diff options
Diffstat (limited to '')
| -rw-r--r-- | Equal.thy | 59 | ||||
| -rw-r--r-- | HoTT.thy | 7 | ||||
| -rw-r--r-- | HoTT_Base.thy | 2 |
3 files changed, 28 insertions, 40 deletions
@@ -1,66 +1,51 @@ -(* Title: HoTT/Equal.thy - Author: Josh Chen +(* +Title: Equal.thy +Author: Joshua Chen +Date: 2018 Equality type *) theory Equal - imports HoTT_Base +imports HoTT_Base HoTT_Methods + begin -section \<open>Constants and syntax\<close> +section \<open>Basic definitions\<close> axiomatization - Equal :: "[t, t, t] \<Rightarrow> t" and - refl :: "t \<Rightarrow> t" and + Equal :: "[t, t, t] \<Rightarrow> t" and + refl :: "t \<Rightarrow> t" and indEqual :: "[t \<Rightarrow> t, t] \<Rightarrow> t" ("(1ind\<^sub>=)") syntax - "_EQUAL" :: "[t, t, t] \<Rightarrow> t" ("(3_ =\<^sub>_/ _)" [101, 0, 101] 100) - "_EQUAL_ASCII" :: "[t, t, t] \<Rightarrow> t" ("(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" +axiomatization where + Equal_form: "\<lbrakk>A: U i; a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U i" and -section \<open>Type rules\<close> + Equal_intro: "a : A \<Longrightarrow> (refl a): a =\<^sub>A a" and -axiomatization where - 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" -and Equal_elim: "\<lbrakk> + p: x =\<^sub>A y; 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 + \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U i; + \<And>x. x: A \<Longrightarrow> f x: C x x (refl x) \<rbrakk> \<Longrightarrow> ind\<^sub>= (\<lambda>x. f x) 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 y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U i; + \<And>x. x: A \<Longrightarrow> f x: C x x (refl x) \<rbrakk> \<Longrightarrow> ind\<^sub>= (\<lambda>x. f x) (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" -and - Equal_wellform2: "a =\<^sub>A b: U i \<Longrightarrow> a: A" -and - Equal_wellform3: "a =\<^sub>A b: U i \<Longrightarrow> b: A" - - -text "Rule attribute declarations:" - -lemmas Equal_comp [comp] -lemmas Equal_wellform [wellform] = Equal_wellform1 Equal_wellform2 Equal_wellform3 lemmas Equal_routine [intro] = Equal_form Equal_intro Equal_elim +lemmas Equal_comp [comp] end @@ -6,7 +6,6 @@ Homotopy type theory theory HoTT imports - (* Basic theories *) HoTT_Base HoTT_Methods @@ -22,18 +21,22 @@ Unit (* Derived definitions and properties *) EqualProps -ProdProps Proj begin + lemmas forms = Nat_form Prod_form Sum_form Coprod_form Equal_form Unit_form Empty_form + lemmas intros = Nat_intro_0 Nat_intro_succ Prod_intro Sum_intro Equal_intro Coprod_intro_inl Coprod_intro_inr Unit_intro + lemmas elims = Nat_elim Prod_elim Sum_elim Equal_elim Coprod_elim Unit_elim Empty_elim + lemmas routines = Nat_routine Prod_routine Sum_routine Equal_routine Coprod_routine Unit_routine Empty_routine + end diff --git a/HoTT_Base.thy b/HoTT_Base.thy index efc6182..17e3142 100644 --- a/HoTT_Base.thy +++ b/HoTT_Base.thy @@ -77,7 +77,7 @@ Declare named theorems to be used by proof methods defined in @{file HoTT_Method These are used by the \<open>compute\<close> method, and may also be passed to invocations of the method \<open>subst\<close> to perform equational rewriting. \<close> -(* Todo: Set up the simplifier! *) +(* Todo: Set up the Simplifier! *) end |