--- a/src/HOL/Groebner_Basis.thy Wed Apr 15 15:30:38 2009 +0200
+++ b/src/HOL/Groebner_Basis.thy Wed Apr 15 15:30:39 2009 +0200
@@ -5,7 +5,7 @@
header {* Semiring normalization and Groebner Bases *}
theory Groebner_Basis
-imports NatBin
+imports Nat_Numeral
uses
"Tools/Groebner_Basis/misc.ML"
"Tools/Groebner_Basis/normalizer_data.ML"
--- a/src/HOL/Import/HOL/arithmetic.imp Wed Apr 15 15:30:38 2009 +0200
+++ b/src/HOL/Import/HOL/arithmetic.imp Wed Apr 15 15:30:39 2009 +0200
@@ -43,7 +43,7 @@
"TWO" > "HOL4Base.arithmetic.TWO"
"TIMES2" > "NatSimprocs.nat_mult_2"
"SUC_SUB1" > "HOL4Base.arithmetic.SUC_SUB1"
- "SUC_ONE_ADD" > "NatBin.Suc_eq_add_numeral_1_left"
+ "SUC_ONE_ADD" > "Nat_Numeral.Suc_eq_add_numeral_1_left"
"SUC_NOT" > "Nat.nat.simps_2"
"SUC_ELIM_THM" > "HOL4Base.arithmetic.SUC_ELIM_THM"
"SUC_ADD_SYM" > "HOL4Base.arithmetic.SUC_ADD_SYM"
@@ -233,7 +233,7 @@
"EVEN_AND_ODD" > "HOL4Base.arithmetic.EVEN_AND_ODD"
"EVEN_ADD" > "HOL4Base.arithmetic.EVEN_ADD"
"EVEN" > "HOL4Base.arithmetic.EVEN"
- "EQ_MULT_LCANCEL" > "NatBin.nat_mult_eq_cancel_disj"
+ "EQ_MULT_LCANCEL" > "Nat_Numeral.nat_mult_eq_cancel_disj"
"EQ_MONO_ADD_EQ" > "Nat.nat_add_right_cancel"
"EQ_LESS_EQ" > "Orderings.order_eq_iff"
"EQ_ADD_RCANCEL" > "Nat.nat_add_right_cancel"
--- a/src/HOL/Import/HOL/real.imp Wed Apr 15 15:30:38 2009 +0200
+++ b/src/HOL/Import/HOL/real.imp Wed Apr 15 15:30:39 2009 +0200
@@ -99,7 +99,7 @@
"REAL_POW_INV" > "Power.power_inverse"
"REAL_POW_DIV" > "Power.power_divide"
"REAL_POW_ADD" > "Power.power_add"
- "REAL_POW2_ABS" > "NatBin.power2_abs"
+ "REAL_POW2_ABS" > "Nat_Numeral.power2_abs"
"REAL_POS_NZ" > "HOL4Real.real.REAL_POS_NZ"
"REAL_POS" > "RealDef.real_of_nat_ge_zero"
"REAL_POASQ" > "HOL4Real.real.REAL_POASQ"
@@ -210,7 +210,7 @@
"REAL_LE_RDIV_EQ" > "Ring_and_Field.pos_le_divide_eq"
"REAL_LE_RDIV" > "Ring_and_Field.mult_imp_le_div_pos"
"REAL_LE_RADD" > "OrderedGroup.add_le_cancel_right"
- "REAL_LE_POW2" > "NatBin.zero_compare_simps_12"
+ "REAL_LE_POW2" > "Nat_Numeral.zero_compare_simps_12"
"REAL_LE_NEGTOTAL" > "HOL4Real.real.REAL_LE_NEGTOTAL"
"REAL_LE_NEGR" > "OrderedGroup.le_minus_self_iff"
"REAL_LE_NEGL" > "OrderedGroup.minus_le_self_iff"
@@ -313,7 +313,7 @@
"POW_ONE" > "Power.power_one"
"POW_NZ" > "Power.field_power_not_zero"
"POW_MUL" > "Power.power_mult_distrib"
- "POW_MINUS1" > "NatBin.power_minus1_even"
+ "POW_MINUS1" > "Nat_Numeral.power_minus1_even"
"POW_M1" > "HOL4Real.real.POW_M1"
"POW_LT" > "HOL4Real.real.POW_LT"
"POW_LE" > "Power.power_mono"
@@ -323,7 +323,7 @@
"POW_ABS" > "Power.power_abs"
"POW_2_LT" > "RealPow.two_realpow_gt"
"POW_2_LE1" > "RealPow.two_realpow_ge_one"
- "POW_2" > "NatBin.power2_eq_square"
+ "POW_2" > "Nat_Numeral.power2_eq_square"
"POW_1" > "Power.power_one_right"
"POW_0" > "Power.power_0_Suc"
"ABS_ZERO" > "OrderedGroup.abs_eq_0"
@@ -335,7 +335,7 @@
"ABS_SIGN2" > "HOL4Real.real.ABS_SIGN2"
"ABS_SIGN" > "HOL4Real.real.ABS_SIGN"
"ABS_REFL" > "HOL4Real.real.ABS_REFL"
- "ABS_POW2" > "NatBin.abs_power2"
+ "ABS_POW2" > "Nat_Numeral.abs_power2"
"ABS_POS" > "OrderedGroup.abs_ge_zero"
"ABS_NZ" > "OrderedGroup.zero_less_abs_iff"
"ABS_NEG" > "OrderedGroup.abs_minus_cancel"
--- a/src/HOL/Import/HOLLight/hollight.imp Wed Apr 15 15:30:38 2009 +0200
+++ b/src/HOL/Import/HOLLight/hollight.imp Wed Apr 15 15:30:39 2009 +0200
@@ -1515,7 +1515,7 @@
"EQ_REFL_T" > "HOL.simp_thms_6"
"EQ_REFL" > "Presburger.fm_modd_pinf"
"EQ_MULT_RCANCEL" > "Nat.mult_cancel2"
- "EQ_MULT_LCANCEL" > "NatBin.nat_mult_eq_cancel_disj"
+ "EQ_MULT_LCANCEL" > "Nat_Numeral.nat_mult_eq_cancel_disj"
"EQ_IMP_LE" > "HOLLight.hollight.EQ_IMP_LE"
"EQ_EXT" > "HOL.meta_eq_to_obj_eq"
"EQ_CLAUSES" > "HOLLight.hollight.EQ_CLAUSES"
--- a/src/HOL/IsaMakefile Wed Apr 15 15:30:38 2009 +0200
+++ b/src/HOL/IsaMakefile Wed Apr 15 15:30:39 2009 +0200
@@ -216,7 +216,7 @@
List.thy \
Main.thy \
Map.thy \
- NatBin.thy \
+ Nat_Numeral.thy \
Presburger.thy \
Recdef.thy \
Relation_Power.thy \
--- a/src/HOL/NatBin.thy Wed Apr 15 15:30:38 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,975 +0,0 @@
-(* Title: HOL/NatBin.thy
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1999 University of Cambridge
-*)
-
-header {* Binary arithmetic for the natural numbers *}
-
-theory NatBin
-imports IntDiv
-uses ("Tools/nat_simprocs.ML")
-begin
-
-text {*
- Arithmetic for naturals is reduced to that for the non-negative integers.
-*}
-
-instantiation nat :: number
-begin
-
-definition
- nat_number_of_def [code inline, code del]: "number_of v = nat (number_of v)"
-
-instance ..
-
-end
-
-lemma [code post]:
- "nat (number_of v) = number_of v"
- unfolding nat_number_of_def ..
