From 80edbd08e13200d2c080ac281d19948bbbcd92e0 Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Mon, 25 May 2020 17:09:54 +0200
Subject: Reorganize theory structure. In particular, the identity type moves
 out from under Spartan to HoTT. Spartan now only has Pi and Sigma.

---
 spartan/theories/Equivalence.thy | 416 ---------------------------------
 spartan/theories/Identity.thy    | 464 -------------------------------------
 spartan/theories/Spartan.thy     | 481 ---------------------------------------
 3 files changed, 1361 deletions(-)
 delete mode 100644 spartan/theories/Equivalence.thy
 delete mode 100644 spartan/theories/Identity.thy
 delete mode 100644 spartan/theories/Spartan.thy

(limited to 'spartan/theories')

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
diff --git a/spartan/theories/Spartan.thy b/spartan/theories/Spartan.thy
deleted file mode 100644
index d235041..0000000
--- a/spartan/theories/Spartan.thy
+++ /dev/null
@@ -1,481 +0,0 @@
-text \<open>Spartan type theory\<close>
-
-theory Spartan
-imports
-  Pure
-  "HOL-Eisbach.Eisbach"
-  "HOL-Eisbach.Eisbach_Tools"
-keywords
-  "Theorem" "Lemma" "Corollary" "Proposition" :: thy_goal_stmt and
-  "focus" "\<guillemotright>" "\<^item>" "\<^enum>" "~" :: prf_script_goal % "proof" and
-  "derive" "prems" "vars":: quasi_command and
-  "print_coercions" :: thy_decl
-
-begin
-
-
-section \<open>Preamble\<close>
-
-declare [[eta_contract=false]]
-
-
-section \<open>Metatype setup\<close>
-
-typedecl o
-
-
-section \<open>Judgments\<close>
-
-judgment has_type :: \<open>o \<Rightarrow> o \<Rightarrow> prop\<close> ("(2_:/ _)" 999)
-
-
-section \<open>Universes\<close>
-
-typedecl lvl \<comment> \<open>Universe levels\<close>
-
-axiomatization
-  O  :: \<open>lvl\<close> and
-  S  :: \<open>lvl \<Rightarrow> lvl\<close> and
-  lt :: \<open>lvl \<Rightarrow> lvl \<Rightarrow> prop\<close> (infix "<" 900)
-  where
-  O_min: "O < S i" and
-  lt_S: "i < S i" and
-  lt_trans: "i < j \<Longrightarrow> j < k \<Longrightarrow> i < k"
-
-axiomatization U :: \<open>lvl \<Rightarrow> o\<close> where
-  U_hierarchy: "i < j \<Longrightarrow> U i: U j" and
-  U_cumulative: "A: U i \<Longrightarrow> i < j \<Longrightarrow> A: U j"
-
-lemma U_in_U:
-  "U i: U (S i)"
-  by (rule U_hierarchy, rule lt_S)
-
-lemma lift_universe:
-  "A: U i \<Longrightarrow> A: U (S i)"
-  by (erule U_cumulative, rule lt_S)
-
-
-section \<open>\<Prod>-type\<close>
-
-axiomatization
-  Pi  :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o\<close> and
-  lam :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o\<close> and
-  app :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> ("(1_ `_)" [120, 121] 120)
-
-syntax
-  "_Pi"   :: \<open>idt \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<Prod>_: _./ _)" 30)
-  "_lam"  :: \<open>pttrns \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<lambda>_: _./ _)" 30)
-  "_lam2" :: \<open>pttrns \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close>
-translations
-  "\<Prod>x: A. B"    \<rightleftharpoons> "CONST Pi A (\<lambda>x. B)"
-  "\<lambda>x xs: A. b" \<rightharpoonup> "CONST lam A (\<lambda>x. _lam2 xs A b)"
-  "_lam2 x A b" \<rightharpoonup> "\<lambda>x: A. b"
-  "\<lambda>x: A. b"    \<rightleftharpoons> "CONST lam A (\<lambda>x. b)"
-
-abbreviation Fn (infixr "\<rightarrow>" 40) where "A \<rightarrow> B \<equiv> \<Prod>_: A. B"
-
