src/HOL/NatBin.thy
author huffman
Wed Jun 13 03:31:11 2007 +0200 (2007-06-13)
changeset 23365 f31794033ae1
parent 23307 2fe3345035c7
child 23389 aaca6a8e5414
permissions -rw-r--r--
removed constant int :: nat => int;
int is now an abbreviation for of_nat :: nat => int
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(*  Title:      HOL/NatBin.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1999  University of Cambridge
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*)
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header {* Binary arithmetic for the natural numbers *}
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theory NatBin
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imports IntDiv
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begin
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text {*
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  Arithmetic for naturals is reduced to that for the non-negative integers.
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*}
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instance nat :: number
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  nat_number_of_def [code inline]: "number_of v == nat (number_of (v\<Colon>int))" ..
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abbreviation (xsymbols)
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  square :: "'a::power => 'a"  ("(_\<twosuperior>)" [1000] 999) where
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  "x\<twosuperior> == x^2"
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notation (latex output)
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  square  ("(_\<twosuperior>)" [1000] 999)
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notation (HTML output)
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  square  ("(_\<twosuperior>)" [1000] 999)
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subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*}
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declare nat_0 [simp] nat_1 [simp]
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lemma nat_number_of [simp]: "nat (number_of w) = number_of w"
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by (simp add: nat_number_of_def)
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lemma nat_numeral_0_eq_0 [simp]: "Numeral0 = (0::nat)"
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by (simp add: nat_number_of_def)
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lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
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by (simp add: nat_1 nat_number_of_def)
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lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
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by (simp add: nat_numeral_1_eq_1)
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lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
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apply (unfold nat_number_of_def)
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apply (rule nat_2)
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done
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text{*Distributive laws for type @{text nat}.  The others are in theory
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   @{text IntArith}, but these require div and mod to be defined for type
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   "int".  They also need some of the lemmas proved above.*}
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lemma nat_div_distrib: "(0::int) <= z ==> nat (z div z') = nat z div nat z'"
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apply (case_tac "0 <= z'")
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apply (auto simp add: div_nonneg_neg_le0 DIVISION_BY_ZERO_DIV)
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apply (case_tac "z' = 0", simp add: DIVISION_BY_ZERO)
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apply (auto elim!: nonneg_eq_int)
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apply (rename_tac m m')
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apply (subgoal_tac "0 <= int m div int m'")
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 prefer 2 apply (simp add: nat_numeral_0_eq_0 pos_imp_zdiv_nonneg_iff) 
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apply (rule of_nat_eq_iff [where 'a=int, THEN iffD1], simp)
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apply (rule_tac r = "int (m mod m') " in quorem_div)
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 prefer 2 apply force
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apply (simp add: nat_less_iff [symmetric] quorem_def nat_numeral_0_eq_0
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                 of_nat_add [symmetric] of_nat_mult [symmetric]
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            del: of_nat_add of_nat_mult)
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done
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(*Fails if z'<0: the LHS collapses to (nat z) but the RHS doesn't*)
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lemma nat_mod_distrib:
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     "[| (0::int) <= z;  0 <= z' |] ==> nat (z mod z') = nat z mod nat z'"
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apply (case_tac "z' = 0", simp add: DIVISION_BY_ZERO)
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apply (auto elim!: nonneg_eq_int)
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apply (rename_tac m m')
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apply (subgoal_tac "0 <= int m mod int m'")
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 prefer 2 apply (simp add: nat_less_iff nat_numeral_0_eq_0 pos_mod_sign)
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apply (rule int_int_eq [THEN iffD1], simp)
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apply (rule_tac q = "int (m div m') " in quorem_mod)
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 prefer 2 apply force
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apply (simp add: nat_less_iff [symmetric] quorem_def nat_numeral_0_eq_0
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                 of_nat_add [symmetric] of_nat_mult [symmetric]
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            del: of_nat_add of_nat_mult)
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done
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text{*Suggested by Matthias Daum*}
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lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
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apply (subgoal_tac "nat x div nat k < nat x")
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 apply (simp (asm_lr) add: nat_div_distrib [symmetric])
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apply (rule Divides.div_less_dividend, simp_all) 
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done
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subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}
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(*"neg" is used in rewrite rules for binary comparisons*)
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lemma int_nat_number_of [simp]:
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     "int (number_of v) =  
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         (if neg (number_of v :: int) then 0  
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          else (number_of v :: int))"
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by (simp del: nat_number_of
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	 add: neg_nat nat_number_of_def not_neg_nat add_assoc)
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subsubsection{*Successor *}
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lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
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apply (rule sym)
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apply (simp add: nat_eq_iff int_Suc)
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done
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lemma Suc_nat_number_of_add:
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     "Suc (number_of v + n) =  
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        (if neg (number_of v :: int) then 1+n else number_of (Numeral.succ v) + n)" 
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by (simp del: nat_number_of 
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         add: nat_number_of_def neg_nat
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              Suc_nat_eq_nat_zadd1 number_of_succ) 
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lemma Suc_nat_number_of [simp]:
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     "Suc (number_of v) =  
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        (if neg (number_of v :: int) then 1 else number_of (Numeral.succ v))"
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apply (cut_tac n = 0 in Suc_nat_number_of_add)
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apply (simp cong del: if_weak_cong)
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done
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subsubsection{*Addition *}
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(*"neg" is used in rewrite rules for binary comparisons*)
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lemma add_nat_number_of [simp]:
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     "(number_of v :: nat) + number_of v' =  
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         (if neg (number_of v :: int) then number_of v'  
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          else if neg (number_of v' :: int) then number_of v  
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          else number_of (v + v'))"
