src/HOL/Nat_Numeral.thy
author haftmann
Mon Apr 27 10:11:44 2009 +0200 (2009-04-27)
changeset 31001 7e6ffd8f51a9
parent 30960 fec1a04b7220
child 31002 bc4117fe72ab
permissions -rw-r--r--
cleaned up theory power further
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(*  Title:      HOL/Nat_Numeral.thy
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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 numerals for the natural numbers *}
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theory Nat_Numeral
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imports IntDiv
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uses ("Tools/nat_simprocs.ML")
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begin
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text {*
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  Arithmetic for naturals is reduced to that for the non-negative integers.
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*}
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instantiation nat :: number
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begin
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definition
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  nat_number_of_def [code inline, code del]: "number_of v = nat (number_of v)"
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instance ..
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end
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lemma [code post]:
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  "nat (number_of v) = number_of v"
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  unfolding nat_number_of_def ..
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context recpower
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begin
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abbreviation (xsymbols)
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  power2 :: "'a \<Rightarrow> 'a"  ("(_\<twosuperior>)" [1000] 999) where
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  "x\<twosuperior> \<equiv> x ^ 2"
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notation (latex output)
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  power2  ("(_\<twosuperior>)" [1000] 999)
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notation (HTML output)
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  power2  ("(_\<twosuperior>)" [1000] 999)
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end
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subsection {* Predicate for negative binary numbers *}
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definition neg  :: "int \<Rightarrow> bool" where
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  "neg Z \<longleftrightarrow> Z < 0"
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lemma not_neg_int [simp]: "~ neg (of_nat n)"
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by (simp add: neg_def)
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lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))"
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by (simp add: neg_def neg_less_0_iff_less del: of_nat_Suc)
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lemmas neg_eq_less_0 = neg_def
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lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
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by (simp add: neg_def linorder_not_less)
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text{*To simplify inequalities when Numeral1 can get simplified to 1*}
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lemma not_neg_0: "~ neg 0"
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by (simp add: One_int_def neg_def)
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lemma not_neg_1: "~ neg 1"
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by (simp add: neg_def linorder_not_less zero_le_one)
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lemma neg_nat: "neg z ==> nat z = 0"
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by (simp add: neg_def order_less_imp_le) 
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lemma not_neg_nat: "~ neg z ==> of_nat (nat z) = z"
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by (simp add: linorder_not_less neg_def)
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text {*
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  If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
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  @{term Numeral0} IS @{term "number_of Pls"}
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*}
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lemma not_neg_number_of_Pls: "~ neg (number_of Int.Pls)"
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  by (simp add: neg_def)
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lemma neg_number_of_Min: "neg (number_of Int.Min)"
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  by (simp add: neg_def)
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lemma neg_number_of_Bit0:
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  "neg (number_of (Int.Bit0 w)) = neg (number_of w)"
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  by (simp add: neg_def)
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lemma neg_number_of_Bit1:
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  "neg (number_of (Int.Bit1 w)) = neg (number_of w)"
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  by (simp add: neg_def)
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lemmas neg_simps [simp] =
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  not_neg_0 not_neg_1
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  not_neg_number_of_Pls neg_number_of_Min
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  neg_number_of_Bit0 neg_number_of_Bit1
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subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*}
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declare nat_0 [simp] nat_1 [simp]
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lemma nat_number_of [simp]: "nat (number_of w) = number_of w"
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by (simp add: nat_number_of_def)
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lemma nat_numeral_0_eq_0 [simp]: "Numeral0 = (0::nat)"
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by (simp add: nat_number_of_def)
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lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
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by (simp add: nat_1 nat_number_of_def)
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lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
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by (simp add: nat_numeral_1_eq_1)
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lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
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apply (unfold nat_number_of_def)
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apply (rule nat_2)
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done
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subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}
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lemma int_nat_number_of [simp]:
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     "int (number_of v) =  
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         (if neg (number_of v :: int) then 0  
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          else (number_of v :: int))"
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  unfolding nat_number_of_def number_of_is_id neg_def
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  by simp
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subsubsection{*Successor *}
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lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
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apply (rule sym)
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apply (simp add: nat_eq_iff int_Suc)
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done
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lemma Suc_nat_number_of_add:
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     "Suc (number_of v + n) =  
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        (if neg (number_of v :: int) then 1+n else number_of (Int.succ v) + n)"
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  unfolding nat_number_of_def number_of_is_id neg_def numeral_simps
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  by (simp add: Suc_nat_eq_nat_zadd1 add_ac)
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lemma Suc_nat_number_of [simp]:
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     "Suc (number_of v) =  
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        (if neg (number_of v :: int) then 1 else number_of (Int.succ v))"
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apply (cut_tac n = 0 in Suc_nat_number_of_add)
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apply (simp cong del: if_weak_cong)
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done
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subsubsection{*Addition *}
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lemma add_nat_number_of [simp]:
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     "(number_of v :: nat) + number_of v' =  
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         (if v < Int.Pls then number_of v'  
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          else if v' < Int.Pls then number_of v  
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          else number_of (v + v'))"
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  unfolding nat_number_of_def number_of_is_id numeral_simps
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  by (simp add: nat_add_distrib)
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lemma nat_number_of_add_1 [simp]:
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  "number_of v + (1::nat) =
