diff options
| -rw-r--r-- | Equality.thy | 215 | ||||
| -rw-r--r-- | Prod.thy | 2 |
2 files changed, 216 insertions, 1 deletions
diff --git a/Equality.thy b/Equality.thy new file mode 100644 index 0000000..f639734 --- /dev/null +++ b/Equality.thy @@ -0,0 +1,215 @@ +(* +Title: Equality.thy +Author: Joshua Chen +Date: 2018 + +Properties of equality +*) + +theory Equality +imports HoTT_Methods Equal Prod + +begin + + +section \<open>Symmetry of equality/Path inverse\<close> + +definition inv :: "t \<Rightarrow> t" ("_\<inverse>" [1000] 1000) where "p\<inverse> \<equiv> ind\<^sub>= (\<lambda>x. refl x) p" + +lemma inv_type: "\<lbrakk>A: U i; x: A; y: A; p: x =\<^sub>A y\<rbrakk> \<Longrightarrow> p\<inverse>: y =\<^sub>A x" +unfolding inv_def by (elim Equal_elim) routine + +lemma inv_comp: "\<lbrakk>A: U i; a: A\<rbrakk> \<Longrightarrow> (refl a)\<inverse> \<equiv> refl a" +unfolding inv_def by compute routine + +declare + inv_type [intro] + inv_comp [comp] + + +section \<open>Transitivity of equality/Path composition\<close> + +text \<open> +Composition function, of type @{term "x =\<^sub>A y \<rightarrow> (y =\<^sub>A z) \<rightarrow> (x =\<^sub>A z)"} polymorphic over @{term A}, @{term x}, @{term y}, and @{term z}. +\<close> + +definition pathcomp :: t where "pathcomp \<equiv> \<^bold>\<lambda>p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>q. ind\<^sub>= (\<lambda>x. (refl x)) q) p" + +syntax "_pathcomp" :: "[t, t] \<Rightarrow> t" (infixl "\<bullet>" 120) +translations "p \<bullet> q" \<rightleftharpoons> "CONST pathcomp`p`q" + +lemma pathcomp_type: + assumes "A: U i" "x: A" "y: A" "z: A" + shows "pathcomp: x =\<^sub>A y \<rightarrow> (y =\<^sub>A z) \<rightarrow> (x =\<^sub>A z)" +unfolding pathcomp_def by rule (elim Equal_elim, routine add: assms) + +corollary pathcomp_trans: + assumes "A: U i" and "x: A" "y: A" "z: A" and "p: x =\<^sub>A y" "q: y =\<^sub>A z" + shows "p \<bullet> q: x =\<^sub>A z" +by (routine add: assms pathcomp_type) + +lemma pathcomp_comp: + assumes "A: U i" and "a: A" + shows "(refl a) \<bullet> (refl a) \<equiv> refl a" +unfolding pathcomp_def proof compute + show "(ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>q. ind\<^sub>= (\<lambda>x. refl x) q) (refl a))`(refl a) \<equiv> refl a" + proof compute + show "\<^bold>\<lambda>q. (ind\<^sub>= (\<lambda>x. refl x) q): a =\<^sub>A a \<rightarrow> a =\<^sub>A a" + by (routine add: assms) + + show "(\<^bold>\<lambda>q. (ind\<^sub>= (\<lambda>x. refl x) q))`(refl a) \<equiv> refl a" + proof compute + show "\<And>q. q: a =\<^sub>A a \<Longrightarrow> ind\<^sub>= (\<lambda>x. refl x) q: a =\<^sub>A a" by (routine add: assms) + qed (derive lems: assms) + qed (routine add: assms) + + show "\<And>p. p: a =\<^sub>A a \<Longrightarrow> ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>q. ind\<^sub>= (\<lambda>x. refl x) q) p: a =\<^sub>A a \<rightarrow> a =\<^sub>A a" + by (routine add: assms) +qed (routine add: assms) + +declare + pathcomp_type [intro] + pathcomp_trans [intro] + pathcomp_comp [comp] + + +section \<open>Higher groupoid structure of types\<close> + +schematic_goal pathcomp_right_id: + assumes "A: U(i)" "x: A" "y: A" "p: x =\<^sub>A y" + shows "?a: p \<bullet> (refl y) =[x =\<^sub>A y] p" +proof (rule Equal_elim[where ?x=x and ?y=y and ?p=p]) \<comment> \<open>@{method elim} does not seem to work with schematic goals.