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by (force dest!: neg_nat
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          simp del: nat_number_of
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          simp add: nat_number_of_def nat_add_distrib [symmetric]) 
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subsubsection{*Subtraction *}
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lemma diff_nat_eq_if:
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     "nat z - nat z' =  
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        (if neg z' then nat z   
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         else let d = z-z' in     
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              if neg d then 0 else nat d)"
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apply (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
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done
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lemma diff_nat_number_of [simp]: 
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     "(number_of v :: nat) - number_of v' =  
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        (if neg (number_of v' :: int) then number_of v  
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         else let d = number_of (v + uminus v') in     
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              if neg d then 0 else nat d)"
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by (simp del: nat_number_of add: diff_nat_eq_if nat_number_of_def) 
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subsubsection{*Multiplication *}
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lemma mult_nat_number_of [simp]:
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     "(number_of v :: nat) * number_of v' =  
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       (if neg (number_of v :: int) then 0 else number_of (v * v'))"
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by (force dest!: neg_nat
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          simp del: nat_number_of
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          simp add: nat_number_of_def nat_mult_distrib [symmetric]) 
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subsubsection{*Quotient *}
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lemma div_nat_number_of [simp]:
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     "(number_of v :: nat)  div  number_of v' =  
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          (if neg (number_of v :: int) then 0  
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           else nat (number_of v div number_of v'))"
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by (force dest!: neg_nat
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          simp del: nat_number_of
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          simp add: nat_number_of_def nat_div_distrib [symmetric]) 
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lemma one_div_nat_number_of [simp]:
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     "(Suc 0)  div  number_of v' = (nat (1 div number_of v'))" 
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by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
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subsubsection{*Remainder *}
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lemma mod_nat_number_of [simp]:
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     "(number_of v :: nat)  mod  number_of v' =  
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        (if neg (number_of v :: int) then 0  
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         else if neg (number_of v' :: int) then number_of v  
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         else nat (number_of v mod number_of v'))"
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by (force dest!: neg_nat
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          simp del: nat_number_of
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          simp add: nat_number_of_def nat_mod_distrib [symmetric]) 
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lemma one_mod_nat_number_of [simp]:
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     "(Suc 0)  mod  number_of v' =  
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        (if neg (number_of v' :: int) then Suc 0
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         else nat (1 mod number_of v'))"
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by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
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subsubsection{* Divisibility *}
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lemmas dvd_eq_mod_eq_0_number_of =
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  dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
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declare dvd_eq_mod_eq_0_number_of [simp]
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ML
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{*
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val nat_number_of_def = thm"nat_number_of_def";
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val nat_number_of = thm"nat_number_of";
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val nat_numeral_0_eq_0 = thm"nat_numeral_0_eq_0";
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val nat_numeral_1_eq_1 = thm"nat_numeral_1_eq_1";
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val numeral_1_eq_Suc_0 = thm"numeral_1_eq_Suc_0";
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val numeral_2_eq_2 = thm"numeral_2_eq_2";
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val nat_div_distrib = thm"nat_div_distrib";
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val nat_mod_distrib = thm"nat_mod_distrib";
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val int_nat_number_of = thm"int_nat_number_of";
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val Suc_nat_eq_nat_zadd1 = thm"Suc_nat_eq_nat_zadd1";
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val Suc_nat_number_of_add = thm"Suc_nat_number_of_add";
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val Suc_nat_number_of = thm"Suc_nat_number_of";
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val add_nat_number_of = thm"add_nat_number_of";
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val diff_nat_eq_if = thm"diff_nat_eq_if";
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val diff_nat_number_of = thm"diff_nat_number_of";
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val mult_nat_number_of = thm"mult_nat_number_of";
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val div_nat_number_of = thm"div_nat_number_of";
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val mod_nat_number_of = thm"mod_nat_number_of";
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*}
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subsection{*Comparisons*}
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subsubsection{*Equals (=) *}
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lemma eq_nat_nat_iff:
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     "[| (0::int) <= z;  0 <= z' |] ==> (nat z = nat z') = (z=z')"
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by (auto elim!: nonneg_eq_int)
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(*"neg" is used in rewrite rules for binary comparisons*)
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lemma eq_nat_number_of [simp]:
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     "((number_of v :: nat) = number_of v') =  
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      (if neg (number_of v :: int) then (iszero (number_of v' :: int) | neg (number_of v' :: int))  
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       else if neg (number_of v' :: int) then iszero (number_of v :: int)  
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       else iszero (number_of (v + uminus v') :: int))"
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apply (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def
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                  eq_nat_nat_iff eq_number_of_eq nat_0 iszero_def
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            split add: split_if cong add: imp_cong)
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apply (simp only: nat_eq_iff nat_eq_iff2)
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apply (simp add: not_neg_eq_ge_0 [symmetric])
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done
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subsubsection{*Less-than (<) *}
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(*"neg" is used in rewrite rules for binary comparisons*)
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lemma less_nat_number_of [simp]:
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     "((number_of v :: nat) < number_of v') =  
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         (if neg (number_of v :: int) then neg (number_of (uminus v') :: int)  
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          else neg (number_of (v + uminus v') :: int))"
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by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def
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                nat_less_eq_zless less_number_of_eq_neg zless_nat_eq_int_zless
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         cong add: imp_cong, simp add: Pls_def)
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(*Maps #n to n for n = 0, 1, 2*)
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lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2
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subsection{*Powers with Numeric Exponents*}
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text{*We cannot refer to the number @{term 2} in @{text Ring_and_Field.thy}.
