Merge pull request #8 from jaycech3n/dev

Merge big diff on dev

16 files changed, 603 insertions, 431 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
diff --git a/spartan/core/Spartan.thy b/spartan/core/Spartan.thy
index 8fa6cfe..a4ad300 100644
--- a/spartan/core/Spartan.thy
+++ b/spartan/core/Spartan.thy
@@ -10,22 +10,29 @@ keywords
   "focus" "\<guillemotright>" "\<^item>" "\<^enum>" "~" :: prf_script_goal % "proof" and
   "congruence" "print_coercions" :: thy_decl and
   "rhs" "derive" "prems" "vars" :: quasi_command
-  
 
 begin
 
 
 section \<open>Prelude\<close>
 
+paragraph \<open>Set up the meta-logic just so.\<close>
+
+paragraph \<open>Notation.\<close>
+
 declare [[eta_contract=false]]
 
+text \<open>
+  Rebind notation for meta-lambdas since we want to use `\<lambda>` for the object
+  lambdas. Meta-functions now use the binder `fn`.
+\<close>
 setup \<open>
 let
   val typ = Simple_Syntax.read_typ
   fun mixfix (sy, ps, p) = Mixfix (Input.string sy, ps, p, Position.no_range)
 in
   Sign.del_syntax (Print_Mode.ASCII, true)
-    [("_lambda", typ "pttrns \<Rightarrow> 'a \<Rightarrow> logic", mixfix ("(3fn _./ _)", [0, 3], 3))]
+    [("_lambda", typ "pttrns \<Rightarrow> 'a \<Rightarrow> logic", mixfix ("(3%_./ _)", [0, 3], 3))]
   #> Sign.del_syntax Syntax.mode_default
     [("_lambda", typ "pttrns \<Rightarrow> 'a \<Rightarrow> logic", mixfix ("(3\<lambda>_./ _)", [0, 3], 3))]
   #> Sign.add_syntax Syntax.mode_default
@@ -38,11 +45,16 @@ translations "a $ b" \<rightharpoonup> "a (b)"
 
 abbreviation (input) K where "K x \<equiv> fn _. x"
 
+paragraph \<open>
+  Deeply embed dependent type theory: meta-type of expressions, and typing
+  judgment.
+\<close>
+
 typedecl o
 
 judgment has_type :: \<open>o \<Rightarrow> o \<Rightarrow> prop\<close> ("(2_:/ _)" 999)
 
-text \<open>Type annotations:\<close>
+text \<open>Type annotations for type-checking and inference.\<close>
 
 definition anno :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> ("'(_ : _')")
   where "(a : A) \<equiv> a" if "a: A"
@@ -51,7 +63,9 @@ lemma anno: "a: A \<Longrightarrow> (a : A): A"
   by (unfold anno_def)
 
 
-section \<open>Universes\<close>
+section \<open>Axioms\<close>
+
+subsection \<open>Universes\<close>
 
 typedecl lvl \<comment> \<open>Universe levels\<close>
 
@@ -65,19 +79,19 @@ axiomatization
   lt_trans: "i < j \<Longrightarrow> j < k \<Longrightarrow> i < k"
 
 axiomatization U :: \<open>lvl \<Rightarrow> o\<close> where
-  U_hierarchy: "i < j \<Longrightarrow> U i: U j" and
-  U_cumulative: "A: U i \<Longrightarrow> i < j \<Longrightarrow> A: U j"
+  Ui_in_Uj: "i < j \<Longrightarrow> U i: U j" and
+  in_Uj_if_in_Ui: "A: U i \<Longrightarrow> i < j \<Longrightarrow> A: U j"
 
-lemma U_in_U:
+lemma Ui_in_USi:
   "U i: U (S i)"
-  by (rule U_hierarchy, rule lt_S)
+  by (rule Ui_in_Uj, rule lt_S)
 
-lemma lift_U:
+lemma in_USi_if_in_Ui:
   "A: U i \<Longrightarrow> A: U (S i)"
-  by (erule U_cumulative, rule lt_S)
+  by (erule in_Uj_if_in_Ui, rule lt_S)
 
 
-section \<open>\<Prod>-type\<close>
+subsection \<open>\<Prod>-type\<close>
 
 axiomatization
   Pi  :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o\<close> and
@@ -100,9 +114,9 @@ translations
 abbreviation Fn (infixr "\<rightarrow>" 40) where "A \<rightarrow> B \<equiv> \<Prod>_: A. B"
 
 axiomatization where
-  PiF: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; A: U i\<rbrakk> \<Longrightarrow> \<Prod>x: A. B x: U i" and
+  PiF: "\<lbrakk>A: U i; \<And>x. x: A \<Longrightarrow> B x: U i\<rbrakk> \<Longrightarrow> \<Prod>x: A. B x: U i" and
 
-  PiI: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; A: U i\<rbrakk> \<Longrightarrow> \<lambda>x: A. b x: \<Prod>x: A. B x" and
+  PiI: "\<lbrakk>A: U i; \<And>x. x: A \<Longrightarrow> b x: B x\<rbrakk> \<Longrightarrow> \<lambda>x: A. b x: \<Prod>x: A. B x" and
 
   PiE: "\<lbrakk>f: \<Prod>x: A. B x; a: A\<rbrakk> \<Longrightarrow> f `a: B a" and
 
@@ -113,14 +127,14 @@ axiomatization where
   Pi_cong: "\<lbrakk>
     \<And>x. x: A \<Longrightarrow> B x \<equiv> B' x;
     A: U i;
-    \<And>x. x: A \<Longrightarrow> B x: U i;
-    \<And>x. x: A \<Longrightarrow> B' x: U i
+    \<And>x. x: A \<Longrightarrow> B x: U j;
+    \<And>x. x: A \<Longrightarrow> B' x: U j
     \<rbrakk> \<Longrightarrow> \<Prod>x: A. B x \<equiv> \<Prod>x: A. B' x" and
 
   lam_cong: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x \<equiv> c x; A: U i\<rbrakk> \<Longrightarrow> \<lambda>x: A. b x \<equiv> \<lambda>x: A. c x"
 
 
-section \<open>\<Sum>-type\<close>
+subsection \<open>\<Sum>-type\<close>
 
 axiomatization
   Sig    :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o\<close> and
@@ -138,16 +152,16 @@ abbreviation "and" (infixl "\<and>" 60)
   where "A \<and> B \<equiv> A \<times> B"
 
 axiomatization where
-  SigF: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; A: U i\<rbrakk> \<Longrightarrow> \<Sum>x: A. B x: U i" and
+  SigF: "\<lbrakk>A: U i; \<And>x. x: A \<Longrightarrow> B x: U i\<rbrakk> \<Longrightarrow> \<Sum>x: A. B x: U i" and
 
   SigI: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; a: A; b: B a\<rbrakk> \<Longrightarrow> <a, b>: \<Sum>x: A. B x" and
 
   SigE: "\<lbrakk>
     p: \<Sum>x: A. B x;
     A: U i;
-    \<And>x. x : A \<Longrightarrow> B x: U i;
-    \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x, y>;
-    \<And>p. p: \<Sum>x: A. B x \<Longrightarrow> C p: U i
+    \<And>x. x : A \<Longrightarrow> B x: U j;
+    \<And>p. p: \<Sum>x: A. B x \<Longrightarrow> C p: U k;
+    \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x, y>
     \<rbrakk> \<Longrightarrow> SigInd A (fn x. B x) (fn p. C p) f p: C p" and
 
   Sig_comp: "\<lbrakk>
@@ -161,22 +175,23 @@ axiomatization where
   Sig_cong: "\<lbrakk>
     \<And>x. x: A \<Longrightarrow> B x \<equiv> B' x;
     A: U i;
-    \<And>x. x : A \<Longrightarrow> B x: U i;
-    \<And>x. x : A \<Longrightarrow> B' x: U i
+    \<And>x. x : A \<Longrightarrow> B x: U j;
+    \<And>x. x : A \<Longrightarrow> B' x: U j
     \<rbrakk> \<Longrightarrow> \<Sum>x: A. B x \<equiv> \<Sum>x: A. B' x"
 
 
 section \<open>Infrastructure\<close>
 
 ML_file \<open>lib.ML\<close>
-ML_file \<open>typechecking.ML\<close>
+ML_file \<open>context_tactical.ML\<close>
 
 subsection \<open>Proof commands\<close>
 
+ML_file \<open>focus.ML\<close>
+
 named_theorems typechk
 
 ML_file \<open>goals.ML\<close>
-ML_file \<open>focus.ML\<close>
 
 subsection \<open>Congruence automation\<close>
 
@@ -184,72 +199,75 @@ consts "rhs" :: \<open>'a\<close> ("..")
 
