(FEAT) Term elaboration of assumption and goal statements.

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

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

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

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

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

8 files changed, 250 insertions, 105 deletions

diff --git a/spartan/core/Spartan.thy b/spartan/core/Spartan.thy
index faa692d..27edd51 100644
--- a/spartan/core/Spartan.thy
+++ b/spartan/core/Spartan.thy
@@ -8,9 +8,9 @@ imports
 keywords
   "Theorem" "Lemma" "Corollary" "Proposition" "Definition" :: thy_goal_stmt and
   "assuming" :: prf_asm % "proof" and
-  "focus" "\<guillemotright>" "\<^item>" "\<^enum>" "~" :: prf_script_goal % "proof" and
+  "focus" "\<^item>" "\<^enum>" "\<circ>" "\<diamondop>" "~" :: prf_script_goal % "proof" and
   "congruence" "print_coercions" :: thy_decl and
-  "rhs" "derive" "prems" "vars" :: quasi_command
+  "rhs" "def" "vars" :: quasi_command
 
 begin
 
@@ -198,8 +198,8 @@ lemmas
   [form] = PiF SigF and
   [intro] = PiI SigI and
   [intros] = PiI[rotated] SigI and
-  [elim "?f"] = PiE and
-  [elim "?p"] = SigE and
+  [elim ?f] = PiE and
+  [elim ?p] = SigE and
   [comp] = beta Sig_comp and
   [cong] = Pi_cong lam_cong Sig_cong
 
@@ -217,7 +217,7 @@ subsection \<open>Statement commands\<close>
 
 ML_file \<open>focus.ML\<close>
 ML_file \<open>elaboration.ML\<close>
-ML_file \<open>elaborated_assumption.ML\<close>
+ML_file \<open>elaborated_statement.ML\<close>
 ML_file \<open>goals.ML\<close>
 
 subsection \<open>Proof methods\<close>
@@ -229,10 +229,6 @@ method_setup rule =
   \<open>Attrib.thms >> (fn ths => K (CONTEXT_METHOD (
     CHEADGOAL o SIDE_CONDS (rule_ctac ths))))\<close>
 
-method_setup rules =
-  \<open>Attrib.thms >> (fn ths => K (CONTEXT_METHOD (
-    CREPEAT o 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 (
@@ -411,10 +407,10 @@ lemma funcompI [typechk]:
 
 lemma funcomp_assoc [comp]:
   assumes
+    "A: U i"
     "f: A \<rightarrow> B"
     "g: B \<rightarrow> C"
     "h: \<Prod>x: C. D x"
-    "A: U i"
   shows
     "(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
@@ -430,10 +426,9 @@ lemma funcomp_lambda_comp [comp]:
 
 lemma funcomp_apply_comp [comp]:
   assumes
+    "A: U i" "B: U i" "\<And>x y. x: B \<Longrightarrow> C x: U i"
     "f: A \<rightarrow> B" "g: \<Prod>x: B. C x"
     "x: A"
-    "A: U i" "B: U i"
-    "\<And>x y. x: B \<Longrightarrow> C x: U i"
   shows "(g \<circ>\<^bsub>A\<^esub> f) x \<equiv> g (f x)"
   unfolding funcomp_def by reduce
 
@@ -449,17 +444,17 @@ subsection \<open>Identity function\<close>
 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_type [typechk]: "A: U i \<Longrightarrow> id A: A \<rightarrow> A" and
   id_comp [comp]: "x: A \<Longrightarrow> (id A) x \<equiv> x" \<comment> \<open>for the occasional manual rewrite\<close>
   by reduce+
 
 lemma id_left [comp]:
-  assumes "f: A \<rightarrow> B" "A: U i" "B: U i"
+  assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
   shows "(id B) \<circ>\<^bsub>A\<^esub> f \<equiv> f"
   by (subst eta_exp[of f]) (reduce, rule eta)
 
 lemma id_right [comp]:
-  assumes "f: A \<rightarrow> B" "A: U i" "B: U i"
+  assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
   shows "f \<circ>\<^bsub>A\<^esub> (id A) \<equiv> f"
   by (subst eta_exp[of f]) (reduce, rule eta)
 
