8 files changed, 602 insertions, 79 deletions

diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy
index d976677..d844b59 100644
--- a/hott/Equivalence.thy
+++ b/hott/Equivalence.thy
@@ -33,13 +33,16 @@ Lemma (derive) hsym:
     "g: \<Prod>x: A. B x"
     "A: U i"
     "\<And>x. x: A \<Longrightarrow> B x: U i"
-  shows "H: f ~ g \<Longrightarrow> g ~ f"
-  unfolding homotopy_def
-  apply intro
-    apply (rule pathinv)
-      \<guillemotright> by (elim H)
-      \<guillemotright> by typechk
-  done
+    "H: f ~ g"
+  shows "g ~ f"
+unfolding homotopy_def
+proof intro
+  fix x assume "x: A" then have "H x: f x = g x"
+    using \<open>H:_\<close>[unfolded homotopy_def]
+      \<comment> \<open>this should become unnecessary when definitional unfolding is implemented\<close>
+    by typechk
+  thus "g x = f x" by (rule pathinv) fact
+qed typechk
 
 Lemma (derive) htrans:
   assumes
@@ -71,9 +74,9 @@ Lemma (derive) commute_homotopy:
   assumes
     "A: U i" "B: U i"
     "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
+    "p: x = y"
     "f: A \<rightarrow> B" "g: A \<rightarrow> B"
-    "H: homotopy A (fn _. B) f g"
+    "H: f ~ g"
   shows "(H x) \<bullet> g[p] = f[p] \<bullet> (H y)"
   \<comment> \<open>Need this assumption unfolded for typechecking\<close>
   supply assms(8)[unfolded homotopy_def]
@@ -94,7 +97,7 @@ Corollary (derive) commute_homotopy':
     "A: U i"
     "x: A"
     "f: A \<rightarrow> A"
-    "H: homotopy A (fn _. A) f (id A)"
+    "H: f ~ (id A)"
   shows "H (f x) = f [H x]"
 oops
 
diff --git a/hott/Identity.thy b/hott/Identity.thy
index 29ce26a..728537c 100644
--- a/hott/Identity.thy
+++ b/hott/Identity.thy
@@ -135,14 +135,14 @@ lemmas
 section \<open>Basic propositional equalities\<close>
 
 Lemma (derive) refl_pathcomp:
-  assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
+  assumes "A: U i" "x: A" "y: A" "p: x = y"
   shows "(refl x) \<bullet> p = p"
   apply (eq p)
     apply (reduce; intro)
   done
 
 Lemma (derive) pathcomp_refl:
-  assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
+  assumes "A: U i" "x: A" "y: A" "p: x = y"
   shows "p \<bullet> (refl y) = p"
   apply (eq p)
     apply (reduce; intro)
@@ -166,24 +166,24 @@ Lemma ru_refl [comp]:
   unfolding pathcomp_refl_def by reduce
 
 Lemma (derive) inv_pathcomp:
-  assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
+  assumes "A: U i" "x: A" "y: A" "p: x = y"
   shows "p\<inverse> \<bullet> p = refl y"
   by (eq p) (reduce; intro)
 
 Lemma (derive) pathcomp_inv:
-  assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
+  assumes "A: U i" "x: A" "y: A" "p: x = y"
   shows "p \<bullet> p\<inverse> = refl x"
   by (eq p) (reduce; intro)
 
 Lemma (derive) pathinv_pathinv:
-  assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
+  assumes "A: U i" "x: A" "y: A" "p: x = y"
   shows "p\<inverse>\<inverse> = p"
   by (eq p) (reduce; intro)
 
 Lemma (derive) pathcomp_assoc:
   assumes
     "A: U i" "x: A" "y: A" "z: A" "w: A"
-    "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z" "r: z =\<^bsub>A\<^esub> w"
+    "p: x = y" "q: y = z" "r: z = w"
   shows "p \<bullet> (q \<bullet> r) = p \<bullet> q \<bullet> r"
   apply (eq p)
     focus prems vars x p
@@ -203,7 +203,7 @@ Lemma (derive) ap:
     "A: U i" "B: U i"
     "x: A" "y: A"
     "f: A \<rightarrow> B"
-    "p: x =\<^bsub>A\<^esub> y"
+    "p: x = y"
   shows "f x = f y"
   by (eq p) intro
 
@@ -222,7 +222,7 @@ Lemma (derive) ap_pathcomp:
     "A: U i" "B: U i"
     "x: A" "y: A" "z: A"
     "f: A \<rightarrow> B"
-    "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z"
+    "p: x = y" "q: y = z"
   shows
     "f[p \<bullet> q] = f[p] \<bullet> f[q]"
   apply (eq p)
@@ -237,7 +237,7 @@ Lemma (derive) ap_pathinv:
     "A: U i" "B: U i"
     "x: A" "y: A"
     "f: A \<rightarrow> B"
-    "p: x =\<^bsub>A\<^esub> y"
+    "p: x = y"
   shows "f[p\<inverse>] = f[p]\<inverse>"
   by (eq p) (reduce; intro)
 
@@ -248,7 +248,7 @@ Lemma (derive) ap_funcomp:
     "A: U i" "B: U i" "C: U i"
     "x: A" "y: A"
     "f: A \<rightarrow> B" "g: B \<rightarrow> C"
-    "p: x =\<^bsub>A\<^esub> y"
+    "p: x = y"
   shows "(g \<circ> f)[p] = g[f[p]]"
   apply (eq p)
     \<guillemotright> by reduce
@@ -256,7 +256,7 @@ Lemma (derive) ap_funcomp:
   done
 
