(FEAT) Term elaboration of assumption and goal statements.

  . spartan/core/goals.ML
  . spartan/core/elaboration.ML
  . spartan/core/elaborated_statement.ML

(FEAT) More context tacticals and search tacticals.
  . spartan/core/context_tactical.ML

(FEAT) Improved subgoal focus.
  Moves fully elaborated assumptions into the context (MINOR INCOMPATIBILITY).
  . spartan/core/focus.ML

(FIX) Normalize facts in order to be able to resolve properly.
  . spartan/core/typechecking.ML

(MAIN) New definitions.
(MAIN) Renamed theories and theorems.
(MAIN) Refactor theories to fit new features.

5 files changed, 400 insertions, 402 deletions

diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy
index d844b59..66c64f6 100644
--- a/hott/Equivalence.thy
+++ b/hott/Equivalence.thy
@@ -16,53 +16,68 @@ 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"
+    "f: \<Prod>x: A. B x"
+    "g: \<Prod>x: A. B x"
   shows "f ~ g: U i"
-  unfolding homotopy_def by typechk
+  unfolding homotopy_def
+  by typechk
 
-Lemma (derive) homotopy_refl [refl]:
+Lemma apply_homotopy:
+  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"
+    "H: f ~ g"
+    "x: A"
+  shows "H x: f x = g x"
+  using \<open>H:_\<close> unfolding homotopy_def
+  by typechk
+
+method htpy for H::o = rule apply_homotopy[where ?H=H]
+
+Lemma (def) homotopy_refl [refl]:
   assumes
     "A: U i"
     "f: \<Prod>x: A. B x"
   shows "f ~ f"
-  unfolding homotopy_def by intros
+  unfolding homotopy_def
+  by intros
 
-Lemma (derive) hsym:
+Lemma (def) 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"
+    "f: \<Prod>x: A. B x"
+    "g: \<Prod>x: A. B x"
     "H: f ~ g"
   shows "g ~ f"
-unfolding homotopy_def
-proof intro
-  fix x assume "x: A" then have "H x: f x = g x"
-    using \<open>H:_\<close>[unfolded homotopy_def]
-      \<comment> \<open>this should become unnecessary when definitional unfolding is implemented\<close>
-    by typechk
-  thus "g x = f x" by (rule pathinv) fact
-qed typechk
-
-Lemma (derive) htrans:
+  unfolding homotopy_def
+  proof intro
+    fix x assume "x: A" then have "f x = g x"
+      by (htpy H)
+    thus "g x = f x"
+      by (rule pathinv) fact
+  qed
+
+Lemma (def) htrans:
   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"
     "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"
+    "H1: f ~ g"
+    "H2: g ~ h"
+  shows "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
+  proof intro
+    fix x assume "x: A"
+    have *: "f x = g x" "g x = h x"
+      by (htpy H1, htpy H2)
+    show "f x = h x"
+      by (rule pathcomp; (rule *)?) typechk
+  qed
 
-text \<open>For calculations:\<close>
+section \<open>Rewriting homotopies\<close>
 
 congruence "f ~ g" rhs g
 
@@ -70,229 +85,258 @@ 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 = y"
-    "f: A \<rightarrow> B" "g: A \<rightarrow> B"
-    "H: 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_refl)
-          \<guillemotright> by (rule pathinv) (rule refl_pathcomp)
-          \<guillemotright> by typechk
-    done
-  done
-
-Corollary (derive) commute_homotopy':
-  assumes
-    "A: U i"
-    "x: A"
-    "f: A \<rightarrow> A"
-    "H: f ~ (id A)"
-  shows "H (f x) = f [H x]"
-oops
-
-Lemma homotopy_id_left [typechk]:
+Lemma id_funcomp_htpy:
   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
+  by reduce
 
-Lemma homotopy_id_right [typechk]:
+Lemma funcomp_id_htpy:
   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
+  by reduce
 
-Lemma homotopy_funcomp_left:
+Lemma funcomp_left_htpy:
   assumes
-    "H: homotopy B C g g'"
+    "A: U i" "B: U i"
+    "\<And>x. x: B \<Longrightarrow> C x: U i"
     "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)"
+    "H: g ~ g'"
+  shows "(g \<circ> f) ~ (g' \<circ> f)"
   unfolding homotopy_def
-  apply (intro; reduce)
-    apply (insert \<open>H: _\<close>[unfolded homotopy_def])
-      apply (elim H)
+  apply (intro, reduce)
+  apply (htpy H)
   done
 
-Lemma homotopy_funcomp_right:
+Lemma funcomp_right_htpy:
   assumes
-    "H: homotopy A (fn _. B) f f'"
+    "A: U i" "B: U i" "C: U i"
     "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')"
+    "H: f ~ f'"
+  shows "(g \<circ> f) ~ (g \<circ> f')"
   unfolding homotopy_def
-  apply (intro; reduce)
-    apply (insert \<open>H: _\<close>[unfolded homotopy_def])
-      apply (dest PiE, assumption)
-      apply (rule ap, assumption)
-  done
+  proof (intro, reduce)
+    fix x assume "x: A"
+    have *: "f x = f' x"
+      by (htpy H)
+    show "g (f x) = g (f' x)"
+      by (transport eq: *) refl
+  qed
 
-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)
+method lhtpy = rule funcomp_left_htpy[rotated 6]
+method rhtpy = rule funcomp_right_htpy[rotated 6]
+
+
+Lemma (def) commute_homotopy:
+  assumes
+    "A: U i" "B: U i"
+    "x: A" "y: A"
+    "p: x = y"
+    "f: A \<rightarrow> B" "g: A \<rightarrow> B"
+    "H: f ~ g"
+  shows "(H x) \<bullet> g[p] = f[p] \<bullet> (H y)"
+  using \<open>H:_\<close>
+  unfolding homotopy_def
+  apply (eq p, reduce)
+  apply (transport eq: pathcomp_refl, transport eq: refl_pathcomp)
+  by refl
+
+Corollary (def) commute_homotopy':
+  assumes
+    "A: U i"
+    "x: A"
+    "f: A \<rightarrow> A"
+    "H: f ~ (id A)"
+  shows "H (f x) = f [H x]"
+oops
 
 
 section \<open>Quasi-inverse and bi-invertibility\<close>
 
 subsection \<open>Quasi-inverses\<close>
 
-definition "qinv A B f \<equiv> \<Sum>g: B \<rightarrow> A.
+definition "is_qinv A B f \<equiv> \<Sum>g: B \<rightarrow> A.
   homotopy A (fn _. A) (g \<circ>\<^bsub>A\<^esub> f) (id A) \<times> homotopy B (fn _. B) (f \<circ>\<^bsub>B\<^esub> g) (id B)"
 