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We cannot prove general results about the numeral @{term "-1"}, so we have to
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use @{term "- 1"} instead.*}
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lemma power2_eq_square: "(a::'a::recpower)\<twosuperior> = a * a"
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  by (simp add: numeral_2_eq_2 Power.power_Suc)
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lemma zero_power2 [simp]: "(0::'a::{semiring_1,recpower})\<twosuperior> = 0"
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  by (simp add: power2_eq_square)
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lemma one_power2 [simp]: "(1::'a::{semiring_1,recpower})\<twosuperior> = 1"
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  by (simp add: power2_eq_square)
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lemma power3_eq_cube: "(x::'a::recpower) ^ 3 = x * x * x"
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  apply (subgoal_tac "3 = Suc (Suc (Suc 0))")
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  apply (erule ssubst)
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  apply (simp add: power_Suc mult_ac)
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  apply (unfold nat_number_of_def)
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  apply (subst nat_eq_iff)
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  apply simp
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done
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text{*Squares of literal numerals will be evaluated.*}
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lemmas power2_eq_square_number_of =
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    power2_eq_square [of "number_of w", standard]
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declare power2_eq_square_number_of [simp]
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lemma zero_le_power2[simp]: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,recpower})"
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  by (simp add: power2_eq_square)
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lemma zero_less_power2[simp]:
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     "(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,recpower}))"
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  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
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lemma power2_less_0[simp]:
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   312
  fixes a :: "'a::{ordered_idom,recpower}"
wenzelm@23164
   313
  shows "~ (a\<twosuperior> < 0)"
wenzelm@23164
   314
by (force simp add: power2_eq_square mult_less_0_iff) 
wenzelm@23164
   315
wenzelm@23164
   316
lemma zero_eq_power2[simp]:
wenzelm@23164
   317
     "(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))"
wenzelm@23164
   318
  by (force simp add: power2_eq_square mult_eq_0_iff)
wenzelm@23164
   319
wenzelm@23164
   320
lemma abs_power2[simp]:
wenzelm@23164
   321
     "abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})"
wenzelm@23164
   322
  by (simp add: power2_eq_square abs_mult abs_mult_self)
wenzelm@23164
   323
wenzelm@23164
   324
lemma power2_abs[simp]:
wenzelm@23164
   325
     "(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})"
wenzelm@23164
   326
  by (simp add: power2_eq_square abs_mult_self)
wenzelm@23164
   327
wenzelm@23164
   328
lemma power2_minus[simp]:
wenzelm@23164
   329
     "(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})"
wenzelm@23164
   330
  by (simp add: power2_eq_square)
wenzelm@23164
   331
wenzelm@23164
   332
lemma power2_le_imp_le:
wenzelm@23164
   333
  fixes x y :: "'a::{ordered_semidom,recpower}"
wenzelm@23164
   334
  shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y"
wenzelm@23164
   335
unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
wenzelm@23164
   336
wenzelm@23164
   337
lemma power2_less_imp_less:
wenzelm@23164
   338
  fixes x y :: "'a::{ordered_semidom,recpower}"
wenzelm@23164
   339
  shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y"
wenzelm@23164
   340
by (rule power_less_imp_less_base)
wenzelm@23164
   341
wenzelm@23164
   342
lemma power2_eq_imp_eq:
wenzelm@23164
   343
  fixes x y :: "'a::{ordered_semidom,recpower}"
wenzelm@23164
   344
  shows "\<lbrakk>x\<twosuperior> = y\<twosuperior>; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> x = y"
wenzelm@23164
   345
unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base, simp)
wenzelm@23164
   346
wenzelm@23164
   347
lemma power_minus1_even[simp]: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})"
wenzelm@23164
   348
apply (induct "n")
wenzelm@23164
   349
apply (auto simp add: power_Suc power_add)
wenzelm@23164
   350
done
wenzelm@23164
   351
wenzelm@23164
   352
lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2"
wenzelm@23164
   353
by (subst mult_commute) (simp add: power_mult)
wenzelm@23164
   354
wenzelm@23164
   355
lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"
wenzelm@23164
   356
by (simp add: power_even_eq) 
wenzelm@23164
   357
wenzelm@23164
   358
lemma power_minus_even [simp]:
wenzelm@23164
   359
     "(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)"
wenzelm@23164
   360
by (simp add: power_minus1_even power_minus [of a]) 
wenzelm@23164
   361
wenzelm@23164
   362
lemma zero_le_even_power'[simp]:
wenzelm@23164
   363
     "0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)"
wenzelm@23164
   364
proof (induct "n")
wenzelm@23164
   365
  case 0
wenzelm@23164
   366
    show ?case by (simp add: zero_le_one)
wenzelm@23164
   367
next
wenzelm@23164
   368
  case (Suc n)
wenzelm@23164
   369
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
wenzelm@23164
   370
      by (simp add: mult_ac power_add power2_eq_square)
wenzelm@23164
   371
    thus ?case
wenzelm@23164
   372
      by (simp add: prems zero_le_mult_iff)
wenzelm@23164
   373
qed
wenzelm@23164
   374
wenzelm@23164
   375
lemma odd_power_less_zero:
wenzelm@23164
   376
     "(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0"
wenzelm@23164
   377
proof (induct "n")
wenzelm@23164
   378
  case 0
wenzelm@23164
   379
    show ?case by (simp add: Power.power_Suc)
wenzelm@23164
   380
next
wenzelm@23164
   381
  case (Suc n)
wenzelm@23164
   382
    have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)" 
wenzelm@23164
   383
      by (simp add: mult_ac power_add power2_eq_square Power.power_Suc)
wenzelm@23164
   384
    thus ?case
wenzelm@23164
   385
      by (simp add: prems mult_less_0_iff mult_neg_neg)
wenzelm@23164
   386
qed
wenzelm@23164
   387
wenzelm@23164
   388
lemma odd_0_le_power_imp_0_le:
wenzelm@23164
   389
     "0 \<le> a  ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})"
wenzelm@23164
   390
apply (insert odd_power_less_zero [of a n]) 
wenzelm@23164
   391
apply (force simp add: linorder_not_less [symmetric]) 
wenzelm@23164
   392
done
wenzelm@23164
   393
wenzelm@23164
   394
text{*Simprules for comparisons where common factors can be cancelled.*}
wenzelm@23164
   395
lemmas zero_compare_simps =
wenzelm@23164
   396
    add_strict_increasing add_strict_increasing2 add_increasing
wenzelm@23164
   397
    zero_le_mult_iff zero_le_divide_iff 
wenzelm@23164
   398
    zero_less_mult_iff zero_less_divide_iff 
wenzelm@23164
   399
    mult_le_0_iff divide_le_0_iff 
wenzelm@23164
   400
    mult_less_0_iff divide_less_0_iff 
wenzelm@23164
   401
    zero_le_power2 power2_less_0
wenzelm@23164
   402
wenzelm@23164
   403
subsubsection{*Nat *}
wenzelm@23164
   404
wenzelm@23164
   405
lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
wenzelm@23164
   406
by (simp add: numerals)
wenzelm@23164
   407
wenzelm@23164
   408
(*Expresses a natural number constant as the Suc of another one.
