src/HOL/Integ/NatBin.thy
author berghofe
Wed, 21 Sep 2005 12:02:19 +0200
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Declared nat_number_of as code lemma.
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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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defs (overloaded)
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  nat_number_of_def:
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     "(number_of::bin => nat) v == nat ((number_of :: bin => int) v)"
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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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declare nat_number_of [symmetric, THEN eq_reflection, code unfold]
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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 inj_int [THEN injD], 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 zadd_int 
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                 zmult_int)
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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 inj_int [THEN injD], 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 zadd_int zmult_int)
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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 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 :: nat) =  
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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 (bin_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 (bin_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 (bin_add 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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apply (simp add: diff_is_0_eq nat_le_eq_zle)
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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 (bin_add v (bin_minus 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 (bin_mult 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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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 (bin_add v (bin_minus 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 (bin_minus v') :: int)  
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          else neg (number_of (bin_add v (bin_minus 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) 
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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::{comm_semiring_1_cancel,recpower})\<twosuperior> = a * a"
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  by (simp add: numeral_2_eq_2 Power.power_Suc)
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lemma [simp]: "(0::'a::{comm_semiring_1_cancel,recpower})\<twosuperior> = 0"
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  by (simp add: power2_eq_square)
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lemma [simp]: "(1::'a::{comm_semiring_1_cancel,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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16775
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   294
lemma zero_le_power2: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,recpower})"
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  by (simp add: power2_eq_square zero_le_square)
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   296
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lemma zero_less_power2:
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     "(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,recpower}))"
14353
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  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
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diff changeset
   300
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   301
lemma power2_less_0:
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  fixes a :: "'a::{ordered_idom,recpower}"
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   303
  shows "~ (a\<twosuperior> < 0)"
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   304
by (force simp add: power2_eq_square mult_less_0_iff) 
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   305
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   306
lemma zero_eq_power2:
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   307
     "(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))"
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diff changeset
   308
  by (force simp add: power2_eq_square mult_eq_0_iff)
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paulson
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diff changeset
   309
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   310
lemma abs_power2:
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     "abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})"
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   312
  by (simp add: power2_eq_square abs_mult abs_mult_self)
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diff changeset
   313
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diff changeset
   314
lemma power2_abs:
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   315
     "(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})"
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   316
  by (simp add: power2_eq_square abs_mult_self)
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diff changeset
   317
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   318
lemma power2_minus:
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   319
     "(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})"
14353
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   320
  by (simp add: power2_eq_square)
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diff changeset
   321
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   322
lemma power_minus1_even: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})"
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apply (induct "n")
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   324
apply (auto simp add: power_Suc power_add power2_minus)
14353
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diff changeset
   325
done
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diff changeset
   326
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   327
lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2"
14443
75910c7557c5 generic theorems about exponentials; general tidying up
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diff changeset
   328
by (simp add: power_mult power_mult_distrib power2_eq_square)
75910c7557c5 generic theorems about exponentials; general tidying up
paulson
parents: 14430
diff changeset
   329
75910c7557c5 generic theorems about exponentials; general tidying up
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   330
lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"
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paulson
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diff changeset
   331
by (simp add: power_even_eq) 
75910c7557c5 generic theorems about exponentials; general tidying up
paulson
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diff changeset
   332
14353
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   333
lemma power_minus_even [simp]:
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   334
     "(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)"
14353
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parents: 14288
diff changeset
   335
by (simp add: power_minus1_even power_minus [of a]) 
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diff changeset
   336
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   337
lemma zero_le_even_power':
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   338
     "0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)"
14353
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   339
proof (induct "n")
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diff changeset
   340
  case 0
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   341
    show ?case by (simp add: zero_le_one)
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   342
next
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