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    (if v < Int.Pls then 1 else number_of (Int.succ v))"
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  unfolding nat_number_of_def number_of_is_id numeral_simps
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  by (simp add: nat_add_distrib)
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lemma nat_1_add_number_of [simp]:
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  "(1::nat) + number_of v =
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    (if v < Int.Pls then 1 else number_of (Int.succ v))"
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  unfolding nat_number_of_def number_of_is_id numeral_simps
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  by (simp add: nat_add_distrib)
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lemma nat_1_add_1 [simp]: "1 + 1 = (2::nat)"
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  by (rule int_int_eq [THEN iffD1]) simp
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subsubsection{*Subtraction *}
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lemma diff_nat_eq_if:
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     "nat z - nat z' =  
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        (if neg z' then nat z   
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         else let d = z-z' in     
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              if neg d then 0 else nat d)"
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by (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
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lemma diff_nat_number_of [simp]: 
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     "(number_of v :: nat) - number_of v' =  
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        (if v' < Int.Pls then number_of v  
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         else let d = number_of (v + uminus v') in     
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              if neg d then 0 else nat d)"
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  unfolding nat_number_of_def number_of_is_id numeral_simps neg_def
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  by auto
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lemma nat_number_of_diff_1 [simp]:
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  "number_of v - (1::nat) =
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    (if v \<le> Int.Pls then 0 else number_of (Int.pred v))"
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  unfolding nat_number_of_def number_of_is_id numeral_simps
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  by auto
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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 v < Int.Pls then 0 else number_of (v * v'))"
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  unfolding nat_number_of_def number_of_is_id numeral_simps
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  by (simp add: nat_mult_distrib)
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subsubsection{*Quotient *}
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lemma div_nat_number_of [simp]:
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     "(number_of v :: nat)  div  number_of v' =  
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          (if neg (number_of v :: int) then 0  
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           else nat (number_of v div number_of v'))"
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  unfolding nat_number_of_def number_of_is_id neg_def
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  by (simp add: nat_div_distrib)
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lemma one_div_nat_number_of [simp]:
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     "Suc 0 div number_of v' = nat (1 div number_of v')" 
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by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
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subsubsection{*Remainder *}
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lemma mod_nat_number_of [simp]:
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     "(number_of v :: nat)  mod  number_of v' =  
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        (if neg (number_of v :: int) then 0  
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         else if neg (number_of v' :: int) then number_of v  
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         else nat (number_of v mod number_of v'))"
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  unfolding nat_number_of_def number_of_is_id neg_def
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  by (simp add: nat_mod_distrib)
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lemma one_mod_nat_number_of [simp]:
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     "Suc 0 mod number_of v' =  
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        (if neg (number_of v' :: int) then Suc 0
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         else nat (1 mod number_of v'))"
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by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
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subsubsection{* Divisibility *}
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lemmas dvd_eq_mod_eq_0_number_of =
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  dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
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declare dvd_eq_mod_eq_0_number_of [simp]
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ML
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{*
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val nat_number_of_def = thm"nat_number_of_def";
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val nat_number_of = thm"nat_number_of";
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val nat_numeral_0_eq_0 = thm"nat_numeral_0_eq_0";
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val nat_numeral_1_eq_1 = thm"nat_numeral_1_eq_1";
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val numeral_1_eq_Suc_0 = thm"numeral_1_eq_Suc_0";
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val numeral_2_eq_2 = thm"numeral_2_eq_2";
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val nat_div_distrib = thm"nat_div_distrib";
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val nat_mod_distrib = thm"nat_mod_distrib";
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val int_nat_number_of = thm"int_nat_number_of";
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val Suc_nat_eq_nat_zadd1 = thm"Suc_nat_eq_nat_zadd1";
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val Suc_nat_number_of_add = thm"Suc_nat_number_of_add";
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val Suc_nat_number_of = thm"Suc_nat_number_of";
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val add_nat_number_of = thm"add_nat_number_of";
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val diff_nat_eq_if = thm"diff_nat_eq_if";
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val diff_nat_number_of = thm"diff_nat_number_of";
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val mult_nat_number_of = thm"mult_nat_number_of";
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val div_nat_number_of = thm"div_nat_number_of";
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val mod_nat_number_of = thm"mod_nat_number_of";
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*}
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subsection{*Comparisons*}
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subsubsection{*Equals (=) *}
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lemma eq_nat_nat_iff:
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     "[| (0::int) <= z;  0 <= z' |] ==> (nat z = nat z') = (z=z')"
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by (auto elim!: nonneg_eq_int)
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lemma eq_nat_number_of [simp]:
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     "((number_of v :: nat) = number_of v') =  
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      (if neg (number_of v :: int) then (number_of v' :: int) \<le> 0
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       else if neg (number_of v' :: int) then (number_of v :: int) = 0
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       else v = v')"
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  unfolding nat_number_of_def number_of_is_id neg_def
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  by auto
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subsubsection{*Less-than (<) *}
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lemma less_nat_number_of [simp]:
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  "(number_of v :: nat) < number_of v' \<longleftrightarrow>
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    (if v < v' then Int.Pls < v' else False)"
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  unfolding nat_number_of_def number_of_is_id numeral_simps
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  by auto
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subsubsection{*Less-than-or-equal *}
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lemma le_nat_number_of [simp]:
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  "(number_of v :: nat) \<le> number_of v' \<longleftrightarrow>
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    (if v \<le> v' then True else v \<le> Int.Pls)"
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  unfolding nat_number_of_def number_of_is_id numeral_simps
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  by auto
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(*Maps #n to n for n = 0, 1, 2*)
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lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2
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subsection{*Powers with Numeric Exponents*}
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text{*We cannot refer to the number @{term 2} in @{text Ring_and_Field.thy}.