wenzelm@23164
   409
  NOT suitable for rewriting because n recurs in the condition.*)
wenzelm@23164
   410
lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
wenzelm@23164
   411
wenzelm@23164
   412
subsubsection{*Arith *}
wenzelm@23164
   413
wenzelm@23164
   414
lemma Suc_eq_add_numeral_1: "Suc n = n + 1"
wenzelm@23164
   415
by (simp add: numerals)
wenzelm@23164
   416
wenzelm@23164
   417
lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n"
wenzelm@23164
   418
by (simp add: numerals)
wenzelm@23164
   419
wenzelm@23164
   420
(* These two can be useful when m = number_of... *)
wenzelm@23164
   421
wenzelm@23164
   422
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
wenzelm@23164
   423
apply (case_tac "m")
wenzelm@23164
   424
apply (simp_all add: numerals)
wenzelm@23164
   425
done
wenzelm@23164
   426
wenzelm@23164
   427
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
wenzelm@23164
   428
apply (case_tac "m")
wenzelm@23164
   429
apply (simp_all add: numerals)
wenzelm@23164
   430
done
wenzelm@23164
   431
wenzelm@23164
   432
lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
wenzelm@23164
   433
apply (case_tac "m")
wenzelm@23164
   434
apply (simp_all add: numerals)
wenzelm@23164
   435
done
wenzelm@23164
   436
wenzelm@23164
   437
wenzelm@23164
   438
subsection{*Comparisons involving (0::nat) *}
wenzelm@23164
   439
wenzelm@23164
   440
text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
wenzelm@23164
   441
wenzelm@23164
   442
lemma eq_number_of_0 [simp]:
wenzelm@23164
   443
     "(number_of v = (0::nat)) =  
wenzelm@23164
   444
      (if neg (number_of v :: int) then True else iszero (number_of v :: int))"
wenzelm@23164
   445
by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] iszero_0)
wenzelm@23164
   446
wenzelm@23164
   447
lemma eq_0_number_of [simp]:
wenzelm@23164
   448
     "((0::nat) = number_of v) =  
wenzelm@23164
   449
      (if neg (number_of v :: int) then True else iszero (number_of v :: int))"
wenzelm@23164
   450
by (rule trans [OF eq_sym_conv eq_number_of_0])
wenzelm@23164
   451
wenzelm@23164
   452
lemma less_0_number_of [simp]:
wenzelm@23164
   453
     "((0::nat) < number_of v) = neg (number_of (uminus v) :: int)"
wenzelm@23164
   454
by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] Pls_def)
wenzelm@23164
   455
wenzelm@23164
   456
wenzelm@23164
   457
lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
wenzelm@23164
   458
by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] iszero_0)
wenzelm@23164
   459
wenzelm@23164
   460
wenzelm@23164
   461
wenzelm@23164
   462
subsection{*Comparisons involving  @{term Suc} *}
wenzelm@23164
   463
wenzelm@23164
   464
lemma eq_number_of_Suc [simp]:
wenzelm@23164
   465
     "(number_of v = Suc n) =  
wenzelm@23164
   466
        (let pv = number_of (Numeral.pred v) in  
wenzelm@23164
   467
         if neg pv then False else nat pv = n)"
wenzelm@23164
   468
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
wenzelm@23164
   469
                  number_of_pred nat_number_of_def 
wenzelm@23164
   470
            split add: split_if)
wenzelm@23164
   471
apply (rule_tac x = "number_of v" in spec)
wenzelm@23164
   472
apply (auto simp add: nat_eq_iff)
wenzelm@23164
   473
done
wenzelm@23164
   474
wenzelm@23164
   475
lemma Suc_eq_number_of [simp]:
wenzelm@23164
   476
     "(Suc n = number_of v) =  
wenzelm@23164
   477
        (let pv = number_of (Numeral.pred v) in  
wenzelm@23164
   478
         if neg pv then False else nat pv = n)"
wenzelm@23164
   479
by (rule trans [OF eq_sym_conv eq_number_of_Suc])
wenzelm@23164
   480
wenzelm@23164
   481
lemma less_number_of_Suc [simp]:
wenzelm@23164
   482
     "(number_of v < Suc n) =  
wenzelm@23164
   483
        (let pv = number_of (Numeral.pred v) in  
wenzelm@23164
   484
         if neg pv then True else nat pv < n)"
wenzelm@23164
   485
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
wenzelm@23164
   486
                  number_of_pred nat_number_of_def  
wenzelm@23164
   487
            split add: split_if)
wenzelm@23164
   488
apply (rule_tac x = "number_of v" in spec)
wenzelm@23164
   489
apply (auto simp add: nat_less_iff)
wenzelm@23164
   490
done
wenzelm@23164
   491
wenzelm@23164
   492
lemma less_Suc_number_of [simp]:
wenzelm@23164
   493
     "(Suc n < number_of v) =  
wenzelm@23164
   494
        (let pv = number_of (Numeral.pred v) in  
wenzelm@23164
   495
         if neg pv then False else n < nat pv)"
wenzelm@23164
   496
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
wenzelm@23164
   497
                  number_of_pred nat_number_of_def
wenzelm@23164
   498
            split add: split_if)
wenzelm@23164
   499
apply (rule_tac x = "number_of v" in spec)
wenzelm@23164
   500
apply (auto simp add: zless_nat_eq_int_zless)
wenzelm@23164
   501
done
wenzelm@23164
   502
wenzelm@23164
   503
lemma le_number_of_Suc [simp]:
wenzelm@23164
   504
     "(number_of v <= Suc n) =  
wenzelm@23164
   505
        (let pv = number_of (Numeral.pred v) in  
wenzelm@23164
   506
         if neg pv then True else nat pv <= n)"
wenzelm@23164
   507
by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric])
wenzelm@23164
   508
wenzelm@23164
   509
lemma le_Suc_number_of [simp]:
wenzelm@23164
   510
     "(Suc n <= number_of v) =  