-lemma qinv_type [typechk]:
+lemma is_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
+  shows "is_qinv A B f: U i"
+  unfolding is_qinv_def
+  by typechk
 
-definition qinv_i ("qinv")
-  where [implicit]: "qinv f \<equiv> Equivalence.qinv ? ? f"
+definition is_qinv_i ("is'_qinv")
+  where [implicit]: "is_qinv f \<equiv> Equivalence.is_qinv ? ? f"
 
-translations "qinv f" \<leftharpoondown> "CONST Equivalence.qinv A B f"
+no_translations "is_qinv f" \<leftharpoondown> "CONST Equivalence.is_qinv A B f"
 
-Lemma (derive) id_qinv:
+Lemma (def) id_is_qinv:
   assumes "A: U i"
-  shows "qinv (id A)"
-  unfolding qinv_def
+  shows "is_qinv (id A)"
+  unfolding is_qinv_def
   proof intro
     show "id A: A \<rightarrow> A" by typechk
   qed (reduce, intro; refl)
 
-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
+Lemma is_qinvI:
+  assumes
+    "A: U i" "B: U i" "f: A \<rightarrow> B"
+    "g: B \<rightarrow> A"
+    "H1: g \<circ> f ~ id A"
+    "H2: f \<circ> g ~ id B"
+  shows "is_qinv f"
+  unfolding is_qinv_def
+  proof intro
+    show "g: B \<rightarrow> A" by fact
+    show "g \<circ> f ~ id A \<and> f \<circ> g ~ id B" by (intro; fact)
+  qed
+
+Lemma is_qinv_components [typechk]:
+  assumes "A: U i" "B: U i" "f: A \<rightarrow> B" "pf: is_qinv f"
+  shows
+    fst_of_is_qinv: "fst pf: B \<rightarrow> A" and
+    p\<^sub>2\<^sub>1_of_is_qinv: "p\<^sub>2\<^sub>1 pf: fst pf \<circ> f ~ id A" and
+    p\<^sub>2\<^sub>2_of_is_qinv: "p\<^sub>2\<^sub>2 pf: f \<circ> fst pf ~ id B"
+  using assms unfolding is_qinv_def
+  by typechk+
+
+Lemma (def) qinv_is_qinv:
+  assumes "A: U i" "B: U i" "f: A \<rightarrow> B" "pf: is_qinv f"
+  shows "is_qinv (fst pf)"
+using [[debug_typechk]]
+  using [[solve_side_conds=2]]
+  apply (rule is_qinvI)
+  back \<comment> \<open>Typechecking/inference goes too far here. Problem would likely be
+    solved with definitional unfolding\<close>
+  \<^item> by (fact \<open>f:_\<close>)
+  \<^item> by (rule p\<^sub>2\<^sub>2_of_is_qinv)
+  \<^item> by (rule p\<^sub>2\<^sub>1_of_is_qinv)
   done
 
-Lemma (derive) funcomp_qinv:
+Lemma (def) funcomp_is_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 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
+  shows "is_qinv f \<rightarrow> is_qinv g \<rightarrow> is_qinv (g \<circ> f)"
+  apply intros
+  unfolding is_qinv_def apply elims
+  focus vars _ _ finv _ ginv
+    apply intro
+    \<^item> by (rule funcompI[where ?f=ginv and ?g=finv])
+    \<^item> proof intro
+        show "(finv \<circ> ginv) \<circ> g \<circ> f ~ id A"
+        proof -
+          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 (rhtpy, lhtpy) fact
+          also have ".. ~ id A" by reduce fact
+          finally show "{}" by this
+        qed
+    
+        show "(g \<circ> f) \<circ> finv \<circ> ginv ~ id C"
+        proof -
+          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 (rhtpy, lhtpy) fact
+          also have ".. ~ id C" by reduce fact
+          finally show "{}" by this
+        qed
+      qed
+    done
   done
 
 subsection \<open>Bi-invertible maps\<close>
 
-definition "biinv A B f \<equiv>
+definition "is_biinv A B f \<equiv>
   (\<Sum>g: B \<rightarrow> A. homotopy A (fn _. A) (g \<circ>\<^bsub>A\<^esub> f) (id A))
     \<times> (\<Sum>g: B \<rightarrow> A. homotopy B (fn _. B) (f \<circ>\<^bsub>B\<^esub> g) (id B))"
 
-lemma biinv_type [typechk]:
+lemma is_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
+  shows "is_biinv A B f: U i"
+  unfolding is_biinv_def by typechk
 
-definition biinv_i ("biinv")
-  where [implicit]: "biinv f \<equiv> Equivalence.biinv ? ? f"
+definition is_biinv_i ("is'_biinv")
+  where [implicit]: "is_biinv f \<equiv> Equivalence.is_biinv ? ? f"
 
-translations "biinv f" \<leftharpoondown> "CONST Equivalence.biinv A B f"
+translations "is_biinv f" \<leftharpoondown> "CONST Equivalence.is_biinv A B f"
 
-Lemma (derive) qinv_imp_biinv:
+Lemma is_biinvI:
   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>
+    "A: U i" "B: U i" "f: A \<rightarrow> B"
+    "g: B \<rightarrow> A" "h: B \<rightarrow> A"
+    "H1: g \<circ> f ~ id A" "H2: f \<circ> h ~ id B"
+  shows "is_biinv f"
+  unfolding is_biinv_def
+  proof intro
+    show "<g, H1>: {}" by typechk
+    show "<h, H2>: {}" by typechk
+  qed
 