 ML_file \<open>congruence.ML\<close>
 
+subsection \<open>Proof methods and type-checking/inference\<close>
+
+named_theorems form and intro and intros and comp
+\<comment> \<open>`intros` stores the introduction rule variants used by the
+  `intro` and `intros` methods.\<close>
+
 ML_file \<open>elimination.ML\<close> \<comment> \<open>elimination rules\<close>
 ML_file \<open>cases.ML\<close> \<comment> \<open>case reasoning rules\<close>
 
-subsection \<open>Methods\<close>
-
-named_theorems intros and comps
 lemmas
-  [intros] = PiF PiI SigF SigI and
-  [elims "?f"] = PiE and
-  [elims "?p"] = SigE and
-  [comps] = beta Sig_comp and
+  [form] = PiF SigF and
+  [intro] = PiI SigI and
+  [intros] = PiI[rotated] SigI and
+  [elim "?f"] = PiE and
+  [elim "?p"] = SigE and
+  [comp] = beta Sig_comp and
   [cong] = Pi_cong lam_cong Sig_cong
 
+ML_file \<open>typechecking.ML\<close>
 ML_file \<open>tactics.ML\<close>
 
-method_setup assumptions =
-  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (
-    CHANGED (TRYALL (assumptions_tac ctxt))))\<close>
+method_setup typechk =
+  \<open>Scan.succeed (K (CONTEXT_METHOD (
+    CHEADGOAL o Types.check_infer)))\<close>
 
 method_setup known =
-  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (
-    CHANGED (TRYALL (known_tac ctxt))))\<close>
+  \<open>Scan.succeed (K (CONTEXT_METHOD (
+    CHEADGOAL o Types.known_ctac)))\<close>
+
+method_setup rule =
+  \<open>Attrib.thms >> (fn ths => K (CONTEXT_METHOD (
+    CHEADGOAL o SIDE_CONDS (rule_ctac ths))))\<close>
+
+method_setup dest =
+  \<open>Scan.lift (Scan.option (Args.parens Parse.int))
+    -- Attrib.thms >> (fn (n_opt, ths) => K (CONTEXT_METHOD (
+      CHEADGOAL o SIDE_CONDS (dest_ctac n_opt ths))))\<close>
 
 method_setup intro =
-  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (intro_tac ctxt)))\<close>
+  \<open>Scan.succeed (K (CONTEXT_METHOD (
+    CHEADGOAL o SIDE_CONDS (intro_ctac))))\<close>
 
 method_setup intros =
-  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (intros_tac ctxt)))\<close>
+  \<open>Scan.succeed (K (CONTEXT_METHOD (
+    CHEADGOAL o SIDE_CONDS (CREPEAT o intro_ctac))))\<close>
 
 method_setup elim =
-  \<open>Scan.repeat Args.term >> (fn tms => fn ctxt =>
-    CONTEXT_METHOD (K (elim_context_tac tms ctxt 1)))\<close>
+  \<open>Scan.repeat Args.term >> (fn tms => K (CONTEXT_METHOD (
+    CHEADGOAL o SIDE_CONDS (elim_ctac tms))))\<close>
 
 method elims = elim+
 
 method_setup cases =
-  \<open>Args.term >> (fn tm => fn ctxt => SIMPLE_METHOD' (cases_tac tm ctxt))\<close>
-
-(*This could possibly use additional simplification hints via a simp: modifier*)
-method_setup typechk =
-  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (
-    SIDE_CONDS (typechk_tac ctxt) ctxt))
-    (* CHANGED (ALLGOALS (TRY o typechk_tac ctxt)))) *)\<close>
+  \<open>Args.term >> (fn tm => K (CONTEXT_METHOD (
+    CHEADGOAL o SIDE_CONDS (cases_ctac tm))))\<close>
 
-method_setup rule =
-  \<open>Attrib.thms >> (fn ths => fn ctxt =>
-    SIMPLE_METHOD (HEADGOAL (SIDE_CONDS (rule_tac ths ctxt) ctxt)))\<close>
-
-method_setup dest =
-  \<open>Scan.lift (Scan.option (Args.parens Parse.int)) -- Attrib.thms
-    >> (fn (opt_n, ths) => fn ctxt =>
-        SIMPLE_METHOD (HEADGOAL (SIDE_CONDS (dest_tac opt_n ths ctxt) ctxt)))\<close>
 
 subsection \<open>Reflexivity\<close>
 
 named_theorems refl
 method refl = (rule refl)
 
-subsection \<open>Trivial proofs modulo typechecking\<close>
+subsection \<open>Trivial proofs (modulo automatic discharge of side conditions)\<close>
 
 method_setup this =
-  \<open>Scan.succeed (fn ctxt => METHOD (
-    EVERY o map (HEADGOAL o
-      (fn ths => SIDE_CONDS (resolve_tac ctxt ths) ctxt) o
-      single)
-    ))\<close>
+  \<open>Scan.succeed (K (CONTEXT_METHOD (fn facts =>
+    CHEADGOAL (SIDE_CONDS
+      (CONTEXT_TACTIC' (fn ctxt => resolve_tac ctxt facts))
+      facts))))\<close>
 
 subsection \<open>Rewriting\<close>
 
@@ -270,7 +288,7 @@ lemma eta_expand:
 lemma rewr_imp:
   assumes "PROP A \<equiv> PROP B"
   shows "(PROP A \<Longrightarrow> PROP C) \<equiv> (PROP B \<Longrightarrow> PROP C)"
-  apply (rule Pure.equal_intr_rule)
+  apply (Pure.rule Pure.equal_intr_rule)
   apply (drule equal_elim_rule2[OF assms]; assumption)
   apply (drule equal_elim_rule1[OF assms]; assumption)
   done
@@ -288,9 +306,11 @@ ML_file \<open>~~/src/HOL/Library/cconv.ML\<close>
 ML_file \<open>rewrite.ML\<close>
 
 \<comment> \<open>\<open>reduce\<close> computes terms via judgmental equalities\<close>
-setup \<open>map_theory_simpset (fn ctxt => ctxt addSolver (mk_solver "" typechk_tac))\<close>
+setup \<open>map_theory_simpset (fn ctxt =>
+  ctxt addSolver (mk_solver "" (fn ctxt' =>
+    NO_CONTEXT_TACTIC ctxt' o Types.check_infer (Simplifier.prems_of ctxt'))))\<close>
 