 Lemma (derive) ap_id:
-  assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
+  assumes "A: U i" "x: A" "y: A" "p: x = y"
   shows "(id A)[p] = p"
   apply (eq p)
     \<guillemotright> by reduce
@@ -271,7 +271,7 @@ Lemma (derive) transport:
     "A: U i"
     "\<And>x. x: A \<Longrightarrow> P x: U i"
     "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
+    "p: x = y"
   shows "P x \<rightarrow> P y"
   by (eq p) intro
 
@@ -308,7 +308,7 @@ Lemma (derive) transport_left_inv:
     "A: U i"
     "\<And>x. x: A \<Longrightarrow> P x: U i"
     "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
+    "p: x = y"
   shows "(trans P p\<inverse>) \<circ> (trans P p) = id (P x)"
   by (eq p) (reduce; refl)
 
@@ -317,7 +317,7 @@ Lemma (derive) transport_right_inv:
     "A: U i"
     "\<And>x. x: A \<Longrightarrow> P x: U i"
     "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
+    "p: x = y"
   shows "(trans P p) \<circ> (trans P p\<inverse>) = id (P y)"
   by (eq p) (reduce; intro)
 
@@ -327,7 +327,7 @@ Lemma (derive) transport_pathcomp:
     "\<And>x. x: A \<Longrightarrow> P x: U i"
     "x: A" "y: A" "z: A"
     "u: P x"
-    "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z"
+    "p: x = y" "q: y = z"
   shows "trans P q (trans P p u) = trans P (p \<bullet> q) u"
   apply (eq p)
     focus prems vars x p
@@ -342,7 +342,7 @@ Lemma (derive) transport_compose_typefam:
     "\<And>x. x: B \<Longrightarrow> P x: U i"
     "f: A \<rightarrow> B"
     "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
+    "p: x = y"
     "u: P (f x)"
   shows "trans (fn x. P (f x)) p u = trans P f[p] u"
   by (eq p) (reduce; intro)
@@ -355,7 +355,7 @@ Lemma (derive) transport_function_family:
     "f: \<Prod>x: A. P x \<rightarrow> Q x"
     "x: A" "y: A"
     "u: P x"
-    "p: x =\<^bsub>A\<^esub> y"
+    "p: x = y"
   shows "trans Q p ((f x) u) = (f y) (trans P p u)"
   by (eq p) (reduce; intro)
 
@@ -363,7 +363,7 @@ Lemma (derive) transport_const:
   assumes
     "A: U i" "B: U i"
     "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
+    "p: x = y"
   shows "\<Prod>b: B. trans (fn _. B) p b = b"
   by (intro, eq p) (reduce; intro)
 
@@ -384,7 +384,7 @@ Lemma (derive) pathlift:
     "A: U i"
     "\<And>x. x: A \<Longrightarrow> P x: U i"
     "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
+    "p: x = y"
     "u: P x"
   shows "<x, u> = <y, trans P p u>"
   by (eq p) (reduce; intro)
@@ -409,7 +409,7 @@ Lemma (derive) pathlift_fst:
     "\<And>x. x: A \<Longrightarrow> P x: U i"
     "x: A" "y: A"
     "u: P x"
-    "p: x =\<^bsub>A\<^esub> y"
+    "p: x = y"
   shows "fst[lift P p u] = p"
   apply (eq p)
     \<guillemotright> by reduce
@@ -425,7 +425,7 @@ Lemma (derive) apd:
     "\<And>x. x: A \<Longrightarrow> P x: U i"
     "f: \<Prod>x: A. P x"
     "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
+    "p: x = y"
   shows "trans P p (f x) = f y"
   by (eq p) (reduce; intro)
 
@@ -448,7 +448,7 @@ Lemma (derive) apd_ap:
     "A: U i" "B: U i"
     "f: A \<rightarrow> B"
     "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
+    "p: x = y"
   shows "apd f p = trans_const B p (f x) \<bullet> f[p]"
   by (eq p) (reduce; intro)
 
@@ -457,10 +457,9 @@ 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>
+      and "p: a = b" "q: a = b" "r: b = c"
+      and "\<alpha>: p = q"
+  shows "p \<bullet> r = q \<bullet> r"
   apply (eq r)
   focus prems vars x s t
     proof -
@@ -473,7 +472,9 @@ Lemma (derive) right_whisker:
 
 Lemma (derive) left_whisker:
   assumes "A: U i" "a: A" "b: A" "c: A"
-  shows "\<lbrakk>p: b = c; q: b = c; r: a = b; \<alpha>: p =\<^bsub>b = c\<^esub> q\<rbrakk> \<Longrightarrow> r \<bullet> p = r \<bullet> q"
+      and "p: b = c" "q: b = c" "r: a = b"
+      and "\<alpha>: p = q"
+  shows "r \<bullet> p = r \<bullet> q"
   apply (eq r)
   focus prems prms vars x s t
     proof -
@@ -495,15 +496,13 @@ translations
   "r \<bullet>\<^sub>l\<^bsub>c\<^esub> \<alpha>" \<leftharpoondown> "CONST left_whisker A a b c p q r \<alpha>"
 
 Lemma whisker_refl [comp]:
-  assumes "A: U i" "a: A" "b: A"
-  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>"
+  assumes "A: U i" "a: A" "b: A" "p: a = b" "q: a = b" "\<alpha>: p = q"
+  shows "\<alpha> \<bullet>\<^sub>r\<^bsub>a\<^esub> (refl b) \<equiv> ru p \<bullet> \<alpha> \<bullet> (ru q)\<inverse>"
   unfolding right_whisker_def by reduce
 
