:A. B x \<longrightarrow> U i
+    \<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum> f <a,b> \<equiv> f a b"
 
 text "Admissible inference rules for sum formation:"
 
 axiomatization where
-  Sum_wellform1: "(\<Sum>x:A. B(x): U(i)) \<Longrightarrow> A: U(i)"
+  Sum_wellform1: "(\<Sum>x:A. B x: U i) \<Longrightarrow> A: U i"
 and
-  Sum_wellform2: "(\<Sum>x:A. B(x): U(i)) \<Longrightarrow> B: A \<longrightarrow> U(i)"
+  Sum_wellform2: "(\<Sum>x:A. B x: U i) \<Longrightarrow> B: A \<longrightarrow> U i"
 
 
 text "Rule attribute declarations:"
diff --git a/Unit.thy b/Unit.thy
index 6adfb02..6760f27 100644
--- a/Unit.thy
+++ b/Unit.thy
@@ -16,13 +16,13 @@ axiomatization
   pt :: Term    ("\<star>") and
   indUnit :: "[Term, Term] \<Rightarrow> Term"  ("(1ind\<^sub>\<one>)")
 where
-  Unit_form: "\<one>: U(O)"
+  Unit_form: "\<one>: U O"
 and
   Unit_intro: "\<star>: \<one>"
 and
-  Unit_elim: "\<lbrakk>C: \<one> \<longrightarrow> U(i); c: C(\<star>); a: \<one>\<rbrakk> \<Longrightarrow> ind\<^sub>\<one>(c)(a) : C(a)"
+  Unit_elim: "\<lbrakk>C: \<one> \<longrightarrow> U i; c: C \<star>; a: \<one>\<rbrakk> \<Longrightarrow> ind\<^sub>\<one> c a: C a"
 and
-  Unit_comp: "\<lbrakk>C: \<one> \<longrightarrow> U(i); c: C(\<star>)\<rbrakk> \<Longrightarrow> ind\<^sub>\<one>(c)(\<star>) \<equiv> c"
+  Unit_comp: "\<lbrakk>C: \<one> \<longrightarrow> U i; c: C \<star>\<rbrakk> \<Longrightarrow> ind\<^sub>\<one> c \<star> \<equiv> c"
 
