(* Title : HOL/Real/RealPow.thy ID : $Id: RealPow.thy,v 1.25 2004/10/19 16:18:48 paulson Exp $ Author : Jacques D. Fleuriot Copyright : 1998 University of Cambridge Description : Natural powers theory *) theory RealPow imports RealDef begin declare abs_mult_self [simp] instance real :: power .. primrec (realpow) realpow_0: "r ^ 0 = 1" realpow_Suc: "r ^ (Suc n) = (r::real) * (r ^ n)" instance real :: recpower proof fix z :: real fix n :: nat show "z^0 = 1" by simp show "z^(Suc n) = z * (z^n)" by simp qed lemma realpow_not_zero: "r ≠ (0::real) ==> r ^ n ≠ 0" by (rule field_power_not_zero) lemma realpow_zero_zero: "r ^ n = (0::real) ==> r = 0" by simp lemma realpow_two: "(r::real)^ (Suc (Suc 0)) = r * r" by simp text{*Legacy: weaker version of the theorem @{text power_strict_mono}, used 6 times in NthRoot and Transcendental*} lemma realpow_less: "[|(0::real) < x; x < y; 0 < n|] ==> x ^ n < y ^ n" apply (rule power_strict_mono, auto) done lemma realpow_two_le [simp]: "(0::real) ≤ r^ Suc (Suc 0)" by (simp add: real_le_square) lemma abs_realpow_two [simp]: "abs((x::real)^Suc (Suc 0)) = x^Suc (Suc 0)" by (simp add: abs_mult) lemma realpow_two_abs [simp]: "abs(x::real)^Suc (Suc 0) = x^Suc (Suc 0)" by (simp add: power_abs [symmetric] del: realpow_Suc) lemma two_realpow_ge_one [simp]: "(1::real) ≤ 2 ^ n" by (insert power_increasing [of 0 n "2::real"], 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 real_add_less_le_mono) apply (auto simp add: two_realpow_ge_one) done lemma realpow_Suc_le_self: "[| 0 ≤ r; r ≤ (1::real) |] ==> r ^ Suc n ≤ r" by (insert power_decreasing [of 1 "Suc n" r], simp) text{*Used ONCE in Transcendental*} lemma realpow_Suc_less_one: "[| 0 < r; r < (1::real) |] ==> r ^ Suc n < 1" by (insert power_strict_decreasing [of 0 "Suc n" r], simp) text{*Used ONCE in Lim.ML*} 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 ≠ 0 ==> r * inverse r ^Suc (Suc 0) = inverse (r::real)" by (simp add: realpow_two 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: right_distrib left_distrib mult_ac) 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 lemma realpow_increasing: "[|(0::real) ≤ x; 0 ≤ y; x ^ Suc n ≤ y ^ Suc n|] ==> x ≤ y" by (rule power_le_imp_le_base) lemma zero_less_realpow_abs_iff [simp]: "(0 < (abs x)^n) = (x ≠ (0::real) | n=0)" apply (induct "n") apply (auto simp add: zero_less_mult_iff) done lemma zero_le_realpow_abs [simp]: "(0::real) ≤ (abs x)^n" apply (induct "n") apply (auto simp add: zero_le_mult_iff) done 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{*Various Other Theorems*} text{*Used several times in Hyperreal/Transcendental.ML*} lemma real_sum_squares_cancel_a: "x * x = -(y * y) ==> x = (0::real) & y=0" apply (auto dest: real_sum_squares_cancel simp add: real_add_eq_0_iff [symmetric]) apply (auto dest: real_sum_squares_cancel simp add: add_commute) done lemma real_squared_diff_one_factored: "x*x - (1::real) = (x + 1)*(x - 1)" by (auto simp add: left_distrib right_distrib real_diff_def) 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_le_add_half_cancel: "(x + y/2 ≤ (y::real)) = (x ≤ 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 text{*Used once: in Hyperreal/Transcendental.ML*} 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 auto lemma inverse_real_of_nat_ge_zero [simp]: "0 ≤ inverse (real (Suc n))" by auto lemma real_sum_squares_not_zero: "x ~= 0 ==> x * x + y * y ~= (0::real)" by (blast dest!: real_sum_squares_cancel) lemma real_sum_squares_not_zero2: "y ~= 0 ==> x * x + y * y ~= (0::real)" by (blast dest!: real_sum_squares_cancel2) subsection {*Various Other Theorems*} lemma realpow_divide: "(x/y) ^ n = ((x::real) ^ n/ y ^ n)" apply (unfold real_divide_def) apply (auto simp add: power_mult_distrib power_inverse) done lemma realpow_two_sum_zero_iff [simp]: "(x ^ 2 + y ^ 2 = (0::real)) = (x = 0 & y = 0)" apply (auto intro: real_sum_squares_cancel real_sum_squares_cancel2 simp add: power2_eq_square) done lemma realpow_two_le_add_order [simp]: "(0::real) ≤ u ^ 2 + v ^ 2" apply (rule real_le_add_order) apply (auto simp add: power2_eq_square) done lemma realpow_two_le_add_order2 [simp]: "(0::real) ≤ u ^ 2 + v ^ 2 + w ^ 2" apply (rule real_le_add_order)+ apply (auto simp add: power2_eq_square) done lemma real_sum_square_gt_zero: "x ~= 0 ==> (0::real) < x * x + y * y" apply (cut_tac x = x and y = y in real_mult_self_sum_ge_zero) apply (drule real_le_imp_less_or_eq) apply (drule_tac y = y in real_sum_squares_not_zero, auto) done lemma real_sum_square_gt_zero2: "y ~= 0 ==> (0::real) < x * x + y * y" apply (rule real_add_commute [THEN subst]) apply (erule real_sum_square_gt_zero) done lemma real_minus_mult_self_le [simp]: "-(u * u) ≤ (x * (x::real))" by (rule_tac j = 0 in real_le_trans, auto) lemma realpow_square_minus_le [simp]: "-(u ^ 2) ≤ (x::real) ^ 2" by (auto simp add: power2_eq_square) lemma realpow_num_eq_if: "(m::real) ^ n = (if n=0 then 1 else m * m ^ (n - 1))" by (case_tac "n", auto) lemma real_num_zero_less_two_pow [simp]: "0 < (2::real) ^ (4*d)" apply (induct "d") apply (auto simp add: realpow_num_eq_if) done lemma lemma_realpow_num_two_mono: "x * (4::real) < y ==> x * (2 ^ 8) < y * (2 ^ 6)" apply (subgoal_tac " (2::real) ^ 8 = 4 * (2 ^ 6) ") apply (simp (no_asm_simp) add: real_mult_assoc [symmetric]) apply (auto simp add: realpow_num_eq_if) done ML {* val realpow_0 = thm "realpow_0"; val realpow_Suc = thm "realpow_Suc"; val realpow_not_zero = thm "realpow_not_zero"; val realpow_zero_zero = thm "realpow_zero_zero"; val realpow_two = thm "realpow_two"; val realpow_less = thm "realpow_less"; val realpow_two_le = thm "realpow_two_le"; val abs_realpow_two = thm "abs_realpow_two"; val realpow_two_abs = thm "realpow_two_abs"; val two_realpow_ge_one = thm "two_realpow_ge_one"; val two_realpow_gt = thm "two_realpow_gt"; val realpow_Suc_le_self = thm "realpow_Suc_le_self"; val realpow_Suc_less_one = thm "realpow_Suc_less_one"; val realpow_minus_mult = thm "realpow_minus_mult"; val realpow_two_mult_inverse = thm "realpow_two_mult_inverse"; val realpow_two_minus = thm "realpow_two_minus"; val realpow_two_disj = thm "realpow_two_disj"; val realpow_real_of_nat = thm "realpow_real_of_nat"; val realpow_real_of_nat_two_pos = thm "realpow_real_of_nat_two_pos"; val realpow_increasing = thm "realpow_increasing"; val zero_less_realpow_abs_iff = thm "zero_less_realpow_abs_iff"; val zero_le_realpow_abs = thm "zero_le_realpow_abs"; val real_of_int_power = thm "real_of_int_power"; val power_real_number_of = thm "power_real_number_of"; val real_sum_squares_cancel_a = thm "real_sum_squares_cancel_a"; val real_mult_inverse_cancel2 = thm "real_mult_inverse_cancel2"; val real_squared_diff_one_factored = thm "real_squared_diff_one_factored"; val real_mult_is_one = thm "real_mult_is_one"; val real_le_add_half_cancel = thm "real_le_add_half_cancel"; val real_minus_half_eq = thm "real_minus_half_eq"; val real_mult_inverse_cancel = thm "real_mult_inverse_cancel"; val real_mult_inverse_cancel2 = thm "real_mult_inverse_cancel2"; val inverse_real_of_nat_gt_zero = thm "inverse_real_of_nat_gt_zero"; val inverse_real_of_nat_ge_zero = thm "inverse_real_of_nat_ge_zero"; val real_sum_squares_not_zero = thm "real_sum_squares_not_zero"; val real_sum_squares_not_zero2 = thm "real_sum_squares_not_zero2"; val realpow_divide = thm "realpow_divide"; val realpow_two_sum_zero_iff = thm "realpow_two_sum_zero_iff"; val realpow_two_le_add_order = thm "realpow_two_le_add_order"; val realpow_two_le_add_order2 = thm "realpow_two_le_add_order2"; val real_sum_square_gt_zero = thm "real_sum_square_gt_zero"; val real_sum_square_gt_zero2 = thm "real_sum_square_gt_zero2"; val real_minus_mult_self_le = thm "real_minus_mult_self_le"; val realpow_square_minus_le = thm "realpow_square_minus_le"; val realpow_num_eq_if = thm "realpow_num_eq_if"; val real_num_zero_less_two_pow = thm "real_num_zero_less_two_pow"; val lemma_realpow_num_two_mono = thm "lemma_realpow_num_two_mono"; *} end
lemma realpow_not_zero:
r ≠ 0 ==> r ^ n ≠ 0
lemma realpow_zero_zero:
r ^ n = 0 ==> r = 0
lemma realpow_two:
r ^ Suc (Suc 0) = r * r
lemma realpow_less:
[| 0 < x; x < y; 0 < n |] ==> x ^ n < y ^ n
lemma realpow_two_le:
0 ≤ r ^ Suc (Suc 0)
lemma abs_realpow_two:
¦x ^ Suc (Suc 0)¦ = x ^ Suc (Suc 0)
lemma realpow_two_abs:
¦x¦ ^ Suc (Suc 0) = x ^ Suc (Suc 0)
lemma two_realpow_ge_one:
1 ≤ 2 ^ n
lemma two_realpow_gt:
real n < 2 ^ n
lemma realpow_Suc_le_self:
[| 0 ≤ r; r ≤ 1 |] ==> r ^ Suc n ≤ r
lemma realpow_Suc_less_one:
[| 0 < r; r < 1 |] ==> r ^ Suc n < 1
lemma realpow_minus_mult:
0 < n ==> x ^ (n - 1) * x = x ^ n
lemma realpow_two_mult_inverse:
r ≠ 0 ==> r * inverse r ^ Suc (Suc 0) = inverse r
lemma realpow_two_minus:
(- x) ^ Suc (Suc 0) = x ^ Suc (Suc 0)
lemma realpow_two_diff:
x ^ Suc (Suc 0) - y ^ Suc (Suc 0) = (x - y) * (x + y)
lemma realpow_two_disj:
(x ^ Suc (Suc 0) = y ^ Suc (Suc 0)) = (x = y ∨ x = - y)
lemma realpow_real_of_nat:
real m ^ n = real (m ^ n)
lemma realpow_real_of_nat_two_pos:
0 < real (Suc (Suc 0) ^ n)
lemma realpow_increasing:
[| 0 ≤ x; 0 ≤ y; x ^ Suc n ≤ y ^ Suc n |] ==> x ≤ y
lemma zero_less_realpow_abs_iff:
(0 < ¦x¦ ^ n) = (x ≠ 0 ∨ n = 0)
lemma zero_le_realpow_abs:
0 ≤ ¦x¦ ^ n
lemma real_of_int_power:
real x ^ n = real (x ^ n)
lemma power_real_number_of:
number_of v ^ n = real (number_of v ^ n)
lemma real_sum_squares_cancel_a:
x * x = - (y * y) ==> x = 0 ∧ y = 0
lemma real_squared_diff_one_factored:
x * x - 1 = (x + 1) * (x - 1)
lemma real_mult_is_one:
(x * x = 1) = (x = 1 ∨ x = - 1)
lemma real_le_add_half_cancel:
(x + y / 2 ≤ y) = (x ≤ y / 2)
lemma real_minus_half_eq:
x - x / 2 = x / 2
lemma real_mult_inverse_cancel:
[| 0 < x; 0 < x1.0; x1.0 * y < x * u |] ==> inverse x * y < inverse x1.0 * u
lemma real_mult_inverse_cancel2:
[| 0 < x; 0 < x1.0; x1.0 * y < x * u |] ==> y * inverse x < u * inverse x1.0
lemma inverse_real_of_nat_gt_zero:
0 < inverse (real (Suc n))
lemma inverse_real_of_nat_ge_zero:
0 ≤ inverse (real (Suc n))
lemma real_sum_squares_not_zero:
x ≠ 0 ==> x * x + y * y ≠ 0
lemma real_sum_squares_not_zero2:
y ≠ 0 ==> x * x + y * y ≠ 0
lemma realpow_divide:
(x / y) ^ n = x ^ n / y ^ n
lemma realpow_two_sum_zero_iff:
(x² + y² = 0) = (x = 0 ∧ y = 0)
lemma realpow_two_le_add_order:
0 ≤ u² + v²
lemma realpow_two_le_add_order2:
0 ≤ u² + v² + w²
lemma real_sum_square_gt_zero:
x ≠ 0 ==> 0 < x * x + y * y
lemma real_sum_square_gt_zero2:
y ≠ 0 ==> 0 < x * x + y * y
lemma real_minus_mult_self_le:
- (u * u) ≤ x * x
lemma realpow_square_minus_le:
- u² ≤ x²
lemma realpow_num_eq_if:
m ^ n = (if n = 0 then 1 else m * m ^ (n - 1))
lemma real_num_zero_less_two_pow:
0 < 2 ^ (4 * d)
lemma lemma_realpow_num_two_mono:
x * 4 < y ==> x * 2 ^ 8 < y * 2 ^ 6