wenzelm@9435: (*  Title       : HOL/Real/RealPow.thy
paulson@7219:     ID          : $Id$
paulson@7077:     Author      : Jacques D. Fleuriot  
paulson@7077:     Copyright   : 1998  University of Cambridge
paulson@7077:     Description : Natural powers theory
paulson@7077: 
paulson@7077: *)
paulson@7077: 
paulson@14269: theory RealPow = RealArith:
wenzelm@9435: 
wenzelm@10309: instance real :: power ..
paulson@7077: 
wenzelm@8856: primrec (realpow)
paulson@12018:      realpow_0:   "r ^ 0       = 1"
wenzelm@9435:      realpow_Suc: "r ^ (Suc n) = (r::real) * (r ^ n)"
paulson@7077: 
paulson@14265: 
paulson@14268: lemma realpow_zero [simp]: "(0::real) ^ (Suc n) = 0"
paulson@14268: by auto
paulson@14265: 
paulson@14268: lemma realpow_not_zero [rule_format]: "r \<noteq> (0::real) --> r ^ n \<noteq> 0"
paulson@14268: by (induct_tac "n", auto)
paulson@14265: 
paulson@14265: lemma realpow_zero_zero: "r ^ n = (0::real) ==> r = 0"
paulson@14265: apply (rule ccontr)
paulson@14265: apply (auto dest: realpow_not_zero)
paulson@14265: done
paulson@14265: 
paulson@14265: lemma realpow_inverse: "inverse ((r::real) ^ n) = (inverse r) ^ n"
paulson@14265: apply (induct_tac "n")
paulson@14265: apply (auto simp add: real_inverse_distrib)
paulson@14265: done
paulson@14265: 
paulson@14265: lemma realpow_abs: "abs(r ^ n) = abs(r::real) ^ n"
paulson@14265: apply (induct_tac "n")
paulson@14265: apply (auto simp add: abs_mult)
paulson@14265: done
paulson@14265: 
paulson@14265: lemma realpow_add: "(r::real) ^ (n + m) = (r ^ n) * (r ^ m)"
paulson@14265: apply (induct_tac "n")
paulson@14265: apply (auto simp add: real_mult_ac)
paulson@14265: done
paulson@14265: 
paulson@14268: lemma realpow_one [simp]: "(r::real) ^ 1 = r"
paulson@14268: by simp
paulson@14265: 
paulson@14265: lemma realpow_two: "(r::real)^ (Suc (Suc 0)) = r * r"
paulson@14268: by simp
paulson@14265: 
paulson@14268: lemma realpow_gt_zero [rule_format]: "(0::real) < r --> 0 < r ^ n"
paulson@14265: apply (induct_tac "n")
paulson@14265: apply (auto intro: real_mult_order simp add: real_zero_less_one)
paulson@14265: done
paulson@14265: 
paulson@14268: lemma realpow_ge_zero [rule_format]: "(0::real) \<le> r --> 0 \<le> r ^ n"
paulson@14265: apply (induct_tac "n")
paulson@14265: apply (auto simp add: real_0_le_mult_iff)
paulson@14265: done
paulson@14265: 
paulson@14268: lemma realpow_le [rule_format]: "(0::real) \<le> x & x \<le> y --> x ^ n \<le> y ^ n"
paulson@14265: apply (induct_tac "n")
paulson@14265: apply (auto intro!: real_mult_le_mono)
paulson@14265: apply (simp (no_asm_simp) add: realpow_ge_zero)
paulson@14265: done
paulson@14265: 
paulson@14265: lemma realpow_0_left [rule_format, simp]:
paulson@14265:      "0 < n --> 0 ^ n = (0::real)"
paulson@14268: apply (induct_tac "n", auto) 
paulson@14265: done
paulson@14265: 
paulson@14265: lemma realpow_less' [rule_format]:
paulson@14265:      "[|(0::real) \<le> x; x < y |] ==> 0 < n --> x ^ n < y ^ n"
paulson@14265: apply (induct n) 
paulson@14268: apply (auto simp add: real_mult_less_mono' realpow_ge_zero) 
paulson@14265: done
paulson@14265: 
paulson@14265: text{*Legacy: weaker version of the theorem above*}
paulson@14268: lemma realpow_less:
paulson@14265:      "[|(0::real) < x; x < y; 0 < n|] ==> x ^ n < y ^ n"
paulson@14268: apply (rule realpow_less', auto) 
paulson@14265: done
paulson@14265: 
paulson@14268: lemma realpow_eq_one [simp]: "1 ^ n = (1::real)"
paulson@14268: by (induct_tac "n", auto)
paulson@14265: 
paulson@14268: lemma abs_realpow_minus_one [simp]: "abs((-1) ^ n) = (1::real)"
paulson@14265: apply (induct_tac "n")
paulson@14265: apply (auto simp add: abs_mult)
paulson@14265: done
paulson@14265: 
paulson@14265: lemma realpow_mult: "((r::real) * s) ^ n = (r ^ n) * (s ^ n)"