-
-abbreviation (xsymbols)
- power2 :: "'a::power => 'a" ("(_\<twosuperior>)" [1000] 999) where
- "x\<twosuperior> == x^2"
-
-notation (latex output)
- power2 ("(_\<twosuperior>)" [1000] 999)
-
-notation (HTML output)
- power2 ("(_\<twosuperior>)" [1000] 999)
-
-
-subsection {* Predicate for negative binary numbers *}
-
-definition neg :: "int \<Rightarrow> bool" where
- "neg Z \<longleftrightarrow> Z < 0"
-
-lemma not_neg_int [simp]: "~ neg (of_nat n)"
-by (simp add: neg_def)
-
-lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))"
-by (simp add: neg_def neg_less_0_iff_less del: of_nat_Suc)
-
-lemmas neg_eq_less_0 = neg_def
-
-lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
-by (simp add: neg_def linorder_not_less)
-
-text{*To simplify inequalities when Numeral1 can get simplified to 1*}
-
-lemma not_neg_0: "~ neg 0"
-by (simp add: One_int_def neg_def)
-
-lemma not_neg_1: "~ neg 1"
-by (simp add: neg_def linorder_not_less zero_le_one)
-
-lemma neg_nat: "neg z ==> nat z = 0"
-by (simp add: neg_def order_less_imp_le)
-
-lemma not_neg_nat: "~ neg z ==> of_nat (nat z) = z"
-by (simp add: linorder_not_less neg_def)
-
-text {*
- If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
- @{term Numeral0} IS @{term "number_of Pls"}
-*}
-
-lemma not_neg_number_of_Pls: "~ neg (number_of Int.Pls)"
- by (simp add: neg_def)
-
-lemma neg_number_of_Min: "neg (number_of Int.Min)"
- by (simp add: neg_def)
-
-lemma neg_number_of_Bit0:
- "neg (number_of (Int.Bit0 w)) = neg (number_of w)"
- by (simp add: neg_def)
-
-lemma neg_number_of_Bit1:
- "neg (number_of (Int.Bit1 w)) = neg (number_of w)"
- by (simp add: neg_def)
-
-lemmas neg_simps [simp] =
- not_neg_0 not_neg_1
- not_neg_number_of_Pls neg_number_of_Min
- neg_number_of_Bit0 neg_number_of_Bit1
-
-
-subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*}
-
-declare nat_0 [simp] nat_1 [simp]
-
-lemma nat_number_of [simp]: "nat (number_of w) = number_of w"
-by (simp add: nat_number_of_def)
-
-lemma nat_numeral_0_eq_0 [simp]: "Numeral0 = (0::nat)"
-by (simp add: nat_number_of_def)
-
-lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
-by (simp add: nat_1 nat_number_of_def)
-
-lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
-by (simp add: nat_numeral_1_eq_1)
-
-lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
-apply (unfold nat_number_of_def)
-apply (rule nat_2)
-done
-
-
-subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}
-
-lemma int_nat_number_of [simp]:
- "int (number_of v) =
- (if neg (number_of v :: int) then 0
- else (number_of v :: int))"
- unfolding nat_number_of_def number_of_is_id neg_def
- by simp
-
-
-subsubsection{*Successor *}
-
-lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
-apply (rule sym)
-apply (simp add: nat_eq_iff int_Suc)
-done
-
-lemma Suc_nat_number_of_add:
- "Suc (number_of v + n) =
- (if neg (number_of v :: int) then 1+n else number_of (Int.succ v) + n)"
- unfolding nat_number_of_def number_of_is_id neg_def numeral_simps
- by (simp add: Suc_nat_eq_nat_zadd1 add_ac)
-
-lemma Suc_nat_number_of [simp]:
- "Suc (number_of v) =
- (if neg (number_of v :: int) then 1 else number_of (Int.succ v))"
-apply (cut_tac n = 0 in Suc_nat_number_of_add)
-apply (simp cong del: if_weak_cong)
-done
-
-
-subsubsection{*Addition *}
-
-lemma add_nat_number_of [simp]:
- "(number_of v :: nat) + number_of v' =
- (if v < Int.Pls then number_of v'
- else if v' < Int.Pls then number_of v
- else number_of (v + v'))"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by (simp add: nat_add_distrib)
-
-lemma nat_number_of_add_1 [simp]:
- "number_of v + (1::nat) =
- (if v < Int.Pls then 1 else number_of (Int.succ v))"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by (simp add: nat_add_distrib)
-
-lemma nat_1_add_number_of [simp]:
- "(1::nat) + number_of v =
- (if v < Int.Pls then 1 else number_of (Int.succ v))"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by (simp add: nat_add_distrib)
-
-lemma nat_1_add_1 [simp]: "1 + 1 = (2::nat)"
- by (rule int_int_eq [THEN iffD1]) simp
-
-
-subsubsection{*Subtraction *}
-
-lemma diff_nat_eq_if:
- "nat z - nat z' =
- (if neg z' then nat z
- else let d = z-z' in
- if neg d then 0 else nat d)"
-by (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
-
-
-lemma diff_nat_number_of [simp]:
- "(number_of v :: nat) - number_of v' =
- (if v' < Int.Pls then number_of v
- else let d = number_of (v + uminus v') in
- if neg d then 0 else nat d)"
- unfolding nat_number_of_def number_of_is_id numeral_simps neg_def
- by auto
-
-lemma nat_number_of_diff_1 [simp]:
- "number_of v - (1::nat) =
- (if v \<le> Int.Pls then 0 else number_of (Int.pred v))"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by auto
-
-
-subsubsection{*Multiplication *}
-
-lemma mult_nat_number_of [simp]:
- "(number_of v :: nat) * number_of v' =
- (if v < Int.Pls then 0 else number_of (v * v'))"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by (simp add: nat_mult_distrib)
-
-
-subsubsection{*Quotient *}
-
-lemma div_nat_number_of [simp]:
- "(number_of v :: nat) div number_of v' =
- (if neg (number_of v :: int) then 0
- else nat (number_of v div number_of v'))"
- unfolding nat_number_of_def number_of_is_id neg_def
- by (simp add: nat_div_distrib)
-
-lemma one_div_nat_number_of [simp]:
- "Suc 0 div number_of v' = nat (1 div number_of v')"
-by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
-
-
-subsubsection{*Remainder *}
-
-lemma mod_nat_number_of [simp]:
- "(number_of v :: nat) mod number_of v' =
- (if neg (number_of v :: int) then 0
- else if neg (number_of v' :: int) then number_of v
- else nat (number_of v mod number_of v'))"
- unfolding nat_number_of_def number_of_is_id neg_def
- by (simp add: nat_mod_distrib)
-
-lemma one_mod_nat_number_of [simp]:
- "Suc 0 mod number_of v' =
- (if neg (number_of v' :: int) then Suc 0
- else nat (1 mod number_of v'))"
-by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
-
-
-subsubsection{* Divisibility *}
-
-lemmas dvd_eq_mod_eq_0_number_of =
- dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
-
-declare dvd_eq_mod_eq_0_number_of [simp]
-
-ML
-{*
-val nat_number_of_def = thm"nat_number_of_def";
-
-val nat_number_of = thm"nat_number_of";
-val nat_numeral_0_eq_0 = thm"nat_numeral_0_eq_0";
-val nat_numeral_1_eq_1 = thm"nat_numeral_1_eq_1";
-val numeral_1_eq_Suc_0 = thm"numeral_1_eq_Suc_0";
-val numeral_2_eq_2 = thm"numeral_2_eq_2";
-val nat_div_distrib = thm"nat_div_distrib";
-val nat_mod_distrib = thm"nat_mod_distrib";
-val int_nat_number_of = thm"int_nat_number_of";
-val Suc_nat_eq_nat_zadd1 = thm"Suc_nat_eq_nat_zadd1";
-val Suc_nat_number_of_add = thm"Suc_nat_number_of_add";
-val Suc_nat_number_of = thm"Suc_nat_number_of";
-val add_nat_number_of = thm"add_nat_number_of";
-val diff_nat_eq_if = thm"diff_nat_eq_if";
-val diff_nat_number_of = thm"diff_nat_number_of";
-val mult_nat_number_of = thm"mult_nat_number_of";
-val div_nat_number_of = thm"div_nat_number_of";
-val mod_nat_number_of = thm"mod_nat_number_of";
-*}
-
-
-subsection{*Comparisons*}
-
-subsubsection{*Equals (=) *}
-
-lemma eq_nat_nat_iff:
- "[| (0::int) <= z; 0 <= z' |] ==> (nat z = nat z') = (z=z')"
-by (auto elim!: nonneg_eq_int)
-
-lemma eq_nat_number_of [simp]:
- "((number_of v :: nat) = number_of v') =
- (if neg (number_of v :: int) then (number_of v' :: int) \<le> 0
- else if neg (number_of v' :: int) then (number_of v :: int) = 0
- else v = v')"
- unfolding nat_number_of_def number_of_is_id neg_def
- by auto
-
-
-subsubsection{*Less-than (<) *}
-
-lemma less_nat_number_of [simp]:
- "(number_of v :: nat) < number_of v' \<longleftrightarrow>
- (if v < v' then Int.Pls < v' else False)"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by auto
-
-
-subsubsection{*Less-than-or-equal *}
-
-lemma le_nat_number_of [simp]:
- "(number_of v :: nat) \<le> number_of v' \<longleftrightarrow>
- (if v \<le> v' then True else v \<le> Int.Pls)"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by auto
-
-(*Maps #n to n for n = 0, 1, 2*)
-lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2
-
-
-subsection{*Powers with Numeric Exponents*}
-
-text{*We cannot refer to the number @{term 2} in @{text Ring_and_Field.thy}.