@@ -480,10 +475,7 @@ lemma fst_type [typechk]:
 
 lemma fst_comp [comp]:
   assumes
-    "a: A"
-    "b: B a"
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> B x: U i"
+    "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" "a: A" "b: B a"
   shows "fst A B <a, b> \<equiv> a"
   unfolding fst_def by reduce
 
@@ -493,7 +485,7 @@ lemma snd_type [typechk]:
   unfolding snd_def by typechk reduce
 
 lemma snd_comp [comp]:
-  assumes "a: A" "b: B a" "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i"
+  assumes "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" "a: A" "b: B a"
   shows "snd A B <a, b> \<equiv> b"
   unfolding snd_def by reduce
 
@@ -513,36 +505,48 @@ subsection \<open>Projections\<close>
 
 Lemma fst [typechk]:
   assumes
-    "p: \<Sum>x: A. B x"
     "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i"
+    "p: \<Sum>x: A. B x"
   shows "fst p: A"
   by typechk
 
 Lemma snd [typechk]:
   assumes
-    "p: \<Sum>x: A. B x"
     "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i"
+    "p: \<Sum>x: A. B x"
   shows "snd p: B (fst p)"
   by typechk
 
-method fst for p::o = rule fst[of p]
-method snd for p::o = rule snd[of p]
+method fst for p::o = rule fst[where ?p=p]
+method snd for p::o = rule snd[where ?p=p]
 
-subsection \<open>Properties of \<Sigma>\<close>
+text \<open>Double projections:\<close>
 
-Lemma (derive) Sig_dist_exp:
+definition [implicit]: "p\<^sub>1\<^sub>1 p \<equiv> Spartan.fst ? ? (Spartan.fst ? ? p)"
+definition [implicit]: "p\<^sub>1\<^sub>2 p \<equiv> Spartan.snd ? ? (Spartan.fst ? ? p)"
+definition [implicit]: "p\<^sub>2\<^sub>1 p \<equiv> Spartan.fst ? ? (Spartan.snd ? ? p)"
+definition [implicit]: "p\<^sub>2\<^sub>2 p \<equiv> Spartan.snd ? ? (Spartan.snd ? ? p)"
+
+translations
+  "CONST p\<^sub>1\<^sub>1 p" \<leftharpoondown> "fst (fst p)"
+  "CONST p\<^sub>1\<^sub>2 p" \<leftharpoondown> "snd (fst p)"
+  "CONST p\<^sub>2\<^sub>1 p" \<leftharpoondown> "fst (snd p)"
+  "CONST p\<^sub>2\<^sub>2 p" \<leftharpoondown> "snd (snd p)"
+
+Lemma (def) distribute_Sig:
   assumes
-    "p: \<Sum>x: A. B x \<times> C x"
     "A: U i"
     "\<And>x. x: A \<Longrightarrow> B x: U i"
     "\<And>x. x: A \<Longrightarrow> C x: U i"
+    "p: \<Sum>x: A. B x \<times> C x"
   shows "(\<Sum>x: A. B x) \<times> (\<Sum>x: A. C x)"
-proof intro
-  have "fst p: A" and "snd p: B (fst p) \<times> C (fst p)" by typechk+
-  thus "<fst p, fst (snd p)>: \<Sum>x: A. B x"
-   and "<fst p, snd (snd p)>: \<Sum>x: A. C x"
-    by typechk+
-qed
+  proof intro
+    have "fst p: A" and "snd p: B (fst p) \<times> C (fst p)"
+      by typechk+
+    thus "<fst p, fst (snd p)>: \<Sum>x: A. B x"
+     and "<fst p, snd (snd p)>: \<Sum>x: A. C x"
+      by typechk+
+  qed
 
 
 end
diff --git a/spartan/core/context_tactical.ML b/spartan/core/context_tactical.ML
index a5159f6..b5a6c00 100644
--- a/spartan/core/context_tactical.ML
+++ b/spartan/core/context_tactical.ML
@@ -2,6 +2,10 @@
     Author:     Joshua Chen
 
