author  hoelzl 
Thu, 05 Feb 2009 11:45:15 +0100  
changeset 29804  e15b74577368 
parent 29667  src/HOL/Library/Float.thy@53103fc8ffa3 
permissions  rwrr 
29804
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Added new Float theory and moved old Library/Float.thy to ComputeFloat
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(* Title: HOL/Tools/ComputeFloat.thy 
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Author: Steven Obua 
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*) 
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20717  5 
header {* Floating Point Representation of the Reals *} 
6 

29804
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theory ComputeFloat 
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imports Complex_Main 
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uses "~~/src/Tools/float.ML" ("~~/src/HOL/Tools/float_arith.ML") 
20485  10 
begin 
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19765  12 
definition 
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pow2 :: "int \<Rightarrow> real" where 
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"pow2 a = (if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (a)))))" 
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definition 
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float :: "int * int \<Rightarrow> real" where 
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"float x = real (fst x) * pow2 (snd x)" 
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lemma pow2_0[simp]: "pow2 0 = 1" 
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by (simp add: pow2_def) 
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lemma pow2_1[simp]: "pow2 1 = 2" 
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by (simp add: pow2_def) 
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25 

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lemma pow2_neg: "pow2 x = inverse (pow2 (x))" 
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by (simp add: pow2_def) 
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19765  29 
lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)" 
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proof  
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have h: "! n. nat (2 + int n)  Suc 0 = nat (1 + int n)" by arith 
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have g: "! a b. a  1 = a + (1::int)" by arith 
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have pos: "! n. pow2 (int n + 1) = 2 * pow2 (int n)" 
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apply (auto, induct_tac n) 
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apply (simp_all add: pow2_def) 
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apply (rule_tac m1="2" and n1="nat (2 + int na)" in ssubst[OF realpow_num_eq_if]) 
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by (auto simp add: h) 
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show ?thesis 
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proof (induct a) 
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case (1 n) 
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from pos show ?case by (simp add: algebra_simps) 
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next 
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case (2 n) 
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show ?case 
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apply (auto) 
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apply (subst pow2_neg[of " int n"]) 
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apply (subst pow2_neg[of "1  int n"]) 
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apply (auto simp add: g pos) 
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done 
19765  50 
qed 
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qed 
19765  52 

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lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)" 
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proof (induct b) 
19765  55 
case (1 n) 
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show ?case 
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proof (induct n) 
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case 0 
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show ?case by simp 
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next 
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case (Suc m) 
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show ?case by (auto simp add: algebra_simps pow2_add1 prems) 
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qed 
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next 
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case (2 n) 
19765  66 
show ?case 
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proof (induct n) 
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case 0 
19765  69 
show ?case 
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apply (auto) 
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apply (subst pow2_neg[of "a + 1"]) 
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apply (subst pow2_neg[of "1"]) 
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apply (simp) 
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apply (insert pow2_add1[of "a"]) 
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apply (simp add: algebra_simps) 
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apply (subst pow2_neg[of "a"]) 
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apply (simp) 
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done 
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case (Suc m) 
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have a: "int m  (a + 2) = 1 + (int m  a + 1)" by arith 
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have b: "int m  2 = 1 + (int m + 1)" by arith 
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show ?case 
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apply (auto) 
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apply (subst pow2_neg[of "a + (2  int m)"]) 
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apply (subst pow2_neg[of "2  int m"]) 
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apply (auto simp add: algebra_simps) 
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apply (subst a) 
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apply (subst b) 
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apply (simp only: pow2_add1) 
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apply (subst pow2_neg[of "int m  a + 1"]) 
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apply (subst pow2_neg[of "int m + 1"]) 
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apply auto 
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apply (insert prems) 
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apply (auto simp add: algebra_simps) 
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done 
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qed 
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qed 
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19765  99 
lemma "float (a, e) + float (b, e) = float (a + b, e)" 
29667  100 
by (simp add: float_def algebra_simps) 
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19765  102 
definition 
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int_of_real :: "real \<Rightarrow> int" where 
19765  104 
"int_of_real x = (SOME y. real y = x)" 
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definition 
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real_is_int :: "real \<Rightarrow> bool" where 
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"real_is_int x = (EX (u::int). x = real u)" 
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lemma real_is_int_def2: "real_is_int x = (x = real (int_of_real x))" 
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by (auto simp add: real_is_int_def int_of_real_def) 
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lemma float_transfer: "real_is_int ((real a)*(pow2 c)) \<Longrightarrow> float (a, b) = float (int_of_real ((real a)*(pow2 c)), b  c)" 
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by (simp add: float_def real_is_int_def2 pow2_add[symmetric]) 
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26313  116 
lemma pow2_int: "pow2 (int c) = 2^c" 
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by (simp add: pow2_def) 
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lemma float_transfer_nat: "float (a, b) = float (a * 2^c, b  int c)" 
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by (simp add: float_def pow2_int[symmetric] pow2_add[symmetric]) 
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lemma real_is_int_real[simp]: "real_is_int (real (x::int))" 
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by (auto simp add: real_is_int_def int_of_real_def) 
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lemma int_of_real_real[simp]: "int_of_real (real x) = x" 
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by (simp add: int_of_real_def) 
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lemma real_int_of_real[simp]: "real_is_int x \<Longrightarrow> real (int_of_real x) = x" 
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by (auto simp add: int_of_real_def real_is_int_def) 
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lemma real_is_int_add_int_of_real: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> (int_of_real (a+b)) = (int_of_real a) + (int_of_real b)" 
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by (auto simp add: int_of_real_def real_is_int_def) 
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lemma real_is_int_add[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a+b)" 
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apply (subst real_is_int_def2) 
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apply (simp add: real_is_int_add_int_of_real real_int_of_real) 
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done 
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138 