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We cannot prove general results about the numeral @{term "-1"}, so we have to
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use @{term "- 1"} instead.*}
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lemma power2_eq_square: "(a::'a::recpower)\<twosuperior> = a * a"
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  by (simp add: numeral_2_eq_2 Power.power_Suc)
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lemma zero_power2 [simp]: "(0::'a::{semiring_1,recpower})\<twosuperior> = 0"
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  by (simp add: power2_eq_square)
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lemma one_power2 [simp]: "(1::'a::{semiring_1,recpower})\<twosuperior> = 1"
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  by (simp add: power2_eq_square)
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lemma power3_eq_cube: "(x::'a::recpower) ^ 3 = x * x * x"
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  apply (subgoal_tac "3 = Suc (Suc (Suc 0))")
wenzelm@23164
   332
  apply (erule ssubst)
wenzelm@23164
   333
  apply (simp add: power_Suc mult_ac)
wenzelm@23164
   334
  apply (unfold nat_number_of_def)
wenzelm@23164
   335
  apply (subst nat_eq_iff)
wenzelm@23164
   336
  apply simp
wenzelm@23164
   337
done
wenzelm@23164
   338
wenzelm@23164
   339
text{*Squares of literal numerals will be evaluated.*}
wenzelm@23164
   340
lemmas power2_eq_square_number_of =
wenzelm@23164
   341
    power2_eq_square [of "number_of w", standard]
wenzelm@23164
   342
declare power2_eq_square_number_of [simp]
wenzelm@23164
   343
wenzelm@23164
   344
wenzelm@23164
   345
lemma zero_le_power2[simp]: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,recpower})"
wenzelm@23164
   346
  by (simp add: power2_eq_square)
wenzelm@23164
   347
wenzelm@23164
   348
lemma zero_less_power2[simp]:
wenzelm@23164
   349
     "(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,recpower}))"
wenzelm@23164
   350
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
wenzelm@23164
   351
wenzelm@23164
   352
lemma power2_less_0[simp]:
wenzelm@23164
   353
  fixes a :: "'a::{ordered_idom,recpower}"
wenzelm@23164
   354
  shows "~ (a\<twosuperior> < 0)"
wenzelm@23164
   355
by (force simp add: power2_eq_square mult_less_0_iff) 
wenzelm@23164
   356
wenzelm@23164
   357
lemma zero_eq_power2[simp]:
wenzelm@23164
   358
     "(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))"
wenzelm@23164
   359
  by (force simp add: power2_eq_square mult_eq_0_iff)
wenzelm@23164
   360
wenzelm@23164
   361
lemma abs_power2[simp]:
wenzelm@23164
   362
     "abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})"
wenzelm@23164
   363
  by (simp add: power2_eq_square abs_mult abs_mult_self)
wenzelm@23164
   364
wenzelm@23164
   365
lemma power2_abs[simp]:
wenzelm@23164
   366
     "(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})"
wenzelm@23164
   367
  by (simp add: power2_eq_square abs_mult_self)
wenzelm@23164
   368
wenzelm@23164
   369
lemma power2_minus[simp]:
wenzelm@23164
   370
     "(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})"
wenzelm@23164
   371
  by (simp add: power2_eq_square)
wenzelm@23164
   372
wenzelm@23164
   373
lemma power2_le_imp_le:
wenzelm@23164
   374
  fixes x y :: "'a::{ordered_semidom,recpower}"
wenzelm@23164
   375
  shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y"
wenzelm@23164
   376
unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
wenzelm@23164
   377
wenzelm@23164
   378
lemma power2_less_imp_less:
wenzelm@23164
   379
  fixes x y :: "'a::{ordered_semidom,recpower}"
wenzelm@23164
   380
  shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y"
wenzelm@23164
   381
by (rule power_less_imp_less_base)
wenzelm@23164
   382
wenzelm@23164
   383
lemma power2_eq_imp_eq:
wenzelm@23164
   384
  fixes x y :: "'a::{ordered_semidom,recpower}"
wenzelm@23164
   385
  shows "\<lbrakk>x\<twosuperior> = y\<twosuperior>; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> x = y"
wenzelm@23164
   386
unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base, simp)
wenzelm@23164
   387
wenzelm@23164
   388
lemma power_minus1_even[simp]: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})"
paulson@29958
   389
proof (induct n)
paulson@29958
   390
  case 0 show ?case by simp
paulson@29958
   391
next
paulson@29958
   392
  case (Suc n) then show ?case by (simp add: power_Suc power_add)
paulson@29958
   393
qed
paulson@29958
   394
paulson@29958
   395
lemma power_minus1_odd: "(- 1) ^ Suc(2*n) = -(1::'a::{comm_ring_1,recpower})"
paulson@29958
   396
  by (simp add: power_Suc) 
wenzelm@23164
   397
wenzelm@23164
   398
lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2"
wenzelm@23164
   399
by (subst mult_commute) (simp add: power_mult)
wenzelm@23164
   400
wenzelm@23164
   401
lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"
wenzelm@23164
   402
by (simp add: power_even_eq) 
wenzelm@23164
   403
wenzelm@23164
   404
lemma power_minus_even [simp]:
haftmann@30960
   405
  "(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)"
haftmann@30960
   406
  by (simp add: power_minus [of a]) 
wenzelm@23164
   407
wenzelm@23164
   408
lemma zero_le_even_power'[simp]:
wenzelm@23164
   409
     "0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)"
wenzelm@23164
   410
proof (induct "n")
wenzelm@23164
   411
  case 0
wenzelm@23164
   412
    show ?case by (simp add: zero_le_one)
wenzelm@23164
   413
next
wenzelm@23164
   414
  case (Suc n)
wenzelm@23164
   415
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
wenzelm@23164
   416
      by (simp add: mult_ac power_add power2_eq_square)
wenzelm@23164
   417
    thus ?case
wenzelm@23164
   418
      by (simp add: prems zero_le_mult_iff)
wenzelm@23164
   419
qed
wenzelm@23164
   420
wenzelm@23164
   421
lemma odd_power_less_zero:
wenzelm@23164
   422
     "(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0"
wenzelm@23164
   423
proof (induct "n")
wenzelm@23164
   424
  case 0
huffman@30273
   425
  then show ?case by simp
wenzelm@23164
   426
next
wenzelm@23164
   427
  case (Suc n)
huffman@30273
   428
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
huffman@30273
   429
    by (simp add: mult_ac power_add power2_eq_square)
wenzelm@23389
   430
  thus ?case
huffman@30273
   431
    by (simp del: power_Suc add: prems mult_less_0_iff mult_neg_neg)
wenzelm@23164
   432
qed
wenzelm@23164
   433
wenzelm@23164
   434
lemma odd_0_le_power_imp_0_le:
wenzelm@23164
   435
     "0 \<le> a  ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})"
wenzelm@23164
   436
apply (insert odd_power_less_zero [of a n]) 
wenzelm@23164
   437
apply (force simp add: linorder_not_less [symmetric]) 
wenzelm@23164
   438
done
wenzelm@23164
   439
wenzelm@23164
   440
text{*Simprules for comparisons where common factors can be cancelled.*}
wenzelm@23164
   441
lemmas zero_compare_simps =
wenzelm@23164
   442
    add_strict_increasing add_strict_increasing2 add_increasing
wenzelm@23164
   443
    zero_le_mult_iff zero_le_divide_iff 
wenzelm@23164
   444
    zero_less_mult_iff zero_less_divide_iff 
wenzelm@23164
   445
    mult_le_0_iff divide_le_0_iff 
wenzelm@23164
   446
    mult_less_0_iff divide_less_0_iff 
wenzelm@23164
   447
    zero_le_power2 power2_less_0
wenzelm@23164
   448
wenzelm@23164
   449
subsubsection{*Nat *}
wenzelm@23164
   450
wenzelm@23164
   451
lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
wenzelm@23164
   452
by (simp add: numerals)
wenzelm@23164
   453
wenzelm@23164
   454
(*Expresses a natural number constant as the Suc of another one.