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   343
  case (Suc n)
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   344
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
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diff changeset
   345
      by (simp add: mult_ac power_add power2_eq_square)
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
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diff changeset
   346
    thus ?case
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parents: 14288
diff changeset
   347
      by (simp add: prems zero_le_square zero_le_mult_iff)
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
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   348
qed
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
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diff changeset
   349
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
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   350
lemma odd_power_less_zero:
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   351
     "(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0"
14353
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diff changeset
   352
proof (induct "n")
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paulson
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diff changeset
   353
  case 0
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   354
    show ?case by (simp add: Power.power_Suc)
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   355
next
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
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diff changeset
   356
  case (Suc n)
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
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diff changeset
   357
    have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)" 
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   358
      by (simp add: mult_ac power_add power2_eq_square Power.power_Suc)
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
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diff changeset
   359
    thus ?case
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16642
diff changeset
   360
      by (simp add: prems mult_less_0_iff mult_neg_neg)
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
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diff changeset
   361
qed
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   362
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   363
lemma odd_0_le_power_imp_0_le:
15003
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paulson
parents: 14738
diff changeset
   364
     "0 \<le> a  ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})"
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   365
apply (insert odd_power_less_zero [of a n]) 
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   366
apply (force simp add: linorder_not_less [symmetric]) 
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   367
done
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diff changeset
   368
15234
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   369
text{*Simprules for comparisons where common factors can be cancelled.*}
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   370
lemmas zero_compare_simps =
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paulson
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diff changeset
   371
    add_strict_increasing add_strict_increasing2 add_increasing
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paulson
parents: 15140
diff changeset
   372
    zero_le_mult_iff zero_le_divide_iff 
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paulson
parents: 15140
diff changeset
   373
    zero_less_mult_iff zero_less_divide_iff 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15140
diff changeset
   374
    mult_le_0_iff divide_le_0_iff 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15140
diff changeset
   375
    mult_less_0_iff divide_less_0_iff 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
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diff changeset
   376
    zero_le_power2 power2_less_0
14353
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diff changeset
   377
14390
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diff changeset
   378
subsubsection{*Nat *}
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   379
5efbb548107d Tidying of the integer development; towards removing the
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diff changeset
   380
lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   381
by (simp add: numerals)
14272
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paulson
parents: 14194
diff changeset
   382
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   383
(*Expresses a natural number constant as the Suc of another one.
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   384
  NOT suitable for rewriting because n recurs in the condition.*)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   385
lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   386
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   387
subsubsection{*Arith *}
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   388
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   389
lemma Suc_eq_add_numeral_1: "Suc n = n + 1"
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   390
by (simp add: numerals)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   391
14467
bbfa6b01a55f new lemma
paulson
parents: 14443
diff changeset
   392
lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n"
bbfa6b01a55f new lemma
paulson
parents: 14443
diff changeset
   393
by (simp add: numerals)
bbfa6b01a55f new lemma
paulson
parents: 14443
diff changeset
   394
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   395
(* These two can be useful when m = number_of... *)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   396
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   397
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   398
apply (case_tac "m")
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   399
apply (simp_all add: numerals)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   400
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   401
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   402
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   403
apply (case_tac "m")
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   404
apply (simp_all add: numerals)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   405
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   406
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   407
lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   408
apply (case_tac "m")
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   409
apply (simp_all add: numerals)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   410
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   411
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   412
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   413
subsection{*Comparisons involving (0::nat) *}
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   414
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   415
text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   416
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   417
lemma eq_number_of_0 [simp]:
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   418
     "(number_of v = (0::nat)) =  
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   419
      (if neg (number_of v :: int) then True else iszero (number_of v :: int))"
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   420
by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] iszero_0)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   421
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   422
lemma eq_0_number_of [simp]:
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   423
     "((0::nat) = number_of v) =  
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   424
      (if neg (number_of v :: int) then True else iszero (number_of v :: int))"
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   425
by (rule trans [OF eq_sym_conv eq_number_of_0])
