1. Type-checking/inference now more principled, and the implementation is better. 2. Changed most tactics to context tactics.

5 files changed, 109 insertions, 117 deletions

diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy
index 66248a9..88adc8b 100644
--- a/hott/Equivalence.thy
+++ b/hott/Equivalence.thy
@@ -35,7 +35,7 @@ Lemma (derive) hsym:
     "\<And>x. x: A \<Longrightarrow> B x: U i"
   shows "H: f ~ g \<Longrightarrow> g ~ f"
   unfolding homotopy_def
-  apply intros
+  apply intro
     apply (rule pathinv)
       \<guillemotright> by (elim H)
       \<guillemotright> by typechk
@@ -106,7 +106,8 @@ Lemma homotopy_id_left [typechk]:
 Lemma homotopy_id_right [typechk]:
   assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
   shows "homotopy_refl A f: f \<circ> (id A) ~ f"
-  unfolding homotopy_refl_def homotopy_def by reduce
+  unfolding homotopy_refl_def homotopy_def
+ by reduce
 
 Lemma homotopy_funcomp_left:
   assumes
@@ -165,11 +166,9 @@ Lemma (derive) id_qinv:
   assumes "A: U i"
   shows "qinv (id A)"
   unfolding qinv_def
-  apply intro defer
-    apply intro defer
-      apply htpy_id
-      apply id_htpy
-  done
+  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"
@@ -272,7 +271,7 @@ Lemma (derive) biinv_imp_qinv:
         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
+        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
@@ -313,10 +312,9 @@ Lemma (derive) equivalence_refl:
   assumes "A: U i"
   shows "A \<simeq> A"
   unfolding equivalence_def
-  apply intro defer
-    apply (rule qinv_imp_biinv) defer
-      apply (rule id_qinv)
-  done
+  proof intro
+    show "biinv (id A)" by (rule qinv_imp_biinv) (rule id_qinv)
+  qed typechk
 
 text \<open>
   The following could perhaps be easier with transport (but then I think we need
@@ -340,11 +338,14 @@ Lemma (derive) equivalence_symmetric:
 Lemma (derive) equivalence_transitive:
   assumes "A: U i" "B: U i" "C: U i"
   shows "A \<simeq> B \<rightarrow> B \<simeq> C \<rightarrow> A \<simeq> C"
+  (* proof intros
+    fix AB BC assume "AB: A \<simeq> B" "BC: B \<simeq> C"
+    let "?f: {}" = "(fst AB) :: o" *)
   apply intros
   unfolding equivalence_def
     focus vars p q apply (elim p, elim q)
       focus vars f biinv\<^sub>f g biinv\<^sub>g apply intro
-        \<guillemotright> apply (rule funcompI) defer by assumption known
+        \<guillemotright> apply (rule funcompI) defer by assumption+ known
         \<guillemotright> by (rule funcomp_biinv)
       done
     done
@@ -372,32 +373,31 @@ Lemma (derive) id_imp_equiv:
   shows "<trans (id (U i)) p, ?isequiv>: A \<simeq> B"
   unfolding equivalence_def
   apply intros defer
-
-  \<comment> \<open>Switch off auto-typechecking, which messes with universe levels\<close>
-  supply [[auto_typechk=false]]
-
-    \<guillemotright> apply (eq p, typechk)
+    \<guillemotright> apply (eq p)
       \<^item> prems 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])
-        apply (rule transport, rule U_in_U)
-        apply (rule lift_universe_codomain, rule U_in_U)
-        apply (typechk, rule U_in_U)
+        apply (rule transport, rule Ui_in_USi)
+        apply (rule lift_universe_codomain, rule Ui_in_USi)
+        apply (typechk, rule Ui_in_USi)
         done
       \<^item> prems vars A
+        using [[solve_side_conds=1]]
         apply (subst transport_comp)
-          \<^enum> by (rule U_in_U)
-          \<^enum> by reduce (rule lift_U)
+          \<^enum> by (rule Ui_in_USi)
+          \<^enum> by reduce (rule in_USi_if_in_Ui)
           \<^enum> by reduce (rule id_biinv)
         done
       done
 
