Basic experiments adding reduction to the type checker

3 files changed, 26 insertions, 73 deletions

diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy
index a57ed44..99300a0 100644
--- a/hott/Equivalence.thy
+++ b/hott/Equivalence.thy
@@ -144,8 +144,6 @@ Lemma (def) commute_homotopy:
   apply (transport eq: pathcomp_refl, transport eq: refl_pathcomp)
   by refl
 
-\<comment> \<open>TODO: *Really* need normalization during type-checking and better equality
-  type rewriting to do the proof below properly\<close>
 Corollary (def) commute_homotopy':
   assumes
     "A: U i"
@@ -154,21 +152,18 @@ Corollary (def) commute_homotopy':
     "H: f ~ (id A)"
   shows "H (f x) = f [H x]"
   proof -
+    (*FIXME: Bug; if the following type declaration isn't made then bad things
+      happen on the next lines.*)
     from \<open>H: f ~ id A\<close> have [type]: "H: \<Prod>x: A. f x = x"
       by (reduce add: homotopy_def)
 
-    have *: "(id A)[H x]: f x = x"
-      by (rewrite at "\<hole> = _" id_comp[symmetric],
-          rewrite at "_ = \<hole>" id_comp[symmetric])
-
     have "H (f x) \<bullet> H x = H (f x) \<bullet> (id A)[H x]"
-      by (rule left_whisker, fact *, transport eq: ap_id) (reduce+, refl)
+      by (rule left_whisker, transport eq: ap_id, refl)
     also have [simplified id_comp]: "H (f x) \<bullet> (id A)[H x] = f[H x] \<bullet> H x"
       by (rule commute_homotopy)
-    finally have *: "{}" using * by this
+    finally have "{}" by this
 
-    show "H (f x) = f [H x]"
-      by pathcomp_cancelr (fact, typechk+)
+    thus "H (f x) = f [H x]" by pathcomp_cancelr (fact, typechk+)
   qed
 
 
@@ -309,23 +304,11 @@ Lemma (def) is_qinv_if_is_biinv:
   assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
   shows "is_biinv f \<rightarrow> is_qinv f"
   apply intro
-
-  text \<open>Split the hypothesis \<^term>\<open>is_biinv f\<close> into its components and name them.\<close>
   unfolding is_biinv_def apply elims
   focus vars _ _ _ g H1 h H2
-    text \<open>Need to give the required function and homotopies.\<close>
     apply (rule is_qinvI)
-      text \<open>We claim that \<^term>\<open>g\<close> is a quasi-inverse to \<^term>\<open>f\<close>.\<close>
       \<^item> by (fact \<open>g: _\<close>)
-  
-      text \<open>The first part \<^prop>\<open>?H1: g \<circ> f ~ id A\<close> is given by \<^term>\<open>H1\<close>.\<close>
       \<^item> by (fact \<open>H1: _\<close>)
-  
-      text \<open>
-        It remains to prove \<^prop>\<open>?H2: f \<circ> g ~ id B\<close>. First show that \<open>g ~ h\<close>,
-        then the result follows from @{thm \<open>H2: f \<circ> h ~ id B\<close>}. Here a proof
-        block is used for calculational reasoning.
-      \<close>
       \<^item> proof -
           have "g ~ g \<circ> (id B)" by reduce refl
           also have ".. ~ g \<circ> f \<circ> h" by rhtpy (rule \<open>H2:_\<close>[symmetric])
@@ -437,7 +420,7 @@ Lemma (def) equiv_if_equal:
         by (rule lift_universe_codomain, rule Ui_in_USi)
       \<^enum> vars A
         using [[solve_side_conds=1]]
-        apply (subst transport_comp)
+        apply (rewrite transport_comp)
           \<circ> by (rule Ui_in_USi)
           \<circ> by reduce (rule in_USi_if_in_Ui)
           \<circ> by reduce (rule id_is_biinv)
@@ -452,24 +435,5 @@ Lemma (def) equiv_if_equal:
       by (rule lift_universe_codomain, rule Ui_in_USi)
   done
 