 More context tactics, and context tactic combinators.
+
+Contains code modified from
+  ~~/Pure/search.ML
+  ~~/Pure/tactical.ML
 *)
 
 infix 1 CTHEN CTHEN' CTHEN_ALL_NEW CTHEN_ALL_NEW_FWD
@@ -23,6 +27,7 @@ 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 CREPEAT1: 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'
@@ -34,6 +39,16 @@ 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'
+val CFIRST: context_tactic list -> context_tactic
+val CFIRST': context_tactic' list -> context_tactic'
+val CTHEN_BEST_FIRST: context_tactic -> (context_state -> bool) ->
+  (context_state -> int) -> context_tactic -> context_tactic
+val CBEST_FIRST: (context_state -> bool) -> (context_state -> int) ->
+  context_tactic -> context_tactic
+val CTHEN_ASTAR: context_tactic -> (context_state -> bool) ->
+  (int -> context_state -> int) -> context_tactic -> context_tactic
+val CASTAR: (context_state -> bool) -> (int -> context_state -> int) ->
+  context_tactic -> context_tactic
 
 end = struct
 
@@ -74,6 +89,8 @@ fun CREPEAT ctac =
           | SOME (cst, q) => rep (q :: qs) cst);
   in fn cst => Seq.make_results (Seq.make (fn () => rep [] cst)) end
 
+fun CREPEAT1 ctac = ctac CTHEN CREPEAT ctac
+
 fun CFILTER pred ctac cst =
   ctac cst
   |> Seq.filter_results
@@ -147,6 +164,89 @@ fun CSOMEGOAL ctac (cst as (_, st)) =
 fun CRANGE [] _ = all_ctac
   | CRANGE (ctac :: ctacs) i = CRANGE ctacs (i + 1) CTHEN ctac i
 