\<close> + show "\<And>x. x: A \<Longrightarrow> refl(refl x): (refl x) \<bullet> (refl x) =[x =\<^sub>A x] (refl x)" + by (derive lems: assms) +qed (routine add: assms) + +schematic_goal pathcomp_left_id: + assumes "A: U(i)" "x: A" "y: A" "p: x =\<^sub>A y" + shows "?a: (refl x) \<bullet> p =[x =\<^sub>A y] p" +proof (rule Equal_elim[where ?x=x and ?y=y and ?p=p]) + show "\<And>x. x: A \<Longrightarrow> refl(refl x): (refl x) \<bullet> (refl x) =[x =\<^sub>A x] (refl x)" + by (derive lems: assms) +qed (routine add: assms) + +schematic_goal pathcomp_left_inv: + assumes "A: U(i)" "x: A" "y: A" "p: x =\<^sub>A y" + shows "?a: (p\<inverse> \<bullet> p) =[y =\<^sub>A y] refl(y)" +proof (rule Equal_elim[where ?x=x and ?y=y and ?p=p]) + show "\<And>x. x: A \<Longrightarrow> refl(refl x): (refl x)\<inverse> \<bullet> (refl x) =[x =\<^sub>A x] (refl x)" + by (derive lems: assms) +qed (routine add: assms) + +schematic_goal pathcomp_right_inv: + assumes "A: U(i)" "x: A" "y: A" "p: x =\<^sub>A y" + shows "?a: (p \<bullet> p\<inverse>) =[x =\<^sub>A x] refl(x)" +proof (rule Equal_elim[where ?x=x and ?y=y and ?p=p]) + show "\<And>x. x: A \<Longrightarrow> refl(refl x): (refl x) \<bullet> (refl x)\<inverse> =[x =\<^sub>A x] (refl x)" + by (derive lems: assms) +qed (routine add: assms) + +schematic_goal inv_involutive: + assumes "A: U(i)" "x: A" "y: A" "p: x =\<^sub>A y" + shows "?a: p\<inverse>\<inverse> =[x =\<^sub>A y] p" +proof (rule Equal_elim[where ?x=x and ?y=y and ?p=p]) + show "\<And>x. x: A \<Longrightarrow> refl(refl x): (refl x)\<inverse>\<inverse> =[x =\<^sub>A x] (refl x)" + by (derive lems: assms) +qed (routine add: assms) + +text \<open>All of the propositions above have the same proof term, which we abbreviate here.\<close> +abbreviation \<iota> :: "t \<Rightarrow> t" where "\<iota> p \<equiv> ind\<^sub>= (\<lambda>x. refl (refl x)) p" + +text \<open>The next proof has a triply-nested path induction.\<close> + +lemma pathcomp_assoc: + assumes "A: U i" "x: A" "y: A" "z: A" "w: A" + shows "\<^bold>\<lambda>p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>q. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>r. \<iota> r) q) p: + \<Prod>p: x =\<^sub>A y. \<Prod>q: y =\<^sub>A z. \<Prod>r: z =\<^sub>A w. p \<bullet> (q \<bullet> r) =[x =\<^sub>A w] (p \<bullet> q) \<bullet> r" +proof + show "\<And>p. p: x =\<^sub>A y \<Longrightarrow> ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>q. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>r. \<iota> r) q) p: + \<Prod>q: y =\<^sub>A z. \<Prod>r: z =\<^sub>A w. p \<bullet> (q \<bullet> r) =[x =\<^sub>A w] p \<bullet> q \<bullet> r" + proof (elim Equal_elim) + fix x assume 1: "x: A" + show "\<^bold>\<lambda>q. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>r. \<iota> r) q: + \<Prod>q: x =\<^sub>A z. \<Prod>r: z =\<^sub>A w. refl x \<bullet> (q \<bullet> r) =[x =\<^sub>A w] refl x \<bullet> q \<bullet> r" + proof + show "\<And>q. q: x =\<^sub>A z \<Longrightarrow> ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>r. \<iota> r) q: + \<Prod>r: z =\<^sub>A w. refl x \<bullet> (q \<bullet> r) =[x =\<^sub>A w] refl x \<bullet> q \<bullet> r" + proof (elim Equal_elim) + fix x assume *: "x: A" + show "\<^bold>\<lambda>r. \<iota> r: \<Prod>r: x =\<^sub>A w. refl x \<bullet> (refl x \<bullet> r) =[x =\<^sub>A w] refl x \<bullet> refl x \<bullet> r" + proof + show "\<And>r. r: x =[A] w \<Longrightarrow> \<iota> r: refl x \<bullet> (refl x \<bullet> r) =[x =[A] w] refl x \<bullet> refl x \<bullet> r" + by (elim Equal_elim, derive lems: * assms) + qed (routine add: * assms) + qed (routine add: 1 assms) + qed (routine add: 1 assms) + + text \<open> + In