src/HOL/Library/Parity.thy

-(*  Title:      Parity.thy
+(*  Title:      HOL/Library/Parity.thy
     ID:         $Id$
     Author:     Jeremy Avigad
 *)
 
 header {* Even and Odd for int and nat *}

   "odd x == \<not> even x"
 
 
 subsection {* Even and odd are mutually exclusive *}
 
-lemma int_pos_lt_two_imp_zero_or_one: 
+lemma int_pos_lt_two_imp_zero_or_one:
     "0 <= x ==> (x::int) < 2 ==> x = 0 | x = 1"
   by auto
 
 lemma neq_one_mod_two [simp]: "((x::int) mod 2 ~= 0) = (x mod 2 = 1)"
-  apply (subgoal_tac "x mod 2 = 0 | x mod 2 = 1", force)
-  apply (rule int_pos_lt_two_imp_zero_or_one, auto)
-  done
+proof -
+  have "x mod 2 = 0 | x mod 2 = 1"
+    by (rule int_pos_lt_two_imp_zero_or_one) auto
+  then show ?thesis by force
+qed
+
 
 subsection {* Behavior under integer arithmetic operations *}
 
 lemma even_times_anything: "even (x::int) ==> even (x * y)"
   by (simp add: even_def zmod_zmult1_eq')

 
 lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)"
   by (simp add: even_def zmod_zmult1_eq)
 
 lemma even_product: "even((x::int) * y) = (even x | even y)"
-  apply (auto simp add: even_times_anything anything_times_even) 
+  apply (auto simp add: even_times_anything anything_times_even)
   apply (rule ccontr)
   apply (auto simp add: odd_times_odd)
   done
 
 lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)"

   done
 
 lemma even_neg: "even (-(x::int)) = even x"
   by (auto simp add: even_def zmod_zminus1_eq_if)
 
-lemma even_difference: 
-  "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))"
+lemma even_difference:
+    "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))"
   by (simp only: diff_minus even_sum even_neg)
 
-lemma even_pow_gt_zero [rule_format]: 
-    "even (x::int) ==> 0 < n --> even (x^n)"
-  apply (induct n)
-  apply (auto simp add: even_product)
-  done
+lemma even_pow_gt_zero:
+    "even (x::int) ==> 0 < n ==> even (x^n)"
+  by (induct n) (auto simp add: even_product)
 
 lemma odd_pow: "odd x ==> odd((x::int)^n)"
   apply (induct n)
-  apply (simp add: even_def)
+   apply (simp add: even_def)
   apply (simp add: even_product)
   done
 
 lemma even_power: "even ((x::int)^n) = (even x & 0 < n)"
-  apply (auto simp add: even_pow_gt_zero) 
+  apply (auto simp add: even_pow_gt_zero)
   apply (erule contrapos_pp, erule odd_pow)
   apply (erule contrapos_pp, simp add: even_def)
   done
 
 lemma even_zero: "even (0::int)"
   by (simp add: even_def)
 
 lemma odd_one: "odd (1::int)"
   by (simp add: even_def)
 
-lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero 
+lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero
   odd_one even_product even_sum even_neg even_difference even_power
 
 
 subsection {* Equivalent definitions *}
 
-lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x" 
+lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x"
   by (auto simp add: even_def)
 
-lemma two_times_odd_div_two_plus_one: "odd (x::int) ==> 
+lemma two_times_odd_div_two_plus_one: "odd (x::int) ==>
     2 * (x div 2) + 1 = x"
-  apply (insert zmod_zdiv_equality [of x 2, THEN sym])
-  by (simp add: even_def)
+  apply (insert zmod_zdiv_equality [of x 2, symmetric])
+  apply (simp add: even_def)
+  done
 
 lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)"
   apply auto
   apply (rule exI)
-  by (erule two_times_even_div_two [THEN sym])
+  apply (erule two_times_even_div_two [symmetric])
+  done
 
 lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)"
   apply auto
   apply (rule exI)
-  by (erule two_times_odd_div_two_plus_one [THEN sym])
+  apply (erule two_times_odd_div_two_plus_one [symmetric])
+  done
 
 
 subsection {* even and odd for nats *}
 
 lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)"
   by (simp add: even_nat_def)
 
 lemma even_nat_product: "even((x::nat) * y) = (even x | even y)"
   by (simp add: even_nat_def int_mult)
 
