src/HOL/NatBin.thy
author huffman
Fri Dec 05 17:26:16 2008 -0800 (2008-12-05)
changeset 29010 5cd646abf6bc
parent 28984 060832a1f087
child 29011 a47003001699
permissions -rw-r--r--
add lemma le_nat_number_of
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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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instantiation nat :: number
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begin
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definition
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  nat_number_of_def [code inline, code del]: "number_of v = nat (number_of v)"
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instance ..
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end
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lemma [code post]:
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  "nat (number_of v) = number_of v"
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  unfolding nat_number_of_def ..
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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)
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apply (case_tac "z' = 0", simp)
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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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  unfolding nat_number_of_def number_of_is_id neg_def
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  by simp
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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 (Int.succ v) + n)"
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  unfolding nat_number_of_def number_of_is_id neg_def numeral_simps
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  by (simp add: Suc_nat_eq_nat_zadd1 add_ac)
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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 (Int.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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  unfolding nat_number_of_def number_of_is_id neg_def
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  by (simp add: nat_add_distrib)
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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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by (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
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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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  unfolding nat_number_of_def number_of_is_id neg_def
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  by (simp add: nat_mult_distrib)
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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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  unfolding nat_number_of_def number_of_is_id neg_def
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  by (simp add: nat_div_distrib)
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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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  unfolding nat_number_of_def number_of_is_id neg_def
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  by (simp add: nat_mod_distrib)
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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 (number_of v' :: int) \<le> 0
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       else if neg (number_of v' :: int) then (number_of v :: int) = 0
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       else v = v')"
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  unfolding nat_number_of_def number_of_is_id neg_def
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  by auto
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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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  unfolding neg_def nat_number_of_def number_of_is_id
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  by auto
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subsubsection{*Less-than-or-equal *}
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lemma le_nat_number_of [simp]:
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  "(number_of v :: nat) \<le> number_of v' \<longleftrightarrow>
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    (if v \<le> v' then True else v \<le> Int.Pls)"
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  unfolding nat_number_of_def number_of_is_id numeral_simps
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  by auto
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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]:
wenzelm@23164
   320
  fixes a :: "'a::{ordered_idom,recpower}"
wenzelm@23164
   321
  shows "~ (a\<twosuperior> < 0)"
wenzelm@23164
   322
by (force simp add: power2_eq_square mult_less_0_iff) 
wenzelm@23164
   323
wenzelm@23164
   324
lemma zero_eq_power2[simp]:
wenzelm@23164
   325
     "(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))"
wenzelm@23164
   326
  by (force simp add: power2_eq_square mult_eq_0_iff)
wenzelm@23164
   327
wenzelm@23164
   328
lemma abs_power2[simp]:
wenzelm@23164
   329
     "abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})"
wenzelm@23164
   330
  by (simp add: power2_eq_square abs_mult abs_mult_self)
wenzelm@23164
   331
wenzelm@23164
   332
lemma power2_abs[simp]:
wenzelm@23164
   333
     "(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})"
wenzelm@23164
   334
  by (simp add: power2_eq_square abs_mult_self)
wenzelm@23164
   335
wenzelm@23164
   336
lemma power2_minus[simp]:
wenzelm@23164
   337
     "(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})"
wenzelm@23164
   338
  by (simp add: power2_eq_square)
wenzelm@23164
   339
wenzelm@23164
   340
lemma power2_le_imp_le:
wenzelm@23164
   341
  fixes x y :: "'a::{ordered_semidom,recpower}"
wenzelm@23164
   342
  shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y"
wenzelm@23164
   343
unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
wenzelm@23164
   344
wenzelm@23164
   345
lemma power2_less_imp_less:
wenzelm@23164
   346
  fixes x y :: "'a::{ordered_semidom,recpower}"
wenzelm@23164
   347
  shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y"
wenzelm@23164
   348
by (rule power_less_imp_less_base)
wenzelm@23164
   349
wenzelm@23164
   350
lemma power2_eq_imp_eq:
wenzelm@23164
   351
  fixes x y :: "'a::{ordered_semidom,recpower}"
wenzelm@23164
   352
  shows "\<lbrakk>x\<twosuperior> = y\<twosuperior>; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> x = y"
wenzelm@23164
   353
unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base, simp)
wenzelm@23164
   354
wenzelm@23164
   355
lemma power_minus1_even[simp]: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})"
wenzelm@23164
   356
apply (induct "n")
wenzelm@23164
   357
apply (auto simp add: power_Suc power_add)
wenzelm@23164
   358
done
wenzelm@23164
   359
wenzelm@23164
   360
lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2"
wenzelm@23164
   361
by (subst mult_commute) (simp add: power_mult)
wenzelm@23164
   362
wenzelm@23164
   363
lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"
wenzelm@23164
   364
by (simp add: power_even_eq) 
wenzelm@23164
   365
wenzelm@23164
   366
lemma power_minus_even [simp]:
wenzelm@23164
   367
     "(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)"
wenzelm@23164
   368
by (simp add: power_minus1_even power_minus [of a]) 
wenzelm@23164
   369
wenzelm@23164
   370
lemma zero_le_even_power'[simp]:
wenzelm@23164
   371
     "0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)"
wenzelm@23164
   372
proof (induct "n")
wenzelm@23164
   373
  case 0
wenzelm@23164
   374
    show ?case by (simp add: zero_le_one)
wenzelm@23164
   375
next
wenzelm@23164
   376
  case (Suc n)
wenzelm@23164
   377
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
wenzelm@23164
   378
      by (simp add: mult_ac power_add power2_eq_square)
wenzelm@23164
   379
    thus ?case
wenzelm@23164
   380
      by (simp add: prems zero_le_mult_iff)
wenzelm@23164
   381
qed
wenzelm@23164
   382
wenzelm@23164
   383
lemma odd_power_less_zero:
wenzelm@23164
   384
     "(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0"
wenzelm@23164
   385
proof (induct "n")
wenzelm@23164
   386
  case 0
wenzelm@23389
   387
  then show ?case by (simp add: Power.power_Suc)
wenzelm@23164
   388
next
wenzelm@23164
   389
  case (Suc n)
wenzelm@23389
   390
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)" 
wenzelm@23389
   391
    by (simp add: mult_ac power_add power2_eq_square Power.power_Suc)
wenzelm@23389
   392
  thus ?case
wenzelm@23389
   393
    by (simp add: prems mult_less_0_iff mult_neg_neg)
wenzelm@23164
   394
qed
wenzelm@23164
   395
wenzelm@23164
   396
lemma odd_0_le_power_imp_0_le:
wenzelm@23164
   397
     "0 \<le> a  ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})"
wenzelm@23164
   398
apply (insert odd_power_less_zero [of a n]) 
wenzelm@23164
   399
apply (force simp add: linorder_not_less [symmetric]) 
wenzelm@23164
   400
done
wenzelm@23164
   401
wenzelm@23164
   402
text{*Simprules for comparisons where common factors can be cancelled.*}
wenzelm@23164
   403
lemmas zero_compare_simps =
wenzelm@23164
   404
    add_strict_increasing add_strict_increasing2 add_increasing
wenzelm@23164
   405
    zero_le_mult_iff zero_le_divide_iff 
wenzelm@23164
   406
    zero_less_mult_iff zero_less_divide_iff 
wenzelm@23164
   407
    mult_le_0_iff divide_le_0_iff 
wenzelm@23164
   408
    mult_less_0_iff divide_less_0_iff 
wenzelm@23164
   409
    zero_le_power2 power2_less_0
wenzelm@23164
   410
wenzelm@23164
   411
subsubsection{*Nat *}
wenzelm@23164
   412
wenzelm@23164
   413
lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
wenzelm@23164
   414
by (simp add: numerals)
wenzelm@23164
   415
wenzelm@23164
   416
(*Expresses a natural number constant as the Suc of another one.