-We cannot prove general results about the numeral @{term "-1"}, so we have to
-use @{term "- 1"} instead.*}
-
-lemma power2_eq_square: "(a::'a::recpower)\<twosuperior> = a * a"
- by (simp add: numeral_2_eq_2 Power.power_Suc)
-
-lemma zero_power2 [simp]: "(0::'a::{semiring_1,recpower})\<twosuperior> = 0"
- by (simp add: power2_eq_square)
-
-lemma one_power2 [simp]: "(1::'a::{semiring_1,recpower})\<twosuperior> = 1"
- by (simp add: power2_eq_square)
-
-lemma power3_eq_cube: "(x::'a::recpower) ^ 3 = x * x * x"
- apply (subgoal_tac "3 = Suc (Suc (Suc 0))")
- apply (erule ssubst)
- apply (simp add: power_Suc mult_ac)
- apply (unfold nat_number_of_def)
- apply (subst nat_eq_iff)
- apply simp
-done
-
-text{*Squares of literal numerals will be evaluated.*}
-lemmas power2_eq_square_number_of =
- power2_eq_square [of "number_of w", standard]
-declare power2_eq_square_number_of [simp]
-
-
-lemma zero_le_power2[simp]: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,recpower})"
- by (simp add: power2_eq_square)
-
-lemma zero_less_power2[simp]:
- "(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,recpower}))"
- by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
-
-lemma power2_less_0[simp]:
- fixes a :: "'a::{ordered_idom,recpower}"
- shows "~ (a\<twosuperior> < 0)"
-by (force simp add: power2_eq_square mult_less_0_iff)
-
-lemma zero_eq_power2[simp]:
- "(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))"
- by (force simp add: power2_eq_square mult_eq_0_iff)
-
-lemma abs_power2[simp]:
- "abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})"
- by (simp add: power2_eq_square abs_mult abs_mult_self)
-
-lemma power2_abs[simp]:
- "(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})"
- by (simp add: power2_eq_square abs_mult_self)
-
-lemma power2_minus[simp]:
- "(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})"
- by (simp add: power2_eq_square)
-
-lemma power2_le_imp_le:
- fixes x y :: "'a::{ordered_semidom,recpower}"
- shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y"
-unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
-
-lemma power2_less_imp_less:
- fixes x y :: "'a::{ordered_semidom,recpower}"
- shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y"
-by (rule power_less_imp_less_base)
-
-lemma power2_eq_imp_eq:
- fixes x y :: "'a::{ordered_semidom,recpower}"
- shows "\<lbrakk>x\<twosuperior> = y\<twosuperior>; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> x = y"
-unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base, simp)
-
-lemma power_minus1_even[simp]: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})"
-proof (induct n)
- case 0 show ?case by simp
-next
- case (Suc n) then show ?case by (simp add: power_Suc power_add)
-qed
-
-lemma power_minus1_odd: "(- 1) ^ Suc(2*n) = -(1::'a::{comm_ring_1,recpower})"
- by (simp add: power_Suc)
-
-lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2"
-by (subst mult_commute) (simp add: power_mult)
-
-lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"
-by (simp add: power_even_eq)
-
-lemma power_minus_even [simp]:
- "(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)"
-by (simp add: power_minus1_even power_minus [of a])
-
-lemma zero_le_even_power'[simp]:
- "0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)"
-proof (induct "n")
- case 0
- show ?case by (simp add: zero_le_one)
-next
- case (Suc n)
- have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
- by (simp add: mult_ac power_add power2_eq_square)
- thus ?case
- by (simp add: prems zero_le_mult_iff)
-qed
-
-lemma odd_power_less_zero:
- "(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0"
-proof (induct "n")
- case 0
- then show ?case by simp
-next
- case (Suc n)
- have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
- by (simp add: mult_ac power_add power2_eq_square)
- thus ?case
- by (simp del: power_Suc add: prems mult_less_0_iff mult_neg_neg)
-qed
-
-lemma odd_0_le_power_imp_0_le:
- "0 \<le> a ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})"
-apply (insert odd_power_less_zero [of a n])
-apply (force simp add: linorder_not_less [symmetric])
-done
-
-text{*Simprules for comparisons where common factors can be cancelled.*}
-lemmas zero_compare_simps =
- add_strict_increasing add_strict_increasing2 add_increasing
- zero_le_mult_iff zero_le_divide_iff
- zero_less_mult_iff zero_less_divide_iff
- mult_le_0_iff divide_le_0_iff
- mult_less_0_iff divide_less_0_iff
- zero_le_power2 power2_less_0
-
-subsubsection{*Nat *}
-
-lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
-by (simp add: numerals)
-
-(*Expresses a natural number constant as the Suc of another one.
- NOT suitable for rewriting because n recurs in the condition.*)
-lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
-
-subsubsection{*Arith *}
-
-lemma Suc_eq_add_numeral_1: "Suc n = n + 1"
-by (simp add: numerals)
-
-lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n"
-by (simp add: numerals)
-
-(* These two can be useful when m = number_of... *)
-
-lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
- unfolding One_nat_def by (cases m) simp_all
-
-lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
- unfolding One_nat_def by (cases m) simp_all
-
-lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
- unfolding One_nat_def by (cases m) simp_all
-
-
-subsection{*Comparisons involving (0::nat) *}
-
-text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
-
-lemma eq_number_of_0 [simp]:
- "number_of v = (0::nat) \<longleftrightarrow> v \<le> Int.Pls"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by auto
-
-lemma eq_0_number_of [simp]:
- "(0::nat) = number_of v \<longleftrightarrow> v \<le> Int.Pls"
-by (rule trans [OF eq_sym_conv eq_number_of_0])
-
-lemma less_0_number_of [simp]:
- "(0::nat) < number_of v \<longleftrightarrow> Int.Pls < v"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by simp
-
-lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
-by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric])
-
-
-
-subsection{*Comparisons involving @{term Suc} *}
-
-lemma eq_number_of_Suc [simp]:
- "(number_of v = Suc n) =
- (let pv = number_of (Int.pred v) in
- if neg pv then False else nat pv = n)"
-apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
- number_of_pred nat_number_of_def
- split add: split_if)
-apply (rule_tac x = "number_of v" in spec)
-apply (auto simp add: nat_eq_iff)
-done
-
-lemma Suc_eq_number_of [simp]:
- "(Suc n = number_of v) =
- (let pv = number_of (Int.pred v) in
- if neg pv then False else nat pv = n)"
-by (rule trans [OF eq_sym_conv eq_number_of_Suc])
-
-lemma less_number_of_Suc [simp]:
- "(number_of v < Suc n) =
- (let pv = number_of (Int.pred v) in
- if neg pv then True else nat pv < n)"
-apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
- number_of_pred nat_number_of_def
- split add: split_if)
-apply (rule_tac x = "number_of v" in spec)
-apply (auto simp add: nat_less_iff)
-done
-
-lemma less_Suc_number_of [simp]:
- "(Suc n < number_of v) =
- (let pv = number_of (Int.pred v) in
- if neg pv then False else n < nat pv)"
-apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
- number_of_pred nat_number_of_def
- split add: split_if)
-apply (rule_tac x = "number_of v" in spec)
-apply (auto simp add: zless_nat_eq_int_zless)
-done
-
-lemma le_number_of_Suc [simp]:
- "(number_of v <= Suc n) =
- (let pv = number_of (Int.pred v) in
- if neg pv then True else nat pv <= n)"
-by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric])
-
-lemma le_Suc_number_of [simp]:
- "(Suc n <= number_of v) =
- (let pv = number_of (Int.pred v) in
- if neg pv then False else n <= nat pv)"
-by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric])
-
-
-lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Int.Min"
-by auto
-
-
-
-subsection{*Max and Min Combined with @{term Suc} *}
-
-lemma max_number_of_Suc [simp]:
- "max (Suc n) (number_of v) =
- (let pv = number_of (Int.pred v) in
- if neg pv then Suc n else Suc(max n (nat pv)))"
-apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
- split add: split_if nat.split)
-apply (rule_tac x = "number_of v" in spec)
-apply auto
-done
-
-lemma max_Suc_number_of [simp]:
- "max (number_of v) (Suc n) =
- (let pv = number_of (Int.pred v) in
- if neg pv then Suc n else Suc(max (nat pv) n))"
-apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
- split add: split_if nat.split)
-apply (rule_tac x = "number_of v" in spec)
-apply auto
-done
-
-lemma min_number_of_Suc [simp]:
- "min (Suc n) (number_of v) =
- (let pv = number_of (Int.pred v) in
- if neg pv then 0 else Suc(min n (nat pv)))"
-apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
- split add: split_if nat.split)
-apply (rule_tac x = "number_of v" in spec)
-apply auto
-done
-
-lemma min_Suc_number_of [simp]:
- "min (number_of v) (Suc n) =
- (let pv = number_of (Int.pred v) in
- if neg pv then 0 else Suc(min (nat pv) n))"
-apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
- split add: split_if nat.split)
-apply (rule_tac x = "number_of v" in spec)
-apply auto
-done
-
-subsection{*Literal arithmetic involving powers*}
-
-lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n"
