diff options
| -rw-r--r-- | Equivalence.thy | 197 | ||||
| -rw-r--r-- | HoTT_Methods.thy | 4 | ||||
| -rw-r--r-- | Prod.thy | 2 | ||||
| -rw-r--r-- | Projections.thy | 4 |
4 files changed, 202 insertions, 5 deletions
diff --git a/Equivalence.thy b/Equivalence.thy new file mode 100644 index 0000000..b8df15d --- /dev/null +++ b/Equivalence.thy @@ -0,0 +1,197 @@ +(******** +Isabelle/HoTT: Quasi-inverse and equivalence +Mar 2019 + +********) + +theory Equivalence +imports Type_Families + +begin + +section \<open>Homotopy\<close> + +definition homotopy :: "[t, t \<Rightarrow> t, t, t] \<Rightarrow> t" ("(2homotopy[_, _] _ _)" [0, 0, 1000, 1000]) +where "homotopy[A, B] f g \<equiv> \<Prod>x: A. f`x =[B x] g`x" + +declare homotopy_def [comp] + +syntax "_homotopy" :: "[t, idt, t, t, t] \<Rightarrow> t" ("(1_ ~[_: _. _]/ _)" [101, 0, 0, 0, 101] 100) +translations "f ~[x: A. B] g" \<rightleftharpoons> "(CONST homotopy) A (\<lambda>x. B) f g" + +lemma homotopy_type: + assumes [intro]: "A: U i" "B: A \<leadsto> U i" "f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x" + shows "f ~[x: A. B x] g: U i" +by derive + +declare homotopy_type [intro] + +text \<open>Homotopy inverse and composition (symmetry and transitivity):\<close> + +definition hominv :: "[t, t \<Rightarrow> t, t, t] \<Rightarrow> t" ("(2hominv[_, _, _, _])") +where "hominv[A, B, f, g] \<equiv> \<lambda>H: f ~[x: A. B x] g. \<lambda>x: A. inv[B x, f`x, g`x]`(H`x)" + +lemma hominv_type: + assumes [intro]: "A: U i" "B: A \<leadsto> U i" "f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x" + shows "hominv[A, B, f, g]: f ~[x: A. B x] g \<rightarrow> g ~[x: A. B x] f" +unfolding hominv_def by (derive, fold homotopy_def)+ derive + +definition homcomp :: "[t, t \<Rightarrow> t, t, t, t] \<Rightarrow> t" ("(2homcomp[_, _, _, _, _])") where + "homcomp[A, B, f, g, h] \<equiv> \<lambda>H: f ~[x: A. B x] g. \<lambda>H': g ~[x: A. B x] h. + \<lambda>x: A. pathcomp[B x, f`x, g`x, h`x]`(H`x)`(H'`x)" + +lemma homcomp_type: + assumes [intro]: + "A: U i" "B: A \<leadsto> U i" + "f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x" "h: \<Prod>x: A. B x" + shows "homcomp[A, B, f, g, h]: f ~[x: A. B x] g \<rightarrow> g ~[x: A. B x] h \<rightarrow> f ~[x: A. B x] h" +unfolding homcomp_def by (derive, fold homotopy_def)+ derive + +schematic_goal fun_eq_imp_homotopy: + assumes [intro]: + "p: f =[\<Prod>x: A. B x] g" + "f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x" + "A: U i" "B: A \<leadsto> U i" + shows "?prf: f ~[x: A. B x] g" +proof (path_ind' f g p) + show "\<And>f. f : \<Prod>(x: A). B x \<Longrightarrow> \<lambda>x: A. refl(f`x): f ~[x: A. B x] f" by derive +qed routine + +definition happly :: "[t, t \<Rightarrow> t, t, t, t] \<Rightarrow> t" +where "happly A B f g p \<equiv> indEq (\<lambda>f g. & f ~[x: A. B x] g) (\<lambda>f. \<lambda>(x: A). refl(f`x)) f g p" + +syntax "_happly" :: "[idt, t, t, t, t, t] \<Rightarrow> t" + ("(2happly[_: _. _] _ _ _)" [0, 0, 0, 1000, 1000, 1000]) +translations "happly[x: A. B] f g p" \<rightleftharpoons> "(CONST happly) A (\<lambda>x. B) f g p" + +corollary happly_type: + assumes [intro]: + "p: f =[\<Prod>x: A. B x] g" + "f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x" + "A: U i" "B: A \<leadsto> U i" + shows "happly[x: A. B x] f g p: f ~[x: A. B x] g" +unfolding happly_def by (derive lems: fun_eq_imp_homotopy) + +text \<open>Homotopy and function composition:\<close> + +declare[[pretty_compose=false]] + +schematic_goal composition_homotopyl: + assumes [intro]: + "H: f ~[x: A. B] g" + "f: A \<rightarrow> B" "g: A \<rightarrow> B" "h: B \<rightarrow> C" + "A: U i" "B: U i" "C: U i" + shows "?prf: h o[A] f ~[x: A. C] h o[A] g" +unfolding homotopy_def compose_def proof (rule Prod_routine, subst (0 1) comp) + fix x assume [intro]: "x: A" + show "ap[h, B, C, f`x, g`x]`(H`x): h`(f`x) =[C] h`(g`x)" by (derive, fold homotopy_def, fact+) +qed routine + +section \<open>Bi-invertibility\<close> + +definition biinv :: "[t, t, t] \<Rightarrow> t" ("(2biinv[_, _]/ _)") +where "biinv[A, B] f \<equiv> + (\<Sum>g: B \<rightarrow> A. g o[A] f ~[x:A. A] id A) \<times> (\<Sum>g: B \<rightarrow> A. f o[B] g ~[x: B. B] id B)" + +text \<open> +The meanings of the syntax defined above are: +\<^item> @{term "f ~[x: A. B x] g"} expresses that @{term f} and @{term g} are homotopy functions of type @{term "\<Prod>x:A. B x"}. +\<^item> @{term "biinv[A, B] f"} expresses that the function @{term f} of type @{term "A \<rightarrow> B"} is bi-invertible. +\<close> + +lemma biinv_type: + assumes [intro]: "A: U i" "B: U i" "f: A \<rightarrow> B" + shows "biinv[A, B] f: U i" +unfolding biinv_def by derive + +declare biinv_type [intro] + +schematic_goal id_is_biinv: + assumes [intro]: "A: U i" + shows "?prf: biinv[A, A] (id A)" +unfolding biinv_def proof (rule Sum_routine, compute) + show "<id A, \<lambda>x: A. refl x>: \<Sum>(g: A \<rightarrow> A). (g o[A] id A) ~[x: A. A] (id A)" by derive + show "<id A, \<lambda>x: A. refl x>: \<Sum>(g: A \<rightarrow> A). (id A o[A] g) ~[x: A. A] (id A)" by derive +qed routine + +definition equivalence :: "[t, t] \<Rightarrow> t" (infix "\<cong>" 100) +where "A \<cong> B \<equiv> \<Sum>f: A \<rightarrow> B. biinv[A, B] f" + +schematic_goal equivalence_symmetric: + assumes [intro]: "A: U i" + shows "?prf: A \<cong> A" +unfolding equivalence_def proof (rule Sum_routine) + show "\<And>f. f : A \<rightarrow> A \<Longrightarrow> biinv[A, A] f : U i" unfolding biinv_def by derive + show "id A: A \<rightarrow> A" by routine +qed (routine add: id_is_biinv) + + +section \<open>Quasi-inverse\<close> + +definition qinv :: "[t, t, t] \<Rightarrow> t" ("(2qinv[_, _]/ _)") +where "qinv[A, B] f \<equiv> \<Sum>g: B \<rightarrow> A. (g o[A] f ~[x: A. A] id A) \<times> (f o[B] g ~[x: B. B] id B)" + +schematic_goal biinv_imp_qinv: + assumes [intro]: "A: U i" "B: U i" "f: A \<rightarrow> B" + shows "?prf: (biinv[A, B] f) \<rightarrow> (qinv[A,B] f)" +proof (rule Prod_routine) +assume "b: biinv[A, B] f" +define g H g' H' where + "g \<equiv> fst[B \<rightarrow> A, \<lambda>g. g o[A] f ~[x: A. A] id A] ` + (fst[\<Sum>g: B \<rightarrow> A. g o[A] f ~[x: A. A] id A, &(\<Sum>g: B \<rightarrow> A. f o[B] g ~[x: A. A] id B)] ` b)" +and + "H \<equiv> snd[B \<rightarrow> A, \<lambda>g. g o[A] f ~[x: A. A] id A] ` + (fst[\<Sum>g: B \<rightarrow> A. g o[A] f ~[x: A. A] id A, &(\<Sum>g: B \<rightarrow> A. f o[B] g ~[x: A. A] id B)] ` b)" +and + "g' \<equiv> fst[B \<rightarrow> A, \<lambda>g. f o[B] g ~[x: B. B] id B] ` + (snd[\<Sum>g: B \<rightarrow> A. g o[A] f ~[x: A. A] id A, &(\<Sum>g: B \<rightarrow> A. f o[B] g ~[x: A. A] id B)] ` b)" +and + "H' \<equiv> snd[B \<rightarrow> A, \<lambda>g. f o[B] g ~[x: B. B] id B] ` + (snd[\<Sum>g: B \<rightarrow> A. g o[A] f ~[x: A. A] id A, &(\<Sum>g: B \<rightarrow> A. f o[B] g ~[x: A. A] id B)] ` b)" + +have "g o[B] (f o[B] g') \<equiv> g" +unfolding g_def g'_def proof compute + + +section \<open>Transport, homotopy, and bi-invertibility\<close> + +schematic_goal transport_invl_hom: + assumes [intro]: + "P: A \<leadsto> U j" "A: U i" + "x: A" "y: A" "p: x =[A] y" + shows "?prf: + (transport[A, P, y, x]`(inv[A, x, y]`p)) o[P x] (transport[A, P, x, y]`p) ~[w: P x. P x] id P x" +by (rule happly_type[OF transport_invl], derive) + +schematic_goal transport_invr_hom: + assumes [intro]: + "A: U i" "P: A \<leadsto> U j" + "y: A" "x: A" "p: x =[A] y" + shows "?prf: + (transport[A, P, x, y]`p) o[P y] (transport[A, P, y, x]`(inv[A, x, y]`p)) ~[w: P y. P y] id P y" +by (rule happly_type[OF transport_invr], derive) + +declare + transport_invl_hom [intro] + transport_invr_hom [intro] + +text \<open> +The following result states that the transport of an equality @{term p} is bi-invertible, with inverse given by the transport of the inverse @{text "~p"}. +\<close> + +schematic_goal transport_biinv: + assumes [intro]: "p: A =[U i] B" "A: U i" "B: U i" + shows "?prf: biinv[A, B] (transport[U i, Id, A, B]`p)" +unfolding biinv_def +apply (rule Sum_routine) +prefer 2 + apply (rule Sum_routine) + prefer 3 apply (rule transport_invl_hom) +prefer 9 + apply (rule Sum_routine) + prefer 3 apply (rule transport_invr_hom) +\<comment> \<open>The remaining subgoals are now handled easily\<close> +by derive + + +end diff --git a/HoTT_Methods.thy b/HoTT_Methods.thy index e45608e..0199a49 100644 --- a/HoTT_Methods.thy +++ b/HoTT_Methods.thy @@ -68,8 +68,8 @@ It also handles universes using @{method cumulativity}. The method @{method hierarchy} has been observed to cause looping in previous versions, and is hence no longer part of @{theory_text derive}. \<close> -method derive uses lems declares comp = - (routine add: lems | compute add: lems comp: comp | cumulativity form: lems)+ +method derive uses lems unfold declares comp = + (unfold unfold | routine add: lems | compute add: lems comp: comp | cumulativity form: lems)+ section \<open>Additional method combinators\<close> @@ -15,7 +15,7 @@ section \<open>Basic type definitions\<close> axiomatization Prod :: "[t, t \<Rightarrow> t] \<Rightarrow> t" and lam :: "[t, t \<Rightarrow> t] \<Rightarrow> t" and - app :: "[t, t] \<Rightarrow> t" ("(2_`_)" [120, 121] 120) + app :: "[t, t] \<Rightarrow> t" ("(2_`/_)" [120, 121] 120) \<comment> \<open>Application should bind tighter than abstraction.\<close> syntax diff --git a/Projections.thy b/Projections.thy index a28c66b..c89b569 100644 --- a/Projections.thy +++ b/Projections.thy @@ -11,10 +11,10 @@ imports Prod Sum begin -definition fst ("(2fst[_, _])") +definition fst ("(2fst[_,/ _])") where "fst[A, B] \<equiv> \<lambda>(p: \<Sum>x: A. B x). indSum (\<lambda>_. A) (\<lambda>x y. x) p" -definition snd ("(2snd[_, _])") +definition snd ("(2snd[_,/ _])") where "snd[A, B] \<equiv> \<lambda>(p: \<Sum>x: A. B x). indSum (\<lambda>p. B (fst[A, B]`p)) (\<lambda>x y. y) p" lemma fst_type: |