wenzelm@23164
   417
  NOT suitable for rewriting because n recurs in the condition.*)
wenzelm@23164
   418
lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
wenzelm@23164
   419
wenzelm@23164
   420
subsubsection{*Arith *}
wenzelm@23164
   421
wenzelm@23164
   422
lemma Suc_eq_add_numeral_1: "Suc n = n + 1"
wenzelm@23164
   423
by (simp add: numerals)
wenzelm@23164
   424
wenzelm@23164
   425
lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n"
wenzelm@23164
   426
by (simp add: numerals)
wenzelm@23164
   427
wenzelm@23164
   428
(* These two can be useful when m = number_of... *)
wenzelm@23164
   429
wenzelm@23164
   430
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
wenzelm@23164
   431
apply (case_tac "m")
wenzelm@23164
   432
apply (simp_all add: numerals)
wenzelm@23164
   433
done
wenzelm@23164
   434
wenzelm@23164
   435
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
wenzelm@23164
   436
apply (case_tac "m")
wenzelm@23164
   437
apply (simp_all add: numerals)
wenzelm@23164
   438
done
wenzelm@23164
   439
wenzelm@23164
   440
lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
wenzelm@23164
   441
apply (case_tac "m")
wenzelm@23164
   442
apply (simp_all add: numerals)
wenzelm@23164
   443
done
wenzelm@23164
   444
wenzelm@23164
   445
wenzelm@23164
   446
subsection{*Comparisons involving (0::nat) *}
wenzelm@23164
   447
wenzelm@23164
   448
text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
wenzelm@23164
   449
wenzelm@23164
   450
lemma eq_number_of_0 [simp]:
huffman@28968
   451
  "number_of v = (0::nat) \<longleftrightarrow> number_of v \<le> (0::int)"  
huffman@28968
   452
  unfolding nat_number_of_def number_of_is_id by auto
wenzelm@23164
   453
wenzelm@23164
   454
lemma eq_0_number_of [simp]:
huffman@28968
   455
  "(0::nat) = number_of v \<longleftrightarrow> number_of v \<le> (0::int)"  
wenzelm@23164
   456
by (rule trans [OF eq_sym_conv eq_number_of_0])
wenzelm@23164
   457
wenzelm@23164
   458
lemma less_0_number_of [simp]:
wenzelm@23164
   459
     "((0::nat) < number_of v) = neg (number_of (uminus v) :: int)"
wenzelm@23164
   460
by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] Pls_def)
wenzelm@23164
   461
wenzelm@23164
   462
wenzelm@23164
   463
lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
huffman@28969
   464
by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric])
wenzelm@23164
   465
wenzelm@23164
   466
wenzelm@23164
   467
wenzelm@23164
   468
subsection{*Comparisons involving  @{term Suc} *}
wenzelm@23164
   469
wenzelm@23164
   470
lemma eq_number_of_Suc [simp]:
wenzelm@23164
   471
     "(number_of v = Suc n) =  
haftmann@25919
   472
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   473
         if neg pv then False else nat pv = n)"
wenzelm@23164
   474
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
wenzelm@23164
   475
                  number_of_pred nat_number_of_def 
wenzelm@23164
   476
            split add: split_if)
wenzelm@23164
   477
apply (rule_tac x = "number_of v" in spec)
wenzelm@23164
   478
apply (auto simp add: nat_eq_iff)
wenzelm@23164
   479
done
wenzelm@23164
   480
wenzelm@23164
   481
lemma Suc_eq_number_of [simp]:
wenzelm@23164
   482
     "(Suc n = number_of v) =  
haftmann@25919
   483
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   484
         if neg pv then False else nat pv = n)"
wenzelm@23164
   485
by (rule trans [OF eq_sym_conv eq_number_of_Suc])
wenzelm@23164
   486
wenzelm@23164
   487
lemma less_number_of_Suc [simp]:
wenzelm@23164
   488
     "(number_of v < Suc n) =  
haftmann@25919
   489
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   490
         if neg pv then True else nat pv < n)"
wenzelm@23164
   491
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
wenzelm@23164
   492
                  number_of_pred nat_number_of_def  