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   426
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   427
lemma less_0_number_of [simp]:
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   428
     "((0::nat) < number_of v) = neg (number_of (bin_minus v) :: int)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   429
by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric])
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   430
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   431
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   432
lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   433
by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] iszero_0)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   434
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   435
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   436
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   437
subsection{*Comparisons involving Suc *}
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   438
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   439
lemma eq_number_of_Suc [simp]:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   440
     "(number_of v = Suc n) =  
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   441
        (let pv = number_of (bin_pred v) in  
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   442
         if neg pv then False else nat pv = n)"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   443
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   444
                  number_of_pred nat_number_of_def 
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   445
            split add: split_if)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   446
apply (rule_tac x = "number_of v" in spec)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   447
apply (auto simp add: nat_eq_iff)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   448
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   449
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   450
lemma Suc_eq_number_of [simp]:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   451
     "(Suc n = number_of v) =  
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   452
        (let pv = number_of (bin_pred v) in  
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   453
         if neg pv then False else nat pv = n)"
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   454
by (rule trans [OF eq_sym_conv eq_number_of_Suc])
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   455
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   456
lemma less_number_of_Suc [simp]:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   457
     "(number_of v < Suc n) =  
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   458
        (let pv = number_of (bin_pred v) in  
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   459
         if neg pv then True else nat pv < n)"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   460
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   461
                  number_of_pred nat_number_of_def  
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   462
            split add: split_if)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   463
apply (rule_tac x = "number_of v" in spec)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   464
apply (auto simp add: nat_less_iff)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   465
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   466
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   467
lemma less_Suc_number_of [simp]:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   468
     "(Suc n < number_of v) =  
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   469
        (let pv = number_of (bin_pred v) in  
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   470
         if neg pv then False else n < nat pv)"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   471
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   472
                  number_of_pred nat_number_of_def
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   473
            split add: split_if)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   474
apply (rule_tac x = "number_of v" in spec)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   475
apply (auto simp add: zless_nat_eq_int_zless)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   476
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   477
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   478
lemma le_number_of_Suc [simp]:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   479
     "(number_of v <= Suc n) =  
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   480
        (let pv = number_of (bin_pred v) in  
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   481
         if neg pv then True else nat pv <= n)"
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   482
by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric])
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   483
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   484
lemma le_Suc_number_of [simp]:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   485
     "(Suc n <= number_of v) =  
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   486
        (let pv = number_of (bin_pred v) in  
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   487
         if neg pv then False else n <= nat pv)"
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   488
by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric])
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   489
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   490
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   491
(* Push int(.) inwards: *)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   492
declare zadd_int [symmetric, simp]
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   493
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   494
lemma lemma1: "(m+m = n+n) = (m = (n::int))"
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   495
by auto
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   496
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   497
lemma lemma2: "m+m ~= (1::int) + (n + n)"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   498
apply auto
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   499
apply (drule_tac f = "%x. x mod 2" in arg_cong)
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   500
apply (simp add: zmod_zadd1_eq)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   501
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   502
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   503
lemma eq_number_of_BIT_BIT:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   504
     "((number_of (v BIT x) ::int) = number_of (w BIT y)) =  
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   505
      (x=y & (((number_of v) ::int) = number_of w))"
15620
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   506
apply (simp only: number_of_BIT lemma1 lemma2 eq_commute
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14467
diff changeset
   507
               OrderedGroup.add_left_cancel add_assoc OrderedGroup.add_0
15620
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   508
            split add: bit.split) 
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   509
apply simp
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   510
done
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   511
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   512
lemma eq_number_of_BIT_Pls:
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   513
     "((number_of (v BIT x) ::int) = Numeral0) =  
15620
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   514
      (x=bit.B0 & (((number_of v) ::int) = Numeral0))"
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   515
apply (simp only: simp_thms  add: number_of_BIT number_of_Pls eq_commute
15620
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   516
            split add: bit.split cong: imp_cong)
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   517
apply (rule_tac x = "number_of v" in spec, safe)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   518
apply (simp_all (no_asm_use))
