diff options
Diffstat (limited to '')
| -rw-r--r-- | Coprod.thy | 42 | ||||
| -rw-r--r-- | Empty.thy | 8 | ||||
| -rw-r--r-- | Equal.thy | 32 | ||||
| -rw-r--r-- | EqualProps.thy | 135 | ||||
| -rw-r--r-- | HoTT_Base.thy | 87 | ||||
| -rw-r--r-- | HoTT_Methods.thy | 6 | ||||
| -rw-r--r-- | LICENSE | 165 | ||||
| -rw-r--r-- | Nat.thy | 36 | ||||
| -rw-r--r-- | Prod.thy | 42 | ||||
| -rw-r--r-- | ProdProps.thy | 19 | ||||
| -rw-r--r-- | Proj.thy | 28 | ||||
| -rw-r--r-- | README.md | 10 | ||||
| -rw-r--r-- | Sum.thy | 38 | ||||
| -rw-r--r-- | Unit.thy | 6 | ||||
| -rw-r--r-- | Univalence.thy | 176 | ||||
| -rw-r--r-- | ex/HoTT book/Ch1.thy | 12 |
16 files changed, 635 insertions, 207 deletions
@@ -12,45 +12,45 @@ begin section \<open>Constants and type rules\<close> axiomatization - Coprod :: "[Term, Term] \<Rightarrow> Term" (infixr "+" 50) and - inl :: "Term \<Rightarrow> Term" and - inr :: "Term \<Rightarrow> Term" and - indCoprod :: "[Term \<Rightarrow> Term, Term \<Rightarrow> Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>+)") + Coprod :: "[t, t] \<Rightarrow> t" (infixr "+" 50) and + inl :: "t \<Rightarrow> t" and + inr :: "t \<Rightarrow> t" and + indCoprod :: "[t \<Rightarrow> t, t \<Rightarrow> t, t] \<Rightarrow> t" ("(1ind\<^sub>+)") where - Coprod_form: "\<lbrakk>A: U(i); B: U(i)\<rbrakk> \<Longrightarrow> A + B: U(i)" + Coprod_form: "\<lbrakk>A: U i; B: U i\<rbrakk> \<Longrightarrow> A + B: U i" and - Coprod_intro_inl: "\<lbrakk>a: A; B: U(i)\<rbrakk> \<Longrightarrow> inl(a): A + B" + Coprod_intro_inl: "\<lbrakk>a: A; B: U i\<rbrakk> \<Longrightarrow> inl a: A + B" and - Coprod_intro_inr: "\<lbrakk>b: B; A: U(i)\<rbrakk> \<Longrightarrow> inr(b): A + B" + Coprod_intro_inr: "\<lbrakk>b: B; A: U i\<rbrakk> \<Longrightarrow> inr b: A + B" and Coprod_elim: "\<lbrakk> - C: A + B \<longrightarrow> U(i); - \<And>x. x: A \<Longrightarrow> c(x): C(inl(x)); - \<And>y. y: B \<Longrightarrow> d(y): C(inr(y)); + C: A + B \<longrightarrow> U i; + \<And>x. x: A \<Longrightarrow> c x: C (inl x); + \<And>y. y: B \<Longrightarrow> d y: C (inr y); u: A + B - \<rbrakk> \<Longrightarrow> ind\<^sub>+(c)(d)(u) : C(u)" + \<rbrakk> \<Longrightarrow> ind\<^sub>+ c d u: C u" and Coprod_comp_inl: "\<lbrakk> - C: A + B \<longrightarrow> U(i); - \<And>x. x: A \<Longrightarrow> c(x): C(inl(x)); - \<And>y. y: B \<Longrightarrow> d(y): C(inr(y)); + C: A + B \<longrightarrow> U i; + \<And>x. x: A \<Longrightarrow> c x: C (inl x); + \<And>y. y: B \<Longrightarrow> d y: C (inr y); a: A - \<rbrakk> \<Longrightarrow> ind\<^sub>+(c)(d)(inl(a)) \<equiv> c(a)" + \<rbrakk> \<Longrightarrow> ind\<^sub>+ c d (inl a) \<equiv> c a" and Coprod_comp_inr: "\<lbrakk> - C: A + B \<longrightarrow> U(i); - \<And>x. x: A \<Longrightarrow> c(x): C(inl(x)); - \<And>y. y: B \<Longrightarrow> d(y): C(inr(y)); + C: A + B \<longrightarrow> U i; + \<And>x. x: A \<Longrightarrow> c x: C (inl x); + \<And>y. y: B \<Longrightarrow> d y: C (inr y); b: B - \<rbrakk> \<Longrightarrow> ind\<^sub>+(c)(d)(inr(b)) \<equiv> d(b)" + \<rbrakk> \<Longrightarrow> ind\<^sub>+ c d (inr b) \<equiv> d b" text "Admissible formation inference rules:" axiomatization where - Coprod_wellform1: "A + B: U(i) \<Longrightarrow> A: U(i)" + Coprod_wellform1: "A + B: U i \<Longrightarrow> A: U i" and - Coprod_wellform2: "A + B: U(i) \<Longrightarrow> B: U(i)" + Coprod_wellform2: "A + B: U i \<Longrightarrow> B: U i" text "Rule attribute declarations:" @@ -14,12 +14,12 @@ section \<open>Constants and type rules\<close> section \<open>Empty type\<close> axiomatization - Empty :: Term ("\<zero>") and - indEmpty :: "Term \<Rightarrow> Term" ("(1ind\<^sub>\<zero>)") + Empty :: t ("\<zero>") and + indEmpty :: "t \<Rightarrow> t" ("(1ind\<^sub>\<zero>)") where - Empty_form: "\<zero> : U(O)" + Empty_form: "\<zero>: U O" and - Empty_elim: "\<lbrakk>C: \<zero> \<longrightarrow> U(i); z: \<zero>\<rbrakk> \<Longrightarrow> ind\<^sub>\<zero>(z): C(z)" + Empty_elim: "\<lbrakk>C: \<zero> \<longrightarrow> U i; z: \<zero>\<rbrakk> \<Longrightarrow> ind\<^sub>\<zero> z: C z" text "Rule attribute declarations:" @@ -12,13 +12,13 @@ begin section \<open>Constants and syntax\<close> axiomatization - Equal :: "[Term, Term, Term] \<Rightarrow> Term" and - refl :: "Term \<Rightarrow> Term" and - indEqual :: "[Term \<Rightarrow> Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>=)") + Equal :: "[t, t, t] \<Rightarrow> t" and + refl :: "t \<Rightarrow> t" and + indEqual :: "[t \<Rightarrow> t, t] \<Rightarrow> t" ("(1ind\<^sub>=)") syntax - "_EQUAL" :: "[Term, Term, Term] \<Rightarrow> Term" ("(3_ =\<^sub>_/ _)" [101, 0, 101] 100) - "_EQUAL_ASCII" :: "[Term, Term, Term] \<Rightarrow> Term" ("(3_ =[_]/ _)" [101, 0, 101] 100) + "_EQUAL" :: "[t, t, t] \<Rightarrow> t" ("(3_ =\<^sub>_/ _)" [101, 0, 101] 100) + "_EQUAL_ASCII" :: "[t, t, t] \<Rightarrow> t" ("(3_ =[_]/ _)" [101, 0, 101] 100) translations "a =[A] b" \<rightleftharpoons> "CONST Equal A a b" "a =\<^sub>A b" \<rightharpoonup> "CONST Equal A a b" @@ -27,33 +27,33 @@ translations section \<open>Type rules\<close> axiomatization where - Equal_form: "\<lbrakk>A: U(i); a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U(i)" + Equal_form: "\<lbrakk>A: U i; a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U i" and - Equal_intro: "a : A \<Longrightarrow> refl(a): a =\<^sub>A a" + Equal_intro: "a : A \<Longrightarrow> (refl a): a =\<^sub>A a" and Equal_elim: "\<lbrakk> x: A; y: A; p: x =\<^sub>A y; - \<And>x. x: A \<Longrightarrow> f(x) : C(x)(x)(refl x); - \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C(x)(y): x =\<^sub>A y \<longrightarrow> U(i) - \<rbrakk> \<Longrightarrow> ind\<^sub>=(f)(p) : C(x)(y)(p)" + \<And>x. x: A \<Longrightarrow> f x: C x x (refl x); + \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U i + \<rbrakk> \<Longrightarrow> ind\<^sub>= f p : C x y p" and Equal_comp: "\<lbrakk> a: A; - \<And>x. x: A \<Longrightarrow> f(x) : C(x)(x)(refl x); - \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C(x)(y): x =\<^sub>A y \<longrightarrow> U(i) - \<rbrakk> \<Longrightarrow> ind\<^sub>=(f)(refl(a)) \<equiv> f(a)" + \<And>x. x: A \<Longrightarrow> f x: C x x (refl x); + \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U i + \<rbrakk> \<Longrightarrow> ind\<^sub>= f (refl a) \<equiv> f a" text "Admissible inference rules for equality type formation:" axiomatization where - Equal_wellform1: "a =\<^sub>A b: U(i) \<Longrightarrow> A: U(i)" + Equal_wellform1: "a =\<^sub>A b: U i \<Longrightarrow> A: U i" and - Equal_wellform2: "a =\<^sub>A b: U(i) \<Longrightarrow> a: A" + Equal_wellform2: "a =\<^sub>A b: U i \<Longrightarrow> a: A" and - Equal_wellform3: "a =\<^sub>A b: U(i) \<Longrightarrow> b: A" + Equal_wellform3: "a =\<^sub>A b: U i \<Longrightarrow> b: A" text "Rule attribute declarations:" diff --git a/EqualProps.thy b/EqualProps.thy index be8e53e..19c788c 100644 --- a/EqualProps.thy +++ b/EqualProps.thy @@ -14,7 +14,7 @@ begin section \<open>Symmetry / Path inverse\<close> -definition inv :: "Term \<Rightarrow> Term" ("_\<inverse>" [1000] 1000) where "p\<inverse> \<equiv> ind\<^sub>= (\<lambda>x. refl(x)) p" +definition inv :: "t \<Rightarrow> t" ("_\<inverse>" [1000] 1000) where "p\<inverse> \<equiv> ind\<^sub>= (\<lambda>x. (refl x)) p" text " In the proof below we begin by using path induction on \<open>p\<close> with the application of \<open>rule Equal_elim\<close>, telling Isabelle the specific substitutions to use. @@ -22,7 +22,7 @@ text " " lemma inv_type: - assumes "A : U(i)" and "x : A" and "y : A" and "p: x =\<^sub>A y" shows "p\<inverse>: y =\<^sub>A x" + assumes "A : U i" and "x : A" and "y : A" and "p: x =\<^sub>A y" shows "p\<inverse>: y =\<^sub>A x" unfolding inv_def by (rule Equal_elim[where ?x=x and ?y=y]) (routine lems: assms) \<comment> \<open>The type doesn't depend on \<open>p\<close> so we don't need to specify \<open>?p\<close> in the \<open>where\<close> clause above.