     \<guillemotright> \<comment> \<open>Similar proof as in the first subgoal above\<close>
       apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric])
+      using [[solve_side_conds=1]]
       apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric])
-      apply (rule transport, rule U_in_U)
-      apply (rule lift_universe_codomain, rule U_in_U)
-      apply (typechk, rule U_in_U)
+      apply (rule transport, rule Ui_in_USi)
+      apply (rule lift_universe_codomain, rule Ui_in_USi)
+      apply (typechk, rule Ui_in_USi)
       done
   done
 
diff --git a/hott/Identity.thy b/hott/Identity.thy
index dd2d6a0..1cb3946 100644
--- a/hott/Identity.thy
+++ b/hott/Identity.thy
@@ -20,7 +20,8 @@ syntax "_Id" :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close>
 translations "a =\<^bsub>A\<^esub> b" \<rightleftharpoons> "CONST Id A a b"
 
 axiomatization where
-  IdF: "\<lbrakk>A: U i; a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^bsub>A\<^esub> b: U i" and
+  \<comment> \<open>Here `A: U i` comes last because A is often implicit\<close>
+  IdF: "\<lbrakk>a: A; b: A; A: U i\<rbrakk> \<Longrightarrow> a =\<^bsub>A\<^esub> b: U i" and
 
   IdI: "a: A \<Longrightarrow> refl a: a =\<^bsub>A\<^esub> a" and
 
@@ -39,35 +40,36 @@ axiomatization where
     \<rbrakk> \<Longrightarrow> IdInd A (fn x y p. C x y p) (fn x. f x) a a (refl a) \<equiv> f a"
 
 lemmas
-  [intros] = IdF IdI and
-  [elims "?p" "?a" "?b"] = IdE and
-  [comps] = Id_comp and
+  [form] = IdF and
+  [intro, intros] = IdI and
+  [elim "?p" "?a" "?b"] = IdE and
+  [comp] = Id_comp and
   [refl] = IdI
 
 
 section \<open>Path induction\<close>
 
-method_setup eq = \<open>
-Args.term >> (fn tm => fn ctxt => CONTEXT_METHOD (K (
-  CONTEXT_SUBGOAL (fn (goal, i) =>
-    let
-      val facts = Proof_Context.facts_of ctxt
-      val prems = Logic.strip_assums_hyp goal
-      val template = \<^const>\<open>has_type\<close>
-        $ tm
-        $ (\<^const>\<open>Id\<close> $ Var (("*?A", 0), \<^typ>\<open>o\<close>)
-          $ Var (("*?x", 0), \<^typ>\<open>o\<close>)
-          $ Var (("*?y", 0), \<^typ>\<open>o\<close>))
-      val types =
-        map (Thm.prop_of o #1) (Facts.could_unify facts template)
-        @ filter (fn prem => Term.could_unify (template, prem)) prems
-        |> map Lib.type_of_typing
-    in case types of
-        (\<^const>\<open>Id\<close> $ _ $ x $ y)::_ =>
-          elim_context_tac [tm, x, y] ctxt i
-      | _ => Context_Tactic.CONTEXT_TACTIC no_tac
-    end) 1)))
-\<close>
+method_setup eq =
+  \<open>Args.term >> (fn tm => K (CONTEXT_METHOD (
+    CHEADGOAL o SIDE_CONDS (
+      CONTEXT_SUBGOAL (fn (goal, i) => fn cst as (ctxt, st) =>
+        let
+          val facts = Proof_Context.facts_of ctxt
+          val prems = Logic.strip_assums_hyp goal
+          val template = \<^const>\<open>has_type\<close>
+            $ tm
+            $ (\<^const>\<open>Id\<close> $ Var (("*?A", 0), \<^typ>\<open>o\<close>)
+              $ Var (("*?x", 0), \<^typ>\<open>o\<close>)
+              $ Var (("*?y", 0), \<^typ>\<open>o\<close>))
+          val types =
+            map (Thm.prop_of o #1) (Facts.could_unify facts template)
+            @ filter (fn prem => Term.could_unify (template, prem)) prems
+            |> map Lib.type_of_typing
+        in case types of
+            (Const (\<^const_name>\<open>Id\<close>, _) $ _ $ x $ y) :: _ =>
+              elim_ctac [tm, x, y] i cst
+          | _ => no_ctac cst
+        end)))))\<close>
 