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lemma int_of_real_sub: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> (int_of_real (ab)) = (int_of_real a)  (int_of_real b)" 
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by (auto simp add: int_of_real_def real_is_int_def) 
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141 

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lemma real_is_int_sub[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (ab)" 
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apply (subst real_is_int_def2) 
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apply (simp add: int_of_real_sub real_int_of_real) 
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done 
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146 

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lemma real_is_int_rep: "real_is_int x \<Longrightarrow> ?! (a::int). real a = x" 
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by (auto simp add: real_is_int_def) 
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149 

19765  150 
lemma int_of_real_mult: 
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assumes "real_is_int a" "real_is_int b" 
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shows "(int_of_real (a*b)) = (int_of_real a) * (int_of_real b)" 
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proof  
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parents:
diff
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154 
from prems have a: "?! (a'::int). real a' = a" by (rule_tac real_is_int_rep, auto) 
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obua
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diff
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155 
from prems have b: "?! (b'::int). real b' = b" by (rule_tac real_is_int_rep, auto) 
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obua
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diff
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156 
from a obtain a'::int where a':"a = real a'" by auto 
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obua
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157 
from b obtain b'::int where b':"b = real b'" by auto 
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obua
parents:
diff
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158 
have r: "real a' * real b' = real (a' * b')" by auto 
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obua
parents:
diff
changeset

159 
show ?thesis 
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obua
parents:
diff
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160 
apply (simp add: a' b') 
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obua
parents:
diff
changeset

161 
apply (subst r) 
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obua
parents:
diff
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162 
apply (simp only: int_of_real_real) 
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obua
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diff
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163 
done 
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obua
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diff
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164 
qed 
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obua
parents:
diff
changeset

165 

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diff
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166 
lemma real_is_int_mult[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a*b)" 
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diff
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167 
apply (subst real_is_int_def2) 
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obua
parents:
diff
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168 
apply (simp add: int_of_real_mult) 
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obua
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169 
done 
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obua
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diff
changeset

170 

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obua
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diff
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171 
lemma real_is_int_0[simp]: "real_is_int (0::real)" 
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172 
by (simp add: real_is_int_def int_of_real_def) 
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obua
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diff
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173 

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174 
lemma real_is_int_1[simp]: "real_is_int (1::real)" 
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175 
proof  
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obua
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176 
have "real_is_int (1::real) = real_is_int(real (1::int))" by auto 
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obua
parents:
diff
changeset

177 
also have "\<dots> = True" by (simp only: real_is_int_real) 
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obua
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178 
ultimately show ?thesis by auto 
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obua
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179 
qed 
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obua
parents:
diff
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180 

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obua
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181 
lemma real_is_int_n1: "real_is_int (1::real)" 
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182 
proof  
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obua
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183 
have "real_is_int (1::real) = real_is_int(real (1::int))" by auto 
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obua
parents:
diff
changeset