paulson@14265: apply (induct_tac "n")
paulson@14265: apply (auto simp add: real_mult_ac)
paulson@14265: done
paulson@14265: 
paulson@14268: lemma realpow_two_le [simp]: "(0::real) \<le> r^ Suc (Suc 0)"
paulson@14268: by (simp add: real_le_square)
paulson@14265: 
paulson@14268: lemma abs_realpow_two [simp]: "abs((x::real)^Suc (Suc 0)) = x^Suc (Suc 0)"
paulson@14268: by (simp add: abs_eqI1 real_le_square)
paulson@14265: 
paulson@14268: lemma realpow_two_abs [simp]: "abs(x::real)^Suc (Suc 0) = x^Suc (Suc 0)"
paulson@14268: by (simp add: realpow_abs [symmetric] abs_eqI1 del: realpow_Suc)
paulson@14265: 
paulson@14265: lemma realpow_two_gt_one: "(1::real) < r ==> 1 < r^ (Suc (Suc 0))"
paulson@14265: apply auto
paulson@14265: apply (cut_tac real_zero_less_one)
paulson@14268: apply (frule_tac x = 0 in order_less_trans, assumption)
paulson@14268: apply (drule_tac  z = r and x = 1 in real_mult_less_mono1)
paulson@14265: apply (auto intro: order_less_trans)
paulson@14265: done
paulson@14265: 
paulson@14268: lemma realpow_ge_one [rule_format]: "(1::real) < r --> 1 \<le> r ^ n"
paulson@14268: apply (induct_tac "n", auto)
paulson@14265: apply (subgoal_tac "1*1 \<le> r * r^n")
paulson@14268: apply (rule_tac [2] real_mult_le_mono, auto)
paulson@14265: done
paulson@14265: 
paulson@14265: lemma realpow_ge_one2: "(1::real) \<le> r ==> 1 \<le> r ^ n"
paulson@14265: apply (drule order_le_imp_less_or_eq)
paulson@14265: apply (auto dest: realpow_ge_one)
paulson@14265: done
paulson@14265: 
paulson@14268: lemma two_realpow_ge_one [simp]: "(1::real) \<le> 2 ^ n"
paulson@14265: apply (rule_tac y = "1 ^ n" in order_trans)
paulson@14265: apply (rule_tac [2] realpow_le)
paulson@14265: apply (auto intro: order_less_imp_le)
paulson@14265: done
paulson@14265: 
paulson@14268: lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n"
paulson@14265: apply (induct_tac "n")
paulson@14265: apply (auto simp add: real_of_nat_Suc)
paulson@14265: apply (subst real_mult_2)
paulson@14265: apply (rule real_add_less_le_mono)
paulson@14265: apply (auto simp add: two_realpow_ge_one)
paulson@14265: done
paulson@14265: 
paulson@14268: lemma realpow_minus_one [simp]: "(-1) ^ (2*n) = (1::real)"
paulson@14268: by (induct_tac "n", auto)
paulson@14268: 
paulson@14268: lemma realpow_minus_one_odd [simp]: "(-1) ^ Suc (2*n) = -(1::real)"
paulson@14268: by auto
paulson@14265: 
paulson@14268: lemma realpow_minus_one_even [simp]: "(-1) ^ Suc (Suc (2*n)) = (1::real)"
paulson@14268: by auto
paulson@14265: 
paulson@14268: lemma realpow_Suc_less [rule_format]:
paulson@14268:      "(0::real) < r & r < (1::real) --> r ^ Suc n < r ^ n"
paulson@14288:   by (induct_tac "n", auto simp add: mult_less_cancel_left)
paulson@14265: 
paulson@14288: lemma realpow_Suc_le [rule_format]:
paulson@14288:      "0 \<le> r & r < (1::real) --> r ^ Suc n \<le> r ^ n"
paulson@14265: apply (induct_tac "n")
paulson@14265: apply (auto intro: order_less_imp_le dest!: order_le_imp_less_or_eq)
paulson@14265: done
paulson@14265: 
paulson@14268: lemma realpow_zero_le [simp]: "(0::real) \<le> 0 ^ n"
paulson@14268: by (case_tac "n", auto)
paulson@14265: 
paulson@14268: lemma realpow_Suc_le2 [rule_format]: "0 < r & r < (1::real) --> r ^ Suc n \<le> r ^ n"
paulson@14268: by (blast intro!: order_less_imp_le realpow_Suc_less)
paulson@14265: 
paulson@14265: lemma realpow_Suc_le3: "[| 0 \<le> r; r < (1::real) |] ==> r ^ Suc n \<le> r ^ n"
paulson@14265: apply (erule order_le_imp_less_or_eq [THEN disjE])
paulson@14268: apply (rule realpow_Suc_le2, auto)
paulson@14265: done
paulson@14265: 
paulson@14268: lemma realpow_less_le [rule_format]: "0 \<le> r & r < (1::real) & n < N --> r ^ N \<le> r ^ n"
paulson@14265: apply (induct_tac "N")
paulson@14265: apply (simp_all (no_asm_simp))
