14 files changed, 15 insertions, 889 deletions
diff --git a/spartan/theories/Spartan.thy b/spartan/core/Spartan.thy
index d235041..b4f7772 100644
--- a/spartan/theories/Spartan.thy
+++ b/spartan/core/Spartan.thy
@@ -142,20 +142,20 @@ section \<open>Proof commands\<close>
 
 named_theorems typechk
 
-ML_file \<open>../lib/lib.ML\<close>
-ML_file \<open>../lib/goals.ML\<close>
-ML_file \<open>../lib/focus.ML\<close>
+ML_file \<open>lib.ML\<close>
+ML_file \<open>goals.ML\<close>
+ML_file \<open>focus.ML\<close>
 
 
 section \<open>Congruence automation\<close>
 
 (*Potential to be retired*)
-ML_file \<open>../lib/congruence.ML\<close>
+ML_file \<open>congruence.ML\<close>
 
 
 section \<open>Methods\<close>
 
-ML_file \<open>../lib/elimination.ML\<close> \<comment> \<open>declares the [elims] attribute\<close>
+ML_file \<open>elimination.ML\<close> \<comment> \<open>declares the [elims] attribute\<close>
 
 named_theorems intros and comps
 lemmas
@@ -165,7 +165,7 @@ lemmas
   [comps] = beta Sig_comp and
   [cong] = Pi_cong lam_cong Sig_cong
 
-ML_file \<open>../lib/tactics.ML\<close>
+ML_file \<open>tactics.ML\<close>
 
 method_setup assumptions =
   \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (
@@ -227,7 +227,7 @@ ML_file "~~/src/Tools/misc_legacy.ML"
 ML_file "~~/src/Tools/IsaPlanner/isand.ML"
 ML_file "~~/src/Tools/IsaPlanner/rw_inst.ML"
 ML_file "~~/src/Tools/IsaPlanner/zipper.ML"
-ML_file "../lib/eqsubst.ML"
+ML_file "eqsubst.ML"
 
 \<comment> \<open>\<open>rewrite\<close> method\<close>
 consts rewrite_HOLE :: "'a::{}"  ("\<hole>")
@@ -254,7 +254,7 @@ lemma imp_cong_eq:
   done
 
 ML_file \<open>~~/src/HOL/Library/cconv.ML\<close>
-ML_file \<open>../lib/rewrite.ML\<close>
+ML_file \<open>rewrite.ML\<close>
 
 \<comment> \<open>\<open>reduce\<close> computes terms via judgmental equalities\<close>
 setup \<open>map_theory_simpset (fn ctxt => ctxt addSolver (mk_solver "" typechk_tac))\<close>
@@ -273,7 +273,7 @@ consts
   iarg :: \<open>'a\<close> ("?")
   hole :: \<open>'b\<close> ("{}")
 
-ML_file \<open>../lib/implicits.ML\<close>
+ML_file \<open>implicits.ML\<close>
 
 attribute_setup implicit = \<open>Scan.succeed Implicits.implicit_defs_attr\<close>
 