\<close> @@ -33,7 +33,7 @@ text " " lemma inv_comp: - assumes "A : U(i)" and "a : A" shows "(refl(a))\<inverse> \<equiv> refl(a)" + assumes "A : U i " and "a : A" shows "(refl a)\<inverse> \<equiv> refl a" unfolding inv_def proof compute show "\<And>x. x: A \<Longrightarrow> refl x: x =\<^sub>A x" .. @@ -46,14 +46,14 @@ text " Raw composition function, of type \<open>\<Prod>x:A. \<Prod>y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)\<close> polymorphic over the type \<open>A\<close>. " -definition rpathcomp :: Term where "rpathcomp \<equiv> \<^bold>\<lambda>_ _ p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>_ q. ind\<^sub>= (\<lambda>x. refl(x)) q) p" +definition rpathcomp :: t where "rpathcomp \<equiv> \<^bold>\<lambda>_ _ p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>_ q. ind\<^sub>= (\<lambda>x. (refl x)) q) p" text " More complicated proofs---the nested path inductions require more explicit step-by-step rule applications: " lemma rpathcomp_type: - assumes "A: U(i)" + assumes "A: U i" shows "rpathcomp: \<Prod>x:A. \<Prod>y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)" unfolding rpathcomp_def proof @@ -81,7 +81,7 @@ proof qed fact corollary - assumes "A: U(i)" "x: A" "y: A" "z: A" "p: x =\<^sub>A y" "q: y =\<^sub>A z" + assumes "A: U i" "x: A" "y: A" "z: A" "p: x =\<^sub>A y" "q: y =\<^sub>A z" shows "rpathcomp`x`y`p`z`q: x =\<^sub>A z" by (routine lems: assms rpathcomp_type) @@ -90,11 +90,11 @@ text " " lemma rpathcomp_comp: - assumes "A: U(i)" and "a: A" - shows "rpathcomp`a`a`refl(a)`a`refl(a) \<equiv> refl(a)" + assumes "A: U i" and "a: A" + shows "rpathcomp`a`a`(refl a)`a`(refl a) \<equiv> refl a" unfolding rpathcomp_def proof compute - { fix x assume 1: "x: A" + fix x assume 1: "x: A" show "\<^bold>\<lambda>y p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p: \<Prod>y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)" proof fix y assume 2: "y: A" @@ -114,11 +114,11 @@ proof compute qed (rule assms) qed (routine lems: assms 1 2 3) qed (routine lems: assms 1 2) - qed (rule assms) } + qed (rule assms) - show "(\<^bold>\<lambda>y p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p)`a`refl(a)`a`refl(a) \<equiv> refl(a)" + next show "(\<^bold>\<lambda>y p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p)`a`(refl a)`a`(refl a) \<equiv> refl a" proof compute - { fix y assume 1: "y: A" + fix y assume 1: "y: A" show "\<^bold>\<lambda>p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p: a =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> a =\<^sub>A z)" proof fix p assume 2: "p: a =\<^sub>A y" @@ -135,11 +135,11 @@ proof compute qed (routine lems: assms 3 4) qed fact qed (routine lems: assms 1 2) - qed (routine lems: assms 1) } + qed (routine lems: assms 1) - show "(\<^bold>\<lambda>p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z. \<^bold>\<lambda>q. ind\<^sub>= refl q) p)`refl(a)`a`refl(a) \<equiv> refl(a)" + next show "(\<^bold>\<lambda>p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z. \<^bold>\<lambda>q. ind\<^sub>= refl q) p)`(refl a)`a`(refl a) \<equiv> refl a" proof compute - { fix p assume 1: "p: a =\<^sub>A a" + fix p assume 1: "p: a =\<^sub>A a" show "ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p: \<Prod>z:A. a =\<^sub>A z \<rightarrow> a =\<^sub>A z" proof (rule Equal_elim[where ?x=a and ?y=a]) fix u assume 2: "u: A" @@ -152,11 +152,11 @@ proof compute by (rule Equal_elim[where ?x=u and ?y=z]) (routine lems: assms 2 3) qed (routine lems: assms 2 3) qed fact - qed (routine lems: assms 1) } + qed (routine lems: assms 1) - show "(ind\<^sub>=(\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q)(refl(a)))`a`refl(a) \<equiv> refl(a)" + next show "(ind\<^sub>=(\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q)(refl a))`a`(refl a) \<equiv> refl a" proof compute - { fix u assume 1: "u: A" + fix u assume 1: "u: A" show "\<^bold>\<lambda>z q. ind\<^sub>= refl q: \<Prod>z:A. u =\<^sub>A z \<rightarrow> u =\<^sub>A z" proof fix z assume 2: "z: A" @@ -165,22 +165,22 @@ proof compute show "\<And>q. q: u =\<^sub>A z \<Longrightarrow> ind\<^sub>= refl q: u =\<^sub>A z" by (rule Equal_elim[where ?x=u and ?y=z]) (routine lems: assms 1 2) qed (routine lems: assms 1 2) - qed fact } + qed fact - show "(\<^bold>\<lambda>z q. ind\<^sub>= refl q)`a`refl(a) \<equiv> refl(a)" + next show "(\<^bold>\<lambda>z q. ind\<^sub>= refl q)`a`(refl a) \<equiv> refl a" proof compute - { fix a assume 1: "a: A" + fix a assume 1: "a: A" show "\<^bold>\<lambda>q. ind\<^sub>= refl q: a =\<^sub>A a \<rightarrow> a =\<^sub>A a" proof show "\<And>q. q: a =\<^sub>A a \<Longrightarrow> ind\<^sub>= refl q: a =\<^sub>A a" by (rule Equal_elim[where ?x=a and ?y=a]) (routine lems: assms 1) - qed (routine lems: assms 1) } + qed (routine lems: assms 1) - show "(\<^bold>\<lambda>q. ind\<^sub>= refl q)`refl(a) \<equiv> refl(a)" + next show "(\<^bold>\<lambda>q. ind\<^sub>= refl q)`(refl a) \<equiv> refl a" proof compute show "\<And>p. p: a =\<^sub>A a \<Longrightarrow> ind\<^sub>= refl p: a =\<^sub>A a" by (rule Equal_elim[where ?x=a and ?y=a]) (routine lems: assms) - show "ind\<^sub>= refl (refl(a)) \<equiv> refl(a)" + show "ind\<^sub>= refl (refl a) \<equiv> refl a" proof compute show "\<And>x. x: A \<Longrightarrow> refl(x): x =\<^sub>A x" .. qed (routine lems: assms) @@ -194,23 +194,25 @@ qed fact text "The raw object lambda term is cumbersome to use, so we define a simpler constant instead." -axiomatization pathcomp :: "[Term, Term] \<Rightarrow> Term" (infixl "\<bullet>" 120) where +axiomatization pathcomp :: "[t, t] \<Rightarrow> t" (infixl "\<bullet>" 120) where pathcomp_def: "\<lbrakk> - A: U(i); + A: U i; x: A; y: A; z: A; p: x =\<^sub>A y; q: y =\<^sub>A z \<rbrakk> \<Longrightarrow> p \<bullet> q \<equiv> rpathcomp`x`y`p`z`q" lemma pathcomp_type: - assumes "A: U(i)" "x: A" "y: A" "z: A" "p: x =\<^sub>A y" "q: y =\<^sub>A z" + assumes "A: U i" "x: A" "y: A" "z: A" "p: x =\<^sub>A y" "q: y =\<^sub>A z" shows "p \<bullet> q: x =\<^sub>A z" + proof (subst