-apply (induct "n")
-apply (simp_all (no_asm_simp) add: nat_mult_distrib)
-done
-
-lemma power_nat_number_of:
- "(number_of v :: nat) ^ n =
- (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
-by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
- split add: split_if cong: imp_cong)
-
-
-lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]
-declare power_nat_number_of_number_of [simp]
-
-
-
-text{*For arbitrary rings*}
-
-lemma power_number_of_even:
- fixes z :: "'a::{number_ring,recpower}"
- shows "z ^ number_of (Int.Bit0 w) = (let w = z ^ (number_of w) in w * w)"
-unfolding Let_def nat_number_of_def number_of_Bit0
-apply (rule_tac x = "number_of w" in spec, clarify)
-apply (case_tac " (0::int) <= x")
-apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square)
-done
-
-lemma power_number_of_odd:
- fixes z :: "'a::{number_ring,recpower}"
- shows "z ^ number_of (Int.Bit1 w) = (if (0::int) <= number_of w
- then (let w = z ^ (number_of w) in z * w * w) else 1)"
-unfolding Let_def nat_number_of_def number_of_Bit1
-apply (rule_tac x = "number_of w" in spec, auto)
-apply (simp only: nat_add_distrib nat_mult_distrib)
-apply simp
-apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat power_Suc)
-done
-
-lemmas zpower_number_of_even = power_number_of_even [where 'a=int]
-lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int]
-
-lemmas power_number_of_even_number_of [simp] =
- power_number_of_even [of "number_of v", standard]
-
-lemmas power_number_of_odd_number_of [simp] =
- power_number_of_odd [of "number_of v", standard]
-
-
-
-ML
-{*
-val numeral_ss = @{simpset} addsimps @{thms numerals};
-
-val nat_bin_arith_setup =
- Lin_Arith.map_data
- (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
- {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
- inj_thms = inj_thms,
- lessD = lessD, neqE = neqE,
- simpset = simpset addsimps @{thms neg_simps} @
- [@{thm Suc_nat_number_of}, @{thm int_nat_number_of}]})
-*}
-
-declaration {* K nat_bin_arith_setup *}
-
-(* Enable arith to deal with div/mod k where k is a numeral: *)
-declare split_div[of _ _ "number_of k", standard, arith_split]
-declare split_mod[of _ _ "number_of k", standard, arith_split]
-
-lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
- by (simp add: number_of_Pls nat_number_of_def)
-
-lemma nat_number_of_Min: "number_of Int.Min = (0::nat)"
- apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
- done
-
-lemma nat_number_of_Bit0:
- "number_of (Int.Bit0 w) = (let n::nat = number_of w in n + n)"
- unfolding nat_number_of_def number_of_is_id numeral_simps Let_def
- by auto
-
-lemma nat_number_of_Bit1:
- "number_of (Int.Bit1 w) =
- (if neg (number_of w :: int) then 0
- else let n = number_of w in Suc (n + n))"
- unfolding nat_number_of_def number_of_is_id numeral_simps neg_def Let_def
- by auto
-
-lemmas nat_number =
- nat_number_of_Pls nat_number_of_Min
- nat_number_of_Bit0 nat_number_of_Bit1
-
-lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
- by (simp add: Let_def)
-
-lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})"
-by (simp add: power_mult power_Suc);
-
-lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})"
-by (simp add: power_mult power_Suc);
-
-
-subsection{*Literal arithmetic and @{term of_nat}*}
-
-lemma of_nat_double:
- "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
-by (simp only: mult_2 nat_add_distrib of_nat_add)
-
-lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
-by (simp only: nat_number_of_def)
-
-lemma of_nat_number_of_lemma:
- "of_nat (number_of v :: nat) =
- (if 0 \<le> (number_of v :: int)
- then (number_of v :: 'a :: number_ring)
- else 0)"
-by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat);
-
-lemma of_nat_number_of_eq [simp]:
- "of_nat (number_of v :: nat) =
- (if neg (number_of v :: int) then 0
- else (number_of v :: 'a :: number_ring))"
-by (simp only: of_nat_number_of_lemma neg_def, simp)
-
-
-subsection {*Lemmas for the Combination and Cancellation Simprocs*}
-
-lemma nat_number_of_add_left:
- "number_of v + (number_of v' + (k::nat)) =
- (if neg (number_of v :: int) then number_of v' + k
- else if neg (number_of v' :: int) then number_of v + k
- else number_of (v + v') + k)"
- unfolding nat_number_of_def number_of_is_id neg_def
- by auto
-
-lemma nat_number_of_mult_left:
- "number_of v * (number_of v' * (k::nat)) =
- (if v < Int.Pls then 0
- else number_of (v * v') * k)"
-by simp
-
-
-subsubsection{*For @{text combine_numerals}*}
-
-lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
-by (simp add: add_mult_distrib)
-
-
-subsubsection{*For @{text cancel_numerals}*}
-
-lemma nat_diff_add_eq1:
- "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
-by (simp split add: nat_diff_split add: add_mult_distrib)
-
-lemma nat_diff_add_eq2:
- "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
-by (simp split add: nat_diff_split add: add_mult_distrib)
-
-lemma nat_eq_add_iff1:
- "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_eq_add_iff2:
- "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_less_add_iff1:
- "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_less_add_iff2:
- "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_le_add_iff1:
- "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_le_add_iff2:
- "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-
-subsubsection{*For @{text cancel_numeral_factors} *}
-
-lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
-by auto
-
-lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
-by auto
-
-lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
-by auto
-
-lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
-by auto
-
-lemma nat_mult_dvd_cancel_disj[simp]:
- "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
-by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])
-
-lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
-by(auto)
-
-
-subsubsection{*For @{text cancel_factor} *}
-
-lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
-by auto
-
-lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
-by auto
-
-lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
-by auto
-
-lemma nat_mult_div_cancel_disj[simp]:
- "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
-by (simp add: nat_mult_div_cancel1)
-
-
-subsection {* Simprocs for the Naturals *}
-
-use "Tools/nat_simprocs.ML"
-declaration {* K nat_simprocs_setup *}
-
-subsubsection{*For simplifying @{term "Suc m - K"} and @{term "K - Suc m"}*}
-
-text{*Where K above is a literal*}
-
-lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
-by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split)
-
-text {*Now just instantiating @{text n} to @{text "number_of v"} does
- the right simplification, but with some redundant inequality
- tests.*}
-lemma neg_number_of_pred_iff_0:
- "neg (number_of (Int.pred v)::int) = (number_of v = (0::nat))"
-apply (subgoal_tac "neg (number_of (Int.pred v)) = (number_of v < Suc 0) ")
-apply (simp only: less_Suc_eq_le le_0_eq)
-apply (subst less_number_of_Suc, simp)
-done
-
-text{*No longer required as a simprule because of the @{text inverse_fold}
- simproc*}
-lemma Suc_diff_number_of:
- "Int.Pls < v ==>
- Suc m - (number_of v) = m - (number_of (Int.pred v))"
-apply (subst Suc_diff_eq_diff_pred)
-apply simp
-apply (simp del: nat_numeral_1_eq_1)
-apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
- neg_number_of_pred_iff_0)
-done
-
-lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
-by (simp add: numerals split add: nat_diff_split)
-
-
-subsubsection{*For @{term nat_case} and @{term nat_rec}*}
-
-lemma nat_case_number_of [simp]:
- "nat_case a f (number_of v) =
- (let pv = number_of (Int.pred v) in
- if neg pv then a else f (nat pv))"
-by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)
-
-lemma nat_case_add_eq_if [simp]:
- "nat_case a f ((number_of v) + n) =
- (let pv = number_of (Int.pred v) in
- if neg pv then nat_case a f n else f (nat pv + n))"
-apply (subst add_eq_if)
-apply (simp split add: nat.split
- del: nat_numeral_1_eq_1
- add: nat_numeral_1_eq_1 [symmetric]
- numeral_1_eq_Suc_0 [symmetric]
- neg_number_of_pred_iff_0)
-done
-
-lemma nat_rec_number_of [simp]:
- "nat_rec a f (number_of v) =
- (let pv = number_of (Int.pred v) in
- if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
-apply (case_tac " (number_of v) ::nat")
-apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
-apply (simp split add: split_if_asm)
-done
-
-lemma nat_rec_add_eq_if [simp]:
- "nat_rec a f (number_of v + n) =
- (let pv = number_of (Int.pred v) in
- if neg pv then nat_rec a f n
- else f (nat pv + n) (nat_rec a f (nat pv + n)))"
-apply (subst add_eq_if)
-apply (simp split add: nat.split
- del: nat_numeral_1_eq_1
- add: nat_numeral_1_eq_1 [symmetric]
- numeral_1_eq_Suc_0 [symmetric]
- neg_number_of_pred_iff_0)
-done
-
-
-subsubsection{*Various Other Lemmas*}
-
-text {*Evens and Odds, for Mutilated Chess Board*}
-
-text{*Lemmas for specialist use, NOT as default simprules*}
-lemma nat_mult_2: "2 * z = (z+z::nat)"
-proof -
- have "2*z = (1 + 1)*z" by simp
- also have "... = z+z" by (simp add: left_distrib)
- finally show ?thesis .