wenzelm@23164
   455
  NOT suitable for rewriting because n recurs in the condition.*)
wenzelm@23164
   456
lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
wenzelm@23164
   457
wenzelm@23164
   458
subsubsection{*Arith *}
wenzelm@23164
   459
wenzelm@23164
   460
lemma Suc_eq_add_numeral_1: "Suc n = n + 1"
wenzelm@23164
   461
by (simp add: numerals)
wenzelm@23164
   462
wenzelm@23164
   463
lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n"
wenzelm@23164
   464
by (simp add: numerals)
wenzelm@23164
   465
wenzelm@23164
   466
(* These two can be useful when m = number_of... *)
wenzelm@23164
   467
wenzelm@23164
   468
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
huffman@30079
   469
  unfolding One_nat_def by (cases m) simp_all
wenzelm@23164
   470
wenzelm@23164
   471
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
huffman@30079
   472
  unfolding One_nat_def by (cases m) simp_all
wenzelm@23164
   473
wenzelm@23164
   474
lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
huffman@30079
   475
  unfolding One_nat_def by (cases m) simp_all
wenzelm@23164
   476
wenzelm@23164
   477
wenzelm@23164
   478
subsection{*Comparisons involving (0::nat) *}
wenzelm@23164
   479
wenzelm@23164
   480
text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
wenzelm@23164
   481
wenzelm@23164
   482
lemma eq_number_of_0 [simp]:
huffman@29012
   483
  "number_of v = (0::nat) \<longleftrightarrow> v \<le> Int.Pls"
huffman@29012
   484
  unfolding nat_number_of_def number_of_is_id numeral_simps
huffman@29012
   485
  by auto
wenzelm@23164
   486
wenzelm@23164
   487
lemma eq_0_number_of [simp]:
huffman@29012
   488
  "(0::nat) = number_of v \<longleftrightarrow> v \<le> Int.Pls"
wenzelm@23164
   489
by (rule trans [OF eq_sym_conv eq_number_of_0])
wenzelm@23164
   490
wenzelm@23164
   491
lemma less_0_number_of [simp]:
huffman@29012
   492
   "(0::nat) < number_of v \<longleftrightarrow> Int.Pls < v"
huffman@29012
   493
  unfolding nat_number_of_def number_of_is_id numeral_simps
huffman@29012
   494
  by simp
wenzelm@23164
   495
wenzelm@23164
   496
lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
huffman@28969
   497
by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric])
wenzelm@23164
   498
wenzelm@23164
   499
wenzelm@23164
   500
wenzelm@23164
   501
subsection{*Comparisons involving  @{term Suc} *}
wenzelm@23164
   502
wenzelm@23164
   503
lemma eq_number_of_Suc [simp]:
wenzelm@23164
   504
     "(number_of v = Suc n) =  
haftmann@25919
   505
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   506
         if neg pv then False else nat pv = n)"
wenzelm@23164
   507
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
wenzelm@23164
   508
                  number_of_pred nat_number_of_def 
wenzelm@23164
   509
            split add: split_if)
wenzelm@23164
   510
apply (rule_tac x = "number_of v" in spec)
wenzelm@23164
   511
apply (auto simp add: nat_eq_iff)
wenzelm@23164
   512
done
wenzelm@23164
   513
wenzelm@23164
   514
lemma Suc_eq_number_of [simp]:
wenzelm@23164
   515
     "(Suc n = number_of v) =  
haftmann@25919
   516
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   517
         if neg pv then False else nat pv = n)"
wenzelm@23164
   518
by (rule trans [OF eq_sym_conv eq_number_of_Suc])
wenzelm@23164
   519
wenzelm@23164
   520
lemma less_number_of_Suc [simp]:
wenzelm@23164
   521
     "(number_of v < Suc n) =  
haftmann@25919
   522
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   523
         if neg pv then True else nat pv < n)"
wenzelm@23164
   524
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
wenzelm@23164
   525
                  number_of_pred nat_number_of_def  
wenzelm@23164
   526
            split add: split_if)
wenzelm@23164
   527
apply (rule_tac x = "number_of v" in spec)
wenzelm@23164
   528
apply (auto simp add: nat_less_iff)
wenzelm@23164
   529
done
wenzelm@23164
   530
wenzelm@23164
   531
lemma less_Suc_number_of [simp]:
wenzelm@23164
   532
     "(Suc n < number_of v) =  
haftmann@25919
   533
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   534
         if neg pv then False else n < nat pv)"
wenzelm@23164
   535
apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
wenzelm@23164
   536
                  number_of_pred nat_number_of_def
wenzelm@23164
   537
            split add: split_if)
wenzelm@23164
   538
apply (rule_tac x = "number_of v" in spec)
wenzelm@23164
   539
apply (auto simp add: zless_nat_eq_int_zless)
wenzelm@23164
   540
done
wenzelm@23164
   541
wenzelm@23164
   542
lemma le_number_of_Suc [simp]:
wenzelm@23164
   543
     "(number_of v <= Suc n) =  
haftmann@25919
   544
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   545
         if neg pv then True else nat pv <= n)"
wenzelm@23164
   546
by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric])
wenzelm@23164
   547
wenzelm@23164
   548
lemma le_Suc_number_of [simp]:
wenzelm@23164
   549
     "(Suc n <= number_of v) =  
haftmann@25919
   550
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   551
         if neg pv then False else n <= nat pv)"
wenzelm@23164
   552
by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric])
wenzelm@23164
   553
wenzelm@23164
   554
haftmann@25919
   555
lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Int.Min"
wenzelm@23164
   556
by auto
wenzelm@23164
   557
wenzelm@23164
   558
wenzelm@23164
   559
wenzelm@23164
   560