wenzelm@23164
   493
            split add: split_if)
wenzelm@23164
   494
apply (rule_tac x = "number_of v" in spec)
wenzelm@23164
   495
apply (auto simp add: nat_less_iff)
wenzelm@23164
   496
done
wenzelm@23164
   497
wenzelm@23164
   498
lemma less_Suc_number_of [simp]:
wenzelm@23164
   499
     "(Suc n < number_of v) =  
haftmann@25919
   500
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   501
         if neg pv then False else n < nat pv)"
wenzelm@23164
   502
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
wenzelm@23164
   503
                  number_of_pred nat_number_of_def
wenzelm@23164
   504
            split add: split_if)
wenzelm@23164
   505
apply (rule_tac x = "number_of v" in spec)
wenzelm@23164
   506
apply (auto simp add: zless_nat_eq_int_zless)
wenzelm@23164
   507
done
wenzelm@23164
   508
wenzelm@23164
   509
lemma le_number_of_Suc [simp]:
wenzelm@23164
   510
     "(number_of v <= Suc n) =  
haftmann@25919
   511
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   512
         if neg pv then True else nat pv <= n)"
wenzelm@23164
   513
by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric])
wenzelm@23164
   514
wenzelm@23164
   515
lemma le_Suc_number_of [simp]:
wenzelm@23164
   516
     "(Suc n <= number_of v) =  
haftmann@25919
   517
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   518
         if neg pv then False else n <= nat pv)"
wenzelm@23164
   519
by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric])
wenzelm@23164
   520
wenzelm@23164
   521
haftmann@25919
   522
lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Int.Min"
wenzelm@23164
   523
by auto
wenzelm@23164
   524
wenzelm@23164
   525
wenzelm@23164
   526
wenzelm@23164
   527
subsection{*Max and Min Combined with @{term Suc} *}
wenzelm@23164
   528
wenzelm@23164
   529
lemma max_number_of_Suc [simp]:
wenzelm@23164
   530
     "max (Suc n) (number_of v) =  
haftmann@25919
   531
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   532
         if neg pv then Suc n else Suc(max n (nat pv)))"
wenzelm@23164
   533
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
wenzelm@23164
   534
            split add: split_if nat.split)
wenzelm@23164
   535
apply (rule_tac x = "number_of v" in spec) 
wenzelm@23164
   536
apply auto
wenzelm@23164
   537
done
wenzelm@23164
   538
 
wenzelm@23164
   539
lemma max_Suc_number_of [simp]:
wenzelm@23164
   540
     "max (number_of v) (Suc n) =  
haftmann@25919
   541
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   542
         if neg pv then Suc n else Suc(max (nat pv) n))"
wenzelm@23164
   543
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
wenzelm@23164
   544
            split add: split_if nat.split)
wenzelm@23164
   545
apply (rule_tac x = "number_of v" in spec) 
wenzelm@23164
   546
apply auto
wenzelm@23164
   547
done
wenzelm@23164
   548
 
wenzelm@23164
   549
lemma min_number_of_Suc [simp]:
wenzelm@23164
   550
     "min (Suc n) (number_of v) =  
haftmann@25919
   551
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   552
         if neg pv then 0 else Suc(min n (nat pv)))"
wenzelm@23164
   553
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
wenzelm@23164
   554
            split add: split_if nat.split)
wenzelm@23164
   555
apply (rule_tac x = "number_of v" in spec) 
wenzelm@23164
   556
apply auto
wenzelm@23164
   557
done
wenzelm@23164
   558
 
wenzelm@23164
   559
lemma min_Suc_number_of [simp]:
wenzelm@23164
   560
     "min (number_of v) (Suc n) =  
haftmann@25919
   561
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   562
         if neg pv then 0 else Suc(min (nat pv) n))"
wenzelm@23164
   563
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
wenzelm@23164
   564
            split add: split_if nat.split)
wenzelm@23164
   565
apply (rule_tac x = "number_of v" in spec) 