-Lemma (derive) biinv_imp_qinv:
+Lemma is_biinv_components [typechk]:
+  assumes "A: U i" "B: U i" "f: A \<rightarrow> B" "pf: is_biinv f"
+  shows
+    p\<^sub>1\<^sub>1_of_is_biinv: "p\<^sub>1\<^sub>1 pf: B \<rightarrow> A" and
+    p\<^sub>2\<^sub>1_of_is_biinv: "p\<^sub>2\<^sub>1 pf: B \<rightarrow> A" and
+    p\<^sub>1\<^sub>2_of_is_biinv: "p\<^sub>1\<^sub>2 pf: p\<^sub>1\<^sub>1 pf \<circ> f ~ id A" and
+    p\<^sub>2\<^sub>2_of_is_biinv: "p\<^sub>2\<^sub>2 pf: f \<circ> p\<^sub>2\<^sub>1 pf ~ id B"
+  using assms unfolding is_biinv_def
+  by typechk+
+
+Lemma (def) is_qinv_imp_is_biinv:
   assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
-  shows "biinv f \<rightarrow> qinv f"
+  shows "is_qinv f \<rightarrow> is_biinv f"
+  apply intros
+  unfolding is_qinv_def is_biinv_def
+  by (rule distribute_Sig)
 
-  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
+Lemma (def) is_biinv_imp_is_qinv:
+  assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
+  shows "is_biinv f \<rightarrow> is_qinv f"
   apply intro
-    \<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 (fact \<open>H1: _\<close>)
 
+  text \<open>Split the hypothesis \<^term>\<open>is_biinv f\<close> into its components and name them.\<close>
+  unfolding is_biinv_def apply elims
+  focus vars _ _ _ g H1 h H2
+    text \<open>Need to give the required function and homotopies.\<close>
+    apply (rule is_qinvI)
+      text \<open>We claim that \<^term>\<open>g\<close> is a quasi-inverse to \<^term>\<open>f\<close>.\<close>
+      \<^item> by (fact \<open>g: _\<close>)
+  
+      text \<open>The first part \<^prop>\<open>?H1: g \<circ> f ~ id A\<close> is given by \<^term>\<open>H1\<close>.\<close>
+      \<^item> by (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>,
         then the result follows from @{thm \<open>H2: f \<circ> h ~ id B\<close>}. Here a proof
-        block is used to calculate "forward".
+        block is used for calculational reasoning.
       \<close>
-      proof -
-        have "g ~ g \<circ> (id B)" by reduce refl
-        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 (o_htpy, this)
-        also note \<open>H2:_\<close>
-        finally show "f \<circ> g ~ id B" by this
-      qed
+      \<^item> proof -
+          have "g ~ g \<circ> (id B)" by reduce refl
+          also have ".. ~ g \<circ> f \<circ> h" by rhtpy (rule \<open>H2:_\<close>[symmetric])
+          also have ".. ~ (id A) \<circ> h" by (rewrite funcomp_assoc[symmetric]) (lhtpy, fact)
+          also have ".. ~ h" by reduce refl
+          finally have "g ~ h" by this
+          then have "f \<circ> g ~ f \<circ> h" by (rhtpy, this)
+          also note \<open>H2:_\<close>
+          finally show "f \<circ> g ~ id B" by this
+        qed
     done
   done
 
-Lemma (derive) id_biinv:
-  "A: U i \<Longrightarrow> biinv (id A)"
-  by (rule qinv_imp_biinv) (rule id_qinv)
+Lemma (def) id_is_biinv:
+  "A: U i \<Longrightarrow> is_biinv (id A)"
+  by (rule is_qinv_imp_is_biinv) (rule id_is_qinv)
 
-Lemma (derive) funcomp_biinv:
+Lemma (def) funcomp_is_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)"
+  shows "is_biinv f \<rightarrow> is_biinv g \<rightarrow> is_biinv (g \<circ> f)"
   apply intros
-  focus vars biinv\<^sub>f biinv\<^sub>g
-    by (rule qinv_imp_biinv) (rule funcomp_qinv; rule biinv_imp_qinv)
+  focus vars pf pg
+    by (rule is_qinv_imp_is_biinv)
+       (rule funcomp_is_qinv; rule is_biinv_imp_is_qinv, fact)
   done
 
 
@@ -304,19 +348,19 @@ text \<open>
 \<close>
 
 definition equivalence (infix "\<simeq>" 110)
-  where "A \<simeq> B \<equiv> \<Sum>f: A \<rightarrow> B. Equivalence.biinv A B f"
+  where "A \<simeq> B \<equiv> \<Sum>f: A \<rightarrow> B. Equivalence.is_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:
+Lemma (def) equivalence_refl:
   assumes "A: U i"
   shows "A \<simeq> A"
   unfolding equivalence_def
   proof intro
-    show "biinv (id A)" by (rule qinv_imp_biinv) (rule id_qinv)
+    show "is_biinv (id A)" by (rule is_qinv_imp_is_biinv) (rule id_is_qinv)
   qed typechk
 
 text \<open>
@@ -324,32 +368,30 @@ text \<open>
   univalence?)...
 \<close>
 
-Lemma (derive) equivalence_symmetric:
+Lemma (def) 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
+  apply (dest (4) is_biinv_imp_is_qinv)
+  apply intro
+    \<^item> by (rule fst_of_is_qinv) facts
+    \<^item> by (rule is_qinv_imp_is_biinv) (rule qinv_is_qinv)
   done
 
-Lemma (derive) equivalence_transitive:
+Lemma (def) equivalence_transitive:
   assumes "A: U i" "B: U i" "C: U i"
   shows "A \<simeq> B \<rightarrow> B \<simeq> C \<rightarrow> A \<simeq> C"
   proof intros
     fix AB BC assume *: "AB: A \<simeq> B" "BC: B \<simeq> C"
     then have
-      "fst AB: A \<rightarrow> B" and 1: "snd AB: biinv (fst AB)"
-      "fst BC: B \<rightarrow> C" and 2: "snd BC: biinv (fst BC)"
+      "fst AB: A \<rightarrow> B" and 1: "snd AB: is_biinv (fst AB)"
+      "fst BC: B \<rightarrow> C" and 2: "snd BC: is_biinv (fst BC)"
       unfolding equivalence_def by typechk+
     then have "fst BC \<circ> fst AB: A \<rightarrow> C" by typechk
-    moreover have "biinv (fst BC \<circ> fst AB)"
-      using * unfolding equivalence_def by (rules funcomp_biinv 1 2)
+    moreover have "is_biinv (fst BC \<circ> fst AB)"
+      using * unfolding equivalence_def by (rule funcomp_is_biinv 1 2) facts
     ultimately show "A \<simeq> C"
       unfolding equivalence_def by intro facts
   qed
@@ -370,14 +412,14 @@ text \<open>
   (2) we don't yet have universe automation.
 \<close>
 