subsection{*Max and Min Combined with @{term Suc} *}
wenzelm@23164
   561
wenzelm@23164
   562
lemma max_number_of_Suc [simp]:
wenzelm@23164
   563
     "max (Suc n) (number_of v) =  
haftmann@25919
   564
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   565
         if neg pv then Suc n else Suc(max n (nat pv)))"
wenzelm@23164
   566
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
wenzelm@23164
   567
            split add: split_if nat.split)
wenzelm@23164
   568
apply (rule_tac x = "number_of v" in spec) 
wenzelm@23164
   569
apply auto
wenzelm@23164
   570
done
wenzelm@23164
   571
 
wenzelm@23164
   572
lemma max_Suc_number_of [simp]:
wenzelm@23164
   573
     "max (number_of v) (Suc n) =  
haftmann@25919
   574
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   575
         if neg pv then Suc n else Suc(max (nat pv) n))"
wenzelm@23164
   576
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
wenzelm@23164
   577
            split add: split_if nat.split)
wenzelm@23164
   578
apply (rule_tac x = "number_of v" in spec) 
wenzelm@23164
   579
apply auto
wenzelm@23164
   580
done
wenzelm@23164
   581
 
wenzelm@23164
   582
lemma min_number_of_Suc [simp]:
wenzelm@23164
   583
     "min (Suc n) (number_of v) =  
haftmann@25919
   584
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   585
         if neg pv then 0 else Suc(min n (nat pv)))"
wenzelm@23164
   586
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
wenzelm@23164
   587
            split add: split_if nat.split)
wenzelm@23164
   588
apply (rule_tac x = "number_of v" in spec) 
wenzelm@23164
   589
apply auto
wenzelm@23164
   590
done
wenzelm@23164
   591
 
wenzelm@23164
   592
lemma min_Suc_number_of [simp]:
wenzelm@23164
   593
     "min (number_of v) (Suc n) =  
haftmann@25919
   594
        (let pv = number_of (Int.pred v) in  
wenzelm@23164
   595
         if neg pv then 0 else Suc(min (nat pv) n))"
wenzelm@23164
   596
apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
wenzelm@23164
   597
            split add: split_if nat.split)
wenzelm@23164
   598
apply (rule_tac x = "number_of v" in spec) 
wenzelm@23164
   599
apply auto
wenzelm@23164
   600
done
wenzelm@23164
   601
 
wenzelm@23164
   602
subsection{*Literal arithmetic involving powers*}
wenzelm@23164
   603
wenzelm@23164
   604
lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n"
wenzelm@23164
   605
apply (induct "n")
wenzelm@23164
   606
apply (simp_all (no_asm_simp) add: nat_mult_distrib)
wenzelm@23164
   607
done
wenzelm@23164
   608
wenzelm@23164
   609
lemma power_nat_number_of:
wenzelm@23164
   610
     "(number_of v :: nat) ^ n =  
wenzelm@23164
   611
       (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
wenzelm@23164
   612
by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
wenzelm@23164
   613
         split add: split_if cong: imp_cong)
wenzelm@23164
   614
wenzelm@23164
   615
wenzelm@23164
   616
lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]
wenzelm@23164
   617
declare power_nat_number_of_number_of [simp]
wenzelm@23164
   618
wenzelm@23164
   619
wenzelm@23164
   620
huffman@23294
   621
text{*For arbitrary rings*}
wenzelm@23164
   622
huffman@23294
   623
lemma power_number_of_even:
huffman@23294
   624
  fixes z :: "'a::{number_ring,recpower}"
huffman@26086
   625
  shows "z ^ number_of (Int.Bit0 w) = (let w = z ^ (number_of w) in w * w)"
huffman@26086
   626
unfolding Let_def nat_number_of_def number_of_Bit0
wenzelm@23164
   627
apply (rule_tac x = "number_of w" in spec, clarify)
wenzelm@23164
   628
apply (case_tac " (0::int) <= x")
wenzelm@23164
   629
apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square)
wenzelm@23164
   630
done
wenzelm@23164
   631
huffman@23294
   632
lemma power_number_of_odd:
huffman@23294
   633
  fixes z :: "'a::{number_ring,recpower}"
huffman@26086
   634
  shows "z ^ number_of (Int.Bit1 w) = (if (0::int) <= number_of w
wenzelm@23164
   635
     then (let w = z ^ (number_of w) in z * w * w) else 1)"
huffman@26086
   636
unfolding Let_def nat_number_of_def number_of_Bit1
wenzelm@23164
   637
apply (rule_tac x = "number_of w" in spec, auto)
wenzelm@23164
   638
apply (simp only: nat_add_distrib nat_mult_distrib)
wenzelm@23164
   639
apply simp
huffman@23294
   640
apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat power_Suc)
wenzelm@23164
   641
done
wenzelm@23164
   642
huffman@23294
   643
lemmas zpower_number_of_even = power_number_of_even [where 'a=int]
huffman@23294
   644
lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int]
wenzelm@23164
   645
huffman@23294
   646
lemmas power_number_of_even_number_of [simp] =
huffman@23294
   647
    power_number_of_even [of "number_of v", standard]
wenzelm@23164
   648
huffman@23294
   649
lemmas power_number_of_odd_number_of [simp] =
huffman@23294
   650
    power_number_of_odd [of "number_of v", standard]
wenzelm@23164
   651
wenzelm@23164
   652
wenzelm@23164
   653
wenzelm@23164
   654
ML
wenzelm@23164
   655
{*
wenzelm@26342
   656
val numeral_ss = @{simpset} addsimps @{thms numerals};
wenzelm@23164
   657
wenzelm@23164
   658
val nat_bin_arith_setup =
haftmann@30685
   659
 Lin_Arith.map_data
wenzelm@23164
   660
   (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
wenzelm@23164
   661
     {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
wenzelm@23164
   662
      inj_thms = inj_thms,
wenzelm@23164
   663
      lessD = lessD, neqE = neqE,
huffman@29039