-method reduce uses add = (simp add: comps add | subst comps)+
+method reduce uses add = changed \<open>((simp add: comp add | sub comp); typechk?)+\<close>
 
 subsection \<open>Implicits\<close>
 
@@ -348,13 +368,19 @@ lemma eta_exp:
   shows "f \<equiv> \<lambda>x: A. f x"
   by (rule eta[symmetric])
 
+lemma refine_codomain:
+  assumes
+    "A: U i"
+    "f: \<Prod>x: A. B x"
+    "\<And>x. x: A \<Longrightarrow> f `x: C x"
+  shows "f: \<Prod>x: A. C x"
+  by (subst eta_exp)
+
 lemma lift_universe_codomain:
   assumes "A: U i" "f: A \<rightarrow> U j"
   shows "f: A \<rightarrow> U (S j)"
-  apply (sub eta_exp)
-    apply known
-    apply (Pure.rule intros; rule lift_U)
-  done
+  using in_USi_if_in_Ui[of "f {}"]
+  by (rule refine_codomain)
 
 subsection \<open>Function composition\<close>
 
@@ -376,7 +402,7 @@ lemma funcompI [typechk]:
     "g \<circ>\<^bsub>A\<^esub> f: \<Prod>x: A. C (f x)"
   unfolding funcomp_def by typechk
 
-lemma funcomp_assoc [comps]:
+lemma funcomp_assoc [comp]:
   assumes
     "f: A \<rightarrow> B"
     "g: B \<rightarrow> C"
@@ -386,7 +412,7 @@ lemma funcomp_assoc [comps]:
     "(h \<circ>\<^bsub>B\<^esub> g) \<circ>\<^bsub>A\<^esub> f \<equiv> h \<circ>\<^bsub>A\<^esub> g \<circ>\<^bsub>A\<^esub> f"
   unfolding funcomp_def by reduce
 
-lemma funcomp_lambda_comp [comps]:
+lemma funcomp_lambda_comp [comp]:
   assumes
     "A: U i"
     "\<And>x. x: A \<Longrightarrow> b x: B"
@@ -395,7 +421,7 @@ lemma funcomp_lambda_comp [comps]:
     "(\<lambda>x: B. c x) \<circ>\<^bsub>A\<^esub> (\<lambda>x: A. b x) \<equiv> \<lambda>x: A. c (b x)"
   unfolding funcomp_def by reduce
 
-lemma funcomp_apply_comp [comps]:
+lemma funcomp_apply_comp [comp]:
   assumes
     "f: A \<rightarrow> B" "g: \<Prod>x: B. C x"
     "x: A"
@@ -417,22 +443,22 @@ abbreviation id where "id A \<equiv> \<lambda>x: A. x"
 
 lemma
   id_type[typechk]: "A: U i \<Longrightarrow> id A: A \<rightarrow> A" and
-  id_comp [comps]: "x: A \<Longrightarrow> (id A) x \<equiv> x" \<comment> \<open>for the occasional manual rewrite\<close>
-  by reduce
+  id_comp [comp]: "x: A \<Longrightarrow> (id A) x \<equiv> x" \<comment> \<open>for the occasional manual rewrite\<close>
+  by reduce+
 
-lemma id_left [comps]:
+lemma id_left [comp]:
   assumes "f: A \<rightarrow> B" "A: U i" "B: U i"
   shows "(id B) \<circ>\<^bsub>A\<^esub> f \<equiv> f"
   by (subst eta_exp[of f]) (reduce, rule eta)
 
-lemma id_right [comps]:
+lemma id_right [comp]:
   assumes "f: A \<rightarrow> B" "A: U i" "B: U i"
   shows "f \<circ>\<^bsub>A\<^esub> (id A) \<equiv> f"
   by (subst eta_exp[of f]) (reduce, rule eta)
 
 lemma id_U [typechk]:
   "id (U i): U i \<rightarrow> U i"
-  by typechk (fact U_in_U)
+  by typechk (rule Ui_in_USi) (*FIXME: Add annotation rule to typechecker*)
 
 
 section \<open>Pairs\<close>
@@ -445,7 +471,7 @@ lemma fst_type [typechk]:
   shows "fst A B: (\<Sum>x: A. B x) \<rightarrow> A"
   unfolding fst_def by typechk
 
-lemma fst_comp [comps]:
+lemma fst_comp [comp]:
   assumes
     "a: A"
     "b: B a"
@@ -459,10 +485,10 @@ lemma snd_type [typechk]:
   shows "snd A B: \<Prod>p: \<Sum>x: A. B x. B (fst A B p)"
   unfolding snd_def by typechk reduce
 
-lemma snd_comp [comps]:
+lemma snd_comp [comp]:
   assumes "a: A" "b: B a" "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i"
   shows "snd A B <a, b> \<equiv> b"
-  unfolding snd_def by reduce
+  unfolding snd_def by reduce+
 
 subsection \<open>Notation\<close>
 
@@ -483,7 +509,7 @@ Lemma fst [typechk]:
     "p: \<Sum>x: A. B x"
     "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i"
   shows "fst p: A"
-  by typechk
+  by typechk    
 
