1 files changed, 431 insertions, 0 deletions
diff --git a/spartan/theories/Equivalence.thy b/spartan/theories/Equivalence.thy
new file mode 100644
index 0000000..44b77dd
--- /dev/null
+++ b/spartan/theories/Equivalence.thy
@@ -0,0 +1,431 @@
+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:
+  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
+
+lemmas homotopy_symmetric = hsym[rotated 4]
+
+text \<open>\<open>hsym\<close> attribute for homotopies:\<close>
+
+ML \<open>
+structure HSym_Attr = Sym_Attr (
+  struct
+    val name = "hsym"
+    val symmetry_rule = @{thm homotopy_symmetric}
+  end
+)
+\<close>
+
+setup \<open>HSym_Attr.setup\<close>
+
+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
+
+lemmas homotopy_transitive = 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 (equality \<open>p:_\<close>)
+    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
+    premises hyps
+    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 premises 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 \<circ> (id B) ~ g \<circ> f \<circ> h"
+          by (rule homotopy_funcomp_right) (rule \<open>H2:_\<close>[hsym])
+
+        moreover have "g \<circ> f \<circ> h ~ (id A) \<circ> h"
+          by (subst funcomp_assoc[symmetric])
+             (rule homotopy_funcomp_left, rule \<open>H1:_\<close>)
+
+        ultimately have "g ~ h"
+          apply (rewrite to "g \<circ> (id B)" id_right[symmetric])
+          apply (rewrite to "(id A) \<circ> h" id_left[symmetric])
+          by (rule homotopy_transitive) (assumption, typechk)
+
+        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_transitive) (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 (equality \<open>p:_\<close>) (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 (equality \<open>p:_\<close>)
+      \<^item> premises 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> premises 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