-qed
-
-lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
-by (subst mult_commute, rule nat_mult_2)
-
-text{*Case analysis on @{term "n<2"}*}
-lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
-by arith
-
-lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)"
-by arith
-
-lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
-by (simp add: nat_mult_2 [symmetric])
-
-lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
-apply (subgoal_tac "m mod 2 < 2")
-apply (erule less_2_cases [THEN disjE])
-apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
-done
-
-lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)"
-apply (subgoal_tac "m mod 2 < 2")
-apply (force simp del: mod_less_divisor, simp)
-done
-
-text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
-
-lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
-by simp
-
-lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
-by simp
-
-text{*Can be used to eliminate long strings of Sucs, but not by default*}
-lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
-by simp
-
-
-text{*These lemmas collapse some needless occurrences of Suc:
- at least three Sucs, since two and fewer are rewritten back to Suc again!
- We already have some rules to simplify operands smaller than 3.*}
-
-lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
-by (simp add: Suc3_eq_add_3)
-
-lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
-by (simp add: Suc3_eq_add_3)
-
-lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
-by (simp add: Suc3_eq_add_3)
-
-lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
-by (simp add: Suc3_eq_add_3)
-
-lemmas Suc_div_eq_add3_div_number_of =
- Suc_div_eq_add3_div [of _ "number_of v", standard]
-declare Suc_div_eq_add3_div_number_of [simp]
-
-lemmas Suc_mod_eq_add3_mod_number_of =
- Suc_mod_eq_add3_mod [of _ "number_of v", standard]
-declare Suc_mod_eq_add3_mod_number_of [simp]
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Nat_Numeral.thy Wed Apr 15 15:30:39 2009 +0200
@@ -0,0 +1,975 @@
+(* Title: HOL/Nat_Numeral.thy
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1999 University of Cambridge
+*)
+
+header {* Binary numerals for the natural numbers *}
+
+theory Nat_Numeral
+imports IntDiv
+uses ("Tools/nat_simprocs.ML")
+begin
+
+text {*
+ Arithmetic for naturals is reduced to that for the non-negative integers.
+*}
+
+instantiation nat :: number
+begin
+
+definition
+ nat_number_of_def [code inline, code del]: "number_of v = nat (number_of v)"
+
+instance ..
+
+end
+
+lemma [code post]:
+ "nat (number_of v) = number_of v"
+ unfolding nat_number_of_def ..
+
+abbreviation (xsymbols)
+ power2 :: "'a::power => 'a" ("(_\<twosuperior>)" [1000] 999) where
+ "x\<twosuperior> == x^2"
+
+notation (latex output)
+ power2 ("(_\<twosuperior>)" [1000] 999)
+
+notation (HTML output)
+ power2 ("(_\<twosuperior>)" [1000] 999)
+
+
+subsection {* Predicate for negative binary numbers *}
+
+definition neg :: "int \<Rightarrow> bool" where
+ "neg Z \<longleftrightarrow> Z < 0"
+
+lemma not_neg_int [simp]: "~ neg (of_nat n)"
+by (simp add: neg_def)
+
+lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))"
+by (simp add: neg_def neg_less_0_iff_less del: of_nat_Suc)
+
+lemmas neg_eq_less_0 = neg_def
+
+lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
+by (simp add: neg_def linorder_not_less)
+
+text{*To simplify inequalities when Numeral1 can get simplified to 1*}
+
+lemma not_neg_0: "~ neg 0"
+by (simp add: One_int_def neg_def)
+
+lemma not_neg_1: "~ neg 1"
+by (simp add: neg_def linorder_not_less zero_le_one)
+
+lemma neg_nat: "neg z ==> nat z = 0"
+by (simp add: neg_def order_less_imp_le)
+
+lemma not_neg_nat: "~ neg z ==> of_nat (nat z) = z"
+by (simp add: linorder_not_less neg_def)
+
+text {*
+ If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
+ @{term Numeral0} IS @{term "number_of Pls"}
+*}
+
+lemma not_neg_number_of_Pls: "~ neg (number_of Int.Pls)"
+ by (simp add: neg_def)
+
+lemma neg_number_of_Min: "neg (number_of Int.Min)"
+ by (simp add: neg_def)
+
+lemma neg_number_of_Bit0:
+ "neg (number_of (Int.Bit0 w)) = neg (number_of w)"
+ by (simp add: neg_def)
+
+lemma neg_number_of_Bit1:
+ "neg (number_of (Int.Bit1 w)) = neg (number_of w)"
+ by (simp add: neg_def)
+
+lemmas neg_simps [simp] =
+ not_neg_0 not_neg_1
+ not_neg_number_of_Pls neg_number_of_Min
+ neg_number_of_Bit0 neg_number_of_Bit1
+
+
+subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*}
+
+declare nat_0 [simp] nat_1 [simp]
+
+lemma nat_number_of [simp]: "nat (number_of w) = number_of w"
+by (simp add: nat_number_of_def)
+
+lemma nat_numeral_0_eq_0 [simp]: "Numeral0 = (0::nat)"
+by (simp add: nat_number_of_def)
+
+lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
+by (simp add: nat_1 nat_number_of_def)
+
+lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
+by (simp add: nat_numeral_1_eq_1)
+
+lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
+apply (unfold nat_number_of_def)
+apply (rule nat_2)
+done
+
+
+subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}
+
+lemma int_nat_number_of [simp]:
+ "int (number_of v) =
+ (if neg (number_of v :: int) then 0
+ else (number_of v :: int))"
+ unfolding nat_number_of_def number_of_is_id neg_def
+ by simp
+
+
+subsubsection{*Successor *}
+
+lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
+apply (rule sym)
+apply (simp add: nat_eq_iff int_Suc)
+done
+
+lemma Suc_nat_number_of_add:
+ "Suc (number_of v + n) =
+ (if neg (number_of v :: int) then 1+n else number_of (Int.succ v) + n)"
+ unfolding nat_number_of_def number_of_is_id neg_def numeral_simps
+ by (simp add: Suc_nat_eq_nat_zadd1 add_ac)
+
+lemma Suc_nat_number_of [simp]:
+ "Suc (number_of v) =
+ (if neg (number_of v :: int) then 1 else number_of (Int.succ v))"
+apply (cut_tac n = 0 in Suc_nat_number_of_add)
+apply (simp cong del: if_weak_cong)
+done
+
+
+subsubsection{*Addition *}
+
+lemma add_nat_number_of [simp]:
+ "(number_of v :: nat) + number_of v' =
+ (if v < Int.Pls then number_of v'
+ else if v' < Int.Pls then number_of v
+ else number_of (v + v'))"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by (simp add: nat_add_distrib)
+
+lemma nat_number_of_add_1 [simp]:
+ "number_of v + (1::nat) =
+ (if v < Int.Pls then 1 else number_of (Int.succ v))"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by (simp add: nat_add_distrib)
+
+lemma nat_1_add_number_of [simp]:
+ "(1::nat) + number_of v =
+ (if v < Int.Pls then 1 else number_of (Int.succ v))"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by (simp add: nat_add_distrib)
+
+lemma nat_1_add_1 [simp]: "1 + 1 = (2::nat)"
+ by (rule int_int_eq [THEN iffD1]) simp
+
+
+subsubsection{*Subtraction *}
+
+lemma diff_nat_eq_if:
+ "nat z - nat z' =
+ (if neg z' then nat z
+ else let d = z-z' in
+ if neg d then 0 else nat d)"
+by (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
+
+
+lemma diff_nat_number_of [simp]:
+ "(number_of v :: nat) - number_of v' =
+ (if v' < Int.Pls then number_of v
+ else let d = number_of (v + uminus v') in
+ if neg d then 0 else nat d)"
+ unfolding nat_number_of_def number_of_is_id numeral_simps neg_def
+ by auto
+
+lemma nat_number_of_diff_1 [simp]:
+ "number_of v - (1::nat) =
+ (if v \<le> Int.Pls then 0 else number_of (Int.pred v))"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by auto
+
+
+subsubsection{*Multiplication *}
+
+lemma mult_nat_number_of [simp]:
+ "(number_of v :: nat) * number_of v' =
+ (if v < Int.Pls then 0 else number_of (v * v'))"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by (simp add: nat_mult_distrib)
+
+
+subsubsection{*Quotient *}
+
+lemma div_nat_number_of [simp]:
+ "(number_of v :: nat) div number_of v' =
+ (if neg (number_of v :: int) then 0
