diff options
Diffstat (limited to 'EqualProps.thy')
| -rw-r--r-- | EqualProps.thy | 48 |
1 files changed, 18 insertions, 30 deletions
diff --git a/EqualProps.thy b/EqualProps.thy index b691133..10d3b17 100644 --- a/EqualProps.thy +++ b/EqualProps.thy @@ -18,6 +18,7 @@ section \<open>Symmetry / Path inverse\<close> definition inv :: "[Term, Term, Term] \<Rightarrow> Term" ("(1inv[_,/ _,/ _])") where "inv[A,x,y] \<equiv> \<^bold>\<lambda>p:x =\<^sub>A y. indEqual[A] (\<lambda>x y _. y =\<^sub>A x) (\<lambda>x. refl(x)) x y p" + lemma inv_type: assumes "p : x =\<^sub>A y" shows "inv[A,x,y]`p : y =\<^sub>A x" @@ -64,45 +65,32 @@ section \<open>Transitivity / Path composition\<close> text "``Raw'' composition function, of type \<open>\<Prod>x,y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)\<close>." definition rcompose :: "Term \<Rightarrow> Term" ("(1rcompose[_])") - where "rcompose[A] \<equiv> \<^bold>\<lambda>x:A. \<^bold>\<lambda>y:A. \<^bold>\<lambda>p:x =\<^sub>A y. indEqual[A] + where "rcompose[A] \<equiv> \<^bold>\<lambda>x:A. \<^bold>\<lambda>y:A. \<^bold>\<lambda>p:(x =\<^sub>A y). indEqual[A] (\<lambda>x y _. \<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z) - (\<lambda>x. \<^bold>\<lambda>z:A. \<^bold>\<lambda>p:x =\<^sub>A z. indEqual[A](\<lambda>x z _. x =\<^sub>A z) (\<lambda>x. refl(x)) x z p) + (\<lambda>x. \<^bold>\<lambda>z:A. \<^bold>\<lambda>p:(x =\<^sub>A z). indEqual[A](\<lambda>x z _. x =\<^sub>A z) (\<lambda>x. refl(x)) x z p) x y p" text "``Natural'' composition function abbreviation, effectively equivalent to a function of type \<open>\<Prod>x,y,z:A. x =\<^sub>A y \<rightarrow> y =\<^sub>A z \<rightarrow> x =\<^sub>A z\<close>." abbreviation compose :: "[Term, Term, Term, Term] \<Rightarrow> Term" ("(1compose[_,/ _,/ _,/ _])") - where "compose[A,x,y,z] \<equiv> \<^bold>\<lambda>p:x =\<^sub>A y. \<^bold>\<lambda>q:y =\<^sub>A z. rcompose[A]`x`y`p`z`q" + where "compose[A,x,y,z] \<equiv> \<^bold>\<lambda>p:(x =\<^sub>A y). \<^bold>\<lambda>q:(y =\<^sub>A z). rcompose[A]`x`y`p`z`q" + + +lemma compose_type: + assumes "p : x =\<^sub>A y" and "q : y =\<^sub>A z" + shows "compose[A,x,y,z]`p`q : x =\<^sub>A z" + +sorry lemma compose_comp: assumes "a : A" shows "compose[A,a,a,a]`refl(a)`refl(a) \<equiv> refl(a)" -proof (unfold rcompose_def) - have "compose[A,a,a,a]`refl(a) \<equiv> \<^bold>\<lambda>q:a =\<^sub>A a. rcompose[A]`a`a`refl(a)`a`q" - proof standard+ (*TODO: Set up the Simplifier to handle this proof at some point.*) - fix p q assume "p : a =\<^sub>A a" and "q : a =\<^sub>A a" - then show "rcompose[A]`a`a`p`a`q : a =\<^sub>A a" - proof (unfold rcompose_def) - have "(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:A. \<^bold>\<lambda>p:x =\<^sub>A y. (indEqual[A] - (\<lambda>x y _. \<Prod>z:A. y =[A] z \<rightarrow> x =[A] z) - (\<lambda>x. \<^bold>\<lambda>z:A. \<^bold>\<lambda>q:x =\<^sub>A z. (indEqual[A] (\<lambda>x z _. x =\<^sub>A z) refl x z q)) - x y p))`a`a`p`a`q \<equiv> ..." (*Okay really need to set up the Simplifier...*) -oops - -text "The above proof is a good candidate for proof automation; in particular we would like the system to be able to automatically find the conditions of the \<open>using\<close> clause in the proof. -This would likely involve something like: - 1. Recognizing that there is a function application that can be simplified. - 2. Noting that the obstruction to applying \<open>Prod_comp\<close> is the requirement that \<open>refl(a) : a =\<^sub>A a\<close>. - 3. Obtaining such a condition, using the known fact \<open>a : A\<close> and the introduction rule \<open>Equal_intro\<close>." - -lemmas Equal_simps [simp] = inv_comp compose_comp - -section \<open>Pretty printing\<close> - -abbreviation inv_pretty :: "[Term, Term, Term, Term] \<Rightarrow> Term" ("(1_\<^sup>-\<^sup>1[_, _, _])" 500) - where "p\<^sup>-\<^sup>1[A,x,y] \<equiv> inv[A,x,y]`p" - -abbreviation compose_pretty :: "[Term, Term, Term, Term, Term, Term] \<Rightarrow> Term" ("(1_ \<bullet>[_, _, _, _]/ _)") - where "p \<bullet>[A,x,y,z] q \<equiv> compose[A,x,y,z]`p`q"
\ No newline at end of file +sorry \<comment> \<open>Long and tedious proof if the Simplifier is not set up.\<close> + + +lemmas Equal_simps [intro] = inv_comp compose_comp + + +end
\ No newline at end of file |