   664
      simpset = simpset addsimps @{thms neg_simps} @
huffman@29039
   665
        [@{thm Suc_nat_number_of}, @{thm int_nat_number_of}]})
wenzelm@23164
   666
*}
wenzelm@23164
   667
wenzelm@24075
   668
declaration {* K nat_bin_arith_setup *}
wenzelm@23164
   669
wenzelm@23164
   670
(* Enable arith to deal with div/mod k where k is a numeral: *)
wenzelm@23164
   671
declare split_div[of _ _ "number_of k", standard, arith_split]
wenzelm@23164
   672
declare split_mod[of _ _ "number_of k", standard, arith_split]
wenzelm@23164
   673
wenzelm@23164
   674
lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
wenzelm@23164
   675
  by (simp add: number_of_Pls nat_number_of_def)
wenzelm@23164
   676
haftmann@25919
   677
lemma nat_number_of_Min: "number_of Int.Min = (0::nat)"
wenzelm@23164
   678
  apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
wenzelm@23164
   679
  done
wenzelm@23164
   680
huffman@26086
   681
lemma nat_number_of_Bit0:
huffman@26086
   682
    "number_of (Int.Bit0 w) = (let n::nat = number_of w in n + n)"
huffman@28969
   683
  unfolding nat_number_of_def number_of_is_id numeral_simps Let_def
huffman@28969
   684
  by auto
huffman@26086
   685
huffman@26086
   686
lemma nat_number_of_Bit1:
huffman@26086
   687
  "number_of (Int.Bit1 w) =
wenzelm@23164
   688
    (if neg (number_of w :: int) then 0
wenzelm@23164
   689
     else let n = number_of w in Suc (n + n))"
huffman@28969
   690
  unfolding nat_number_of_def number_of_is_id numeral_simps neg_def Let_def
huffman@28968
   691
  by auto
wenzelm@23164
   692
wenzelm@23164
   693
lemmas nat_number =
wenzelm@23164
   694
  nat_number_of_Pls nat_number_of_Min
huffman@26086
   695
  nat_number_of_Bit0 nat_number_of_Bit1
wenzelm@23164
   696
wenzelm@23164
   697
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
wenzelm@23164
   698
  by (simp add: Let_def)
wenzelm@23164
   699
wenzelm@23164
   700
lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})"
huffman@23294
   701
by (simp add: power_mult power_Suc); 
wenzelm@23164
   702
wenzelm@23164
   703
lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})"
wenzelm@23164
   704
by (simp add: power_mult power_Suc); 
wenzelm@23164
   705
wenzelm@23164
   706
wenzelm@23164
   707
subsection{*Literal arithmetic and @{term of_nat}*}
wenzelm@23164
   708
wenzelm@23164
   709
lemma of_nat_double:
wenzelm@23164
   710
     "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
wenzelm@23164
   711
by (simp only: mult_2 nat_add_distrib of_nat_add) 
wenzelm@23164
   712
wenzelm@23164
   713
lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
wenzelm@23164
   714
by (simp only: nat_number_of_def)
wenzelm@23164
   715
wenzelm@23164
   716
lemma of_nat_number_of_lemma:
wenzelm@23164
   717
     "of_nat (number_of v :: nat) =  
wenzelm@23164
   718
         (if 0 \<le> (number_of v :: int) 
wenzelm@23164
   719
          then (number_of v :: 'a :: number_ring)
wenzelm@23164
   720
          else 0)"
wenzelm@23164
   721
by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat);
wenzelm@23164
   722
wenzelm@23164
   723
lemma of_nat_number_of_eq [simp]:
wenzelm@23164
   724
     "of_nat (number_of v :: nat) =  
wenzelm@23164
   725
         (if neg (number_of v :: int) then 0  
wenzelm@23164
   726
          else (number_of v :: 'a :: number_ring))"
wenzelm@23164
   727
by (simp only: of_nat_number_of_lemma neg_def, simp) 
wenzelm@23164
   728
wenzelm@23164
   729
wenzelm@23164
   730
subsection {*Lemmas for the Combination and Cancellation Simprocs*}
wenzelm@23164
   731
wenzelm@23164
   732
lemma nat_number_of_add_left:
wenzelm@23164
   733
     "number_of v + (number_of v' + (k::nat)) =  
wenzelm@23164
   734
         (if neg (number_of v :: int) then number_of v' + k  
wenzelm@23164
   735
          else if neg (number_of v' :: int) then number_of v + k  
wenzelm@23164
   736
          else number_of (v + v') + k)"
huffman@28968
   737
  unfolding nat_number_of_def number_of_is_id neg_def
huffman@28968
   738
  by auto
wenzelm@23164
   739
wenzelm@23164
   740
lemma nat_number_of_mult_left:
wenzelm@23164
   741
     "number_of v * (number_of v' * (k::nat)) =  
huffman@29012
   742
         (if v < Int.Pls then 0
wenzelm@23164
   743
          else number_of (v * v') * k)"
wenzelm@23164
   744
by simp
wenzelm@23164
   745
wenzelm@23164
   746
wenzelm@23164
   747
subsubsection{*For @{text combine_numerals}*}
wenzelm@23164
   748
wenzelm@23164
   749
lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
wenzelm@23164
   750
by (simp add: add_mult_distrib)
wenzelm@23164
   751
wenzelm@23164
   752
wenzelm@23164
   753
subsubsection{*For @{text cancel_numerals}*}
wenzelm@23164
   754
wenzelm@23164
   755
lemma nat_diff_add_eq1:
wenzelm@23164
   756
     "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
wenzelm@23164
   757
by (simp split add: nat_diff_split add: add_mult_distrib)
wenzelm@23164
   758
wenzelm@23164
   759
lemma nat_diff_add_eq2:
wenzelm@23164
   760
     "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
wenzelm@23164
   761
by (simp split add: nat_diff_split add: add_mult_distrib)
wenzelm@23164
   762
wenzelm@23164
   763
lemma nat_eq_add_iff1:
wenzelm@23164
   764
     "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
wenzelm@23164
   765
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   766
wenzelm@23164
   767
lemma nat_eq_add_iff2:
wenzelm@23164
   768
     "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