-axiomatization where
-  PiF: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; A: U i\<rbrakk> \<Longrightarrow> \<Prod>x: A. B x: U i" and
-
-  PiI: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; A: U i\<rbrakk> \<Longrightarrow> \<lambda>x: A. b x: \<Prod>x: A. B x" and
-
-  PiE: "\<lbrakk>f: \<Prod>x: A. B x; a: A\<rbrakk> \<Longrightarrow> f `a: B a" and
-
-  beta: "\<lbrakk>a: A; \<And>x. x: A \<Longrightarrow> b x: B x\<rbrakk> \<Longrightarrow> (\<lambda>x: A. b x) `a \<equiv> b a" and
-
-  eta: "f: \<Prod>x: A. B x \<Longrightarrow> \<lambda>x: A. f `x \<equiv> f" and
-
-  Pi_cong: "\<lbrakk>
-    A: U i;
-    \<And>x. x: A \<Longrightarrow> B x: U i;
-    \<And>x. x: A \<Longrightarrow> B' x: U i;
-    \<And>x. x: A \<Longrightarrow> B x \<equiv> B' x
-    \<rbrakk> \<Longrightarrow> \<Prod>x: A. B x \<equiv> \<Prod>x: A. B' x" and
-
-  lam_cong: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x \<equiv> c x; A: U i\<rbrakk> \<Longrightarrow> \<lambda>x: A. b x \<equiv> \<lambda>x: A. c x"
-
-
-section \<open>\<Sum>-type\<close>
-
-axiomatization
-  Sig    :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o\<close> and
-  pair   :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> ("(2<_,/ _>)") and
-  SigInd :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o\<close>
-
-syntax "_Sum" :: \<open>idt \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<Sum>_: _./ _)" 20)
-
-translations "\<Sum>x: A. B" \<rightleftharpoons> "CONST Sig A (\<lambda>x. B)"
-
-abbreviation Prod (infixl "\<times>" 50)
-  where "A \<times> B \<equiv> \<Sum>_: A. B"
-
-axiomatization where
-  SigF: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; A: U i\<rbrakk> \<Longrightarrow> \<Sum>x: A. B x: U i" and
-
-  SigI: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; a: A; b: B a\<rbrakk> \<Longrightarrow> <a, b>: \<Sum>x: A. B x" and
-
-  SigE: "\<lbrakk>
-    p: \<Sum>x: A. B x;
-    A: U i;
-    \<And>x. x : A \<Longrightarrow> B x: U i;
-    \<And>p. p: \<Sum>x: A. B x \<Longrightarrow> C p: U i;
-    \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x, y>
-    \<rbrakk> \<Longrightarrow> SigInd A (\<lambda>x. B x) (\<lambda>p. C p) f p: C p" and
-
-  Sig_comp: "\<lbrakk>
-    a: A;
-    b: B a;
-    \<And>x. x: A \<Longrightarrow> B x: U i;
-    \<And>p. p: \<Sum>x: A. B x \<Longrightarrow> C p: U i;
-    \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x, y>
-    \<rbrakk> \<Longrightarrow> SigInd A (\<lambda>x. B x) (\<lambda>p. C p) f <a, b> \<equiv> f a b" and
-
-  Sig_cong: "\<lbrakk>
-    \<And>x. x: A \<Longrightarrow> B x \<equiv> B' x;
-    A: U i;
-    \<And>x. x : A \<Longrightarrow> B x: U i;
-    \<And>x. x : A \<Longrightarrow> B' x: U i
-    \<rbrakk> \<Longrightarrow> \<Sum>x: A. B x \<equiv> \<Sum>x: A. B' x"
-
-
-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>
-
-
-section \<open>Congruence automation\<close>
-
-(*Potential to be retired*)
-ML_file \<open>../lib/congruence.ML\<close>
-
-
-section \<open>Methods\<close>
-
-ML_file \<open>../lib/elimination.ML\<close> \<comment> \<open>declares the [elims] attribute\<close>
-
-named_theorems intros and comps
-lemmas
-  [intros] = PiF PiI SigF SigI and
-  [elims "?f"] = PiE and
-  [elims "?p"] = SigE and
-  [comps] = beta Sig_comp and
-  [cong] = Pi_cong lam_cong Sig_cong
-
-ML_file \<open>../lib/tactics.ML\<close>
-
-method_setup assumptions =