paulson@14265: apply clarify
paulson@14268: apply (subgoal_tac "r * r ^ na \<le> 1 * r ^ n", simp)
paulson@14265: apply (rule real_mult_le_mono)
paulson@14265: apply (auto simp add: realpow_ge_zero less_Suc_eq)
paulson@14265: done
paulson@14265: 
paulson@14265: lemma realpow_le_le: "[| 0 \<le> r; r < (1::real); n \<le> N |] ==> r ^ N \<le> r ^ n"
paulson@14268: apply (drule_tac n = N in le_imp_less_or_eq)
paulson@14265: apply (auto intro: realpow_less_le)
paulson@14265: done
paulson@14265: 
paulson@14265: lemma realpow_Suc_le_self: "[| 0 < r; r < (1::real) |] ==> r ^ Suc n \<le> r"
paulson@14268: by (drule_tac n = 1 and N = "Suc n" in order_less_imp_le [THEN realpow_le_le], auto)
paulson@14265: 
paulson@14265: lemma realpow_Suc_less_one: "[| 0 < r; r < (1::real) |] ==> r ^ Suc n < 1"
paulson@14268: by (blast intro: realpow_Suc_le_self order_le_less_trans)
paulson@14268: 
paulson@14268: lemma realpow_le_Suc [rule_format]: "(1::real) \<le> r --> r ^ n \<le> r ^ Suc n"
paulson@14268: by (induct_tac "n", auto)
paulson@14268: 
paulson@14268: lemma realpow_less_Suc [rule_format]: "(1::real) < r --> r ^ n < r ^ Suc n"
paulson@14288: by (induct_tac "n", auto simp add: mult_less_cancel_left)
paulson@14265: 
paulson@14268: lemma realpow_le_Suc2 [rule_format]: "(1::real) < r --> r ^ n \<le> r ^ Suc n"
paulson@14268: by (blast intro!: order_less_imp_le realpow_less_Suc)
paulson@14268: 
paulson@14268: (*One use in RealPow.thy*)
paulson@14268: lemma real_mult_self_le2: "[| (1::real) \<le> r; (1::real) \<le> x |]  ==> x \<le> r * x"
paulson@14268: apply (subgoal_tac "1 * x \<le> r * x", simp) 
paulson@14268: apply (rule mult_right_mono, auto) 
paulson@14265: done
paulson@14265: 
paulson@14268: lemma realpow_gt_ge2 [rule_format]: "(1::real) \<le> r & n < N --> r ^ n \<le> r ^ N"
paulson@14268: apply (induct_tac "N", auto)
paulson@14268: apply (frule_tac [!] n = na in realpow_ge_one2)
paulson@14268: apply (drule_tac [!] real_mult_self_le2, assumption)
paulson@14268: prefer 2 apply assumption
paulson@14265: apply (auto intro: order_trans simp add: less_Suc_eq)
paulson@14265: done
paulson@14265: 
paulson@14265: lemma realpow_ge_ge2: "[| (1::real) \<le> r; n \<le> N |] ==> r ^ n \<le> r ^ N"
paulson@14268: apply (drule_tac n = N in le_imp_less_or_eq)
paulson@14265: apply (auto intro: realpow_gt_ge2)
paulson@14265: done
paulson@14265: 
paulson@14268: lemma realpow_Suc_ge_self2: "(1::real) \<le> r ==> r \<le> r ^ Suc n"
paulson@14268: by (drule_tac n = 1 and N = "Suc n" in realpow_ge_ge2, auto)
paulson@14265: 
paulson@14268: (*Used ONCE in Hyperreal/NthRoot.ML*)
paulson@14265: lemma realpow_ge_self2: "[| (1::real) \<le> r; 0 < n |] ==> r \<le> r ^ n"
paulson@14265: apply (drule less_not_refl2 [THEN not0_implies_Suc])
paulson@14265: apply (auto intro!: realpow_Suc_ge_self2)
paulson@14265: done
paulson@14265: 
paulson@14268: lemma realpow_minus_mult [rule_format, simp]:
paulson@14268:      "0 < n --> (x::real) ^ (n - 1) * x = x ^ n"
paulson@14265: apply (induct_tac "n")
paulson@14265: apply (auto simp add: real_mult_commute)
paulson@14265: done
paulson@14265: 
paulson@14268: lemma realpow_two_mult_inverse [simp]: "r \<noteq> 0 ==> r * inverse r ^Suc (Suc 0) = inverse (r::real)"
paulson@14268: by (simp add: realpow_two real_mult_assoc [symmetric])
paulson@14265: 
paulson@14265: (* 05/00 *)
paulson@14268: lemma realpow_two_minus [simp]: "(-x)^Suc (Suc 0) = (x::real)^Suc (Suc 0)"
paulson@14268: by simp
paulson@14265: 
paulson@14265: lemma realpow_two_diff: "(x::real)^Suc (Suc 0) - y^Suc (Suc 0) = (x - y) * (x + y)"
paulson@14265: apply (unfold real_diff_def)
paulson@14265: apply (simp add: real_add_mult_distrib2 real_add_mult_distrib real_mult_ac)
paulson@14265: done
paulson@14265: 
