--- a/src/HOL/Real/RealPow.thy Wed Dec 03 09:53:58 2008 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,282 +0,0 @@
-(* Title : HOL/Real/RealPow.thy
- ID : $Id$
- Author : Jacques D. Fleuriot
- Copyright : 1998 University of Cambridge
-*)
-
-header {* Natural powers theory *}
-
-theory RealPow
-imports RealDef
-uses ("float_syntax.ML")
-begin
-
-declare abs_mult_self [simp]
-
-instantiation real :: recpower
-begin
-
-primrec power_real where
- realpow_0: "r ^ 0 = (1\<Colon>real)"
- | realpow_Suc: "r ^ Suc n = (r\<Colon>real) * r ^ n"
-
-instance proof
- fix z :: real
- fix n :: nat
- show "z^0 = 1" by simp
- show "z^(Suc n) = z * (z^n)" by simp
-qed
-
-end
-
-
-lemma two_realpow_ge_one [simp]: "(1::real) \<le> 2 ^ n"
-by simp
-
-lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n"
-apply (induct "n")
-apply (auto simp add: real_of_nat_Suc)
-apply (subst mult_2)
-apply (rule add_less_le_mono)
-apply (auto simp add: two_realpow_ge_one)
-done
-
-lemma realpow_Suc_le_self: "[| 0 \<le> r; r \<le> (1::real) |] ==> r ^ Suc n \<le> r"
-by (insert power_decreasing [of 1 "Suc n" r], simp)
-
-lemma realpow_minus_mult [rule_format]:
- "0 < n --> (x::real) ^ (n - 1) * x = x ^ n"
-apply (simp split add: nat_diff_split)
-done
-
-lemma realpow_two_mult_inverse [simp]:
- "r \<noteq> 0 ==> r * inverse r ^Suc (Suc 0) = inverse (r::real)"
-by (simp add: real_mult_assoc [symmetric])
-
-lemma realpow_two_minus [simp]: "(-x)^Suc (Suc 0) = (x::real)^Suc (Suc 0)"
-by simp
-
-lemma realpow_two_diff:
- "(x::real)^Suc (Suc 0) - y^Suc (Suc 0) = (x - y) * (x + y)"
-apply (unfold real_diff_def)
-apply (simp add: ring_simps)
-done
-
-lemma realpow_two_disj:
- "((x::real)^Suc (Suc 0) = y^Suc (Suc 0)) = (x = y | x = -y)"
-apply (cut_tac x = x and y = y in realpow_two_diff)
-apply (auto simp del: realpow_Suc)
-done
-
-lemma realpow_real_of_nat: "real (m::nat) ^ n = real (m ^ n)"
-apply (induct "n")
-apply (auto simp add: real_of_nat_one real_of_nat_mult)
-done
-
-lemma realpow_real_of_nat_two_pos [simp] : "0 < real (Suc (Suc 0) ^ n)"
-apply (induct "n")
-apply (auto simp add: real_of_nat_mult zero_less_mult_iff)
-done
-
-(* used by AFP Integration theory *)
-lemma realpow_increasing:
- "[|(0::real) \<le> x; 0 \<le> y; x ^ Suc n \<le> y ^ Suc n|] ==> x \<le> y"
- by (rule power_le_imp_le_base)
-
-
-subsection{*Literal Arithmetic Involving Powers, Type @{typ real}*}
-
-lemma real_of_int_power: "real (x::int) ^ n = real (x ^ n)"
-apply (induct "n")
-apply (simp_all add: nat_mult_distrib)
-done
-declare real_of_int_power [symmetric, simp]
-
-lemma power_real_number_of:
- "(number_of v :: real) ^ n = real ((number_of v :: int) ^ n)"
-by (simp only: real_number_of [symmetric] real_of_int_power)
-
-declare power_real_number_of [of _ "number_of w", standard, simp]
-
-
-subsection {* Properties of Squares *}
-
-lemma sum_squares_ge_zero:
- fixes x y :: "'a::ordered_ring_strict"
