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-rw-r--r--hott/Nat.thy14
-rw-r--r--spartan/data/List.thy35
-rw-r--r--spartan/data/Maybe.thy4
-rw-r--r--spartan/data/More_Types.thy3
4 files changed, 26 insertions, 30 deletions
diff --git a/hott/Nat.thy b/hott/Nat.thy
index c36154e..e8657a4 100644
--- a/hott/Nat.thy
+++ b/hott/Nat.thy
@@ -18,24 +18,24 @@ where
NatE: "\<lbrakk>
n: Nat;
c\<^sub>0: C 0;
- \<And>k c. \<lbrakk>k: Nat; c: C k\<rbrakk> \<Longrightarrow> f k c: C (suc k);
+ \<And>k rec. \<lbrakk>k: Nat; rec: C k\<rbrakk> \<Longrightarrow> f k rec: C (suc k);
\<And>n. n: Nat \<Longrightarrow> C n: U i
- \<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k c. f k c) n: C n" and
+ \<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k rec. f k rec) n: C n" and
Nat_comp_zero: "\<lbrakk>
c\<^sub>0: C 0;
- \<And>k c. \<lbrakk>k: Nat; c: C k\<rbrakk> \<Longrightarrow> f k c: C (suc k);
+ \<And>k rec. \<lbrakk>k: Nat; rec: C k\<rbrakk> \<Longrightarrow> f k rec: C (suc k);
\<And>n. n: Nat \<Longrightarrow> C n: U i
- \<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k c. f k c) 0 \<equiv> c\<^sub>0" and
+ \<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k rec. f k rec) 0 \<equiv> c\<^sub>0" and
Nat_comp_suc: "\<lbrakk>
n: Nat;
c\<^sub>0: C 0;
- \<And>k c. \<lbrakk>k: Nat; c: C k\<rbrakk> \<Longrightarrow> f k c: C (suc k);
+ \<And>k rec. \<lbrakk>k: Nat; rec: C k\<rbrakk> \<Longrightarrow> f k rec: C (suc k);
\<And>n. n: Nat \<Longrightarrow> C n: U i
\<rbrakk> \<Longrightarrow>
- NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k c. f k c) (suc n) \<equiv>
- f n (NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k c. f k c) n)"
+ NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k rec. f k rec) (suc n) \<equiv>
+ f n (NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k rec. f k rec) n)"
lemmas
[intros] = NatF Nat_zero Nat_suc and
diff --git a/spartan/data/List.thy b/spartan/data/List.thy
index 3399da6..52f042b 100644
--- a/spartan/data/List.thy
+++ b/spartan/data/List.thy
@@ -1,3 +1,5 @@
+chapter \<open>Lists\<close>
+
theory List
imports Maybe
@@ -22,24 +24,24 @@ where
ListE: "\<lbrakk>
xs: List A;
c\<^sub>0: C (nil A);
- \<And>x xs c. \<lbrakk>x: A; xs: List A; c: C xs\<rbrakk> \<Longrightarrow> f x xs c: C (cons A x xs);
+ \<And>x xs rec. \<lbrakk>x: A; xs: List A; rec: C xs\<rbrakk> \<Longrightarrow> f x xs rec: C (cons A x xs);
\<And>xs. xs: List A \<Longrightarrow> C xs: U i
- \<rbrakk> \<Longrightarrow> ListInd A (\<lambda>xs. C xs) c\<^sub>0 (\<lambda>x xs c. f x xs c) xs: C xs" and
+ \<rbrakk> \<Longrightarrow> ListInd A (\<lambda>xs. C xs) c\<^sub>0 (\<lambda>x xs rec. f x xs rec) xs: C xs" and
List_comp_nil: "\<lbrakk>
c\<^sub>0: C (nil A);
- \<And>x xs c. \<lbrakk>x: A; xs: List A; c: C xs\<rbrakk> \<Longrightarrow> f x xs c: C (cons A x xs);
+ \<And>x xs rec. \<lbrakk>x: A; xs: List A; rec: C xs\<rbrakk> \<Longrightarrow> f x xs rec: C (cons A x xs);
\<And>xs. xs: List A \<Longrightarrow> C xs: U i
- \<rbrakk> \<Longrightarrow> ListInd A (\<lambda>xs. C xs) c\<^sub>0 (\<lambda>x xs c. f x xs c) (nil A) \<equiv> c\<^sub>0" and
