rename things + some small changes

3 files changed, 0 insertions, 417 deletions

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
diff --git a/spartan/lib/Maybe.thy b/spartan/lib/Maybe.thy
deleted file mode 100644
index 452acc2..0000000
--- a/spartan/lib/Maybe.thy
+++ /dev/null
@@ -1,75 +0,0 @@
-chapter \<open>Maybe type\<close>
-
-theory Maybe
-imports Prelude
-
-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"
-
-lemma
-  MaybeF: "A: U i \<Longrightarrow> Maybe A: U i" and
-  Maybe_none: "A: U i \<Longrightarrow> none A: Maybe A" and
-  Maybe_some: "a: A \<Longrightarrow> some A a: Maybe A"
-  unfolding Maybe_def none_def some_def by typechk+
-
-Definition MaybeInd:
-  assumes
-    "A: U i"
-    "\<And>m. m: Maybe A \<Longrightarrow> C m: U i"
-    "c\<^sub>0: C (none A)"
-    "\<And>a. a: A \<Longrightarrow> f a: C (some A a)"
-    "m: Maybe A"
-  shows "C m"
-  using assms[unfolded Maybe_def none_def some_def, type]
-  apply (elim m)
-    apply fact
-    apply (elim, fact)
-  done
-
-Lemma Maybe_comp_none:
-  assumes
-    "A: U i"
-    "c\<^sub>0: C (none A)"
-    "\<And>a. a: A \<Longrightarrow> f a: C (some A a)"
-    "\<And>m. m: Maybe A \<Longrightarrow> C m: U i"
-  shows "MaybeInd A C c\<^sub>0 f (none A) \<equiv> c\<^sub>0"
-  using assms
-  unfolding Maybe_def MaybeInd_def none_def some_def
-  by compute
-
-Lemma Maybe_comp_some:
-  assumes
-    "A: U i"
-    "a: A"
-    "c\<^sub>0: C (none A)"
-    "\<And>a. a: A \<Longrightarrow> f a: C (some A a)"
-    "\<And>m. m: Maybe A \<Longrightarrow> C m: U i"
-  shows "MaybeInd A C c\<^sub>0 f (some A a) \<equiv> f a"
-  using assms
-  unfolding Maybe_def MaybeInd_def none_def some_def
-  by compute
-
-lemmas
-  [form] = MaybeF and
-  [intr, intro] = Maybe_none Maybe_some and
-  [comp] = Maybe_comp_none Maybe_comp_some and
-  MaybeE [elim "?m"] = MaybeInd[rotated 4]
-lemmas
-  Maybe_cases [cases] = MaybeE
-
-abbreviation "MaybeRec A C \<equiv> MaybeInd A (K C)"
-
-definition none_i ("none") where [implicit]: "none \<equiv> Maybe.none {}"
-definition some_i ("some") where [implicit]: "some a \<equiv> Maybe.some {} a"
-
-translations
-  "none" \<leftharpoondown> "CONST Maybe.none A"
-  "some a" \<leftharpoondown> "CONST Maybe.some A a"
-
-
-end
diff --git a/spartan/lib/Prelude.thy b/spartan/lib/Prelude.thy
deleted file mode 100644
index 56adbfc..0000000
--- a/spartan/lib/Prelude.thy
+++ /dev/null
@@ -1,151 +0,0 @@
-theory Prelude
-imports Spartan
-
-begin
-
-section \<open>Sum type\<close>
-
-axiomatization
-  Sum    :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> and
-  inl    :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> and
-  inr    :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> and
-  SumInd :: \<open>o \<Rightarrow> o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o\<close>
-
-notation Sum (infixl "\<or>" 50)
-
-axiomatization where
-  SumF: "\<lbrakk>A: U i; B: U i\<rbrakk> \<Longrightarrow> A \<or> B: U i" and
-
-  Sum_inl: "\<lbrakk>B: U i; a: A\<rbrakk> \<Longrightarrow> inl A B a: A \<or> B" and
-
-  Sum_inr: "\<lbrakk>A: U i; b: B\<rbrakk> \<Longrightarrow> inr A B b: A \<or> B" and
-
-  SumE: "\<lbrakk>
-    s: A \<or> B;
-    \<And>s. s: A \<or> B \<Longrightarrow> C s: U i;
-    \<And>a. a: A \<Longrightarrow> c a: C (inl A B a);
-    \<And>b. b: B \<Longrightarrow> d b: C (inr A B b)
-    \<rbrakk> \<Longrightarrow> SumInd A B (fn s. C s) (fn a. c a) (fn b. d b) s: C s" and
-
-  Sum_comp_inl: "\<lbrakk>
-    a: A;
-    \<And>s. s: A \<or> B \<Longrightarrow> C s: U i;