 Lemma snd [typechk]:
   assumes
diff --git a/spartan/core/context_tactical.ML b/spartan/core/context_tactical.ML
new file mode 100644
index 0000000..a5159f6
--- /dev/null
+++ b/spartan/core/context_tactical.ML
@@ -0,0 +1,152 @@
+(*  Title:      context_tactical.ML
+    Author:     Joshua Chen
+
+More context tactics, and context tactic combinators.
+*)
+
+infix 1 CTHEN CTHEN' CTHEN_ALL_NEW CTHEN_ALL_NEW_FWD
+infix 0 CORELSE CAPPEND CORELSE' CAPPEND'
+
+structure Context_Tactical:
+sig
+
+type context_tactic' = int -> context_tactic
+val CONTEXT_TACTIC': (Proof.context -> int -> tactic) -> context_tactic'
+val all_ctac: context_tactic
+val no_ctac: context_tactic
+val print_ctac: (Proof.context -> string) -> context_tactic
+val CTHEN: context_tactic * context_tactic -> context_tactic
+val CORELSE: context_tactic * context_tactic -> context_tactic
+val CAPPEND: context_tactic * context_tactic -> context_tactic
+val CTHEN': context_tactic' * context_tactic' -> context_tactic'
+val CORELSE': context_tactic' * context_tactic' -> context_tactic'
+val CAPPEND': context_tactic' * context_tactic' -> context_tactic'
+val CTRY: context_tactic -> context_tactic
+val CREPEAT: context_tactic -> context_tactic
+val CFILTER: (context_state -> bool) -> context_tactic -> context_tactic
+val CCHANGED: context_tactic -> context_tactic
+val CTHEN_ALL_NEW: context_tactic' * context_tactic' -> context_tactic'
+val CREPEAT_IN_RANGE: int -> int -> context_tactic' -> context_tactic
+val CREPEAT_ALL_NEW: context_tactic' -> context_tactic'
+val CTHEN_ALL_NEW_FWD: context_tactic' * context_tactic' -> context_tactic'
+val CREPEAT_ALL_NEW_FWD: context_tactic' -> context_tactic'
+val CHEADGOAL: context_tactic' -> context_tactic
+val CALLGOALS: context_tactic' -> context_tactic
+val CSOMEGOAL: context_tactic' -> context_tactic
+val CRANGE: context_tactic' list -> context_tactic'
+
+end = struct
+
+type context_tactic' = int -> context_tactic
+
+fun CONTEXT_TACTIC' tac i (ctxt, st) = TACTIC_CONTEXT ctxt ((tac ctxt i) st)
+
+val all_ctac = Seq.make_results o Seq.single
+val no_ctac = K Seq.empty
+fun print_ctac f (ctxt, st) = CONTEXT_TACTIC (print_tac ctxt (f ctxt)) (ctxt, st)
+
+fun (ctac1 CTHEN ctac2) cst =  Seq.maps_results ctac2 (ctac1 cst)
+
+fun (ctac1 CORELSE ctac2) cst =
+  (case Seq.pull (ctac1 cst) of
+    NONE => ctac2 cst
+  | some => Seq.make (fn () => some))
+
+fun (ctac1 CAPPEND ctac2) cst =
+  Seq.append (ctac1 cst) (Seq.make (fn () => Seq.pull (ctac2 cst)))
+
+fun (ctac1 CTHEN' ctac2) x = ctac1 x CTHEN ctac2 x
+fun (ctac1 CORELSE' ctac2) x = ctac1 x CORELSE ctac2 x
+fun (ctac1 CAPPEND' ctac2) x = ctac1 x CAPPEND ctac2 x
+
+fun CTRY ctac = ctac CORELSE all_ctac
+
+fun CREPEAT ctac =
+  let
+    fun rep qs cst =
+      (case Seq.pull (Seq.filter_results (ctac cst)) of
+        NONE => SOME (cst, Seq.make (fn () => repq qs))
+      | SOME (cst', q) => rep (q :: qs) cst')
+    and repq [] = NONE
+      | repq (q :: qs) =
+          (case Seq.pull q of
+            NONE => repq qs
+          | SOME (cst, q) => rep (q :: qs) cst);
+  in fn cst => Seq.make_results (Seq.make (fn () => rep [] cst)) end
+
+fun CFILTER pred ctac cst =
+  ctac cst
+  |> Seq.filter_results
+  |> Seq.filter pred
+  |> Seq.make_results
+
+(*Only accept next states where the subgoals have changed*)
+fun CCHANGED ctac (cst as (_, st)) =
+  CFILTER (fn (_, st') => not (Thm.eq_thm (st, st'))) ctac cst
+
+local
+  fun op THEN (f, g) x = Seq.maps_results g (f x)
+
+  fun INTERVAL f i j x =
+    if i > j then Seq.make_results (Seq.single x)
+    else op THEN (f j, INTERVAL f i (j - 1)) x
+
+  (*By Peter Lammich: apply tactic to subgoals in interval in a forward manner,
+    skipping over emerging subgoals*)
+  fun INTERVAL_FWD ctac l u (cst as (_, st)) = cst |>
+    (if l > u then all_ctac
+    else (ctac l CTHEN (fn cst' as (_, st') =>
+      let val ofs = Thm.nprems_of st' - Thm.nprems_of st in
+        if ofs < ~1
+        then raise THM (
+          "INTERVAL_FWD: tactic solved more than one goal", ~1, [st, st'])
+        else INTERVAL_FWD ctac (l + 1 + ofs) (u + ofs) cst'
+      end)))
+in
+
+fun (ctac1 CTHEN_ALL_NEW ctac2) i (cst as (_, st)) =
+  cst |> (ctac1 i CTHEN (fn cst' as (_, st') =>
+    INTERVAL ctac2 i (i + Thm.nprems_of st' - Thm.nprems_of st) cst'))
+
+(*By Peter Lammich: apply ctac2 to all subgoals emerging from ctac1, in forward
+  manner*)
+fun (ctac1 CTHEN_ALL_NEW_FWD ctac2) i (cst as (_, st)) =
+  cst |> (ctac1 i CTHEN (fn cst' as (_, st') =>
+    INTERVAL_FWD ctac2 i (i + Thm.nprems_of st' - Thm.nprems_of st) cst'))
+
+(*Repeatedly apply ctac to the i-th until the k-th-from-last subgoals
+  (i.e. leave the last k subgoals alone), until no more changes appear in the
+  goal state.*)
+fun CREPEAT_IN_RANGE i k ctac =
+  let fun interval_ctac (cst as (_, st)) =
+    INTERVAL_FWD ctac i (Thm.nprems_of st - k) cst
+  in CREPEAT (CCHANGED interval_ctac) end
+
+end
+
+fun CREPEAT_ALL_NEW ctac =
+  ctac CTHEN_ALL_NEW (CTRY o (fn i => CREPEAT_ALL_NEW ctac i))
+
+fun CREPEAT_ALL_NEW_FWD ctac =
+  ctac CTHEN_ALL_NEW_FWD (CTRY o (fn i => CREPEAT_ALL_NEW_FWD ctac i))
+
+fun CHEADGOAL ctac = ctac 1
+
+fun CALLGOALS ctac (cst as (_, st)) =
+  let
+    fun doall 0 = all_ctac
+      | doall n = ctac n CTHEN doall (n - 1);
+  in doall (Thm.nprems_of st) cst end
+
+fun CSOMEGOAL ctac (cst as (_, st)) =
+  let
+    fun find 0 = no_ctac
+      | find n = ctac n CORELSE find (n - 1);
+  in find (Thm.nprems_of st) cst end
+
+fun CRANGE [] _ = all_ctac
+  | CRANGE (ctac :: ctacs) i = CRANGE ctacs (i + 1) CTHEN ctac i
+
+end
+
+open Context_Tactical
diff --git a/spartan/core/elimination.ML b/spartan/core/elimination.ML
index 11b3af9..cd4e414 100644
--- a/spartan/core/elimination.ML
+++ b/spartan/core/elimination.ML
@@ -34,13 +34,13 @@ fun register_rule tms rl =
   in Rules.map (Termtab.update (hd, (rl, map (#1 o dest_Var) tms))) end
 
 
-(* [elims] attribute *)
+(* [elim] attribute *)
 val _ = Theory.setup (
-  Attrib.setup \<^binding>\<open>elims\<close>
+  Attrib.setup \<^binding>\<open>elim\<close>
     (Scan.repeat Args.term_pattern >>
       (Thm.declaration_attribute o register_rule))
     ""
-  #> Global_Theory.add_thms_dynamic (\<^binding>\<open>elims\<close>,
+  #> Global_Theory.add_thms_dynamic (\<^binding>\<open>elim\<close>,
       fn context => (map #1 (rules (Context.proof_of context))))
 )
 
diff --git a/spartan/core/eqsubst.ML b/spartan/core/eqsubst.ML
index ea6f098..e7ecf63 100644
--- a/spartan/core/eqsubst.ML
+++ b/spartan/core/eqsubst.ML
@@ -416,19 +416,27 @@ fun eqsubst_asm_tac ctxt occs thms i st =
    should be done to an assumption, false = apply to the conclusion of
    the goal) as well as the theorems to use *)
 val _ =
-  Theory.setup
-    (Method.setup \<^binding>\<open>sub\<close>
-      (Scan.lift (Args.mode "asm" -- Scan.optional (Args.parens (Scan.repeat Parse.nat)) [0]) --
-        Attrib.thms >> (fn ((asm, occs), inthms) => fn ctxt =>
-          SIMPLE_METHOD' ((if asm then eqsubst_asm_tac else eqsubst_tac) ctxt occs inthms)))
-      "single-step substitution"
-    #>
-    (Method.setup \<^binding>\<open>subst\<close>
-      (Scan.lift (Args.mode "asm" -- Scan.optional (Args.parens (Scan.repeat Parse.nat)) [0]) --
-        Attrib.thms >> (fn ((asm, occs), inthms) => fn ctxt =>
-          SIMPLE_METHOD' (SIDE_CONDS
-            ((if asm then eqsubst_asm_tac else eqsubst_tac) ctxt occs inthms)
-            ctxt)))
-      "single-step substitution with auto-typechecking"))
+  let
+    val parser =
+      Scan.lift (Args.mode "asm"
+        -- Scan.optional (Args.parens (Scan.repeat Parse.nat)) [0])
+      -- Attrib.thms
+    fun eqsubst_asm_ctac occs inthms =
+      CONTEXT_TACTIC' (fn ctxt => eqsubst_asm_tac ctxt occs inthms)
+    fun eqsubst_ctac occs inthms =
+      CONTEXT_TACTIC' (fn ctxt => eqsubst_tac ctxt occs inthms)
+  in
+    Theory.setup (
+      Method.setup \<^binding>\<open>sub\<close>
+        (parser >> (fn ((asm, occs), inthms) => fn ctxt => SIMPLE_METHOD' (
+          (if asm then eqsubst_asm_tac else eqsubst_tac) ctxt occs inthms)))
+        "single-step substitution" #>
+      Method.setup \<^binding>\<open>subst\<close>
+        (parser >> (fn ((asm, occs), inthms) => K (CONTEXT_METHOD (
+          CHEADGOAL o SIDE_CONDS
+            ((if asm then eqsubst_asm_ctac else eqsubst_ctac) occs inthms)))))
+        "single-step substitution with automatic discharge of side conditions"
+    )
+  end
 