-lemma even_nat_sum: "even ((x::nat) + y) = 
+lemma even_nat_sum: "even ((x::nat) + y) =
     ((even x & even y) | (odd x & odd y))"
   by (unfold even_nat_def, simp)
 
-lemma even_nat_difference: 
+lemma even_nat_difference:
     "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"
-  apply (auto simp add: even_nat_def zdiff_int [THEN sym])
-  apply (case_tac "x < y", auto simp add: zdiff_int [THEN sym])
-  apply (case_tac "x < y", auto simp add: zdiff_int [THEN sym])
+  apply (auto simp add: even_nat_def zdiff_int [symmetric])
+  apply (case_tac "x < y", auto simp add: zdiff_int [symmetric])
+  apply (case_tac "x < y", auto simp add: zdiff_int [symmetric])
   done
 
 lemma even_nat_Suc: "even (Suc x) = odd x"
   by (simp add: even_nat_def)
 

   by (simp add: even_nat_def int_power)
 
 lemma even_nat_zero: "even (0::nat)"
   by (simp add: even_nat_def)
 
-lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard] 
+lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard]
   even_nat_zero even_nat_Suc even_nat_product even_nat_sum even_nat_power
 
 
 subsection {* Equivalent definitions *}
 
-lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==> 
+lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==>
     x = 0 | x = Suc 0"
   by auto
 
 lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"
-  apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])
+  apply (insert mod_div_equality [of x "Suc (Suc 0)", symmetric])
   apply (drule subst, assumption)
   apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0")
   apply force
   apply (subgoal_tac "0 < Suc (Suc 0)")
   apply (frule mod_less_divisor [of "Suc (Suc 0)" x])
   apply (erule nat_lt_two_imp_zero_or_one, auto)
   done
 
 lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"
-  apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])
+  apply (insert mod_div_equality [of x "Suc (Suc 0)", symmetric])
   apply (drule subst, assumption)
   apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0")
-  apply force 
+  apply force
   apply (subgoal_tac "0 < Suc (Suc 0)")
   apply (frule mod_less_divisor [of "Suc (Suc 0)" x])
   apply (erule nat_lt_two_imp_zero_or_one, auto)
   done
 
-lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)" 
+lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)"
   apply (rule iffI)
   apply (erule even_nat_mod_two_eq_zero)
   apply (insert odd_nat_mod_two_eq_one [of x], auto)
   done
 

   apply (auto simp add: even_nat_equiv_def)
   apply (subgoal_tac "x mod (Suc (Suc 0)) < Suc (Suc 0)")
   apply (frule nat_lt_two_imp_zero_or_one, auto)
   done
 
-lemma even_nat_div_two_times_two: "even (x::nat) ==> 
+lemma even_nat_div_two_times_two: "even (x::nat) ==>
     Suc (Suc 0) * (x div Suc (Suc 0)) = x"
-  apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])
+  apply (insert mod_div_equality [of x "Suc (Suc 0)", symmetric])
   apply (drule even_nat_mod_two_eq_zero, simp)
   done
 
-lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==> 
-    Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x"  
-  apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])
+lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==>
+    Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x"
+  apply (insert mod_div_equality [of x "Suc (Suc 0)", symmetric])
   apply (drule odd_nat_mod_two_eq_one, simp)
   done
 
 lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)"
   apply (rule iffI, rule exI)
-  apply (erule even_nat_div_two_times_two [THEN sym], auto)
+  apply (erule even_nat_div_two_times_two [symmetric], auto)
   done
 
 lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"
   apply (rule iffI, rule exI)
-  apply (erule odd_nat_div_two_times_two_plus_one [THEN sym], auto)
+  apply (erule odd_nat_div_two_times_two_plus_one [symmetric], auto)
   done
 
 subsection {* Parity and powers *}
 
-lemma minus_one_even_odd_power:
-     "(even x --> (- 1::'a::{comm_ring_1,recpower})^x = 1) & 
+lemma  minus_one_even_odd_power:
+     "(even x --> (- 1::'a::{comm_ring_1,recpower})^x = 1) &
       (odd x --> (- 1::'a)^x = - 1)"
   apply (induct x)
   apply (rule conjI)
   apply simp
   apply (insert even_nat_zero, blast)
   apply (simp add: power_Suc)
-done
+  done
 
 lemma minus_one_even_power [simp]:
-     "even x ==> (- 1::'a::{comm_ring_1,recpower})^x = 1"
-  by (rule minus_one_even_odd_power [THEN conjunct1, THEN mp], assumption)
+    "even x ==> (- 1::'a::{comm_ring_1,recpower})^x = 1"
+  using minus_one_even_odd_power by blast
 
 lemma minus_one_odd_power [simp]:
-     "odd x ==> (- 1::'a::{comm_ring_1,recpower})^x = - 1"
-  by (rule minus_one_even_odd_power [THEN conjunct2, THEN mp], assumption)
+    "odd x ==> (- 1::'a::{comm_ring_1,recpower})^x = - 1"
+  using minus_one_even_odd_power by blast
 
 lemma neg_one_even_odd_power:
-     "(even x --> (-1::'a::{number_ring,recpower})^x = 1) & 
+     "(even x --> (-1::'a::{number_ring,recpower})^x = 1) &
       (odd x --> (-1::'a)^x = -1)"
   apply (induct x)
   apply (simp, simp add: power_Suc)
   done
 
 lemma neg_one_even_power [simp]:
-     "even x ==> (-1::'a::{number_ring,recpower})^x = 1"
-  by (rule neg_one_even_odd_power [THEN conjunct1, THEN mp], assumption)
+    "even x ==> (-1::'a::{number_ring,recpower})^x = 1"
+  using neg_one_even_odd_power by blast
 
 lemma neg_one_odd_power [simp]:
-     "odd x ==> (-1::'a::{number_ring,recpower})^x = -1"
-  by (rule neg_one_even_odd_power [THEN conjunct2, THEN mp], assumption)
+    "odd x ==> (-1::'a::{number_ring,recpower})^x = -1"
+  using neg_one_even_odd_power by blast
 
 lemma neg_power_if:
-     "(-x::'a::{comm_ring_1,recpower}) ^ n = 
+     "(-x::'a::{comm_ring_1,recpower}) ^ n =
       (if even n then (x ^ n) else -(x ^ n))"
-  by (induct n, simp_all split: split_if_asm add: power_Suc) 
-
-lemma zero_le_even_power: "even n ==> 
+  apply (induct n)
+  apply (simp_all split: split_if_asm add: power_Suc)
+  done
+
+lemma zero_le_even_power: "even n ==>
     0 <= (x::'a::{recpower,ordered_ring_strict}) ^ n"
   apply (simp add: even_nat_equiv_def2)
   apply (erule exE)
   apply (erule ssubst)
   apply (subst power_add)
   apply (rule zero_le_square)
   done
 
