Checkpoint. THIS BUILD WILL FAIL

6 files changed, 451 insertions, 51 deletions

diff --git a/spartan/core/Spartan.thy b/spartan/core/Spartan.thy
index 8fa6cfe..1b9093b 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>
 
@@ -77,7 +91,7 @@ lemma lift_U:
   by (erule U_cumulative, 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>
 
@@ -184,19 +199,22 @@ consts "rhs" :: \<open>'a\<close> ("..")
 
 ML_file \<open>congruence.ML\<close>
 
+subsection \<open>Theorem attributes, type-checking and proof methods\<close>
+
+named_theorems intros and comps
+
 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
+  [intros] = anno PiF PiI SigF SigI and
   [elims "?f"] = PiE and
   [elims "?p"] = SigE and
   [comps] = beta Sig_comp and
   [cong] = Pi_cong lam_cong Sig_cong
 
+ML_file \<open>typechecking.ML\<close>
+ML_file \<open>tactics2.ML\<close>
 ML_file \<open>tactics.ML\<close>
 
 method_setup assumptions =
@@ -223,12 +241,12 @@ 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 =
+method_setup typechk' =
   \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (
     SIDE_CONDS (typechk_tac ctxt) ctxt))
     (* CHANGED (ALLGOALS (TRY o typechk_tac ctxt)))) *)\<close>
 
-method_setup rule =
+method_setup rule' =
   \<open>Attrib.thms >> (fn ths => fn ctxt =>
     SIMPLE_METHOD (HEADGOAL (SIDE_CONDS (rule_tac ths ctxt) ctxt)))\<close>
 
@@ -237,6 +255,16 @@ method_setup dest =
     >> (fn (opt_n, ths) => fn ctxt =>
         SIMPLE_METHOD (HEADGOAL (SIDE_CONDS (dest_tac opt_n ths ctxt) ctxt)))\<close>
 
+(*NEW*)
+method_setup typechk =
+  \<open>Scan.succeed (K (CONTEXT_METHOD (
+    CHEADGOAL o Types.check_infer)))\<close>
+
+method_setup rule =
+  \<open>Attrib.thms >> (fn ths => K (CONTEXT_METHOD (
+    CHEADGOAL o Tactics2.SIDE_CONDS (Tactics2.rule_tac ths))))\<close>
+
+
 subsection \<open>Reflexivity\<close>
 
 named_theorems refl
@@ -348,12 +376,21 @@ 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)
+  (*FIXME: Temporary; should be fixed once all methods are rewritten to use
+    the new typechk*)
+  apply (Pure.rule refine_codomain, typechk, typechk)
+  apply (Pure.rule lift_U, typechk)
   done
 
 subsection \<open>Function composition\<close>
@@ -472,7 +509,7 @@ definition fst_i ("fst")
 definition snd_i ("snd")
   where [implicit]: "snd \<equiv> Spartan.snd ? ?"
 
-translations
+no_translations
   "fst" \<leftharpoondown> "CONST Spartan.fst A B"
   "snd" \<leftharpoondown> "CONST Spartan.snd A B"
 
@@ -483,14 +520,22 @@ 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
+  \<comment> \<open>This can't be solved by a single application of `typechk`; it needs
+    multiple (two) passes. Something to do with constraint/subgoal reordering.\<close>
+  apply (Pure.rule elims)
+    apply (Pure.rule typechk)
+    apply known [1]
+    defer \<comment> \<open>The deferred subgoal is an inhabitation problem.\<close>
+    apply known [1]
+  by known
+    
 
 Lemma snd [typechk]:
   assumes
     "p: \<Sum>x: A. B x"
     "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i"
   shows "snd p: B (fst p)"
-  by typechk
+  by typechk+
 
 method fst for p::o = rule fst[of p]
 method snd for p::o = rule snd[of p]
diff --git a/spartan/core/context_tactical.ML b/spartan/core/context_tactical.ML
new file mode 100644
index 0000000..78991a9
--- /dev/null
+++ b/spartan/core/context_tactical.ML
@@ -0,0 +1,129 @@
+(*  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 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_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
+
+end = struct
+
+type context_tactic' = int -> context_tactic
+
+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)) =
+    if l > u then all_ctac cst
+    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)) cst
+in
+
+fun (ctac1 CTHEN_ALL_NEW ctac2) i (cst as (_, st)) =
+  cst |> (ctac1 i CTHEN (fn cst' as (_, st') =>
+    cst' |> INTERVAL ctac2 i (i + Thm.nprems_of st' - Thm.nprems_of st)))
+
+(*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') =>
+    cst' |> INTERVAL_FWD ctac2 i (i + Thm.nprems_of st' - Thm.nprems_of st)))
+
+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
+
+end
+
+open Context_Tactical
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/tactics.ML b/spartan/core/tactics.ML
index 0c71665..172ae90 100644
--- a/spartan/core/tactics.ML
+++ b/spartan/core/tactics.ML
@@ -64,13 +64,6 @@ fun typechk_tac ctxt =
     REPEAT_ALL_NEW (known_tac ctxt ORELSE' tac)
   end
 