wenzelm@23164
   566
apply auto
wenzelm@23164
   567
done
wenzelm@23164
   568
 
wenzelm@23164
   569
subsection{*Literal arithmetic involving powers*}
wenzelm@23164
   570
wenzelm@23164
   571
lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n"
wenzelm@23164
   572
apply (induct "n")
wenzelm@23164
   573
apply (simp_all (no_asm_simp) add: nat_mult_distrib)
wenzelm@23164
   574
done
wenzelm@23164
   575
wenzelm@23164
   576
lemma power_nat_number_of:
wenzelm@23164
   577
     "(number_of v :: nat) ^ n =  
wenzelm@23164
   578
       (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
wenzelm@23164
   579
by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
wenzelm@23164
   580
         split add: split_if cong: imp_cong)
wenzelm@23164
   581
wenzelm@23164
   582
wenzelm@23164
   583
lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]
wenzelm@23164
   584
declare power_nat_number_of_number_of [simp]
wenzelm@23164
   585
wenzelm@23164
   586
wenzelm@23164
   587
huffman@23294
   588
text{*For arbitrary rings*}
wenzelm@23164
   589
huffman@23294
   590
lemma power_number_of_even:
huffman@23294
   591
  fixes z :: "'a::{number_ring,recpower}"
huffman@26086
   592
  shows "z ^ number_of (Int.Bit0 w) = (let w = z ^ (number_of w) in w * w)"
huffman@26086
   593
unfolding Let_def nat_number_of_def number_of_Bit0
wenzelm@23164
   594
apply (rule_tac x = "number_of w" in spec, clarify)
wenzelm@23164
   595
apply (case_tac " (0::int) <= x")
wenzelm@23164
   596
apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square)
wenzelm@23164
   597
done
wenzelm@23164
   598
huffman@23294
   599
lemma power_number_of_odd:
huffman@23294
   600
  fixes z :: "'a::{number_ring,recpower}"
huffman@26086
   601
  shows "z ^ number_of (Int.Bit1 w) = (if (0::int) <= number_of w
wenzelm@23164
   602
     then (let w = z ^ (number_of w) in z * w * w) else 1)"
huffman@26086
   603
unfolding Let_def nat_number_of_def number_of_Bit1
wenzelm@23164
   604
apply (rule_tac x = "number_of w" in spec, auto)
wenzelm@23164
   605
apply (simp only: nat_add_distrib nat_mult_distrib)
wenzelm@23164
   606
apply simp
huffman@23294
   607
apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat power_Suc)
wenzelm@23164
   608
done
wenzelm@23164
   609
huffman@23294
   610
lemmas zpower_number_of_even = power_number_of_even [where 'a=int]
huffman@23294
   611
lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int]
wenzelm@23164
   612
huffman@23294
   613
lemmas power_number_of_even_number_of [simp] =
huffman@23294
   614
    power_number_of_even [of "number_of v", standard]
wenzelm@23164
   615
huffman@23294
   616
lemmas power_number_of_odd_number_of [simp] =
huffman@23294
   617
    power_number_of_odd [of "number_of v", standard]
wenzelm@23164
   618
wenzelm@23164
   619
wenzelm@23164
   620
wenzelm@23164
   621
ML
wenzelm@23164
   622
{*
wenzelm@26342
   623
val numeral_ss = @{simpset} addsimps @{thms numerals};
wenzelm@23164
   624
wenzelm@23164
   625
val nat_bin_arith_setup =
wenzelm@24093
   626
 LinArith.map_data
wenzelm@23164
   627
   (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
wenzelm@23164
   628
     {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
wenzelm@23164
   629
      inj_thms = inj_thms,
wenzelm@23164
   630
      lessD = lessD, neqE = neqE,
wenzelm@23164
   631
      simpset = simpset addsimps [Suc_nat_number_of, int_nat_number_of,
haftmann@25481
   632
        @{thm not_neg_number_of_Pls}, @{thm neg_number_of_Min},
huffman@26086
   633
        @{thm neg_number_of_Bit0}, @{thm neg_number_of_Bit1}]})
wenzelm@23164
   634
*}
wenzelm@23164
   635
wenzelm@24075
   636
declaration {* K nat_bin_arith_setup *}
wenzelm@23164
   637
wenzelm@23164
   638
(* Enable arith to deal with div/mod k where k is a numeral: *)
wenzelm@23164
   639