wenzelm@23164
   511
        (let pv = number_of (Numeral.pred v) in  
wenzelm@23164
   512
         if neg pv then False else n <= nat pv)"
wenzelm@23164
   513
by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric])
wenzelm@23164
   514
wenzelm@23164
   515
wenzelm@23164
   516
lemma lemma1: "(m+m = n+n) = (m = (n::int))"
wenzelm@23164
   517
by auto
wenzelm@23164
   518
wenzelm@23164
   519
lemma lemma2: "m+m ~= (1::int) + (n + n)"
wenzelm@23164
   520
apply auto
wenzelm@23164
   521
apply (drule_tac f = "%x. x mod 2" in arg_cong)
wenzelm@23164
   522
apply (simp add: zmod_zadd1_eq)
wenzelm@23164
   523
done
wenzelm@23164
   524
wenzelm@23164
   525
lemma eq_number_of_BIT_BIT:
wenzelm@23164
   526
     "((number_of (v BIT x) ::int) = number_of (w BIT y)) =  
wenzelm@23164
   527
      (x=y & (((number_of v) ::int) = number_of w))"
wenzelm@23164
   528
apply (simp only: number_of_BIT lemma1 lemma2 eq_commute
wenzelm@23164
   529
               OrderedGroup.add_left_cancel add_assoc OrderedGroup.add_0_left
wenzelm@23164
   530
            split add: bit.split)
wenzelm@23164
   531
apply simp
wenzelm@23164
   532
done
wenzelm@23164
   533
wenzelm@23164
   534
lemma eq_number_of_BIT_Pls:
wenzelm@23164
   535
     "((number_of (v BIT x) ::int) = Numeral0) =  
wenzelm@23164
   536
      (x=bit.B0 & (((number_of v) ::int) = Numeral0))"
wenzelm@23164
   537
apply (simp only: simp_thms  add: number_of_BIT number_of_Pls eq_commute
wenzelm@23164
   538
            split add: bit.split cong: imp_cong)
wenzelm@23164
   539
apply (rule_tac x = "number_of v" in spec, safe)
wenzelm@23164
   540
apply (simp_all (no_asm_use))
wenzelm@23164
   541
apply (drule_tac f = "%x. x mod 2" in arg_cong)
wenzelm@23164
   542
apply (simp add: zmod_zadd1_eq)
wenzelm@23164
   543
done
wenzelm@23164
   544
wenzelm@23164
   545
lemma eq_number_of_BIT_Min:
wenzelm@23164
   546
     "((number_of (v BIT x) ::int) = number_of Numeral.Min) =  
wenzelm@23164
   547
      (x=bit.B1 & (((number_of v) ::int) = number_of Numeral.Min))"
wenzelm@23164
   548
apply (simp only: simp_thms  add: number_of_BIT number_of_Min eq_commute
wenzelm@23164
   549
            split add: bit.split cong: imp_cong)
wenzelm@23164
   550
apply (rule_tac x = "number_of v" in spec, auto)
wenzelm@23164
   551
apply (drule_tac f = "%x. x mod 2" in arg_cong, auto)
wenzelm@23164
   552
done
wenzelm@23164
   553
wenzelm@23164
   554
lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Numeral.Min"
wenzelm@23164
   555
by auto
wenzelm@23164
   556
wenzelm@23164
   557
wenzelm@23164
   558
wenzelm@23164
   559
subsection{*Max and Min Combined with @{term Suc} *}
wenzelm@23164
   560
wenzelm@23164
   561
lemma max_number_of_Suc [simp]:
wenzelm@23164
   562
     "max (Suc n) (number_of v) =  
wenzelm@23164
   563
        (let pv = number_of (Numeral.pred v) in  
wenzelm@23164
   564
         if neg pv then Suc n else Suc(max n (nat pv)))"
wenzelm@23164
   565
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
wenzelm@23164
   566
            split add: split_if nat.split)
wenzelm@23164
   567
apply (rule_tac x = "number_of v" in spec) 
wenzelm@23164
   568
apply auto
wenzelm@23164
   569
done
wenzelm@23164
   570
 
wenzelm@23164
   571
lemma max_Suc_number_of [simp]:
wenzelm@23164
   572
     "max (number_of v) (Suc n) =  
wenzelm@23164
   573
        (let pv = number_of (Numeral.pred v) in  
wenzelm@23164
   574
         if neg pv then Suc n else Suc(max (nat pv) n))"
wenzelm@23164
   575
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
wenzelm@23164
   576
            split add: split_if nat.split)
wenzelm@23164
   577
apply (rule_tac x = "number_of v" in spec) 
wenzelm@23164
   578
apply auto
wenzelm@23164
   579
done
wenzelm@23164
   580
 
wenzelm@23164
   581
lemma min_number_of_Suc [simp]:
wenzelm@23164
   582
     "min (Suc n) (number_of v) =  
wenzelm@23164
   583
        (let pv = number_of (Numeral.pred v) in  
wenzelm@23164
   584
         if neg pv then 0 else Suc(min n (nat pv)))"
wenzelm@23164
   585
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
wenzelm@23164
   586
            split add: split_if nat.split)
wenzelm@23164
   587
apply (rule_tac x = "number_of v" in spec) 
wenzelm@23164
   588
apply auto
wenzelm@23164
   589
done
wenzelm@23164
   590
 
wenzelm@23164
   591
lemma min_Suc_number_of [simp]:
wenzelm@23164
   592
     "min (number_of v) (Suc n) =  
wenzelm@23164
   593
        (let pv = number_of (Numeral.pred v) in  
wenzelm@23164
   594
         if neg pv then 0 else Suc(min (nat pv) n))"
wenzelm@23164
   595
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
wenzelm@23164
   596
            split add: split_if nat.split)
wenzelm@23164
   597
apply (rule_tac x = "number_of v" in spec) 
wenzelm@23164
   598
apply auto
wenzelm@23164
   599
done
wenzelm@23164
   600
 
wenzelm@23164
   601
subsection{*Literal arithmetic involving powers*}
wenzelm@23164
   602
wenzelm@23164
   603
lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n"
wenzelm@23164
   604
apply (induct "n")
wenzelm@23164
   605
apply (simp_all (no_asm_simp) add: nat_mult_distrib)
wenzelm@23164
   606
done
wenzelm@23164
   607
wenzelm@23164
   608
lemma power_nat_number_of:
wenzelm@23164
   609
     "(number_of v :: nat) ^ n =  
wenzelm@23164
   610
       (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