diff --git a/spartan/lib/congruence.ML b/spartan/core/congruence.ML
index bb7348c..bb7348c 100644
--- a/spartan/lib/congruence.ML
+++ b/spartan/core/congruence.ML
diff --git a/spartan/lib/elimination.ML b/spartan/core/elimination.ML
index 617f83e..617f83e 100644
--- a/spartan/lib/elimination.ML
+++ b/spartan/core/elimination.ML
diff --git a/spartan/lib/eqsubst.ML b/spartan/core/eqsubst.ML
index ea6f098..ea6f098 100644
--- a/spartan/lib/eqsubst.ML
+++ b/spartan/core/eqsubst.ML
diff --git a/spartan/lib/equality.ML b/spartan/core/equality.ML
index 023147b..023147b 100644
--- a/spartan/lib/equality.ML
+++ b/spartan/core/equality.ML
diff --git a/spartan/lib/focus.ML b/spartan/core/focus.ML
index 1d8de78..1d8de78 100644
--- a/spartan/lib/focus.ML
+++ b/spartan/core/focus.ML
diff --git a/spartan/lib/goals.ML b/spartan/core/goals.ML
index 9f394f0..9f394f0 100644
--- a/spartan/lib/goals.ML
+++ b/spartan/core/goals.ML
diff --git a/spartan/lib/implicits.ML b/spartan/core/implicits.ML
index 04fc825..04fc825 100644
--- a/spartan/lib/implicits.ML
+++ b/spartan/core/implicits.ML
diff --git a/spartan/lib/lib.ML b/spartan/core/lib.ML
index 615f601..615f601 100644
--- a/spartan/lib/lib.ML
+++ b/spartan/core/lib.ML
diff --git a/spartan/lib/rewrite.ML b/spartan/core/rewrite.ML
index f9c5d8e..f9c5d8e 100644
--- a/spartan/lib/rewrite.ML
+++ b/spartan/core/rewrite.ML
diff --git a/spartan/lib/tactics.ML b/spartan/core/tactics.ML
index f23bfee..f23bfee 100644
--- a/spartan/lib/tactics.ML
+++ b/spartan/core/tactics.ML
diff --git a/spartan/data/List.thy b/spartan/data/List.thy
new file mode 100644
index 0000000..71a879b
--- /dev/null
+++ b/spartan/data/List.thy
@@ -0,0 +1,6 @@
+theory List
+imports Spartan
+
+begin
+
+end
diff --git a/spartan/theories/Equivalence.thy b/spartan/theories/Equivalence.thy
deleted file mode 100644
index 2975738..0000000
--- a/spartan/theories/Equivalence.thy
+++ /dev/null
@@ -1,416 +0,0 @@
-theory Equivalence
-imports Identity
-
-begin
-
-section \<open>Homotopy\<close>
-
-definition "homotopy A B f g \<equiv> \<Prod>x: A. f `x =\<^bsub>B x\<^esub> g `x"
-
-definition homotopy_i (infix "~" 100)
-  where [implicit]: "f ~ g \<equiv> homotopy ? ? f g"
-
-translations "f ~ g" \<leftharpoondown> "CONST homotopy A B f g"
-
-Lemma homotopy_type [typechk]:
-  assumes
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> B x: U i"
-    "f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x"
-  shows "f ~ g: U i"
-  unfolding homotopy_def by typechk
-
-Lemma (derive) homotopy_refl [refl]:
-  assumes
-    "A: U i"
-    "f: \<Prod>x: A. B x"
-  shows "f ~ f"
-  unfolding homotopy_def by intros
-
-Lemma (derive) hsym:
-  assumes
-    "f: \<Prod>x: A. B x"
-    "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 intros
-    apply (rule pathinv)
-      \<guillemotright> by (elim H)
-      \<guillemotright> by typechk
-  done
-
-Lemma (derive) htrans:
-  assumes
-    "f: \<Prod>x: A. B x"
-    "g: \<Prod>x: A. B x"
-    "h: \<Prod>x: A. B x"
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> B x: U i"
-  shows "\<lbrakk>H1: f ~ g; H2: g ~ h\<rbrakk> \<Longrightarrow> f ~ h"
-  unfolding homotopy_def
-  apply intro
-    \<guillemotright> vars x
-      apply (rule pathcomp[where ?y="g x"])
-        \<^item> by (elim H1)
-        \<^item> by (elim H2)
-      done
-    \<guillemotright> by typechk
-  done
-
-text \<open>For calculations:\<close>
-
-lemmas
-  homotopy_sym [sym] = hsym[rotated 4] and
-  homotopy_trans [trans] = htrans[rotated 5]
-
-Lemma (derive) commute_homotopy:
-  assumes
-    "A: U i" "B: U i"
-    "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
-    "f: A \<rightarrow> B" "g: A \<rightarrow> B"
-    "H: homotopy A (\<lambda>_. B) 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]
-  apply (eq p)