-lemma zero_le_odd_power: "odd n ==> 
+lemma zero_le_odd_power: "odd n ==>
     (0 <= (x::'a::{recpower,ordered_idom}) ^ n) = (0 <= x)"
   apply (simp add: odd_nat_equiv_def2)
   apply (erule exE)
   apply (erule ssubst)
   apply (subst power_Suc)
   apply (subst power_add)
   apply (subst zero_le_mult_iff)
   apply auto
   apply (subgoal_tac "x = 0 & 0 < y")
   apply (erule conjE, assumption)
-  apply (subst power_eq_0_iff [THEN sym])
+  apply (subst power_eq_0_iff [symmetric])
   apply (subgoal_tac "0 <= x^y * x^y")
   apply simp
   apply (rule zero_le_square)+
-done
-
-lemma zero_le_power_eq: "(0 <= (x::'a::{recpower,ordered_idom}) ^ n) = 
+  done
+
+lemma zero_le_power_eq: "(0 <= (x::'a::{recpower,ordered_idom}) ^ n) =
     (even n | (odd n & 0 <= x))"
   apply auto
-  apply (subst zero_le_odd_power [THEN sym])
+  apply (subst zero_le_odd_power [symmetric])
   apply assumption+
   apply (erule zero_le_even_power)
-  apply (subst zero_le_odd_power) 
+  apply (subst zero_le_odd_power)
   apply assumption+
-done
-
-lemma zero_less_power_eq: "(0 < (x::'a::{recpower,ordered_idom}) ^ n) = 
+  done
+
+lemma zero_less_power_eq: "(0 < (x::'a::{recpower,ordered_idom}) ^ n) =
     (n = 0 | (even n & x ~= 0) | (odd n & 0 < x))"
   apply (rule iffI)
   apply clarsimp
   apply (rule conjI)
   apply clarsimp
   apply (rule ccontr)
   apply (subgoal_tac "~ (0 <= x^n)")
   apply simp
   apply (subst zero_le_odd_power)
-  apply assumption 
+  apply assumption
   apply simp
   apply (rule notI)
   apply (simp add: power_0_left)
   apply (rule notI)
   apply (simp add: power_0_left)

   apply (frule order_le_imp_less_or_eq)
   apply auto
   apply (subst zero_le_odd_power)
   apply assumption
   apply (erule order_less_imp_le)
-done
+  done
 
 lemma power_less_zero_eq: "((x::'a::{recpower,ordered_idom}) ^ n < 0) =
-    (odd n & x < 0)" 
-  apply (subst linorder_not_le [THEN sym])+
+    (odd n & x < 0)"
+  apply (subst linorder_not_le [symmetric])+
   apply (subst zero_le_power_eq)
   apply auto
-done
+  done
 
 lemma power_le_zero_eq: "((x::'a::{recpower,ordered_idom}) ^ n <= 0) =
     (n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))"
-  apply (subst linorder_not_less [THEN sym])+
+  apply (subst linorder_not_less [symmetric])+
   apply (subst zero_less_power_eq)
   apply auto
-done
-
-lemma power_even_abs: "even n ==> 
+  done
+
+lemma power_even_abs: "even n ==>
     (abs (x::'a::{recpower,ordered_idom}))^n = x^n"
-  apply (subst power_abs [THEN sym])
+  apply (subst power_abs [symmetric])
   apply (simp add: zero_le_even_power)
-done
+  done
 
 lemma zero_less_power_nat_eq: "(0 < (x::nat) ^ n) = (n = 0 | 0 < x)"
-  by (induct n, auto)
-
-lemma power_minus_even [simp]: "even n ==> 
+  by (induct n) auto
+
+lemma power_minus_even [simp]: "even n ==>
     (- x)^n = (x^n::'a::{recpower,comm_ring_1})"
   apply (subst power_minus)
   apply simp
-done
-
-lemma power_minus_odd [simp]: "odd n ==> 
+  done
+
+lemma power_minus_odd [simp]: "odd n ==>
     (- x)^n = - (x^n::'a::{recpower,comm_ring_1})"
   apply (subst power_minus)
   apply simp
-done
-
-(* Simplify, when the exponent is a numeral *)
+  done
+
+
+text {* Simplify, when the exponent is a numeral *}
 
 lemmas power_0_left_number_of = power_0_left [of "number_of w", standard]
 declare power_0_left_number_of [simp]
 