wenzelm@23164
   611
by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
wenzelm@23164
   612
         split add: split_if cong: imp_cong)
wenzelm@23164
   613
wenzelm@23164
   614
wenzelm@23164
   615
lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]
wenzelm@23164
   616
declare power_nat_number_of_number_of [simp]
wenzelm@23164
   617
wenzelm@23164
   618
wenzelm@23164
   619
huffman@23294
   620
text{*For arbitrary rings*}
wenzelm@23164
   621
huffman@23294
   622
lemma power_number_of_even:
huffman@23294
   623
  fixes z :: "'a::{number_ring,recpower}"
huffman@23294
   624
  shows "z ^ number_of (w BIT bit.B0) = (let w = z ^ (number_of w) in w * w)"
wenzelm@23164
   625
unfolding Let_def nat_number_of_def number_of_BIT bit.cases
wenzelm@23164
   626
apply (rule_tac x = "number_of w" in spec, clarify)
wenzelm@23164
   627
apply (case_tac " (0::int) <= x")
wenzelm@23164
   628
apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square)
wenzelm@23164
   629
done
wenzelm@23164
   630
huffman@23294
   631
lemma power_number_of_odd:
huffman@23294
   632
  fixes z :: "'a::{number_ring,recpower}"
huffman@23294
   633
  shows "z ^ number_of (w BIT bit.B1) = (if (0::int) <= number_of w
wenzelm@23164
   634
     then (let w = z ^ (number_of w) in z * w * w) else 1)"
wenzelm@23164
   635
unfolding Let_def nat_number_of_def number_of_BIT bit.cases
wenzelm@23164
   636
apply (rule_tac x = "number_of w" in spec, auto)
wenzelm@23164
   637
apply (simp only: nat_add_distrib nat_mult_distrib)
wenzelm@23164
   638
apply simp
huffman@23294
   639
apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat power_Suc)
wenzelm@23164
   640
done
wenzelm@23164
   641
huffman@23294
   642
lemmas zpower_number_of_even = power_number_of_even [where 'a=int]
huffman@23294
   643
lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int]
wenzelm@23164
   644
huffman@23294
   645
lemmas power_number_of_even_number_of [simp] =
huffman@23294
   646
    power_number_of_even [of "number_of v", standard]
wenzelm@23164
   647
huffman@23294
   648
lemmas power_number_of_odd_number_of [simp] =
huffman@23294
   649
    power_number_of_odd [of "number_of v", standard]
wenzelm@23164
   650
wenzelm@23164
   651
wenzelm@23164
   652
wenzelm@23164
   653
ML
wenzelm@23164
   654
{*
wenzelm@23164
   655
val numerals = thms"numerals";
wenzelm@23164
   656
val numeral_ss = simpset() addsimps numerals;
wenzelm@23164
   657
wenzelm@23164
   658
val nat_bin_arith_setup =
wenzelm@23164
   659
 Fast_Arith.map_data
wenzelm@23164
   660
   (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
wenzelm@23164
   661
     {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
wenzelm@23164
   662
      inj_thms = inj_thms,
wenzelm@23164
   663
      lessD = lessD, neqE = neqE,
wenzelm@23164
   664
      simpset = simpset addsimps [Suc_nat_number_of, int_nat_number_of,
wenzelm@23164
   665
                                  not_neg_number_of_Pls,
wenzelm@23164
   666
                                  neg_number_of_Min,neg_number_of_BIT]})
wenzelm@23164
   667
*}
wenzelm@23164
   668
wenzelm@23164
   669
setup nat_bin_arith_setup
wenzelm@23164
   670
wenzelm@23164
   671
(* Enable arith to deal with div/mod k where k is a numeral: *)
wenzelm@23164
   672
declare split_div[of _ _ "number_of k", standard, arith_split]
wenzelm@23164
   673
declare split_mod[of _ _ "number_of k", standard, arith_split]
wenzelm@23164
   674
wenzelm@23164
   675
lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
wenzelm@23164
   676
  by (simp add: number_of_Pls nat_number_of_def)
wenzelm@23164
   677
wenzelm@23164
   678
lemma nat_number_of_Min: "number_of Numeral.Min = (0::nat)"
wenzelm@23164
   679
  apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
wenzelm@23164
   680
  done
wenzelm@23164
   681
wenzelm@23164
   682
lemma nat_number_of_BIT_1:
wenzelm@23164
   683
  "number_of (w BIT bit.B1) =
wenzelm@23164
   684
    (if neg (number_of w :: int) then 0
wenzelm@23164
   685
     else let n = number_of w in Suc (n + n))"
wenzelm@23164
   686
  apply (simp only: nat_number_of_def Let_def split: split_if)
wenzelm@23164
   687
  apply (intro conjI impI)
wenzelm@23164
   688
   apply (simp add: neg_nat neg_number_of_BIT)
wenzelm@23164
   689
  apply (rule int_int_eq [THEN iffD1])
wenzelm@23164
   690
  apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms)
wenzelm@23164
   691
  apply (simp only: number_of_BIT zadd_assoc split: bit.split)
wenzelm@23164
   692
  apply simp
wenzelm@23164
   693
  done
wenzelm@23164
   694
wenzelm@23164
   695
lemma nat_number_of_BIT_0:
wenzelm@23164
   696
    "number_of (w BIT bit.B0) = (let n::nat = number_of w in n + n)"
wenzelm@23164
   697
  apply (simp only: nat_number_of_def Let_def)
wenzelm@23164
   698
  apply (cases "neg (number_of w :: int)")
wenzelm@23164
   699
   apply (simp add: neg_nat neg_number_of_BIT)
wenzelm@23164
   700
  apply (rule int_int_eq [THEN iffD1])
wenzelm@23164
   701
  apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms)
wenzelm@23164
   702
  apply (simp only: number_of_BIT zadd_assoc)
wenzelm@23164
   703
  apply simp
wenzelm@23164
   704
  done
wenzelm@23164
   705
wenzelm@23164
   706
lemmas nat_number =
wenzelm@23164
   707
  nat_number_of_Pls nat_number_of_Min
wenzelm@23164
   708