-    focus vars x
-      apply reduce
-        \<comment> \<open>Here it would really be nice to have automation for transport and
-          propositional equality.\<close>
-        apply (rule use_transport[where ?y="H x \<bullet> refl (g x)"])
-          \<guillemotright> by (rule pathcomp_right_refl)
-          \<guillemotright> by (rule pathinv) (rule pathcomp_left_refl)
-          \<guillemotright> by typechk
-    done
-  done
-
-Corollary (derive) commute_homotopy':
-  assumes
-    "A: U i"
-    "x: A"
-    "f: A \<rightarrow> A"
-    "H: homotopy A (\<lambda>_. A) f (id A)"
-  shows "H (f x) = f [H x]"
-oops
-
-Lemma homotopy_id_left [typechk]:
-  assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
-  shows "homotopy_refl A f: (id B) \<circ> f ~ f"
-  unfolding homotopy_refl_def homotopy_def by reduce
-
-Lemma homotopy_id_right [typechk]:
-  assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
-  shows "homotopy_refl A f: f \<circ> (id A) ~ f"
-  unfolding homotopy_refl_def homotopy_def by reduce
-
-Lemma homotopy_funcomp_left:
-  assumes
-    "H: homotopy B C g g'"
-    "f: A \<rightarrow> B"
-    "g: \<Prod>x: B. C x"
-    "g': \<Prod>x: B. C x"
-    "A: U i" "B: U i"
-    "\<And>x. x: B \<Longrightarrow> C x: U i"
-  shows "homotopy A {} (g \<circ>\<^bsub>A\<^esub> f) (g' \<circ>\<^bsub>A\<^esub> f)"
-  unfolding homotopy_def
-  apply (intro; reduce)
-    apply (insert \<open>H: _\<close>[unfolded homotopy_def])
-      apply (elim H)
-  done
-
-Lemma homotopy_funcomp_right:
-  assumes
-    "H: homotopy A (\<lambda>_. B) f f'"
-    "f: A \<rightarrow> B"
-    "f': A \<rightarrow> B"
-    "g: B \<rightarrow> C"
-    "A: U i" "B: U i" "C: U i"
-  shows "homotopy A {} (g \<circ>\<^bsub>A\<^esub> f) (g \<circ>\<^bsub>A\<^esub> f')"
-  unfolding homotopy_def
-  apply (intro; reduce)
-    apply (insert \<open>H: _\<close>[unfolded homotopy_def])
-      apply (dest PiE, assumption)
-      apply (rule ap, assumption)
-  done
-
-
-section \<open>Quasi-inverse and bi-invertibility\<close>
-
-subsection \<open>Quasi-inverses\<close>
-
-definition "qinv A B f \<equiv> \<Sum>g: B \<rightarrow> A.
-  homotopy A (\<lambda>_. A) (g \<circ>\<^bsub>A\<^esub> f) (id A) \<times> homotopy B (\<lambda>_. B) (f \<circ>\<^bsub>B\<^esub> g) (id B)"
-
-lemma qinv_type [typechk]:
-  assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
-  shows "qinv A B f: U i"
-  unfolding qinv_def by typechk
-
-definition qinv_i ("qinv")
-  where [implicit]: "qinv f \<equiv> Equivalence.qinv ? ? f"
-
-translations "qinv f" \<leftharpoondown> "CONST Equivalence.qinv A B f"
-
-Lemma (derive) id_qinv:
-  assumes "A: U i"
-  shows "qinv (id A)"
-  unfolding qinv_def
-  apply intro defer
-    apply intro defer
-      apply (rule homotopy_id_right)
-      apply (rule homotopy_id_left)
-  done
-
-Lemma (derive) quasiinv_qinv:
-  assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
-  shows "prf: qinv f \<Longrightarrow> qinv (fst prf)"
-  unfolding qinv_def
-  apply intro
-    \<guillemotright> by (rule \<open>f:_\<close>)
-    \<guillemotright> apply (elim "prf")
-        focus vars g HH
-          apply intro
-            \<^item> by reduce (snd HH)
-            \<^item> by reduce (fst HH)
-        done
-      done
-  done
-
-Lemma (derive) funcomp_qinv:
-  assumes
-    "A: U i" "B: U i" "C: U i"
-    "f: A \<rightarrow> B" "g: B \<rightarrow> C"
-  shows "qinv f \<rightarrow> qinv g \<rightarrow> qinv (g \<circ> f)"
-  apply (intros, unfold qinv_def, elims)
-  focus
-    prems prms
-    vars _ _ finv _ ginv _ HfA HfB HgB HgC
-
-    apply intro
-    apply (rule funcompI[where ?f=ginv and ?g=finv])
-
-    text \<open>Now a whole bunch of rewrites and we're done.\<close>
-oops
-
-subsection \<open>Bi-invertible maps\<close>
-