184 
also have "\<dots> = True" by (simp only: real_is_int_real) 
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obua
parents:
diff
changeset

185 
ultimately show ?thesis by auto 
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obua
parents:
diff
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186 
qed 
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obua
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187 

20485  188 
lemma real_is_int_number_of[simp]: "real_is_int ((number_of \<Colon> int \<Rightarrow> real) x)" 
16782
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189 
proof  
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obua
parents:
diff
changeset

190 
have neg1: "real_is_int (1::real)" 
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obua
parents:
diff
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191 
proof  
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obua
parents:
diff
changeset

192 
have "real_is_int (1::real) = real_is_int(real (1::int))" by auto 
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obua
parents:
diff
changeset

193 
also have "\<dots> = True" by (simp only: real_is_int_real) 
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obua
parents:
diff
changeset

194 
ultimately show ?thesis by auto 
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195 
qed 
19765  196 

197 
{ 

20485  198 
fix x :: int 
199 
have "real_is_int ((number_of \<Colon> int \<Rightarrow> real) x)" 

200 
unfolding number_of_eq 

16782
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201 
apply (induct x) 
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obua
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diff
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202 
apply (induct_tac n) 
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obua
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diff
changeset

203 
apply (simp) 
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obua
parents:
diff
changeset

204 
apply (simp) 
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obua
parents:
diff
changeset

205 
apply (induct_tac n) 
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obua
parents:
diff
changeset

206 
apply (simp add: neg1) 
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obua
parents:
diff
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207 
proof  
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obua
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208 
fix n :: nat 
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209 
assume rn: "(real_is_int (of_int ( (int (Suc n)))))" 
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210 
have s: "(int (Suc (Suc n))) = 1 +  (int (Suc n))" by simp 
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obua
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211 
show "real_is_int (of_int ( (int (Suc (Suc n)))))" 
19765  212 
apply (simp only: s of_int_add) 
213 
apply (rule real_is_int_add) 

214 
apply (simp add: neg1) 

215 
apply (simp only: rn) 

216 
done 

16782
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217 
qed 
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218 
} 
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obua
parents:
diff
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219 
note Abs_Bin = this 
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220 
{ 
20485  221 
fix x :: int 
222 
have "? u. x = u" 

223 
apply (rule exI[where x = "x"]) 

224 
apply (simp) 

16782
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225 
done 
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226 
} 
20485  227 
then obtain u::int where "x = u" by auto 
16782
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228 
with Abs_Bin show ?thesis by auto 
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229 
qed 
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obua
parents:
diff
changeset

230 

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obua
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diff
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231 
lemma int_of_real_0[simp]: "int_of_real (0::real) = (0::int)" 
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obua
parents:
diff
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232 
by (simp add: int_of_real_def) 
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obua
parents:
diff
changeset

233 

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obua
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diff
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234 
lemma int_of_real_1[simp]: "int_of_real (1::real) = (1::int)" 
19765  235 
proof  
16782
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obua
parents:
diff
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236 
have 1: "(1::real) = real (1::int)" by auto 
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obua
parents:
diff
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237 
show ?thesis by (simp only: 1 int_of_real_real) 
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obua
parents:
diff
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238 
qed 
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obua
parents:
diff
changeset

239 

b214f21ae396
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obua
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diff
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240 
lemma int_of_real_number_of[simp]: "int_of_real (number_of b) = number_of b" 
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obua
parents:
diff
changeset

241 
proof  
b214f21ae396
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obua
parents:
diff
changeset

242 
have "real_is_int (number_of b)" by simp 
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obua
parents:
diff
changeset

243 
then have uu: "?! u::int. number_of b = real u" by (auto simp add: real_is_int_rep) 
b214f21ae396
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obua
parents:
diff
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244 
then obtain u::int where u:"number_of b = real u" by auto 
19765  245 
have "number_of b = real ((number_of b)::int)" 
16782
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246 
by (simp add: number_of_eq real_of_int_def) 
19765  247 
have ub: "number_of b = real ((number_of b)::int)" 
16782
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obua
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diff
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248 
by (simp add: number_of_eq real_of_int_def) 
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obua
parents:
diff
changeset

249 
from uu u ub have unb: "u = number_of b" 
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obua
parents:
diff
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250 
by blast 
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obua
parents:
diff
changeset