-Lemma (derive) id_imp_equiv:
+Lemma (def) 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
-    \<guillemotright> apply (eq p)
-      \<^item> prems vars A B
+    \<^item> apply (eq p)
+      \<^enum> vars A B
         apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric])
         using [[solve_side_conds=1]]
         apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric])
@@ -385,16 +427,16 @@ Lemma (derive) id_imp_equiv:
         apply (rule lift_universe_codomain, rule Ui_in_USi)
         apply (typechk, rule Ui_in_USi)
         done
-      \<^item> prems vars A
+      \<^enum> vars A
         using [[solve_side_conds=1]]
         apply (subst transport_comp)
-          \<^enum> by (rule Ui_in_USi)
-          \<^enum> by reduce (rule in_USi_if_in_Ui)
-          \<^enum> by reduce (rule id_biinv)
+          \<circ> by (rule Ui_in_USi)
+          \<circ> by reduce (rule in_USi_if_in_Ui)
+          \<circ> by reduce (rule id_is_biinv)
         done
       done
 
-    \<guillemotright> \<comment> \<open>Similar proof as in the first subgoal above\<close>
+    \<^item> \<comment> \<open>Similar proof as in the first subgoal above\<close>
       apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric])
       using [[solve_side_conds=1]]
       apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric])
@@ -419,8 +461,8 @@ no_translations
   "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"
+  "is_qinv f" \<leftharpoondown> "CONST Equivalence.is_qinv A B f"
+  "is_biinv f" \<leftharpoondown> "CONST Equivalence.is_biinv A B f"
 *)
 
 
diff --git a/hott/Identity.thy b/hott/Identity.thy
index 728537c..d6efbf8 100644
--- a/hott/Identity.thy
+++ b/hott/Identity.thy
@@ -42,7 +42,7 @@ axiomatization where
 lemmas
   [form] = IdF and
   [intro, intros] = IdI and
-  [elim "?p" "?a" "?b"] = IdE and
+  [elim ?p ?a ?b] = IdE and
   [comp] = Id_comp and
   [refl] = IdI
 
@@ -77,34 +77,34 @@ method_setup eq =
 
 section \<open>Symmetry and transitivity\<close>
 
-Lemma (derive) pathinv:
+Lemma (def) pathinv:
   assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
   shows "y =\<^bsub>A\<^esub> x"
   by (eq p) intro
 
 lemma pathinv_comp [comp]:
-  assumes "x: A" "A: U i"
+  assumes "A: U i" "x: A"
   shows "pathinv A x x (refl x) \<equiv> refl x"
   unfolding pathinv_def by reduce
 
-Lemma (derive) pathcomp:
+Lemma (def) pathcomp:
   assumes
     "A: U i" "x: A" "y: A" "z: A"
     "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z"
   shows
     "x =\<^bsub>A\<^esub> z"
-  apply (eq p)
-    focus prems vars x p
-      apply (eq p)
-        apply intro
-    done
-  done
+  proof (eq p)
+    fix x q assume "x: A" and "q: x =\<^bsub>A\<^esub> z"
+    show "x =\<^bsub>A\<^esub> z" by (eq q) refl
+  qed
 
 lemma pathcomp_comp [comp]:
-  assumes "a: A" "A: U i"
+  assumes "A: U i" "a: A"
   shows "pathcomp A a a a (refl a) (refl a) \<equiv> refl a"
   unfolding pathcomp_def by reduce
 
+method pathcomp for p q :: o = rule pathcomp[where ?p=p and ?q=q]
+
 
 section \<open>Notation\<close>
 
@@ -134,26 +134,22 @@ lemmas
 
 section \<open>Basic propositional equalities\<close>
 
-Lemma (derive) refl_pathcomp:
+Lemma (def) refl_pathcomp:
   assumes "A: U i" "x: A" "y: A" "p: x = y"
   shows "(refl x) \<bullet> p = p"
-  apply (eq p)
-    apply (reduce; intro)
-  done
+  by (eq p) (reduce, refl)
 
-Lemma (derive) pathcomp_refl:
+Lemma (def) pathcomp_refl:
   assumes "A: U i" "x: A" "y: A" "p: x = y"
   shows "p \<bullet> (refl y) = p"
-  apply (eq p)
-    apply (reduce; intro)
-  done
+  by (eq p) (reduce, refl)
 
-definition [implicit]: "ru p \<equiv> pathcomp_refl ? ? ? p"
 definition [implicit]: "lu p \<equiv> refl_pathcomp ? ? ? p"
+definition [implicit]: "ru p \<equiv> pathcomp_refl ? ? ? p"
 
 translations
-  "CONST ru p" \<leftharpoondown> "CONST pathcomp_refl A x y p"
   "CONST lu p" \<leftharpoondown> "CONST refl_pathcomp A x y p"
+  "CONST ru p" \<leftharpoondown> "CONST pathcomp_refl A x y p"
 
 Lemma lu_refl [comp]:
   assumes "A: U i" "x: A"
@@ -165,40 +161,40 @@ Lemma ru_refl [comp]:
   shows "ru (refl x) \<equiv> refl (refl x)"
   unfolding pathcomp_refl_def by reduce
 
-Lemma (derive) inv_pathcomp:
+Lemma (def) inv_pathcomp:
   assumes "A: U i" "x: A" "y: A" "p: x = y"
   shows "p\<inverse> \<bullet> p = refl y"
-  by (eq p) (reduce; intro)
+  by (eq p) (reduce, refl)
 
-Lemma (derive) pathcomp_inv:
+Lemma (def) pathcomp_inv:
   assumes "A: U i" "x: A" "y: A" "p: x = y"
   shows "p \<bullet> p\<inverse> = refl x"
-  by (eq p) (reduce; intro)
+  by (eq p) (reduce, refl)
 
-Lemma (derive) pathinv_pathinv:
+Lemma (def) pathinv_pathinv:
   assumes "A: U i" "x: A" "y: A" "p: x = y"
   shows "p\<inverse>\<inverse> = p"
-  by (eq p) (reduce; intro)
+  by (eq p) (reduce, refl)
 