-definition "biinv A B f \<equiv>
-  (\<Sum>g: B \<rightarrow> A. homotopy A (\<lambda>_. A) (g \<circ>\<^bsub>A\<^esub> f) (id A))
-    \<times> (\<Sum>g: B \<rightarrow> A. homotopy B (\<lambda>_. B) (f \<circ>\<^bsub>B\<^esub> g) (id B))"
-
-lemma biinv_type [typechk]:
-  assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
-  shows "biinv A B f: U i"
-  unfolding biinv_def by typechk
-
-definition biinv_i ("biinv")
-  where [implicit]: "biinv f \<equiv> Equivalence.biinv ? ? f"
-
-translations "biinv f" \<leftharpoondown> "CONST Equivalence.biinv A B f"
-
-Lemma (derive) qinv_imp_biinv:
-  assumes
-    "A: U i" "B: U i"
-    "f: A \<rightarrow> B"
-  shows "?prf: qinv f \<rightarrow> biinv f"
-  apply intros
-  unfolding qinv_def biinv_def
-  by (rule Sig_dist_exp)
-
-text \<open>
-  Show that bi-invertible maps are quasi-inverses, as a demonstration of how to
-  work in this system.
-\<close>
-
-Lemma (derive) biinv_imp_qinv:
-  assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
-  shows "biinv f \<rightarrow> qinv f"
-
-  text \<open>Split the hypothesis \<^term>\<open>biinv f\<close> into its components:\<close>
-  apply intro
-  unfolding biinv_def
-  apply elims
-
-  text \<open>Name the components:\<close>
-  focus prems vars _ _ _ g H1 h H2
-  thm \<open>g:_\<close> \<open>h:_\<close> \<open>H1:_\<close> \<open>H2:_\<close>
-
-  text \<open>
-    \<^term>\<open>g\<close> is a quasi-inverse to \<^term>\<open>f\<close>, so the proof will be a triple
-    \<^term>\<open><g, <?H1, ?H2>>\<close>.
-  \<close>
-  unfolding qinv_def
-  apply intro
-    \<guillemotright> by (rule \<open>g: _\<close>)
-    \<guillemotright> apply intro
-      text \<open>The first part \<^prop>\<open>?H1: g \<circ> f ~ id A\<close> is given by \<^term>\<open>H1\<close>.\<close>
-      apply (rule \<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 to calculate "forward".
-      \<close>
-      proof -
-        have "g ~ g \<circ> (id B)" by reduce refl
-        also have "g \<circ> (id B) ~ g \<circ> f \<circ> h"
-          by (rule homotopy_funcomp_right) (rule \<open>H2:_\<close>[symmetric])
-        also have "g \<circ> f \<circ> h ~ (id A) \<circ> h"
-          by (subst funcomp_assoc[symmetric])
-             (rule homotopy_funcomp_left, rule \<open>H1:_\<close>)
-        also have "(id A) \<circ> h ~ h" by reduce refl
-        finally have "g ~ h" by this
-
-        then have "f \<circ> g ~ f \<circ> h" by (rule homotopy_funcomp_right)
-
-        with \<open>H2:_\<close>
-        show "f \<circ> g ~ id B"
-          by (rule homotopy_trans) (assumption+, typechk)
-      qed
-    done
-  done
-
-Lemma (derive) id_biinv:
-  "A: U i \<Longrightarrow> biinv (id A)"
-    by (rule qinv_imp_biinv) (rule id_qinv)
-
-Lemma (derive) funcomp_biinv:
-  assumes
-    "A: U i" "B: U i" "C: U i"
-    "f: A \<rightarrow> B" "g: B \<rightarrow> C"
-  shows "biinv f \<rightarrow> biinv g \<rightarrow> biinv (g \<circ> f)"
-  apply intros
-  focus vars biinv\<^sub>f biinv\<^sub>g
-
-  text \<open>Follows from \<open>funcomp_qinv\<close>.\<close>
-oops
-
-
-section \<open>Equivalence\<close>
-
-text \<open>
-  Following the HoTT book, we first define equivalence in terms of
-  bi-invertibility.
-\<close>
-
-definition equivalence (infix "\<simeq>" 110)
-  where "A \<simeq> B \<equiv> \<Sum>f: A \<rightarrow> B. Equivalence.biinv A B f"
-
-lemma equivalence_type [typechk]:
-  assumes "A: U i" "B: U i"
-  shows "A \<simeq> B: U i"
-  unfolding equivalence_def by typechk
-
-Lemma (derive) equivalence_refl:
-  assumes "A: U i"
-  shows "A \<simeq> A"
-  unfolding equivalence_def
-  apply intro defer
-    apply (rule qinv_imp_biinv) defer
-      apply (rule id_qinv)
-  done
-
-text \<open>
-  The following could perhaps be easier with transport (but then I think we need