declare split_div[of _ _ "number_of k", standard, arith_split]
wenzelm@23164
   640
declare split_mod[of _ _ "number_of k", standard, arith_split]
wenzelm@23164
   641
wenzelm@23164
   642
lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
wenzelm@23164
   643
  by (simp add: number_of_Pls nat_number_of_def)
wenzelm@23164
   644
haftmann@25919
   645
lemma nat_number_of_Min: "number_of Int.Min = (0::nat)"
wenzelm@23164
   646
  apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
wenzelm@23164
   647
  done
wenzelm@23164
   648
huffman@26086
   649
lemma nat_number_of_Bit0:
huffman@26086
   650
    "number_of (Int.Bit0 w) = (let n::nat = number_of w in n + n)"
huffman@28969
   651
  unfolding nat_number_of_def number_of_is_id numeral_simps Let_def
huffman@28969
   652
  by auto
huffman@26086
   653
huffman@26086
   654
lemma nat_number_of_Bit1:
huffman@26086
   655
  "number_of (Int.Bit1 w) =
wenzelm@23164
   656
    (if neg (number_of w :: int) then 0
wenzelm@23164
   657
     else let n = number_of w in Suc (n + n))"
huffman@28969
   658
  unfolding nat_number_of_def number_of_is_id numeral_simps neg_def Let_def
huffman@28968
   659
  by auto
wenzelm@23164
   660
wenzelm@23164
   661
lemmas nat_number =
wenzelm@23164
   662
  nat_number_of_Pls nat_number_of_Min
huffman@26086
   663
  nat_number_of_Bit0 nat_number_of_Bit1
wenzelm@23164
   664
wenzelm@23164
   665
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
wenzelm@23164
   666
  by (simp add: Let_def)
wenzelm@23164
   667
wenzelm@23164
   668
lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})"
huffman@23294
   669
by (simp add: power_mult power_Suc); 
wenzelm@23164
   670
wenzelm@23164
   671
lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})"
wenzelm@23164
   672
by (simp add: power_mult power_Suc); 
wenzelm@23164
   673
wenzelm@23164
   674
wenzelm@23164
   675
subsection{*Literal arithmetic and @{term of_nat}*}
wenzelm@23164
   676
wenzelm@23164
   677
lemma of_nat_double:
wenzelm@23164
   678
     "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
wenzelm@23164
   679
by (simp only: mult_2 nat_add_distrib of_nat_add) 
wenzelm@23164
   680
wenzelm@23164
   681
lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
wenzelm@23164
   682
by (simp only: nat_number_of_def)
wenzelm@23164
   683
wenzelm@23164
   684
lemma of_nat_number_of_lemma:
wenzelm@23164
   685
     "of_nat (number_of v :: nat) =  
wenzelm@23164
   686
         (if 0 \<le> (number_of v :: int) 
wenzelm@23164
   687
          then (number_of v :: 'a :: number_ring)
wenzelm@23164
   688
          else 0)"
wenzelm@23164
   689
by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat);
wenzelm@23164
   690
wenzelm@23164
   691
lemma of_nat_number_of_eq [simp]:
wenzelm@23164
   692
     "of_nat (number_of v :: nat) =  
wenzelm@23164
   693
         (if neg (number_of v :: int) then 0  
wenzelm@23164
   694
          else (number_of v :: 'a :: number_ring))"
wenzelm@23164
   695
by (simp only: of_nat_number_of_lemma neg_def, simp) 
wenzelm@23164
   696
wenzelm@23164
   697
wenzelm@23164
   698
subsection {*Lemmas for the Combination and Cancellation Simprocs*}
wenzelm@23164
   699
wenzelm@23164
   700
lemma nat_number_of_add_left:
wenzelm@23164
   701
     "number_of v + (number_of v' + (k::nat)) =  
wenzelm@23164
   702
         (if neg (number_of v :: int) then number_of v' + k  
wenzelm@23164
   703
          else if neg (number_of v' :: int) then number_of v + k  
wenzelm@23164
   704
          else number_of (v + v') + k)"
huffman@28968
   705
  unfolding nat_number_of_def number_of_is_id neg_def
huffman@28968
   706
  by auto
wenzelm@23164
   707
wenzelm@23164
   708
lemma nat_number_of_mult_left:
wenzelm@23164
   709
     "number_of v * (number_of v' * (k::nat)) =  
wenzelm@23164
   710
         (if neg (number_of v :: int) then 0
wenzelm@23164
   711