-Lemma (derive) pathcomp_assoc:
+Lemma (def) pathcomp_assoc:
   assumes
     "A: U i" "x: A" "y: A" "z: A" "w: A"
     "p: x = y" "q: y = z" "r: z = w"
   shows "p \<bullet> (q \<bullet> r) = p \<bullet> q \<bullet> r"
-  apply (eq p)
-    focus prems vars x p
-      apply (eq p)
-        focus prems vars x p
-          apply (eq p)
-            apply (reduce; intro)
-        done
-    done
-  done
+  proof (eq p)
+    fix x q assuming "x: A" "q: x = z"
+    show "refl x \<bullet> (q \<bullet> r) = refl x \<bullet> q \<bullet> r"
+    proof (eq q)
+      fix x r assuming "x: A" "r: x = w"
+      show "refl x \<bullet> (refl x \<bullet> r) = refl x \<bullet> refl x \<bullet> r"
+        by (eq r) (reduce, refl)
+    qed
+  qed
 
 
 section \<open>Functoriality of functions\<close>
 
-Lemma (derive) ap:
+Lemma (def) ap:
   assumes
     "A: U i" "B: U i"
     "x: A" "y: A"
@@ -213,11 +209,11 @@ definition ap_i ("_[_]" [1000, 0])
 translations "f[p]" \<leftharpoondown> "CONST ap A B x y f p"
 
 Lemma ap_refl [comp]:
-  assumes "f: A \<rightarrow> B" "x: A" "A: U i" "B: U i"
+  assumes "A: U i" "B: U i" "f: A \<rightarrow> B" "x: A"
   shows "f[refl x] \<equiv> refl (f x)"
   unfolding ap_def by reduce
 
-Lemma (derive) ap_pathcomp:
+Lemma (def) ap_pathcomp:
   assumes
     "A: U i" "B: U i"
     "x: A" "y: A" "z: A"
@@ -225,48 +221,45 @@ Lemma (derive) ap_pathcomp:
     "p: x = y" "q: y = z"
   shows
     "f[p \<bullet> q] = f[p] \<bullet> f[q]"
-  apply (eq p)
-    focus prems vars x p
-      apply (eq p)
-        apply (reduce; intro)
-    done
-  done
+  proof (eq p)
+    fix x q assuming "x: A" "q: x = z"
+    show "f[refl x \<bullet> q] = f[refl x] \<bullet> f[q]"
+      by (eq q) (reduce, refl)
+  qed
 
-Lemma (derive) ap_pathinv:
+Lemma (def) ap_pathinv:
   assumes
     "A: U i" "B: U i"
     "x: A" "y: A"
     "f: A \<rightarrow> B"
     "p: x = y"
   shows "f[p\<inverse>] = f[p]\<inverse>"
-  by (eq p) (reduce; intro)
-
-text \<open>The next two proofs currently use some low-level rewriting.\<close>
+  by (eq p) (reduce, refl)
 
-Lemma (derive) ap_funcomp:
+Lemma (def) ap_funcomp:
   assumes
     "A: U i" "B: U i" "C: U i"
     "x: A" "y: A"
     "f: A \<rightarrow> B" "g: B \<rightarrow> C"
     "p: x = y"
-  shows "(g \<circ> f)[p] = g[f[p]]"
+  shows "(g \<circ> f)[p] = g[f[p]]" thm ap
   apply (eq p)
-    \<guillemotright> by reduce
-    \<guillemotright> by reduce intro
+  \<^item> by reduce
+  \<^item> by reduce refl
   done
 
-Lemma (derive) ap_id:
+Lemma (def) ap_id:
   assumes "A: U i" "x: A" "y: A" "p: x = y"
   shows "(id A)[p] = p"
   apply (eq p)
-    \<guillemotright> by reduce
-    \<guillemotright> by reduce intro
+  \<^item> by reduce
+  \<^item> by reduce refl
   done
 
 
 section \<open>Transport\<close>
 
-Lemma (derive) transport:
+Lemma (def) transport:
   assumes
     "A: U i"
     "\<And>x. x: A \<Longrightarrow> P x: U i"
@@ -288,22 +281,18 @@ Lemma transport_comp [comp]:
   shows "trans P (refl a) \<equiv> id (P a)"
   unfolding transport_def by reduce
 
-\<comment> \<open>TODO: Build transport automation!\<close>
-
-Lemma use_transport:
+Lemma apply_transport:
   assumes
+    "A: U i" "\<And>x. x: A \<Longrightarrow> P x: U i"
+    "x: A" "y: A"
     "p: y =\<^bsub>A\<^esub> x"
     "u: P x"
-    "x: A" "y: A"
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> P x: U i"
   shows "trans P p\<inverse> u: P y"
-  unfolding transport_def pathinv_def
   by typechk
 
-method transport uses eq = (rule use_transport[OF eq])
+method transport uses eq = (rule apply_transport[OF _ _ _ _ eq])
 
-Lemma (derive) transport_left_inv:
+Lemma (def) transport_left_inv:
   assumes
     "A: U i"
     "\<And>x. x: A \<Longrightarrow> P x: U i"
@@ -312,16 +301,16 @@ Lemma (derive) transport_left_inv:
   shows "(trans P p\<inverse>) \<circ> (trans P p) = id (P x)"
   by (eq p) (reduce; refl)
 
-Lemma (derive) transport_right_inv:
+Lemma (def) transport_right_inv:
   assumes
     "A: U i"
     "\<And>x. x: A \<Longrightarrow> P x: U i"
     "x: A" "y: A"
     "p: x = y"
   shows "(trans P p) \<circ> (trans P p\<inverse>) = id (P y)"
-  by (eq p) (reduce; intro)
+  by (eq p) (reduce, refl)
 
-Lemma (derive) transport_pathcomp:
+Lemma (def) transport_pathcomp:
   assumes
     "A: U i"
     "\<And>x. x: A \<Longrightarrow> P x: U i"
@@ -329,14 +318,14 @@ Lemma (derive) transport_pathcomp:
     "u: P x"
     "p: x = y" "q: y = z"
   shows "trans P q (trans P p u) = trans P (p \<bullet> q) u"
-  apply (eq p)
-    focus prems vars x p
-      apply (eq p)
-        apply (reduce; intro)
-    done
-  done
-
-Lemma (derive) transport_compose_typefam:
+proof (eq p)
+  fix x q u
+  assuming "x: A" "q: x = z" "u: P x"
+  show "trans P q (trans P (refl x) u) = trans P ((refl x) \<bullet> q) u"
+    by (eq q) (reduce, refl)
+qed
+
+Lemma (def) transport_compose_typefam:
   assumes
     "A: U i" "B: U i"
     "\<And>x. x: B \<Longrightarrow> P x: U i"
@@ -345,9 +334,9 @@ Lemma (derive) transport_compose_typefam:
     "p: x = y"
     "u: P (f x)"
   shows "trans (fn x. P (f x)) p u = trans P f[p] u"
-  by (eq p) (reduce; intro)
+  by (eq p) (reduce, refl)
 