pathcomp_def) - show "A: U(i)" "x: A" "y: A" "z: A" "p: x =\<^sub>A y" "q: y =\<^sub>A z" by fact+ + show "A: U i" "x: A" "y: A" "z: A" "p: x =\<^sub>A y" "q: y =\<^sub>A z" by fact+ qed (routine lems: assms rpathcomp_type) + lemma pathcomp_comp: - assumes "A : U(i)" and "a : A" shows "refl(a) \<bullet> refl(a) \<equiv> refl(a)" + assumes "A : U i" and "a : A" shows "(refl a) \<bullet> (refl a) \<equiv> refl a" by (subst pathcomp_def) (routine lems: assms rpathcomp_comp) @@ -281,4 +283,79 @@ proof (rule Equal_elim[where ?x=x and ?y=y and ?p=p]) proof (compute lems: whg1b) +section \<open>Higher groupoid structure of types\<close> + +lemma + assumes "A: U i" "x: A" "y: A" "p: x =\<^sub>A y" + shows + "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p =[x =\<^sub>A y] p \<bullet> (refl y)" and + "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p =[x =\<^sub>A y] (refl x) \<bullet> p" + +proof - + show "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p =[x =[A] y] p \<bullet> (refl y)" + by (rule Equal_elim[where ?p=p and ?x=x and ?y=y]) (derive lems: assms)+ + + show "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p =[x =[A] y] (refl x) \<bullet> p" + by (rule Equal_elim[where ?p=p and ?x=x and ?y=y]) (derive lems: assms)+ +qed + + +lemma + assumes "A: U i" "x: A" "y: A" "p: x =\<^sub>A y" + shows + "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p\<inverse> \<bullet> p =[y =\<^sub>A y] (refl y)" and + "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p \<bullet> p\<inverse> =[x =\<^sub>A x] (refl x)" + +proof - + show "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p\<inverse> \<bullet> p =[y =\<^sub>A y] (refl y)" + by (rule Equal_elim[where ?p=p and ?x=x and ?y=y]) (derive lems: assms)+ + + show "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p \<bullet> p\<inverse> =[x =\<^sub>A x] (refl x)" + by (rule Equal_elim[where ?p=p and ?x=x and ?y=y]) (derive lems: assms)+ +qed + + +lemma + assumes "A: U i" "x: A" "y: A" "p: x =\<^sub>A y" + shows "ind\<^sub>= (\<lambda>u. refl (refl u)) p: p\<inverse>\<inverse> =[x =\<^sub>A y] p" +by (rule Equal_elim[where ?p=p and ?x=x and ?y=y]) (derive lems: assms) + +text "Next we construct a proof term of associativity of path composition." + +schematic_goal + assumes + "A: U i" + "x: A" "y: A" "z: A" "w: A" + "p: x =\<^sub>A y" "q: y =\<^sub>A z" "r: z =\<^sub>A w" + shows + "?a: p \<bullet> (q \<bullet> r) =[x =\<^sub>A z] (p \<bullet> q) \<bullet> r" + +apply (rule Equal_elim[where ?p=p and ?x=x and ?y=y]) +apply (rule assms)+ +\<comment> \<open>Continue by substituting \<open>refl x \<bullet> q = q\<close> etc.\<close> +sorry + + +section \<open>Transport\<close> + +definition transport :: "t \<Rightarrow> t" where + "transport p \<equiv> ind\<^sub>= (\<lambda>x. (\<^bold>\<lambda>x. x)) p" + +text "Note that \<open>transport\<close> is a polymorphic function in our formulation." + +lemma transport_type: + assumes + "A: U i" "P: A \<longrightarrow> U i" + "x: A" "y: A" + "p: x =\<^sub>A y" + shows "transport p: P x \<rightarrow> P y" +unfolding transport_def +by (rule Equal_elim[where ?p=p and ?x=x and ?y=y]) (routine lems: assms) + +lemma transport_comp: + assumes "A: U i" and "x: A" + shows "transport (refl x) \<equiv> id" +unfolding transport_def by (derive lems: assms) + + end diff --git a/HoTT_Base.thy b/HoTT_Base.thy index 8ea767f..07fbfc4 100644 --- a/HoTT_Base.thy +++ b/HoTT_Base.thy @@ -1,84 +1,77 @@ (* Title: HoTT/HoTT_Base.thy Author: Joshua Chen -Basic setup and definitions of a homotopy type theory object logic with a cumulative universe hierarchy à la Russell. +Basic setup of a homotopy type theory object logic with a Russell-style cumulative universe hierarchy. *) theory HoTT_Base -imports Pure +imports + Pure + "HOL-Eisbach.Eisbach" begin -section \<open>Foundational definitions\<close> +section \<open>Basic setup\<close> -text "Meta syntactic type for object-logic types and terms." - -typedecl t +typedecl t \<comment> \<open>Type of object types and terms\<close> +typedecl ord \<comment> \<open>Type of meta-level numerals\<close> +axiomatization + O :: ord and + Suc :: "ord \<Rightarrow> ord" and + lt :: "[ord, ord] \<Rightarrow> prop" (infix "<" 999) +where + lt_Suc [intro]: "n < (Suc n)" and + lt_trans [intro]: "\<lbrakk>m1 < m2; m2 < m3\<rbrakk> \<Longrightarrow> m1 < m3" and + Suc_monotone [simp]: "m < n \<Longrightarrow> (Suc m) < (Suc n)" -section \<open>Judgments\<close> +method proveSuc = (rule lt_Suc | (rule lt_trans, (rule lt_Suc)+)+) -text " - Formalize the typing judgment \<open>a: A\<close>. - For judgmental/definitional equality we use the existing Pure equality \<open>\<equiv>\<close> and hence do not need to define a separate judgment for it. -" +text \<open>Method @{method proveSuc} proves statements of the form \<open>n < (Suc (... (Suc n) ...))\<close>.\<close> -judgment hastype :: "[t, t] \<Rightarrow> prop" ("(3_:/ _)") +section \<open>Judgment\<close> -section \<open>Universe hierarchy\<close> +judgment hastype :: "[t, t] \<Rightarrow> prop" ("(3_:/ _)") -text "Meta-numerals to index the universes." -typedecl ord +section \<open>Universes\<close> axiomatization - O :: ord and - S :: "ord \<Rightarrow> ord" ("S_") and - lt :: "[ord, ord] \<Rightarrow> prop" (infix "<-" 999) + U :: "ord \<Rightarrow> t" where - Ord_min: "\<And>n. O <- S(n)" -and - Ord_monotone: "\<And>m n. m <- n \<Longrightarrow> S(m) <- S(n)" + U_hierarchy: "i < j \<Longrightarrow> U i: U j" and + U_cumulative: "\<lbrakk>A: U i; i < j\<rbrakk> \<Longrightarrow> A: U j" -lemmas Ord_rules [intro] = Ord_min Ord_monotone - \<comment> \<open>Enables \<open>standard\<close> to automatically solve inequalities.\<close> +text \<open> +Using method @{method rule} with @{thm U_cumulative} is unsafe, if applied blindly it will typically lead to non-termination. +One should instead use @{method elim}, or instantiate @{thm U_cumulative} suitably. +\<close> -text "Define the universe types." - -axiomatization - U :: "Ord \<Rightarrow> Term" -where - U_hierarchy: "\<And>i j. i <- j \<Longrightarrow> U(i): U(j)" -and - U_cumulative: "\<And>A i j. \<lbrakk>A: U(i); i <- j\<rbrakk> \<Longrightarrow> A: U(j)" - \<comment> \<open>WARNING: \<open>rule Universe_cumulative\<close> can result in an infinite rewrite loop!\<close> +method cumulativity = (elim U_cumulative, proveSuc) \<comment> \<open>Proves \<open>A: U i \<Longrightarrow> A: U (Suc (... (Suc i) ...))\<close>.\<close> +method hierarchy = (rule U_hierarchy, proveSuc) \<comment> \<open>Proves \<open>U i: U (Suc (... (Suc i) ...)).