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   519
apply (drule_tac f = "%x. x mod 2" in arg_cong)
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   520
apply (simp add: zmod_zadd1_eq)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   521
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   522
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   523
lemma eq_number_of_BIT_Min:
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   524
     "((number_of (v BIT x) ::int) = number_of Numeral.Min) =  
15620
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   525
      (x=bit.B1 & (((number_of v) ::int) = number_of Numeral.Min))"
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   526
apply (simp only: simp_thms  add: number_of_BIT number_of_Min eq_commute
15620
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   527
            split add: bit.split cong: imp_cong)
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   528
apply (rule_tac x = "number_of v" in spec, auto)
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   529
apply (drule_tac f = "%x. x mod 2" in arg_cong, auto)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   530
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   531
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   532
lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Numeral.Min"
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   533
by auto
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   534
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   535
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   536
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   537
subsection{*Literal arithmetic involving powers*}
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   538
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   539
lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
   540
apply (induct "n")
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   541
apply (simp_all (no_asm_simp) add: nat_mult_distrib)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   542
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   543
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   544
lemma power_nat_number_of:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   545
     "(number_of v :: nat) ^ n =  
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   546
       (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   547
by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   548
         split add: split_if cong: imp_cong)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   549
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   550
17085
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
   551
lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
   552
declare power_nat_number_of_number_of [simp]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
   553
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   554
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   555
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   556
text{*For the integers*}
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   557
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   558
lemma zpower_number_of_even:
15620
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   559
     "(z::int) ^ number_of (w BIT bit.B0) =  
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   560
      (let w = z ^ (number_of w) in  w*w)"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   561
apply (simp del: nat_number_of  add: nat_number_of_def number_of_BIT Let_def)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   562
apply (simp only: number_of_add) 
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   563
apply (rule_tac x = "number_of w" in spec, clarify)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   564
apply (case_tac " (0::int) <= x")
14443
75910c7557c5 generic theorems about exponentials; general tidying up
paulson
parents: 14430
diff changeset
   565
apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   566
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   567
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   568
lemma zpower_number_of_odd:
15620
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   569
     "(z::int) ^ number_of (w BIT bit.B1) =  
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   570
          (if (0::int) <= number_of w                    
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   571
           then (let w = z ^ (number_of w) in  z*w*w)    
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   572
           else 1)"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   573
apply (simp del: nat_number_of  add: nat_number_of_def number_of_BIT Let_def)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   574
apply (simp only: number_of_add nat_numeral_1_eq_1 not_neg_eq_ge_0 neg_eq_less_0) 
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   575
apply (rule_tac x = "number_of w" in spec, clarify)
14443
75910c7557c5 generic theorems about exponentials; general tidying up
paulson
parents: 14430
diff changeset
   576
apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   577
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   578
17085
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
   579
lemmas zpower_number_of_even_number_of =
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
   580
    zpower_number_of_even [of "number_of v", standard]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
   581
declare zpower_number_of_even_number_of [simp]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
   582
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
   583
lemmas zpower_number_of_odd_number_of =
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
   584
    zpower_number_of_odd [of "number_of v", standard]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
   585
declare zpower_number_of_odd_number_of [simp]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
   586
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   587
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   588
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   589
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   590
ML
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   591
{*
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   592
val numerals = thms"numerals";
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   593
val numeral_ss = simpset() addsimps numerals;
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   594
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   595
val nat_bin_arith_setup =
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   596
 [Fast_Arith.map_data 
15921
b6e345548913 Fixing a problem with lin.arith.
nipkow
parents: 15620
diff changeset
   597
   (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   598
     {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   599
      inj_thms = inj_thms,
15921
b6e345548913 Fixing a problem with lin.arith.
nipkow
parents: 15620
diff changeset
   600
      lessD = lessD, neqE = neqE,
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   601
      simpset = simpset addsimps [Suc_nat_number_of, int_nat_number_of,
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   602
                                  not_neg_number_of_Pls,
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   603
                                  neg_number_of_Min,neg_number_of_BIT]})]
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   604
*}
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14194
diff changeset
   605
12838
wenzelm
parents: 12440
diff changeset
   606
setup nat_bin_arith_setup
wenzelm
parents: 12440
diff changeset
   607
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13154
diff changeset
   608
(* Enable arith to deal with div/mod k where k is a numeral: *)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13154
diff changeset
   609
declare split_div[of _ _ "number_of k", standard, arith_split]
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13154
diff changeset
   610
declare split_mod[of _ _ "number_of k", standard, arith_split]
13154
f1097ea60ba4 Set up arith to deal with div 2 and mod 2.