-lemmas zero_le_power_eq_number_of =
+lemmas zero_le_power_eq_number_of [simp] =
     zero_le_power_eq [of _ "number_of w", standard]
-declare zero_le_power_eq_number_of [simp]
-
-lemmas zero_less_power_eq_number_of =
+
+lemmas zero_less_power_eq_number_of [simp] =
     zero_less_power_eq [of _ "number_of w", standard]
-declare zero_less_power_eq_number_of [simp]
-
-lemmas power_le_zero_eq_number_of =
+
+lemmas power_le_zero_eq_number_of [simp] =
     power_le_zero_eq [of _ "number_of w", standard]
-declare power_le_zero_eq_number_of [simp]
-
-lemmas power_less_zero_eq_number_of =
+
+lemmas power_less_zero_eq_number_of [simp] =
     power_less_zero_eq [of _ "number_of w", standard]
-declare power_less_zero_eq_number_of [simp]
-
-lemmas zero_less_power_nat_eq_number_of =
+
+lemmas zero_less_power_nat_eq_number_of [simp] =
     zero_less_power_nat_eq [of _ "number_of w", standard]
-declare zero_less_power_nat_eq_number_of [simp]
-
-lemmas power_eq_0_iff_number_of = power_eq_0_iff [of _ "number_of w", standard]
-declare power_eq_0_iff_number_of [simp]
-
-lemmas power_even_abs_number_of = power_even_abs [of "number_of w" _, standard]
-declare power_even_abs_number_of [simp]
+
+lemmas power_eq_0_iff_number_of [simp] = power_eq_0_iff [of _ "number_of w", standard]
+
+lemmas power_even_abs_number_of [simp] = power_even_abs [of "number_of w" _, standard]
 
 
 subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *}
 
 lemma even_power_le_0_imp_0:
-     "a ^ (2*k) \<le> (0::'a::{ordered_idom,recpower}) ==> a=0"
-apply (induct k) 
-apply (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc)  
-done
+    "a ^ (2*k) \<le> (0::'a::{ordered_idom,recpower}) ==> a=0"
+  by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc)
 
 lemma zero_le_power_iff:
-     "(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom,recpower}) | even n)"
-      (is "?P n")
+  "(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom,recpower}) | even n)"
 proof cases
   assume even: "even n"
   then obtain k where "n = 2*k"
     by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2)
-  thus ?thesis by (simp add: zero_le_even_power even) 
+  thus ?thesis by (simp add: zero_le_even_power even)
 next
   assume odd: "odd n"
   then obtain k where "n = Suc(2*k)"
     by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2)
   thus ?thesis
-    by (auto simp add: power_Suc zero_le_mult_iff zero_le_even_power 
-             dest!: even_power_le_0_imp_0) 
-qed 
+    by (auto simp add: power_Suc zero_le_mult_iff zero_le_even_power
+             dest!: even_power_le_0_imp_0)
+qed
+
 
 subsection {* Miscellaneous *}
 
 lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2"
   apply (subst zdiv_zadd1_eq)

 lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1"
   apply (subst zdiv_zadd1_eq)
   apply (simp add: even_def)
   done
 
-lemma div_Suc: "Suc a div c = a div c + Suc 0 div c + 
+lemma div_Suc: "Suc a div c = a div c + Suc 0 div c +
     (a mod c + Suc 0 mod c) div c"
   apply (subgoal_tac "Suc a = a + Suc 0")
   apply (erule ssubst)
   apply (rule div_add1_eq, simp)
   done
 
-lemma even_nat_plus_one_div_two: "even (x::nat) ==> 
-   (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)"
+lemma even_nat_plus_one_div_two: "even (x::nat) ==>
+    (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)"
   apply (subst div_Suc)
   apply (simp add: even_nat_equiv_def)
   done
 
-lemma odd_nat_plus_one_div_two: "odd (x::nat) ==> 
+lemma odd_nat_plus_one_div_two: "odd (x::nat) ==>
     (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))"
   apply (subst div_Suc)
   apply (simp add: odd_nat_equiv_def)
   done