paulson@14265: lemma realpow_two_disj: "((x::real)^Suc (Suc 0) = y^Suc (Suc 0)) = (x = y | x = -y)"
paulson@14268: apply (cut_tac x = x and y = y in realpow_two_diff)
paulson@14265: apply (auto simp del: realpow_Suc)
paulson@14265: done
paulson@14265: 
paulson@14265: (* used in Transc *)
paulson@14265: lemma realpow_diff: "[|(x::real) \<noteq> 0; m \<le> n |] ==> x ^ (n - m) = x ^ n * inverse (x ^ m)"
paulson@14268: by (auto simp add: le_eq_less_or_eq less_iff_Suc_add realpow_add realpow_not_zero real_mult_ac)
paulson@14265: 
paulson@14265: lemma realpow_real_of_nat: "real (m::nat) ^ n = real (m ^ n)"
paulson@14265: apply (induct_tac "n")
paulson@14265: apply (auto simp add: real_of_nat_one real_of_nat_mult)
paulson@14265: done
paulson@14265: 
paulson@14268: lemma realpow_real_of_nat_two_pos [simp] : "0 < real (Suc (Suc 0) ^ n)"
paulson@14265: apply (induct_tac "n")
paulson@14265: apply (auto simp add: real_of_nat_mult real_0_less_mult_iff)
paulson@14265: done
paulson@14265: 
paulson@14265: lemma realpow_increasing:
paulson@14265:   assumes xnonneg: "(0::real) \<le> x"
paulson@14265:       and ynonneg: "0 \<le> y"
paulson@14265:       and le: "x ^ Suc n \<le> y ^ Suc n"
paulson@14265:   shows "x \<le> y"
paulson@14265:  proof (rule ccontr)
paulson@14265:    assume "~ x \<le> y"
paulson@14265:    then have "y<x" by simp
paulson@14265:    then have "y ^ Suc n < x ^ Suc n"
paulson@14265:      by (simp only: prems realpow_less') 
paulson@14265:    from le and this show "False"
paulson@14265:      by simp
paulson@14265:  qed
paulson@14265:   
paulson@14265: lemma realpow_Suc_cancel_eq: "[| (0::real) \<le> x; 0 \<le> y; x ^ Suc n = y ^ Suc n |] ==> x = y"
paulson@14268: by (blast intro: realpow_increasing order_antisym order_eq_refl sym)
paulson@14265: 
paulson@14265: 
paulson@14265: (*** Logical equivalences for inequalities ***)
paulson@14265: 
paulson@14268: lemma realpow_eq_0_iff [simp]: "(x^n = 0) = (x = (0::real) & 0<n)"
paulson@14268: by (induct_tac "n", auto)
paulson@14265: 
paulson@14268: lemma zero_less_realpow_abs_iff [simp]: "(0 < (abs x)^n) = (x \<noteq> (0::real) | n=0)"
paulson@14265: apply (induct_tac "n")
paulson@14265: apply (auto simp add: real_0_less_mult_iff)
paulson@14265: done
paulson@14265: 
paulson@14268: lemma zero_le_realpow_abs [simp]: "(0::real) \<le> (abs x)^n"
paulson@14265: apply (induct_tac "n")
paulson@14265: apply (auto simp add: real_0_le_mult_iff)
paulson@14265: done
paulson@14265: 
paulson@14265: 
paulson@14265: (*** Literal arithmetic involving powers, type real ***)
paulson@14265: 
paulson@14265: lemma real_of_int_power: "real (x::int) ^ n = real (x ^ n)"
paulson@14265: apply (induct_tac "n")
paulson@14265: apply (simp_all (no_asm_simp) add: nat_mult_distrib)
paulson@14265: done
paulson@14265: declare real_of_int_power [symmetric, simp]
paulson@14265: 
paulson@14265: lemma power_real_number_of: "(number_of v :: real) ^ n = real ((number_of v :: int) ^ n)"
paulson@14268: by (simp only: real_number_of_def real_of_int_power)
paulson@14265: 
paulson@14265: declare power_real_number_of [of _ "number_of w", standard, simp]
paulson@14265: 
paulson@14265: 
paulson@14265: lemma real_power_two: "(r::real)\<twosuperior> = r * r"
paulson@14265:   by (simp add: numeral_2_eq_2)
paulson@14265: 
paulson@14265: lemma real_sqr_ge_zero [iff]: "0 \<le> (r::real)\<twosuperior>"
paulson@14265:   by (simp add: real_power_two)
paulson@14265: 
paulson@14265: lemma real_sqr_gt_zero: "(r::real) \<noteq> 0 ==> 0 < r\<twosuperior>"
paulson@14265: proof -
paulson@14265:   assume "r \<noteq> 0"
paulson@14265:   hence "0 \<noteq> r\<twosuperior>" by simp
paulson@14265:   also have "0 \<le> r\<twosuperior>" by (simp add: real_sqr_ge_zero)
paulson@14265:   finally show ?thesis .