+ \<rbrakk> \<Longrightarrow> ListInd A (\<lambda>xs. C xs) c\<^sub>0 (\<lambda>x xs rec. f x xs rec) (nil A) \<equiv> c\<^sub>0" and
List_comp_cons: "\<lbrakk>
xs: List A;
c\<^sub>0: C (nil A);
- \<And>x xs c. \<lbrakk>x: A; xs: List A; c: C xs\<rbrakk> \<Longrightarrow> f x xs c: C (cons A x xs);
+ \<And>x xs rec. \<lbrakk>x: A; xs: List A; rec: C xs\<rbrakk> \<Longrightarrow> f x xs rec: C (cons A x xs);
\<And>xs. xs: List A \<Longrightarrow> C xs: U i
\<rbrakk> \<Longrightarrow>
- ListInd A (\<lambda>xs. C xs) c\<^sub>0 (\<lambda>x xs c. f x xs c) (cons A x xs) \<equiv>
- f x xs (ListInd A (\<lambda>xs. C xs) c\<^sub>0 (\<lambda>x xs c. f x xs c) xs)"
+ ListInd A (\<lambda>xs. C xs) c\<^sub>0 (\<lambda>x xs rec. f x xs rec) (cons A x xs) \<equiv>
+ f x xs (ListInd A (\<lambda>xs. C xs) c\<^sub>0 (\<lambda>x xs rec. f x xs rec) xs)"
lemmas
[intros] = ListF List_nil List_cons and
@@ -70,11 +72,6 @@ translations
section \<open>Standard functions\<close>
-\<comment> \<open>
-definition "head A \<equiv> \<lambda>xs: List A.
- ListRec A (Maybe A) (Maybe.none A) (\<lambda>x _ _. Maybe.some A x) xs"
-\<close>
-
Lemma (derive) head:
assumes "A: U i" "xs: List A"
shows "Maybe A"
@@ -83,11 +80,8 @@ proof (elim xs)
show "\<And>x. x: A \<Longrightarrow> some x: Maybe A" by intro
qed
-thm head_def \<comment> \<open>head ?A ?xs \<equiv> ListRec ?A (Maybe ?A) none (\<lambda>x xs c. some x) ?xs\<close>
-
-\<comment> \<open>
-definition "tail A \<equiv> \<lambda>xs: List A. ListRec A (List A) (nil A) (\<lambda>x xs _. xs) xs"
-\<close>
+thm head_def
+ \<comment> \<open>head ?A ?xs \<equiv> ListRec ?A (Maybe ?A) none (\<lambda>x xs c. some x) ?xs\<close>
Lemma (derive) tail:
assumes "A: U i" "xs: List A"
@@ -98,11 +92,7 @@ proof (elim xs)
qed
thm tail_def
-
-\<comment> \<open>
-definition "map A B \<equiv> \<lambda>f: A \<rightarrow> B. \<lambda>xs: List A.
- ListRec A (List B) (nil B) (\<lambda>x _ c. cons B (f `x) c) xs"
-\<close>
+ \<comment> \<open>tail ?A ?xs \<equiv> ListRec ?A (List ?A) [] (\<lambda>x xs c. xs) ?xs\<close>
Lemma (derive) map:
assumes "A: U i" "B: U i" "f: A \<rightarrow> B" "xs: List A"
@@ -115,6 +105,7 @@ proof (elim xs)
qed
thm map_def
+ \<comment> \<open>map ?A ?B ?f ?xs \<equiv> ListRec ?A (List ?B) [] (\<lambda>x xs. cons ?B (?f x)) ?xs\<close>
definition head_i ("head")
where [implicit]: "head xs \<equiv> List.head ? xs"
diff --git a/spartan/data/Maybe.thy b/spartan/data/Maybe.thy
index dc3ad04..98f5283 100644
--- a/spartan/data/Maybe.thy
+++ b/spartan/data/Maybe.thy
@@ -1,8 +1,12 @@
+chapter \<open>Maybe type\<close>
+
theory Maybe
imports More_Types
begin
+text \<open>Defined as a sum.\<close>
+
definition "Maybe A \<equiv> A \<or> \<top>"
definition "none A \<equiv> inr A \<top> tt"
definition "some A a \<equiv> inl A \<top> a"
diff --git a/spartan/data/More_Types.thy b/spartan/data/More_Types.thy
index f0cee6d..fecf378 100644
--- a/spartan/data/More_Types.thy
+++ b/spartan/data/More_Types.thy
@@ -1,9 +1,10 @@
+chapter \<open>Some standard types\<close>
+
theory More_Types
imports Spartan
begin
-
section \<open>Sum type\<close>
axiomatization