- shows "0 \<le> x * x + y * y"
-by (intro add_nonneg_nonneg zero_le_square)
-
-lemma not_sum_squares_lt_zero:
- fixes x y :: "'a::ordered_ring_strict"
- shows "\<not> x * x + y * y < 0"
-by (simp add: linorder_not_less sum_squares_ge_zero)
-
-lemma sum_nonneg_eq_zero_iff:
- fixes x y :: "'a::pordered_ab_group_add"
- assumes x: "0 \<le> x" and y: "0 \<le> y"
- shows "(x + y = 0) = (x = 0 \<and> y = 0)"
-proof (auto)
- from y have "x + 0 \<le> x + y" by (rule add_left_mono)
- also assume "x + y = 0"
- finally have "x \<le> 0" by simp
- thus "x = 0" using x by (rule order_antisym)
-next
- from x have "0 + y \<le> x + y" by (rule add_right_mono)
- also assume "x + y = 0"
- finally have "y \<le> 0" by simp
- thus "y = 0" using y by (rule order_antisym)
-qed
-
-lemma sum_squares_eq_zero_iff:
- fixes x y :: "'a::ordered_ring_strict"
- shows "(x * x + y * y = 0) = (x = 0 \<and> y = 0)"
-by (simp add: sum_nonneg_eq_zero_iff)
-
-lemma sum_squares_le_zero_iff:
- fixes x y :: "'a::ordered_ring_strict"
- shows "(x * x + y * y \<le> 0) = (x = 0 \<and> y = 0)"
-by (simp add: order_le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
-
-lemma sum_squares_gt_zero_iff:
- fixes x y :: "'a::ordered_ring_strict"
- shows "(0 < x * x + y * y) = (x \<noteq> 0 \<or> y \<noteq> 0)"
-by (simp add: order_less_le sum_squares_ge_zero sum_squares_eq_zero_iff)
-
-lemma sum_power2_ge_zero:
- fixes x y :: "'a::{ordered_idom,recpower}"
- shows "0 \<le> x\<twosuperior> + y\<twosuperior>"
-unfolding power2_eq_square by (rule sum_squares_ge_zero)
-
-lemma not_sum_power2_lt_zero:
- fixes x y :: "'a::{ordered_idom,recpower}"
- shows "\<not> x\<twosuperior> + y\<twosuperior> < 0"
-unfolding power2_eq_square by (rule not_sum_squares_lt_zero)
-
-lemma sum_power2_eq_zero_iff:
- fixes x y :: "'a::{ordered_idom,recpower}"
- shows "(x\<twosuperior> + y\<twosuperior> = 0) = (x = 0 \<and> y = 0)"
-unfolding power2_eq_square by (rule sum_squares_eq_zero_iff)
-
-lemma sum_power2_le_zero_iff:
- fixes x y :: "'a::{ordered_idom,recpower}"
- shows "(x\<twosuperior> + y\<twosuperior> \<le> 0) = (x = 0 \<and> y = 0)"
-unfolding power2_eq_square by (rule sum_squares_le_zero_iff)
-
-lemma sum_power2_gt_zero_iff:
- fixes x y :: "'a::{ordered_idom,recpower}"
- shows "(0 < x\<twosuperior> + y\<twosuperior>) = (x \<noteq> 0 \<or> y \<noteq> 0)"
-unfolding power2_eq_square by (rule sum_squares_gt_zero_iff)
-
-
-subsection{* Squares of Reals *}
-
-lemma real_two_squares_add_zero_iff [simp]:
- "(x * x + y * y = 0) = ((x::real) = 0 \<and> y = 0)"
-by (rule sum_squares_eq_zero_iff)
-
-lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)"
-by simp
-
-lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)"
-by simp
-
-lemma real_mult_self_sum_ge_zero: "(0::real) \<le> x*x + y*y"
-by (rule sum_squares_ge_zero)
-
-lemma real_sum_squares_cancel_a: "x * x = -(y * y) ==> x = (0::real) & y=0"