paulson@14265: qed
paulson@14265: 
paulson@14265: lemma real_sqr_not_zero: "r \<noteq> 0 ==> (r::real)\<twosuperior> \<noteq> 0"
paulson@14265:   by simp
paulson@14265: 
paulson@14265: 
paulson@14268: subsection{*Various Other Theorems*}
paulson@14268: 
paulson@14268: text{*Used several times in Hyperreal/Transcendental.ML*}
paulson@14268: lemma real_sum_squares_cancel_a: "x * x = -(y * y) ==> x = (0::real) & y=0"
paulson@14268:   by (auto intro: real_sum_squares_cancel)
paulson@14268: 
paulson@14268: lemma real_squared_diff_one_factored: "x*x - (1::real) = (x + 1)*(x - 1)"
paulson@14268: apply (auto simp add: real_add_mult_distrib real_add_mult_distrib2 real_diff_def)
paulson@14268: done
paulson@14268: 
paulson@14268: lemma real_mult_is_one: "(x*x = (1::real)) = (x = 1 | x = - 1)"
paulson@14268: apply auto
paulson@14268: apply (drule right_minus_eq [THEN iffD2]) 
paulson@14268: apply (auto simp add: real_squared_diff_one_factored)
paulson@14268: done
paulson@14268: declare real_mult_is_one [iff]
paulson@14268: 
paulson@14304: lemma real_le_add_half_cancel: "(x + y/2 \<le> (y::real)) = (x \<le> y /2)"
paulson@14268: apply auto
paulson@14268: done
paulson@14268: declare real_le_add_half_cancel [simp]
paulson@14268: 
paulson@14268: lemma real_minus_half_eq: "(x::real) - x/2 = x/2"
paulson@14268: apply auto
paulson@14268: done
paulson@14268: declare real_minus_half_eq [simp]
paulson@14268: 
paulson@14268: lemma real_mult_inverse_cancel:
paulson@14268:      "[|(0::real) < x; 0 < x1; x1 * y < x * u |] 
paulson@14268:       ==> inverse x * y < inverse x1 * u"
paulson@14268: apply (rule_tac c=x in mult_less_imp_less_left) 
paulson@14268: apply (auto simp add: real_mult_assoc [symmetric])
paulson@14268: apply (simp (no_asm) add: real_mult_ac)
paulson@14268: apply (rule_tac c=x1 in mult_less_imp_less_right) 
paulson@14268: apply (auto simp add: real_mult_ac)
paulson@14268: done
paulson@14268: 
paulson@14268: text{*Used once: in Hyperreal/Transcendental.ML*}
paulson@14268: lemma real_mult_inverse_cancel2: "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
paulson@14268: apply (auto dest: real_mult_inverse_cancel simp add: real_mult_ac)
paulson@14268: done
paulson@14268: 
paulson@14268: lemma inverse_real_of_nat_gt_zero: "0 < inverse (real (Suc n))"
paulson@14268: apply auto
paulson@14268: done
paulson@14268: declare inverse_real_of_nat_gt_zero [simp]
paulson@14268: 
paulson@14304: lemma inverse_real_of_nat_ge_zero: "0 \<le> inverse (real (Suc n))"
paulson@14268: apply auto
paulson@14268: done
paulson@14268: declare inverse_real_of_nat_ge_zero [simp]
paulson@14268: 
paulson@14268: lemma real_sum_squares_not_zero: "x ~= 0 ==> x * x + y * y ~= (0::real)"
paulson@14268: apply (blast dest!: real_sum_squares_cancel) 
paulson@14268: done
paulson@14268: 
paulson@14268: lemma real_sum_squares_not_zero2: "y ~= 0 ==> x * x + y * y ~= (0::real)"
paulson@14268: apply (blast dest!: real_sum_squares_cancel2) 
paulson@14268: done
paulson@14268: 
paulson@14268: (* nice theorem *)
paulson@14268: lemma abs_mult_abs: "abs x * abs x = x * (x::real)"
paulson@14268: apply (insert linorder_less_linear [of x 0]) 
paulson@14268: apply (auto simp add: abs_eqI2 abs_minus_eqI2)
paulson@14268: done
paulson@14268: declare abs_mult_abs [simp]
paulson@14268: 
paulson@14268: 
paulson@14268: subsection {*Various Other Theorems*}
paulson@14268: 
paulson@14268: lemma realpow_divide: 
paulson@14268:     "(x/y) ^ n = ((x::real) ^ n/ y ^ n)"
paulson@14268: apply (unfold real_divide_def)
