# Theory HyperPow

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theory HyperPow
imports HyperArith HyperNat
begin

    (*  Title       : HyperPow.thy
        Author      : Jacques D. Fleuriot
        Copyright   : 1998  University of Cambridge
        Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
    *)

    header{*Exponentials on the Hyperreals*}

    theory HyperPow
    imports HyperArith HyperNat
    begin

    (* consts hpowr :: "[hypreal,nat] => hypreal" *)

    lemma hpowr_0 [simp]:   "r ^ 0       = (1::hypreal)"
    by (rule power_0)

    lemma hpowr_Suc [simp]: "r ^ (Suc n) = (r::hypreal) * (r ^ n)"
    by (rule power_Suc)

    consts
      "pow"  :: "[hypreal,hypnat] => hypreal"     (infixr "pow" 80)

    defs

      (* hypernatural powers of hyperreals *)
      hyperpow_def [transfer_unfold]:
      "(R::hypreal) pow (N::hypnat) == ( *f2* op ^) R N"

    lemma hrealpow_two: "(r::hypreal) ^ Suc (Suc 0) = r * r"
    by simp

    lemma hrealpow_two_le [simp]: "(0::hypreal) ≤ r ^ Suc (Suc 0)"
    by (auto simp add: zero_le_mult_iff)

    lemma hrealpow_two_le_add_order [simp]:
         "(0::hypreal) ≤ u ^ Suc (Suc 0) + v ^ Suc (Suc 0)"
    by (simp only: hrealpow_two_le add_nonneg_nonneg)

    lemma hrealpow_two_le_add_order2 [simp]:
         "(0::hypreal) ≤ u ^ Suc (Suc 0) + v ^ Suc (Suc 0) + w ^ Suc (Suc 0)"
    by (simp only: hrealpow_two_le add_nonneg_nonneg)

    lemma hypreal_add_nonneg_eq_0_iff:
         "[| 0 ≤ x; 0 ≤ y |] ==> (x+y = 0) = (x = 0 & y = (0::hypreal))"
    by arith

    text{*FIXME: DELETE THESE*}
    lemma hypreal_three_squares_add_zero_iff:
         "(x*x + y*y + z*z = 0) = (x = 0 & y = 0 & z = (0::hypreal))"
    apply (simp only: zero_le_square add_nonneg_nonneg hypreal_add_nonneg_eq_0_iff, auto)
    done

    lemma hrealpow_three_squares_add_zero_iff [simp]:
         "(x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + z ^ Suc (Suc 0) = (0::hypreal)) =
          (x = 0 & y = 0 & z = 0)"
    by (simp only: hypreal_three_squares_add_zero_iff hrealpow_two)

    (*FIXME: This and RealPow.abs_realpow_two should be replaced by an abstract
      result proved in Ring_and_Field*)
    lemma hrabs_hrealpow_two [simp]:
         "abs(x ^ Suc (Suc 0)) = (x::hypreal) ^ Suc (Suc 0)"
    by (simp add: abs_mult)

    lemma two_hrealpow_ge_one [simp]: "(1::hypreal) ≤ 2 ^ n"
    by (insert power_increasing [of 0 n "2::hypreal"], simp)

    lemma two_hrealpow_gt [simp]: "hypreal_of_nat n < 2 ^ n"
    apply (induct_tac "n")
    apply (auto simp add: hypreal_of_nat_Suc left_distrib)
    apply (cut_tac n = n in two_hrealpow_ge_one, arith)
    done

    lemma hrealpow:
        "star_n X ^ m = star_n (%n. (X n::real) ^ m)"
    apply (induct_tac "m")
    apply (auto simp add: star_n_one_num star_n_mult power_0)
    done

    lemma hrealpow_sum_square_expand:
         "(x + (y::hypreal)) ^ Suc (Suc 0) =
          x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + (hypreal_of_nat (Suc (Suc 0)))*x*y"
    by (simp add: right_distrib left_distrib hypreal_of_nat_Suc)

    subsection{*Literal Arithmetic Involving Powers and Type @{typ hypreal}*}

    lemma power_hypreal_of_real_number_of:
         "(number_of v :: hypreal) ^ n = hypreal_of_real ((number_of v) ^ n)"
    by simp
    (* why is this a simp rule? - BH *)
    declare power_hypreal_of_real_number_of [of _ "number_of w", standard, simp]