 
 section \<open>Symmetry and transitivity\<close>
@@ -77,7 +79,7 @@ Lemma (derive) pathinv:
   shows "y =\<^bsub>A\<^esub> x"
   by (eq p) intro
 
-lemma pathinv_comp [comps]:
+lemma pathinv_comp [comp]:
   assumes "x: A" "A: U i"
   shows "pathinv A x x (refl x) \<equiv> refl x"
   unfolding pathinv_def by reduce
@@ -95,7 +97,7 @@ Lemma (derive) pathcomp:
     done
   done
 
-lemma pathcomp_comp [comps]:
+lemma pathcomp_comp [comp]:
   assumes "a: A" "A: U i"
   shows "pathcomp A a a a (refl a) (refl a) \<equiv> refl a"
   unfolding pathcomp_def by reduce
@@ -133,14 +135,14 @@ Lemma (derive) refl_pathcomp:
   assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
   shows "(refl x) \<bullet> p = p"
   apply (eq p)
-    apply (reduce; intros)
+    apply (reduce; intro)
   done
 
 Lemma (derive) pathcomp_refl:
   assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
   shows "p \<bullet> (refl y) = p"
   apply (eq p)
-    apply (reduce; intros)
+    apply (reduce; intro)
   done
 
 definition [implicit]: "ru p \<equiv> pathcomp_refl ? ? ? p"
@@ -150,36 +152,30 @@ translations
   "CONST ru p" \<leftharpoondown> "CONST pathcomp_refl A x y p"
   "CONST lu p" \<leftharpoondown> "CONST refl_pathcomp A x y p"
 
-Lemma lu_refl [comps]:
+Lemma lu_refl [comp]:
   assumes "A: U i" "x: A"
   shows "lu (refl x) \<equiv> refl (refl x)"
-  unfolding refl_pathcomp_def by reduce
+  unfolding refl_pathcomp_def by reduce+
 
-Lemma ru_refl [comps]:
+Lemma ru_refl [comp]:
   assumes "A: U i" "x: A"
   shows "ru (refl x) \<equiv> refl (refl x)"
-  unfolding pathcomp_refl_def by reduce
+  unfolding pathcomp_refl_def by reduce+
 
 Lemma (derive) inv_pathcomp:
   assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
   shows "p\<inverse> \<bullet> p = refl y"
-  apply (eq p)
-    apply (reduce; intros)
-  done
+  by (eq p) (reduce; intro)
 
 Lemma (derive) pathcomp_inv:
   assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
   shows "p \<bullet> p\<inverse> = refl x"
-  apply (eq p)
-    apply (reduce; intros)
-  done
+  by (eq p) (reduce; intro)
 
 Lemma (derive) pathinv_pathinv:
   assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
   shows "p\<inverse>\<inverse> = p"
-  apply (eq p)
-    apply (reduce; intros)
-  done
+  by (eq p) (reduce; intro)
 
 Lemma (derive) pathcomp_assoc:
   assumes
@@ -191,7 +187,7 @@ Lemma (derive) pathcomp_assoc:
       apply (eq p)
         focus prems vars x p
           apply (eq p)
-            apply (reduce; intros)
+            apply (reduce; intro)
         done
     done
   done
@@ -206,16 +202,14 @@ Lemma (derive) ap:
     "f: A \<rightarrow> B"
     "p: x =\<^bsub>A\<^esub> y"
   shows "f x = f y"
-  apply (eq p)
-    apply intro
-  done
+  by (eq p) intro
 
 definition ap_i ("_[_]" [1000, 0])
   where [implicit]: "ap_i f p \<equiv> ap ? ? ? ? f p"
 
 translations "f[p]" \<leftharpoondown> "CONST ap A B x y f p"
 