nipkow
parents: 13043
diff changeset
   611
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   612
lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
12838
wenzelm
parents: 12440
diff changeset
   613
  by (simp add: number_of_Pls nat_number_of_def)
wenzelm
parents: 12440
diff changeset
   614
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   615
lemma nat_number_of_Min: "number_of Numeral.Min = (0::nat)"
12838
wenzelm
parents: 12440
diff changeset
   616
  apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
wenzelm
parents: 12440
diff changeset
   617
  apply (simp add: neg_nat)
wenzelm
parents: 12440
diff changeset
   618
  done
7032
d6efb3b8e669 NatBin: binary arithmetic for the naturals
paulson
parents:
diff changeset
   619
15620
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   620
lemma nat_number_of_BIT_1:
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   621
  "number_of (w BIT bit.B1) =
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   622
    (if neg (number_of w :: int) then 0
12838
wenzelm
parents: 12440
diff changeset
   623
     else let n = number_of w in Suc (n + n))"
wenzelm
parents: 12440
diff changeset
   624
  apply (simp only: nat_number_of_def Let_def split: split_if)
wenzelm
parents: 12440
diff changeset
   625
  apply (intro conjI impI)
wenzelm
parents: 12440
diff changeset
   626
   apply (simp add: neg_nat neg_number_of_BIT)
wenzelm
parents: 12440
diff changeset
   627
  apply (rule int_int_eq [THEN iffD1])
wenzelm
parents: 12440
diff changeset
   628
  apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms)
15620
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   629
  apply (simp only: number_of_BIT zadd_assoc split: bit.split)
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   630
  apply simp
12838
wenzelm
parents: 12440
diff changeset
   631
  done
7032
d6efb3b8e669 NatBin: binary arithmetic for the naturals
paulson
parents:
diff changeset
   632
15620
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   633
lemma nat_number_of_BIT_0:
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   634
    "number_of (w BIT bit.B0) = (let n::nat = number_of w in n + n)"
12838
wenzelm
parents: 12440
diff changeset
   635
  apply (simp only: nat_number_of_def Let_def)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   636
  apply (cases "neg (number_of w :: int)")
12838
wenzelm
parents: 12440
diff changeset
   637
   apply (simp add: neg_nat neg_number_of_BIT)
wenzelm
parents: 12440
diff changeset
   638
  apply (rule int_int_eq [THEN iffD1])
wenzelm
parents: 12440
diff changeset
   639
  apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms)
15620
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   640
  apply (simp only: number_of_BIT zadd_assoc)
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   641
  apply simp
12838
wenzelm
parents: 12440
diff changeset
   642
  done
wenzelm
parents: 12440
diff changeset
   643
13043
ad1828b479b7 renamed nat_number_of to nat_number (avoid clash with separate theorem);
wenzelm
parents: 12933
diff changeset
   644
lemmas nat_number =
12838
wenzelm
parents: 12440
diff changeset
   645
  nat_number_of_Pls nat_number_of_Min
15620
8ccdc8bc66a2 replaced bool by a new datatype "bit" for binary numerals
paulson
parents: 15531
diff changeset
   646
  nat_number_of_BIT_1 nat_number_of_BIT_0
12838
wenzelm
parents: 12440
diff changeset
   647
wenzelm
parents: 12440
diff changeset
   648
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
wenzelm
parents: 12440
diff changeset
   649
  by (simp add: Let_def)
10574
8f98f0301d67 Linear arithmetic now copes with mixed nat/int formulae.
nipkow
parents: 9509
diff changeset
   650
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14738
diff changeset
   651
lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})"
14443
75910c7557c5 generic theorems about exponentials; general tidying up
paulson
parents: 14430
diff changeset
   652
by (simp add: power_mult); 
75910c7557c5 generic theorems about exponentials; general tidying up
paulson
parents: 14430
diff changeset
   653
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14738
diff changeset
   654
lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})"
14443
75910c7557c5 generic theorems about exponentials; general tidying up
paulson
parents: 14430
diff changeset
   655
by (simp add: power_mult power_Suc); 
75910c7557c5 generic theorems about exponentials; general tidying up
paulson
parents: 14430
diff changeset
   656
12440
fb5851b71a82 Added code generator setup.