-Lemma (derive) transport_function_family:
+Lemma (def) transport_function_family:
   assumes
     "A: U i"
     "\<And>x. x: A \<Longrightarrow> P x: U i"
@@ -357,15 +346,15 @@ Lemma (derive) transport_function_family:
     "u: P x"
     "p: x = y"
   shows "trans Q p ((f x) u) = (f y) (trans P p u)"
-  by (eq p) (reduce; intro)
+  by (eq p) (reduce, refl)
 
-Lemma (derive) transport_const:
+Lemma (def) transport_const:
   assumes
     "A: U i" "B: U i"
     "x: A" "y: A"
     "p: x = y"
   shows "\<Prod>b: B. trans (fn _. B) p b = b"
-  by (intro, eq p) (reduce; intro)
+  by intro (eq p, reduce, refl)
 
 definition transport_const_i ("trans'_const")
   where [implicit]: "trans_const B p \<equiv> transport_const ? B ? ? p"
@@ -376,10 +365,10 @@ Lemma transport_const_comp [comp]:
   assumes
     "x: A" "b: B"
     "A: U i" "B: U i"
-  shows "trans_const B (refl x) b\<equiv> refl b"
+  shows "trans_const B (refl x) b \<equiv> refl b"
   unfolding transport_const_def by reduce
 
-Lemma (derive) pathlift:
+Lemma (def) pathlift:
   assumes
     "A: U i"
     "\<And>x. x: A \<Longrightarrow> P x: U i"
@@ -387,7 +376,7 @@ Lemma (derive) pathlift:
     "p: x = y"
     "u: P x"
   shows "<x, u> = <y, trans P p u>"
-  by (eq p) (reduce; intro)
+  by (eq p) (reduce, refl)
 
 definition pathlift_i ("lift")
   where [implicit]: "lift P p u \<equiv> pathlift ? P ? ? p u"
@@ -403,7 +392,7 @@ Lemma pathlift_comp [comp]:
   shows "lift P (refl x) u \<equiv> refl <x, u>"
   unfolding pathlift_def by reduce
 
-Lemma (derive) pathlift_fst:
+Lemma (def) pathlift_fst:
   assumes
     "A: U i"
     "\<And>x. x: A \<Longrightarrow> P x: U i"
@@ -412,14 +401,14 @@ Lemma (derive) pathlift_fst:
     "p: x = y"
   shows "fst[lift P p u] = p"
   apply (eq p)
-    \<guillemotright> by reduce
-    \<guillemotright> by reduce intro
+  \<^item> by reduce
+  \<^item> by reduce refl
   done
 
 
 section \<open>Dependent paths\<close>
 
-Lemma (derive) apd:
+Lemma (def) apd:
   assumes
     "A: U i"
     "\<And>x. x: A \<Longrightarrow> P x: U i"
@@ -427,7 +416,7 @@ Lemma (derive) apd:
     "x: A" "y: A"
     "p: x = y"
   shows "trans P p (f x) = f y"
-  by (eq p) (reduce; intro)
+  by (eq p) (reduce, refl)
 
 definition apd_i ("apd")
   where [implicit]: "apd f p \<equiv> Identity.apd ? ? f ? ? p"
@@ -443,26 +432,25 @@ Lemma dependent_map_comp [comp]:
   shows "apd f (refl x) \<equiv> refl (f x)"
   unfolding apd_def by reduce
 
-Lemma (derive) apd_ap:
+Lemma (def) apd_ap:
   assumes
     "A: U i" "B: U i"
     "f: A \<rightarrow> B"
     "x: A" "y: A"
     "p: x = y"
   shows "apd f p = trans_const B p (f x) \<bullet> f[p]"
-  by (eq p) (reduce; intro)
+  by (eq p) (reduce, refl)
 
 
 section \<open>Whiskering\<close>
 
-Lemma (derive) right_whisker:
+Lemma (def) right_whisker:
   assumes "A: U i" "a: A" "b: A" "c: A"
       and "p: a = b" "q: a = b" "r: b = c"
       and "\<alpha>: p = q"
   shows "p \<bullet> r = q \<bullet> r"
   apply (eq r)
-  focus prems vars x s t
-    proof -
+    focus 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])
@@ -470,14 +458,13 @@ Lemma (derive) right_whisker:
     qed
   done
 
-Lemma (derive) left_whisker:
+Lemma (def) left_whisker:
   assumes "A: U i" "a: A" "b: A" "c: A"
       and "p: b = c" "q: b = c" "r: a = b"
       and "\<alpha>: p = q"
   shows "r \<bullet> p = r \<bullet> q"
   apply (eq r)
-  focus prems prms vars x s t
-    proof -
+    focus 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])
@@ -485,24 +472,24 @@ Lemma (derive) left_whisker:
     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 right_whisker_i (infix "\<bullet>\<^sub>r" 121)
+  where [implicit]: "\<alpha> \<bullet>\<^sub>r r \<equiv> right_whisker ? ? ? ? ? ? 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>"
+definition left_whisker_i (infix "\<bullet>\<^sub>l" 121)
+  where [implicit]: "r \<bullet>\<^sub>l \<alpha> \<equiv> left_whisker ? ? ? ? ? ? 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>"
+  "\<alpha> \<bullet>\<^sub>r r" \<leftharpoondown> "CONST right_whisker A a b c p q r \<alpha>"
+  "r \<bullet>\<^sub>l \<alpha>" \<leftharpoondown> "CONST left_whisker A a b c p q r \<alpha>"
 
 Lemma whisker_refl [comp]:
   assumes "A: U i" "a: A" "b: A" "p: a = b" "q: a = b" "\<alpha>: p = q"
-  shows "\<alpha> \<bullet>\<^sub>r\<^bsub>a\<^esub> (refl b) \<equiv> ru p \<bullet> \<alpha> \<bullet> (ru q)\<inverse>"
+  shows "\<alpha> \<bullet>\<^sub>r (refl b) \<equiv> ru p \<bullet> \<alpha> \<bullet> (ru q)\<inverse>"
   unfolding right_whisker_def by reduce
 