    lemma hrealpow_HFinite: "x ∈ HFinite ==> x ^ n ∈ HFinite"
    apply (induct_tac "n")
    apply (auto intro: HFinite_mult)
    done

    subsection{*Powers with Hypernatural Exponents*}

    lemma hyperpow: "star_n X pow star_n Y = star_n (%n. X n ^ Y n)"
    by (simp add: hyperpow_def starfun2_star_n)

    lemma hyperpow_zero [simp]: "!!n. (0::hypreal) pow (n + (1::hypnat)) = 0"
    by (transfer, simp)

    lemma hyperpow_not_zero: "!!r n. r ≠ (0::hypreal) ==> r pow n ≠ 0"
    by (transfer, simp)

    lemma hyperpow_inverse:
         "!!r n. r ≠ (0::hypreal) ==> inverse(r pow n) = (inverse r) pow n"
    by (transfer, rule power_inverse)

    lemma hyperpow_hrabs: "!!r n. abs r pow n = abs (r pow n)"
    by (transfer, rule power_abs [symmetric])

    lemma hyperpow_add: "!!r n m. r pow (n + m) = (r pow n) * (r pow m)"
    by (transfer, rule power_add)

    lemma hyperpow_one [simp]: "!!r. r pow (1::hypnat) = r"
    by (transfer, simp)

    lemma hyperpow_two:
         "!!r. r pow ((1::hypnat) + (1::hypnat)) = r * r"
    by (transfer, simp)

    lemma hyperpow_gt_zero: "!!r n. (0::hypreal) < r ==> 0 < r pow n"
    by (transfer, rule zero_less_power)

    lemma hyperpow_ge_zero: "!!r n. (0::hypreal) ≤ r ==> 0 ≤ r pow n"
    by (transfer, rule zero_le_power)

    lemma hyperpow_le:
      "!!x y n. [|(0::hypreal) < x; x ≤ y|] ==> x pow n ≤ y pow n"
    by (transfer, rule power_mono [OF _ order_less_imp_le])

    lemma hyperpow_eq_one [simp]: "!!n. 1 pow n = (1::hypreal)"
    by (transfer, simp)

    lemma hrabs_hyperpow_minus_one [simp]: "!!n. abs(-1 pow n) = (1::hypreal)"
    by (transfer, simp)

    lemma hyperpow_mult: "!!r s n. (r * s) pow n = (r pow n) * (s pow n)"
    by (transfer, rule power_mult_distrib)

    lemma hyperpow_two_le [simp]: "0 ≤ r pow (1 + 1)"
    by (auto simp add: hyperpow_two zero_le_mult_iff)

    lemma hrabs_hyperpow_two [simp]: "abs(x pow (1 + 1)) = x pow (1 + 1)"
    by (simp add: abs_if hyperpow_two_le linorder_not_less)

    lemma hyperpow_two_hrabs [simp]: "abs(x) pow (1 + 1)  = x pow (1 + 1)"
    by (simp add: hyperpow_hrabs)

    text{*The precondition could be weakened to @{term "0≤x"}*}
    lemma hypreal_mult_less_mono:
         "[| u<v;  x<y;  (0::hypreal) < v;  0 < x |] ==> u*x < v* y"
     by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le)

    lemma hyperpow_two_gt_one: "1 < r ==> 1 < r pow (1 + 1)"
    apply (auto simp add: hyperpow_two)
    apply (rule_tac y = "1*1" in order_le_less_trans)
    apply (rule_tac [2] hypreal_mult_less_mono, auto)
    done

    lemma hyperpow_two_ge_one:
         "1 ≤ r ==> 1 ≤ r pow (1 + 1)"
    by (auto dest!: order_le_imp_less_or_eq intro: hyperpow_two_gt_one order_less_imp_le)

    lemma two_hyperpow_ge_one [simp]: "(1::hypreal) ≤ 2 pow n"
    apply (rule_tac y = "1 pow n" in order_trans)
    apply (rule_tac [2] hyperpow_le, auto)
    done

