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diff --git a/mltt/lib/Prelude.thy b/mltt/lib/Prelude.thy
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+theory Prelude
+imports MLTT
+
+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
+
+abbreviation (input) Not ("\<not>_" [1000] 1000) where "\<not>A \<equiv> A \<rightarrow> \<bottom>"
+
+
+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