 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>"
+  assumes "A: U i" "a: A" "b: A" "p: a = b" "q: a = b" "\<alpha>: p = q"
+  shows "(refl a) \<bullet>\<^sub>l\<^bsub>b\<^esub> \<alpha> \<equiv> (lu p) \<bullet> \<alpha> \<bullet> (lu q)\<inverse>"
   unfolding left_whisker_def by reduce
 
 method left_whisker = (rule left_whisker)
@@ -524,20 +523,22 @@ begin
 
 Lemma (derive) horiz_pathcomp:
   notes assums
-  shows "\<lbrakk>\<alpha>: p = q; \<beta>: r = s\<rbrakk> \<Longrightarrow> ?prf \<alpha> \<beta>: p \<bullet> r = q \<bullet> s"
+  assumes "\<alpha>: p = q" "\<beta>: r = s"
+  shows "p \<bullet> r = q \<bullet> s"
 proof (rule pathcomp)
-  show "\<alpha>: p = q \<Longrightarrow> p \<bullet> r = q \<bullet> r" by right_whisker
-  show "\<beta>: r = s \<Longrightarrow> .. = q \<bullet> s" by left_whisker
+  show "p \<bullet> r = q \<bullet> r" by right_whisker fact
+  show ".. = q \<bullet> s" by left_whisker fact
 qed typechk
 
 text \<open>A second horizontal composition:\<close>
 
 Lemma (derive) horiz_pathcomp':
   notes assums
-  shows "\<lbrakk>\<alpha>: p = q; \<beta>: r = s\<rbrakk> \<Longrightarrow> ?prf \<alpha> \<beta>: p \<bullet> r = q \<bullet> s"
+  assumes "\<alpha>: p = q" "\<beta>: r = s"
+  shows "p \<bullet> r = q \<bullet> s"
 proof (rule pathcomp)
-  show "\<beta>: r = s \<Longrightarrow> p \<bullet> r = p \<bullet> s" by left_whisker
-  show "\<alpha>: p = q \<Longrightarrow> .. = q \<bullet> s" by right_whisker
+  show "p \<bullet> r = p \<bullet> s" by left_whisker fact
+  show ".. = q \<bullet> s" by right_whisker fact
 qed typechk
 
 notation horiz_pathcomp (infix "\<star>" 121)
@@ -545,7 +546,8 @@ notation horiz_pathcomp' (infix "\<star>''" 121)
 
 Lemma (derive) horiz_pathcomp_eq_horiz_pathcomp':
   notes assums
-  shows "\<lbrakk>\<alpha>: p = q; \<beta>: r = s\<rbrakk> \<Longrightarrow> \<alpha> \<star> \<beta> = \<alpha> \<star>' \<beta>"
+  assumes "\<alpha>: p = q" "\<beta>: r = s"
+  shows "\<alpha> \<star> \<beta> = \<alpha> \<star>' \<beta>"
   unfolding horiz_pathcomp_def horiz_pathcomp'_def
   apply (eq \<alpha>, eq \<beta>)
     focus vars p apply (eq p)
diff --git a/spartan/core/Spartan.thy b/spartan/core/Spartan.thy
index 2c7216e..11bdc2b 100644
--- a/spartan/core/Spartan.thy
+++ b/spartan/core/Spartan.thy
@@ -215,6 +215,8 @@ method_setup known =
 subsection \<open>Statement commands\<close>
 
 ML_file \<open>focus.ML\<close>
+ML_file \<open>elaboration.ML\<close>
+ML_file \<open>elaborated_expression.ML\<close>
 ML_file \<open>goals.ML\<close>
 