 end;
diff --git a/spartan/core/lib.ML b/spartan/core/lib.ML
index 615f601..7b93a08 100644
--- a/spartan/core/lib.ML
+++ b/spartan/core/lib.ML
@@ -7,6 +7,7 @@ val maxint: int list -> int
 
 (*Terms*)
 val is_rigid: term -> bool
+val no_vars: term -> bool
 val dest_eq: term -> term * term
 val mk_Var: string -> int -> typ -> term
 val lambda_var: term -> term -> term
@@ -50,6 +51,8 @@ val maxint = max (op >)
 
 val is_rigid = not o is_Var o head_of
 
+val no_vars = not o exists_subterm is_Var
+
 fun dest_eq (Const (\<^const_name>\<open>Pure.eq\<close>, _) $ t $ def) = (t, def)
   | dest_eq _ = error "dest_eq"
 
diff --git a/spartan/core/rewrite.ML b/spartan/core/rewrite.ML
index f9c5d8e..eba0e81 100644
--- a/spartan/core/rewrite.ML
+++ b/spartan/core/rewrite.ML
@@ -15,8 +15,8 @@ patterns but has diverged significantly during its development.
 We also allow introduction of identifiers for bound variables,
 which can then be used to match arbitrary subterms inside abstractions.
 
-This code is slightly modified from the original at HOL/Library/rewrite.ML,
-to incorporate auto-typechecking for type theory.
+This code has been slightly modified from the original at HOL/Library/rewrite.ML
+to incorporate automatic discharge of type-theoretic side conditions.
 *)
 
 infix 1 then_pconv;
@@ -448,18 +448,18 @@ val _ =
     val subst_parser =
       let val scan = raw_pattern -- to_parser -- Parse.thms1
       in context_lift scan prep_args end
+
+    fun rewrite_export_ctac inputs inthms =
+      CONTEXT_TACTIC' (fn ctxt => rewrite_export_tac ctxt inputs inthms)
   in
     Method.setup \<^binding>\<open>rewr\<close> (subst_parser >>
-      (fn (pattern, inthms, (to, pat_ctxt)) => fn orig_ctxt =>
-        SIMPLE_METHOD'
-          (rewrite_export_tac orig_ctxt ((pattern, to), SOME pat_ctxt) inthms)))
-      "single-step rewriting, allowing subterm selection via patterns"
-    #>
-    (Method.setup \<^binding>\<open>rewrite\<close> (subst_parser >>
-      (fn (pattern, inthms, (to, pat_ctxt)) => fn orig_ctxt =>
-        SIMPLE_METHOD' (SIDE_CONDS
-          (rewrite_export_tac orig_ctxt ((pattern, to), SOME pat_ctxt) inthms)
-          orig_ctxt)))
-      "single-step rewriting with auto-typechecking")
+      (fn (pattern, inthms, (to, pat_ctxt)) => fn orig_ctxt => SIMPLE_METHOD'
+        (rewrite_export_tac orig_ctxt ((pattern, to), SOME pat_ctxt) inthms)))
+      "single-step rewriting, allowing subterm selection via patterns" #>
+    Method.setup \<^binding>\<open>rewrite\<close> (subst_parser >>
+      (fn (pattern, inthms, (to, pat_ctxt)) => K (CONTEXT_METHOD (
+        CHEADGOAL o SIDE_CONDS
+          (rewrite_export_ctac ((pattern, to), SOME pat_ctxt) inthms)))))
+      "single-step rewriting with auto-typechecking"
   end
 end
diff --git a/spartan/core/tactics.ML b/spartan/core/tactics.ML
index 0c71665..3922ae0 100644
--- a/spartan/core/tactics.ML
+++ b/spartan/core/tactics.ML
@@ -7,109 +7,66 @@ General tactics for dependent type theory.
 structure Tactics:
 sig
 
-val assumptions_tac: Proof.context -> int -> tactic
-val known_tac: Proof.context -> int -> tactic
-val typechk_tac: Proof.context -> int -> tactic
-val auto_typechk: bool Config.T
-val SIDE_CONDS: (int -> tactic) -> Proof.context -> int -> tactic
-val rule_tac: thm list -> Proof.context -> int -> tactic
-val dest_tac: int option -> thm list -> Proof.context -> int -> tactic
-val intro_tac: Proof.context -> int -> tactic
-val intros_tac: Proof.context -> int -> tactic
-val elim_context_tac: term list -> Proof.context -> int -> context_tactic
-val cases_tac: term -> Proof.context -> int -> tactic
+val solve_side_conds: int Config.T
+val SIDE_CONDS: context_tactic' -> thm list -> context_tactic'
+val rule_ctac: thm list -> context_tactic'
+val dest_ctac: int option -> thm list -> context_tactic'
+val intro_ctac: context_tactic'
+val elim_ctac: term list -> context_tactic'
+val cases_ctac: term -> context_tactic'
 
 end = struct
 
-(*An assumption tactic that only solves typing goals with rigid terms and
-  judgmental equalities without schematic variables*)
-fun assumptions_tac ctxt = SUBGOAL (fn (goal, i) =>
-  let
-    val concl = Logic.strip_assums_concl goal
-  in
-    if
-      Lib.is_typing concl andalso Lib.is_rigid (Lib.term_of_typing concl)
-      orelse not ((exists_subterm is_Var) concl)
-    then assume_tac ctxt i
-    else no_tac
-  end)
-
-(*Solves typing goals with rigid term by resolving with context facts and
-  simplifier premises, or arbitrary goals by *non-unifying* assumption*)
-fun known_tac ctxt = SUBGOAL (fn (goal, i) =>
-  let
-    val concl = Logic.strip_assums_concl goal
-  in
-    ((if Lib.is_typing concl andalso Lib.is_rigid (Lib.term_of_typing concl)
-      then
-        let val ths = map fst (Facts.props (Proof_Context.facts_of ctxt))
-        in resolve_tac ctxt (ths @ Simplifier.prems_of ctxt) end
-      else K no_tac)
-    ORELSE' assumptions_tac ctxt) i
-  end)
-
-(*Typechecking: try to solve goals of the form "a: A" where a is rigid*)
-fun typechk_tac ctxt =
-  let
-    val tac = SUBGOAL (fn (goal, i) =>
-      if Lib.rigid_typing_concl goal
-      then
-        let val net = Tactic.build_net
-          ((Named_Theorems.get ctxt \<^named_theorems>\<open>typechk\<close>)
-          @(Named_Theorems.get ctxt \<^named_theorems>\<open>intros\<close>)
-          @(map #1 (Elim.rules ctxt)))
-        in (resolve_from_net_tac ctxt net) i end
-      else no_tac)
-  in
-    REPEAT_ALL_NEW (known_tac ctxt ORELSE' tac)
-  end
 