 Lemma refl_whisker [comp]:
   assumes "A: U i" "a: A" "b: A" "p: a = b" "q: a = b" "\<alpha>: p = q"
-  shows "(refl a) \<bullet>\<^sub>l\<^bsub>b\<^esub> \<alpha> \<equiv> (lu p) \<bullet> \<alpha> \<bullet> (lu q)\<inverse>"
+  shows "(refl a) \<bullet>\<^sub>l \<alpha> \<equiv> (lu p) \<bullet> \<alpha> \<bullet> (lu q)\<inverse>"
   unfolding left_whisker_def by reduce
 
 method left_whisker = (rule left_whisker)
@@ -521,7 +508,7 @@ assumes assums:
   "r: b =\<^bsub>A\<^esub> c" "s: b =\<^bsub>A\<^esub> c"
 begin
 
-Lemma (derive) horiz_pathcomp:
+Lemma (def) horiz_pathcomp:
   notes assums
   assumes "\<alpha>: p = q" "\<beta>: r = s"
   shows "p \<bullet> r = q \<bullet> s"
@@ -532,7 +519,7 @@ qed typechk
 
 text \<open>A second horizontal composition:\<close>
 
-Lemma (derive) horiz_pathcomp':
+Lemma (def) horiz_pathcomp':
   notes assums
   assumes "\<alpha>: p = q" "\<beta>: r = s"
   shows "p \<bullet> r = q \<bullet> s"
@@ -544,14 +531,14 @@ qed typechk
 notation horiz_pathcomp (infix "\<star>" 121)
 notation horiz_pathcomp' (infix "\<star>''" 121)
 
-Lemma (derive) horiz_pathcomp_eq_horiz_pathcomp':
+Lemma (def) horiz_pathcomp_eq_horiz_pathcomp':
   notes assums
   assumes "\<alpha>: p = q" "\<beta>: r = s"
   shows "\<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)
+      focus vars a q by (eq q) (reduce, refl)
     done
   done
 
@@ -580,12 +567,20 @@ interpretation \<Omega>2:
 notation \<Omega>2.horiz_pathcomp (infix "\<star>" 121)
 notation \<Omega>2.horiz_pathcomp' (infix "\<star>''" 121)
 
-Lemma (derive) pathcomp_eq_horiz_pathcomp:
+Lemma [typechk]:
+  assumes "\<alpha>: \<Omega>2 A a" and "\<beta>: \<Omega>2 A a"
+  shows horiz_pathcomp_type: "\<alpha> \<star> \<beta>: \<Omega>2 A a"
+    and horiz_pathcomp'_type: "\<alpha> \<star>' \<beta>: \<Omega>2 A a"
+  using assms
+  unfolding \<Omega>2.horiz_pathcomp_def \<Omega>2.horiz_pathcomp'_def \<Omega>2_alt_def
+  by reduce+
+
+Lemma (def) 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) 
+  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)
@@ -596,12 +591,12 @@ Lemma (derive) pathcomp_eq_horiz_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>)
+      = \<alpha> \<bullet> {}" by right_whisker (fact eq\<alpha>)
+    also have ".. = \<alpha> \<bullet> \<beta>" by left_whisker (fact eq\<beta>)
     finally show "{} = \<alpha> \<bullet> \<beta>" by this
   qed
 
-Lemma (derive) pathcomp_eq_horiz_pathcomp':
+Lemma (def) 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
@@ -616,12 +611,12 @@ Lemma (derive) pathcomp_eq_horiz_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>)
+      = \<beta> \<bullet> {}" by right_whisker (fact eq\<beta>)
+    also have ".. = \<beta> \<bullet> \<alpha>" by left_whisker (fact eq\<alpha>)
     finally show "{} = \<beta> \<bullet> \<alpha>" by this
   qed
 
-Lemma (derive) eckmann_hilton:
+Lemma (def) 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]
@@ -629,11 +624,15 @@ Lemma (derive) eckmann_hilton:
     have "\<alpha> \<bullet> \<beta> = \<alpha> \<star> \<beta>"
       by (rule pathcomp_eq_horiz_pathcomp)
     also have [simplified comp]: ".. = \<alpha> \<star>' \<beta>"
+      \<comment> \<open>Danger, Will Robinson! (Inferred implicit has an equivalent form but
+          needs to be manually simplified.)\<close>
       by (transport eq: \<Omega>2.horiz_pathcomp_eq_horiz_pathcomp') refl
     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)+
+      by (this; reduce add: \<Omega>2_alt_def[symmetric])
+      \<comment> \<open>TODO: The finishing call to `reduce` should be unnecessary with some
+          kind of definitional unfolding.\<close>
   qed
 
 end
diff --git a/hott/More_List.thy b/hott/List+.thy
index 1b8034f..b73cc24 100644
--- a/hott/More_List.thy
+++ b/hott/List+.thy
@@ -1,4 +1,4 @@
-theory More_List
+theory "List+"
 imports
   Spartan.List
   Nat
@@ -10,8 +10,8 @@ section \<open>Length\<close>
 definition [implicit]: "len \<equiv> ListRec ? Nat 0 (fn _ _ rec. suc rec)"
 
 experiment begin
-  Lemma "len [] \<equiv> ?n" by (subst comp)+
-  Lemma "len [0, suc 0, suc (suc 0)] \<equiv> ?n" by (subst comp)+
+  Lemma "len [] \<equiv> ?n" by (subst comp)
+  Lemma "len [0, suc 0, suc (suc 0)] \<equiv> ?n" by (subst comp)
 end
 
 
diff --git a/hott/More_Nat.thy b/hott/More_Nat.thy
deleted file mode 100644
index 59c8d54..0000000
--- a/hott/More_Nat.thy
+++ /dev/null
@@ -1,43 +0,0 @@
-theory More_Nat
-imports Nat Spartan.More_Types
-
-begin
-
-section \<open>Equality on Nat\<close>
-
-text \<open>Via the encode-decode method.\<close>
-
-context begin
-
-Lemma (derive) eq: shows "Nat \<rightarrow> Nat \<rightarrow> U O"
-  apply intro focus vars m apply elim
-    \<comment> \<open>m \<equiv> 0\<close>
-    apply intro focus vars n apply elim
-      \<guillemotright> by (rule TopF) \<comment> \<open>n \<equiv> 0\<close>
-      \<guillemotright> by (rule BotF) \<comment> \<open>n \<equiv> suc _\<close>
-      \<guillemotright> by (rule Ui_in_USi)
-    done
-    \<comment> \<open>m \<equiv> suc k\<close>
-    apply intro focus vars k k_eq n apply (elim n)
-      \<guillemotright> by (rule BotF) \<comment> \<open>n \<equiv> 0\<close>
-      \<guillemotright> prems vars l proof - show "k_eq l: U O" by typechk qed
-      \<guillemotright> by (rule Ui_in_USi)
-    done
-    by (rule Ui_in_USi)
-  done
-
-text \<open>
-\<^term>\<open>eq\<close> is defined by
-  eq 0 0 \<equiv> \<top>
-  eq 0 (suc k) \<equiv> \<bottom>
-  eq (suc k) 0 \<equiv> \<bottom>
-  eq (suc k) (suc l) \<equiv> eq k l
-\<close>
-
-
-
-
-end
-
-
-end
diff --git a/hott/Nat.thy b/hott/Nat.thy
index 177ec47..fd567f3 100644
--- a/hott/Nat.thy
+++ b/hott/Nat.thy
@@ -77,43 +77,43 @@ lemma add_suc [comp]:
   shows "m + suc n \<equiv> suc (m + n)"
   unfolding add_def by reduce
 