paulson@14268: apply (auto simp add: realpow_mult realpow_inverse)
paulson@14268: done
paulson@14268: 
paulson@14304: lemma realpow_ge_zero2 [rule_format (no_asm)]: "(0::real) \<le> r --> 0 \<le> r ^ n"
paulson@14268: apply (induct_tac "n")
paulson@14268: apply (auto simp add: real_0_le_mult_iff)
paulson@14268: done
paulson@14268: 
paulson@14304: lemma realpow_le2 [rule_format (no_asm)]: "(0::real) \<le> x & x \<le> y --> x ^ n \<le> y ^ n"
paulson@14268: apply (induct_tac "n")
paulson@14268: apply (auto intro!: real_mult_le_mono simp add: realpow_ge_zero2)
paulson@14268: done
paulson@14268: 
paulson@14268: lemma realpow_Suc_gt_one: "(1::real) < r ==> 1 < r ^ (Suc n)"
paulson@14268: apply (frule_tac n = "n" in realpow_ge_one)
paulson@14268: apply (drule real_le_imp_less_or_eq, safe)
paulson@14268: apply (frule real_zero_less_one [THEN real_less_trans])
paulson@14268: apply (drule_tac y = "r ^ n" in real_mult_less_mono2)
paulson@14268: apply assumption
paulson@14268: apply (auto dest: real_less_trans)
paulson@14268: done
paulson@14268: 
paulson@14268: lemma realpow_two_sum_zero_iff: "(x ^ 2 + y ^ 2 = (0::real)) = (x = 0 & y = 0)"
paulson@14268: apply (auto intro: real_sum_squares_cancel real_sum_squares_cancel2 simp add: numeral_2_eq_2)
paulson@14268: done
paulson@14268: declare realpow_two_sum_zero_iff [simp]
paulson@14268: 
paulson@14304: lemma realpow_two_le_add_order: "(0::real) \<le> u ^ 2 + v ^ 2"
paulson@14268: apply (rule real_le_add_order)
paulson@14268: apply (auto simp add: numeral_2_eq_2)
paulson@14268: done
paulson@14268: declare realpow_two_le_add_order [simp]
paulson@14268: 
paulson@14304: lemma realpow_two_le_add_order2: "(0::real) \<le> u ^ 2 + v ^ 2 + w ^ 2"
paulson@14268: apply (rule real_le_add_order)+
paulson@14268: apply (auto simp add: numeral_2_eq_2)
paulson@14268: done
paulson@14268: declare realpow_two_le_add_order2 [simp]
paulson@14268: 
paulson@14268: lemma real_sum_square_gt_zero: "x ~= 0 ==> (0::real) < x * x + y * y"
paulson@14268: apply (cut_tac x = "x" and y = "y" in real_mult_self_sum_ge_zero)
paulson@14268: apply (drule real_le_imp_less_or_eq)
paulson@14268: apply (drule_tac y = "y" in real_sum_squares_not_zero)
paulson@14268: apply auto
paulson@14268: done
paulson@14268: 
paulson@14268: lemma real_sum_square_gt_zero2: "y ~= 0 ==> (0::real) < x * x + y * y"
paulson@14268: apply (rule real_add_commute [THEN subst])
paulson@14268: apply (erule real_sum_square_gt_zero)
paulson@14268: done
paulson@14268: 
paulson@14304: lemma real_minus_mult_self_le: "-(u * u) \<le> (x * (x::real))"
paulson@14268: apply (rule_tac j = "0" in real_le_trans)
paulson@14268: apply auto
paulson@14268: done
paulson@14268: declare real_minus_mult_self_le [simp]
paulson@14268: 
paulson@14304: lemma realpow_square_minus_le: "-(u ^ 2) \<le> (x::real) ^ 2"
paulson@14268: apply (auto simp add: numeral_2_eq_2)
paulson@14268: done
paulson@14268: declare realpow_square_minus_le [simp]
paulson@14268: 
paulson@14268: lemma realpow_num_eq_if: "(m::real) ^ n = (if n=0 then 1 else m * m ^ (n - 1))"
paulson@14268: apply (case_tac "n")
paulson@14268: apply auto
paulson@14268: done
paulson@14268: 
paulson@14268: lemma real_num_zero_less_two_pow: "0 < (2::real) ^ (4*d)"
paulson@14268: apply (induct_tac "d")
paulson@14268: apply (auto simp add: realpow_num_eq_if)
paulson@14268: done
paulson@14268: declare real_num_zero_less_two_pow [simp]
paulson@14268: 
paulson@14268: lemma lemma_realpow_num_two_mono: "x * (4::real)   < y ==> x * (2 ^ 8) < y * (2 ^ 6)"
paulson@14268: apply (subgoal_tac " (2::real) ^ 8 = 4 * (2 ^ 6) ")