 
 text "Rule attribute declarations:"
diff --git a/Univalence.thy b/Univalence.thy
new file mode 100644
index 0000000..001ee33
--- /dev/null
+++ b/Univalence.thy
@@ -0,0 +1,176 @@
+(*  Title:  HoTT/Univalence.thy
+    Author: Joshua Chen
+
+Definitions of homotopy, equivalence and the univalence axiom.
+*)
+
+theory Univalence
+imports
+  HoTT_Methods
+  EqualProps
+  ProdProps
+  Sum
+
+begin
+
+
+section \<open>Homotopy and equivalence\<close>
+
+axiomatization homotopic :: "[t, t] \<Rightarrow> t"  (infix "~" 100) where
+  homotopic_def: "\<lbrakk>
+    f: \<Prod>x:A. B x;
+    g: \<Prod>x:A. B x
+  \<rbrakk> \<Longrightarrow> f ~ g \<equiv> \<Prod>x:A. (f`x) =[B x] (g`x)"
+
+axiomatization isequiv :: "t \<Rightarrow> t" where
+  isequiv_def: "f: A \<rightarrow> B \<Longrightarrow> isequiv f \<equiv> (\<Sum>g: B \<rightarrow> A. g \<circ> f ~ id) \<times> (\<Sum>g: B \<rightarrow> A. f \<circ> g ~ id)"
+
+definition equivalence :: "[t, t] \<Rightarrow> t"  (infix "\<simeq>" 100)
+  where "A \<simeq> B \<equiv> \<Sum>f: A \<rightarrow> B. isequiv f"
+
+
+text "The identity function is an equivalence:"
+
+lemma isequiv_id:
+  assumes "A: U i" and "id: A \<rightarrow> A"
+  shows "<<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>: isequiv id"
+proof (derive lems: assms isequiv_def homotopic_def)
+  fix g assume asm: "g: A \<rightarrow> A"
+  show "id \<circ> g: A \<rightarrow> A"
+    unfolding compose_def by (routine lems: asm assms)
+
+  show "\<Prod>x:A. ((id \<circ> g)`x) =\<^sub>A (id`x): U i"
+    unfolding compose_def by (routine lems: asm assms)
+  next
+  
+  show "<\<^bold>\<lambda>x. x, \<^bold>\<lambda>x. refl x>: \<Sum>g:A \<rightarrow> A. (g \<circ> id) ~ id"
+    unfolding compose_def by (derive lems: assms homotopic_def)
+
+  show "<\<^bold>\<lambda>x. x, lambda refl>: \<Sum>g:A \<rightarrow> A. (id \<circ> g) ~ id"
+    unfolding compose_def by (derive lems: assms homotopic_def)
+qed (rule assms)
+
+
+text "We use the following lemma in a few proofs:"
+
+lemma isequiv_type:
+  assumes "A: U i" and "B: U i" and "f: A \<rightarrow> B"
+  shows "isequiv f: U i"
+  by (derive lems: assms isequiv_def homotopic_def compose_def)
+
+
+text "The equivalence relation \<open>\<simeq>\<close> is symmetric:"
+
+lemma equiv_sym:
+  assumes "A: U i" and "id: A \<rightarrow> A"
+  shows "<id, <<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>>: A \<simeq> A"
+unfolding equivalence_def proof
+  show "<<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>: isequiv id" using assms by (rule isequiv_id)
+  
+  fix f assume "f: A \<rightarrow> A"
+  with assms(1) assms(1) show "isequiv f: U i" by (rule isequiv_type)
+qed (rule assms)
+
+
+section \<open>idtoeqv and the univalence axiom\<close>
+
+definition idtoeqv :: t
+  where "idtoeqv \<equiv> \<^bold>\<lambda>p. <transport p, ind\<^sub>= (\<lambda>A. <<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>) p>"
+
+
+text "We prove that equal types are equivalent. The proof is long and uses universes."
+
+theorem
+  assumes "A: U i" and "B: U i"
+  shows "idtoeqv: (A =[U i] B) \<rightarrow> A \<simeq> B"
+unfolding idtoeqv_def equivalence_def
+proof
+  fix p assume "p: A =[U i] B"
+  show "<transport p, ind\<^sub>= (\<lambda>A. <<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>) p>: \<Sum>f: A \<rightarrow> B. isequiv f"
+  proof
+    { fix f assume "f: A \<rightarrow> B"
+    with assms show "isequiv f: U i" by (rule isequiv_type)
+    }
+    
+    show "transport p: A \<rightarrow> B"
+    proof (rule transport_type[where ?P="\<lambda>x. x" and ?A="U i" and ?i="Suc i"])
+      show "\<And>x. x: U i \<Longrightarrow> x: U (Suc i)" by cumulativity
+      show "U i: U (Suc i)" by hierarchy
+    qed fact+
+    
+    show "ind\<^sub>= (\<lambda>A. <<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>) p: isequiv (transport p)"
+    proof (rule Equal_elim[where ?C="\<lambda>_ _ p. isequiv (transport p)"])
+      fix A assume asm: "A: U i"
