Reorganize theory structure. In particular, the identity type moves out from under Spartan to HoTT. Spartan now only has Pi and Sigma.

3 files changed, 881 insertions, 1 deletions

diff --git a/hott/Base.thy b/hott/Base.thy
index 2a4ff9c..610a373 100644
--- a/hott/Base.thy
+++ b/hott/Base.thy
@@ -1,5 +1,5 @@
 theory Base
-imports Spartan.Equivalence
+imports Equivalence
 
 begin
 
diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy
new file mode 100644
index 0000000..2975738
--- /dev/null
+++ b/hott/Equivalence.thy
@@ -0,0 +1,416 @@
+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/hott/Identity.thy b/hott/Identity.thy
new file mode 100644
index 0000000..3a982f6
--- /dev/null
+++ b/hott/Identity.thy
@@ -0,0 +1,464 @@
+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