-fun typechk_context_tac (ctxt, st) =
-  let
-    
-  in
-    ()
-  end
-
 (*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)
diff --git a/spartan/core/tactics2.ML b/spartan/core/tactics2.ML
new file mode 100644
index 0000000..801424f
--- /dev/null
+++ b/spartan/core/tactics2.ML
@@ -0,0 +1,191 @@
+(*  Title:      tactics.ML
+    Author:     Joshua Chen
+
+General tactics for dependent type theory.
+*)
+
+structure Tactics2:
+sig
+
+val solve_side_conds: bool Config.T
+val SIDE_CONDS: context_tactic' -> thm list -> context_tactic'
+val rule_tac: thm list -> context_tactic'
+val dest_tac: int option -> thm list -> context_tactic'
+val intro_tac: context_tactic'
+val intros_tac: context_tactic'
+(*
+val elim_context_tac: term list -> Proof.context -> int -> context_tactic
+val cases_tac: term -> Proof.context -> int -> tactic
+*)
+
+end = struct
+
+(*Automatically try to solve side conditions?*)
+val solve_side_conds =
+  Attrib.setup_config_bool \<^binding>\<open>solve_side_conds\<close> (K true)
+
+local
+  fun side_cond_ctac facts i (cst as (ctxt, _)) =
+    if Config.get ctxt solve_side_conds
+    then (Types.check_infer facts i) cst
+    else all_ctac cst
+in
+
+(*Combinator runs tactic and tries to discharge newly arising side conditions*)
+fun SIDE_CONDS ctac facts = ctac CTHEN_ALL_NEW (CTRY o side_cond_ctac facts)
+
+end
+
+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
+in
+
+(*Resolves with given rules*)
+fun rule_tac 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 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 i (ctxt, st) =
+  TACTIC_CONTEXT ctxt (resolve_tac ctxt
+    (Named_Theorems.get ctxt \<^named_theorems>\<open>intros\<close>) i st)
+
+val intros_tac = CONTEXT_SUBGOAL (fn (_, i) =>
+  (CCHANGED o CREPEAT o CCHANGED o intro_tac) i)
+
+(*
+(* Induction/elimination *)
+
+(*Pushes a context/goal premise typing t:T into a \<Prod>-type*)
+fun internalize_fact_tac t =
+  Subgoal.FOCUS_PARAMS (fn {context = ctxt, concl = raw_concl, ...} =>
+    let
+      val concl = Logic.strip_assums_concl (Thm.term_of raw_concl)
+      val C = Lib.type_of_typing concl
+      val B = Thm.cterm_of ctxt (Lib.lambda_var t C)
+      val a = Thm.cterm_of ctxt t
+      (*The resolvent is PiE[where ?B=B and ?a=a]*)
+      val resolvent =
+        Drule.infer_instantiate' ctxt [NONE, NONE, SOME B, SOME a] @{thm PiE}
+    in
+      HEADGOAL (resolve_tac ctxt [resolvent])
+      (*known_tac infers the correct type T inferred by unification*)
+      THEN SOMEGOAL (known_tac ctxt)
+    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 (
+  let
+    val rule_insts = map ((Elim.lookup_rule ctxt) o Term.head_of) types
+    val rules = flat (map
+      (fn rule_inst => case rule_inst of
+          NONE => []
+        | SOME (rl, idxnames) => [Drule.infer_instantiate ctxt
+          (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))
+
+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)
+
+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)
+
+end
+*)
+
+end
+
+open Tactics2
diff --git a/spartan/core/typechecking.ML b/spartan/core/typechecking.ML
index 437a2dc..13ca440 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 +33,60 @@ 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 *)
-
-local
+(* Context tactics for type-checking *)
 
-fun checkable prop = Lib.is_typing prop
-  andalso not (exists_subterm is_Var (Lib.type_of_typing prop))
+(*For debugging*)
+structure Probe = Proof_Data (
+  type T = int
+  val init = K 0
+)
 
+(*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 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>intros\<close>)
+            @(map #1 (Elim.rules ctxt)))
+          in (resolve_from_net_tac ctxt net) i end
+        else no_tac)
+      val ctxt' = Probe.map (fn i => i + 1) 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 =
+  let val probe = K (print_ctac (Int.toString o Probe.get))
+  in
+    CREPEAT_ALL_NEW_FWD (
+      probe CTHEN' known_ctac facts
+      CORELSE' probe CTHEN' check_infer_step facts)
+  end
 
 end
 
-
 end