paulson@14268: apply (simp (no_asm_simp) add: real_mult_assoc [symmetric])
paulson@14268: apply (auto simp add: realpow_num_eq_if)
paulson@14268: done
paulson@14268: 
paulson@14268: lemma lemma_realpow_4: "2 ^ 2 = (4::real)"
paulson@14268: apply (simp (no_asm) add: realpow_num_eq_if)
paulson@14268: done
paulson@14268: declare lemma_realpow_4 [simp]
paulson@14268: 
paulson@14268: lemma lemma_realpow_16: "2 ^ 4 = (16::real)"
paulson@14268: apply (simp (no_asm) add: realpow_num_eq_if)
paulson@14268: done
paulson@14268: declare lemma_realpow_16 [simp]
paulson@14268: 
paulson@14304: lemma zero_le_x_squared: "(0::real) \<le> x^2"
paulson@14268: apply (simp add: numeral_2_eq_2)
paulson@14268: done
paulson@14268: declare zero_le_x_squared [simp]
paulson@14268: 
paulson@14268: 
paulson@14268: 
paulson@14265: ML
paulson@14265: {*
paulson@14265: val realpow_0 = thm "realpow_0";
paulson@14265: val realpow_Suc = thm "realpow_Suc";
paulson@14265: 
paulson@14265: val realpow_zero = thm "realpow_zero";
paulson@14265: val realpow_not_zero = thm "realpow_not_zero";
paulson@14265: val realpow_zero_zero = thm "realpow_zero_zero";
paulson@14265: val realpow_inverse = thm "realpow_inverse";
paulson@14265: val realpow_abs = thm "realpow_abs";
paulson@14265: val realpow_add = thm "realpow_add";
paulson@14265: val realpow_one = thm "realpow_one";
paulson@14265: val realpow_two = thm "realpow_two";
paulson@14265: val realpow_gt_zero = thm "realpow_gt_zero";
paulson@14265: val realpow_ge_zero = thm "realpow_ge_zero";
paulson@14265: val realpow_le = thm "realpow_le";
paulson@14265: val realpow_0_left = thm "realpow_0_left";
paulson@14265: val realpow_less = thm "realpow_less";
paulson@14265: val realpow_eq_one = thm "realpow_eq_one";
paulson@14265: val abs_realpow_minus_one = thm "abs_realpow_minus_one";
paulson@14265: val realpow_mult = thm "realpow_mult";
paulson@14265: val realpow_two_le = thm "realpow_two_le";
paulson@14265: val abs_realpow_two = thm "abs_realpow_two";
paulson@14265: val realpow_two_abs = thm "realpow_two_abs";
paulson@14265: val realpow_two_gt_one = thm "realpow_two_gt_one";
paulson@14265: val realpow_ge_one = thm "realpow_ge_one";
paulson@14265: val realpow_ge_one2 = thm "realpow_ge_one2";
paulson@14265: val two_realpow_ge_one = thm "two_realpow_ge_one";
paulson@14265: val two_realpow_gt = thm "two_realpow_gt";
paulson@14265: val realpow_minus_one = thm "realpow_minus_one";
paulson@14265: val realpow_minus_one_odd = thm "realpow_minus_one_odd";
paulson@14265: val realpow_minus_one_even = thm "realpow_minus_one_even";
paulson@14265: val realpow_Suc_less = thm "realpow_Suc_less";
paulson@14265: val realpow_Suc_le = thm "realpow_Suc_le";
paulson@14265: val realpow_zero_le = thm "realpow_zero_le";
paulson@14265: val realpow_Suc_le2 = thm "realpow_Suc_le2";
paulson@14265: val realpow_Suc_le3 = thm "realpow_Suc_le3";
paulson@14265: val realpow_less_le = thm "realpow_less_le";
paulson@14265: val realpow_le_le = thm "realpow_le_le";
paulson@14265: val realpow_Suc_le_self = thm "realpow_Suc_le_self";
paulson@14265: val realpow_Suc_less_one = thm "realpow_Suc_less_one";
paulson@14265: val realpow_le_Suc = thm "realpow_le_Suc";
paulson@14265: val realpow_less_Suc = thm "realpow_less_Suc";
paulson@14265: val realpow_le_Suc2 = thm "realpow_le_Suc2";
paulson@14265: val realpow_gt_ge2 = thm "realpow_gt_ge2";
paulson@14265: val realpow_ge_ge2 = thm "realpow_ge_ge2";
paulson@14265: val realpow_Suc_ge_self2 = thm "realpow_Suc_ge_self2";