          else number_of (v * v') * k)"
wenzelm@23164
   712
by simp
wenzelm@23164
   713
wenzelm@23164
   714
wenzelm@23164
   715
subsubsection{*For @{text combine_numerals}*}
wenzelm@23164
   716
wenzelm@23164
   717
lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
wenzelm@23164
   718
by (simp add: add_mult_distrib)
wenzelm@23164
   719
wenzelm@23164
   720
wenzelm@23164
   721
subsubsection{*For @{text cancel_numerals}*}
wenzelm@23164
   722
wenzelm@23164
   723
lemma nat_diff_add_eq1:
wenzelm@23164
   724
     "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
wenzelm@23164
   725
by (simp split add: nat_diff_split add: add_mult_distrib)
wenzelm@23164
   726
wenzelm@23164
   727
lemma nat_diff_add_eq2:
wenzelm@23164
   728
     "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
wenzelm@23164
   729
by (simp split add: nat_diff_split add: add_mult_distrib)
wenzelm@23164
   730
wenzelm@23164
   731
lemma nat_eq_add_iff1:
wenzelm@23164
   732
     "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
wenzelm@23164
   733
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   734
wenzelm@23164
   735
lemma nat_eq_add_iff2:
wenzelm@23164
   736
     "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
wenzelm@23164
   737
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   738
wenzelm@23164
   739
lemma nat_less_add_iff1:
wenzelm@23164
   740
     "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
wenzelm@23164
   741
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   742
wenzelm@23164
   743
lemma nat_less_add_iff2:
wenzelm@23164
   744
     "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
wenzelm@23164
   745
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   746
wenzelm@23164
   747
lemma nat_le_add_iff1:
wenzelm@23164
   748
     "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
wenzelm@23164
   749
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   750
wenzelm@23164
   751
lemma nat_le_add_iff2:
wenzelm@23164
   752
     "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
wenzelm@23164
   753
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   754
wenzelm@23164
   755
wenzelm@23164
   756
subsubsection{*For @{text cancel_numeral_factors} *}
wenzelm@23164
   757
wenzelm@23164
   758
lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
wenzelm@23164
   759
by auto
wenzelm@23164
   760
wenzelm@23164
   761
lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
wenzelm@23164
   762
by auto
wenzelm@23164
   763
wenzelm@23164
   764
lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
wenzelm@23164
   765
by auto
wenzelm@23164
   766
wenzelm@23164
   767
lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
wenzelm@23164
   768
by auto
wenzelm@23164
   769
nipkow@23969
   770
lemma nat_mult_dvd_cancel_disj[simp]:
nipkow@23969
   771
  "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
nipkow@23969
   772
by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])
nipkow@23969
   773
nipkow@23969
   774
lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
nipkow@23969
   775
by(auto)
nipkow@23969
   776
wenzelm@23164
   777
wenzelm@23164
   778
subsubsection{*For @{text cancel_factor} *}
wenzelm@23164
   779
wenzelm@23164
   780
lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
wenzelm@23164
   781
by auto
wenzelm@23164
   782
wenzelm@23164
   783
lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
wenzelm@23164
   784
by auto
wenzelm@23164
   785
wenzelm@23164
   786
lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
wenzelm@23164
   787
by auto
wenzelm@23164
   788
nipkow@23969
   789
lemma nat_mult_div_cancel_disj[simp]:
wenzelm@23164
   790
     "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
wenzelm@23164
   791
by (simp add: nat_mult_div_cancel1)
wenzelm@23164
   792
wenzelm@23164
   793
end