src/ZF/Bool.thy
 author paulson Tue, 09 Jul 2002 23:05:26 +0200 changeset 13328 703de709a64b parent 13269 3ba9be497c33 child 13356 c9cfe1638bf2 permissions -rw-r--r--
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```
(*  Title:      ZF/bool.thy
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

*)

theory Bool = pair:

syntax
"1"         :: i             ("1")
"2"         :: i             ("2")

translations
"1"  == "succ(0)"
"2"  == "succ(1)"

text{*2 is equal to bool, but is used as a number rather than a type.*}

constdefs
bool        :: i
"bool == {0,1}"

cond        :: "[i,i,i]=>i"
"cond(b,c,d) == if(b=1,c,d)"

not         :: "i=>i"
"not(b) == cond(b,0,1)"

"and"       :: "[i,i]=>i"      (infixl "and" 70)
"a and b == cond(a,b,0)"

or          :: "[i,i]=>i"      (infixl "or" 65)
"a or b == cond(a,1,b)"

xor         :: "[i,i]=>i"      (infixl "xor" 65)
"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, standard]

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, standard, simp]
lemmas not_0 = not_def [THEN def_cond_0, standard, simp]

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

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

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

(*** Laws for '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)

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)

(** binary '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)

constdefs bool_of_o :: "o=>i"
"bool_of_o(P) == (if P then 1 else 0)"

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

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

lemma [simp,TC]: "bool_of_o(P) \<in> bool"

lemma [simp]: "(bool_of_o(P) = 1) <-> P"

lemma [simp]: "(bool_of_o(P) = 0) <-> ~P"

ML
{*
val bool_def = thm "bool_def";

val bool_defs = thms "bool_defs";
val singleton_0 = thm "singleton_0";
val bool_1I = thm "bool_1I";
val bool_0I = thm "bool_0I";
val one_not_0 = thm "one_not_0";
val one_neq_0 = thm "one_neq_0";
val boolE = thm "boolE";
val cond_1 = thm "cond_1";
val cond_0 = thm "cond_0";
val cond_type = thm "cond_type";
val cond_simple_type = thm "cond_simple_type";
val def_cond_1 = thm "def_cond_1";
val def_cond_0 = thm "def_cond_0";
val not_1 = thm "not_1";
val not_0 = thm "not_0";
val and_1 = thm "and_1";
val and_0 = thm "and_0";
val or_1 = thm "or_1";
val or_0 = thm "or_0";
val xor_1 = thm "xor_1";
val xor_0 = thm "xor_0";
val not_type = thm "not_type";
val and_type = thm "and_type";
val or_type = thm "or_type";
val xor_type = thm "xor_type";
val bool_typechecks = thms "bool_typechecks";
val not_not = thm "not_not";
val not_and = thm "not_and";
val not_or = thm "not_or";
val and_absorb = thm "and_absorb";
val and_commute = thm "and_commute";
val and_assoc = thm "and_assoc";
val and_or_distrib = thm "and_or_distrib";
val or_absorb = thm "or_absorb";
val or_commute = thm "or_commute";
val or_assoc = thm "or_assoc";
val or_and_distrib = thm "or_and_distrib";
*}

end
```