berghofe
parents: 11468
diff changeset
   657
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   658
subsection{*Literal arithmetic and @{term of_nat}*}
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   659
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   660
lemma of_nat_double:
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   661
     "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   662
by (simp only: mult_2 nat_add_distrib of_nat_add) 
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   663
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   664
lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   665
by (simp only:  nat_number_of_def, simp)
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   666
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   667
lemma of_nat_number_of_lemma:
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   668
     "of_nat (number_of v :: nat) =  
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   669
         (if 0 \<le> (number_of v :: int) 
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   670
          then (number_of v :: 'a :: number_ring)
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   671
          else 0)"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   672
by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat);
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   673
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   674
lemma of_nat_number_of_eq [simp]:
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   675
     "of_nat (number_of v :: nat) =  
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   676
         (if neg (number_of v :: int) then 0  
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   677
          else (number_of v :: 'a :: number_ring))"
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   678
by (simp only: of_nat_number_of_lemma neg_def, simp) 
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   679
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   680
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   681
subsection {*Lemmas for the Combination and Cancellation Simprocs*}
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   682
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   683
lemma nat_number_of_add_left:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   684
     "number_of v + (number_of v' + (k::nat)) =  
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   685
         (if neg (number_of v :: int) then number_of v' + k  
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14365
diff changeset
   686
          else if neg (number_of v' :: int) then number_of v + k  
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   687
          else number_of (bin_add v v') + k)"
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   688
by simp
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   689
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14417
diff changeset
   690
lemma nat_number_of_mult_left:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14417
diff changeset
   691
     "number_of v * (number_of v' * (k::nat)) =  
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14417
diff changeset
   692
         (if neg (number_of v :: int) then 0
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14417
diff changeset
   693
          else number_of (bin_mult v v') * k)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14417
diff changeset
   694
by simp
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14417
diff changeset
   695
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   696
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   697
subsubsection{*For @{text combine_numerals}*}
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   698
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   699
lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   700
by (simp add: add_mult_distrib)
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   701
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   702
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   703
subsubsection{*For @{text cancel_numerals}*}
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   704
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   705
lemma nat_diff_add_eq1:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   706
     "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   707
by (simp split add: nat_diff_split add: add_mult_distrib)
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   708
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   709
lemma nat_diff_add_eq2:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   710
     "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   711
by (simp split add: nat_diff_split add: add_mult_distrib)
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   712
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   713
lemma nat_eq_add_iff1:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   714
     "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   715
by (auto split add: nat_diff_split simp add: add_mult_distrib)
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   716
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   717
lemma nat_eq_add_iff2:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   718
     "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   719
by (auto split add: nat_diff_split simp add: add_mult_distrib)
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   720
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   721
lemma nat_less_add_iff1:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   722
     "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   723
by (auto split add: nat_diff_split simp add: add_mult_distrib)
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   724
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   725
lemma nat_less_add_iff2:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   726
     "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   727
by (auto split add: nat_diff_split simp add: add_mult_distrib)
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   728
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   729
lemma nat_le_add_iff1:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   730
     "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   731
by (auto split add: nat_diff_split simp add: add_mult_distrib)
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   732
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   733
lemma nat_le_add_iff2:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   734
     "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   735
by (auto split add: nat_diff_split simp add: add_mult_distrib)
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   736
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   737
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   738
subsubsection{*For @{text cancel_numeral_factors} *}
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   739
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   740
lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   741
by auto
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   742
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   743
lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   744
by auto
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   745
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   746
lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   747
by auto
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   748
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   749
lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   750
by auto
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   751
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   752
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   753
subsubsection{*For @{text cancel_factor} *}
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   754
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   755
lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   756
by auto
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   757
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   758
lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   759
by auto
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   760