251 
have "int_of_real (number_of b) = u" by (simp add: u) 
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obua
parents:
diff
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252 
with unb show ?thesis by simp 
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obua
parents:
diff
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253 
qed 
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obua
parents:
diff
changeset

254 

b214f21ae396
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obua
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diff
changeset

255 
lemma float_transfer_even: "even a \<Longrightarrow> float (a, b) = float (a div 2, b+1)" 
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obua
parents:
diff
changeset

256 
apply (subst float_transfer[where a="a" and b="b" and c="1", simplified]) 
29667  257 
apply (simp_all add: pow2_def even_def real_is_int_def algebra_simps) 
16782
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obua
parents:
diff
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258 
apply (auto) 
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obua
parents:
diff
changeset

259 
proof  
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obua
parents:
diff
changeset

260 
fix q::int 
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obua
parents:
diff
changeset

261 
have a:"b  (1\<Colon>int) = (1\<Colon>int) + b" by arith 
19765  262 
show "(float (q, (b  (1\<Colon>int)))) = (float (q, ((1\<Colon>int) + b)))" 
16782
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obua
parents:
diff
changeset

263 
by (simp add: a) 
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obua
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diff
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264 
qed 
19765  265 

16782
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diff
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266 
lemma int_div_zdiv: "int (a div b) = (int a) div (int b)" 
23431
25ca91279a9b
change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents:
23365
diff
changeset

267 
by (rule zdiv_int) 
16782
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obua
parents:
diff
changeset

268 

b214f21ae396
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obua
parents:
diff
changeset

269 
lemma int_mod_zmod: "int (a mod b) = (int a) mod (int b)" 
23431
25ca91279a9b
change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents:
23365
diff
changeset

270 
by (rule zmod_int) 
16782
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obua
parents:
diff
changeset

271 

b214f21ae396
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obua
parents:
diff
changeset

272 
lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> abs((a::int) div 2) < abs a" 
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obua
parents:
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273 
by arith 
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obua
parents:
diff
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274 

27366  275 
function norm_float :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where 
276 
"norm_float a b = (if a \<noteq> 0 \<and> even a then norm_float (a div 2) (b + 1) 

277 
else if a = 0 then (0, 0) else (a, b))" 

278 
by auto 

16782
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279 

27366  280 
termination by (relation "measure (nat o abs o fst)") 
281 
(auto intro: abs_div_2_less) 

16782
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282 

27366  283 
lemma norm_float: "float x = float (split norm_float x)" 
16782
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parents:
diff
changeset

284 
proof  
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obua
parents:
diff
changeset

285 
{ 
19765  286 
fix a b :: int 
27366  287 
have norm_float_pair: "float (a, b) = float (norm_float a b)" 
16782
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obua
parents:
diff
changeset

288 
proof (induct a b rule: norm_float.induct) 
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parents:
diff
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289 
case (1 u v) 
19765  290 
show ?case 
16782
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obua
parents:
diff
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291 
proof cases 
19765  292 
assume u: "u \<noteq> 0 \<and> even u" 
27366  293 
with prems have ind: "float (u div 2, v + 1) = float (norm_float (u div 2) (v + 1))" by auto 
19765  294 
with u have "float (u,v) = float (u div 2, v+1)" by (simp add: float_transfer_even) 
295 
then show ?thesis 

296 
apply (subst norm_float.simps) 

297 
apply (simp add: ind) 

298 
done 

16782
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obua
parents:
diff
changeset

299 
next 
19765  300 
assume "~(u \<noteq> 0 \<and> even u)" 
301 
then show ?thesis 

302 
by (simp add: prems float_def) 

16782
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parents:
diff
changeset

303 
qed 
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obua
parents:
diff
changeset

304 
qed 
b214f21ae396
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obua
parents:
diff
changeset

305 
} 
b214f21ae396
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obua
parents:
diff
changeset

306 
note helper = this 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

307 
have "? a b. x = (a,b)" by auto 
b214f21ae396
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obua
parents:
diff
changeset

308 
then obtain a b where "x = (a, b)" by blast 
27366  309 
then show ?thesis by (simp add: helper) 
16782
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obua
parents:
diff
changeset

310 
qed 
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obua
parents:
diff
changeset

311 

24301  312 
lemma float_add_l0: "float (0, e) + x = x" 
313 
by (simp add: float_def) 