-  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (
-    CHANGED (TRYALL (assumptions_tac ctxt))))\<close>
-
-method_setup known =
-  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (
-    CHANGED (TRYALL (known_tac ctxt))))\<close>
-
-method_setup intro =
-  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (intro_tac ctxt)))\<close>
-
-method_setup intros =
-  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (intros_tac ctxt)))\<close>
-
-method_setup old_elim =
-  \<open>Scan.option Args.term >> (fn tm => fn ctxt =>
-    SIMPLE_METHOD' (SIDE_CONDS (old_elims_tac tm ctxt) ctxt))\<close>
-
-method_setup elim =
-  \<open>Scan.repeat Args.term >> (fn tms => fn ctxt =>
-    CONTEXT_METHOD (K (elim_context_tac tms ctxt 1)))\<close>
-
-method elims = elim+
-
-(*This could possibly use additional simplification hints via a simp: modifier*)
-method_setup typechk =
-  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (
-    SIDE_CONDS (typechk_tac ctxt) ctxt))
-    (* CHANGED (ALLGOALS (TRY o typechk_tac ctxt)))) *)\<close>
-
-method_setup rule =
-  \<open>Attrib.thms >> (fn ths => fn ctxt =>
-    SIMPLE_METHOD (HEADGOAL (SIDE_CONDS (rule_tac ths ctxt) ctxt)))\<close>
-
-method_setup dest =
-  \<open>Scan.lift (Scan.option (Args.parens Parse.int)) -- Attrib.thms
-    >> (fn (opt_n, ths) => fn ctxt =>
-        SIMPLE_METHOD (HEADGOAL (SIDE_CONDS (dest_tac opt_n ths ctxt) ctxt)))\<close>
-
-subsection \<open>Reflexivity\<close>
-
-named_theorems refl
-method refl = (rule refl)
-
-subsection \<open>Trivial proofs modulo typechecking\<close>
-
-method_setup this =
-  \<open>Scan.succeed (fn ctxt => METHOD (
-    EVERY o map (HEADGOAL o
-      (fn ths => SIDE_CONDS (resolve_tac ctxt ths) ctxt) o
-      single)
-    ))\<close>
-
-subsection \<open>Rewriting\<close>
-
-\<comment> \<open>\<open>subst\<close> method\<close>
-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"
-
-\<comment> \<open>\<open>rewrite\<close> method\<close>
-consts rewrite_HOLE :: "'a::{}"  ("\<hole>")
-
-lemma eta_expand:
-  fixes f :: "'a::{} \<Rightarrow> 'b::{}"
-  shows "f \<equiv> \<lambda>x. f x" .
-
-lemma rewr_imp:
-  assumes "PROP A \<equiv> PROP B"
-  shows "(PROP A \<Longrightarrow> PROP C) \<equiv> (PROP B \<Longrightarrow> PROP C)"
-  apply (rule Pure.equal_intr_rule)
-  apply (drule equal_elim_rule2[OF assms]; assumption)
-  apply (drule equal_elim_rule1[OF assms]; assumption)
-  done
-
-lemma imp_cong_eq:
-  "(PROP A \<Longrightarrow> (PROP B \<Longrightarrow> PROP C) \<equiv> (PROP B' \<Longrightarrow> PROP C')) \<equiv>
-    ((PROP B \<Longrightarrow> PROP A \<Longrightarrow> PROP C) \<equiv> (PROP B' \<Longrightarrow> PROP A \<Longrightarrow> PROP C'))"
-  apply (Pure.intro Pure.equal_intr_rule)
-    apply (drule (1) cut_rl; drule Pure.equal_elim_rule1 Pure.equal_elim_rule2;
-      assumption)+
-    apply (drule Pure.equal_elim_rule1 Pure.equal_elim_rule2; assumption)+
-  done
-
-ML_file \<open>~~/src/HOL/Library/cconv.ML\<close>
-ML_file \<open>../lib/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>
-
-method reduce uses add = (simp add: comps add | subst comps)+
-
-
-section \<open>Implicit notations\<close>
-
-text \<open>
-  \<open>?\<close> is used to mark implicit arguments in definitions, while \<open>{}\<close> is expanded
-  immediately for elaboration in statements.
-\<close>
-
-consts
-  iarg :: \<open>'a\<close> ("?")