  nat_number_of_BIT_1 nat_number_of_BIT_0
wenzelm@23164
   709
wenzelm@23164
   710
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
wenzelm@23164
   711
  by (simp add: Let_def)
wenzelm@23164
   712
wenzelm@23164
   713
lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})"
huffman@23294
   714
by (simp add: power_mult power_Suc); 
wenzelm@23164
   715
wenzelm@23164
   716
lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})"
wenzelm@23164
   717
by (simp add: power_mult power_Suc); 
wenzelm@23164
   718
wenzelm@23164
   719
wenzelm@23164
   720
subsection{*Literal arithmetic and @{term of_nat}*}
wenzelm@23164
   721
wenzelm@23164
   722
lemma of_nat_double:
wenzelm@23164
   723
     "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
wenzelm@23164
   724
by (simp only: mult_2 nat_add_distrib of_nat_add) 
wenzelm@23164
   725
wenzelm@23164
   726
lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
wenzelm@23164
   727
by (simp only: nat_number_of_def)
wenzelm@23164
   728
wenzelm@23164
   729
lemma of_nat_number_of_lemma:
wenzelm@23164
   730
     "of_nat (number_of v :: nat) =  
wenzelm@23164
   731
         (if 0 \<le> (number_of v :: int) 
wenzelm@23164
   732
          then (number_of v :: 'a :: number_ring)
wenzelm@23164
   733
          else 0)"
wenzelm@23164
   734
by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat);
wenzelm@23164
   735
wenzelm@23164
   736
lemma of_nat_number_of_eq [simp]:
wenzelm@23164
   737
     "of_nat (number_of v :: nat) =  
wenzelm@23164
   738
         (if neg (number_of v :: int) then 0  
wenzelm@23164
   739
          else (number_of v :: 'a :: number_ring))"
wenzelm@23164
   740
by (simp only: of_nat_number_of_lemma neg_def, simp) 
wenzelm@23164
   741
wenzelm@23164
   742
wenzelm@23164
   743
subsection {*Lemmas for the Combination and Cancellation Simprocs*}
wenzelm@23164
   744
wenzelm@23164
   745
lemma nat_number_of_add_left:
wenzelm@23164
   746
     "number_of v + (number_of v' + (k::nat)) =  
wenzelm@23164
   747
         (if neg (number_of v :: int) then number_of v' + k  
wenzelm@23164
   748
          else if neg (number_of v' :: int) then number_of v + k  
wenzelm@23164
   749
          else number_of (v + v') + k)"
wenzelm@23164
   750
by simp
wenzelm@23164
   751
wenzelm@23164
   752
lemma nat_number_of_mult_left:
wenzelm@23164
   753
     "number_of v * (number_of v' * (k::nat)) =  
wenzelm@23164
   754
         (if neg (number_of v :: int) then 0
wenzelm@23164
   755
          else number_of (v * v') * k)"
wenzelm@23164
   756
by simp
wenzelm@23164
   757
wenzelm@23164
   758
wenzelm@23164
   759
subsubsection{*For @{text combine_numerals}*}
wenzelm@23164
   760
wenzelm@23164
   761
lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
wenzelm@23164
   762
by (simp add: add_mult_distrib)
wenzelm@23164
   763
wenzelm@23164
   764
wenzelm@23164
   765
subsubsection{*For @{text cancel_numerals}*}
wenzelm@23164
   766
wenzelm@23164
   767
lemma nat_diff_add_eq1:
wenzelm@23164
   768
     "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
wenzelm@23164
   769
by (simp split add: nat_diff_split add: add_mult_distrib)
wenzelm@23164
   770
wenzelm@23164
   771
lemma nat_diff_add_eq2:
wenzelm@23164
   772
     "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
wenzelm@23164
   773
by (simp split add: nat_diff_split add: add_mult_distrib)
wenzelm@23164
   774
wenzelm@23164
   775
lemma nat_eq_add_iff1:
wenzelm@23164
   776
     "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
wenzelm@23164
   777
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   778
wenzelm@23164
   779
lemma nat_eq_add_iff2:
wenzelm@23164
   780
     "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
wenzelm@23164
   781
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   782
wenzelm@23164
   783
lemma nat_less_add_iff1:
wenzelm@23164
   784
     "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
wenzelm@23164
   785
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   786
wenzelm@23164
   787
lemma nat_less_add_iff2:
wenzelm@23164
   788
     "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
wenzelm@23164
   789
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   790
wenzelm@23164
   791
lemma nat_le_add_iff1:
wenzelm@23164
   792
     "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
wenzelm@23164
   793
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   794
wenzelm@23164
   795
lemma nat_le_add_iff2:
wenzelm@23164
   796
     "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
wenzelm@23164
   797
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   798
wenzelm@23164
   799
wenzelm@23164
   800
subsubsection{*For @{text cancel_numeral_factors} *}
wenzelm@23164
   801
wenzelm@23164
   802
lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
wenzelm@23164
   803
by auto
wenzelm@23164
   804
wenzelm@23164
   805
lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
wenzelm@23164
   806
by auto
wenzelm@23164
   807
wenzelm@23164
   808
lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
wenzelm@23164
   809
by auto
wenzelm@23164
   810
wenzelm@23164
   811
lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
wenzelm@23164
   812
by auto
wenzelm@23164
   813
wenzelm@23164
   814
wenzelm@23164
   815
subsubsection{*For @{text cancel_factor} *}