+fun CFIRST ctacs = fold_rev (curry op CORELSE) ctacs no_ctac
+
+(*FIRST' [tac1,...,tacn] i  equals    tac1 i ORELSE ... ORELSE tacn i*)
+fun CFIRST' ctacs = fold_rev (curry op CORELSE') ctacs (K no_ctac)
+
+
+(** Search tacticals **)
+
+(* Best-first search *)
+
+structure Thm_Heap = Heap (
+  type elem = int * thm;
+  val ord = prod_ord int_ord (Term_Ord.term_ord o apply2 Thm.prop_of)
+)
+
+structure Context_State_Heap = Heap (
+  type elem = int * context_state;
+  val ord = prod_ord int_ord (Term_Ord.term_ord o apply2 (Thm.prop_of o #2))
+)
+
+fun some_of_list [] = NONE
+  | some_of_list (x :: l) = SOME (x, Seq.make (fn () => some_of_list l))
+
+(*Check for and delete duplicate proof states*)
+fun delete_all_min (cst as (_, st)) heap =
+  if Context_State_Heap.is_empty heap then heap
+  else if Thm.eq_thm (st, #2 (#2 (Context_State_Heap.min heap)))
+  then delete_all_min cst (Context_State_Heap.delete_min heap)
+  else heap
+
+(*Best-first search for a state that satisfies satp (incl initial state)
+  Function sizef estimates size of problem remaining (smaller means better).
+  tactic tac0 sets up the initial priority queue, while tac1 searches it. *)
+fun CTHEN_BEST_FIRST ctac0 satp sizef ctac =
+  let
+    fun pairsize cst = (sizef cst, cst);
+    fun bfs (news, nst_heap) =
+      (case List.partition satp news of
+        ([], nonsats) => next (fold_rev Context_State_Heap.insert (map pairsize nonsats) nst_heap)
+      | (sats, _)  => some_of_list sats)
+    and next nst_heap =
+      if Context_State_Heap.is_empty nst_heap then NONE
+      else
+        let
+          val (n, cst) = Context_State_Heap.min nst_heap;
+        in
+          bfs (Seq.list_of (Seq.filter_results (ctac cst)), delete_all_min cst (Context_State_Heap.delete_min nst_heap))
+        end;
+    fun btac cst = bfs (Seq.list_of (Seq.filter_results (ctac0 cst)), Context_State_Heap.empty)
+  in fn cst => Seq.make_results (Seq.make (fn () => btac cst)) end
+
+(*Ordinary best-first search, with no initial tactic*)
+val CBEST_FIRST = CTHEN_BEST_FIRST all_ctac
+
+
+(* A*-like search *)
+
+(*Insertion into priority queue of states, marked with level*)
+fun insert_with_level (lnth: int * int * context_state) [] = [lnth]
+  | insert_with_level (l, m, cst) ((l', n, cst') :: csts) =
+      if  n < m then (l', n, cst') :: insert_with_level (l, m, cst) csts
+      else if n = m andalso Thm.eq_thm (#2 cst, #2 cst') then (l', n, cst') :: csts
+      else (l, m, cst) :: (l', n, cst') :: csts;
+
+fun CTHEN_ASTAR ctac0 satp costf ctac =
+  let
+    fun bfs (news, nst, level) =
+      let fun cost cst = (level, costf level cst, cst) in
+        (case List.partition satp news of
+          ([], nonsats) => next (fold_rev (insert_with_level o cost) nonsats nst)
+        | (sats, _) => some_of_list sats)
+      end
+    and next [] = NONE
+      | next ((level, n, cst) :: nst) =
+          bfs (Seq.list_of (Seq.filter_results (ctac cst)), nst, level + 1)
+  in fn cst => Seq.make_results
+    (Seq.make (fn () => bfs (Seq.list_of (Seq.filter_results (ctac0 cst)), [], 0)))
+  end
+
+(*Ordinary ASTAR, with no initial tactic*)
+val CASTAR = CTHEN_ASTAR all_ctac;
+
+
 end
 
 open Context_Tactical
diff --git a/spartan/core/elaborated_assumption.ML b/spartan/core/elaborated_statement.ML
index af3a384..81f4a7d 100644
--- a/spartan/core/elaborated_assumption.ML
+++ b/spartan/core/elaborated_statement.ML
@@ -1,7 +1,7 @@
-(*  Title:      elaborated_assumption.ML
+(*  Title:      elaborated_statement.ML
     Author:     Joshua Chen
 
-Term elaboration for goal and proof assumptions.
+Term elaboration for goal statements and proof commands.
 
 Contains code from parts of
   ~~/Pure/Isar/element.ML and
@@ -9,7 +9,7 @@ Contains code from parts of
 in both verbatim and modified forms.
 *)
 
-structure Elaborated_Assumption: sig
+structure Elaborated_Statement: sig
 
 val read_goal_statement:
   (string, string, Facts.ref) Element.ctxt list ->
@@ -20,24 +20,7 @@ val read_goal_statement:
 end = struct
 
 
-(*Apply elaboration to the list format assumptions are given in*)
-fun elaborate ctxt assms =
-  let
-    fun subst_term env = Envir.subst_term (Envir.type_env env, Envir.term_env env)
-    fun elab_fact (fact, xs) assums =
-      let val (subst, fact') = Elab.elab_stmt ctxt assums fact in
-        ((fact', map (subst_term subst) xs), Thm.cterm_of ctxt fact' :: assums)
-      end
-    fun elab (b, facts) assums =
-      let val (facts', assums') = fold_map elab_fact facts assums
-      in ((b, facts'), assums') end
-  in #1 (fold_map elab assms []) end
-
-
-(* Goal assumptions *)
-
-(*Most of the code in this section is copied verbatim from the originals; the
-  only change is a modification to`activate_i` incorporating elaboration.*)
+(* Elaborated goal statements *)
 
 local
 
@@ -371,16 +354,28 @@ val read_full_context_statement = prep_full_context_statement
   Proof_Context.read_var parse_inst Syntax.read_props Attrib.check_src
   Proof_Context.read_var check_expr
 