-  hole :: \<open>'b\<close> ("{}")
-
-ML_file \<open>../lib/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>
-
-text \<open>Automatically insert inhabitation judgments where needed:\<close>
-
-consts inhabited :: \<open>o \<Rightarrow> prop\<close> ("(_)")
-translations "CONST inhabited A" \<rightharpoonup> "CONST has_type {} A"
-
-
-section \<open>Lambda coercion\<close>
-
-\<comment> \<open>Coerce object lambdas to meta-lambdas\<close>
-abbreviation (input) lambda :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close>
-  where "lambda f \<equiv> \<lambda>x. f `x"
-
-ML_file \<open>~~/src/Tools/subtyping.ML\<close>
-declare [[coercion_enabled, coercion lambda]]
-
-translations "f x" \<leftharpoondown> "f `x"
-
-
-section \<open>Functions\<close>
-
-lemma eta_exp:
-  assumes "f: \<Prod>x: A. B x"
-  shows "f \<equiv> \<lambda>x: A. f x"
-  by (rule eta[symmetric])
-
-lemma lift_universe_codomain:
-  assumes "A: U i" "f: A \<rightarrow> U j"
-  shows "f: A \<rightarrow> U (S j)"
-  apply (sub eta_exp)
-    apply known
-    apply (Pure.rule intros; rule lift_universe)
-  done
-
-subsection \<open>Function composition\<close>
-
-definition "funcomp A g f \<equiv> \<lambda>x: A. g `(f `x)"
-
-syntax
-  "_funcomp" :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2_ \<circ>\<^bsub>_\<^esub>/ _)" [111, 0, 110] 110)
-translations
-  "g \<circ>\<^bsub>A\<^esub> f" \<rightleftharpoons> "CONST funcomp A g f"
-
-lemma funcompI [typechk]:
-  assumes
-    "A: U i"
-    "B: U i"
-    "\<And>x. x: B \<Longrightarrow> C x: U i"
-    "f: A \<rightarrow> B"
-    "g: \<Prod>x: B. C x"
-  shows
-    "g \<circ>\<^bsub>A\<^esub> f: \<Prod>x: A. C (f x)"
-  unfolding funcomp_def by typechk
-
-lemma funcomp_assoc [comps]:
-  assumes
-    "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
-
-lemma funcomp_lambda_comp [comps]:
-  assumes
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> b x: B"
-    "\<And>x. x: B \<Longrightarrow> c x: C x"
-  shows
-    "(\<lambda>x: B. c x) \<circ>\<^bsub>A\<^esub> (\<lambda>x: A. b x) \<equiv> \<lambda>x: A. c (b x)"
-  unfolding funcomp_def by reduce
-
-lemma funcomp_apply_comp [comps]:
-  assumes
-    "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
-
-text \<open>Notation:\<close>
-
-definition funcomp_i (infixr "\<circ>" 120)
-  where [implicit]: "funcomp_i g f \<equiv> g \<circ>\<^bsub>?\<^esub> f"
-
-translations "g \<circ> f" \<leftharpoondown> "g \<circ>\<^bsub>A\<^esub> f"
-
-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_comp [comps]: "x: A \<Longrightarrow> (id A) x \<equiv> x" \<comment> \<open>for the occasional manual rewrite\<close>
-  by reduce
-
-lemma id_left [comps]:
-  assumes "f: A \<rightarrow> B" "A: U i" "B: U i"
-  shows "(id B) \<circ>\<^bsub>A\<^esub> f \<equiv> f"
-  by (subst eta_exp[of f]) (reduce, rule eta)
-
-lemma id_right [comps]:
-  assumes "f: A \<rightarrow> B" "A: U i" "B: U i"
-  shows "f \<circ>\<^bsub>A\<^esub> (id A) \<equiv> f"
-  by (subst eta_exp[of f]) (reduce, rule eta)
-
-lemma id_U [typechk]:
-  "id (U i): U i \<rightarrow> U i"
-  by typechk (fact U_in_U)
-
-
-section \<open>Pairs\<close>
-
-definition "fst A B \<equiv> \<lambda>p: \<Sum>x: A. B x. SigInd A B (\<lambda>_. A) (\<lambda>x y. x) p"
-definition "snd A B \<equiv> \<lambda>p: \<Sum>x: A. B x. SigInd A B (\<lambda>p. B (fst A B p)) (\<lambda>x y. y) p"
-
-lemma fst_type [typechk]:
-  assumes "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i"
-  shows "fst A B: (\<Sum>x: A. B x) \<rightarrow> A"
-  unfolding fst_def by typechk
-
-lemma fst_comp [comps]:
-  assumes
-    "a: A"
-    "b: B a"
-    "A: U i"
-    "\<And>x. x: A \<Longrightarrow> B x: U i"
-  shows "fst A B <a, b> \<equiv> a"
-  unfolding fst_def by reduce
-
-lemma snd_type [typechk]:
-  assumes "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i"
-  shows "snd A B: \<Prod>p: \<Sum>x: A. B x. B (fst A B p)"
-  unfolding snd_def by typechk reduce
-
-lemma snd_comp [comps]:
-  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
-
-subsection \<open>Notation\<close>
-
-definition fst_i ("fst")
-  where [implicit]: "fst \<equiv> Spartan.fst ? ?"
-
-definition snd_i ("snd")
-  where [implicit]: "snd \<equiv> Spartan.snd ? ?"
-
-translations
-  "fst" \<leftharpoondown> "CONST Spartan.fst A B"
-  "snd" \<leftharpoondown> "CONST Spartan.snd A B"
-
-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"
-  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"
-  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]
-
-subsection \<open>Properties of \<Sigma>\<close>
-
-Lemma (derive) Sig_dist_exp:
-  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"
-  shows "(\<Sum>x: A. B x) \<times> (\<Sum>x: A. C x)"
-  apply (elim p)
-    focus vars x y
-      apply intro
-        \<guillemotright> apply intro
-            apply assumption
-            apply (fst y)
-          done
-        \<guillemotright> apply intro
-            apply assumption
-            apply (snd y)
-          done
-    done
-  done
-
-
-end
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