# Theory Bool

```(*  Title:      ZF/Bool.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section‹Booleans in Zermelo-Fraenkel Set Theory›

theory Bool imports pair begin

abbreviation
one  (‹1›) where
"1 ≡ succ(0)"

abbreviation
two  (‹2›) where
"2 ≡ succ(1)"

text‹2 is equal to bool, but is used as a number rather than a type.›

definition "bool ≡ {0,1}"

definition "cond(b,c,d) ≡ if(b=1,c,d)"

definition "not(b) ≡ cond(b,0,1)"

definition
"and"       :: "[i,i]⇒i"      (infixl ‹and› 70)  where
"a and b ≡ cond(a,b,0)"

definition
or          :: "[i,i]⇒i"      (infixl ‹or› 65)  where
"a or b ≡ cond(a,1,b)"

definition
xor         :: "[i,i]⇒i"      (infixl ‹xor› 65) where
"a xor b ≡ cond(a,not(b),b)"

lemmas bool_defs = bool_def cond_def

lemma singleton_0: "{0} = 1"

(* Introduction rules *)

lemma bool_1I [simp,TC]: "1 ∈ bool"

lemma bool_0I [simp,TC]: "0 ∈ bool"

lemma one_not_0: "1≠0"

(** 1=0 ⟹ R **)
lemmas one_neq_0 = one_not_0 [THEN notE]

lemma boolE:
"⟦c: bool;  c=1 ⟹ P;  c=0 ⟹ P⟧ ⟹ P"

(** cond **)

(*1 means true*)
lemma cond_1 [simp]: "cond(1,c,d) = c"

(*0 means false*)
lemma cond_0 [simp]: "cond(0,c,d) = d"

lemma cond_type [TC]: "⟦b: bool;  c: A(1);  d: A(0)⟧ ⟹ cond(b,c,d): A(b)"

(*For Simp_tac and Blast_tac*)
lemma cond_simple_type: "⟦b: bool;  c: A;  d: A⟧ ⟹ cond(b,c,d): A"

lemma def_cond_1: "⟦⋀b. j(b)≡cond(b,c,d)⟧ ⟹ j(1) = c"
by simp

lemma def_cond_0: "⟦⋀b. j(b)≡cond(b,c,d)⟧ ⟹ j(0) = d"
by simp

lemmas not_1 = not_def [THEN def_cond_1, simp]
lemmas not_0 = not_def [THEN def_cond_0, simp]

lemmas and_1 = and_def [THEN def_cond_1, simp]
lemmas and_0 = and_def [THEN def_cond_0, simp]

lemmas or_1 = or_def [THEN def_cond_1, simp]
lemmas or_0 = or_def [THEN def_cond_0, simp]

lemmas xor_1 = xor_def [THEN def_cond_1, simp]
lemmas xor_0 = xor_def [THEN def_cond_0, simp]

lemma not_type [TC]: "a:bool ⟹ not(a) ∈ bool"

lemma and_type [TC]: "⟦a:bool;  b:bool⟧ ⟹ a and b ∈ bool"

lemma or_type [TC]: "⟦a:bool;  b:bool⟧ ⟹ a or b ∈ bool"

lemma xor_type [TC]: "⟦a:bool;  b:bool⟧ ⟹ a xor b ∈ bool"

lemmas bool_typechecks = bool_1I bool_0I cond_type not_type and_type
or_type xor_type

lemma not_not [simp]: "a:bool ⟹ not(not(a)) = a"
by (elim boolE, auto)

lemma not_and [simp]: "a:bool ⟹ not(a and b) = not(a) or not(b)"
by (elim boolE, auto)

lemma not_or [simp]: "a:bool ⟹ not(a or b) = not(a) and not(b)"
by (elim boolE, auto)

lemma and_absorb [simp]: "a: bool ⟹ a and a = a"
by (elim boolE, auto)

lemma and_commute: "⟦a: bool; b:bool⟧ ⟹ a and b = b and a"
by (elim boolE, auto)

lemma and_assoc: "a: bool ⟹ (a and b) and c  =  a and (b and c)"
by (elim boolE, auto)

lemma and_or_distrib: "⟦a: bool; b:bool; c:bool⟧ ⟹
(a or b) and c  =  (a and c) or (b and c)"
by (elim boolE, auto)

lemma or_absorb [simp]: "a: bool ⟹ a or a = a"
by (elim boolE, auto)

lemma or_commute: "⟦a: bool; b:bool⟧ ⟹ a or b = b or a"
by (elim boolE, auto)

lemma or_assoc: "a: bool ⟹ (a or b) or c  =  a or (b or c)"
by (elim boolE, auto)

lemma or_and_distrib: "⟦a: bool; b: bool; c: bool⟧ ⟹
(a and b) or c  =  (a or c) and (b or c)"
by (elim boolE, auto)

definition
bool_of_o :: "o⇒i" where
"bool_of_o(P) ≡ (if P then 1 else 0)"

lemma [simp]: "bool_of_o(True) = 1"

lemma [simp]: "bool_of_o(False) = 0"