+fun filter_assumes ((x as Element.Assumes _) :: xs) = x :: filter_assumes xs
+  | filter_assumes (_ :: xs) = filter_assumes xs
+  | filter_assumes [] = []
+
 fun prep_statement prep activate raw_elems raw_stmt ctxt =
   let
     val ((_, _, _, elems, concl), _) =
       prep {strict = true, do_close = false, fixed_frees = true}
-        ([], []) I raw_elems raw_stmt ctxt;
-    val ctxt' = ctxt
+        ([], []) I raw_elems raw_stmt ctxt
+
+    val (elems', ctxt') = ctxt
       |> Proof_Context.set_stmt true
-      |> fold_map activate elems |> #2
-      |> Proof_Context.restore_stmt ctxt;
-  in (concl, ctxt') end
+      |> fold_map activate elems
+      |> apsnd (Proof_Context.restore_stmt ctxt)
+
+    val assumes = filter_assumes elems'
+    val assms = flat (flat (map
+      (fn (Element.Assumes asms) =>
+        map (fn (_, facts) => map (Thm.cterm_of ctxt' o #1) facts) asms)
+      assumes))
+    val concl' = Elab.elaborate ctxt' assms concl handle error => concl
+  in (concl', ctxt') end
 
 fun activate_i elem ctxt =
   let
@@ -391,7 +386,7 @@ fun activate_i elem ctxt =
             ((Thm.def_binding_optional
               (Binding.name (#1 (#1 (Local_Defs.cert_def ctxt (K []) t)))) a, atts),
               (t, ps))))
-      | Element.Assumes assms => Element.Assumes (elaborate ctxt assms)
+      | Element.Assumes assms => Element.Assumes (Elab.elaborate ctxt [] assms)
       | e => e);
     val ctxt' = Context.proof_map (Element.init elem') ctxt;
   in ((Element.map_ctxt_attrib o map) Token.closure elem', ctxt') end
@@ -437,7 +432,7 @@ val _ = Outer_Syntax.command \<^command_keyword>\<open>assuming\<close> "elabora
       val b' = map (map read_terms) b
       val c' = c |> map (fn ((b, atts), ss) =>
         ((b, map (Attrib.attribute_cmd ctxt) atts), map read_terms ss))
-      val c'' = elaborate ctxt c'
+      val c'' = Elab.elaborate ctxt [] c'
     in Proof.assume a' b' c'' state end)))
 
 end
diff --git a/spartan/core/elaboration.ML b/spartan/core/elaboration.ML
index 27b6bb0..9e5e0bd 100644
--- a/spartan/core/elaboration.ML
+++ b/spartan/core/elaboration.ML
@@ -1,13 +1,14 @@
 (*  Title:      elaboration.ML
     Author:     Joshua Chen
 
-Basic elaboration.
+Basic term elaboration.
 *)
 
 structure Elab: sig
 
 val elab: Proof.context -> cterm list -> term -> Envir.env
 val elab_stmt: Proof.context -> cterm list -> term -> Envir.env * term
+val elaborate: Proof.context -> cterm list -> ('a * (term * term list) list) list -> ('a * (term * term list) list) list
 
 end = struct
 
@@ -72,5 +73,19 @@ fun elab_stmt ctxt assums stmt =
       in (subst', subst_term subst' stmt) end
   end
 
+(*Apply elaboration to the list format that assumptions and goal statements are
+  given in*)
+fun elaborate ctxt known assms =
+  let
+    fun subst_term env = Envir.subst_term (Envir.type_env env, Envir.term_env env)
+    fun elab_fact (fact, xs) assums =
+      let val (subst, fact') = elab_stmt ctxt assums fact in
+        ((fact', map (subst_term subst) xs), Thm.cterm_of ctxt fact' :: assums)
+      end
+    fun elab (b, facts) assums =
+      let val (facts', assums') = fold_map elab_fact facts assums
+      in ((b, facts'), assums') end
+  in #1 (fold_map elab assms known) end
+
 
 end
diff --git a/spartan/core/focus.ML b/spartan/core/focus.ML
index 1d8de78..2ad4c8a 100644
--- a/spartan/core/focus.ML
+++ b/spartan/core/focus.ML
@@ -1,14 +1,51 @@
 (*  Title:      focus.ML
-    Author:     Makarius Wenzel, Joshua Chen
+    Author:     Joshua Chen
 