+ else nat (number_of v div number_of v'))"
+ unfolding nat_number_of_def number_of_is_id neg_def
+ by (simp add: nat_div_distrib)
+
+lemma one_div_nat_number_of [simp]:
+ "Suc 0 div number_of v' = nat (1 div number_of v')"
+by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
+
+
+subsubsection{*Remainder *}
+
+lemma mod_nat_number_of [simp]:
+ "(number_of v :: nat) mod number_of v' =
+ (if neg (number_of v :: int) then 0
+ else if neg (number_of v' :: int) then number_of v
+ else nat (number_of v mod number_of v'))"
+ unfolding nat_number_of_def number_of_is_id neg_def
+ by (simp add: nat_mod_distrib)
+
+lemma one_mod_nat_number_of [simp]:
+ "Suc 0 mod number_of v' =
+ (if neg (number_of v' :: int) then Suc 0
+ else nat (1 mod number_of v'))"
+by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
+
+
+subsubsection{* Divisibility *}
+
+lemmas dvd_eq_mod_eq_0_number_of =
+ dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
+
+declare dvd_eq_mod_eq_0_number_of [simp]
+
+ML
+{*
+val nat_number_of_def = thm"nat_number_of_def";
+
+val nat_number_of = thm"nat_number_of";
+val nat_numeral_0_eq_0 = thm"nat_numeral_0_eq_0";
+val nat_numeral_1_eq_1 = thm"nat_numeral_1_eq_1";
+val numeral_1_eq_Suc_0 = thm"numeral_1_eq_Suc_0";
+val numeral_2_eq_2 = thm"numeral_2_eq_2";
+val nat_div_distrib = thm"nat_div_distrib";
+val nat_mod_distrib = thm"nat_mod_distrib";
+val int_nat_number_of = thm"int_nat_number_of";
+val Suc_nat_eq_nat_zadd1 = thm"Suc_nat_eq_nat_zadd1";
+val Suc_nat_number_of_add = thm"Suc_nat_number_of_add";
+val Suc_nat_number_of = thm"Suc_nat_number_of";
+val add_nat_number_of = thm"add_nat_number_of";
+val diff_nat_eq_if = thm"diff_nat_eq_if";
+val diff_nat_number_of = thm"diff_nat_number_of";
+val mult_nat_number_of = thm"mult_nat_number_of";
+val div_nat_number_of = thm"div_nat_number_of";
+val mod_nat_number_of = thm"mod_nat_number_of";
+*}
+
+
+subsection{*Comparisons*}
+
+subsubsection{*Equals (=) *}
+
+lemma eq_nat_nat_iff:
+ "[| (0::int) <= z; 0 <= z' |] ==> (nat z = nat z') = (z=z')"
+by (auto elim!: nonneg_eq_int)
+
+lemma eq_nat_number_of [simp]:
+ "((number_of v :: nat) = number_of v') =
+ (if neg (number_of v :: int) then (number_of v' :: int) \<le> 0
+ else if neg (number_of v' :: int) then (number_of v :: int) = 0
+ else v = v')"
+ unfolding nat_number_of_def number_of_is_id neg_def
+ by auto
+
+
+subsubsection{*Less-than (<) *}
+
+lemma less_nat_number_of [simp]:
+ "(number_of v :: nat) < number_of v' \<longleftrightarrow>
+ (if v < v' then Int.Pls < v' else False)"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by auto
+
+
+subsubsection{*Less-than-or-equal *}
+
+lemma le_nat_number_of [simp]:
+ "(number_of v :: nat) \<le> number_of v' \<longleftrightarrow>
+ (if v \<le> v' then True else v \<le> Int.Pls)"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by auto
+
+(*Maps #n to n for n = 0, 1, 2*)
+lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2
+
+
+subsection{*Powers with Numeric Exponents*}
+
+text{*We cannot refer to the number @{term 2} in @{text Ring_and_Field.thy}.
+We cannot prove general results about the numeral @{term "-1"}, so we have to
+use @{term "- 1"} instead.*}
+
+lemma power2_eq_square: "(a::'a::recpower)\<twosuperior> = a * a"
+ by (simp add: numeral_2_eq_2 Power.power_Suc)
+
+lemma zero_power2 [simp]: "(0::'a::{semiring_1,recpower})\<twosuperior> = 0"
+ by (simp add: power2_eq_square)
+
+lemma one_power2 [simp]: "(1::'a::{semiring_1,recpower})\<twosuperior> = 1"
+ by (simp add: power2_eq_square)
+
+lemma power3_eq_cube: "(x::'a::recpower) ^ 3 = x * x * x"
+ apply (subgoal_tac "3 = Suc (Suc (Suc 0))")
+ apply (erule ssubst)
+ apply (simp add: power_Suc mult_ac)
+ apply (unfold nat_number_of_def)
+ apply (subst nat_eq_iff)
+ apply simp
+done
+
+text{*Squares of literal numerals will be evaluated.*}
+lemmas power2_eq_square_number_of =
+ power2_eq_square [of "number_of w", standard]
+declare power2_eq_square_number_of [simp]
+
+
+lemma zero_le_power2[simp]: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,recpower})"
+ by (simp add: power2_eq_square)
+
+lemma zero_less_power2[simp]:
+ "(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,recpower}))"
+ by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
+
+lemma power2_less_0[simp]:
+ fixes a :: "'a::{ordered_idom,recpower}"
+ shows "~ (a\<twosuperior> < 0)"
+by (force simp add: power2_eq_square mult_less_0_iff)
+
+lemma zero_eq_power2[simp]:
+ "(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))"
+ by (force simp add: power2_eq_square mult_eq_0_iff)
+
+lemma abs_power2[simp]:
+ "abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})"
+ by (simp add: power2_eq_square abs_mult abs_mult_self)
+
+lemma power2_abs[simp]:
+ "(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})"
+ by (simp add: power2_eq_square abs_mult_self)
+
+lemma power2_minus[simp]:
+ "(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})"
+ by (simp add: power2_eq_square)
+
+lemma power2_le_imp_le:
+ fixes x y :: "'a::{ordered_semidom,recpower}"
+ shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y"
+unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
+
+lemma power2_less_imp_less:
+ fixes x y :: "'a::{ordered_semidom,recpower}"
+ shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y"
+by (rule power_less_imp_less_base)
+
+lemma power2_eq_imp_eq:
+ fixes x y :: "'a::{ordered_semidom,recpower}"
+ shows "\<lbrakk>x\<twosuperior> = y\<twosuperior>; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> x = y"
+unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base, simp)
+
+lemma power_minus1_even[simp]: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})"
+proof (induct n)
+ case 0 show ?case by simp
+next
+ case (Suc n) then show ?case by (simp add: power_Suc power_add)
+qed
+
+lemma power_minus1_odd: "(- 1) ^ Suc(2*n) = -(1::'a::{comm_ring_1,recpower})"
+ by (simp add: power_Suc)
+
+lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2"
+by (subst mult_commute) (simp add: power_mult)
+
+lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"
+by (simp add: power_even_eq)
+
+lemma power_minus_even [simp]:
+ "(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)"
+by (simp add: power_minus1_even power_minus [of a])
+
+lemma zero_le_even_power'[simp]:
+ "0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)"
+proof (induct "n")
+ case 0
+ show ?case by (simp add: zero_le_one)
+next
+ case (Suc n)
+ have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
+ by (simp add: mult_ac power_add power2_eq_square)
+ thus ?case
+ by (simp add: prems zero_le_mult_iff)
+qed
+
+lemma odd_power_less_zero:
+ "(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0"
+proof (induct "n")
+ case 0
+ then show ?case by simp
+next
+ case (Suc n)
+ have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
+ by (simp add: mult_ac power_add power2_eq_square)
+ thus ?case
+ by (simp del: power_Suc add: prems mult_less_0_iff mult_neg_neg)
+qed
+
+lemma odd_0_le_power_imp_0_le:
+ "0 \<le> a ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})"
+apply (insert odd_power_less_zero [of a n])
+apply (force simp add: linorder_not_less [symmetric])
+done
+
+text{*Simprules for comparisons where common factors can be cancelled.*}
+lemmas zero_compare_simps =
+ add_strict_increasing add_strict_increasing2 add_increasing
+ zero_le_mult_iff zero_le_divide_iff
+ zero_less_mult_iff zero_less_divide_iff
+ mult_le_0_iff divide_le_0_iff
+ mult_less_0_iff divide_less_0_iff
+ zero_le_power2 power2_less_0
+
+subsubsection{*Nat *}
+
+lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
+by (simp add: numerals)
+
+(*Expresses a natural number constant as the Suc of another one.