-    \<And>a. a: A \<Longrightarrow> c a: C (inl A B a);
-    \<And>b. b: B \<Longrightarrow> d b: C (inr A B b)
-    \<rbrakk> \<Longrightarrow> SumInd A B (fn s. C s) (fn a. c a) (fn b. d b) (inl A B a) \<equiv> c a" and
-
-  Sum_comp_inr: "\<lbrakk>
-    b: B;
-    \<And>s. s: A \<or> B \<Longrightarrow> C s: U i;
-    \<And>a. a: A \<Longrightarrow> c a: C (inl A B a);
-    \<And>b. b: B \<Longrightarrow> d b: C (inr A B b)
-    \<rbrakk> \<Longrightarrow> SumInd A B (fn s. C s) (fn a. c a) (fn b. d b) (inr A B b) \<equiv> d b"
-
-lemmas
-  [form] = SumF and
-  [intr] = Sum_inl Sum_inr and
-  [intro] = Sum_inl[rotated] Sum_inr[rotated] and
-  [elim ?s] = SumE and
-  [comp] = Sum_comp_inl Sum_comp_inr
-
-method left = rule Sum_inl
-method right = rule Sum_inr
-
-
-section \<open>Empty and unit types\<close>
-
-axiomatization
-  Top    :: \<open>o\<close> and
-  tt     :: \<open>o\<close> and
-  TopInd :: \<open>(o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close>
-and
-  Bot    :: \<open>o\<close> and
-  BotInd :: \<open>(o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o\<close>
-
-notation Top ("\<top>") and Bot ("\<bottom>")
-
-axiomatization where
-  TopF: "\<top>: U i" and
-
-  TopI: "tt: \<top>" and
-
-  TopE: "\<lbrakk>a: \<top>; \<And>x. x: \<top> \<Longrightarrow> C x: U i; c: C tt\<rbrakk> \<Longrightarrow> TopInd (fn x. C x) c a: C a" and
-
-  Top_comp: "\<lbrakk>\<And>x. x: \<top> \<Longrightarrow> C x: U i; c: C tt\<rbrakk> \<Longrightarrow> TopInd (fn x. C x) c tt \<equiv> c"
-and
-  BotF: "\<bottom>: U i" and
-
-  BotE: "\<lbrakk>x: \<bottom>; \<And>x. x: \<bottom> \<Longrightarrow> C x: U i\<rbrakk> \<Longrightarrow> BotInd (fn x. C x) x: C x"
-
-lemmas
-  [form] = TopF BotF and
-  [intr, intro] = TopI and
-  [elim ?a] = TopE and
-  [elim ?x] = BotE and
-  [comp] = Top_comp
-
-
-section \<open>Booleans\<close>
-
-definition "Bool \<equiv> \<top> \<or> \<top>"
-definition "true \<equiv> inl \<top> \<top> tt"
-definition "false \<equiv> inr \<top> \<top> tt"
-
-Lemma
-  BoolF: "Bool: U i" and
-  Bool_true: "true: Bool" and
-  Bool_false: "false: Bool"
-  unfolding Bool_def true_def false_def by typechk+
-
-\<comment> \<open>Definitions like these should be handled by a future function package\<close>
-Definition ifelse [rotated 1]:
-  assumes *[unfolded Bool_def true_def false_def]:
-    "\<And>x. x: Bool \<Longrightarrow> C x: U i"
-    "x: Bool"
-    "a: C true"
-    "b: C false"
-  shows "C x"
-  using assms[unfolded Bool_def true_def false_def, type]
-  by (elim x) (elim, fact)+
-
-Lemma if_true:
-  assumes
-    "\<And>x. x: Bool \<Longrightarrow> C x: U i"
-    "a: C true"
-    "b: C false"
-  shows "ifelse C true a b \<equiv> a"
-  unfolding ifelse_def true_def
-  using assms unfolding Bool_def true_def false_def
-  by compute
-
-Lemma if_false:
-  assumes
-    "\<And>x. x: Bool \<Longrightarrow> C x: U i"
-    "a: C true"
-    "b: C false"
-  shows "ifelse C false a b \<equiv> b"
-  unfolding ifelse_def false_def
-  using assms unfolding Bool_def true_def false_def
-  by compute
-
-lemmas
-  [form] = BoolF and
-  [intr, intro] = Bool_true Bool_false and
-  [comp] = if_true if_false and
-  [elim ?x] = ifelse
-lemmas
-  BoolE = ifelse
-
-subsection \<open>Notation\<close>
-
-definition ifelse_i ("if _ then _ else _")
-  where [implicit]: "if x then a else b \<equiv> ifelse {} x a b"
-
-translations "if x then a else b" \<leftharpoondown> "CONST ifelse C x a b"
-
-subsection \<open>Logical connectives\<close>
-
-definition not ("!_") where "!x \<equiv> ifelse (K Bool) x false true"
-
-
-end