wenzelm@23164
   769
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   770
wenzelm@23164
   771
lemma nat_less_add_iff1:
wenzelm@23164
   772
     "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
wenzelm@23164
   773
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   774
wenzelm@23164
   775
lemma nat_less_add_iff2:
wenzelm@23164
   776
     "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
wenzelm@23164
   777
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   778
wenzelm@23164
   779
lemma nat_le_add_iff1:
wenzelm@23164
   780
     "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
wenzelm@23164
   781
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   782
wenzelm@23164
   783
lemma nat_le_add_iff2:
wenzelm@23164
   784
     "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
wenzelm@23164
   785
by (auto split add: nat_diff_split simp add: add_mult_distrib)
wenzelm@23164
   786
wenzelm@23164
   787
wenzelm@23164
   788
subsubsection{*For @{text cancel_numeral_factors} *}
wenzelm@23164
   789
wenzelm@23164
   790
lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
wenzelm@23164
   791
by auto
wenzelm@23164
   792
wenzelm@23164
   793
lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
wenzelm@23164
   794
by auto
wenzelm@23164
   795
wenzelm@23164
   796
lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
wenzelm@23164
   797
by auto
wenzelm@23164
   798
wenzelm@23164
   799
lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
wenzelm@23164
   800
by auto
wenzelm@23164
   801
nipkow@23969
   802
lemma nat_mult_dvd_cancel_disj[simp]:
nipkow@23969
   803
  "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
nipkow@23969
   804
by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])
nipkow@23969
   805
nipkow@23969
   806
lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
nipkow@23969
   807
by(auto)
nipkow@23969
   808
wenzelm@23164
   809
wenzelm@23164
   810
subsubsection{*For @{text cancel_factor} *}
wenzelm@23164
   811
wenzelm@23164
   812
lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
wenzelm@23164
   813
by auto
wenzelm@23164
   814
wenzelm@23164
   815
lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
wenzelm@23164
   816
by auto
wenzelm@23164
   817
wenzelm@23164
   818
lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
wenzelm@23164
   819
by auto
wenzelm@23164
   820
nipkow@23969
   821
lemma nat_mult_div_cancel_disj[simp]:
wenzelm@23164
   822
     "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
wenzelm@23164
   823
by (simp add: nat_mult_div_cancel1)
wenzelm@23164
   824
haftmann@30652
   825
haftmann@30652
   826
subsection {* Simprocs for the Naturals *}
haftmann@30652
   827
haftmann@30652
   828
use "Tools/nat_simprocs.ML"
haftmann@30652
   829
declaration {* K nat_simprocs_setup *}
haftmann@30652
   830
haftmann@30652
   831
subsubsection{*For simplifying @{term "Suc m - K"} and  @{term "K - Suc m"}*}
haftmann@30652
   832
haftmann@30652
   833
text{*Where K above is a literal*}
haftmann@30652
   834
haftmann@30652
   835
lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
haftmann@30652
   836
by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split)
haftmann@30652
   837
haftmann@30652
   838
text {*Now just instantiating @{text n} to @{text "number_of v"} does
haftmann@30652
   839
  the right simplification, but with some redundant inequality
haftmann@30652
   840
  tests.*}
haftmann@30652
   841
lemma neg_number_of_pred_iff_0:
haftmann@30652
   842
  "neg (number_of (Int.pred v)::int) = (number_of v = (0::nat))"
haftmann@30652
   843
apply (subgoal_tac "neg (number_of (Int.pred v)) = (number_of v < Suc 0) ")
haftmann@30652
   844
apply (simp only: less_Suc_eq_le le_0_eq)
haftmann@30652
   845
apply (subst less_number_of_Suc, simp)
haftmann@30652
   846
done
haftmann@30652
   847
haftmann@30652
   848
text{*No longer required as a simprule because of the @{text inverse_fold}
haftmann@30652
   849
   simproc*}
haftmann@30652
   850
lemma Suc_diff_number_of:
haftmann@30652
   851
     "Int.Pls < v ==>
haftmann@30652
   852
      Suc m - (number_of v) = m - (number_of (Int.pred v))"
haftmann@30652
   853
apply (subst Suc_diff_eq_diff_pred)
haftmann@30652
   854
apply simp
haftmann@30652
   855
apply (simp del: nat_numeral_1_eq_1)
haftmann@30652
   856
apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
haftmann@30652
   857
                        neg_number_of_pred_iff_0)
haftmann@30652
   858
done
haftmann@30652
   859
haftmann@30652
   860
lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
haftmann@30652
   861
by (simp add: numerals split add: nat_diff_split)
haftmann@30652
   862
haftmann@30652
   863
haftmann@30652
   864
subsubsection{*For @{term nat_case} and @{term nat_rec}*}
haftmann@30652
   865
haftmann@30652
   866
lemma nat_case_number_of [simp]:
haftmann@30652
   867
     "nat_case a f (number_of v) =
haftmann@30652
   868
        (let pv = number_of (Int.pred v) in
haftmann@30652
   869
         if neg pv then a else f (nat pv))"
haftmann@30652
   870
by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)
haftmann@30652
   871
haftmann@30652
   872
lemma nat_case_add_eq_if [simp]:
haftmann@30652
   873
     "nat_case a f ((number_of v) + n) =
haftmann@30652
   874
       (let pv = number_of (Int.pred v) in
haftmann@30652
   875
         if neg pv then nat_case a f n else f (nat pv + n))"
haftmann@30652
