# Theory Bool

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

sectionBooleans in Zermelo-Fraenkel Set Theory

theory Bool imports pair begin

abbreviation
one  (1) where
"1  succ(0)"

abbreviation
two  (2) where
"2  succ(1)"

text2 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: "10"

(** 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 :: "oi" 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"