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-rw-r--r--EqualProps.thy135
1 files changed, 106 insertions, 29 deletions
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
generated by cgit v1.2.3 (git 2.25.1) at 2024-07-28 17:15:03 +0000