paulson@14265: val realpow_ge_self2 = thm "realpow_ge_self2";
paulson@14265: val realpow_minus_mult = thm "realpow_minus_mult";
paulson@14265: val realpow_two_mult_inverse = thm "realpow_two_mult_inverse";
paulson@14265: val realpow_two_minus = thm "realpow_two_minus";
paulson@14265: val realpow_two_disj = thm "realpow_two_disj";
paulson@14265: val realpow_diff = thm "realpow_diff";
paulson@14265: val realpow_real_of_nat = thm "realpow_real_of_nat";
paulson@14265: val realpow_real_of_nat_two_pos = thm "realpow_real_of_nat_two_pos";
paulson@14265: val realpow_increasing = thm "realpow_increasing";
paulson@14265: val realpow_Suc_cancel_eq = thm "realpow_Suc_cancel_eq";
paulson@14265: val realpow_eq_0_iff = thm "realpow_eq_0_iff";
paulson@14265: val zero_less_realpow_abs_iff = thm "zero_less_realpow_abs_iff";
paulson@14265: val zero_le_realpow_abs = thm "zero_le_realpow_abs";
paulson@14265: val real_of_int_power = thm "real_of_int_power";
paulson@14265: val power_real_number_of = thm "power_real_number_of";
paulson@14265: val real_power_two = thm "real_power_two";
paulson@14265: val real_sqr_ge_zero = thm "real_sqr_ge_zero";
paulson@14265: val real_sqr_gt_zero = thm "real_sqr_gt_zero";
paulson@14265: val real_sqr_not_zero = thm "real_sqr_not_zero";
paulson@14268: val real_sum_squares_cancel_a = thm "real_sum_squares_cancel_a";
paulson@14268: val real_mult_inverse_cancel2 = thm "real_mult_inverse_cancel2";
paulson@14268: val real_squared_diff_one_factored = thm "real_squared_diff_one_factored";
paulson@14268: val real_mult_is_one = thm "real_mult_is_one";
paulson@14268: val real_le_add_half_cancel = thm "real_le_add_half_cancel";
paulson@14268: val real_minus_half_eq = thm "real_minus_half_eq";
paulson@14268: val real_mult_inverse_cancel = thm "real_mult_inverse_cancel";
paulson@14268: val real_mult_inverse_cancel2 = thm "real_mult_inverse_cancel2";
paulson@14268: val inverse_real_of_nat_gt_zero = thm "inverse_real_of_nat_gt_zero";
paulson@14268: val inverse_real_of_nat_ge_zero = thm "inverse_real_of_nat_ge_zero";
paulson@14268: val real_sum_squares_not_zero = thm "real_sum_squares_not_zero";
paulson@14268: val real_sum_squares_not_zero2 = thm "real_sum_squares_not_zero2";
paulson@14268: val abs_mult_abs = thm "abs_mult_abs";
paulson@14268: 
paulson@14268: val realpow_divide = thm "realpow_divide";
paulson@14268: val realpow_ge_zero2 = thm "realpow_ge_zero2";
paulson@14268: val realpow_le2 = thm "realpow_le2";
paulson@14268: val realpow_Suc_gt_one = thm "realpow_Suc_gt_one";
paulson@14268: val realpow_two_sum_zero_iff = thm "realpow_two_sum_zero_iff";
paulson@14268: val realpow_two_le_add_order = thm "realpow_two_le_add_order";
paulson@14268: val realpow_two_le_add_order2 = thm "realpow_two_le_add_order2";
paulson@14268: val real_sum_square_gt_zero = thm "real_sum_square_gt_zero";
paulson@14268: val real_sum_square_gt_zero2 = thm "real_sum_square_gt_zero2";
paulson@14268: val real_minus_mult_self_le = thm "real_minus_mult_self_le";
paulson@14268: val realpow_square_minus_le = thm "realpow_square_minus_le";
paulson@14268: val realpow_num_eq_if = thm "realpow_num_eq_if";
paulson@14268: val real_num_zero_less_two_pow = thm "real_num_zero_less_two_pow";
paulson@14268: val lemma_realpow_num_two_mono = thm "lemma_realpow_num_two_mono";
paulson@14268: val lemma_realpow_4 = thm "lemma_realpow_4";
paulson@14268: val lemma_realpow_16 = thm "lemma_realpow_16";
paulson@14268: val zero_le_x_squared = thm "zero_le_x_squared";
paulson@14265: *}
paulson@14265: 
paulson@14265: 
paulson@7077: end