 subsection \<open>Proof methods\<close>
diff --git a/spartan/core/elaborated_expression.ML b/spartan/core/elaborated_expression.ML
new file mode 100644
index 0000000..49b7758
--- /dev/null
+++ b/spartan/core/elaborated_expression.ML
@@ -0,0 +1,402 @@
+(*  Title:      elaborated_expression.ML
+    Author:     Joshua Chen
+
+A modification of parts of ~~/Pure/Isar/expression.ML to incorporate elaboration
+into the assumptions mechanism.
+
+Most of this file is copied verbatim from the original; the only changes are the
+addition of `elaborate` and a modification to `activate_i`.
+*)
+
+structure Elaborated_Expression = struct
+
+local
+
+open Element
+
+fun mk_type T = (Logic.mk_type T, [])
+fun mk_term t = (t, [])
+fun mk_propp (p, pats) = (Type.constraint propT p, pats)
+
+fun dest_type (T, []) = Logic.dest_type T
+fun dest_term (t, []) = t
+fun dest_propp (p, pats) = (p, pats)
+
+fun extract_inst (_, (_, ts)) = map mk_term ts
+fun restore_inst ((l, (p, _)), cs) = (l, (p, map dest_term cs))
+
+fun extract_eqns es = map (mk_term o snd) es
+fun restore_eqns (es, cs) = map2 (fn (b, _) => fn c => (b, dest_term c)) es cs
+
+fun extract_elem (Fixes fixes) = map (#2 #> the_list #> map mk_type) fixes
+  | extract_elem (Constrains csts) = map (#2 #> single #> map mk_type) csts
+  | extract_elem (Assumes asms) = map (#2 #> map mk_propp) asms
+  | extract_elem (Defines defs) = map (fn (_, (t, ps)) => [mk_propp (t, ps)]) defs
+  | extract_elem (Notes _) = []
+  | extract_elem (Lazy_Notes _) = []
+
+fun restore_elem (Fixes fixes, css) =
+      (fixes ~~ css) |> map (fn ((x, _, mx), cs) =>
+        (x, cs |> map dest_type |> try hd, mx)) |> Fixes
+  | restore_elem (Constrains csts, css) =
+      (csts ~~ css) |> map (fn ((x, _), cs) =>
+        (x, cs |> map dest_type |> hd)) |> Constrains
+  | restore_elem (Assumes asms, css) =
+      (asms ~~ css) |> map (fn ((b, _), cs) => (b, map dest_propp cs)) |> Assumes
+  | restore_elem (Defines defs, css) =
+      (defs ~~ css) |> map (fn ((b, _), [c]) => (b, dest_propp c)) |> Defines
+  | restore_elem (elem as Notes _, _) = elem
+  | restore_elem (elem as Lazy_Notes _, _) = elem
+
+fun prep (_, pats) (ctxt, t :: ts) =
+  let val ctxt' = Proof_Context.augment t ctxt
+  in
+    ((t, Syntax.check_props
+        (Proof_Context.set_mode Proof_Context.mode_pattern ctxt') pats),
+      (ctxt', ts))
+  end
+
+fun check cs ctxt =
+  let
+    val (cs', (ctxt', _)) = fold_map prep cs
+      (ctxt, Syntax.check_terms
+        (Proof_Context.set_mode Proof_Context.mode_schematic ctxt) (map fst cs))
+  in (cs', ctxt') end
+
+fun inst_morphism params ((prfx, mandatory), insts') ctxt =
+  let
+    (* parameters *)
+    val parm_types = map #2 params;
+    val type_parms = fold Term.add_tfreesT parm_types [];
+
+    (* type inference *)
+    val parm_types' = map (Type_Infer.paramify_vars o Logic.varifyT_global) parm_types;
+    val type_parms' = fold Term.add_tvarsT parm_types' [];
+    val checked =
+      (map (Logic.mk_type o TVar) type_parms' @ map2 Type.constraint parm_types' insts')
+      |> Syntax.check_terms (Config.put Type_Infer.object_logic false ctxt)
+    val (type_parms'', insts'') = chop (length type_parms') checked;
+
+    (* context *)
+    val ctxt' = fold Proof_Context.augment checked ctxt;
+    val certT = Thm.trim_context_ctyp o Thm.ctyp_of ctxt';
+    val cert = Thm.trim_context_cterm o Thm.cterm_of ctxt';
+
+    (* instantiation *)
+    val instT =
+      (type_parms ~~ map Logic.dest_type type_parms'')
+      |> map_filter (fn (v, T) => if TFree v = T then NONE else SOME (v, T));
+    val cert_inst =
+      ((map #1 params ~~ map (Term_Subst.instantiateT_frees instT) parm_types) ~~ insts'')
+      |> map_filter (fn (v, t) => if Free v = t then NONE else SOME (v, cert t));
+  in
+    (Element.instantiate_normalize_morphism (map (apsnd certT) instT, cert_inst) $>
+      Morphism.binding_morphism "Expression.inst" (Binding.prefix mandatory prfx), ctxt')
+  end;
+
+fun abs_def ctxt =
+  Thm.cterm_of ctxt #> Assumption.assume ctxt #> Local_Defs.abs_def_rule ctxt #> Thm.prop_of;
+
+fun declare_elem prep_var (Element.Fixes fixes) ctxt =
+      let val (vars, _) = fold_map prep_var fixes ctxt
+      in ctxt |> Proof_Context.add_fixes vars |> snd end
+  | declare_elem prep_var (Element.Constrains csts) ctxt =
+      ctxt |> fold_map (fn (x, T) => prep_var (Binding.name x, SOME T, NoSyn)) csts |> snd
+  | declare_elem _ (Element.Assumes _) ctxt = ctxt
+  | declare_elem _ (Element.Defines _) ctxt = ctxt
+  | declare_elem _ (Element.Notes _) ctxt = ctxt
+  | declare_elem _ (Element.Lazy_Notes _) ctxt = ctxt;
+
+fun parameters_of thy strict (expr, fixed) =
+  let
+    val ctxt = Proof_Context.init_global thy;
+
+    fun reject_dups message xs =
+      (case duplicates (op =) xs of
+        [] => ()
+      | dups => error (message ^ commas dups));
+
+    fun parm_eq ((p1, mx1), (p2, mx2)) =
