author  berghofe 
Tue, 25 Jun 1996 13:11:29 +0200  
changeset 1824  44254696843a 
parent 1798  c055505f36d1 
child 1875  54c0462f8fb2 
permissions  rwrr 
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(* Title: HOL/Hoare/Examples.thy 
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ID: $Id$ 
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Author: Norbert Galm 
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Copyright 1995 TUM 
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Various arithmetic examples. 

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*) 

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open Examples; 

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(*** multiplication by successive addition ***) 

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goal thy 

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"{m=0 & s=0} \ 

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\ WHILE m ~= a DO {s = m*b} s := s+b; m := Suc(m) END\ 

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\ {s = a*b}"; 

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by(hoare_tac 1); 

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by(ALLGOALS (asm_full_simp_tac (!simpset addsimps add_ac))); 

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qed "multiply_by_add"; 

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(*** Euclid's algorithm for GCD ***) 

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goal thy 

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" {0<A & 0<B & a=A & b=B} \ 

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\ WHILE a ~= b \ 

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\ DO {0<a & 0<b & gcd A B = gcd a b} \ 

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\ IF a<b \ 

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\ THEN \ 

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\ b:=ba \ 

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\ ELSE \ 

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\ a:=ab \ 

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\ END \ 

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\ END \ 

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\ {a = gcd A B}"; 

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by (hoare_tac 1); 

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by (safe_tac HOL_cs); 

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by (etac less_imp_diff_positive 1); 
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by (etac gcd_diff_r 1); 

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by (cut_facts_tac [less_linear] 1); 

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by (cut_facts_tac [less_linear] 2); 

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by (rtac less_imp_diff_positive 1); 
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by (rtac gcd_diff_l 2); 

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by (dtac gcd_nnn 3); 
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by (ALLGOALS (fast_tac HOL_cs)); 

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qed "Euclid_GCD"; 

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(*** Power by interated squaring and multiplication ***) 

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goal thy 

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" {a=A & b=B} \ 

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\ c:=1; \ 

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\ WHILE b~=0 \ 

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\ DO {A pow B = c * a pow b} \ 

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\ WHILE b mod 2=0 \ 

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\ DO {A pow B = c * a pow b} \ 

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\ a:=a*a; \ 

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\ b:=b div 2 \ 

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\ END; \ 

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\ c:=c*a; \ 

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\ b:=b1 \ 

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\ END \ 

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\ {c = A pow B}"; 

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by (hoare_tac 1); 

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by (simp_tac ((simpset_of "Arith") addsimps [pow_0]) 3); 

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by (safe_tac HOL_cs); 

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by (subgoal_tac "a*a=a pow 2" 1); 

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by (Asm_simp_tac 1); 

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by (rtac (pow_pow_reduce RS ssubst) 1); 
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by (subgoal_tac "(b div 2)*2=b" 1); 

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by (subgoal_tac "0<2" 2); 

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by (dres_inst_tac [("m","b")] mod_div_equality 2); 

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by (ALLGOALS (asm_full_simp_tac ((simpset_of "Arith") addsimps [pow_0,pow_Suc,mult_assoc]))); 

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by (subgoal_tac "b~=0" 1); 

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by (res_inst_tac [("n","b")] natE 1); 

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by (res_inst_tac [("Q","b mod 2 ~= 0")] not_imp_swap 3); 

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by (assume_tac 4); 
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by (ALLGOALS (asm_full_simp_tac ((simpset_of "Arith") addsimps [pow_0,pow_Suc,mult_assoc]))); 

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by (rtac mod_less 1); 
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by (Simp_tac 1); 
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qed "power_by_mult"; 

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(*** factorial ***) 

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goal thy 

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" {a=A} \ 

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\ b:=1; \ 

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\ WHILE a~=0 \ 

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\ DO {fac A = b*fac a} \ 

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\ b:=b*a; \ 

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\ a:=a1 \ 

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\ END \ 

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\ {b = fac A}"; 

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by (hoare_tac 1); 

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by (safe_tac HOL_cs); 

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by (res_inst_tac [("n","a")] natE 1); 

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by (ALLGOALS 
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(asm_simp_tac 
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(!simpset addsimps [add_mult_distrib,add_mult_distrib2,mult_assoc]))); 
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by (fast_tac HOL_cs 1); 
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qed"factorial"; 