+ NOT suitable for rewriting because n recurs in the condition.*)
+lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
+
+subsubsection{*Arith *}
+
+lemma Suc_eq_add_numeral_1: "Suc n = n + 1"
+by (simp add: numerals)
+
+lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n"
+by (simp add: numerals)
+
+(* These two can be useful when m = number_of... *)
+
+lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
+ unfolding One_nat_def by (cases m) simp_all
+
+lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
+ unfolding One_nat_def by (cases m) simp_all
+
+lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
+ unfolding One_nat_def by (cases m) simp_all
+
+
+subsection{*Comparisons involving (0::nat) *}
+
+text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
+
+lemma eq_number_of_0 [simp]:
+ "number_of v = (0::nat) \<longleftrightarrow> v \<le> Int.Pls"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by auto
+
+lemma eq_0_number_of [simp]:
+ "(0::nat) = number_of v \<longleftrightarrow> v \<le> Int.Pls"
+by (rule trans [OF eq_sym_conv eq_number_of_0])
+
+lemma less_0_number_of [simp]:
+ "(0::nat) < number_of v \<longleftrightarrow> Int.Pls < v"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by simp
+
+lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
+by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric])
+
+
+
+subsection{*Comparisons involving @{term Suc} *}
+
+lemma eq_number_of_Suc [simp]:
+ "(number_of v = Suc n) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then False else nat pv = n)"
+apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
+ number_of_pred nat_number_of_def
+ split add: split_if)
+apply (rule_tac x = "number_of v" in spec)
+apply (auto simp add: nat_eq_iff)
+done
+
+lemma Suc_eq_number_of [simp]:
+ "(Suc n = number_of v) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then False else nat pv = n)"
+by (rule trans [OF eq_sym_conv eq_number_of_Suc])
+
+lemma less_number_of_Suc [simp]:
+ "(number_of v < Suc n) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then True else nat pv < n)"
+apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
+ number_of_pred nat_number_of_def
+ split add: split_if)
+apply (rule_tac x = "number_of v" in spec)
+apply (auto simp add: nat_less_iff)
+done
+
+lemma less_Suc_number_of [simp]:
+ "(Suc n < number_of v) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then False else n < nat pv)"
+apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
+ number_of_pred nat_number_of_def
+ split add: split_if)
+apply (rule_tac x = "number_of v" in spec)
+apply (auto simp add: zless_nat_eq_int_zless)
+done
+
+lemma le_number_of_Suc [simp]:
+ "(number_of v <= Suc n) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then True else nat pv <= n)"
+by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric])
+
+lemma le_Suc_number_of [simp]:
+ "(Suc n <= number_of v) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then False else n <= nat pv)"
+by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric])
+
+
+lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Int.Min"
+by auto
+
+
+
+subsection{*Max and Min Combined with @{term Suc} *}
+
+lemma max_number_of_Suc [simp]:
+ "max (Suc n) (number_of v) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then Suc n else Suc(max n (nat pv)))"
+apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
+ split add: split_if nat.split)
+apply (rule_tac x = "number_of v" in spec)
+apply auto
+done
+
+lemma max_Suc_number_of [simp]:
+ "max (number_of v) (Suc n) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then Suc n else Suc(max (nat pv) n))"
+apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
+ split add: split_if nat.split)
+apply (rule_tac x = "number_of v" in spec)
+apply auto
+done
+
+lemma min_number_of_Suc [simp]:
+ "min (Suc n) (number_of v) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then 0 else Suc(min n (nat pv)))"
+apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
+ split add: split_if nat.split)
+apply (rule_tac x = "number_of v" in spec)
+apply auto
+done
+
+lemma min_Suc_number_of [simp]:
+ "min (number_of v) (Suc n) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then 0 else Suc(min (nat pv) n))"
+apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
+ split add: split_if nat.split)
+apply (rule_tac x = "number_of v" in spec)
+apply auto
+done
+
+subsection{*Literal arithmetic involving powers*}
+
+lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n"
+apply (induct "n")
+apply (simp_all (no_asm_simp) add: nat_mult_distrib)
+done
+
+lemma power_nat_number_of:
+ "(number_of v :: nat) ^ n =
+ (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
+by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
+ split add: split_if cong: imp_cong)
+
+
+lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]
+declare power_nat_number_of_number_of [simp]
+
+
+
+text{*For arbitrary rings*}
+
+lemma power_number_of_even:
+ fixes z :: "'a::{number_ring,recpower}"
+ shows "z ^ number_of (Int.Bit0 w) = (let w = z ^ (number_of w) in w * w)"
+unfolding Let_def nat_number_of_def number_of_Bit0
+apply (rule_tac x = "number_of w" in spec, clarify)
+apply (case_tac " (0::int) <= x")
+apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square)
+done
+
+lemma power_number_of_odd:
+ fixes z :: "'a::{number_ring,recpower}"
+ shows "z ^ number_of (Int.Bit1 w) = (if (0::int) <= number_of w
+ then (let w = z ^ (number_of w) in z * w * w) else 1)"
+unfolding Let_def nat_number_of_def number_of_Bit1
+apply (rule_tac x = "number_of w" in spec, auto)
+apply (simp only: nat_add_distrib nat_mult_distrib)
+apply simp
+apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat power_Suc)
+done
+
+lemmas zpower_number_of_even = power_number_of_even [where 'a=int]
+lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int]
+
+lemmas power_number_of_even_number_of [simp] =
+ power_number_of_even [of "number_of v", standard]
+
+lemmas power_number_of_odd_number_of [simp] =
+ power_number_of_odd [of "number_of v", standard]
+
+
+
+ML
+{*
+val numeral_ss = @{simpset} addsimps @{thms numerals};
+
+val nat_bin_arith_setup =
+ Lin_Arith.map_data
+ (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
+ {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
+ inj_thms = inj_thms,
+ lessD = lessD, neqE = neqE,
+ simpset = simpset addsimps @{thms neg_simps} @
+ [@{thm Suc_nat_number_of}, @{thm int_nat_number_of}]})
+*}
+
+declaration {* K nat_bin_arith_setup *}
+
+(* Enable arith to deal with div/mod k where k is a numeral: *)
+declare split_div[of _ _ "number_of k", standard, arith_split]
+declare split_mod[of _ _ "number_of k", standard, arith_split]
+
+lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
+ by (simp add: number_of_Pls nat_number_of_def)
+
+lemma nat_number_of_Min: "number_of Int.Min = (0::nat)"
+ apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
+ done
+
+lemma nat_number_of_Bit0:
+ "number_of (Int.Bit0 w) = (let n::nat = number_of w in n + n)"
+ unfolding nat_number_of_def number_of_is_id numeral_simps Let_def
+ by auto
+
+lemma nat_number_of_Bit1:
+ "number_of (Int.Bit1 w) =
+ (if neg (number_of w :: int) then 0
+ else let n = number_of w in Suc (n + n))"
+ unfolding nat_number_of_def number_of_is_id numeral_simps neg_def Let_def
+ by auto
+
+lemmas nat_number =
+ nat_number_of_Pls nat_number_of_Min
+ nat_number_of_Bit0 nat_number_of_Bit1
+
+lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
+ by (simp add: Let_def)
+
+lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})"
+by (simp add: power_mult power_Suc);
+
+lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})"
+by (simp add: power_mult power_Suc);
+
+
+subsection{*Literal arithmetic and @{term of_nat}*}
+
+lemma of_nat_double:
+ "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
+by (simp only: mult_2 nat_add_distrib of_nat_add)
+
+lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
