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-rw-r--r--EqualProps.thy66
1 files changed, 33 insertions, 33 deletions
diff --git a/EqualProps.thy b/EqualProps.thy
index 8b83c5b..a1d4c45 100644
--- a/EqualProps.thy
+++ b/EqualProps.thy
@@ -24,7 +24,7 @@ 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"
unfolding inv_def
-by (rule Equal_elim[where ?x=x and ?y=y]) (simple lems: assms)
+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>
text "
@@ -35,9 +35,9 @@ text "
lemma inv_comp:
assumes "A : U(i)" and "a : A" shows "(refl a)\<inverse> \<equiv> refl(a)"
unfolding inv_def
-proof
+proof compute
show "\<And>x. x: A \<Longrightarrow> refl x: x =\<^sub>A x" ..
-qed (simple lems: assms)
+qed (routine lems: assms)
section \<open>Transitivity / Path composition\<close>
@@ -72,18 +72,18 @@ proof
proof
fix u z q assume asm: "u: A" "z: A" "q: u =\<^sub>A z"
show "ind\<^sub>= refl q: u =\<^sub>A z"
- by (rule Equal_elim[where ?x=u and ?y=z]) (simple lems: assms asm)
- qed (simple lems: assms)
+ by (rule Equal_elim[where ?x=u and ?y=z]) (routine lems: assms asm)
+ qed (routine lems: assms)
qed (rule assms)
- qed (simple lems: assms 1 2 3)
- qed (simple lems: assms 1 2)
+ qed (routine lems: assms 1 2 3)
+ qed (routine lems: assms 1 2)
qed (rule assms)
qed fact
corollary
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 (simple lems: assms rpathcomp_type)
+ by (routine lems: assms rpathcomp_type)
text "
The following proof is very long, chiefly because for every application of \<open>`\<close> we have to show the wellformedness of the type family appearing in the equality computation rule.
@@ -109,11 +109,11 @@ proof compute
proof
fix u z assume asm: "u: A" "z: A"
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]) (simple lems: assms asm)
- qed (simple lems: assms)
+ by (rule Equal_elim[where ?x=u and ?y=z]) (routine lems: assms asm)
+ qed (routine lems: assms)
qed (rule assms)
- qed (simple lems: assms 1 2 3)
- qed (simple lems: assms 1 2)
+ qed (routine lems: assms 1 2 3)
+ qed (routine lems: assms 1 2)
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)"
@@ -131,11 +131,11 @@ proof compute
show "\<^bold>\<lambda>q. ind\<^sub>= refl q: u =\<^sub>A z \<rightarrow> u =\<^sub>A z"
proof
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]) (simple lems: assms 3 4)
- qed (simple lems: assms 3 4)
+ by (rule Equal_elim[where ?x=u and ?y=z]) (routine lems: assms 3 4)
+ qed (routine lems: assms 3 4)
qed fact
- qed (simple lems: assms 1 2)
- qed (simple lems: assms 1) }
+ qed (routine lems: assms 1 2)
+ 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)"
proof compute
@@ -149,10 +149,10 @@ proof compute
show "\<^bold>\<lambda>q. ind\<^sub>= refl q: u =\<^sub>A z \<rightarrow> u =\<^sub>A z"
proof
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]) (simple lems: assms 2 3)
- qed (simple lems: assms 2 3)
+ by (rule Equal_elim[where ?x=u and ?y=z]) (routine lems: assms 2 3)
+ qed (routine lems: assms 2 3)
qed fact
- qed (simple 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)"
proof compute
@@ -163,8 +163,8 @@ proof compute
show "\<^bold>\<lambda>q. ind\<^sub>= refl q: u =\<^sub>A z \<rightarrow> u =\<^sub>A z"
proof
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]) (simple lems: assms 1 2)
- qed (simple lems: assms 1 2)
+ by (rule Equal_elim[where ?x=u and ?y=z]) (routine lems: assms 1 2)
+ qed (routine lems: assms 1 2)
qed fact }
show "(\<^bold>\<lambda>z q. ind\<^sub>= refl q)`a`refl(a) \<equiv> refl(a)"
@@ -173,21 +173,21 @@ proof compute
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]) (simple lems: assms 1)
- qed (simple lems: assms 1) }
+ by (rule Equal_elim[where ?x=a and ?y=a]) (routine lems: assms 1)
+ qed (routine lems: assms 1) }
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]) (simple lems: assms)
+ by (rule Equal_elim[where ?x=a and ?y=a]) (routine lems: assms)
show "ind\<^sub>= refl (refl(a)) \<equiv> refl(a)"
- proof
+ proof compute
show "\<And>x. x: A \<Longrightarrow> refl(x): x =\<^sub>A x" ..
- qed (simple lems: assms)
- qed (simple lems: assms)
+ qed (routine lems: assms)
+ qed (routine lems: assms)
qed fact
- qed (simple lems: assms)
- qed (simple lems: assms)
+ qed (routine lems: assms)
+ qed (routine lems: assms)
qed fact
qed fact
@@ -207,15 +207,15 @@ lemma pathcomp_type:
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+
-qed (simple lems: assms rpathcomp_type)
+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)"
-by (subst pathcomp_def) (simple lems: assms rpathcomp_comp)
+by (subst pathcomp_def) (routine lems: assms rpathcomp_comp)
-lemmas EqualProps_rules [intro] = inv_type pathcomp_type
-lemmas EqualProps_comps [comp] = inv_comp pathcomp_comp
+lemmas EqualProps_type [intro] = inv_type pathcomp_type
+lemmas EqualProps_comp [comp] = inv_comp pathcomp_comp
end
generated by cgit v1.2.3 (git 2.25.1) at 2024-07-05 22:42:50 +0000