author  clasohm 
Wed, 07 Dec 1994 13:12:04 +0100  
changeset 760  f0200e91b272 
parent 129  dc50a4b96d7b 
child 1461  6bcb44e4d6e5 
permissions  rwrr 
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(* Title: ZF/bool 
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ID: $Id$ 

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Author: Martin D Coen, Cambridge University Computer Laboratory 

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Copyright 1992 University of Cambridge 

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For ZF/bool.thy. Booleans in ZermeloFraenkel Set Theory 

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*) 

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open Bool; 

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1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
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val bool_defs = [bool_def,cond_def]; 
0  12 

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(* Introduction rules *) 

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goalw Bool.thy bool_defs "1 : bool"; 

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by (rtac (consI1 RS consI2) 1); 

760  17 
qed "bool_1I"; 
0  18 

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goalw Bool.thy bool_defs "0 : bool"; 

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by (rtac consI1 1); 

760  21 
qed "bool_0I"; 
0  22 

37  23 
goalw Bool.thy bool_defs "1~=0"; 
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by (rtac succ_not_0 1); 
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qed "one_not_0"; 
0  26 

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(** 1=0 ==> R **) 

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val one_neq_0 = one_not_0 RS notE; 

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val major::prems = goalw Bool.thy bool_defs 
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"[ c: bool; c=1 ==> P; c=0 ==> P ] ==> P"; 
129  32 
by (rtac (major RS consE) 1); 
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by (REPEAT (eresolve_tac (singletonE::prems) 1)); 
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qed "boolE"; 
0  35 

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(** cond **) 

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(*1 means true*) 

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goalw Bool.thy bool_defs "cond(1,c,d) = c"; 

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by (rtac (refl RS if_P) 1); 

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qed "cond_1"; 
0  42 

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(*0 means false*) 

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goalw Bool.thy bool_defs "cond(0,c,d) = d"; 

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by (rtac (succ_not_0 RS not_sym RS if_not_P) 1); 

760  46 
qed "cond_0"; 
0  47 

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val major::prems = goal Bool.thy 

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"[ b: bool; c: A(1); d: A(0) ] ==> cond(b,c,d): A(b)"; 

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by (rtac (major RS boolE) 1); 

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by (asm_simp_tac (ZF_ss addsimps (cond_1::prems)) 1); 
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by (asm_simp_tac (ZF_ss addsimps (cond_0::prems)) 1); 
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qed "cond_type"; 
0  54 

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val [rew] = goal Bool.thy "[ !!b. j(b)==cond(b,c,d) ] ==> j(1) = c"; 

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by (rewtac rew); 

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by (rtac cond_1 1); 

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qed "def_cond_1"; 
0  59 

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val [rew] = goal Bool.thy "[ !!b. j(b)==cond(b,c,d) ] ==> j(0) = d"; 

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by (rewtac rew); 

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by (rtac cond_0 1); 

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qed "def_cond_0"; 
0  64 

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fun conds def = [standard (def RS def_cond_1), standard (def RS def_cond_0)]; 

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val [not_1,not_0] = conds not_def; 

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val [and_1,and_0] = conds and_def; 

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val [or_1,or_0] = conds or_def; 

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val [xor_1,xor_0] = conds xor_def; 

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qed_goalw "not_type" Bool.thy [not_def] 
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"a:bool ==> not(a) : bool" 
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(fn prems=> [ (typechk_tac (prems@[bool_1I, bool_0I, cond_type])) ]); 

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760  79 
qed_goalw "and_type" Bool.thy [and_def] 
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"[ a:bool; b:bool ] ==> a and b : bool" 
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(fn prems=> [ (typechk_tac (prems@[bool_1I, bool_0I, cond_type])) ]); 

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760  83 
qed_goalw "or_type" Bool.thy [or_def] 
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"[ a:bool; b:bool ] ==> a or b : bool" 
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(fn prems=> [ (typechk_tac (prems@[bool_1I, bool_0I, cond_type])) ]); 

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760  87 
qed_goalw "xor_type" Bool.thy [xor_def] 
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"[ a:bool; b:bool ] ==> a xor b : bool" 
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(fn prems=> [ (typechk_tac(prems@[bool_1I, bool_0I, cond_type, not_type])) ]); 

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val bool_typechecks = [bool_1I, bool_0I, cond_type, not_type, and_type, 

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or_type, xor_type] 

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val bool_simps = [cond_1,cond_0,not_1,not_0,and_1,and_0,or_1,or_0,xor_1,xor_0]; 
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val bool_ss0 = ZF_ss addsimps bool_simps; 
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fun bool0_tac i = 
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EVERY [etac boolE i, asm_simp_tac bool_ss0 (i+1), asm_simp_tac bool_ss0 i]; 
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(*** Laws for 'not' ***) 
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goal Bool.thy "!!a. a:bool ==> not(not(a)) = a"; 
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by (bool0_tac 1); 
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qed "not_not"; 
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goal Bool.thy "!!a b. a:bool ==> not(a and b) = not(a) or not(b)"; 
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by (bool0_tac 1); 
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qed "not_and"; 
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goal Bool.thy "!!a b. a:bool ==> not(a or b) = not(a) and not(b)"; 
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by (bool0_tac 1); 
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qed "not_or"; 
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(*** Laws about 'and' ***) 
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goal Bool.thy "!!a. a: bool ==> a and a = a"; 
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by (bool0_tac 1); 
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qed "and_absorb"; 
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goal Bool.thy "!!a b. [ a: bool; b:bool ] ==> a and b = b and a"; 
129  122 
by (etac boolE 1); 
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by (bool0_tac 1); 
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by (bool0_tac 1); 
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qed "and_commute"; 
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goal Bool.thy 
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"!!a. a: bool ==> (a and b) and c = a and (b and c)"; 
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by (bool0_tac 1); 
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qed "and_assoc"; 
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goal Bool.thy 
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"!!a. [ a: bool; b:bool; c:bool ] ==> \ 
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\ (a or b) and c = (a and c) or (b and c)"; 
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by (REPEAT (bool0_tac 1)); 
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qed "and_or_distrib"; 
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(** binary orion **) 
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goal Bool.thy "!!a. a: bool ==> a or a = a"; 
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by (bool0_tac 1); 
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qed "or_absorb"; 
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goal Bool.thy "!!a b. [ a: bool; b:bool ] ==> a or b = b or a"; 
129  145 
by (etac boolE 1); 
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by (bool0_tac 1); 
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by (bool0_tac 1); 
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qed "or_commute"; 
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goal Bool.thy "!!a. a: bool ==> (a or b) or c = a or (b or c)"; 
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by (bool0_tac 1); 
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qed "or_assoc"; 
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goal Bool.thy 
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"!!a b c. [ a: bool; b: bool; c: bool ] ==> \ 
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\ (a and b) or c = (a or c) and (b or c)"; 
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by (REPEAT (bool0_tac 1)); 
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qed "or_and_distrib"; 
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