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-rw-r--r--Equal.thy86
1 files changed, 58 insertions, 28 deletions
diff --git a/Equal.thy b/Equal.thy
index c10e1c2..17c1c99 100644
--- a/Equal.thy
+++ b/Equal.thy
@@ -23,33 +23,33 @@ translations
"a =[A] b" \<rightleftharpoons> "(CONST Eq) A a b"
axiomatization where
- Equal_form: "\<lbrakk>A: U i; a: A; b: A\<rbrakk> \<Longrightarrow> a =[A] b: U i" and
+ Eq_form: "\<lbrakk>A: U i; a: A; b: A\<rbrakk> \<Longrightarrow> a =[A] b: U i" and
- Equal_intro: "a: A \<Longrightarrow> (refl a): a =[A] a" and
+ Eq_intro: "a: A \<Longrightarrow> (refl a): a =[A] a" and
- Equal_elim: "\<lbrakk>
+ Eq_elim: "\<lbrakk>
p: x =[A] y;
- x: A;
- y: A;
+ a: A;
+ b: 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 =[A] y \<leadsto> U i \<rbrakk> \<Longrightarrow> indEq C f x y p : C x y p" and
+ \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C x y: x =[A] y \<leadsto> U i \<rbrakk> \<Longrightarrow> indEq C f a b p : C a b p" and
- Equal_comp: "\<lbrakk>
+ Eq_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 =[A] y \<leadsto> U i \<rbrakk> \<Longrightarrow> indEq C f a a (refl a) \<equiv> f a"
-lemmas Equal_form [form]
-lemmas Equal_routine [intro] = Equal_form Equal_intro Equal_elim
-lemmas Equal_comp [comp]
+lemmas Eq_form [form]
+lemmas Eq_routine [intro] = Eq_form Eq_intro Eq_elim
+lemmas Eq_comp [comp]
section \<open>Path induction\<close>
text \<open>We set up rudimentary automation of path induction:\<close>
-method path_ind for x y p :: t =
- (rule Equal_elim[where ?x=x and ?y=y and ?p=p]; (fact | assumption)?)
+method path_ind for a b p :: t =
+ (rule Eq_elim[where ?a=a and ?b=b and ?p=p]; (fact | assumption)?)
section \<open>Properties of equality\<close>
@@ -120,14 +120,13 @@ lemma pathcomp_type:
shows "pathcomp A x y z p q: x =[A] z"
unfolding pathcomp_def by (routine add: transitivity assms)
-lemma pathcomp_comp:
+lemma pathcomp_cmp:
assumes "A: U i" and "a: A"
shows "pathcomp A a a a (refl a) (refl a) \<equiv> refl a"
unfolding pathcomp_def by (derive lems: assms)
-declare
- pathcomp_type [intro]
- pathcomp_comp [comp]
+lemmas pathcomp_type [intro]
+lemmas pathcomp_comp [comp] = pathcomp_cmp
section \<open>Higher groupoid structure of types\<close>
@@ -148,23 +147,23 @@ proof (path_ind x y p)
by (derive lems: assms)
qed (routine add: assms)
-schematic_goal pathcomp_invl:
+schematic_goal pathcomp_invr:
assumes "A: U i" "x: A" "y: A" "p: x =[A] y"
- shows "?a: pathcomp A y x y (inv A x y p) p =[y =[A] y] refl(y)"
+ shows "?a: 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 (inv A x x (refl 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 lems: assms)
qed (routine add: assms)
-schematic_goal pathcomp_right_inv:
+schematic_goal pathcomp_invl:
assumes "A: U i" "x: A" "y: A" "p: x =[A] y"
- shows "?a: pathcomp A x y x p (inv A x y p) =[x =[A] x] (refl x)"
+ shows "?a: 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 (refl x) (inv A x 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 lems: assms)
qed (routine add: assms)
@@ -176,14 +175,45 @@ proof (path_ind x y p)
by (derive lems: assms)
qed (routine add: assms)
+declare[[pretty_pathcomp=false]]
+
schematic_goal pathcomp_assoc:
- fixes x y p
- assumes "A: U i" "x: A" "y: A" "z: A" "w: A" and "p: x =[A] y" "q: y =[A] z" "r: z =[A] w"
- shows
- "?a:
+ assumes "A: U i" "x: A" "y: A" "z: A" "w: A" "p: x =[A] y"
+ shows
+ "?a: \<Prod>q: y =[A] z. \<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"
-proof
- fix x y p
+proof (path_ind x y p)
+ oops
+ schematic_goal l:
+ assumes "A: U i" "x: A" "z: A" "w: A" "q: x =[A] z"
+ shows
+ "?a: \<Prod>r: z =[A] w. pathcomp A x x w (refl x) (pathcomp A y z w q r) =[x =[A] w]
+ pathcomp A x z w (pathcomp A x x z (refl x) q) r"
+ proof (path_ind x z q)
+ oops
+ schematic_goal l':
+ assumes "A: U i" "x: A" "w: A" "r: x =[A] w"
+ shows
+ "?a: 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"
+ proof (path_ind x w r)
+ 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)"
+ by (derive lems: assms)
+ qed (routine add: assms)
+
+(* Possible todo:
+ implement a custom version of schematic_goal/theorem that exports the derived proof term.
+*)
+
+definition l'_prftm :: "[t, t, t, t] \<Rightarrow> t"
+where "l'_prftm A x w r\<equiv>
+ indEq
+ (\<lambda>a b c. pathcomp A a a b (refl a) (pathcomp A a a b (refl a) c) =[a =[A] b]
+ pathcomp A a a b (pathcomp A a a a (refl a) (refl a)) c)
+ (\<lambda>x. refl (refl x)) x w r"
+
end