-fun typechk_context_tac (ctxt, st) =
-  let
-    
-  in
-    ()
-  end
+(* Side conditions *)
+val solve_side_conds =
+  Attrib.setup_config_int \<^binding>\<open>solve_side_conds\<close> (K 2)
 
-(*Many methods try to automatically discharge side conditions by typechecking.
-  Switch this flag off to discharge by non-unifying assumption instead.*)
-val auto_typechk = Attrib.setup_config_bool \<^binding>\<open>auto_typechk\<close> (K true)
+fun SIDE_CONDS ctac facts i (cst as (ctxt, st)) = cst |> (ctac i CTHEN
+  (case Config.get ctxt solve_side_conds of
+    1 => CALLGOALS (CTRY o Types.known_ctac facts)
+  | 2 => CREPEAT_IN_RANGE i (Thm.nprems_of st - i)
+      (CTRY o CREPEAT_ALL_NEW_FWD (Types.check_infer facts))
+  | _ => all_ctac))
 
-fun side_cond_tac ctxt = CHANGED o REPEAT o
-  (if Config.get ctxt auto_typechk then typechk_tac ctxt else known_tac ctxt)
 
-(*Combinator runs tactic and tries to discharge all new typing side conditions*)
-fun SIDE_CONDS tac ctxt = tac THEN_ALL_NEW (TRY o side_cond_tac ctxt)
+(* rule, dest, intro *)
 
 local
-fun mk_rules _ ths [] = ths
-  | mk_rules n ths ths' =
-      let val ths'' = foldr1 (op @)
-        (map (fn th => [rotate_prems n (th RS @{thm PiE})] handle THM _ => []) ths')
-      in
-        mk_rules n (ths @ ths') ths''
-      end
+  fun mk_rules _ ths [] = ths
+    | mk_rules n ths ths' =
+        let val ths'' = foldr1 (op @)
+          (map
+            (fn th => [rotate_prems n (th RS @{thm PiE})] handle THM _ => [])
+            ths')
+        in
+          mk_rules n (ths @ ths') ths''
+        end
 in
 
-(*Resolves with given rules, discharging as many side conditions as possible*)
-fun rule_tac ths ctxt = resolve_tac ctxt (mk_rules 0 [] ths)
+(*Resolves with given rules*)
+fun rule_ctac ths i (ctxt, st) =
+  TACTIC_CONTEXT ctxt (resolve_tac ctxt (mk_rules 0 [] ths) i st)
 
 (*Attempts destruct-resolution with the n-th premise of the given rules*)
-fun dest_tac opt_n ths ctxt = dresolve_tac ctxt
-  (mk_rules (case opt_n of NONE => 0 | SOME 0 => 0 | SOME n => n-1) [] ths)
+fun dest_ctac opt_n ths i (ctxt, st) =
+  TACTIC_CONTEXT ctxt (dresolve_tac ctxt
+    (mk_rules (case opt_n of NONE => 0 | SOME 0 => 0 | SOME n => n-1) [] ths)
+    i st)
 
 end
 
 (*Applies some introduction rule*)
-fun intro_tac ctxt = SUBGOAL (fn (_, i) => SIDE_CONDS
-  (resolve_tac ctxt (Named_Theorems.get ctxt \<^named_theorems>\<open>intros\<close>)) ctxt i)
+fun intro_ctac i (ctxt, st) = TACTIC_CONTEXT ctxt (resolve_tac ctxt
+  (Named_Theorems.get ctxt \<^named_theorems>\<open>intros\<close>) i st)
 
-fun intros_tac ctxt = SUBGOAL (fn (_, i) =>
-  (CHANGED o REPEAT o CHANGED o intro_tac ctxt) i)
 
 (* Induction/elimination *)
 
-(*Pushes a context/goal premise typing t:T into a \<Prod>-type*)
+(*Pushes a known typing t:T into a \<Prod>-type.
+  This tactic is well-behaved only when t is sufficiently well specified
+  (otherwise there might be multiple possible judgments t:T that unify, and
+  which is chosen is undefined).*)
 fun internalize_fact_tac t =
   Subgoal.FOCUS_PARAMS (fn {context = ctxt, concl = raw_concl, ...} =>
     let
@@ -123,18 +80,10 @@ fun internalize_fact_tac t =
     in
       HEADGOAL (resolve_tac ctxt [resolvent])
       (*known_tac infers the correct type T inferred by unification*)
-      THEN SOMEGOAL (known_tac ctxt)
+      THEN SOMEGOAL (NO_CONTEXT_TACTIC ctxt o Types.known_ctac [])
     end)
 
-(*Premises that have already been pushed into the \<Prod>-type*)
-structure Inserts = Proof_Data (
-  type T = term Item_Net.T
-  val init = K (Item_Net.init Term.aconv_untyped single)
-)
-
-local
-
-fun elim_core_tac tms types ctxt = SUBGOAL (K (
+fun elim_core_tac tms types ctxt =
   let
     val rule_insts = map ((Elim.lookup_rule ctxt) o Term.head_of) types
     val rules = flat (map
@@ -144,84 +93,86 @@ fun elim_core_tac tms types ctxt = SUBGOAL (K (
           (idxnames ~~ map (Thm.cterm_of ctxt) tms) rl])
       rule_insts)
   in
-    HEADGOAL (resolve_tac ctxt rules)
-    THEN RANGE (replicate (length tms) (typechk_tac ctxt)) 1
-  end handle Option => no_tac))
+    resolve_tac ctxt rules
+    THEN' RANGE (replicate (length tms) (NO_CONTEXT_TACTIC ctxt o Types.check_infer []))
+  end handle Option => K no_tac
 
-in
-
-fun elim_context_tac tms ctxt = case tms of
-    [] => CONTEXT_SUBGOAL (K (Context_Tactic.CONTEXT_TACTIC (HEADGOAL (
-      SIDE_CONDS (eresolve_tac ctxt (map #1 (Elim.rules ctxt))) ctxt))))
-  | major::_  => CONTEXT_SUBGOAL (fn (goal, _) =>
-    let
-      val facts = Proof_Context.facts_of ctxt
-      val prems = Logic.strip_assums_hyp goal
-      val template = Lib.typing_of_term major
-      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
-        [] => Context_Tactic.CONTEXT_TACTIC no_tac
-      | _ =>
-        let
-          val inserts = map (Thm.prop_of o fst) (Facts.props facts) @ prems
-            |> filter Lib.is_typing
-            |> map Lib.dest_typing
-            |> filter_out (fn (t, _) =>
-              Term.aconv (t, major) orelse Item_Net.member (Inserts.get ctxt) t)
-            |> map (fn (t, T) => ((t, T), Lib.subterm_count_distinct tms T))
-            |> filter (fn (_, i) => i > 0)
-            (*`t1: T1` comes before `t2: T2` if T1 contains t2 as subterm.
-              If they are incomparable, then order by decreasing
-              `subterm_count [p, x, y] T`*)
-            |> sort (fn (((t1, _), i), ((_, T2), j)) =>
-                Lib.cond_order (Lib.subterm_order T2 t1) (int_ord (j, i)))
-            |> map (#1 o #1)
-          val record_inserts = Inserts.map (fold Item_Net.update inserts)
-          val tac =
-            (*Push premises having a subterm in `tms` into a \<Prod>*)
-            fold (fn t => fn tac =>
-              tac THEN HEADGOAL (internalize_fact_tac t ctxt))
-              inserts all_tac
-            (*Apply elimination rule*)
-            THEN (HEADGOAL (
-              elim_core_tac tms types ctxt
-              (*Pull pushed premises back out*)
-              THEN_ALL_NEW (SUBGOAL (fn (_, i) =>
-                REPEAT_DETERM_N (length inserts)
-                  (resolve_tac ctxt @{thms PiI} i)))
-            ))
-            (*Side conditions*)
-            THEN ALLGOALS (TRY o side_cond_tac ctxt)
-        in
-          fn (ctxt, st) => Context_Tactic.TACTIC_CONTEXT
-            (record_inserts ctxt) (tac st)
-        end
-    end)
+(*Premises that have already been pushed into the \<Prod>-type*)
+structure Inserts = Proof_Data (
+  type T = term Item_Net.T
+  val init = K (Item_Net.init Term.aconv_untyped single)
+)
 