-Lemma ap_refl [comps]:
+Lemma ap_refl [comp]:
   assumes "f: A \<rightarrow> B" "x: A" "A: U i" "B: U i"
   shows "f[refl x] \<equiv> refl (f x)"
   unfolding ap_def by reduce
@@ -254,17 +248,16 @@ Lemma (derive) ap_funcomp:
     "p: x =\<^bsub>A\<^esub> y"
   shows "(g \<circ> f)[p] = g[f[p]]"
   apply (eq p)
-    apply (simp only: funcomp_apply_comp[symmetric])
-    apply (reduce; intro)
+    \<guillemotright> by reduce
+    \<guillemotright> by reduce intro
   done
 
 Lemma (derive) ap_id:
   assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
   shows "(id A)[p] = p"
   apply (eq p)
-    apply (rewrite at "\<hole> = _" id_comp[symmetric])
-    apply (rewrite at "_ = \<hole>" id_comp[symmetric])
-    apply (reduce; intros)
+    \<guillemotright> by reduce
+    \<guillemotright> by reduce intro
   done
 
 
@@ -284,7 +277,7 @@ definition transport_i ("trans")
 
 translations "trans P p" \<leftharpoondown> "CONST transport A P x y p"
 
-Lemma transport_comp [comps]:
+Lemma transport_comp [comp]:
   assumes
     "a: A"
     "A: U i"
@@ -302,6 +295,7 @@ Lemma use_transport:
     "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])
@@ -322,7 +316,7 @@ Lemma (derive) transport_right_inv:
     "x: A" "y: A"
     "p: x =\<^bsub>A\<^esub> y"
   shows "(trans P p) \<circ> (trans P p\<inverse>) = id (P y)"
-  by (eq p) (reduce; intros)
+  by (eq p) (reduce; intro)
 
 Lemma (derive) transport_pathcomp:
   assumes
@@ -335,7 +329,7 @@ Lemma (derive) transport_pathcomp:
   apply (eq p)
     focus prems vars x p
       apply (eq p)
-        apply (reduce; intros)
+        apply (reduce; intro)
     done
   done
 
@@ -348,7 +342,7 @@ Lemma (derive) transport_compose_typefam:
     "p: x =\<^bsub>A\<^esub> y"
     "u: P (f x)"
   shows "trans (fn x. P (f x)) p u = trans P f[p] u"
-  by (eq p) (reduce; intros)
+  by (eq p) (reduce; intro)
 
 Lemma (derive) transport_function_family:
   assumes
@@ -360,7 +354,7 @@ Lemma (derive) transport_function_family:
     "u: P x"
     "p: x =\<^bsub>A\<^esub> y"
   shows "trans Q p ((f x) u) = (f y) (trans P p u)"
-  by (eq p) (reduce; intros)
+  by (eq p) (reduce; intro)
 
 Lemma (derive) transport_const:
   assumes
@@ -375,12 +369,12 @@ definition transport_const_i ("trans'_const")
 
 translations "trans_const B p" \<leftharpoondown> "CONST transport_const A B x y p"
 
-Lemma transport_const_comp [comps]:
+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"
-  unfolding transport_const_def by reduce
+  unfolding transport_const_def by reduce+
 
 Lemma (derive) pathlift:
   assumes
@@ -390,21 +384,21 @@ Lemma (derive) pathlift:
     "p: x =\<^bsub>A\<^esub> y"
     "u: P x"
   shows "<x, u> = <y, trans P p u>"
-  by (eq p) (reduce; intros)
+  by (eq p) (reduce; intro)
 
 definition pathlift_i ("lift")
   where [implicit]: "lift P p u \<equiv> pathlift ? P ? ? p u"
 
 translations "lift P p u" \<leftharpoondown> "CONST pathlift A P x y p u"
 
