diff options
Diffstat (limited to 'Eq.thy')
| -rw-r--r-- | Eq.thy | 159 |
1 files changed, 91 insertions, 68 deletions
@@ -74,7 +74,10 @@ subsection \<open>Symmetry (path inverse)\<close> definition inv :: "[t, t, t, t] \<Rightarrow> t" where "inv A x y p \<equiv> indEq (\<lambda>x y. ^(y =[A] x)) (\<lambda>x. refl x) x y p" -syntax "_inv" :: "t \<Rightarrow> t" ("~_" [1000]) +syntax "_inv" :: "[t, t, t, t] \<Rightarrow> t" ("(2inv[_, _, _] _)" [0, 0, 0, 1000]) +translations "inv[A, x, y] p" \<rightleftharpoons> "(CONST inv) A x y p" + +syntax "_inv'" :: "t \<Rightarrow> t" ("~_" [1000]) text \<open>Pretty-printing switch for path inverse:\<close> @@ -83,17 +86,17 @@ ML \<open>val pretty_inv = Attrib.setup_config_bool @{binding "pretty_inv"} (K t print_translation \<open> let fun inv_tr' ctxt [A, x, y, p] = if Config.get ctxt pretty_inv - then Syntax.const @{syntax_const "_inv"} $ p - else @{const inv} $ A $ x $ y $ p + then Syntax.const @{syntax_const "_inv'"} $ p + else Syntax.const @{syntax_const "_inv"} $ A $ x $ y $ p in [(@{const_syntax inv}, inv_tr')] end \<close> -lemma inv_type: "\<lbrakk>A: U i; x: A; y: A; p: x =[A] y\<rbrakk> \<Longrightarrow> inv A x y p: y =[A] x" +lemma inv_type: "\<lbrakk>A: U i; x: A; y: A; p: x =[A] y\<rbrakk> \<Longrightarrow> inv[A, x, y] p: y =[A] x" unfolding inv_def by derive -lemma inv_comp: "\<lbrakk>A: U i; a: A\<rbrakk> \<Longrightarrow> inv A a a (refl a) \<equiv> refl a" +lemma inv_comp: "\<lbrakk>A: U i; a: A\<rbrakk> \<Longrightarrow> inv[A, a, a] (refl a) \<equiv> refl a" unfolding inv_def by derive declare @@ -118,7 +121,10 @@ where (\<lambda>x. \<lambda>z: A. \<lambda>q: x =[A] z. indEq (\<lambda>x z _. x =[A] z) (\<lambda>x. refl x) x z q) x y p)`z`q" -syntax "_pathcomp" :: "[t, t] \<Rightarrow> t" (infixl "*" 110) +syntax "_pathcomp" :: "[t, t, t, t, t, t] \<Rightarrow> t" ("(2pathcomp[_, _, _, _] _ _)" [0, 0, 0, 0, 1000, 1000]) +translations "pathcomp[A, x, y, z] p q" \<rightleftharpoons> "(CONST pathcomp) A x y z p q" + +syntax "_pathcomp'" :: "[t, t] \<Rightarrow> t" (infixl "^" 110) ML \<open>val pretty_pathcomp = Attrib.setup_config_bool @{binding "pretty_pathcomp"} (K true)\<close> \<comment> \<open>Pretty-printing switch for path composition\<close> @@ -126,8 +132,8 @@ ML \<open>val pretty_pathcomp = Attrib.setup_config_bool @{binding "pretty_pathc print_translation \<open> let fun pathcomp_tr' ctxt [A, x, y, z, p, q] = if Config.get ctxt pretty_pathcomp - then Syntax.const @{syntax_const "_pathcomp"} $ p $ q - else @{const pathcomp} $ A $ x $ y $ z $ p $ q + then Syntax.const @{syntax_const "_pathcomp'"} $ p $ q + else Syntax.const @{syntax_const "_pathcomp"} $ A $ x $ y $ z $ p $ q in [(@{const_syntax pathcomp}, pathcomp_tr')] end @@ -135,12 +141,12 @@ end lemma pathcomp_type: assumes [intro]: "A: U i" "x: A" "y: A" "z: A" "p: x =[A] y" "q: y =[A] z" - shows "pathcomp A x y z p q: x =[A] z" + shows "pathcomp[A, x, y, z] p q: x =[A] z" unfolding pathcomp_def by (derive lems: transitivity) lemma pathcomp_comp: assumes [intro]: "A: U i" "a: A" - shows "pathcomp A a a a (refl a) (refl a) \<equiv> refl a" + shows "pathcomp[A, a, a, a] (refl a) (refl a) \<equiv> refl a" unfolding pathcomp_def by (derive lems: transitivity) declare @@ -152,43 +158,43 @@ section \<open>Higher groupoid structure of types\<close> schematic_goal pathcomp_idr: assumes [intro]: "A: U i" "x: A" "y: A" "p: x =[A] y" - shows "?prf: pathcomp A x y y p (refl y) =[x =[A] y] p" + shows "?prf: pathcomp[A, x, y, y] p (refl y) =[x =[A] y] p" proof (path_ind' x y p) - show "\<And>x. x: A \<Longrightarrow> refl(refl x): pathcomp A x x x (refl x) (refl x) =[x =[A] x] (refl x)" + show "\<And>x. x: A \<Longrightarrow> refl(refl x): pathcomp[A, x, x, x] (refl x) (refl x) =[x =[A] x] (refl x)" by derive qed routine schematic_goal pathcomp_idl: assumes [intro]: "A: U i" "x: A" "y: A" "p: x =[A] y" - shows "?prf: pathcomp A x x y (refl x) p =[x =[A] y] p" + shows "?prf: pathcomp[A, x, x, y] (refl x) p =[x =[A] y] p" proof (path_ind' x y p) - show "\<And>x. x: A \<Longrightarrow> refl(refl x): pathcomp A x x x (refl x) (refl x) =[x =[A] x] (refl x)" + show "\<And>x. x: A \<Longrightarrow> refl(refl x): pathcomp[A, x, x, x] (refl x) (refl x) =[x =[A] x] (refl x)" by derive qed routine schematic_goal pathcomp_invr: assumes [intro]: "A: U i" "x: A" "y: A" "p: x =[A] y" - shows "?prf: pathcomp A x y x p (inv A x y p) =[x =[A] x] (refl x)" + shows "?prf: pathcomp[A, x, y, x] p (inv[A, x, y] p) =[x =[A] x] (refl x)" proof (path_ind' x y p) show "\<And>x. x: A \<Longrightarrow> refl(refl x): - pathcomp A x x x (refl x) (inv A x x (refl x)) =[x =[A] x] (refl x)" + pathcomp[A, x, x, x] (refl x) (inv[A, x, x] (refl x)) =[x =[A] x] (refl x)" by derive qed routine schematic_goal pathcomp_invl: assumes [intro]: "A: U i" "x: A" "y: A" "p: x =[A] y" - shows "?prf: pathcomp A y x y (inv A x y p) p =[y =[A] y] refl(y)" + shows "?prf: pathcomp[A, y, x, y] (inv[A, x, y] p) p =[y =[A] y] refl(y)" proof (path_ind' x y p) show "\<And>x. x: A \<Longrightarrow> refl(refl x): - pathcomp A x x x (inv A x x (refl x)) (refl x) =[x =[A] x] (refl x)" + pathcomp[A, x, x, x] (inv[A, x, x] (refl x)) (refl x) =[x =[A] x] (refl x)" by derive qed routine schematic_goal inv_involutive: assumes [intro]: "A: U i" "x: A" "y: A" "p: x =[A] y" - shows "?prf: inv A y x (inv A x y p) =[x =[A] y] p" + shows "?prf: inv[A, y, x] (inv[A, x, y] p) =[x =[A] y] p" proof (path_ind' x y p) show "\<And>x. x: A \<Longrightarrow> refl(refl x): inv A x x (inv A x x (refl x)) =[x =[A] x] (refl x)" by derive @@ -208,31 +214,31 @@ schematic_goal pathcomp_assoc: assumes [intro]: "A: U i" shows "?prf: \<Prod>x: A. \<Prod>y: A. \<Prod>p: x =[A] y. \<Prod>z: A. \<Prod>q: y =[A] z. \<Prod>w: A. \<Prod>r: z =[A] w. - pathcomp A x y w p (pathcomp A y z w q r) =[x =[A] w] - pathcomp A x z w (pathcomp A x y z p q) r" + pathcomp[A, x, y, w] p (pathcomp[A, y, z, w] q r) =[x =[A] w] + pathcomp[A, x, z, w] (pathcomp[A, x, y, z] p q) r" apply (quantify 3) apply (path_ind "{x, y, p} \<Prod>(z: A). \<Prod>(q: y =[A] z). \<Prod>(w: A). \<Prod>(r: z =[A] w). - pathcomp A x y w p (pathcomp A y z w q r) =[x =[A] w] - pathcomp A x z w (pathcomp A x y z p q) r") + pathcomp[A, x, y, w] p (pathcomp[A, y, z, w] q r) =[x =[A] w] + pathcomp[A, x, z, w] (pathcomp[A, x, y, z] p q) r") apply (quantify 2) -apply (path_ind "{xa, z, q} +apply (path_ind "{x, z, q} \<Prod>(w: A). \<Prod>(r: z =[A] w). - pathcomp A xa xa w (refl xa) (pathcomp A xa z w q r) =[xa =[A] w] - pathcomp A xa z w (pathcomp A xa xa z (refl xa) q) r") + pathcomp[A, x, x, w] (refl x) (pathcomp[A, x, z, w] q r) =[x =[A] w] + pathcomp[A, x, z, w] (pathcomp[A, x, x, z] (refl x) q) r") apply (quantify 2) -apply (path_ind "{xb, w, r} - pathcomp A xb xb w (refl xb) (pathcomp A xb xb w (refl xb) r) =[xb =[A] w] - pathcomp A xb xb w (pathcomp A xb xb xb (refl xb) (refl xb)) r") +apply (path_ind "{x, w, r} + pathcomp[A, x, x, w] (refl x) (pathcomp[A, x, x, w] (refl x) r) =[x =[A] w] + pathcomp[A, x, x, w] (pathcomp[A, x, x, x] (refl x) (refl x)) r") text \<open>The rest is now routine.\<close> proof - show "\<And>x. x: A \<Longrightarrow> refl(refl x): - pathcomp A x x x (refl x) (pathcomp A x x x (refl x) (refl x)) =[x =[A] x] - pathcomp A x x x (pathcomp A x x x (refl x) (refl x)) (refl x)" + pathcomp[A, x, x, x] (refl x) (pathcomp[A, x, x, x] (refl x) (refl x)) =[x =[A] x] + pathcomp[A, x, x, x] (pathcomp[A, x, x, x] (refl x) (refl x)) (refl x)" by derive qed routine @@ -253,15 +259,18 @@ by (path_ind' x y p, rule Eq_routine, routine) definition ap :: "[t, t, t, t, t] \<Rightarrow> t" where "ap B f x y p \<equiv> indEq ({x, y, _} f`x =[B] f`y) (\<lambda>x. refl (f`x)) x y p" -syntax "_ap" :: "[t, t] \<Rightarrow> t" ("(_{_})" [1000, 0] 1000) +syntax "_ap" :: "[t, t, t, t, t] \<Rightarrow> t" ("(2ap[_, _, _] _ _)" [0, 0, 0, 1000, 1000]) +translations "ap[B, x, y] f p" \<rightleftharpoons> "(CONST ap) B f x y p" + +syntax "_ap'" :: "[t, t] \<Rightarrow> t" ("(_{_})" [1000, 0] 1000) ML \<open>val pretty_ap = Attrib.setup_config_bool @{binding "pretty_ap"} (K true)\<close> print_translation \<open> let fun ap_tr' ctxt [B, f, x, y, p] = if Config.get ctxt pretty_ap - then Syntax.const @{syntax_const "_ap"} $ f $ p - else @{const ap} $ B $ f $ x $ y $ p + then Syntax.const @{syntax_const "_ap'"} $ f $ p + else Syntax.const @{syntax_const "_ap"} $ B $ x $ y $ f $ p in [(@{const_syntax ap}, ap_tr')] end @@ -271,12 +280,12 @@ lemma ap_type: assumes "A: U i" "B: U i" "f: A \<rightarrow> B" "x: A" "y: A" "p: x =[A] y" - shows "ap B f x y p: f`x =[B] f`y" + shows "ap[B, x, y] f p: f`x =[B] f`y" unfolding ap_def using assms by (rule transfer) lemma ap_comp: assumes [intro]: "A: U i" "B: U i" "f: A \<rightarrow> B" "x: A" - shows "ap B f x x (refl x) \<equiv> refl (f`x)" + shows "ap[B, x, x] f (refl x) \<equiv> refl (f`x)" unfolding ap_def by derive declare @@ -288,23 +297,24 @@ schematic_goal ap_func_pathcomp: assumes [intro]: "A: U i" "B: U i" "f: A \<rightarrow> B" shows "?prf: \<Prod>x: A. \<Prod>y: A. \<Prod>p: x =[A] y. \<Prod>z: A. \<Prod>q: y =[A] z. - ap B f x z (pathcomp A x y z p q) =[f`x =[B] f`z] - pathcomp B (f`x) (f`y) (f`z) (ap B f x y p) (ap B f y z q)" + ap[B, x, z] f (pathcomp[A, x, y, z] p q) =[f`x =[B] f`z] + pathcomp[B, f`x, f`y, f`z] (ap[B, x, y] f p) (ap[B, y, z] f q)" apply (quantify 3) apply (path_ind "{x, y, p} \<Prod>z: A. \<Prod>q: y =[A] z. - ap B f x z (pathcomp A x y z p q) =[f`x =[B] f`z] - pathcomp B (f`x) (f`y) (f`z) (ap B f x y p) (ap B f y z q)") + ap[B, x, z] f (pathcomp[A, x, y, z] p q) =[f`x =[B] f`z] + pathcomp[B, f`x, f`y, f`z] (ap[B, x, y] f p) (ap[B, y, z] f q)") apply (quantify 2) apply (path_ind "{x, z, q} - ap B f x z (pathcomp A x x z (refl x) q) =[f`x =[B] f`z] - pathcomp B (f`x) (f`x) (f`z) (ap B f x x (refl x)) (ap B f x z q)") + ap[B, x, z] f (pathcomp[A, x, x, z] (refl x) q) =[f`x =[B] f`z] + pathcomp[B, f`x, f`x, f`z] (ap[B, x, x] f (refl x)) (ap[B, x, z] f q)") proof - show - "\<And>x. x: A \<Longrightarrow> refl(refl(f`x)): ap B f x x (pathcomp A x x x (refl x) (refl x)) =[f`x =[B] f`x] - pathcomp B (f`x) (f`x) (f`x) (ap B f x x (refl x)) (ap B f x x (refl x))" + "\<And>x. x: A \<Longrightarrow> refl(refl(f`x)) : + ap[B, x, x] f (pathcomp[A, x, x, x] (refl x) (refl x)) =[f`x =[B] f`x] + pathcomp[B, f`x, f`x, f`x] (ap[B, x, x] f (refl x)) (ap[B, x, x] f (refl x))" by derive -qed derive +qed routine schematic_goal ap_func_compose: @@ -313,40 +323,39 @@ schematic_goal ap_func_compose: "f: A \<rightarrow> B" "g: B \<rightarrow> C" shows "?prf: \<Prod>x: A. \<Prod>y: A. \<Prod>p: x =[A] y. - ap C g (f`x) (f`y) (ap B f x y p) =[g`(f`x) =[C] g`(f`y)] - ap C (compose A g f) x y p" + ap[C, f`x, f`y] g (ap[B, x, y] f p) =[g`(f`x) =[C] g`(f`y)] + ap[C, x, y] (g o[A] f) p" apply (quantify 3) apply (path_ind "{x, y, p} - ap C g (f`x) (f`y) (ap B f x y p) =[g`(f`x) =[C] g`(f`y)] - ap C (compose A g f) x y p") + ap[C, f`x, f`y] g (ap[B, x, y] f p) =[g`(f`x) =[C] g`(f`y)] + ap[C, x, y] (g o[A] f) p") proof - show "\<And>x. x: A \<Longrightarrow> refl(refl (g`(f`x))) : - ap C g (f`x) (f`x) (ap B f x x (refl x)) =[g`(f`x) =[C] g`(f`x)] - ap C (compose A g f) x x (refl x)" + ap[C, f`x, f`x] g (ap[B, x, x] f (refl x)) =[g`(f`x) =[C] g`(f`x)] + ap[C, x, x] (g o[A] f) (refl x)" unfolding compose_def by derive fix x y p assume [intro]: "x: A" "y: A" "p: x =[A] y" - show "ap C g (f`x) (f`y) (ap B f x y p) =[g`(f`x) =[C] g`(f`y)] ap C (compose A g f) x y p: U i" + show "ap[C, f`x, f`y] g (ap[B, x, y] f p) =[g`(f`x) =[C] g`(f`y)] ap[C, x, y] (g o[A] f) p: U i" proof have - "(\<lambda>x: A. g`(f`x))`x =[C] (\<lambda>x: A. g`(f`x))`y \<equiv> g`(f`x) =[C] g`(f`y)" by derive - moreover have - "ap C (compose A g f) x y p : (\<lambda>x: A. g`(f`x))`x =[C] (\<lambda>x: A. g`(f`x))`y" + "ap[C, x, y] (g o[A] f) p: (\<lambda>x: A. g`(f`x))`x =[C] (\<lambda>x: A. g`(f`x))`y" unfolding compose_def by derive + moreover have + "(\<lambda>x: A. g`(f`x))`x =[C] (\<lambda>x: A. g`(f`x))`y \<equiv> g`(f`x) =[C] g`(f`y)" by derive ultimately show - "ap C (compose A g f) x y p : g`(f`x) =[C] g`(f`y)" by simp + "ap[C, x, y] (g o[A] f) p: g`(f`x) =[C] g`(f`y)" by simp qed derive qed routine - - +declare[[pretty_inv=false, pretty_ap=false]] schematic_goal ap_func_inv: assumes [intro]: "A: U i" "B: U i" "f: A \<rightarrow> B" "x: A" "y: A" "p: x =[A] y" - shows "?prf: ap B f y x (inv A x y p) =[f`y =[B] f`x] inv B (f`x) (f`y) (ap B f x y p)" + shows "?prf: ap[B, y, x] f (inv[A, x, y] p) =[f`y =[B] f`x] inv[B, f`x, f`y] (ap[B, x, y] f p)" proof (path_ind' x y p) show "\<And>x. x: A \<Longrightarrow> refl(refl(f`x)): - ap B f x x (inv A x x (refl x)) =[f`x =[B] f`x] inv B (f`x) (f`x) (ap B f x x (refl x))" + ap[B, x, x] f (inv[A, x, x] (refl x)) =[f`x =[B] f`x] inv[B, f`x, f`x] (ap[B, x, x] f (refl x))" by derive qed routine @@ -355,14 +364,14 @@ schematic_goal ap_func_id: shows "?prf: ap A (id A) x y p =[x =[A] y] p" proof (path_ind' x y p) fix x assume [intro]: "x: A" - show "refl(refl x): ap A (id A) x x (refl x) =[x =[A] x] refl x" by derive + show "refl(refl x): ap[A, x, x] (id A) (refl x) =[x =[A] x] refl x" by derive fix y p assume [intro]: "y: A" "p: x =[A] y" - have "ap A (id A) x y p: (id A)`x =[A] (id A)`y" by derive + have "ap[A, x, y] (id A) p: (id A)`x =[A] (id A)`y" by derive moreover have "(id A)`x =[A] (id A)`y \<equiv> x =[A] y" by derive - ultimately have [intro]: "ap A (id A) x y p: x =[A] y" by simp + ultimately have [intro]: "ap[A, x, y] (id A) p: x =[A] y" by simp - show "ap A (id A) x y p =[x =[A] y] p: U i" by derive + show "ap[A, x, y] (id A) p =[x =[A] y] p: U i" by derive qed @@ -374,15 +383,29 @@ schematic_goal transport: "x: A" "y: A" "p: x =[A] y" shows "?prf: P x \<rightarrow> P y" proof (path_ind' x y p) - show "\<And>x. x: A \<Longrightarrow> id (P x): P x \<rightarrow> P x" by derive + show "\<And>x. x: A \<Longrightarrow> id(P x): P x \<rightarrow> P x" by derive qed routine -definition transport :: "[t \<Rightarrow> t, t, t, t] \<Rightarrow> t" -where "transport P x y p \<equiv> indEq (\<lambda>a b _. P a \<rightarrow> P b) (\<lambda>x. id (P x)) x y p" +definition transport :: "[t \<Rightarrow> t, t, t, t] \<Rightarrow> t" ("(2transport[_, _, _] _)" [0, 0, 0, 1000]) +where "transport[P, x, y] p \<equiv> indEq (\<lambda>a b _. P a \<rightarrow> P b) (\<lambda>x. id (P x)) x y p" + +syntax "_transport'" :: "[t, t] \<Rightarrow> t" ("(2_*)" [1000]) + +ML \<open>val pretty_transport = Attrib.setup_config_bool @{binding "pretty_transport"} (K true)\<close> + +print_translation \<open> +let fun transport_tr' ctxt [P, x, y, p] = + if Config.get ctxt pretty_transport + then Syntax.const @{syntax_const "_transport'"} $ p + else @{const transport} $ P $ x $ y $ p +in + [(@{const_syntax transport}, transport_tr')] +end +\<close> lemma transport_type: assumes "A: U i" "P: A \<leadsto> U i" "x: A" "y: A" "p: x =[A] y" - shows "transport P x y p: P x \<rightarrow> P y" + shows "transport[P, x, y] p: P x \<rightarrow> P y" unfolding transport_def using assms by (rule transport) lemma transport_comp: |