+      p1 = p2 andalso
+        (Mixfix.equal (mx1, mx2) orelse
+          error ("Conflicting syntax for parameter " ^ quote p1 ^ " in expression" ^
+            Position.here_list [Mixfix.pos_of mx1, Mixfix.pos_of mx2]));
+
+    fun params_loc loc = Locale.params_of thy loc |> map (apfst #1);
+    fun params_inst (loc, (prfx, (Expression.Positional insts, eqns))) =
+          let
+            val ps = params_loc loc;
+            val d = length ps - length insts;
+            val insts' =
+              if d < 0 then
+                error ("More arguments than parameters in instantiation of locale " ^
+                  quote (Locale.markup_name ctxt loc))
+              else insts @ replicate d NONE;
+            val ps' = (ps ~~ insts') |>
+              map_filter (fn (p, NONE) => SOME p | (_, SOME _) => NONE);
+          in (ps', (loc, (prfx, (Expression.Positional insts', eqns)))) end
+      | params_inst (loc, (prfx, (Expression.Named insts, eqns))) =
+          let
+            val _ =
+              reject_dups "Duplicate instantiation of the following parameter(s): "
+                (map fst insts);
+            val ps' = (insts, params_loc loc) |-> fold (fn (p, _) => fn ps =>
+              if AList.defined (op =) ps p then AList.delete (op =) p ps
+              else error (quote p ^ " not a parameter of instantiated expression"));
+          in (ps', (loc, (prfx, (Expression.Named insts, eqns)))) end;
+    fun params_expr is =
+      let
+        val (is', ps') = fold_map (fn i => fn ps =>
+          let
+            val (ps', i') = params_inst i;
+            val ps'' = distinct parm_eq (ps @ ps');
+          in (i', ps'') end) is []
+      in (ps', is') end;
+
+    val (implicit, expr') = params_expr expr;
+
+    val implicit' = map #1 implicit;
+    val fixed' = map (Variable.check_name o #1) fixed;
+    val _ = reject_dups "Duplicate fixed parameter(s): " fixed';
+    val implicit'' =
+      if strict then []
+      else
+        let
+          val _ =
+            reject_dups
+              "Parameter(s) declared simultaneously in expression and for clause: "
+              (implicit' @ fixed');
+        in map (fn (x, mx) => (Binding.name x, NONE, mx)) implicit end;
+  in (expr', implicit'' @ fixed) end;
+
+fun parse_elem prep_typ prep_term ctxt =
+  Element.map_ctxt
+   {binding = I,
+    typ = prep_typ ctxt,
+    term = prep_term (Proof_Context.set_mode Proof_Context.mode_schematic ctxt),
+    pattern = prep_term (Proof_Context.set_mode Proof_Context.mode_pattern ctxt),
+    fact = I,
+    attrib = I};
+
+fun prepare_stmt prep_prop prep_obtains ctxt stmt =
+  (case stmt of
+    Element.Shows raw_shows =>
+      raw_shows |> (map o apsnd o map) (fn (t, ps) =>
+        (prep_prop (Proof_Context.set_mode Proof_Context.mode_schematic ctxt) t,
+          map (prep_prop (Proof_Context.set_mode Proof_Context.mode_pattern ctxt)) ps))
+  | Element.Obtains raw_obtains =>
+      let
+        val ((_, thesis), thesis_ctxt) = Obtain.obtain_thesis ctxt;
+        val obtains = prep_obtains thesis_ctxt thesis raw_obtains;
+      in map (fn (b, t) => ((b, []), [(t, [])])) obtains end);
+
+fun finish_fixes (parms: (string * typ) list) = map (fn (binding, _, mx) =>
+  let val x = Binding.name_of binding
+  in (binding, AList.lookup (op =) parms x, mx) end)
+
+fun finish_inst ctxt (loc, (prfx, inst)) =
+  let
+    val thy = Proof_Context.theory_of ctxt;
+    val (morph, _) = inst_morphism (map #1 (Locale.params_of thy loc)) (prfx, inst) ctxt;
+  in (loc, morph) end
+
+fun closeup _ _ false elem = elem
+  | closeup (outer_ctxt, ctxt) parms true elem =
+      let
+        (* FIXME consider closing in syntactic phase -- before type checking *)
+        fun close_frees t =
+          let
+            val rev_frees =
+              Term.fold_aterms (fn Free (x, T) =>
+                if Variable.is_fixed outer_ctxt x orelse AList.defined (op =) parms x then I
+                else insert (op =) (x, T) | _ => I) t [];
+          in fold (Logic.all o Free) rev_frees t end;
+
+        fun no_binds [] = []
+          | no_binds _ = error "Illegal term bindings in context element";
+      in
+        (case elem of
+          Assumes asms => Assumes (asms |> map (fn (a, propps) =>
+            (a, map (fn (t, ps) => (close_frees t, no_binds ps)) propps)))
+        | Defines defs => Defines (defs |> map (fn ((name, atts), (t, ps)) =>
+            let val ((c, _), t') = Local_Defs.cert_def ctxt (K []) (close_frees t)
+            in ((Thm.def_binding_optional (Binding.name c) name, atts), (t', no_binds ps)) end))
+        | e => e)
+      end
+
+fun finish_elem _ parms _ (Fixes fixes) = Fixes (finish_fixes parms fixes)
+  | finish_elem _ _ _ (Constrains _) = Constrains []
+  | finish_elem ctxts parms do_close (Assumes asms) = closeup ctxts parms do_close (Assumes asms)
+  | finish_elem ctxts parms do_close (Defines defs) = closeup ctxts parms do_close (Defines defs)