-Lemma pathlift_comp [comps]:
+Lemma pathlift_comp [comp]:
   assumes
     "u: P x"
     "x: A"
     "\<And>x. x: A \<Longrightarrow> P x: U i"
     "A: U i"
   shows "lift P (refl x) u \<equiv> refl <x, u>"
-  unfolding pathlift_def by reduce
+  unfolding pathlift_def by reduce+
 
 Lemma (derive) pathlift_fst:
   assumes
@@ -415,13 +409,7 @@ Lemma (derive) pathlift_fst:
     "p: x =\<^bsub>A\<^esub> y"
   shows "fst[lift P p u] = p"
   apply (eq p)
-    text \<open>Some rewriting needed here:\<close>
-    \<guillemotright> vars x y
-      (*Here an automatic reordering tactic would be helpful*)
-      apply (rewrite at x in "x = y" fst_comp[symmetric])
-        prefer 4
-        apply (rewrite at y in "_ = y" fst_comp[symmetric])
-      done
+    \<guillemotright> by reduce
     \<guillemotright> by reduce intro
   done
 
@@ -436,21 +424,21 @@ Lemma (derive) apd:
     "x: A" "y: A"
     "p: x =\<^bsub>A\<^esub> y"
   shows "trans P p (f x) = f y"
-  by (eq p) (reduce; intros; typechk)
+  by (eq p) (reduce; intro)
 
 definition apd_i ("apd")
   where [implicit]: "apd f p \<equiv> Identity.apd ? ? f ? ? p"
 
 translations "apd f p" \<leftharpoondown> "CONST Identity.apd A P f x y p"
 
-Lemma dependent_map_comp [comps]:
+Lemma dependent_map_comp [comp]:
   assumes
     "f: \<Prod>x: A. P x"
     "x: A"
     "A: U i"
     "\<And>x. x: A \<Longrightarrow> P x: U i"
   shows "apd f (refl x) \<equiv> refl (f x)"
-  unfolding apd_def by reduce
+  unfolding apd_def by reduce+
 
 Lemma (derive) apd_ap:
   assumes
@@ -467,6 +455,9 @@ section \<open>Whiskering\<close>
 Lemma (derive) right_whisker:
   assumes "A: U i" "a: A" "b: A" "c: A"
   shows "\<lbrakk>p: a = b; q: a = b; r: b = c; \<alpha>: p =\<^bsub>a = b\<^esub> q\<rbrakk> \<Longrightarrow> p \<bullet> r = q \<bullet> r"
+  \<comment> \<open>TODO: In the above we need to annotate the type of \<alpha> with the type `a = b`
+    in order for the `eq` method to work correctly. This should be fixed with a
+    pre-proof elaborator.\<close>
   apply (eq r)
   focus prems vars x s t
     proof -
@@ -500,17 +491,17 @@ translations
   "\<alpha> \<bullet>\<^sub>r\<^bsub>a\<^esub> r" \<leftharpoondown> "CONST right_whisker A a b c p q r \<alpha>"
   "r \<bullet>\<^sub>l\<^bsub>c\<^esub> \<alpha>" \<leftharpoondown> "CONST left_whisker A a b c p q r \<alpha>"
 
-Lemma whisker_refl [comps]:
+Lemma whisker_refl [comp]:
   assumes "A: U i" "a: A" "b: A"
   shows "\<lbrakk>p: a = b; q: a = b; \<alpha>: p =\<^bsub>a = b\<^esub> q\<rbrakk> \<Longrightarrow>
     \<alpha> \<bullet>\<^sub>r\<^bsub>a\<^esub> (refl b) \<equiv> ru p \<bullet> \<alpha> \<bullet> (ru q)\<inverse>"
-  unfolding right_whisker_def by reduce
+  unfolding right_whisker_def by reduce+
 