\<close>\<close> section \<open>Type families\<close> -text " - The following abbreviation constrains the output type of a meta lambda expression when given input of certain type. -" - -abbreviation (input) constrained :: "[Term \<Rightarrow> Term, Term, Term] \<Rightarrow> prop" ("(1_: _ \<longrightarrow> _)") - where "f: A \<longrightarrow> B \<equiv> (\<And>x. x : A \<Longrightarrow> f(x): B)" +abbreviation (input) constrained :: "[t \<Rightarrow> t, t, t] \<Rightarrow> prop" ("(1_:/ _ \<longrightarrow> _)") + where "f: A \<longrightarrow> B \<equiv> (\<And>x. x : A \<Longrightarrow> f x: B)" -text " - The above is used to define type families, which are constrained meta-lambdas \<open>P: A \<longrightarrow> B\<close> where \<open>A\<close> and \<open>B\<close> are small types. -" +text \<open> +The abbreviation @{abbrev constrained} is used to define type families, which are constrained expressions @{term "P: A \<longrightarrow> B"} where @{term "A::t"} and @{term "B::t"} are small types. +\<close> -type_synonym Typefam = "Term \<Rightarrow> Term" +type_synonym tf = "t \<Rightarrow> t" \<comment> \<open>Type of type families.\<close> section \<open>Named theorems\<close> -text " - Named theorems to be used by proof methods later (see HoTT_Methods.thy). - - \<open>wellform\<close> declares necessary wellformedness conditions for type and inhabitation judgments, while \<open>comp\<close> declares computation rules, which are usually passed to invocations of the method \<open>subst\<close> to perform equational rewriting. -" +text \<open> +Declare named theorems to be used by proof methods defined in @{file HoTT_Methods.thy}. +\<open>wellform\<close> declares necessary well-formedness conditions for type and inhabitation judgments. +\<open>comp\<close> declares computation rules, which are usually passed to invocations of the method \<open>subst\<close> to perform equational rewriting. +\<close> named_theorems wellform named_theorems comp diff --git a/HoTT_Methods.thy b/HoTT_Methods.thy index 32e412b..abb6dda 100644 --- a/HoTT_Methods.thy +++ b/HoTT_Methods.thy @@ -14,12 +14,12 @@ begin section \<open>Deriving typing judgments\<close> +method routine uses lems = (assumption | rule lems | standard)+ + text " - \<open>routine\<close> proves routine type judgments \<open>a : A\<close> using the rules declared [intro] in the respective theory files, as well as additional provided lemmas. + @{method routine} proves routine type judgments \<open>a : A\<close> using the rules declared [intro] in the respective theory files, as well as additional provided lemmas. " -method routine uses lems = (assumption | rule lems | standard)+ - text " \<open>wellformed'\<close> finds a proof of any valid typing judgment derivable from the judgment passed as \<open>jdmt\<close>. If no judgment is passed, it will try to resolve with the theorems declared \<open>wellform\<close>. @@ -0,0 +1,165 @@ + GNU LESSER GENERAL PUBLIC LICENSE + Version 3, 29 June 2007 + + Copyright (C) 2007 Free Software Foundation, Inc. <http://fsf.org/> + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + + This version of the GNU Lesser General Public License incorporates +the terms and conditions of version 3 of the GNU General Public +License, supplemented by the additional permissions listed below. + + 0. Additional Definitions. + + As used herein, "this License" refers to version 3 of the GNU Lesser +General Public License, and the "GNU GPL" refers to version 3 of the GNU +General Public License. + + "The Library" refers to a covered work governed by this License, +other than an Application or a Combined Work as defined below. + + An "Application" is any work that makes use of an interface provided +by the Library, but which is not otherwise based on the Library. +Defining a subclass of a class defined by the Library is deemed a mode +of using an interface provided by the Library. + + A "Combined Work" is a work produced by combining or linking an +Application with the Library. The particular version of the Library +with which the Combined Work was made is also called the "Linked +Version". + + The "Minimal Corresponding Source" for a Combined Work means the +Corresponding Source for the Combined Work, excluding any source code +for portions of the Combined Work that, considered in isolation, are +based on the Application, and not on the Linked Version. + + The "Corresponding Application Code" for a Combined Work means the +object code and/or source code for the Application, including any data +and utility programs needed for reproducing the Combined Work from the +Application, but excluding the System Libraries of the Combined Work. + + 1. Exception to Section 3 of the GNU GPL. + + You may convey a covered work under sections 3 and 4 of this License +without being bound by section 3 of the GNU GPL. + + 2. Conveying Modified Versions. + + If you modify a copy of the Library, and, in your modifications, a +facility refers to a function or data to be supplied by an Application +that uses the facility (other than as an argument passed when the +facility is invoked), then you may convey a copy of the modified +version: + + a) under this License, provided that you make a good faith effort to + ensure that, in the event an Application does not supply the + function or data, the facility still operates, and performs + whatever part of its purpose remains meaningful, or + + b) under the GNU GPL, with none of the additional permissions of + this License applicable to that copy. + + 3. Object Code Incorporating Material from Library Header Files. + + The object code form of an Application may incorporate material from +a header file that is part of the Library. You may convey such object +code under terms of your choice, provided that, if the incorporated +material is not limited to numerical parameters, data structure +layouts and accessors, or small macros, inline functions and templates +(ten or fewer lines in length), you do both of the following: + + a) Give prominent notice with each copy of the object code that the + Library is used in it and that the Library and its use are + covered by this License. + + b) Accompany the object code with a copy of the GNU GPL and this license + document. + + 4. Combined Works. + + You may convey a Combined Work under terms of your choice that, +taken together, effectively do not restrict modification of the +portions of the Library contained in the Combined Work and reverse +engineering for debugging such modifications, if you also do each of +the following: + + a) Give prominent notice with each copy of the Combined Work that + the Library is used in it and that the Library and its use are + covered by this License. + + b) Accompany the Combined Work with a copy of the GNU GPL and this license + document. + + c) For a Combined Work that displays copyright notices during + execution, include the copyright notice for the Library among + these notices, as well as a reference directing the user to the + copies of the GNU GPL and this license document. + + d) Do one of the following: + + 0) Convey the Minimal Corresponding Source under the terms of this + License, and the Corresponding Application Code in a form + suitable for, and under terms that permit, the user to + recombine or relink the Application with a modified version of + the Linked Version