-  univalence?)...
-\<close>
-
-Lemma (derive) equivalence_symmetric:
-  assumes "A: U i" "B: U i"
-  shows "A \<simeq> B \<rightarrow> B \<simeq> A"
-  apply intros
-  unfolding equivalence_def
-  apply elim
-  \<guillemotright> vars _ f "prf"
-    apply (dest (4) biinv_imp_qinv)
-    apply intro
-      \<^item> unfolding qinv_def by (rule fst[of "(biinv_imp_qinv A B f) prf"])
-      \<^item> by (rule qinv_imp_biinv) (rule quasiinv_qinv)
-    done
-  done
-
-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"
-  apply intros
-  unfolding equivalence_def
-
-  text \<open>Use \<open>funcomp_biinv\<close>.\<close>
-oops
-
-text \<open>
-  Equal types are equivalent. We give two proofs: the first by induction, and
-  the second by following the HoTT book and showing that transport is an
-  equivalence.
-\<close>
-
-Lemma
-  assumes
-    "A: U i" "B: U i" "p: A =\<^bsub>U i\<^esub> B"
-  shows "A \<simeq> B"
-  by (eq p) (rule equivalence_refl)
-
-text \<open>
-  The following proof is wordy because (1) the typechecker doesn't rewrite, and
-  (2) we don't yet have universe automation.
-\<close>
-
-Lemma (derive) id_imp_equiv:
-  assumes
-    "A: U i" "B: U i" "p: A =\<^bsub>U i\<^esub> B"
-  shows "<trans (id (U i)) p, ?isequiv>: A \<simeq> B"
-  unfolding equivalence_def
-  apply intros defer
-
-  \<comment> \<open>Switch off auto-typechecking, which messes with universe levels\<close>
-  supply [[auto_typechk=false]]
-
-    \<guillemotright> apply (eq p, typechk)
-      \<^item> prems vars A B
-        apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric])
-        apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric])
-        apply (rule transport, rule U_in_U)
-        apply (rule lift_universe_codomain, rule U_in_U)
-        apply (typechk, rule U_in_U)
-        done
-      \<^item> prems vars A
-        apply (subst transport_comp)
-          \<^enum> by (rule U_in_U)
-          \<^enum> by reduce (rule lift_universe)
-          \<^enum> by reduce (rule id_biinv)
-        done
-      done
-
-    \<guillemotright> \<comment> \<open>Similar proof as in the first subgoal above\<close>
-      apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric])
-      apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric])
-      apply (rule transport, rule U_in_U)
-      apply (rule lift_universe_codomain, rule U_in_U)
-      apply (typechk, rule U_in_U)
-      done
-  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"
-  "qinv f" \<leftharpoondown> "CONST Equivalence.qinv A B f"
-  "biinv f" \<leftharpoondown> "CONST Equivalence.biinv A B f"
-*)
-
-
-end
diff --git a/spartan/theories/Identity.thy b/spartan/theories/Identity.thy
deleted file mode 100644
index 3a982f6..0000000
--- a/spartan/theories/Identity.thy
+++ /dev/null
@@ -1,464 +0,0 @@
-chapter \<open>The identity type\<close>
-
-text \<open>
-  The identity type, the higher groupoid structure of types,
-  and type families as fibrations.
-\<close>
-
-theory Identity
-imports Spartan
-
-begin
-
-axiomatization
-  Id    :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> and
-  refl  :: \<open>o \<Rightarrow> o\<close> and
-  IdInd :: \<open>o \<Rightarrow> (o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close>
-
-syntax "_Id" :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2_ =\<^bsub>_\<^esub>/ _)" [111, 0, 111] 110)
-
-translations "a =\<^bsub>A\<^esub> b" \<rightleftharpoons> "CONST Id A a b"
-
-axiomatization where
-  IdF: "\<lbrakk>A: U i; a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^bsub>A\<^esub> b: U i" and
-
-  IdI: "a: A \<Longrightarrow> refl a: a =\<^bsub>A\<^esub> a" and
-
-  IdE: "\<lbrakk>
-    p: a =\<^bsub>A\<^esub> b;
-    a: A;
-    b: A;
