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