-A modified version of the Isar `subgoal` command
-that keeps schematic variables in the goal state.
+Focus on head subgoal, with optional variable renaming.
 
-Modified from code originally written by Makarius Wenzel.
+Modified from code contained in ~~/Pure/Isar/subgoal.ML.
 *)
 
 local
 
+fun reverse_prems imp =
+  let val (prems, concl) = (Drule.strip_imp_prems imp, Drule.strip_imp_concl imp)
+  in fold (curry mk_implies) prems concl end
+
+fun gen_focus ctxt i bindings raw_st =
+  let
+    val st = raw_st
+      |> Thm.solve_constraints
+      |> Thm.transfer' ctxt
+      |> Raw_Simplifier.norm_hhf_protect ctxt
+
+    val ((schematic_types, [st']), ctxt1) = Variable.importT [st] ctxt
+
+    val ((params, goal), ctxt2) =
+      Variable.focus_cterm bindings (Thm.cprem_of st' i) ctxt1
+
+    val (asms, concl) =
+      (Drule.strip_imp_prems goal, Drule.strip_imp_concl goal)
+
+    fun intern_var_assms asm (asms, concl) =
+      if Lib.no_vars (Thm.term_of asm)
+      then (asm :: asms, concl)
+      else (asms, Drule.mk_implies (asm, concl))
+
+    val (asms', concl') = fold intern_var_assms asms ([], concl)
+      |> apfst rev |> apsnd reverse_prems
+
+    val (inst, ctxt3) = Variable.import_inst true (map Thm.term_of (asms')) ctxt2
+    val schematic_terms = map (apsnd (Thm.cterm_of ctxt3)) (#2 inst)
+    val schematics = (schematic_types, schematic_terms)
+    val asms' = map (Thm.instantiate_cterm schematics) asms'
+    val concl' = Thm.instantiate_cterm schematics concl'
+    val (prems, context) = Assumption.add_assumes asms' ctxt3
+  in
+    ({context = context, params = params, prems = prems,
+      asms = asms', concl = concl', schematics = schematics}, Goal.init concl')
+  end
+
 fun param_bindings ctxt (param_suffix, raw_param_specs) st =
   let
     val _ = if Thm.no_prems st then error "No subgoals!" else ()
@@ -24,8 +61,8 @@ fun param_bindings ctxt (param_suffix, raw_param_specs) st =
         error ("Excessive subgoal parameter specification" ^
           Position.here_list (map snd (drop n raw_param_specs)))
 
-    val param_specs =
-      raw_param_specs |> map
+    val param_specs = raw_param_specs
+      |> map
         (fn (NONE, _) => NONE
           | (SOME x, pos) =>
               let
@@ -39,7 +76,7 @@ fun param_bindings ctxt (param_suffix, raw_param_specs) st =
       | bindings _ ys = map Binding.name ys
   in bindings param_specs subgoal_params end
 
-fun gen_schematic_subgoal prep_atts raw_result_binding raw_prems_binding param_specs state =
+fun gen_schematic_subgoal prep_atts raw_result_binding param_specs state =
   let
     val _ = Proof.assert_backward state
 
@@ -47,17 +84,13 @@ fun gen_schematic_subgoal prep_atts raw_result_binding raw_prems_binding param_s
       |> Proof.map_context (Proof_Context.set_mode Proof_Context.mode_schematic)
       |> Proof.refine_insert []
 
