1. Bugfix: implicits now properly name schematic variables. Fixes problems caused by variable name clashes. 2. reduce method now more principled: restricts to repeating on first subgoal. 3. An example declarative proof in Equivalence.thy.

4 files changed, 45 insertions, 35 deletions

diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy
index 88adc8b..d976677 100644
--- a/hott/Equivalence.thy
+++ b/hott/Equivalence.thy
@@ -338,18 +338,18 @@ 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> by (rule funcomp_biinv)
-      done
-    done
-  done
+  proof intros
+    fix AB BC assume *: "AB: A \<simeq> B" "BC: B \<simeq> C"
+    then have
+      "fst AB: A \<rightarrow> B" and 1: "snd AB: biinv (fst AB)"
+      "fst BC: B \<rightarrow> C" and 2: "snd BC: biinv (fst BC)"
+      unfolding equivalence_def by typechk+
+    then have "fst BC \<circ> fst AB: A \<rightarrow> C" by typechk
+    moreover have "biinv (fst BC \<circ> fst AB)"
+      using * unfolding equivalence_def by (rules funcomp_biinv 1 2)
+    ultimately show "A \<simeq> C"
+      unfolding equivalence_def by intro facts
+  qed
 
 text \<open>
   Equal types are equivalent. We give two proofs: the first by induction, and
diff --git a/hott/Identity.thy b/hott/Identity.thy
index 1cb3946..29ce26a 100644
--- a/hott/Identity.thy
+++ b/hott/Identity.thy
@@ -49,6 +49,9 @@ lemmas
 
 section \<open>Path induction\<close>
 
+\<comment> \<open>With `p: x = y` in the context the invokation `eq p` is essentially
+  `elim p x y`, with some extra bits before and after.\<close>
+
 method_setup eq =
   \<open>Args.term >> (fn tm => K (CONTEXT_METHOD (
     CHEADGOAL o SIDE_CONDS (
@@ -155,12 +158,12 @@ translations
 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 [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"
@@ -374,7 +377,7 @@ Lemma transport_const_comp [comp]:
     "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
@@ -398,7 +401,7 @@ Lemma pathlift_comp [comp]:
     "\<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
@@ -438,7 +441,7 @@ Lemma dependent_map_comp [comp]:
     "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
@@ -495,13 +498,13 @@ 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 [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)
diff --git a/spartan/core/Spartan.thy b/spartan/core/Spartan.thy
index a4ad300..fa6148f 100644
--- a/spartan/core/Spartan.thy
+++ b/spartan/core/Spartan.thy
@@ -52,7 +52,7 @@ paragraph \<open>
 
 typedecl o
 
-judgment has_type :: \<open>o \<Rightarrow> o \<Rightarrow> prop\<close> ("(2_:/ _)" 999)
+consts has_type :: \<open>o \<Rightarrow> o \<Rightarrow> prop\<close> ("(2_:/ _)" 999)
 
 text \<open>Type annotations for type-checking and inference.\<close>
 
@@ -232,6 +232,10 @@ 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 (
@@ -255,6 +259,8 @@ method_setup cases =
   \<open>Args.term >> (fn tm => K (CONTEXT_METHOD (
     CHEADGOAL o SIDE_CONDS (cases_ctac tm))))\<close>
 
+method facts = fact+
+
 
 subsection \<open>Reflexivity\<close>
 
@@ -310,7 +316,8 @@ 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 = changed \<open>((simp add: comp add | sub comp); typechk?)+\<close>
+method reduce uses add =
+  changed \<open>repeat_new \<open>(simp add: comp add | sub comp); typechk?\<close>\<close>
 
 subsection \<open>Implicits\<close>
 
@@ -327,10 +334,7 @@ ML_file \<open>implicits.ML\<close>
 
 attribute_setup implicit = \<open>Scan.succeed Implicits.implicit_defs_attr\<close>
 
-ML \<open>
-val _ = Context.>>
-  (Syntax_Phases.term_check 1 "" (fn ctxt => map (Implicits.make_holes ctxt)))
-\<close>
+ML \<open>val _ = Context.>> (Syntax_Phases.term_check 1 "" Implicits.make_holes)\<close>
 
 text \<open>Automatically insert inhabitation judgments where needed:\<close>
 
@@ -488,7 +492,7 @@ lemma snd_type [typechk]:
 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>
 
diff --git a/spartan/core/implicits.ML b/spartan/core/implicits.ML
index 4d73c8d..ab64b0f 100644
--- a/spartan/core/implicits.ML
+++ b/spartan/core/implicits.ML
@@ -3,7 +3,7 @@ sig
 
 val implicit_defs: Proof.context -> (term * term) Symtab.table
 val implicit_defs_attr: attribute
-val make_holes: Proof.context -> term -> term
+val make_holes: Proof.context -> term list -> term list
 
 end = struct
 
@@ -25,16 +25,15 @@ val implicit_defs_attr = Thm.declaration_attribute (fn th =>
     Defs.map (Symtab.update (Term.term_name head, (head, def')))
   end)
 
-fun make_holes ctxt =
+fun make_holes_single ctxt tm name_ctxt =
   let
     fun iarg_to_hole (Const (\<^const_name>\<open>iarg\<close>, T)) =
           Const (\<^const_name>\<open>hole\<close>, T)
       | iarg_to_hole t = t
 
     fun expand head args =
-      let
-        fun betapplys (head', args') =
-          Term.betapplys (map_aterms iarg_to_hole head', args')
+      let fun betapplys (head', args') =
+        Term.betapplys (map_aterms iarg_to_hole head', args')
       in
         case head of
           Abs (x, T, t) =>
@@ -66,13 +65,17 @@ fun make_holes ctxt =
               in subst t (take n vs) Ts $ subst u (drop n vs) Ts end
           | subst t _ _ = t
 
-        val vars = map (fn n => Var ((n, 0), dummyT))
-          (Name.invent (Variable.names_of ctxt) "*" (count t))
+        val names = Name.invent name_ctxt "*" (count t)
+        val vars = map (fn n => Var ((n, 0), dummyT)) names
       in
-        subst t vars []
+        (subst t vars [], fold Name.declare names name_ctxt)
       end
   in
-    Lib.traverse_term expand #> holes_to_vars
+    holes_to_vars (Lib.traverse_term expand tm)
   end
 
+fun make_holes ctxt tms = #1
+  (fold_map (make_holes_single ctxt) tms (Variable.names_of ctxt))
+
+
 end