+  | finish_elem _ _ _ (elem as Notes _) = elem
+  | finish_elem _ _ _ (elem as Lazy_Notes _) = elem
+
+fun check_autofix insts eqnss elems concl ctxt =
+  let
+    val inst_cs = map extract_inst insts;
+    val eqns_cs = map extract_eqns eqnss;
+    val elem_css = map extract_elem elems;
+    val concl_cs = (map o map) mk_propp (map snd concl);
+    (* Type inference *)
+    val (inst_cs' :: eqns_cs' :: css', ctxt') =
+      (fold_burrow o fold_burrow) check (inst_cs :: eqns_cs :: elem_css @ [concl_cs]) ctxt;
+    val (elem_css', [concl_cs']) = chop (length elem_css) css';
+  in
+    ((map restore_inst (insts ~~ inst_cs'),
+      map restore_eqns (eqnss ~~ eqns_cs'),
+      map restore_elem (elems ~~ elem_css'),
+      map fst concl ~~ concl_cs'), ctxt')
+  end
+
+fun prep_full_context_statement
+  parse_typ parse_prop
+  prep_obtains prep_var_elem prep_inst prep_eqns prep_attr prep_var_inst prep_expr
+  {strict, do_close, fixed_frees} raw_import init_body raw_elems raw_stmt
+  ctxt1
+  =
+  let
+    val thy = Proof_Context.theory_of ctxt1
+    val (raw_insts, fixed) = parameters_of thy strict (apfst (prep_expr thy) raw_import)
+    fun prep_insts_cumulative (loc, (prfx, (inst, eqns))) (i, insts, eqnss, ctxt) =
+      let
+        val params = map #1 (Locale.params_of thy loc)
+        val inst' = prep_inst ctxt (map #1 params) inst
+        val parm_types' =
+          params |> map (#2 #> Logic.varifyT_global #>
+              Term.map_type_tvar (fn ((x, _), S) => TVar ((x, i), S)) #>
+              Type_Infer.paramify_vars)
+        val inst'' = map2 Type.constraint parm_types' inst'
+        val insts' = insts @ [(loc, (prfx, inst''))]
+        val ((insts'', _, _, _), ctxt2) = check_autofix insts' [] [] [] ctxt
+        val inst''' = insts'' |> List.last |> snd |> snd
+        val (inst_morph, _) = inst_morphism params (prfx, inst''') ctxt
+        val ctxt' = Locale.activate_declarations (loc, inst_morph) ctxt2
+          handle ERROR msg => if null eqns then error msg else
+            (Locale.tracing ctxt1
+             (msg ^ "\nFalling back to reading rewrites clause before activation.");
+             ctxt2)
+        val attrss = map (apsnd (map (prep_attr ctxt)) o fst) eqns
+        val eqns' = (prep_eqns ctxt' o map snd) eqns
+        val eqnss' = [attrss ~~ eqns']
+        val ((_, [eqns''], _, _), _) = check_autofix insts'' eqnss' [] [] ctxt'
+        val rewrite_morph = eqns'
+          |> map (abs_def ctxt')
+          |> Variable.export_terms ctxt' ctxt
+          |> Element.eq_term_morphism (Proof_Context.theory_of ctxt)
+          |> the_default Morphism.identity
+       val ctxt'' = Locale.activate_declarations (loc, inst_morph $> rewrite_morph) ctxt
+       val eqnss' = eqnss @ [attrss ~~ Variable.export_terms ctxt' ctxt eqns']
+      in (i + 1, insts', eqnss', ctxt'') end
+
+    fun prep_elem raw_elem ctxt =
+      let
+        val ctxt' = ctxt
+          |> Context_Position.set_visible false
+          |> declare_elem prep_var_elem raw_elem
+          |> Context_Position.restore_visible ctxt
+        val elems' = parse_elem parse_typ parse_prop ctxt' raw_elem
+      in (elems', ctxt') end
+
+    val fors = fold_map prep_var_inst fixed ctxt1 |> fst
+    val ctxt2 = ctxt1 |> Proof_Context.add_fixes fors |> snd
+    val (_, insts', eqnss', ctxt3) = fold prep_insts_cumulative raw_insts (0, [], [], ctxt2)
+
+    fun prep_stmt elems ctxt =
+      check_autofix insts' [] elems (prepare_stmt parse_prop prep_obtains ctxt raw_stmt) ctxt
+
+    val _ =
+      if fixed_frees then ()
+      else
+        (case fold (fold (Variable.add_frees ctxt3) o snd o snd) insts' [] of
+          [] => ()
+        | frees => error ("Illegal free variables in expression: " ^
+            commas_quote (map (Syntax.string_of_term ctxt3 o Free) (rev frees))))
+
+    val ((insts, _, elems', concl), ctxt4) = ctxt3
+      |> init_body
+      |> fold_map prep_elem raw_elems
+      |-> prep_stmt
+
+    (* parameters from expression and elements *)
+    val xs = maps (fn Element.Fixes fixes => map (Variable.check_name o #1) fixes | _ => [])
+      (Element.Fixes fors :: elems')
+    val (parms, ctxt5) = fold_map Proof_Context.inferred_param xs ctxt4
+    val fors' = finish_fixes parms fors
+    val fixed = map (fn (b, SOME T, mx) => ((Binding.name_of b, T), mx)) fors'
+    val deps = map (finish_inst ctxt5) insts
+    val elems'' = map (finish_elem (ctxt1, ctxt5) parms do_close) elems'
+  in ((fixed, deps, eqnss', elems'', concl), (parms, ctxt5)) end
+
+fun prep_inst prep_term ctxt parms (Expression.Positional insts) =
+      (insts ~~ parms) |> map
+        (fn (NONE, p) => Free (p, dummyT)
+          | (SOME t, _) => prep_term ctxt t)
+  | prep_inst prep_term ctxt parms (Expression.Named insts) =
+      parms |> map (fn p =>
+        (case AList.lookup (op =) insts p of
+          SOME t => prep_term ctxt t |