-Lemma refl_whisker [comps]:
+Lemma refl_whisker [comp]:
   assumes "A: U i" "a: A" "b: A"
   shows "\<lbrakk>p: a = b; q: a = b; \<alpha>: p = q\<rbrakk> \<Longrightarrow>
     (refl a) \<bullet>\<^sub>l\<^bsub>b\<^esub> \<alpha> \<equiv> (lu p) \<bullet> \<alpha> \<bullet> (lu q)\<inverse>"
-  unfolding left_whisker_def by reduce
+  unfolding left_whisker_def by reduce+
 
 method left_whisker = (rule left_whisker)
 method right_whisker = (rule right_whisker)
@@ -632,12 +623,12 @@ Lemma (derive) eckmann_hilton:
   proof -
     have "\<alpha> \<bullet> \<beta> = \<alpha> \<star> \<beta>"
       by (rule pathcomp_eq_horiz_pathcomp)
-    also have [simplified comps]: ".. = \<alpha> \<star>' \<beta>"
+    also have [simplified comp]: ".. = \<alpha> \<star>' \<beta>"
       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.horiz_pathcomp_def \<Omega>2.horiz_pathcomp'_def)+
   qed
 
 end
diff --git a/hott/More_List.thy b/hott/More_List.thy
index 460dc7b..1b8034f 100644
--- a/hott/More_List.thy
+++ b/hott/More_List.thy
@@ -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 comps)+
-  Lemma "len [0, suc 0, suc (suc 0)] \<equiv> ?n" by (subst comps)+
+  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
index 8177170..59c8d54 100644
--- a/hott/More_Nat.thy
+++ b/hott/More_Nat.thy
@@ -15,15 +15,15 @@ Lemma (derive) eq: shows "Nat \<rightarrow> Nat \<rightarrow> U O"
     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 U_in_U)
+      \<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 U_in_U)
+      \<guillemotright> by (rule Ui_in_USi)
     done
-    by (rule U_in_U)
+    by (rule Ui_in_USi)
   done
 
 text \<open>
diff --git a/hott/Nat.thy b/hott/Nat.thy
index 4abd315..177ec47 100644
--- a/hott/Nat.thy
+++ b/hott/Nat.thy
@@ -38,9 +38,10 @@ where
         f n (NatInd (fn n. C n) c\<^sub>0 (fn k rec. f k rec) n)"
 
 lemmas
-  [intros] = NatF Nat_zero Nat_suc and
-  [elims "?n"] = NatE and
-  [comps] = Nat_comp_zero Nat_comp_suc
+  [form] = NatF and
+  [intro, intros] = Nat_zero Nat_suc and
+  [elim "?n"] = NatE and
+  [comp] = Nat_comp_zero Nat_comp_suc
 
 abbreviation "NatRec C \<equiv> NatInd (fn _. C)"
 
@@ -66,12 +67,12 @@ lemma add_type [typechk]:
   shows "m + n: Nat"
   unfolding add_def by typechk
 
-lemma add_zero [comps]:
+lemma add_zero [comp]:
   assumes "m: Nat"
   shows "m + 0 \<equiv> m"
   unfolding add_def by reduce
 
-lemma add_suc [comps]:
+lemma add_suc [comp]:
   assumes "m: Nat" "n: Nat"
   shows "m + suc n \<equiv> suc (m + n)"
   unfolding add_def by reduce
@@ -127,17 +128,17 @@ lemma mul_type [typechk]:
   shows "m * n: Nat"
   unfolding mul_def by typechk
 
-lemma mul_zero [comps]:
+lemma mul_zero [comp]:
   assumes "n: Nat"
   shows "n * 0 \<equiv> 0"
   unfolding mul_def by reduce
 
-lemma mul_one [comps]:
+lemma mul_one [comp]:
   assumes "n: Nat"
   shows "n * 1 \<equiv> n"
   unfolding mul_def by reduce
 
-lemma mul_suc [comps]:
+lemma mul_suc [comp]:
   assumes "m: Nat" "n: Nat"
   shows "m * suc n \<equiv> m + m * n"
   unfolding mul_def by reduce