-Lemma (derive) zero_add:
+Lemma (def) zero_add:
   assumes "n: Nat"
   shows "0 + n = n"
   apply (elim n)
-    \<guillemotright> by (reduce; intro)
-    \<guillemotright> vars _ ih by reduce (eq ih; intro)
+    \<^item> by (reduce; intro)
+    \<^item> vars _ ih by reduce (eq ih; refl)
   done
 
-Lemma (derive) suc_add:
+Lemma (def) suc_add:
   assumes "m: Nat" "n: Nat"
   shows "suc m + n = suc (m + n)"
   apply (elim n)
-    \<guillemotright> by reduce refl
-    \<guillemotright> vars _ ih by reduce (eq ih; intro)
+    \<^item> by reduce refl
+    \<^item> vars _ ih by reduce (eq ih; refl)
   done
 
-Lemma (derive) suc_eq:
+Lemma (def) suc_eq:
   assumes "m: Nat" "n: Nat"
   shows "p: m = n \<Longrightarrow> suc m = suc n"
   by (eq p) intro
 
-Lemma (derive) add_assoc:
+Lemma (def) add_assoc:
   assumes "l: Nat" "m: Nat" "n: Nat"
   shows "l + (m + n) = l + m+ n"
   apply (elim n)
-    \<guillemotright> by reduce intro
-    \<guillemotright> vars _ ih by reduce (eq ih; intro)
+    \<^item> by reduce intro
+    \<^item> vars _ ih by reduce (eq ih; refl)
   done
 
-Lemma (derive) add_comm:
+Lemma (def) add_comm:
   assumes "m: Nat" "n: Nat"
   shows "m + n = n + m"
   apply (elim n)
-    \<guillemotright> by (reduce; rule zero_add[symmetric])
-    \<guillemotright> prems vars n ih
+    \<^item> by (reduce; rule zero_add[symmetric])
+    \<^item> vars n ih
       proof reduce
-        have "suc (m + n) = suc (n + m)" by (eq ih) intro
+        have "suc (m + n) = suc (n + m)" by (eq ih) refl
         also have ".. = suc n + m" by (transport eq: suc_add) refl
         finally show "{}" by this
       qed
@@ -143,12 +143,12 @@ lemma mul_suc [comp]:
   shows "m * suc n \<equiv> m + m * n"
   unfolding mul_def by reduce
 
-Lemma (derive) zero_mul:
+Lemma (def) zero_mul:
   assumes "n: Nat"
   shows "0 * n = 0"
   apply (elim n)
-  \<guillemotright> by reduce refl
-  \<guillemotright> prems vars n ih
+  \<^item> by reduce refl
+  \<^item> vars n ih
     proof reduce
       have "0 + 0 * n = 0 + 0 " by (eq ih) refl
       also have ".. = 0" by reduce refl
@@ -156,12 +156,12 @@ Lemma (derive) zero_mul:
     qed
   done
 
-Lemma (derive) suc_mul:
+Lemma (def) suc_mul:
   assumes "m: Nat" "n: Nat"
   shows "suc m * n = m * n + n"
   apply (elim n)
-  \<guillemotright> by reduce refl
-  \<guillemotright> prems vars n ih
+  \<^item> by reduce refl
+  \<^item> vars n ih
     proof (reduce, transport eq: \<open>ih:_\<close>)
       have "suc m + (m * n + n) = suc (m + {})" by (rule suc_add)
       also have ".. = suc (m + m * n + n)" by (transport eq: add_assoc) refl
@@ -169,12 +169,12 @@ Lemma (derive) suc_mul:
     qed
   done
 
-Lemma (derive) mul_dist_add:
+Lemma (def) mul_dist_add:
   assumes "l: Nat" "m: Nat" "n: Nat"
   shows "l * (m + n) = l * m + l * n"
   apply (elim n)
-  \<guillemotright> by reduce refl
-  \<guillemotright> prems prms vars n ih
+  \<^item> by reduce refl
+  \<^item> vars n ih
     proof reduce
       have "l + l * (m + n) = l + (l * m + l * n)" by (eq ih) refl
       also have ".. = l + l * m + l * n" by (rule add_assoc)
@@ -184,12 +184,12 @@ Lemma (derive) mul_dist_add:
     qed
   done
 
-Lemma (derive) mul_assoc:
+Lemma (def) mul_assoc:
   assumes "l: Nat" "m: Nat" "n: Nat"
   shows "l * (m * n) = l * m * n"
   apply (elim n)
-  \<guillemotright> by reduce refl
-  \<guillemotright> prems vars n ih
+  \<^item> by reduce refl
+  \<^item> vars n ih
     proof reduce
       have "l * (m + m * n) = l * m + l * (m * n)" by (rule mul_dist_add)
       also have ".. = l * m + l * m * n" by (transport eq: \<open>ih:_\<close>) refl
@@ -197,12 +197,12 @@ Lemma (derive) mul_assoc:
     qed
   done
 
-Lemma (derive) mul_comm:
+Lemma (def) mul_comm:
   assumes "m: Nat" "n: Nat"
   shows "m * n = n * m"
   apply (elim n)
-  \<guillemotright> by reduce (transport eq: zero_mul, refl)
-  \<guillemotright> prems vars n ih
+  \<^item> by reduce (transport eq: zero_mul, refl)
+  \<^item> vars n ih
     proof (reduce, rule pathinv)
       have "suc n * m = n * m + m" by (rule suc_mul)
       also have ".. = m + m * n"