the following part @{method derive} fails to obtain the correct subgoals, so we have to prove the statement manually. + \<close> + fix y p assume 2: "y: A" "p: x =\<^sub>A y" + show "\<Prod>q: y =\<^sub>A z. \<Prod>r: z =\<^sub>A w. p \<bullet> (q \<bullet> r) =[x =\<^sub>A w] p \<bullet> q \<bullet> r: U i" + proof (routine add: assms) + fix q assume 3: "q: y =\<^sub>A z" + show "\<Prod>r: z =\<^sub>A w. p \<bullet> (q \<bullet> r) =[x =\<^sub>A w] p \<bullet> q \<bullet> r: U i" + proof (routine add: assms) + show "\<And>r. r: z =[A] w \<Longrightarrow> p \<bullet> (q \<bullet> r): x =[A] w" and "\<And>r. r: z =[A] w \<Longrightarrow> p \<bullet> q \<bullet> r: x =[A] w" + by (routine add: 1 2 3 assms) + qed (routine add: 1 assms) + qed fact+ + qed fact+ +qed (routine add: assms) + + +section \<open>Functoriality of functions on types\<close> + +schematic_goal transfer_lemma: + assumes + "f: A \<rightarrow> B" "A: U i" "B: U i" + "p: x =\<^sub>A y" "x: A" "y: A" + shows "?a: (f`x =\<^sub>B f`y)" +proof (rule Equal_elim[where ?C="\<lambda>x y _. f`x =\<^sub>B f`y"]) + show "\<And>x. x: A \<Longrightarrow> refl (f`x): f`x =\<^sub>B f`x" by (routine add: assms) +qed (routine add: assms) + +definition ap :: "t \<Rightarrow> t" ("(ap\<^sub>_)") where "ap f \<equiv> \<^bold>\<lambda>p. ind\<^sub>= (\<lambda>x. refl (f`x)) p" + +syntax "_ap_ascii" :: "t \<Rightarrow> t" ("(ap[_])") +translations "ap[f]" \<rightleftharpoons> "CONST ap f" + +corollary ap_type: + assumes "f: A \<rightarrow> B" "A: U i" "B: U i" + shows "\<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> ap f: x =\<^sub>A y \<rightarrow> f`x =\<^sub>B f`y" +unfolding ap_def by (derive lems: assms transfer_lemma) + +schematic_goal functions_are_functorial: + assumes + "f: A \<rightarrow> B" "g: B \<rightarrow> C" + "A: U i" "B: U i" "C: U i" + "x: A" "y: A" "z: A" + "p: x =\<^sub>A y" "q: y =\<^sub>A z" + shows + 1: "?a: ap\<^sub>f`(p \<bullet> q) =[?A] (ap\<^sub>f`p) \<bullet> (ap\<^sub>f`q)" and + 2: "?b: ap\<^sub>f`(p\<inverse>) =[?B] (ap\<^sub>f`p)\<inverse>" and + 3: "?c: ap\<^sub>g`(ap\<^sub>f`p) =[?C] ap[g \<circ> f]`p" and + 4: "?d: ap[id]`p =[?D] p" +oops + + +section \<open>Transport\<close> + +definition transport :: "t \<Rightarrow> t" where "transport p \<equiv> ind\<^sub>= (\<lambda>_. (\<^bold>\<lambda>x. x)) p" + +text \<open>Note that @{term transport} is a polymorphic function in our formulation.\<close> + +lemma transport_type: "\<lbrakk>p: x =\<^sub>A y; x: A; y: A; A: U i; P: A \<longrightarrow> U i\<rbrakk> \<Longrightarrow> transport p: P x \<rightarrow> P y" +unfolding transport_def by (elim Equal_elim) routine + +corollary transport_elim: "\<lbrakk>x: P a; P: A \<longrightarrow> U i; p: a =\<^sub>A b; a: A; b: A; A: U i\<rbrakk> \<Longrightarrow> (transport p)`x: P b" +by (routine add: transport_type) + +lemma transport_comp: "\<lbrakk>A: U i; x: A\<rbrakk> \<Longrightarrow> transport (refl x) \<equiv> id" +unfolding transport_def by derive + + +end @@ -17,7 +17,7 @@ section \<open>Basic definitions\<close> axiomatization Prod :: "[t, tf] \<Rightarrow> t" and lambda :: "(t \<Rightarrow> t) \<Rightarrow> t" (binder "\<^bold>\<lambda>" 30) and - appl :: "[t, t] \<Rightarrow> t" ("(1_`/_)" [105, 106] 105) \<comment> \<open>Application binds tighter than abstraction.\<close> + appl :: "[t, t] \<Rightarrow> t" (infixl "`" 105) \<comment> \<open>Application binds tighter than abstraction.\<close> syntax "_prod" :: "[idt, t, t] \<Rightarrow> t" ("(3\<Prod>_: _./ _)" 30) |