distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly

--- a/src/HOL/Decision_Procs/Dense_Linear_Order.thy	Sun Mar 22 11:56:32 2009 +0100
+++ b/src/HOL/Decision_Procs/Dense_Linear_Order.thy	Sun Mar 22 20:46:10 2009 +0100
@@ -299,7 +299,7 @@
 *} "Langford's algorithm for quantifier elimination in dense linear orders"
 
 
-section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields -- see @{text "Arith_Tools.thy"} *}
+section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields *}
 
 text {* Linear order without upper bounds *}
 

--- a/src/HOL/Int.thy	Sun Mar 22 11:56:32 2009 +0100
+++ b/src/HOL/Int.thy	Sun Mar 22 20:46:10 2009 +0100
@@ -1256,14 +1256,14 @@
 by (simp add: algebra_simps diff_number_of_eq [symmetric])
 
 
+
+
 subsection {* The Set of Integers *}
 
 context ring_1
 begin
 
-definition
-  Ints  :: "'a set"
-where
+definition Ints  :: "'a set" where
   [code del]: "Ints = range of_int"
 
 end
@@ -1854,7 +1854,7 @@
 qed
 
 
-subsection{*Integer Powers*} 
+subsection {* Integer Powers *} 
 
 instantiation int :: recpower
 begin
@@ -1896,6 +1896,161 @@
 
 lemmas zpower_int = int_power [symmetric]
 
+
+subsection {* Further theorems on numerals *}
+
+subsubsection{*Special Simplification for Constants*}
+
+text{*These distributive laws move literals inside sums and differences.*}
+
+lemmas left_distrib_number_of [simp] =
+  left_distrib [of _ _ "number_of v", standard]
+
+lemmas right_distrib_number_of [simp] =
+  right_distrib [of "number_of v", standard]
+
+lemmas left_diff_distrib_number_of [simp] =
+  left_diff_distrib [of _ _ "number_of v", standard]
+
+lemmas right_diff_distrib_number_of [simp] =
+  right_diff_distrib [of "number_of v", standard]
+
+text{*These are actually for fields, like real: but where else to put them?*}
+
+lemmas zero_less_divide_iff_number_of [simp, noatp] =
+  zero_less_divide_iff [of "number_of w", standard]
+
+lemmas divide_less_0_iff_number_of [simp, noatp] =
+  divide_less_0_iff [of "number_of w", standard]
+
+lemmas zero_le_divide_iff_number_of [simp, noatp] =
+  zero_le_divide_iff [of "number_of w", standard]
+
+lemmas divide_le_0_iff_number_of [simp, noatp] =
+  divide_le_0_iff [of "number_of w", standard]
+
+
+text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
+  strange, but then other simprocs simplify the quotient.*}
+
+lemmas inverse_eq_divide_number_of [simp] =
+  inverse_eq_divide [of "number_of w", standard]
+
+text {*These laws simplify inequalities, moving unary minus from a term
+into the literal.*}
+
+lemmas less_minus_iff_number_of [simp, noatp] =
+  less_minus_iff [of "number_of v", standard]
+
+lemmas le_minus_iff_number_of [simp, noatp] =
+  le_minus_iff [of "number_of v", standard]
+
+lemmas equation_minus_iff_number_of [simp, noatp] =
+  equation_minus_iff [of "number_of v", standard]
+
+lemmas minus_less_iff_number_of [simp, noatp] =
+  minus_less_iff [of _ "number_of v", standard]
+
+lemmas minus_le_iff_number_of [simp, noatp] =
+  minus_le_iff [of _ "number_of v", standard]
+
+lemmas minus_equation_iff_number_of [simp, noatp] =
+  minus_equation_iff [of _ "number_of v", standard]
+
+
+text{*To Simplify Inequalities Where One Side is the Constant 1*}
+
+lemma less_minus_iff_1 [simp,noatp]:
+  fixes b::"'b::{ordered_idom,number_ring}"
+  shows "(1 < - b) = (b < -1)"
+by auto
+
+lemma le_minus_iff_1 [simp,noatp]:
+  fixes b::"'b::{ordered_idom,number_ring}"
+  shows "(1 \<le> - b) = (b \<le> -1)"
+by auto
+
+lemma equation_minus_iff_1 [simp,noatp]:
+  fixes b::"'b::number_ring"
+  shows "(1 = - b) = (b = -1)"
+by (subst equation_minus_iff, auto)
+
+lemma minus_less_iff_1 [simp,noatp]:
+  fixes a::"'b::{ordered_idom,number_ring}"
+  shows "(- a < 1) = (-1 < a)"
+by auto
+
+lemma minus_le_iff_1 [simp,noatp]:
+  fixes a::"'b::{ordered_idom,number_ring}"
+  shows "(- a \<le> 1) = (-1 \<le> a)"
+by auto
+
+lemma minus_equation_iff_1 [simp,noatp]:
+  fixes a::"'b::number_ring"
+  shows "(- a = 1) = (a = -1)"
+by (subst minus_equation_iff, auto)
+
+
+text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
+
+lemmas mult_less_cancel_left_number_of [simp, noatp] =
+  mult_less_cancel_left [of "number_of v", standard]
+
+lemmas mult_less_cancel_right_number_of [simp, noatp] =
+  mult_less_cancel_right [of _ "number_of v", standard]
+
+lemmas mult_le_cancel_left_number_of [simp, noatp] =
+  mult_le_cancel_left [of "number_of v", standard]
+
+lemmas mult_le_cancel_right_number_of [simp, noatp] =
+  mult_le_cancel_right [of _ "number_of v", standard]
+
+
+text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
+
+lemmas le_divide_eq_number_of1 [simp] = le_divide_eq [of _ _ "number_of w", standard]
+lemmas divide_le_eq_number_of1 [simp] = divide_le_eq [of _ "number_of w", standard]
+lemmas less_divide_eq_number_of1 [simp] = less_divide_eq [of _ _ "number_of w", standard]
+lemmas divide_less_eq_number_of1 [simp] = divide_less_eq [of _ "number_of w", standard]
+lemmas eq_divide_eq_number_of1 [simp] = eq_divide_eq [of _ _ "number_of w", standard]
+lemmas divide_eq_eq_number_of1 [simp] = divide_eq_eq [of _ "number_of w", standard]
+
+
+subsubsection{*Optional Simplification Rules Involving Constants*}
+
+text{*Simplify quotients that are compared with a literal constant.*}
+
+lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard]
+lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard]
+lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard]
+lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard]
+lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard]
+lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard]
+
+
+text{*Not good as automatic simprules because they cause case splits.*}
+lemmas divide_const_simps =
+  le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of
+  divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of
+  le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
+
+text{*Division By @{text "-1"}*}
+
+lemma divide_minus1 [simp]:
+     "x/-1 = -(x::'a::{field,division_by_zero,number_ring})"
+by simp
+
+lemma minus1_divide [simp]:
+     "-1 / (x::'a::{field,division_by_zero,number_ring}) = - (1/x)"
+by (simp add: divide_inverse inverse_minus_eq)
+
+lemma half_gt_zero_iff:
+     "(0 < r/2) = (0 < (r::'a::{ordered_field,division_by_zero,number_ring}))"
+by auto
+
+lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2, standard]
+
+
 subsection {* Configuration of the code generator *}
 