-(*Uncomment this to see all implicits from here on.
-no_translations
-  "f x" \<leftharpoondown> "f `x"
-  "x = y" \<leftharpoondown> "x =\<^bsub>A\<^esub> y"
-  "g \<circ> f" \<leftharpoondown> "g \<circ>\<^bsub>A\<^esub> f"
-  "p\<inverse>" \<leftharpoondown> "CONST pathinv A x y p"
-  "p \<bullet> q" \<leftharpoondown> "CONST pathcomp A x y z p q"
-  "fst" \<leftharpoondown> "CONST Spartan.fst A B"
-  "snd" \<leftharpoondown> "CONST Spartan.snd A B"
-  "f[p]" \<leftharpoondown> "CONST ap A B x y f p"
-  "trans P p" \<leftharpoondown> "CONST transport A P x y p"
-  "trans_const B p" \<leftharpoondown> "CONST transport_const A B x y p"
-  "lift P p u" \<leftharpoondown> "CONST pathlift A P x y p u"
-  "apd f p" \<leftharpoondown> "CONST Identity.apd A P f x y p"
-  "f ~ g" \<leftharpoondown> "CONST homotopy A B f g"
-  "is_qinv f" \<leftharpoondown> "CONST Equivalence.is_qinv A B f"
-  "is_biinv f" \<leftharpoondown> "CONST Equivalence.is_biinv A B f"
-*)
-
 
 end
diff --git a/hott/Identity.thy b/hott/Identity.thy
index 247d6a4..dc27ee8 100644
--- a/hott/Identity.thy
+++ b/hott/Identity.thy
@@ -242,7 +242,7 @@ Lemma (def) ap_funcomp:
     "x: A" "y: A"
     "f: A \<rightarrow> B" "g: B \<rightarrow> C"
     "p: x = y"
-  shows "(g \<circ> f)[p] = g[f[p]]" thm ap
+  shows "(g \<circ> f)[p] = g[f[p]]"
   apply (eq p)
   \<^item> by reduce
   \<^item> by reduce refl
@@ -251,10 +251,7 @@ Lemma (def) ap_funcomp:
 Lemma (def) ap_id:
   assumes "A: U i" "x: A" "y: A" "p: x = y"
   shows "(id A)[p] = p"
-  apply (eq p)
-  \<^item> by reduce
-  \<^item> by reduce refl
-  done
+  by (eq p) (reduce, refl)
 
 
 section \<open>Transport\<close>
@@ -303,7 +300,7 @@ Lemma (def) pathcomp_cancel_left:
       by (transport eq: inv_pathcomp, transport eq: refl_pathcomp) refl
     also have ".. = p\<inverse> \<bullet> (p \<bullet> r)"
       by (transport eq: pathcomp_assoc[symmetric], transport eq: \<open>\<alpha>:_\<close>) refl
-    also have ".. = r" thm inv_pathcomp
+    also have ".. = r"
       by (transport eq: pathcomp_assoc,
           transport eq: inv_pathcomp,
           transport eq: refl_pathcomp) refl
@@ -339,7 +336,7 @@ Lemma (def) transport_left_inv:
     "x: A" "y: A"
     "p: x = y"
   shows "(trans P p\<inverse>) \<circ> (trans P p) = id (P x)"
-  by (eq p) (reduce; refl)
+  by (eq p) (reduce, refl)
 
 Lemma (def) transport_right_inv:
   assumes
@@ -440,10 +437,7 @@ Lemma (def) pathlift_fst:
     "u: P x"
     "p: x = y"
   shows "fst[lift P p u] = p"
-  apply (eq p)
-  \<^item> by reduce
-  \<^item> by reduce refl
-  done
+  by (eq p) (reduce, refl)
 
 
 section \<open>Dependent paths\<close>
@@ -585,10 +579,10 @@ end
 section \<open>Loop space\<close>
 
 definition \<Omega> where "\<Omega> A a \<equiv> a =\<^bsub>A\<^esub> a"
-definition \<Omega>2 where "\<Omega>2 A a \<equiv> \<Omega> (\<Omega> A a) (refl a)"
+definition \<Omega>2 where "\<Omega>2 A a \<equiv> refl a =\<^bsub>a =\<^bsub>A\<^esub> a\<^esub> refl a"
 
-Lemma \<Omega>2_alt_def:
-  "\<Omega>2 A a \<equiv> refl a = refl a"
+Lemma \<Omega>2_\<Omega>_of_\<Omega>:
+  "\<Omega>2 A a \<equiv> \<Omega> (\<Omega> A a) (refl a)"
   unfolding \<Omega>2_def \<Omega>_def .
 