-    \<And>x y p. \<lbrakk>p: x =\<^bsub>A\<^esub> y; x: A; y: A\<rbrakk> \<Longrightarrow> C x y p: U i;
-    \<And>x. x: A \<Longrightarrow> f x: C x x (refl x)
-    \<rbrakk> \<Longrightarrow> IdInd A (\<lambda>x y p. C x y p) f a b p: C a b p" and
-
-  Id_comp: "\<lbrakk>
-    a: A;
-    \<And>x y p. \<lbrakk>x: A; y: A; p: x =\<^bsub>A\<^esub> y\<rbrakk> \<Longrightarrow> C x y p: U i;
-    \<And>x. x: A \<Longrightarrow> f x: C x x (refl x)
-    \<rbrakk> \<Longrightarrow> IdInd A (\<lambda>x y p. C x y p) f a a (refl a) \<equiv> f a"
-
-lemmas
-  [intros] = IdF IdI and
-  [elims "?p" "?a" "?b"] = IdE and
-  [comps] = Id_comp and
-  [refl] = IdI
-
-
-section \<open>Path induction\<close>
-
-method_setup eq = \<open>
-Args.term >> (fn tm => fn ctxt => CONTEXT_METHOD (K (
-  CONTEXT_SUBGOAL (fn (goal, i) =>
-    let
-      val facts = Proof_Context.facts_of ctxt
-      val prems = Logic.strip_assums_hyp goal
-      val template = \<^const>\<open>has_type\<close>
-        $ tm
-        $ (\<^const>\<open>Id\<close> $ Var (("*?A", 0), \<^typ>\<open>o\<close>)
-          $ Var (("*?x", 0), \<^typ>\<open>o\<close>)
-          $ Var (("*?y", 0), \<^typ>\<open>o\<close>))
-      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
-        (\<^const>\<open>Id\<close> $ _ $ x $ y)::_ =>
-          elim_context_tac [tm, x, y] ctxt i
-      | _ => Context_Tactic.CONTEXT_TACTIC no_tac
-    end) 1)))
-\<close>
-
-
-section \<open>Symmetry and transitivity\<close>
-
-Lemma (derive) pathinv:
-  assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
-  shows "y =\<^bsub>A\<^esub> x"
-  by (eq p) intro
-
-lemma pathinv_comp [comps]:
-  assumes "x: A" "A: U i"
-  shows "pathinv A x x (refl x) \<equiv> refl x"
-  unfolding pathinv_def by reduce
-
-Lemma (derive) pathcomp:
-  assumes
-    "A: U i" "x: A" "y: A" "z: A"
-    "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z"
-  shows
-    "x =\<^bsub>A\<^esub> z"
-  apply (eq p)
-    focus prems vars x p
-      apply (eq p)
-        apply intro
-    done
-  done
-
-lemma pathcomp_comp [comps]:
-  assumes "a: A" "A: U i"
-  shows "pathcomp A a a a (refl a) (refl a) \<equiv> refl a"
-  unfolding pathcomp_def by reduce
-
-
-section \<open>Notation\<close>
-
-definition Id_i (infix "=" 110)
-  where [implicit]: "Id_i x y \<equiv> x =\<^bsub>?\<^esub> y"
-
-definition pathinv_i ("_\<inverse>" [1000])
-  where [implicit]: "pathinv_i p \<equiv> pathinv ? ? ? p"
-
-definition pathcomp_i (infixl "\<bullet>" 121)
-  where [implicit]: "pathcomp_i p q \<equiv> pathcomp ? ? ? ? p q"
-
-translations
-  "x = y" \<leftharpoondown> "x =\<^bsub>A\<^esub> y"
-  "p\<inverse>" \<leftharpoondown> "CONST pathinv A x y p"
-  "p \<bullet> q" \<leftharpoondown> "CONST pathcomp A x y z p q"
-
-
-section \<open>Calculational reasoning\<close>
-
-consts rhs :: \<open>'a\<close> ("''''")
-
-ML \<open>
-local fun last_rhs ctxt =
-  let
-    val this_name = Name_Space.full_name (Proof_Context.naming_of ctxt)
-      (Binding.name Auto_Bind.thisN)
-    val this = #thms (the (Proof_Context.lookup_fact ctxt this_name))
-      handle Option => []
-    val rhs = case map Thm.prop_of this of
-       [\<^const>\<open>has_type\<close> $ _ $ (\<^const>\<open>Id\<close> $ _ $ _ $ y)] => y
-      | _ => Term.dummy
-  in
-    map_aterms (fn t => case t of Const (\<^const_name>\<open>rhs\<close>, _) => rhs | _ => t)
-  end
-in val _ = Context.>>
-  (Syntax_Phases.term_check 5 "" (fn ctxt => map (last_rhs ctxt)))
-end
-\<close>
-
-lemmas
-  [sym] = pathinv[rotated 3] and
-  [trans] = pathcomp[rotated 4]
-
-
-section \<open>Basic propositional equalities\<close>
-
-Lemma (derive) pathcomp_left_refl:
-  assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
-  shows "(refl x) \<bullet> p = p"
-  apply (eq p)
-    apply (reduce; intros)
-  done
-
-Lemma (derive) pathcomp_right_refl:
-  assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
-  shows "p \<bullet> (refl y) = p"
-  apply (eq p)
-    apply (reduce; intros)
-  done
-
-Lemma (derive) pathcomp_left_inv:
-  assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
-  shows "p\<inverse> \<bullet> p = refl y"
-  apply (eq p)
-    apply (reduce; intros)
-  done
-
-Lemma (derive) pathcomp_right_inv:
-  assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
-  shows "p \<bullet> p\<inverse> = refl x"
-  apply (eq p)
-    apply (reduce; intros)
-  done
-
-Lemma (derive) pathinv_pathinv:
-  assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
-  shows "p\<inverse>\<inverse> = p"
-  apply (eq p)
-    apply (reduce; intros)
-  done
-
-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"
-  shows "p \<bullet> (q \<bullet> r) = p \<bullet> q \<bullet> r"
-  apply (eq p)
-    focus prems vars x p
-      apply (eq p)
-        focus prems vars x p
-          apply (eq p)
-            apply (reduce; intros)
-        done
-    done
-  done
-
-
-section \<open>Functoriality of functions\<close>
-
-Lemma (derive) ap:
-  assumes
-    "A: U i" "B: U i"
-    "x: A" "y: A"
-    "f: A \<rightarrow> B"
-    "p: x =\<^bsub>A\<^esub> y"
-  shows "f x = f y"
-  apply (eq p)
-    apply intro
-  done
-
-definition ap_i ("_[_]" [1000, 0])
-  where [implicit]: "ap_i f p \<equiv> ap ? ? ? ? f p"
-
-translations "f[p]" \<leftharpoondown> "CONST ap A B x y f p"
-
-Lemma ap_refl [comps]:
-  assumes "f: A \<rightarrow> B" "x: A" "A: U i" "B: U i"
-  shows "f[refl x] \<equiv> refl (f x)"
-  unfolding ap_def by reduce
-
-Lemma (derive) ap_pathcomp:
-  assumes
-    "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"
-  shows
-    "f[p \<bullet> q] = f[p] \<bullet> f[q]"
-  apply (eq p)
-    focus prems vars x p
-      apply (eq p)
-        apply (reduce; intro)
-    done
-  done
-
-Lemma (derive) ap_pathinv:
-  assumes
-    "A: U i" "B: U i"
-    "x: A" "y: A"
-    "f: A \<rightarrow> B"
-    "p: x =\<^bsub>A\<^esub> y"
-  shows "f[p\<inverse>] = f[p]\<inverse>"
-  by (eq p) (reduce; intro)
-
-text \<open>The next two proofs currently use some low-level rewriting.\<close>
-
-Lemma (derive) ap_funcomp:
-  assumes
-    "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"
-  shows "(g \<circ> f)[p] = g[f[p]]"
-  apply (eq p)
-    apply (simp only: funcomp_apply_comp[symmetric])
-    apply (reduce; intro)
-  done
-
-Lemma (derive) ap_id:
-  assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
-  shows "(id A)[p] = p"
-  apply (eq p)
-    apply (rewrite at "\<hole> = _" id_comp[symmetric])
-    apply (rewrite at "_ = \<hole>" id_comp[symmetric])
-    apply (reduce; intros)
-  done
-
-
-section \<open>Transport\<close>
-
-Lemma (derive) transport:
-  assumes
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> P x: U i"
-    "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
-  shows "P x \<rightarrow> P y"
-  by (eq p) intro
-
-definition transport_i ("trans")
-  where [implicit]: "trans P p \<equiv> transport ? P ? ? p"
-
-translations "trans P p" \<leftharpoondown> "CONST transport A P x y p"
-
-Lemma transport_comp [comps]:
-  assumes
-    "a: A"
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> P x: U i"
-  shows "trans P (refl a) \<equiv> id (P a)"
-  unfolding transport_def by reduce
-
-\<comment> \<open>TODO: Build transport automation!\<close>
-
-Lemma use_transport:
-  assumes
-    "p: y =\<^bsub>A\<^esub> x"
-    "u: P x"
-    "x: A" "y: A"
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> P x: U i"
-  shows "trans P p\<inverse> u: P y"
-  by typechk
-
-Lemma (derive) transport_left_inv:
-  assumes
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> P x: U i"
-    "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