314 

315 
lemma float_add_r0: "x + float (0, e) = x" 

316 
by (simp add: float_def) 

317 

19765  318 
lemma float_add: 
319 
"float (a1, e1) + float (a2, e2) = 

320 
(if e1<=e2 then float (a1+a2*2^(nat(e2e1)), e1) 

16782
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obua
parents:
diff
changeset

321 
else float (a1*2^(nat (e1e2))+a2, e2))" 
29667  322 
apply (simp add: float_def algebra_simps) 
16782
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obua
parents:
diff
changeset

323 
apply (auto simp add: pow2_int[symmetric] pow2_add[symmetric]) 
b214f21ae396
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obua
parents:
diff
changeset

324 
done 
b214f21ae396
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obua
parents:
diff
changeset

325 

24301  326 
lemma float_add_assoc1: 
327 
"(x + float (y1, e1)) + float (y2, e2) = (float (y1, e1) + float (y2, e2)) + x" 

328 
by simp 

329 

330 
lemma float_add_assoc2: 

331 
"(float (y1, e1) + x) + float (y2, e2) = (float (y1, e1) + float (y2, e2)) + x" 

332 
by simp 

333 

334 
lemma float_add_assoc3: 

335 
"float (y1, e1) + (x + float (y2, e2)) = (float (y1, e1) + float (y2, e2)) + x" 

336 
by simp 

337 

338 
lemma float_add_assoc4: 

339 
"float (y1, e1) + (float (y2, e2) + x) = (float (y1, e1) + float (y2, e2)) + x" 

340 
by simp 

341 

342 
lemma float_mult_l0: "float (0, e) * x = float (0, 0)" 

343 
by (simp add: float_def) 

344 

345 
lemma float_mult_r0: "x * float (0, e) = float (0, 0)" 

346 
by (simp add: float_def) 

347 

348 
definition 

349 
lbound :: "real \<Rightarrow> real" 

350 
where 

351 
"lbound x = min 0 x" 

352 

353 
definition 

354 
ubound :: "real \<Rightarrow> real" 

355 
where 

356 
"ubound x = max 0 x" 

357 

358 
lemma lbound: "lbound x \<le> x" 

359 
by (simp add: lbound_def) 

360 

361 
lemma ubound: "x \<le> ubound x" 

362 
by (simp add: ubound_def) 

363 

16782
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obua
parents:
diff
changeset

364 
lemma float_mult: 
19765  365 
"float (a1, e1) * float (a2, e2) = 
16782
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obua
parents:
diff
changeset

366 
(float (a1 * a2, e1 + e2))" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

367 
by (simp add: float_def pow2_add) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

368 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

369 
lemma float_minus: 
b214f21ae396
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obua
parents:
diff
changeset

370 
" (float (a,b)) = float (a, b)" 
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obua
parents:
diff
changeset

371 
by (simp add: float_def) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

372 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

373 
lemma zero_less_pow2: 
b214f21ae396
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obua
parents:
diff
changeset

374 
"0 < pow2 x" 
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obua
parents:
diff
changeset

375 
proof  
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

376 
{ 
b214f21ae396
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obua
parents:
diff
changeset

377 
fix y 
19765  378 
have "0 <= y \<Longrightarrow> 0 < pow2 y" 
16782
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obua
parents:
diff
changeset

379 
by (induct y, induct_tac n, simp_all add: pow2_add) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

380 
} 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

381 
note helper=this 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

382 
show ?thesis 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

383 
apply (case_tac "0 <= x") 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

384 
apply (simp add: helper) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

385 
apply (subst pow2_neg) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

386 
apply (simp add: helper) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

387 
done 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

388 
qed 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

389 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

390 
lemma zero_le_float: 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

391 
"(0 <= float (a,b)) = (0 <= a)" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

392 
apply (auto simp add: float_def) 
19765  393 
apply (auto simp add: zero_le_mult_iff zero_less_pow2) 
16782
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 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

394 
apply (insert zero_less_pow2[of b]) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

395 
apply (simp_all) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

396 
done 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

397 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

398 
lemma float_le_zero: 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

399 
"(float (a,b) <= 0) = (a <= 0)" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

400 
apply (auto simp add: float_def) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

401 
apply (auto simp add: mult_le_0_iff) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