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   761
lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   762
by auto
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   763
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   764
lemma nat_mult_div_cancel_disj:
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   765
     "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   766
by (simp add: nat_mult_div_cancel1)
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   767
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   768
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   769
ML
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   770
{*
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   771
val eq_nat_nat_iff = thm"eq_nat_nat_iff";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   772
val eq_nat_number_of = thm"eq_nat_number_of";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   773
val less_nat_number_of = thm"less_nat_number_of";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   774
val power2_eq_square = thm "power2_eq_square";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   775
val zero_le_power2 = thm "zero_le_power2";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   776
val zero_less_power2 = thm "zero_less_power2";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   777
val zero_eq_power2 = thm "zero_eq_power2";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   778
val abs_power2 = thm "abs_power2";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   779
val power2_abs = thm "power2_abs";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   780
val power2_minus = thm "power2_minus";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   781
val power_minus1_even = thm "power_minus1_even";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   782
val power_minus_even = thm "power_minus_even";
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16642
diff changeset
   783
(* val zero_le_even_power = thm "zero_le_even_power"; *)
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   784
val odd_power_less_zero = thm "odd_power_less_zero";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   785
val odd_0_le_power_imp_0_le = thm "odd_0_le_power_imp_0_le";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   786
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   787
val Suc_pred' = thm"Suc_pred'";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   788
val expand_Suc = thm"expand_Suc";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   789
val Suc_eq_add_numeral_1 = thm"Suc_eq_add_numeral_1";
14467
bbfa6b01a55f new lemma
paulson
parents: 14443
diff changeset
   790
val Suc_eq_add_numeral_1_left = thm"Suc_eq_add_numeral_1_left";
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   791
val add_eq_if = thm"add_eq_if";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   792
val mult_eq_if = thm"mult_eq_if";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   793
val power_eq_if = thm"power_eq_if";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   794
val eq_number_of_0 = thm"eq_number_of_0";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   795
val eq_0_number_of = thm"eq_0_number_of";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   796
val less_0_number_of = thm"less_0_number_of";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   797
val neg_imp_number_of_eq_0 = thm"neg_imp_number_of_eq_0";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   798
val eq_number_of_Suc = thm"eq_number_of_Suc";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   799
val Suc_eq_number_of = thm"Suc_eq_number_of";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   800
val less_number_of_Suc = thm"less_number_of_Suc";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   801
val less_Suc_number_of = thm"less_Suc_number_of";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   802
val le_number_of_Suc = thm"le_number_of_Suc";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   803
val le_Suc_number_of = thm"le_Suc_number_of";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   804
val eq_number_of_BIT_BIT = thm"eq_number_of_BIT_BIT";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   805
val eq_number_of_BIT_Pls = thm"eq_number_of_BIT_Pls";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   806
val eq_number_of_BIT_Min = thm"eq_number_of_BIT_Min";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   807
val eq_number_of_Pls_Min = thm"eq_number_of_Pls_Min";
14390
55fe71faadda further tweaks to the numeric theories
paulson
parents: 14387
diff changeset
   808
val of_nat_number_of_eq = thm"of_nat_number_of_eq";
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   809
val nat_power_eq = thm"nat_power_eq";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   810
val power_nat_number_of = thm"power_nat_number_of";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   811
val zpower_number_of_even = thm"zpower_number_of_even";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   812
val zpower_number_of_odd = thm"zpower_number_of_odd";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   813
val nat_number_of_Pls = thm"nat_number_of_Pls";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   814
val nat_number_of_Min = thm"nat_number_of_Min";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   815
val Let_Suc = thm"Let_Suc";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   816
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   817
val nat_number = thms"nat_number";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   818
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   819
val nat_number_of_add_left = thm"nat_number_of_add_left";
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14417
diff changeset
   820
val nat_number_of_mult_left = thm"nat_number_of_mult_left";
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   821
val left_add_mult_distrib = thm"left_add_mult_distrib";
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   822
val nat_diff_add_eq1 = thm"nat_diff_add_eq1";
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   823
val nat_diff_add_eq2 = thm"nat_diff_add_eq2";
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   824
val nat_eq_add_iff1 = thm"nat_eq_add_iff1";
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   825
val nat_eq_add_iff2 = thm"nat_eq_add_iff2";
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   826
val nat_less_add_iff1 = thm"nat_less_add_iff1";
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   827
val nat_less_add_iff2 = thm"nat_less_add_iff2";
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   828
val nat_le_add_iff1 = thm"nat_le_add_iff1";
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   829
val nat_le_add_iff2 = thm"nat_le_add_iff2";
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   830
val nat_mult_le_cancel1 = thm"nat_mult_le_cancel1";
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   831
val nat_mult_less_cancel1 = thm"nat_mult_less_cancel1";
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   832
val nat_mult_eq_cancel1 = thm"nat_mult_eq_cancel1";
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   833
val nat_mult_div_cancel1 = thm"nat_mult_div_cancel1";
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   834
val nat_mult_le_cancel_disj = thm"nat_mult_le_cancel_disj";
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   835
val nat_mult_less_cancel_disj = thm"nat_mult_less_cancel_disj";
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   836
val nat_mult_eq_cancel_disj = thm"nat_mult_eq_cancel_disj";
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   837
val nat_mult_div_cancel_disj = thm"nat_mult_div_cancel_disj";
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   838
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   839
val power_minus_even = thm"power_minus_even";
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16642
diff changeset
   840
(* val zero_le_even_power = thm"zero_le_even_power"; *)
14273
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   841
*}
e33ffff0123c further simplifications of the integer development; converting more .ML files
paulson
parents: 14272
diff changeset
   842
7032
d6efb3b8e669 NatBin: binary arithmetic for the naturals
paulson
parents:
diff changeset
   843
end