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chapter \<open>Lists\<close>
theory List
imports Maybe
begin
(*TODO: Inductive type and recursive function definitions. The ad-hoc
axiomatization below should be subsumed once general inductive types are
properly implemented.*)
axiomatization
List :: \<open>o \<Rightarrow> o\<close> and
nil :: \<open>o \<Rightarrow> o\<close> and
cons :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> and
ListInd :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> (o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o\<close>
where
ListF: "A: U i \<Longrightarrow> List A: U i" and
List_nil: "A: U i \<Longrightarrow> nil A: List A" and
List_cons: "\<lbrakk>x: A; xs: List A\<rbrakk> \<Longrightarrow> cons A x xs: List A" and
ListE: "\<lbrakk>
xs: List A;
c\<^sub>0: C (nil A);
\<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 (fn xs. C xs) c\<^sub>0 (fn x xs rec. f x xs rec) xs: C xs" and
List_comp_nil: "\<lbrakk>
c\<^sub>0: C (nil A);
\<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 (fn xs. C xs) c\<^sub>0 (fn 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 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 (fn xs. C xs) c\<^sub>0 (fn x xs rec. f x xs rec) (cons A x xs) \<equiv>
f x xs (ListInd A (fn xs. C xs) c\<^sub>0 (fn x xs rec. f x xs rec) xs)"
lemmas
[form] = ListF and
[intr, intro] = List_nil List_cons and
[elim "?xs"] = ListE and
[comp] = List_comp_nil List_comp_cons
abbreviation "ListRec A C \<equiv> ListInd A (fn _. C)"
Lemma list_cases [cases]:
assumes
"xs: List A" and
nil_case: "c\<^sub>0: C (nil A)" and
cons_case: "\<And>x xs. \<lbrakk>x: A; xs: List A\<rbrakk> \<Longrightarrow> f x xs: C (cons A x xs)" and
"\<And>xs. xs: List A \<Longrightarrow> C xs: U i"
shows "C xs"
by (elim xs) (fact nil_case, rule cons_case)
section \<open>Notation\<close>
definition nil_i ("[]")
where [implicit]: "[] \<equiv> nil {}"
definition cons_i (infixr "#" 120)
where [implicit]: "x # xs \<equiv> cons {} x xs"
translations
"[]" \<leftharpoondown> "CONST List.nil A"
"x # xs" \<leftharpoondown> "CONST List.cons A x xs"
syntax
"_list" :: \<open>args \<Rightarrow> o\<close> ("[_]")
translations
"[x, xs]" \<rightleftharpoons> "x # [xs]"
"[x]" \<rightleftharpoons> "x # []"
section \<open>Standard functions\<close>
subsection \<open>Head and tail\<close>
Definition head:
assumes "A: U i" "xs: List A"
shows "Maybe A"
proof (cases xs)
show "none: Maybe A" by intro
show "\<And>x. x: A \<Longrightarrow> some x: Maybe A" by intro
qed
Definition tail:
assumes "A: U i" "xs: List A"
shows "List A"
proof (cases xs)
show "[]: List A" by intro
show "\<And>xs. xs: List A \<Longrightarrow> xs: List A" .
qed
definition head_i ("head") where [implicit]: "head xs \<equiv> List.head {} xs"
definition tail_i ("tail") where [implicit]: "tail xs \<equiv> List.tail {} xs"
translations
"head" \<leftharpoondown> "CONST List.head A"
"tail" \<leftharpoondown> "CONST List.tail A"
Lemma head_type [type]:
assumes "A: U i" "xs: List A"
shows "head xs: Maybe A"
unfolding head_def by typechk
Lemma head_of_cons [comp]:
assumes "A: U i" "x: A" "xs: List A"
shows "head (x # xs) \<equiv> some x"
unfolding head_def by compute
Lemma tail_type [type]:
assumes "A: U i" "xs: List A"
shows "tail xs: List A"
unfolding tail_def by typechk
Lemma tail_of_cons [comp]:
assumes "A: U i" "x: A" "xs: List A"
shows "tail (x # xs) \<equiv> xs"
unfolding tail_def by compute
subsection \<open>Append\<close>
Definition app:
assumes "A: U i" "xs: List A" "ys: List A"
shows "List A"
apply (elim xs)
\<^item> by (fact \<open>ys:_\<close>)
\<^item> vars x _ rec
proof - show "x # rec: List A" by typechk qed
done
definition app_i ("app") where [implicit]: "app xs ys \<equiv> List.app {} xs ys"
translations "app" \<leftharpoondown> "CONST List.app A"
subsection \<open>Map\<close>
Definition map:
assumes "A: U i" "B: U i" "f: A \<rightarrow> B" "xs: List A"
shows "List B"
proof (elim xs)
show "[]: List B" by intro
next fix x ys
assuming "x: A" "ys: List B"
show "f x # ys: List B" by typechk
qed
definition map_i ("map") where [implicit]: "map \<equiv> List.map {} {}"
translations "map" \<leftharpoondown> "CONST List.map A B"
Lemma map_type [type]:
assumes "A: U i" "B: U i" "f: A \<rightarrow> B" "xs: List A"
shows "map f xs: List B"
unfolding map_def by typechk
subsection \<open>Reverse\<close>
Definition rev:
assumes "A: U i" "xs: List A"
shows "List A"
apply (elim xs)
\<^item> by (rule List_nil)
\<^item> vars x _ rec proof - show "app rec [x]: List A" by typechk qed
done
definition rev_i ("rev") where [implicit]: "rev \<equiv> List.rev {}"
translations "rev" \<leftharpoondown> "CONST List.rev A"
Lemma rev_type [type]:
assumes "A: U i" "xs: List A"
shows "rev xs: List A"
unfolding rev_def by typechk
Lemma rev_nil [comp]:
assumes "A: U i"
shows "rev (nil A) \<equiv> nil A"
unfolding rev_def by compute
end
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