-fun cases_tac tm ctxt = SUBGOAL (fn (goal, i) =>
-  let
-    val facts = Proof_Context.facts_of ctxt
-    val prems = Logic.strip_assums_hyp goal
-    val template = Lib.typing_of_term tm
-    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
-    val res = (case types of
-        [typ] => Drule.infer_instantiate' ctxt [SOME (Thm.cterm_of ctxt tm)]
-          (the (Case.lookup_rule ctxt (Term.head_of typ)))
-      | [] => raise Option
-      | _ => raise error (Syntax.string_of_term ctxt tm ^ "not uniquely typed"))
-      handle Option => error ("no case rule known for "
-        ^ (Syntax.string_of_term ctxt tm))
-  in
-    SIDE_CONDS (resolve_tac ctxt [res]) ctxt i
-  end)
+fun elim_ctac tms =
+  case tms of
+    [] => CONTEXT_TACTIC' (fn ctxt => eresolve_tac ctxt (map #1 (Elim.rules ctxt)))
+  | major :: _ => CONTEXT_SUBGOAL (fn (goal, _) => fn cst as (ctxt, st) =>
+      let
+        val facts = Proof_Context.facts_of ctxt
+        val prems = Logic.strip_assums_hyp goal
+        val template = Lib.typing_of_term major
+        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
+          [] => no_ctac cst
+        | _ =>
+            let
+              val inserts = map (Thm.prop_of o fst) (Facts.props facts) @ prems
+                |> filter Lib.is_typing
+                |> map Lib.dest_typing
+                |> filter_out (fn (t, _) =>
+                  Term.aconv (t, major) orelse Item_Net.member (Inserts.get ctxt) t)
+                |> map (fn (t, T) => ((t, T), Lib.subterm_count_distinct tms T))
+                |> filter (fn (_, i) => i > 0)
+                (*`t1: T1` comes before `t2: T2` if T1 contains t2 as subterm.
+                  If they are incomparable, then order by decreasing
+                  `subterm_count [p, x, y] T`*)
+                |> sort (fn (((t1, _), i), ((_, T2), j)) =>
+                    Lib.cond_order (Lib.subterm_order T2 t1) (int_ord (j, i)))
+                |> map (#1 o #1)
+              val record_inserts = Inserts.map (fold Item_Net.update inserts)
+              val tac =
+                (*Push premises having a subterm in `tms` into a \<Prod>*)
+                fold (fn t => fn tac =>
+                  tac THEN HEADGOAL (internalize_fact_tac t ctxt))
+                  inserts all_tac
+                (*Apply elimination rule*)
+                THEN HEADGOAL (
+                  elim_core_tac tms types ctxt
+                  (*Pull pushed premises back out*)
+                  THEN_ALL_NEW (SUBGOAL (fn (_, i) =>
+                    REPEAT_DETERM_N (length inserts)
+                      (resolve_tac ctxt @{thms PiI[rotated]} i))))
+            in
+              TACTIC_CONTEXT (record_inserts ctxt) (tac st)
+            end
+      end)
+
+fun cases_ctac tm =
+  let fun tac ctxt =
+    SUBGOAL (fn (goal, i) =>
+      let
+        val facts = Proof_Context.facts_of ctxt
+        val prems = Logic.strip_assums_hyp goal
+        val template = Lib.typing_of_term tm
+        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
+        val res = (case types of
+            [typ] => Drule.infer_instantiate' ctxt [SOME (Thm.cterm_of ctxt tm)]
+              (the (Case.lookup_rule ctxt (Term.head_of typ)))
+          | [] => raise Option
+          | _ => raise error (Syntax.string_of_term ctxt tm ^ "not uniquely typed"))
+          handle Option =>
+            error ("no case rule known for " ^ (Syntax.string_of_term ctxt tm))
+      in
+        resolve_tac ctxt [res] i
+      end)
+  in CONTEXT_TACTIC' tac end
 
-end
 
 end
 
diff --git a/spartan/core/typechecking.ML b/spartan/core/typechecking.ML
index 437a2dc..946ecd6 100644
--- a/spartan/core/typechecking.ML
+++ b/spartan/core/typechecking.ML
@@ -1,7 +1,7 @@
 (*  Title:      typechecking.ML
     Author:     Joshua Chen
 
-Type information and typechecking infrastructure.
+Type information and type-checking infrastructure.
 *)
 
 structure Types: sig
@@ -11,11 +11,12 @@ val types: Proof.context -> term -> thm list
 val put_type: thm -> Proof.context -> Proof.context
 val put_types: thm list -> Proof.context -> Proof.context
 
-val check: Proof.context -> thm -> int -> tactic
-val infer: Proof.context -> thm -> int -> tactic
+val known_ctac: thm list -> int -> context_tactic
+val check_infer: thm list -> int -> context_tactic
 
 end = struct
 
+
 (* Context data *)
 
 structure Data = Generic_Data (
@@ -32,22 +33,55 @@ fun put_type typing = Context.proof_map (Data.map (Item_Net.update typing))
 fun put_types typings = foldr1 (op o) (map put_type typings)
 
 
-(* Checking and inference *)
-
+(* Context tactics for type-checking *)
+
+(*Solves goals without metavariables and type inference problems by resolving
+  with facts or assumption from inline premises.*)
+fun known_ctac facts = CONTEXT_SUBGOAL (fn (goal, i) => fn (ctxt, st) =>
+  TACTIC_CONTEXT ctxt
+    let val concl = Logic.strip_assums_concl goal in
+      if Lib.no_vars concl orelse
+        (Lib.is_typing concl andalso Lib.no_vars (Lib.term_of_typing concl))
+      then
+        let val ths = facts
+          (*FIXME: Shouldn't pull nameless facts directly from context*)
+          @ map fst (Facts.props (Proof_Context.facts_of ctxt))
+        in (resolve_tac ctxt ths i ORELSE assume_tac ctxt i) st end
+      else Seq.empty
+    end)
+
+(*Simple bidirectional typing tactic, with some nondeterminism from backtracking
+  search over arbitrary `typechk` rules. The current implementation does not
+  perform any normalization.*)
 local
-
-fun checkable prop = Lib.is_typing prop
-  andalso not (exists_subterm is_Var (Lib.type_of_typing prop))
-
+  fun check_infer_step facts i (ctxt, st) =
+    let
+      val tac = SUBGOAL (fn (goal, i) =>
+        if Lib.rigid_typing_concl goal
+        then
+          let val net = Tactic.build_net (facts
+            (*MAYBE FIXME: Remove `typechk` from this list, and instead perform
+              definitional unfolding to (w?)hnf.*)
+            @(Named_Theorems.get ctxt \<^named_theorems>\<open>typechk\<close>)
+            @(Named_Theorems.get ctxt \<^named_theorems>\<open>form\<close>)
+            @(Named_Theorems.get ctxt \<^named_theorems>\<open>intro\<close>)
+            @(map #1 (Elim.rules ctxt)))
+          in (resolve_from_net_tac ctxt net) i end
+        else no_tac)
+      val ctxt' = ctxt
+    in
+      TACTIC_CONTEXT ctxt' (tac i st)
+    end
 in
 