-by (simp add: real_add_eq_0_iff [symmetric])
-
-lemma real_squared_diff_one_factored: "x*x - (1::real) = (x + 1)*(x - 1)"
-by (simp add: left_distrib right_diff_distrib)
-
-lemma real_mult_is_one [simp]: "(x*x = (1::real)) = (x = 1 | x = - 1)"
-apply auto
-apply (drule right_minus_eq [THEN iffD2])
-apply (auto simp add: real_squared_diff_one_factored)
-done
-
-lemma real_sum_squares_not_zero: "x ~= 0 ==> x * x + y * y ~= (0::real)"
-by simp
-
-lemma real_sum_squares_not_zero2: "y ~= 0 ==> x * x + y * y ~= (0::real)"
-by simp
-
-lemma realpow_two_sum_zero_iff [simp]:
- "(x ^ 2 + y ^ 2 = (0::real)) = (x = 0 & y = 0)"
-by (rule sum_power2_eq_zero_iff)
-
-lemma realpow_two_le_add_order [simp]: "(0::real) \<le> u ^ 2 + v ^ 2"
-by (rule sum_power2_ge_zero)
-
-lemma realpow_two_le_add_order2 [simp]: "(0::real) \<le> u ^ 2 + v ^ 2 + w ^ 2"
-by (intro add_nonneg_nonneg zero_le_power2)
-
-lemma real_sum_square_gt_zero: "x ~= 0 ==> (0::real) < x * x + y * y"
-by (simp add: sum_squares_gt_zero_iff)
-
-lemma real_sum_square_gt_zero2: "y ~= 0 ==> (0::real) < x * x + y * y"
-by (simp add: sum_squares_gt_zero_iff)
-
-lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"
-by (rule_tac j = 0 in real_le_trans, auto)
-
-lemma realpow_square_minus_le [simp]: "-(u ^ 2) \<le> (x::real) ^ 2"
-by (auto simp add: power2_eq_square)
-
-(* The following theorem is by Benjamin Porter *)
-lemma real_sq_order:
- fixes x::real
- assumes xgt0: "0 \<le> x" and ygt0: "0 \<le> y" and sq: "x^2 \<le> y^2"
- shows "x \<le> y"
-proof -
- from sq have "x ^ Suc (Suc 0) \<le> y ^ Suc (Suc 0)"
- by (simp only: numeral_2_eq_2)
- thus "x \<le> y" using ygt0
- by (rule power_le_imp_le_base)
-qed
-
-
-subsection {*Various Other Theorems*}
-
-lemma real_le_add_half_cancel: "(x + y/2 \<le> (y::real)) = (x \<le> y /2)"
-by auto
-
-lemma real_minus_half_eq [simp]: "(x::real) - x/2 = x/2"
-by auto
-
-lemma real_mult_inverse_cancel:
- "[|(0::real) < x; 0 < x1; x1 * y < x * u |]
- ==> inverse x * y < inverse x1 * u"
-apply (rule_tac c=x in mult_less_imp_less_left)
-apply (auto simp add: real_mult_assoc [symmetric])
-apply (simp (no_asm) add: mult_ac)
-apply (rule_tac c=x1 in mult_less_imp_less_right)
-apply (auto simp add: mult_ac)
-done
-
-lemma real_mult_inverse_cancel2:
- "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
-apply (auto dest: real_mult_inverse_cancel simp add: mult_ac)
-done
-
-lemma inverse_real_of_nat_gt_zero [simp]: "0 < inverse (real (Suc n))"
-by simp
-
-lemma inverse_real_of_nat_ge_zero [simp]: "0 \<le> inverse (real (Suc n))"
-by simp
-
-lemma realpow_num_eq_if: "(m::real) ^ n = (if n=0 then 1 else m * m ^ (n - 1))"
-by (case_tac "n", auto)
-
-subsection{* Float syntax *}
-
-syntax "_Float" :: "float_const \<Rightarrow> 'a" ("_")
-
-use "float_syntax.ML"
-setup FloatSyntax.setup
-
-text{* Test: *}
-lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::real)"
-by simp
-
-end