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"
by (simp add: succ_def)
lemma bool_1I [simp,TC]: "1 ∈ bool"
by (simp add: bool_defs )
lemma bool_0I [simp,TC]: "0 ∈ bool"
by (simp add: bool_defs)
lemma one_not_0: "1≠0"
by (simp add: bool_defs )
lemmas one_neq_0 = one_not_0 [THEN notE]
lemma boolE:
"[| c: bool; c=1 ==> P; c=0 ==> P |] ==> P"
by (simp add: bool_defs, blast)
lemma cond_1 [simp]: "cond(1,c,d) = c"
by (simp add: bool_defs )
lemma cond_0 [simp]: "cond(0,c,d) = d"
by (simp add: bool_defs )
lemma cond_type [TC]: "[| b: bool; c: A(1); d: A(0) |] ==> cond(b,c,d): A(b)"
by (simp add: bool_defs, blast)
lemma cond_simple_type: "[| b: bool; c: A; d: A |] ==> cond(b,c,d): A"
by (simp add: bool_defs )
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"
by (simp add: not_def)
lemma and_type [TC]: "[| a:bool; b:bool |] ==> a and b ∈ bool"
by (simp add: and_def)
lemma or_type [TC]: "[| a:bool; b:bool |] ==> a or b ∈ bool"
by (simp add: or_def)
lemma xor_type [TC]: "[| a:bool; b:bool |] ==> a xor b ∈ bool"
by (simp add: xor_def)
lemmas bool_typechecks = bool_1I bool_0I cond_type not_type and_type
or_type xor_type
subsection‹Laws About 'not'›
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)
subsection‹Laws About 'and'›
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)
subsection‹Laws About 'or'›
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"
by (simp add: bool_of_o_def)
lemma [simp]: "bool_of_o(False) = 0"
by (simp add: bool_of_o_def)
lemma [simp,TC]: "bool_of_o(P) ∈ bool"
by (simp add: bool_of_o_def)
lemma [simp]: "(bool_of_o(P) = 1) ⟷ P"
by (simp add: bool_of_o_def)
lemma [simp]: "(bool_of_o(P) = 0) ⟷ ~P"
by (simp add: bool_of_o_def)
end