to produce a modified Combined Work, in the + manner specified by section 6 of the GNU GPL for conveying + Corresponding Source. + + 1) Use a suitable shared library mechanism for linking with the + Library. A suitable mechanism is one that (a) uses at run time + a copy of the Library already present on the user's computer + system, and (b) will operate properly with a modified version + of the Library that is interface-compatible with the Linked + Version. + + e) Provide Installation Information, but only if you would otherwise + be required to provide such information under section 6 of the + GNU GPL, and only to the extent that such information is + necessary to install and execute a modified version of the + Combined Work produced by recombining or relinking the + Application with a modified version of the Linked Version. (If + you use option 4d0, the Installation Information must accompany + the Minimal Corresponding Source and Corresponding Application + Code. If you use option 4d1, you must provide the Installation + Information in the manner specified by section 6 of the GNU GPL + for conveying Corresponding Source.) + + 5. Combined Libraries. + + You may place library facilities that are a work based on the +Library side by side in a single library together with other library +facilities that are not Applications and are not covered by this +License, and convey such a combined library under terms of your +choice, if you do both of the following: + + a) Accompany the combined library with a copy of the same work based + on the Library, uncombined with any other library facilities, + conveyed under the terms of this License. + + b) Give prominent notice with the combined library that part of it + is a work based on the Library, and explaining where to find the + accompanying uncombined form of the same work. + + 6. Revised Versions of the GNU Lesser General Public License. + + The Free Software Foundation may publish revised and/or new versions +of the GNU Lesser General Public License from time to time. Such new +versions will be similar in spirit to the present version, but may +differ in detail to address new problems or concerns. + + Each version is given a distinguishing version number. If the +Library as you received it specifies that a certain numbered version +of the GNU Lesser General Public License "or any later version" +applies to it, you have the option of following the terms and +conditions either of that published version or of any later version +published by the Free Software Foundation. If the Library as you +received it does not specify a version number of the GNU Lesser +General Public License, you may choose any version of the GNU Lesser +General Public License ever published by the Free Software Foundation. + + If the Library as you received it specifies that a proxy can decide +whether future versions of the GNU Lesser General Public License shall +apply, that proxy's public statement of acceptance of any version is +permanent authorization for you to choose that version for the +Library. @@ -12,36 +12,36 @@ begin section \<open>Constants and type rules\<close> axiomatization - Nat :: Term ("\<nat>") and - zero :: Term ("0") and - succ :: "Term \<Rightarrow> Term" and - indNat :: "[[Term, Term] \<Rightarrow> Term, Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>\<nat>)") + Nat :: t ("\<nat>") and + zero :: t ("0") and + succ :: "t \<Rightarrow> t" and + indNat :: "[[t, t] \<Rightarrow> t, t, t] \<Rightarrow> t" ("(1ind\<^sub>\<nat>)") where - Nat_form: "\<nat>: U(O)" + Nat_form: "\<nat>: U O" and Nat_intro_0: "0: \<nat>" and - Nat_intro_succ: "n: \<nat> \<Longrightarrow> succ(n): \<nat>" + Nat_intro_succ: "n: \<nat> \<Longrightarrow> succ n: \<nat>" and Nat_elim: "\<lbrakk> - C: \<nat> \<longrightarrow> U(i); - \<And>n c. \<lbrakk>n: \<nat>; c: C(n)\<rbrakk> \<Longrightarrow> f(n)(c): C(succ n); - a: C(0); + C: \<nat> \<longrightarrow> U i; + \<And>n c. \<lbrakk>n: \<nat>; c: C n\<rbrakk> \<Longrightarrow> f n c: C (succ n); + a: C 0; n: \<nat> - \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat>(f)(a)(n): C(n)" + \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat> f a n: C n" and Nat_comp_0: "\<lbrakk> - C: \<nat> \<longrightarrow> U(i); - \<And>n c. \<lbrakk>n: \<nat>; c: C(n)\<rbrakk> \<Longrightarrow> f(n)(c): C(succ n); - a: C(0) - \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat>(f)(a)(0) \<equiv> a" + C: \<nat> \<longrightarrow> U i; + \<And>n c. \<lbrakk>n: \<nat>; c: C(n)\<rbrakk> \<Longrightarrow> f n c: C (succ n); + a: C 0 + \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat> f a 0 \<equiv> a" and Nat_comp_succ: "\<lbrakk> - C: \<nat> \<longrightarrow> U(i); - \<And>n c. \<lbrakk>n: \<nat>; c: C(n)\<rbrakk> \<Longrightarrow> f(n)(c): C(succ n); - a: C(0); + C: \<nat> \<longrightarrow> U i; + \<And>n c. \<lbrakk>n: \<nat>; c: C n\<rbrakk> \<Longrightarrow> f n c: C (succ n); + a: C 0; n: \<nat> - \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat>(f)(a)(succ n) \<equiv> f(n)(ind\<^sub>\<nat> f a n)" + \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat> f a (succ n) \<equiv> f n (ind\<^sub>\<nat> f a n)" text "Rule attribute declarations:" @@ -12,14 +12,14 @@ begin section \<open>Constants and syntax\<close> axiomatization - Prod :: "[Term, Typefam] \<Rightarrow> Term" and - lambda :: "(Term \<Rightarrow> Term) \<Rightarrow> Term" (binder "\<^bold>\<lambda>" 30) and - appl :: "[Term, Term] \<Rightarrow> Term" (infixl "`" 60) + Prod :: "[t, tf] \<Rightarrow> t" and + lambda :: "(t \<Rightarrow> t) \<Rightarrow> t" (binder "\<^bold>\<lambda>" 30) and + appl :: "[t, t] \<Rightarrow> t" (infixl "`" 60) \<comment> \<open>Application binds tighter than abstraction.\<close> syntax - "_PROD" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3\<Prod>_:_./ _)" 30) - "_PROD_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3PROD _:_./ _)" 30) + "_PROD" :: "[idt, t, t] \<Rightarrow> t" ("(3\<Prod>_:_./ _)" 30) + "_PROD_ASCII" :: "[idt, t, t] \<Rightarrow> t" ("(3PROD _:_./ _)" 30) text "The translations below bind the variable \<open>x\<close> in the expressions \<open>B\<close> and \<open>b\<close>." @@ -29,24 +29,24 @@ translations text "Nondependent functions are a special case." -abbreviation Function :: "[Term, Term] \<Rightarrow> Term" (infixr "\<rightarrow>" 40) +abbreviation Function :: "[t, t] \<Rightarrow> t" (infixr "\<rightarrow>" 40) where "A \<rightarrow> B \<equiv> \<Prod>_: A. B" section \<open>Type rules\<close> axiomatization where - Prod_form: "\<lbrakk>A: U(i); B: A \<longrightarrow> U(i)\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x): U(i)" + Prod_form: "\<lbrakk>A: U i; B: A \<longrightarrow> U i\<rbrakk> \<Longrightarrow> \<Prod>x:A. B x: U i" and - Prod_intro: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b(x): B(x); A: U(i)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b(x): \<Prod>x:A. B(x)" + Prod_intro: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; A: U i\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b x: \<Prod>x:A. B x" and - Prod_elim: "\<lbrakk>f: \<Prod>x:A. B(x); a: A\<rbrakk> \<Longrightarrow> f`a: B(a)" + Prod_elim: "\<lbrakk>f: \<Prod>x:A. B x; a: A\<rbrakk> \<Longrightarrow> f`a: B a" and - Prod_appl: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b(x): B(x); a: A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. b(x))`a \<equiv> b(a)" + Prod_appl: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; a: A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. b x)`a \<equiv> b a" and - Prod_uniq: "f : \<Prod>x:A. B(x) \<Longrightarrow> \<^bold>\<lambda>x. (f`x) \<equiv> f" + Prod_uniq: "f : \<Prod>x:A. B x \<Longrightarrow> \<^bold>\<lambda>x. (f`x) \<equiv> f" and - Prod_eq: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b(x) \<equiv> b'(x); A: U(i)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b(x) \<equiv> \<^bold>\<lambda>x. b'(x)" + Prod_eq: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x \<equiv> c x; A: U i\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b x \<equiv> \<^bold>\<lambda>x. c x" text " The Pure rules for \<open>\<equiv>\<close> only let us judge strict syntactic equality of object lambda expressions; Prod_eq is the actual definitional equality rule. @@ -55,15 +55,15 @@ text " " text " - In addition to the usual type rules, it is a meta-theorem that whenever \<open>\<Prod>x:A. B x: U(i)\<close> is derivable from some set of premises \<Gamma>, then so are \<open>A: U(i)\<close> and \<open>B: A \<longrightarrow> U(i)\<close>. + In addition to the usual type rules, it is a meta-theorem that whenever \<open>\<Prod>x:A. B x: U i\<close> is derivable from some set of premises \<Gamma>, then so are \<open>A: U i\<close> and \<open>B: A \<longrightarrow> U i\<close>. That is to say, the following inference rules are admissible, and it simplifies proofs greatly to axiomatize them directly. " axiomatization where - Prod_wellform1: "(\<Prod>x:A. B(x): U(i)) \<Longrightarrow> A: U(i)" + Prod_wellform1: "(\<Prod>x:A. B x: U i) \<Longrightarrow> A: U i" and - Prod_wellform2: "(\<Prod>x:A. B(x): U(i)) \<Longrightarrow> B: A \<longrightarrow> U(i)" + Prod_wellform2: "(\<Prod>x:A. B x: U i) \<Longrightarrow> B: A \<longrightarrow> U i" text "Rule attribute declarations---set up various methods to use the type rules." @@ -75,19 +75,15 @@ lemmas Prod_routine [intro] = Prod_form Prod_intro Prod_elim section \<open>Function composition\<close> -definition compose :: "[Term, Term] \<Rightarrow> Term" (infixr "o" 70) where "g o f \<equiv> \<^bold>\<lambda>x. g`(f`x)" +definition compose :: "[t, t] \<Rightarrow> t" (infixr "o" 110) where "g o f \<equiv> \<^bold>\<lambda>x. g`(f`x)" -syntax "_COMPOSE" :: "[Term, Term] \<Rightarrow> Term" (infixr "\<circ>" 70) +syntax "_COMPOSE" :: "[t, t] \<Rightarrow> t" (infixr "\<circ>" 110) translations "g \<circ> f" \<rightleftharpoons> "g o f" -section \<open>Atomization\<close> +section \<open>Polymorphic identity function\<close> -text " - Universal statements can be internalized within the theory; the following rule is admissible. -" (* PROOF NEEDED *) - -axiomatization where Prod_atomize: "(\<^bold>\<lambda>x. b(x): \<Prod>x:A. B(x)) \<Longrightarrow> (\<And>x. x: A \<Longrightarrow> b(x): B(x))" +abbreviation id :: t where "id \<equiv> \<^bold>\<lambda>x. x" end diff --git a/ProdProps.thy b/ProdProps.thy index 1af6ad3..a68f79b 100644 --- a/ProdProps.thy +++ b/ProdProps.thy @@ -14,11 +14,11 @@ begin section \<open>Composition\<close> text " - The proof of associativity needs some guidance; it involves telling Isabelle to use the correct rule for \<Pi>-type definitional equality, and the correct substitutions in the subgoals thereafter. + The proof of associativity needs some guidance; it involves telling Isabelle to use the correct rule for \<Prod>-type definitional equality, and the correct substitutions in the subgoals thereafter. " lemma compose_assoc: - assumes "A: U(i)" and "f: A \<rightarrow> B" "g: B \<rightarrow> C" "h: \<Prod>x:C. D(x)" + assumes "A: U i" and "f: A \<rightarrow> B" "g: B \<rightarrow> C" "h: \<Prod>x:C. D x" shows "(h \<circ> g) \<circ> f \<equiv> h \<circ> (g \<circ> f)" proof (subst (0 1 2 3) compose_def) show "\<^bold>\<lambda>x. (\<^bold>\<lambda>y. h`(g`y))`(f`x) \<equiv> \<^bold>\<lambda>x. h`((\<^bold>\<lambda>y. g`(f`y))`x)" @@ -31,22 +31,27 @@ proof (subst (0 1 2 3) compose_def) proof compute show "\<And>x. x: A \<Longrightarrow> g`(f`x): C" by (routine lems: assms) qed - show "\<And>x. x: B \<Longrightarrow> h`(g`x): D(g`x)" by (routine lems: assms) + show "\<And>x. x: B \<Longrightarrow> h`(g`x): D (g`x)" by (routine lems: assms) qed (routine lems: assms) qed fact qed lemma compose_comp: - assumes "A: U(i)" and "\<And>x. x: A \<Longrightarrow> b(x): B" and "\<And>x. x: B \<Longrightarrow> c(x): C(x)" - shows "(\<^bold>\<lambda>x. c(x)) \<circ> (\<^bold>\<lambda>x. b(x)) \<equiv> \<^bold>\<lambda>x. c(b(x))" + assumes "A: U i" and "\<And>x. x: A \<Longrightarrow> b x: B" and "\<And>x. x: B \<Longrightarrow> c x: C x" + shows "(\<^bold>\<lambda>x. c x) \<circ> (\<^bold>\<lambda>x. b x) \<equiv> \<^bold>\<lambda>x. c (b x)" proof (subst compose_def, subst Prod_eq) - show "\<And>a. a: A \<Longrightarrow> (\<^bold>\<lambda>x. c(x))`((\<^bold>\<lambda>x. b(x))`a) \<equiv> (\<^bold>\<lambda>x. c (b x))`a" + show "\<And>a. a: A \<Longrightarrow> (\<^bold>\<lambda>x. c x)`((\<^bold>\<lambda>x. b x)`a) \<equiv> (\<^bold>\<lambda>x. c (b x))`a" proof compute - show "\<And>a. a: A \<Longrightarrow> c((\<^bold>\<lambda>x. b(x))`a) \<equiv> (\<^bold>\<lambda>x. c(b(x)))`a" + show "\<And>a. a: A \<Longrightarrow> c ((\<^bold>\<lambda>x. b x)`a) \<equiv> (\<^bold>\<lambda>x. c (b x))`a" by (derive lems: assms) qed (routine lems: assms) qed (derive lems: assms) +text "Set up the \<open>compute\<close> method to automatically simplify function compositions." + +lemmas compose_comp [comp] + + end @@ -12,44 +12,44 @@ theory Proj begin -definition fst :: "Term \<Rightarrow> Term" where "fst(p) \<equiv> ind\<^sub>\<Sum> (\<lambda>x y. x) p" -definition snd :: "Term \<Rightarrow> Term" where "snd(p) \<equiv> ind\<^sub>\<Sum> (\<lambda>x y. y) p" +definition fst :: "Term \<Rightarrow> Term" where "fst p \<equiv> ind\<^sub>\<Sum> (\<lambda>x y. x) p" +definition snd :: "Term \<Rightarrow> Term" where "snd p \<equiv> ind\<^sub>\<Sum> (\<lambda>x y. y) p" text "Typing judgments and computation rules for the dependent and non-dependent projection functions." lemma fst_type: - assumes "\<Sum>x:A. B(x): U(i)" and "p: \<Sum>x:A. B(x)" shows "fst(p): A" + assumes "\<Sum>x:A. B x: U i" and "p: \<Sum>x:A. B x" shows "fst p: A" unfolding fst_def by (derive lems: assms) lemma fst_comp: - assumes "A: U(i)" and "B: A \<longrightarrow> U(i)" and "a: A" and "b: B(a)" shows "fst(<a,b>) \<equiv> a" + assumes "A: U i" and "B: A \<longrightarrow> U i" and "a: A" and "b: B a" shows "fst <a,b> \<equiv> a" unfolding fst_def proof compute - show "a: A" and "b: B(a)" by fact+ + show "a: A" and "b: B a" by fact+ qed (routine lems: assms)+ lemma snd_type: - assumes "\<Sum>x:A. B(x): U(i)" and "p: \<Sum>x:A. B(x)" shows "snd(p): B(fst p)" + assumes "\<Sum>x:A. B x: U i" and "p: \<Sum>x:A. B x" shows "snd p: B (fst p)" unfolding snd_def proof - show "\<And>p. p: \<Sum>x:A. B(x) \<Longrightarrow> B(fst p): U(i)" by (derive lems: assms fst_type) + show "\<And>p. p: \<Sum>x:A. B x \<Longrightarrow> B (fst p): U i" by (derive lems: assms fst_type) fix x y - assume asm: "x: A" "y: B(x)" - show "y: B(fst <x,y>)" + assume asm: "x: A" "y: B x" + show "y: B (fst <x,y>)" proof (subst fst_comp) - show "A: U(i)" by (wellformed lems: assms(1)) - show "\<And>x. x: A \<Longrightarrow> B(x): U(i)" by (wellformed lems: assms(1)) + show "A: U i" by (wellformed lems: assms(1)) + show "\<And>x. x: A \<Longrightarrow> B x: U i" by (wellformed lems: assms(1)) qed fact+ qed fact lemma snd_comp: - assumes "A: U(i)" and "B: A \<longrightarrow> U(i)" and "a: A" and "b: B(a)" shows "snd(<a,b>) \<equiv> b" + assumes "A: U i" and "B: A \<longrightarrow> U i" and "a: A" and "b: B a" shows "snd <a,b> \<equiv> b" unfolding snd_def proof compute - show "\<And>x y. y: B(x) \<Longrightarrow> y: B(x)" . + show "\<And>x y. y: B x \<Longrightarrow> y: B x" . show "a: A" by fact - show "b: B(a)" by fact + show "b: B a" by fact qed (routine lems: assms) @@ -4,15 +4,19 @@ An experimental implementation of [homotopy type theory](https://en.wikipedia.or ### Installation & Usage -Clone the contents of this repository into `<Isabelle root directory>/src/HoTT`. +Clone or copy the contents of this repository into `<Isabelle root directory>/src/Isabelle-HoTT`. To use, set Isabelle's prover to Pure in the Theories panel, and import `HoTT`. +### Some comments on the implementation + +The implementation in the `master` branch is polymorphic without type annotations, and as such has some differences with the standard theory as presented in the [Homotopy Type Theory book](https://homotopytypetheory.org/book/). + ### Collaboration -I've been flying solo on this library as part of my Masters project, and there are very many improvements and developments that have yet to be implemented, so ***collaborators are welcome!*** +I've been flying solo on this library as part of my Masters project, and there are very many improvements and developments that have yet to be implemented, so **collaborators are welcome!** -If you're interested in working together on any part of this do drop me a line at `joshua DOT chen AT uni-bonn DOT de`. +If you're interested in working together on any part of this library do drop me a line at `joshua DOT chen AT uni-bonn DOT de`. ### License @@ -12,13 +12,13 @@ begin section \<open>Constants and syntax\<close> axiomatization - Sum :: "[Term, Typefam] \<Rightarrow> Term" and - pair :: "[Term, Term] \<Rightarrow> Term" ("(1<_,/ _>)") and - indSum :: "[[Term, Term] \<Rightarrow> Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>\<Sum>)") + Sum :: "[t, Typefam] \<Rightarrow> t" and + pair :: "[t, t] \<Rightarrow> t" ("(1<_,/ _>)") and + indSum :: "[[t, t] \<Rightarrow> t, t] \<Rightarrow> t" ("(1ind\<^sub>\<Sum>)") syntax - "_SUM" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3\<Sum>_:_./ _)" 20) - "_SUM_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3SUM _:_./ _)" 20) + "_SUM" :: "[idt, t, t] \<Rightarrow> t" ("(3\<Sum>_:_./ _)" 20) + "_SUM_ASCII" :: "[idt, t, t] \<Rightarrow> t" ("(3SUM _:_./ _)" 20) translations "\<Sum>x:A. B" \<rightleftharpoons> "CONST Sum A (\<lambda>x. B)" @@ -26,37 +26,37 @@ translations text "Nondependent pair." -abbreviation Pair :: "[Term, Term] \<Rightarrow> Term" (infixr "\<times>" 50) +abbreviation Pair :: "[t, t] \<Rightarrow> t" (infixr "\<times>" 50) where "A \<times> B \<equiv> \<Sum>_:A. B" section \<open>Type rules\<close> axiomatization where - Sum_form: "\<lbrakk>A: U(i); B: A \<longrightarrow> U(i)\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x): U(i)" + Sum_form: "\<lbrakk>A: U i; B: A \<longrightarrow> U i\<rbrakk> \<Longrightarrow> \<Sum>x:A. B x: U i" and - Sum_intro: "\<lbrakk>B: A \<longrightarrow> U(i); a: A; b: B(a)\<rbrakk> \<Longrightarrow> <a,b>: \<Sum>x:A. B(x)" + Sum_intro: "\<lbrakk>B: A \<longrightarrow> U i; a: A; b: B a\<rbrakk> \<Longrightarrow> <a,b>: \<Sum>x:A. B x" and Sum_elim: "\<lbrakk> - p: \<Sum>x:A. B(x); - \<And>x y. \<lbrakk>x: A; y: B(x)\<rbrakk> \<Longrightarrow> f(x)(y): C(<x,y>); - C: \<Sum>x:A. B(x) \<longrightarrow> U(i) - \<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum>(f)(p): C(p)" (* What does writing \<lambda>x y. f(x, y) change? *) + p: \<Sum>x:A. B x; + \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x,y>; + C: \<Sum>x:A. B x \<longrightarrow> U i + \<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum> f p: C p" (* What does writing \<lambda>x y. f(x, y) change? *) and Sum_comp: "\<lbrakk> a: A; - b: B(a); - \<And>x y. \<lbrakk>x: A; y: B(x)\<rbrakk> \<Longrightarrow> f(x)(y): C(<x,y>); - B: A \<longrightarrow> U(i); - C: \<Sum>x:A. B(x) \<longrightarrow> U(i) - \<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum>(f)(<a,b>) \<equiv> f(a)(b)" + b: B a; + \<And>x y. \<lbrakk>x: A; y: B(x)\<rbrakk> \<Longrightarrow> f x y: C <x,y>; + B: A \<longrightarrow> U i; + C: \<Sum>x:A. B x \<longrightarrow> U i + \<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum> f <a,b> \<equiv> f a b" text "Admissible inference rules for sum formation:" axiomatization where - Sum_wellform1: "(\<Sum>x:A. B(x): U(i)) \<Longrightarrow> A: U(i)" + Sum_wellform1: "(\<Sum>x:A. B x: U i) \<Longrightarrow> A: U i" and - Sum_wellform2: "(\<Sum>x:A. B(x): U(i)) \<Longrightarrow> B: A \<longrightarrow> U(i)" + Sum_wellform2: "(\<Sum>x:A. B x: U i) \<Longrightarrow> B: A \<longrightarrow> U i" text "Rule attribute declarations:" @@ -16,13 +16,13 @@ axiomatization pt :: Term ("\<star>") and indUnit :: "[Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>\<one>)") where - Unit_form: "\<one>: U(O)" + Unit_form: "\<one>: U O" and Unit_intro: "\<star>: \<one>" and - Unit_elim: "\<lbrakk>C: \<one> \<longrightarrow> U(i); c: C(\<star>); a: \<one>\<rbrakk> \<Longrightarrow> ind\<^sub>\<one>(c)(a) : C(a)" + Unit_elim: "\<lbrakk>C: \<one> \<longrightarrow> U i; c: C \<star>; a: \<one>\<rbrakk> \<Longrightarrow> ind\<^sub>\<one> c a: C a" and - Unit_comp: "\<lbrakk>C: \<one> \<longrightarrow> U(i); c: C(\<star>)\<rbrakk> \<Longrightarrow> ind\<^sub>\<one>(c)(\<star>) \<equiv> c" + Unit_comp: "\<lbrakk>C: \<one> \<longrightarrow> U i; c: C \<star>\<rbrakk> \<Longrightarrow> ind\<^sub>\<one> c \<star> \<equiv> c" text "Rule attribute declarations:" diff --git a/Univalence.thy b/Univalence.thy new file mode 100644 index 0000000..001ee33 --- /dev/null +++ b/Univalence.thy @@ -0,0 +1,176 @@ +(* Title: HoTT/Univalence.thy + Author: Joshua Chen + +Definitions of homotopy, equivalence and