-fun check ctxt rule = Subgoal.FOCUS_PREMS (
-  fn {context = goal_ctxt, prems, concl, ...} => no_tac) ctxt
-
-fun infer ctxt rule = Subgoal.FOCUS_PREMS (
-  fn {context = goal_ctxt, prems, concl, ...} => no_tac) ctxt
+fun check_infer facts i (cst as (_, st)) =
+  let
+    val ctac = known_ctac facts CORELSE' check_infer_step facts
+  in
+    cst |> (ctac i CTHEN
+      CREPEAT_IN_RANGE i (Thm.nprems_of st - i) (CTRY o CREPEAT_ALL_NEW_FWD ctac))
+  end
 
 end
 
-
 end
diff --git a/spartan/lib/List.thy b/spartan/lib/List.thy
index a755859..be86b63 100644
--- a/spartan/lib/List.thy
+++ b/spartan/lib/List.thy
@@ -44,9 +44,10 @@ where
         f x xs (ListInd A (fn xs. C xs) c\<^sub>0 (fn x xs rec. f x xs rec) xs)"
 
 lemmas
-  [intros] = ListF List_nil List_cons and
-  [elims "?xs"] = ListE and
-  [comps] = List_comp_nil List_comp_cons
+  [form] = ListF and
+  [intro, intros] = List_nil List_cons and
+  [elim "?xs"] = ListE and
+  [comp] = List_comp_nil List_comp_cons
 
 abbreviation "ListRec A C \<equiv> ListInd A (fn _. C)"
 
@@ -110,7 +111,7 @@ Lemma head_type [typechk]:
   shows "head xs: Maybe A"
   unfolding head_def by typechk
 
-Lemma head_of_cons [comps]:
+Lemma head_of_cons [comp]:
   assumes "A: U i" "x: A" "xs: List A"
   shows "head (x # xs) \<equiv> some x"
   unfolding head_def by reduce
@@ -120,7 +121,7 @@ Lemma tail_type [typechk]:
   shows "tail xs: List A"
   unfolding tail_def by typechk
 
-Lemma tail_of_cons [comps]:
+Lemma tail_of_cons [comp]:
   assumes "A: U i" "x: A" "xs: List A"
   shows "tail (x # xs) \<equiv> xs"
   unfolding tail_def by reduce
@@ -181,7 +182,7 @@ Lemma rev_type [typechk]:
   shows "rev xs: List A"
   unfolding rev_def by typechk
 
-Lemma rev_nil [comps]:
+Lemma rev_nil [comp]:
   assumes "A: U i"
   shows "rev (nil A) \<equiv> nil A"
   unfolding rev_def by reduce
diff --git a/spartan/lib/Maybe.thy b/spartan/lib/Maybe.thy
index d821920..0ce534c 100644
--- a/spartan/lib/Maybe.thy
+++ b/spartan/lib/Maybe.thy
@@ -54,9 +54,10 @@ Lemma Maybe_comp_some:
   unfolding MaybeInd_def some_def by (reduce add: Maybe_def)
 
 lemmas
-  [intros] = MaybeF Maybe_none Maybe_some and
-  [comps] = Maybe_comp_none Maybe_comp_some and
-  MaybeE [elims "?m"] = MaybeInd[rotated 4]
+  [form] = MaybeF and
+  [intro, intros] = Maybe_none Maybe_some and
+  [comp] = Maybe_comp_none Maybe_comp_some and
+  MaybeE [elim "?m"] = MaybeInd[rotated 4]
 lemmas
   Maybe_cases [cases] = MaybeE
 
diff --git a/spartan/lib/More_Types.thy b/spartan/lib/More_Types.thy
index 0d7096f..55e6554 100644
--- a/spartan/lib/More_Types.thy
+++ b/spartan/lib/More_Types.thy
@@ -16,9 +16,9 @@ notation Sum (infixl "\<or>" 50)
 axiomatization where
   SumF: "\<lbrakk>A: U i; B: U i\<rbrakk> \<Longrightarrow> A \<or> B: U i" and
 
-  Sum_inl: "\<lbrakk>a: A; B: U i\<rbrakk> \<Longrightarrow> inl A B a: A \<or> B" and
+  Sum_inl: "\<lbrakk>B: U i; a: A\<rbrakk> \<Longrightarrow> inl A B a: A \<or> B" and
 
-  Sum_inr: "\<lbrakk>b: B; A: U i\<rbrakk> \<Longrightarrow> inr A B b: A \<or> B" and
+  Sum_inr: "\<lbrakk>A: U i; b: B\<rbrakk> \<Longrightarrow> inr A B b: A \<or> B" and
 
   SumE: "\<lbrakk>
     s: A \<or> B;
@@ -42,9 +42,11 @@ axiomatization where
     \<rbrakk> \<Longrightarrow> SumInd A B (fn s. C s) (fn a. c a) (fn b. d b) (inr A B b) \<equiv> d b"
 
 lemmas
-  [intros] = SumF Sum_inl Sum_inr and
-  [elims ?s] = SumE and
-  [comps] = Sum_comp_inl Sum_comp_inr
+  [form] = SumF and
+  [intro] = Sum_inl Sum_inr and
+  [intros] = Sum_inl[rotated] Sum_inr[rotated] and
+  [elim ?s] = SumE and
+  [comp] = Sum_comp_inl Sum_comp_inr
 
 method left = rule Sum_inl
 method right = rule Sum_inr
@@ -76,10 +78,11 @@ and
   BotE: "\<lbrakk>x: \<bottom>; \<And>x. x: \<bottom> \<Longrightarrow> C x: U i\<rbrakk> \<Longrightarrow> BotInd (fn x. C x) x: C x"
 
 lemmas
-  [intros] = TopF TopI BotF and
-  [elims ?a] = TopE and
-  [elims ?x] = BotE and
-  [comps] = Top_comp
+  [form] = TopF BotF and
+  [intro, intros] = TopI and
+  [elim ?a] = TopE and
+  [elim ?x] = BotE and
+  [comp] = Top_comp
 
 
 section \<open>Booleans\<close>
@@ -125,9 +128,10 @@ Lemma if_false:
   by reduce
 
 lemmas
-  [intros] = BoolF Bool_true Bool_false and
-  [comps] = if_true if_false and
-  [elims ?x] = ifelse
+  [form] = BoolF and
+  [intro, intros] = Bool_true Bool_false and
+  [comp] = if_true if_false and
+  [elim ?x] = ifelse
 lemmas
   BoolE = ifelse
 
@@ -136,7 +140,7 @@ subsection \<open>Notation\<close>
 definition ifelse_i ("if _ then _ else _")
   where [implicit]: "if x then a else b \<equiv> ifelse ? x a b"
 
-no_translations "if x then a else b" \<leftharpoondown> "CONST ifelse C x a b"
+translations "if x then a else b" \<leftharpoondown> "CONST ifelse C x a b"
 
 subsection \<open>Logical connectives\<close>