402 
apply (insert zero_less_pow2[of b]) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

403 
apply auto 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

404 
done 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

405 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

406 
lemma float_abs: 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

407 
"abs (float (a,b)) = (if 0 <= a then (float (a,b)) else (float (a,b)))" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

408 
apply (auto simp add: abs_if) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

409 
apply (simp_all add: zero_le_float[symmetric, of a b] float_minus) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

410 
done 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

411 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

412 
lemma float_zero: 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

413 
"float (0, b) = 0" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

414 
by (simp add: float_def) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

415 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

416 
lemma float_pprt: 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

417 
"pprt (float (a, b)) = (if 0 <= a then (float (a,b)) else (float (0, b)))" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

418 
by (auto simp add: zero_le_float float_le_zero float_zero) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

419 

24301  420 
lemma pprt_lbound: "pprt (lbound x) = float (0, 0)" 
421 
apply (simp add: float_def) 

422 
apply (rule pprt_eq_0) 

423 
apply (simp add: lbound_def) 

424 
done 

425 

426 
lemma nprt_ubound: "nprt (ubound x) = float (0, 0)" 

427 
apply (simp add: float_def) 

428 
apply (rule nprt_eq_0) 

429 
apply (simp add: ubound_def) 

430 
done 

431 

16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

432 
lemma float_nprt: 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

433 
"nprt (float (a, b)) = (if 0 <= a then (float (0,b)) else (float (a, b)))" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

434 
by (auto simp add: zero_le_float float_le_zero float_zero) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

435 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

436 
lemma norm_0_1: "(0::_::number_ring) = Numeral0 & (1::_::number_ring) = Numeral1" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

437 
by auto 
19765  438 

16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

439 
lemma add_left_zero: "0 + a = (a::'a::comm_monoid_add)" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

440 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

441 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

442 
lemma add_right_zero: "a + 0 = (a::'a::comm_monoid_add)" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

443 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

444 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

445 
lemma mult_left_one: "1 * a = (a::'a::semiring_1)" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

446 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

447 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

448 
lemma mult_right_one: "a * 1 = (a::'a::semiring_1)" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

449 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

450 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

451 
lemma int_pow_0: "(a::int)^(Numeral0) = 1" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

452 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

453 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

454 
lemma int_pow_1: "(a::int)^(Numeral1) = a" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

455 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

456 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

457 
lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

458 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

459 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

460 
lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

461 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

462 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

463 
lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

464 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

465 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

466 
lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

467 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

468 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

469 
lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

470 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

471 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

472 
lemma zpower_Min: "(z::int)^((1)::nat) = Numeral1" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

473 
proof  
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

474 
have 1:"((1)::nat) = 0" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

475 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

476 
show ?thesis by (simp add: 1) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

477 
qed 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

478 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

479 
lemma fst_cong: "a=a' \<Longrightarrow> fst (a,b) = fst (a',b)" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

480 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

481 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

482 
lemma snd_cong: "b=b' \<Longrightarrow> snd (a,b) = snd (a,b')" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

483 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

484 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

485 
lemma lift_bool: "x \<Longrightarrow> x=True" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

486 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

487 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

488 
lemma nlift_bool: "~x \<Longrightarrow> x=False" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

489 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

490 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

491 
lemma not_false_eq_true: "(~ False) = True" by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

492 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

493 
lemma not_true_eq_false: "(~ True) = False" by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

494 

19765  495 
lemmas binarith = 
26076
b9c716a9fb5f
added lemma lists {normalize,succ,pred,minus,add,mult}_bin_simps
huffman
parents:
24653
diff
changeset

496 
normalize_bin_simps 
b9c716a9fb5f
added lemma lists {normalize,succ,pred,minus,add,mult}_bin_simps
huffman
parents:
24653
diff
changeset

497 
pred_bin_simps succ_bin_simps 
b9c716a9fb5f
added lemma lists {normalize,succ,pred,minus,add,mult}_bin_simps
huffman
parents:
24653
diff
changeset

498 
add_bin_simps minus_bin_simps mult_bin_simps 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

499 

20485  500 
lemma int_eq_number_of_eq: 
501 
"(((number_of v)::int)=(number_of w)) = iszero ((number_of (v + uminus w))::int)" 

28967
3bdb1eae352c
enable eq_bin_simps for simplifying equalities on numerals
huffman
parents:
28963
diff
changeset