the univalence axiom. +*) + +theory Univalence +imports + HoTT_Methods + EqualProps + ProdProps + Sum + +begin + + +section \<open>Homotopy and equivalence\<close> + +axiomatization homotopic :: "[t, t] \<Rightarrow> t" (infix "~" 100) where + homotopic_def: "\<lbrakk> + f: \<Prod>x:A. B x; + g: \<Prod>x:A. B x + \<rbrakk> \<Longrightarrow> f ~ g \<equiv> \<Prod>x:A. (f`x) =[B x] (g`x)" + +axiomatization isequiv :: "t \<Rightarrow> t" where + isequiv_def: "f: A \<rightarrow> B \<Longrightarrow> isequiv f \<equiv> (\<Sum>g: B \<rightarrow> A. g \<circ> f ~ id) \<times> (\<Sum>g: B \<rightarrow> A. f \<circ> g ~ id)" + +definition equivalence :: "[t, t] \<Rightarrow> t" (infix "\<simeq>" 100) + where "A \<simeq> B \<equiv> \<Sum>f: A \<rightarrow> B. isequiv f" + + +text "The identity function is an equivalence:" + +lemma isequiv_id: + assumes "A: U i" and "id: A \<rightarrow> A" + shows "<<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>: isequiv id" +proof (derive lems: assms isequiv_def homotopic_def) + fix g assume asm: "g: A \<rightarrow> A" + show "id \<circ> g: A \<rightarrow> A" + unfolding compose_def by (routine lems: asm assms) + + show "\<Prod>x:A. ((id \<circ> g)`x) =\<^sub>A (id`x): U i" + unfolding compose_def by (routine lems: asm assms) + next + + show "<\<^bold>\<lambda>x. x, \<^bold>\<lambda>x. refl x>: \<Sum>g:A \<rightarrow> A. (g \<circ> id) ~ id" + unfolding compose_def by (derive lems: assms homotopic_def) + + show "<\<^bold>\<lambda>x. x, lambda refl>: \<Sum>g:A \<rightarrow> A. (id \<circ> g) ~ id" + unfolding compose_def by (derive lems: assms homotopic_def) +qed (rule assms) + + +text "We use the following lemma in a few proofs:" + +lemma isequiv_type: + assumes "A: U i" and "B: U i" and "f: A \<rightarrow> B" + shows "isequiv f: U i" + by (derive lems: assms isequiv_def homotopic_def compose_def) + + +text "The equivalence relation \<open>\<simeq>\<close> is symmetric:" + +lemma equiv_sym: + assumes "A: U i" and "id: A \<rightarrow> A" + shows "<id, <<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>>: A \<simeq> A" +unfolding equivalence_def proof + show "<<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>: isequiv id" using assms by (rule isequiv_id) + + fix f assume "f: A \<rightarrow> A" + with assms(1) assms(1) show "isequiv f: U i" by (rule isequiv_type) +qed (rule assms) + + +section \<open>idtoeqv and the univalence axiom\<close> + +definition idtoeqv :: t + where "idtoeqv \<equiv> \<^bold>\<lambda>p. <transport p, ind\<^sub>= (\<lambda>A. <<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>) p>" + + +text "We prove that equal types are equivalent. The proof is long and uses universes." + +theorem + assumes "A: U i" and "B: U i" + shows "idtoeqv: (A =[U i] B) \<rightarrow> A \<simeq> B" +unfolding idtoeqv_def equivalence_def +proof + fix p assume "p: A =[U i] B" + show "<transport p, ind\<^sub>= (\<lambda>A. <<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>) p>: \<Sum>f: A \<rightarrow> B. isequiv f" + proof + { fix f assume "f: A \<rightarrow> B" + with assms show "isequiv f: U i" by (rule isequiv_type) + } + + show "transport p: A \<rightarrow> B" + proof (rule transport_type[where ?P="\<lambda>x. x" and ?A="U i" and ?i="Suc i"]) + show "\<And>x. x: U i \<Longrightarrow> x: U (Suc i)" by cumulativity + show "U i: U (Suc i)" by hierarchy + qed fact+ + + show "ind\<^sub>= (\<lambda>A. <<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>) p: isequiv (transport p)" + proof (rule Equal_elim[where ?C="\<lambda>_ _ p. isequiv (transport p)"]) + fix A assume asm: "A: U i" + show "<<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>: isequiv (transport (refl A))" + proof (derive lems: isequiv_def) + show "transport (refl A): A \<rightarrow> A" + unfolding transport_def + by (compute lems: Equal_comp[where ?A="U i" and ?C="\<lambda>_ _ _. A \<rightarrow> A"]) (derive lems: asm) + + show "<<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>: + (\<Sum>g:A \<rightarrow> A. g \<circ> (transport (refl A)) ~ id) \<times> + (\<Sum>g:A \<rightarrow> A. (transport (refl A)) \<circ> g ~ id)" + proof (subst (1 2) transport_comp) + show "U i: U (Suc i)" by (rule U_hierarchy) rule + show "U i: U (Suc i)" by (rule U_hierarchy) rule + + show "<<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>: + (\<Sum>g:A \<rightarrow> A. g \<circ> id ~ id) \<times> (\<Sum>g:A \<rightarrow> A. id \<circ> g ~ id)" + proof + show "\<Sum>g:A \<rightarrow> A. id \<circ> g ~ id: U i" + proof (derive lems: asm homotopic_def) + fix g assume asm': "g: A \<rightarrow> A" + show *: "id \<circ> g: A \<rightarrow> A" by (derive lems: asm asm' compose_def) + show "\<Prod>x:A. ((id \<circ> g)`x) =\<^sub>A (id`x): U i" by (derive lems: asm *) + qed (routine lems: asm) + + show "<id, \<^bold>\<lambda>x. refl x>: \<Sum>g:A \<rightarrow> A. id \<circ> g ~ id" + proof + fix g assume asm': "g: A \<rightarrow> A" + show "id \<circ> g ~ id: U i" + proof (derive lems: homotopic_def) + show *: "id \<circ> g: A \<rightarrow> A" by (derive lems: asm asm' compose_def) + show "\<Prod>x:A. ((id \<circ> g)`x) =\<^sub>A (id`x): U i" by (derive lems: asm *) + qed (routine lems: asm) + next + show "\<^bold>\<lambda>x. refl x: id \<circ> id ~ id" + proof compute + show "\<^bold>\<lambda>x. refl x: id ~ id" by (subst homotopic_def) (derive lems: asm) + qed (rule asm) + qed (routine lems: asm) + + show "<id, \<^bold>\<lambda>x. refl x>: \<Sum>g:A \<rightarrow> A. g \<circ> id ~ id" + proof + fix g assume asm': "g: A \<rightarrow> A" + show "g \<circ> id ~ id: U i" by (derive lems: asm asm' homotopic_def compose_def) + next + show "\<^bold>\<lambda>x. refl x: id \<circ> id ~ id" + proof compute + show "\<^bold>\<lambda>x. refl x: id ~ id" by (subst homotopic_def) (derive lems: asm) + qed (rule asm) + qed (routine lems: asm) + qed + qed fact+ + qed + next + + fix A' B' p' assume asm: "A': U i" "B': U i" "p': A' =[U i] B'" + show "isequiv (transport p'): U i" + proof (rule isequiv_type) + show "transport p': A' \<rightarrow> B'" by (derive lems: asm transport_def) + qed fact+ + qed fact+ + qed + next + + show "A =[U i] B: U (Suc i)" proof (derive lems: assms, (rule U_hierarchy, rule lt_Suc)?)+ +qed + + +text "The univalence axiom." + +axiomatization univalence :: t where + UA: "univalence: isequiv idtoeqv" + + +end diff --git a/ex/HoTT book/Ch1.thy b/ex/HoTT book/Ch1.thy index d5f05dd..a577fca 100644 --- a/ex/HoTT book/Ch1.thy +++ b/ex/HoTT book/Ch1.thy @@ -40,4 +40,16 @@ proof (rule Sum_elim[where ?p=p]) qed (derive lems: assms) +section \<open>Exercises\<close> + +text "Exercise 1.13" + +abbreviation "not" ("\<not>'(_')") where "\<not>(A) \<equiv> A \<rightarrow> \<zero>" + +text "This proof requires the use of universe cumulativity." + +proposition assumes "A: U(i)" shows "\<^bold>\<lambda>f. f`(inr(\<^bold>\<lambda>a. f`inl(a))): \<not>(\<not>(A + \<not>(A)))" +by (derive lems: assms U_cumulative[where ?A=\<zero> and ?i=O and ?j=i]) + + end |