+          NONE => Free (p, dummyT)))
+fun parse_inst x = prep_inst Syntax.parse_term x
+fun check_expr thy instances = map (apfst (Locale.check thy)) instances
+
+val read_full_context_statement = prep_full_context_statement
+  Syntax.parse_typ Syntax.parse_prop Obtain.parse_obtains
+  Proof_Context.read_var parse_inst Syntax.read_props Attrib.check_src
+  Proof_Context.read_var check_expr
+
+in
+
+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
+      |> Proof_Context.set_stmt true
+      |> fold_map activate elems |> #2
+      |> Proof_Context.restore_stmt ctxt;
+  in (concl, ctxt') end
+
+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
+
+fun activate_i elem ctxt =
+  let
+    val elem' =
+      (case (map_ctxt_attrib o map) Token.init_assignable elem of
+        Defines defs =>
+          Defines (defs |> map (fn ((a, atts), (t, ps)) =>
+            ((Thm.def_binding_optional
+              (Binding.name (#1 (#1 (Local_Defs.cert_def ctxt (K []) t)))) a, atts),
+              (t, ps))))
+      | Assumes assms => Assumes (elaborate ctxt assms)
+      | e => e);
+    val ctxt' = Context.proof_map (init elem') ctxt;
+  in ((map_ctxt_attrib o map) Token.closure elem', ctxt') end
+
+fun activate raw_elem ctxt =
+  let val elem = raw_elem |> map_ctxt
+   {binding = I,
+    typ = I,
+    term = I,
+    pattern = I,
+    fact = Proof_Context.get_fact ctxt,
+    attrib = Attrib.check_src ctxt}
+  in activate_i elem ctxt end;
+
+fun read_statement x = prep_statement read_full_context_statement activate x
+
+end
+
+end
\ No newline at end of file
diff --git a/spartan/core/elaboration.ML b/spartan/core/elaboration.ML
new file mode 100644
index 0000000..27b6bb0
--- /dev/null
+++ b/spartan/core/elaboration.ML
@@ -0,0 +1,76 @@
+(*  Title:      elaboration.ML
+    Author:     Joshua Chen
+
+Basic elaboration.
+*)
+
+structure Elab: sig
+
+val elab: Proof.context -> cterm list -> term -> Envir.env
+val elab_stmt: Proof.context -> cterm list -> term -> Envir.env * term
+
+end = struct
+
+(*Elaborate `tm` by solving the inference problem `tm: {}`, knowing `assums`,
+  which are fully elaborated, in `ctxt`. Return a substitution.*)
+fun elab ctxt assums tm =
+  if Lib.no_vars tm
+  then Envir.init
+  else
+    let
+      val inf = Goal.init (Thm.cterm_of ctxt (Lib.typing_of_term tm))
+      val res = Types.check_infer (map Thm.assume assums) 1 (ctxt, inf)
+      val tm' =
+        Thm.prop_of (#2 (Seq.hd (Seq.filter_results res)))
+        |> Lib.dest_prop |> Lib.term_of_typing
+        handle TERM ("dest_typing", [t]) =>
+          let val typ = Logic.unprotect (Logic.strip_assums_concl t)
+            |> Lib.term_of_typing
+          in
+            error ("Elaboration of " ^ Syntax.string_of_term ctxt typ ^ " failed")
+          end
+    in
+      Seq.hd (Unify.matchers (Context.Proof ctxt) [(tm, tm')])
+    end
+    handle Option => error
+      ("Elaboration of " ^ Syntax.string_of_term ctxt tm ^ " failed")
+
+(*Recursively elaborate a statement \<And>x ... y. \<lbrakk>...\<rbrakk> \<Longrightarrow> P x ... y by elaborating
+  only the types of typing judgments (in particular, does not look at judgmental
+  equality statements). Could also elaborate the terms of typing judgments, but
+  for now we assume that these are always free variables in all the cases we're
+  interested in.*)
+fun elab_stmt ctxt assums stmt =
+  let
+    val stmt = Lib.dest_prop stmt
+    fun subst_term env = Envir.subst_term (Envir.type_env env, Envir.term_env env)
+  in
+    if Lib.no_vars stmt orelse Lib.is_eq stmt then
+      (Envir.init, stmt)
+    else if Lib.is_typing stmt then
+      let
+        val typ = Lib.type_of_typing stmt
+        val subst = elab ctxt assums typ
+      in (subst, subst_term subst stmt) end
+    else
+      let
+        fun elab' assums (x :: xs) =
+              let
+                val (env, x') = elab_stmt ctxt assums x
+                val assums' =
+                  if Lib.no_vars x' then Thm.cterm_of ctxt x' :: assums else assums
+              in env :: elab' assums' xs end
+          | elab' _ [] = []
+        val (prems, concl) = Lib.decompose_goal ctxt stmt
+        val subst = fold (curry Envir.merge) (elab' assums prems) Envir.init
+        val prems' = map (Thm.cterm_of ctxt o subst_term subst) prems
+        val subst' =
+          if Lib.is_typing concl then
+            let val typ = Lib.type_of_typing concl
+            in Envir.merge (subst, elab ctxt (assums @ prems') typ) end
+          else subst
+      in (subst', subst_term subst' stmt) end
+  end
+
+
+end
diff --git a/spartan/core/goals.ML b/spartan/core/goals.ML
index 51a853e..3e2bd09 100644
--- a/spartan/core/goals.ML
+++ b/spartan/core/goals.ML
@@ -184,7 +184,7 @@ val schematic_theorem_cmd =
   gen_schematic_theorem
     Bundle.includes_cmd
     Attrib.check_src
-    Expression.read_statement
+    Elaborated_Expression.read_statement
 