   876
apply (subst add_eq_if)
haftmann@30652
   877
apply (simp split add: nat.split
haftmann@30652
   878
            del: nat_numeral_1_eq_1
haftmann@30652
   879
            add: nat_numeral_1_eq_1 [symmetric]
haftmann@30652
   880
                 numeral_1_eq_Suc_0 [symmetric]
haftmann@30652
   881
                 neg_number_of_pred_iff_0)
haftmann@30652
   882
done
haftmann@30652
   883
haftmann@30652
   884
lemma nat_rec_number_of [simp]:
haftmann@30652
   885
     "nat_rec a f (number_of v) =
haftmann@30652
   886
        (let pv = number_of (Int.pred v) in
haftmann@30652
   887
         if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
haftmann@30652
   888
apply (case_tac " (number_of v) ::nat")
haftmann@30652
   889
apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
haftmann@30652
   890
apply (simp split add: split_if_asm)
haftmann@30652
   891
done
haftmann@30652
   892
haftmann@30652
   893
lemma nat_rec_add_eq_if [simp]:
haftmann@30652
   894
     "nat_rec a f (number_of v + n) =
haftmann@30652
   895
        (let pv = number_of (Int.pred v) in
haftmann@30652
   896
         if neg pv then nat_rec a f n
haftmann@30652
   897
                   else f (nat pv + n) (nat_rec a f (nat pv + n)))"
haftmann@30652
   898
apply (subst add_eq_if)
haftmann@30652
   899
apply (simp split add: nat.split
haftmann@30652
   900
            del: nat_numeral_1_eq_1
haftmann@30652
   901
            add: nat_numeral_1_eq_1 [symmetric]
haftmann@30652
   902
                 numeral_1_eq_Suc_0 [symmetric]
haftmann@30652
   903
                 neg_number_of_pred_iff_0)
haftmann@30652
   904
done
haftmann@30652
   905
haftmann@30652
   906
haftmann@30652
   907
subsubsection{*Various Other Lemmas*}
haftmann@30652
   908
haftmann@30652
   909
text {*Evens and Odds, for Mutilated Chess Board*}
haftmann@30652
   910
haftmann@30652
   911
text{*Lemmas for specialist use, NOT as default simprules*}
haftmann@30652
   912
lemma nat_mult_2: "2 * z = (z+z::nat)"
haftmann@30652
   913
proof -
haftmann@30652
   914
  have "2*z = (1 + 1)*z" by simp
haftmann@30652
   915
  also have "... = z+z" by (simp add: left_distrib)
haftmann@30652
   916
  finally show ?thesis .
haftmann@30652
   917
qed
haftmann@30652
   918
haftmann@30652
   919
lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
haftmann@30652
   920
by (subst mult_commute, rule nat_mult_2)
haftmann@30652
   921
haftmann@30652
   922
text{*Case analysis on @{term "n<2"}*}
haftmann@30652
   923
lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
haftmann@30652
   924
by arith
haftmann@30652
   925
haftmann@30652
   926
lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)"
haftmann@30652
   927
by arith
haftmann@30652
   928
haftmann@30652
   929
lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
haftmann@30652
   930
by (simp add: nat_mult_2 [symmetric])
haftmann@30652
   931
haftmann@30652
   932
lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
haftmann@30652
   933
apply (subgoal_tac "m mod 2 < 2")
haftmann@30652
   934
apply (erule less_2_cases [THEN disjE])
haftmann@30652
   935
apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
haftmann@30652
   936
done
haftmann@30652
   937
haftmann@30652
   938
lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)"
haftmann@30652
   939
apply (subgoal_tac "m mod 2 < 2")
haftmann@30652
   940
apply (force simp del: mod_less_divisor, simp)
haftmann@30652
   941
done
haftmann@30652
   942
haftmann@30652
   943
text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
haftmann@30652
   944
haftmann@30652
   945
lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
haftmann@30652
   946
by simp
haftmann@30652
   947
haftmann@30652
   948
lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
haftmann@30652
   949
by simp
haftmann@30652
   950
haftmann@30652
   951
text{*Can be used to eliminate long strings of Sucs, but not by default*}
haftmann@30652
   952
lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
haftmann@30652
   953
by simp
haftmann@30652
   954
haftmann@30652
   955
haftmann@30652
   956
text{*These lemmas collapse some needless occurrences of Suc:
haftmann@30652
   957
    at least three Sucs, since two and fewer are rewritten back to Suc again!
haftmann@30652
   958
    We already have some rules to simplify operands smaller than 3.*}
haftmann@30652
   959
haftmann@30652
   960
lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
haftmann@30652
   961
by (simp add: Suc3_eq_add_3)
haftmann@30652
   962
haftmann@30652
   963
lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
haftmann@30652
   964
by (simp add: Suc3_eq_add_3)
haftmann@30652
   965
haftmann@30652
   966
lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
haftmann@30652
   967
by (simp add: Suc3_eq_add_3)
haftmann@30652
   968
haftmann@30652
   969
lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
haftmann@30652
   970
by (simp add: Suc3_eq_add_3)
haftmann@30652
   971
haftmann@30652
   972
lemmas Suc_div_eq_add3_div_number_of =
haftmann@30652
   973
    Suc_div_eq_add3_div [of _ "number_of v", standard]
haftmann@30652
   974
declare Suc_div_eq_add3_div_number_of [simp]
haftmann@30652
   975
haftmann@30652
   976
lemmas Suc_mod_eq_add3_mod_number_of =
haftmann@30652
   977
    Suc_mod_eq_add3_mod [of _ "number_of v", standard]
haftmann@30652
   978
declare Suc_mod_eq_add3_mod_number_of [simp]
haftmann@30652
   979
haftmann@30960
   980
end