From 2feb56660700af107abb5a28a7120052ac405518 Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Sun, 31 Jan 2021 02:54:51 +0000
Subject: rename things + some small changes

---
 spartan/lib/List.thy | 191 ---------------------------------------------------
 1 file changed, 191 deletions(-)
 delete mode 100644 spartan/lib/List.thy

(limited to 'spartan/lib/List.thy')

diff --git a/spartan/lib/List.thy b/spartan/lib/List.thy
deleted file mode 100644
index 4beb9b6..0000000
--- a/spartan/lib/List.thy
+++ /dev/null
@@ -1,191 +0,0 @@
-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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