+      show "<<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>: isequiv (transport (refl A))"
+      proof (derive lems: isequiv_def)
+        show "transport (refl A): A \<rightarrow> A"
+          unfolding transport_def
+          by (compute lems: Equal_comp[where ?A="U i" and ?C="\<lambda>_ _ _. A \<rightarrow> A"]) (derive lems: asm)
+        
+        show "<<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>:
+                (\<Sum>g:A \<rightarrow> A. g \<circ> (transport (refl A)) ~ id) \<times>
+                (\<Sum>g:A \<rightarrow> A. (transport (refl A)) \<circ> g ~ id)"
+        proof (subst (1 2) transport_comp)
+          show "U i: U (Suc i)" by (rule U_hierarchy) rule
+          show "U i: U (Suc i)" by (rule U_hierarchy) rule
+          
+          show "<<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>:
+                  (\<Sum>g:A \<rightarrow> A. g \<circ> id ~ id) \<times> (\<Sum>g:A \<rightarrow> A. id \<circ> g ~ id)"
+          proof
+            show "\<Sum>g:A \<rightarrow> A. id \<circ> g ~ id: U i"
+            proof (derive lems: asm homotopic_def)
+              fix g assume asm': "g: A \<rightarrow> A"
+              show *: "id \<circ> g: A \<rightarrow> A" by (derive lems: asm asm' compose_def)
+              show "\<Prod>x:A. ((id \<circ> g)`x) =\<^sub>A (id`x): U i" by (derive lems: asm *)
+            qed (routine lems: asm)
+
+            show "<id, \<^bold>\<lambda>x. refl x>: \<Sum>g:A \<rightarrow> A. id \<circ> g ~ id"
+            proof
+              fix g assume asm': "g: A \<rightarrow> A"
+              show "id \<circ> g ~ id: U i"
+              proof (derive lems: homotopic_def)
+                show *: "id \<circ> g: A \<rightarrow> A" by (derive lems: asm asm' compose_def)
+                show "\<Prod>x:A. ((id \<circ> g)`x) =\<^sub>A (id`x): U i" by (derive lems: asm *)
+              qed (routine lems: asm)
+              next
+              show "\<^bold>\<lambda>x. refl x: id \<circ> id ~ id"
+              proof compute
+                show "\<^bold>\<lambda>x. refl x: id ~ id" by (subst homotopic_def) (derive lems: asm)
+              qed (rule asm)
+            qed (routine lems: asm)
+
+            show "<id, \<^bold>\<lambda>x. refl x>: \<Sum>g:A \<rightarrow> A. g \<circ> id ~ id"
+            proof
+              fix g assume asm': "g: A \<rightarrow> A"
+              show "g \<circ> id ~ id: U i" by (derive lems: asm asm' homotopic_def compose_def)
+              next
+              show "\<^bold>\<lambda>x. refl x: id \<circ> id ~ id"
+              proof compute
+                show "\<^bold>\<lambda>x. refl x: id ~ id" by (subst homotopic_def) (derive lems: asm)
+              qed (rule asm)
+            qed (routine lems: asm)
+          qed
+        qed fact+
+      qed
+      next
+      
+      fix A' B' p' assume asm:  "A': U i" "B': U i" "p': A' =[U i] B'"
+      show "isequiv (transport p'): U i"
+      proof (rule isequiv_type)
+        show "transport p': A' \<rightarrow> B'" by (derive lems: asm transport_def)
+      qed fact+
+    qed fact+
+  qed
+  next
+
+  show "A =[U i] B: U (Suc i)" proof (derive lems: assms, (rule U_hierarchy, rule lt_Suc)?)+
+qed
+
+
+text "The univalence axiom."
+
+axiomatization univalence :: t where
+  UA: "univalence: isequiv idtoeqv"
+
+
+end
diff --git a/ex/HoTT book/Ch1.thy b/ex/HoTT book/Ch1.thy
index d5f05dd..a577fca 100644
--- a/ex/HoTT book/Ch1.thy
+++ b/ex/HoTT book/Ch1.thy
@@ -40,4 +40,16 @@ proof (rule Sum_elim[where ?p=p])
 qed (derive lems: assms)
 
 
+section \<open>Exercises\<close>
+
+text "Exercise 1.13"
+
+abbreviation "not" ("\<not>'(_')") where "\<not>(A) \<equiv> A \<rightarrow> \<zero>"
+
+text "This proof requires the use of universe cumulativity."
+
+proposition assumes "A: U(i)" shows "\<^bold>\<lambda>f. f`(inr(\<^bold>\<lambda>a. f`inl(a))): \<not>(\<not>(A + \<not>(A)))"
+by (derive lems: assms U_cumulative[where ?A=\<zero> and ?i=O and ?j=i])
+
+
 end