wenzelm@23164
   816
wenzelm@23164
   817
lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
wenzelm@23164
   818
by auto
wenzelm@23164
   819
wenzelm@23164
   820
lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
wenzelm@23164
   821
by auto
wenzelm@23164
   822
wenzelm@23164
   823
lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
wenzelm@23164
   824
by auto
wenzelm@23164
   825
wenzelm@23164
   826
lemma nat_mult_div_cancel_disj:
wenzelm@23164
   827
     "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
wenzelm@23164
   828
by (simp add: nat_mult_div_cancel1)
wenzelm@23164
   829
wenzelm@23164
   830
wenzelm@23164
   831
subsection {* legacy ML bindings *}
wenzelm@23164
   832
wenzelm@23164
   833
ML
wenzelm@23164
   834
{*
wenzelm@23164
   835
val eq_nat_nat_iff = thm"eq_nat_nat_iff";
wenzelm@23164
   836
val eq_nat_number_of = thm"eq_nat_number_of";
wenzelm@23164
   837
val less_nat_number_of = thm"less_nat_number_of";
wenzelm@23164
   838
val power2_eq_square = thm "power2_eq_square";
wenzelm@23164
   839
val zero_le_power2 = thm "zero_le_power2";
wenzelm@23164
   840
val zero_less_power2 = thm "zero_less_power2";
wenzelm@23164
   841
val zero_eq_power2 = thm "zero_eq_power2";
wenzelm@23164
   842
val abs_power2 = thm "abs_power2";
wenzelm@23164
   843
val power2_abs = thm "power2_abs";
wenzelm@23164
   844
val power2_minus = thm "power2_minus";
wenzelm@23164
   845
val power_minus1_even = thm "power_minus1_even";
wenzelm@23164
   846
val power_minus_even = thm "power_minus_even";
wenzelm@23164
   847
val odd_power_less_zero = thm "odd_power_less_zero";
wenzelm@23164
   848
val odd_0_le_power_imp_0_le = thm "odd_0_le_power_imp_0_le";
wenzelm@23164
   849
wenzelm@23164
   850
val Suc_pred' = thm"Suc_pred'";
wenzelm@23164
   851
val expand_Suc = thm"expand_Suc";
wenzelm@23164
   852
val Suc_eq_add_numeral_1 = thm"Suc_eq_add_numeral_1";
wenzelm@23164
   853
val Suc_eq_add_numeral_1_left = thm"Suc_eq_add_numeral_1_left";
wenzelm@23164
   854
val add_eq_if = thm"add_eq_if";
wenzelm@23164
   855
val mult_eq_if = thm"mult_eq_if";
wenzelm@23164
   856
val power_eq_if = thm"power_eq_if";
wenzelm@23164
   857
val eq_number_of_0 = thm"eq_number_of_0";
wenzelm@23164
   858
val eq_0_number_of = thm"eq_0_number_of";
wenzelm@23164
   859
val less_0_number_of = thm"less_0_number_of";
wenzelm@23164
   860
val neg_imp_number_of_eq_0 = thm"neg_imp_number_of_eq_0";
wenzelm@23164
   861
val eq_number_of_Suc = thm"eq_number_of_Suc";
wenzelm@23164
   862
val Suc_eq_number_of = thm"Suc_eq_number_of";
wenzelm@23164
   863
val less_number_of_Suc = thm"less_number_of_Suc";
wenzelm@23164
   864
val less_Suc_number_of = thm"less_Suc_number_of";
wenzelm@23164
   865
val le_number_of_Suc = thm"le_number_of_Suc";
wenzelm@23164
   866
val le_Suc_number_of = thm"le_Suc_number_of";
wenzelm@23164
   867
val eq_number_of_BIT_BIT = thm"eq_number_of_BIT_BIT";
wenzelm@23164
   868
val eq_number_of_BIT_Pls = thm"eq_number_of_BIT_Pls";
wenzelm@23164
   869
val eq_number_of_BIT_Min = thm"eq_number_of_BIT_Min";
wenzelm@23164
   870
val eq_number_of_Pls_Min = thm"eq_number_of_Pls_Min";
wenzelm@23164
   871
val of_nat_number_of_eq = thm"of_nat_number_of_eq";
wenzelm@23164
   872
val nat_power_eq = thm"nat_power_eq";
wenzelm@23164
   873
val power_nat_number_of = thm"power_nat_number_of";
wenzelm@23164
   874
val zpower_number_of_even = thm"zpower_number_of_even";
wenzelm@23164
   875
val zpower_number_of_odd = thm"zpower_number_of_odd";
wenzelm@23164
   876
val nat_number_of_Pls = thm"nat_number_of_Pls";
wenzelm@23164
   877
val nat_number_of_Min = thm"nat_number_of_Min";
wenzelm@23164
   878
val Let_Suc = thm"Let_Suc";
wenzelm@23164
   879
wenzelm@23164
   880
val nat_number = thms"nat_number";
wenzelm@23164
   881
wenzelm@23164
   882
val nat_number_of_add_left = thm"nat_number_of_add_left";
wenzelm@23164
   883
val nat_number_of_mult_left = thm"nat_number_of_mult_left";
wenzelm@23164
   884
val left_add_mult_distrib = thm"left_add_mult_distrib";
wenzelm@23164
   885
val nat_diff_add_eq1 = thm"nat_diff_add_eq1";
wenzelm@23164
   886
val nat_diff_add_eq2 = thm"nat_diff_add_eq2";
wenzelm@23164
   887
val nat_eq_add_iff1 = thm"nat_eq_add_iff1";
wenzelm@23164
   888
val nat_eq_add_iff2 = thm"nat_eq_add_iff2";
wenzelm@23164
   889
val nat_less_add_iff1 = thm"nat_less_add_iff1";
wenzelm@23164
   890
val nat_less_add_iff2 = thm"nat_less_add_iff2";
wenzelm@23164
   891
val nat_le_add_iff1 = thm"nat_le_add_iff1";
wenzelm@23164
   892
val nat_le_add_iff2 = thm"nat_le_add_iff2";
wenzelm@23164
   893
val nat_mult_le_cancel1 = thm"nat_mult_le_cancel1";
wenzelm@23164
   894
val nat_mult_less_cancel1 = thm"nat_mult_less_cancel1";
wenzelm@23164
   895
val nat_mult_eq_cancel1 = thm"nat_mult_eq_cancel1";
wenzelm@23164
   896
val nat_mult_div_cancel1 = thm"nat_mult_div_cancel1";
wenzelm@23164
   897
val nat_mult_le_cancel_disj = thm"nat_mult_le_cancel_disj";
wenzelm@23164
   898
val nat_mult_less_cancel_disj = thm"nat_mult_less_cancel_disj";
wenzelm@23164
   899
val nat_mult_eq_cancel_disj = thm"nat_mult_eq_cancel_disj";
wenzelm@23164
   900
val nat_mult_div_cancel_disj = thm"nat_mult_div_cancel_disj";
wenzelm@23164
   901
wenzelm@23164
   902
val power_minus_even = thm"power_minus_even";
wenzelm@23164
   903
*}
wenzelm@23164
   904
wenzelm@23164
   905
end