-    val {context = ctxt, facts = facts, goal = st} = Proof.raw_goal state1
-
+    val {context = ctxt, facts, goal = st} = Proof.raw_goal state1
     val result_binding = apsnd (map (prep_atts ctxt)) raw_result_binding
-    val (prems_binding, do_prems) =
-      (case raw_prems_binding of
-        SOME (b, raw_atts) => ((b, map (prep_atts ctxt) raw_atts), true)
-      | NONE => (Binding.empty_atts, false))
 
-    val (subgoal_focus, _) =
-      (if do_prems then Subgoal.focus_prems else Subgoal.focus_params) ctxt
-        1 (SOME (param_bindings ctxt param_specs st)) st
+    val subgoal_focus = #1
+      (gen_focus ctxt 1 (SOME (param_bindings ctxt param_specs st)) st)
+
+    val prems = #prems subgoal_focus
 
     fun after_qed (ctxt'', [[result]]) =
       Proof.end_block #> (fn state' =>
@@ -83,31 +116,29 @@ fun gen_schematic_subgoal prep_atts raw_result_binding raw_prems_binding param_s
     |> Proof.begin_block
     |> Proof.map_context (fn _ =>
       #context subgoal_focus
-      |> Proof_Context.note_thmss "" [(prems_binding, [(#prems subgoal_focus, [])])] |> #2)
+      |> Proof_Context.note_thmss "" [((Binding.name "prems", []), [(prems, [])])]
+      |> #2)
     |> Proof.internal_goal (K (K ())) (Proof_Context.get_mode ctxt) true "subgoal"
-        NONE after_qed [] [] [(Binding.empty_atts, [(Thm.term_of (#concl subgoal_focus), [])])] |> #2
-    |> Proof.using_facts facts
+        NONE after_qed [] [] [(Binding.empty_atts, [(Thm.term_of (#concl subgoal_focus), [])])]
+    |> #2
+    |> Proof.using_facts (facts @ prems)
     |> pair subgoal_focus
   end
 
 val opt_fact_binding =
-  Scan.optional (Parse.binding -- Parse.opt_attribs || Parse.attribs >> pair Binding.empty)
+  Scan.optional ((Parse.binding -- Parse.opt_attribs || Parse.attribs >> pair Binding.empty) --| Args.colon)
     Binding.empty_atts
 
-val for_params =
-  Scan.optional
-    (\<^keyword>\<open>vars\<close> |--
-      Parse.!!! ((Scan.option Parse.dots >> is_some) --
-        (Scan.repeat1 (Parse.maybe_position Parse.name_position))))
-    (false, [])
+val for_params = Scan.optional
+  (\<^keyword>\<open>vars\<close> |--
+    Parse.!!! ((Scan.option Parse.dots >> is_some) --
+      (Scan.repeat1 (Parse.maybe_position Parse.name_position))))
+  (false, [])
 
 val schematic_subgoal_cmd = gen_schematic_subgoal Attrib.attribute_cmd
 
-val parser =
-  opt_fact_binding
-  -- (Scan.option (\<^keyword>\<open>prems\<close> |-- Parse.!!! opt_fact_binding))
-  -- for_params >> (fn ((a, b), c) =>
-    Toplevel.proofs (Seq.make_results o Seq.single o #2 o schematic_subgoal_cmd a b c))
+val parser = opt_fact_binding -- for_params >> (fn (fact, params) =>
+  Toplevel.proofs (Seq.make_results o Seq.single o #2 o schematic_subgoal_cmd fact params))
 
 in
 
@@ -117,9 +148,10 @@ val _ = Outer_Syntax.command \<^command_keyword>\<open>focus\<close>
   "focus on first subgoal within backward refinement, without instantiating schematic vars"
   parser
 