-  shows "(trans P p\<inverse>) \<circ> (trans P p) = id (P x)"
-  by (eq p) (reduce; intro)
-
-Lemma (derive) transport_right_inv:
-  assumes
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> P x: U i"
-    "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
-  shows "(trans P p) \<circ> (trans P p\<inverse>) = id (P y)"
-  by (eq p) (reduce; intros)
-
-Lemma (derive) transport_pathcomp:
-  assumes
-    "A: U i"
-    "\<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"
-  shows "trans P q (trans P p u) = trans P (p \<bullet> q) u"
-  apply (eq p)
-    focus prems vars x p
-      apply (eq p)
-        apply (reduce; intros)
-    done
-  done
-
-Lemma (derive) transport_compose_typefam:
-  assumes
-    "A: U i" "B: U i"
-    "\<And>x. x: B \<Longrightarrow> P x: U i"
-    "f: A \<rightarrow> B"
-    "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
-    "u: P (f x)"
-  shows "trans (\<lambda>x. P (f x)) p u = trans P f[p] u"
-  by (eq p) (reduce; intros)
-
-Lemma (derive) transport_function_family:
-  assumes
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> P x: U i"
-    "\<And>x. x: A \<Longrightarrow> Q x: U i"
-    "f: \<Prod>x: A. P x \<rightarrow> Q x"
-    "x: A" "y: A"
-    "u: P x"
-    "p: x =\<^bsub>A\<^esub> y"
-  shows "trans Q p ((f x) u) = (f y) (trans P p u)"
-  by (eq p) (reduce; intros)
-
-Lemma (derive) transport_const:
-  assumes
-    "A: U i" "B: U i"
-    "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
-  shows "\<Prod>b: B. trans (\<lambda>_. B) p b = b"
-  by (intro, eq p) (reduce; intro)
-
-definition transport_const_i ("trans'_const")
-  where [implicit]: "trans_const B p \<equiv> transport_const ? B ? ? p"
-
-translations "trans_const B p" \<leftharpoondown> "CONST transport_const A B x y p"
-
-Lemma transport_const_comp [comps]:
-  assumes
-    "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
-
-Lemma (derive) pathlift:
-  assumes
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> P x: U i"
-    "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
-    "u: P x"
-  shows "<x, u> = <y, trans P p u>"
-  by (eq p) (reduce; intros)
-
-definition pathlift_i ("lift")
-  where [implicit]: "lift P p u \<equiv> pathlift ? P ? ? p u"
-
-translations "lift P p u" \<leftharpoondown> "CONST pathlift A P x y p u"
-
-Lemma pathlift_comp [comps]:
-  assumes
-    "u: P x"
-    "x: A"
-    "\<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
-
-Lemma (derive) pathlift_fst:
-  assumes
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> P x: U i"
-    "x: A" "y: A"
-    "u: P x"
-    "p: x =\<^bsub>A\<^esub> y"
-  shows "fst[lift P p u] = p"
-  apply (eq p)
-    text \<open>Some rewriting needed here:\<close>
-    \<guillemotright> vars x y
-      (*Here an automatic reordering tactic would be helpful*)
-      apply (rewrite at x in "x = y" fst_comp[symmetric])
-        prefer 4
-        apply (rewrite at y in "_ = y" fst_comp[symmetric])
-      done
-    \<guillemotright> by reduce intro
-  done
-
-
-section \<open>Dependent paths\<close>
-
-Lemma (derive) apd:
-  assumes
-    "A: U i"
-    "\<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"
-  shows "trans P p (f x) = f y"
-  by (eq p) (reduce; intros; typechk)
-
-definition apd_i ("apd")
-  where [implicit]: "apd f p \<equiv> Identity.apd ? ? f ? ? p"
-
-translations "apd f p" \<leftharpoondown> "CONST Identity.apd A P f x y p"
-
-Lemma dependent_map_comp [comps]:
-  assumes
-    "f: \<Prod>x: A. P x"
-    "x: A"
-    "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
-
-Lemma (derive) apd_ap:
-  assumes
-    "A: U i" "B: U i"
-    "f: A \<rightarrow> B"
-    "x: A" "y: A"
-    "p: x =\<^bsub>A\<^esub> y"
-  shows "apd f p = trans_const B p (f x) \<bullet> f[p]"
-  by (eq p) (reduce; intro)
-
-
-end