+by (simp only: nat_number_of_def)
+
+lemma of_nat_number_of_lemma:
+ "of_nat (number_of v :: nat) =
+ (if 0 \<le> (number_of v :: int)
+ then (number_of v :: 'a :: number_ring)
+ else 0)"
+by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat);
+
+lemma of_nat_number_of_eq [simp]:
+ "of_nat (number_of v :: nat) =
+ (if neg (number_of v :: int) then 0
+ else (number_of v :: 'a :: number_ring))"
+by (simp only: of_nat_number_of_lemma neg_def, simp)
+
+
+subsection {*Lemmas for the Combination and Cancellation Simprocs*}
+
+lemma nat_number_of_add_left:
+ "number_of v + (number_of v' + (k::nat)) =
+ (if neg (number_of v :: int) then number_of v' + k
+ else if neg (number_of v' :: int) then number_of v + k
+ else number_of (v + v') + k)"
+ unfolding nat_number_of_def number_of_is_id neg_def
+ by auto
+
+lemma nat_number_of_mult_left:
+ "number_of v * (number_of v' * (k::nat)) =
+ (if v < Int.Pls then 0
+ else number_of (v * v') * k)"
+by simp
+
+
+subsubsection{*For @{text combine_numerals}*}
+
+lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
+by (simp add: add_mult_distrib)
+
+
+subsubsection{*For @{text cancel_numerals}*}
+
+lemma nat_diff_add_eq1:
+ "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
+by (simp split add: nat_diff_split add: add_mult_distrib)
+
+lemma nat_diff_add_eq2:
+ "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
+by (simp split add: nat_diff_split add: add_mult_distrib)
+
+lemma nat_eq_add_iff1:
+ "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
+by (auto split add: nat_diff_split simp add: add_mult_distrib)
+
+lemma nat_eq_add_iff2:
+ "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
+by (auto split add: nat_diff_split simp add: add_mult_distrib)
+
+lemma nat_less_add_iff1:
+ "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
+by (auto split add: nat_diff_split simp add: add_mult_distrib)
+
+lemma nat_less_add_iff2:
+ "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
+by (auto split add: nat_diff_split simp add: add_mult_distrib)
+
+lemma nat_le_add_iff1:
+ "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
+by (auto split add: nat_diff_split simp add: add_mult_distrib)
+
+lemma nat_le_add_iff2:
+ "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
+by (auto split add: nat_diff_split simp add: add_mult_distrib)
+
+
+subsubsection{*For @{text cancel_numeral_factors} *}
+
+lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
+by auto
+
+lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
+by auto
+
+lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
+by auto
+
+lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
+by auto
+
+lemma nat_mult_dvd_cancel_disj[simp]:
+ "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
+by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])
+
+lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
+by(auto)
+
+
+subsubsection{*For @{text cancel_factor} *}
+
+lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
+by auto
+
+lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
+by auto
+
+lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
+by auto
+
+lemma nat_mult_div_cancel_disj[simp]:
+ "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
+by (simp add: nat_mult_div_cancel1)
+
+
+subsection {* Simprocs for the Naturals *}
+
+use "Tools/nat_simprocs.ML"
+declaration {* K nat_simprocs_setup *}
+
+subsubsection{*For simplifying @{term "Suc m - K"} and @{term "K - Suc m"}*}
+
+text{*Where K above is a literal*}
+
+lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
+by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split)
+
+text {*Now just instantiating @{text n} to @{text "number_of v"} does
+ the right simplification, but with some redundant inequality
+ tests.*}
+lemma neg_number_of_pred_iff_0:
+ "neg (number_of (Int.pred v)::int) = (number_of v = (0::nat))"
+apply (subgoal_tac "neg (number_of (Int.pred v)) = (number_of v < Suc 0) ")
+apply (simp only: less_Suc_eq_le le_0_eq)
+apply (subst less_number_of_Suc, simp)
+done
+
+text{*No longer required as a simprule because of the @{text inverse_fold}
+ simproc*}
+lemma Suc_diff_number_of:
+ "Int.Pls < v ==>
+ Suc m - (number_of v) = m - (number_of (Int.pred v))"
+apply (subst Suc_diff_eq_diff_pred)
+apply simp
+apply (simp del: nat_numeral_1_eq_1)
+apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
+ neg_number_of_pred_iff_0)
+done
+
+lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
+by (simp add: numerals split add: nat_diff_split)
+
+
+subsubsection{*For @{term nat_case} and @{term nat_rec}*}
+
+lemma nat_case_number_of [simp]:
+ "nat_case a f (number_of v) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then a else f (nat pv))"
+by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)
+
+lemma nat_case_add_eq_if [simp]:
+ "nat_case a f ((number_of v) + n) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then nat_case a f n else f (nat pv + n))"
+apply (subst add_eq_if)
+apply (simp split add: nat.split
+ del: nat_numeral_1_eq_1
+ add: nat_numeral_1_eq_1 [symmetric]
+ numeral_1_eq_Suc_0 [symmetric]
+ neg_number_of_pred_iff_0)
+done
+
+lemma nat_rec_number_of [simp]:
+ "nat_rec a f (number_of v) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
+apply (case_tac " (number_of v) ::nat")
+apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
+apply (simp split add: split_if_asm)
+done
+
+lemma nat_rec_add_eq_if [simp]:
+ "nat_rec a f (number_of v + n) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then nat_rec a f n
+ else f (nat pv + n) (nat_rec a f (nat pv + n)))"
+apply (subst add_eq_if)
+apply (simp split add: nat.split
+ del: nat_numeral_1_eq_1
+ add: nat_numeral_1_eq_1 [symmetric]
+ numeral_1_eq_Suc_0 [symmetric]
+ neg_number_of_pred_iff_0)
+done
+
+
+subsubsection{*Various Other Lemmas*}
+
+text {*Evens and Odds, for Mutilated Chess Board*}
+
+text{*Lemmas for specialist use, NOT as default simprules*}
+lemma nat_mult_2: "2 * z = (z+z::nat)"
+proof -
+ have "2*z = (1 + 1)*z" by simp
+ also have "... = z+z" by (simp add: left_distrib)
+ finally show ?thesis .
+qed
+
+lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
+by (subst mult_commute, rule nat_mult_2)
+
+text{*Case analysis on @{term "n<2"}*}
+lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
+by arith
+
+lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)"
+by arith
+
+lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
+by (simp add: nat_mult_2 [symmetric])
+
+lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
+apply (subgoal_tac "m mod 2 < 2")
+apply (erule less_2_cases [THEN disjE])
+apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
+done
+
+lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)"
+apply (subgoal_tac "m mod 2 < 2")
+apply (force simp del: mod_less_divisor, simp)
+done
+
+text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
+
+lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
+by simp
+
+lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
+by simp
+
+text{*Can be used to eliminate long strings of Sucs, but not by default*}
+lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
+by simp
+
+
+text{*These lemmas collapse some needless occurrences of Suc:
+ at least three Sucs, since two and fewer are rewritten back to Suc again!
+ We already have some rules to simplify operands smaller than 3.*}
+
+lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
+by (simp add: Suc3_eq_add_3)
+
+lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
+by (simp add: Suc3_eq_add_3)
+
+lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
+by (simp add: Suc3_eq_add_3)
+
+lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
+by (simp add: Suc3_eq_add_3)
+
+lemmas Suc_div_eq_add3_div_number_of =
+ Suc_div_eq_add3_div [of _ "number_of v", standard]
+declare Suc_div_eq_add3_div_number_of [simp]
+
+lemmas Suc_mod_eq_add3_mod_number_of =
+ Suc_mod_eq_add3_mod [of _ "number_of v", standard]
+declare Suc_mod_eq_add3_mod_number_of [simp]
+
+end