 
@@ -604,23 +598,20 @@ interpretation \<Omega>2:
 notation \<Omega>2.horiz_pathcomp (infix "\<star>" 121)
 notation \<Omega>2.horiz_pathcomp' (infix "\<star>''" 121)
 
-Lemma [type]:
+Lemma
   assumes "\<alpha>: \<Omega>2 A a" and "\<beta>: \<Omega>2 A a"
-  shows horiz_pathcomp_type: "\<alpha> \<star> \<beta>: \<Omega>2 A a"
-    and horiz_pathcomp'_type: "\<alpha> \<star>' \<beta>: \<Omega>2 A a"
+  shows horiz_pathcomp_type [type]: "\<alpha> \<star> \<beta>: \<Omega>2 A a"
+    and horiz_pathcomp'_type [type]: "\<alpha> \<star>' \<beta>: \<Omega>2 A a"
   using assms
-  unfolding \<Omega>2.horiz_pathcomp_def \<Omega>2.horiz_pathcomp'_def \<Omega>2_alt_def
+  unfolding \<Omega>2.horiz_pathcomp_def \<Omega>2.horiz_pathcomp'_def \<Omega>2_def
   by reduce+
 
 Lemma (def) pathcomp_eq_horiz_pathcomp:
   assumes "\<alpha>: \<Omega>2 A a" "\<beta>: \<Omega>2 A a"
   shows "\<alpha> \<bullet> \<beta> = \<alpha> \<star> \<beta>"
   unfolding \<Omega>2.horiz_pathcomp_def
-  (*FIXME: Definitional unfolding + normalization; shouldn't need explicit
-    unfolding*)
-  using assms[unfolded \<Omega>2_alt_def, type]
+  using assms[unfolded \<Omega>2_def, type] (*TODO: A `noting` keyword that puts the noted theorem into [type]*)
   proof (reduce, rule pathinv)
-    \<comment> \<open>Propositional equality rewriting needs to be improved\<close>
     have "refl (refl a) \<bullet> \<alpha> \<bullet> refl (refl a) = refl (refl a) \<bullet> \<alpha>"
       by (rule pathcomp_refl)
     also have ".. = \<alpha>" by (rule refl_pathcomp)
@@ -641,7 +632,7 @@ Lemma (def) pathcomp_eq_horiz_pathcomp':
   assumes "\<alpha>: \<Omega>2 A a" "\<beta>: \<Omega>2 A a"
   shows "\<alpha> \<star>' \<beta> = \<beta> \<bullet> \<alpha>"
   unfolding \<Omega>2.horiz_pathcomp'_def
-  using assms[unfolded \<Omega>2_alt_def, type]
+  using assms[unfolded \<Omega>2_def, type]
   proof reduce
     have "refl (refl a) \<bullet> \<beta> \<bullet> refl (refl a) = refl (refl a) \<bullet> \<beta>"
       by (rule pathcomp_refl)
@@ -662,20 +653,18 @@ Lemma (def) pathcomp_eq_horiz_pathcomp':
 Lemma (def) eckmann_hilton:
   assumes "\<alpha>: \<Omega>2 A a" "\<beta>: \<Omega>2 A a"
   shows "\<alpha> \<bullet> \<beta> = \<beta> \<bullet> \<alpha>"
-  using assms[unfolded \<Omega>2_alt_def, type]
+  using \<Omega>2_def[comp]
   proof -
     have "\<alpha> \<bullet> \<beta> = \<alpha> \<star> \<beta>"
       by (rule pathcomp_eq_horiz_pathcomp)
     also have [simplified comp]: ".. = \<alpha> \<star>' \<beta>"
-      \<comment> \<open>Danger, Will Robinson! (Inferred implicit has an equivalent form but
-          needs to be manually simplified.)\<close>
+      \<comment> \<open>Danger! Inferred implicit has an equivalent form but needs to be
+          manually simplified.\<close>
       by (transport eq: \<Omega>2.horiz_pathcomp_eq_horiz_pathcomp') refl
     also have ".. = \<beta> \<bullet> \<alpha>"
       by (rule pathcomp_eq_horiz_pathcomp')
     finally show "\<alpha> \<bullet> \<beta> = \<beta> \<bullet> \<alpha>"
-      by (this; reduce add: \<Omega>2_alt_def[symmetric])
-      \<comment> \<open>TODO: The finishing call to `reduce` should be unnecessary with some
-          kind of definitional unfolding.\<close>
+      by this
   qed
 
 end
diff --git a/hott/List+.thy b/hott/List+.thy
index b73cc24..869d667 100644
--- a/hott/List+.thy
+++ b/hott/List+.thy
@@ -10,8 +10,8 @@ section \<open>Length\<close>
 definition [implicit]: "len \<equiv> ListRec ? Nat 0 (fn _ _ rec. suc rec)"
 
 experiment begin
-  Lemma "len [] \<equiv> ?n" by (subst comp)
-  Lemma "len [0, suc 0, suc (suc 0)] \<equiv> ?n" by (subst comp)
+  Lemma "len [] \<equiv> ?n" by (subst comp; typechk?)
+  Lemma "len [0, suc 0, suc (suc 0)] \<equiv> ?n" by (subst comp; typechk?)+
 end