-val _ = Outer_Syntax.command \<^command_keyword>\<open>\<guillemotright>\<close> "focus bullet" parser
 val _ = Outer_Syntax.command \<^command_keyword>\<open>\<^item>\<close> "focus bullet" parser
 val _ = Outer_Syntax.command \<^command_keyword>\<open>\<^enum>\<close> "focus bullet" parser
+val _ = Outer_Syntax.command \<^command_keyword>\<open>\<circ>\<close> "focus bullet" parser
+val _ = Outer_Syntax.command \<^command_keyword>\<open>\<diamondop>\<close> "focus bullet" parser
 val _ = Outer_Syntax.command \<^command_keyword>\<open>~\<close> "focus bullet" parser
 
 end
diff --git a/spartan/core/goals.ML b/spartan/core/goals.ML
index 87ec2b8..540f5c7 100644
--- a/spartan/core/goals.ML
+++ b/spartan/core/goals.ML
@@ -1,9 +1,9 @@
 (*  Title:      goals.ML
-    Author:     Makarius Wenzel, Joshua Chen
+    Author:     Joshua Chen
 
 Goal statements and proof term export.
 
-Modified from code originally written by Makarius Wenzel.
+Modified from code contained in ~~/Pure/Isar/specification.ML.
 *)
 
 local
@@ -184,15 +184,15 @@ val schematic_theorem_cmd =
   gen_schematic_theorem
     Bundle.includes_cmd
     Attrib.check_src
-    Elaborated_Assumption.read_goal_statement
+    Elaborated_Statement.read_goal_statement
 
 fun theorem spec descr =
   Outer_Syntax.local_theory_to_proof' spec ("state " ^ descr) 
-    ( Scan.option (Args.parens (Args.$$$ "derive"))
+    ( Scan.option (Args.parens (Args.$$$ "def"))
       -- (long_statement || short_statement) >>
         (fn (opt_derive, (long, binding, includes, elems, concl)) =>
           schematic_theorem_cmd
-            (case opt_derive of SOME "derive" => true | _ => false)
+            (case opt_derive of SOME "def" => true | _ => false)
             long descr NONE (K I) binding includes elems concl) )
 
 fun definition spec descr =
diff --git a/spartan/core/tactics.ML b/spartan/core/tactics.ML
index 3922ae0..19d3d71 100644
--- a/spartan/core/tactics.ML
+++ b/spartan/core/tactics.ML
@@ -19,8 +19,7 @@ end = struct
 
 
 (* Side conditions *)
-val solve_side_conds =
-  Attrib.setup_config_int \<^binding>\<open>solve_side_conds\<close> (K 2)
+val solve_side_conds = Attrib.setup_config_int \<^binding>\<open>solve_side_conds\<close> (K 2)
 
 fun SIDE_CONDS ctac facts i (cst as (ctxt, st)) = cst |> (ctac i CTHEN
   (case Config.get ctxt solve_side_conds of
diff --git a/spartan/core/typechecking.ML b/spartan/core/typechecking.ML
index 57164a1..a6e1726 100644
--- a/spartan/core/typechecking.ML
+++ b/spartan/core/typechecking.ML
@@ -51,10 +51,9 @@ fun known_ctac facts = CONTEXT_SUBGOAL (fn (goal, i) => fn (ctxt, st) =>
       if Lib.no_vars concl orelse
         (Lib.is_typing concl andalso Lib.no_vars (Lib.term_of_typing concl))
       then
-        let val ths = facts
+        let val ths = map (Simplifier.norm_hhf ctxt)
+          (facts @ map fst (Facts.props (Proof_Context.facts_of ctxt)))
           (*FIXME: Shouldn't pull nameless facts directly from context*)
-          @ map fst (Facts.props (Proof_Context.facts_of ctxt))
-          |> map (Simplifier.norm_hhf ctxt)
         in
           (debug_tac ctxt "resolve" THEN resolve_tac ctxt ths i ORELSE
           debug_tac ctxt "assume" THEN assume_tac ctxt i) st
@@ -70,7 +69,8 @@ fun check_infer_step facts i (ctxt, st) =
     val tac = SUBGOAL (fn (goal, i) =>
       if Lib.rigid_typing_concl goal
       then
-        let val net = Tactic.build_net (facts
+        let val net = Tactic.build_net (
+          map (Simplifier.norm_hhf ctxt) 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>)