 code_datatype Pls Min Bit0 Bit1 "number_of \<Colon> int \<Rightarrow> int"

--- a/src/HOL/IntDiv.thy	Sun Mar 22 11:56:32 2009 +0100
+++ b/src/HOL/IntDiv.thy	Sun Mar 22 20:46:10 2009 +0100
@@ -8,6 +8,10 @@
 
 theory IntDiv
 imports Int Divides FunDef
+uses
+  "~~/src/Provers/Arith/cancel_numeral_factor.ML"
+  "~~/src/Provers/Arith/extract_common_term.ML"
+  ("Tools/int_factor_simprocs.ML")
 begin
 
 definition divmod_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
@@ -920,9 +924,10 @@
 next
   assume "a\<noteq>0" and le_a: "0\<le>a"   
   hence a_pos: "1 \<le> a" by arith
-  hence one_less_a2: "1 < 2*a" by arith
+  hence one_less_a2: "1 < 2 * a" by arith
   hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
-    by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq)
+    unfolding mult_le_cancel_left
+    by (simp add: add1_zle_eq add_commute [of 1])
   with a_pos have "0 \<le> b mod a" by simp
   hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
     by (simp add: mod_pos_pos_trivial one_less_a2)
@@ -1357,6 +1362,11 @@
 qed
 
 
+subsection {* Simproc setup *}
+
+use "Tools/int_factor_simprocs.ML"
+
+
 subsection {* Code generation *}
 
 definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

--- a/src/HOL/NatBin.thy	Sun Mar 22 11:56:32 2009 +0100
+++ b/src/HOL/NatBin.thy	Sun Mar 22 20:46:10 2009 +0100
@@ -7,6 +7,7 @@
 
 theory NatBin
 imports IntDiv
+uses ("Tools/nat_simprocs.ML")
 begin
 
 text {*
@@ -40,9 +41,7 @@
 
 subsection {* Predicate for negative binary numbers *}
 
-definition
-  neg  :: "int \<Rightarrow> bool"
-where
+definition neg  :: "int \<Rightarrow> bool" where
   "neg Z \<longleftrightarrow> Z < 0"
 
 lemma not_neg_int [simp]: "~ neg (of_nat n)"
@@ -818,4 +817,159 @@
      "(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