502 
by (rule eq_number_of_eq) 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

503 

19765  504 
lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)" 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

505 
by (simp only: iszero_number_of_Pls) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

506 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

507 
lemma int_nonzero_number_of_Min: "~(iszero ((1)::int))" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

508 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

509 

26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26076
diff
changeset

510 
lemma int_iszero_number_of_Bit0: "iszero ((number_of (Int.Bit0 w))::int) = iszero ((number_of w)::int)" 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

511 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

512 

26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26076
diff
changeset

513 
lemma int_iszero_number_of_Bit1: "\<not> iszero ((number_of (Int.Bit1 w))::int)" 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

514 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

515 

20485  516 
lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (x + (uminus y)))::int)" 
29040
286c669d3a7a
move all negrelated lemmas to NatBin; make type of neg specific to int
huffman
parents:
28967
diff
changeset

517 
unfolding neg_def number_of_is_id by simp 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

518 

19765  519 
lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))" 
16782
b214f21ae396
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obua
parents:
diff
changeset

520 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

521 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

522 
lemma int_neg_number_of_Min: "neg (1::int)" 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

523 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

524 

26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26076
diff
changeset

525 
lemma int_neg_number_of_Bit0: "neg ((number_of (Int.Bit0 w))::int) = neg ((number_of w)::int)" 
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26076
diff
changeset

526 
by simp 
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26076
diff
changeset

527 

3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26076
diff
changeset

528 
lemma int_neg_number_of_Bit1: "neg ((number_of (Int.Bit1 w))::int) = neg ((number_of w)::int)" 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

529 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

530 

20485  531 
lemma int_le_number_of_eq: "(((number_of x)::int) \<le> number_of y) = (\<not> neg ((number_of (y + (uminus x)))::int))" 
28963
f6d9e0e0b153
fix proofs related to simplification of inequalities on numerals
huffman
parents:
28952
diff
changeset

532 
unfolding neg_def number_of_is_id by (simp add: not_less) 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

533 

19765  534 
lemmas intarithrel = 
535 
int_eq_number_of_eq 

26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26076
diff
changeset

536 
lift_bool[OF int_iszero_number_of_Pls] nlift_bool[OF int_nonzero_number_of_Min] int_iszero_number_of_Bit0 
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26076
diff
changeset

537 
lift_bool[OF int_iszero_number_of_Bit1] int_less_number_of_eq_neg nlift_bool[OF int_not_neg_number_of_Pls] lift_bool[OF int_neg_number_of_Min] 
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26076
diff
changeset

538 
int_neg_number_of_Bit0 int_neg_number_of_Bit1 int_le_number_of_eq 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

539 

20485  540 
lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (v + w)" 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

541 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

542 

20485  543 
lemma int_number_of_diff_sym: "((number_of v)::int)  number_of w = number_of (v + (uminus w))" 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

544 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

545 

20485  546 
lemma int_number_of_mult_sym: "((number_of v)::int) * number_of w = number_of (v * w)" 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

547 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

548 

20485  549 
lemma int_number_of_minus_sym: " ((number_of v)::int) = number_of (uminus v)" 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

550 
by simp 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

551 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

552 
lemmas intarith = int_number_of_add_sym int_number_of_minus_sym int_number_of_diff_sym int_number_of_mult_sym 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

553 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

554 
lemmas natarith = add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

555 

19765  556 
lemmas powerarith = nat_number_of zpower_number_of_even 
557 
zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring] 

16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

558 
zpower_Pls zpower_Min 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

559 

24301  560 
lemmas floatarith[simplified norm_0_1] = float_add float_add_l0 float_add_r0 float_mult float_mult_l0 float_mult_r0 
24653  561 
float_minus float_abs zero_le_float float_pprt float_nprt pprt_lbound nprt_ubound 
16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

562 

b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

563 
(* for use with the compute oracle *) 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

564 
lemmas arith = binarith intarith intarithrel natarith powerarith floatarith not_false_eq_true not_true_eq_false 
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

565 

28952
15a4b2cf8c34
made repository layout more coherent with logical distribution structure; stripped some $Id$s
haftmann
parents:
27366
diff
changeset

566 
use "~~/src/HOL/Tools/float_arith.ML" 
20771  567 

16782
b214f21ae396
 use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff
changeset

568 
end 