 fun theorem spec descr =
   Outer_Syntax.local_theory_to_proof' spec ("state " ^ descr) 
diff --git a/spartan/core/lib.ML b/spartan/core/lib.ML
index 7b93a08..392ae2e 100644
--- a/spartan/core/lib.ML
+++ b/spartan/core/lib.ML
@@ -6,13 +6,16 @@ val max: ('a * 'a -> bool) -> 'a list -> 'a
 val maxint: int list -> int
 
 (*Terms*)
-val is_rigid: term -> bool
 val no_vars: term -> bool
+val is_rigid: term -> bool
+val is_eq: term -> bool
+val dest_prop: term -> term
 val dest_eq: term -> term * term
 val mk_Var: string -> int -> typ -> term
 val lambda_var: term -> term -> term
 
 val is_typing: term -> bool
+val mk_typing: term -> term -> term
 val dest_typing: term -> term * term
 val term_of_typing: term -> term
 val type_of_typing: term -> term
@@ -21,6 +24,7 @@ val mk_Pi: term -> term -> term -> term
 val typing_of_term: term -> term
 
 (*Goals*)
+val decompose_goal: Proof.context -> term -> term list * term
 val rigid_typing_concl: term -> bool
 
 (*Subterms*)
@@ -49,9 +53,15 @@ val maxint = max (op >)
 
 (* Meta *)
 
+val no_vars = not o exists_subterm is_Var
+
 val is_rigid = not o is_Var o head_of
 
-val no_vars = not o exists_subterm is_Var
+fun is_eq (Const (\<^const_name>\<open>Pure.eq\<close>, _) $ _ $ _) = true
+  | is_eq _ = false
+
+fun dest_prop (Const (\<^const_name>\<open>Pure.prop\<close>, _) $ P) = P
+  | dest_prop P = P
 
 fun dest_eq (Const (\<^const_name>\<open>Pure.eq\<close>, _) $ t $ def) = (t, def)
   | dest_eq _ = error "dest_eq"
@@ -72,8 +82,10 @@ fun lambda_var x tm =
 fun is_typing (Const (\<^const_name>\<open>has_type\<close>, _) $ _ $ _) = true
   | is_typing _ = false
 
+fun mk_typing t T = \<^const>\<open>has_type\<close> $ t $ T
+
 fun dest_typing (Const (\<^const_name>\<open>has_type\<close>, _) $ t $ T) = (t, T)
-  | dest_typing _ = error "dest_typing"
+  | dest_typing t = raise TERM ("dest_typing", [t])
 
 val term_of_typing = #1 o dest_typing
 val type_of_typing = #2 o dest_typing
@@ -82,11 +94,28 @@ fun mk_Pi v typ body = Const (\<^const_name>\<open>Pi\<close>, dummyT) $ typ $ l
 
 fun typing_of_term tm = \<^const>\<open>has_type\<close> $ tm $ Var (("*?", 0), \<^typ>\<open>o\<close>)
 (*The above is a bit hacky; basically we need to guarantee that the schematic
-  var is fresh*)
+  var is fresh. This works for now because no other code in the Isabelle system
+  or the current logic uses this identifier.*)
 
 
 (** Goals **)
 
+(*Breaks a goal \<And>x ... y. \<lbrakk>P1; ... Pn\<rbrakk> \<Longrightarrow> Q into ([P1, ..., Pn], Q), fixing
+  \<And>-quantified variables and keeping schematics.*)
+fun decompose_goal ctxt goal =
+  let
+    val focus =
+      #1 (Subgoal.focus_prems ctxt 1 NONE (Thm.trivial (Thm.cterm_of ctxt goal)))
+
+    val schematics = #2 (#schematics focus)
+      |> map (fn (v, ctm) => (Thm.term_of ctm, Var v))
+  in
+    map Thm.prop_of (#prems focus) @ [Thm.term_of (#concl focus)]
+    |> map (subst_free schematics)
+    |> (fn xs => chop (length xs - 1) xs) |> apsnd the_single
+  end
+  handle List.Empty => error "Lib.decompose_goal"
+
 fun rigid_typing_concl goal =
   let val concl = Logic.strip_assums_concl goal
   in is_typing concl andalso is_rigid (term_of_typing concl) end
diff --git a/spartan/core/typechecking.ML b/spartan/core/typechecking.ML
index 946ecd6..57164a1 100644
--- a/spartan/core/typechecking.ML
+++ b/spartan/core/typechecking.ML
@@ -11,6 +11,8 @@ 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 debug_typechk: bool Config.T
+
 val known_ctac: thm list -> int -> context_tactic
 val check_infer: thm list -> int -> context_tactic
 
@@ -35,6 +37,12 @@ fun put_types typings = foldr1 (op o) (map put_type typings)
 
 (* Context tactics for type-checking *)
 
+val debug_typechk =
+  Attrib.setup_config_bool \<^binding>\<open>debug_typechk\<close> (K false)
+
+fun debug_tac ctxt s =
+  if Config.get ctxt debug_typechk then print_tac ctxt s else all_tac
+
 (*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) =>
@@ -46,42 +54,43 @@ fun known_ctac facts = CONTEXT_SUBGOAL (fn (goal, i) => fn (ctxt, st) =>
         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
+          |> 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
+        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>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_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 (*TODO: Use this to store already-derived typing judgments*)
+  in
+    TACTIC_CONTEXT ctxt' (tac i st)
+  end
 
 fun check_infer facts i (cst as (_, st)) =
-  let
-    val ctac = known_ctac facts CORELSE' check_infer_step facts
+  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