    lemma hyperpow_minus_one2 [simp]:
         "!!n. -1 pow ((1 + 1)*n) = (1::hypreal)"
    by (transfer, simp)

    lemma hyperpow_less_le:
         "!!r n N. [|(0::hypreal) ≤ r; r ≤ 1; n < N|] ==> r pow N ≤ r pow n"
    by (transfer, rule power_decreasing [OF order_less_imp_le])

    lemma hyperpow_SHNat_le:
         "[| 0 ≤ r;  r ≤ (1::hypreal);  N ∈ HNatInfinite |]
          ==> ALL n: Nats. r pow N ≤ r pow n"
    by (auto intro!: hyperpow_less_le simp add: HNatInfinite_iff)

    lemma hyperpow_realpow:
          "(hypreal_of_real r) pow (hypnat_of_nat n) = hypreal_of_real (r ^ n)"
    by (simp add: star_of_def hypnat_of_nat_eq hyperpow)

    lemma hyperpow_SReal [simp]:
         "(hypreal_of_real r) pow (hypnat_of_nat n) ∈ Reals"
    by (simp del: star_of_power add: hyperpow_realpow SReal_def)

    lemma hyperpow_zero_HNatInfinite [simp]:
         "N ∈ HNatInfinite ==> (0::hypreal) pow N = 0"
    by (drule HNatInfinite_is_Suc, auto)

    lemma hyperpow_le_le:
         "[| (0::hypreal) ≤ r; r ≤ 1; n ≤ N |] ==> r pow N ≤ r pow n"
    apply (drule order_le_less [of n, THEN iffD1])
    apply (auto intro: hyperpow_less_le)
    done

    lemma hyperpow_Suc_le_self2:
         "[| (0::hypreal) ≤ r; r < 1 |] ==> r pow (n + (1::hypnat)) ≤ r"
    apply (drule_tac n = " (1::hypnat) " in hyperpow_le_le)
    apply auto
    done

    lemma lemma_Infinitesimal_hyperpow:
         "[| x ∈ Infinitesimal; 0 < N |] ==> abs (x pow N) ≤ abs x"
    apply (unfold Infinitesimal_def)
    apply (auto intro!: hyperpow_Suc_le_self2
              simp add: hyperpow_hrabs [symmetric] hypnat_gt_zero_iff2 abs_ge_zero)
    done

    lemma Infinitesimal_hyperpow:
         "[| x ∈ Infinitesimal; 0 < N |] ==> x pow N ∈ Infinitesimal"
    apply (rule hrabs_le_Infinitesimal)
    apply (rule_tac [2] lemma_Infinitesimal_hyperpow, auto)
    done

    lemma hrealpow_hyperpow_Infinitesimal_iff:
         "(x ^ n ∈ Infinitesimal) = (x pow (hypnat_of_nat n) ∈ Infinitesimal)"
    apply (cases x)
    apply (simp add: hrealpow hyperpow hypnat_of_nat_eq)
    done

    lemma Infinitesimal_hrealpow:
         "[| x ∈ Infinitesimal; 0 < n |] ==> x ^ n ∈ Infinitesimal"
    by (simp add: hrealpow_hyperpow_Infinitesimal_iff Infinitesimal_hyperpow)

    ML
    {*
    val hrealpow_two = thm "hrealpow_two";
    val hrealpow_two_le = thm "hrealpow_two_le";
    val hrealpow_two_le_add_order = thm "hrealpow_two_le_add_order";
    val hrealpow_two_le_add_order2 = thm "hrealpow_two_le_add_order2";
    val hypreal_add_nonneg_eq_0_iff = thm "hypreal_add_nonneg_eq_0_iff";
    val hypreal_three_squares_add_zero_iff = thm "hypreal_three_squares_add_zero_iff";
    val hrealpow_three_squares_add_zero_iff = thm "hrealpow_three_squares_add_zero_iff";
    val hrabs_hrealpow_two = thm "hrabs_hrealpow_two";
    val two_hrealpow_ge_one = thm "two_hrealpow_ge_one";
    val two_hrealpow_gt = thm "two_hrealpow_gt";
    val hrealpow = thm "hrealpow";
    val hrealpow_sum_square_expand = thm "hrealpow_sum_square_expand";
    val power_hypreal_of_real_number_of = thm "power_hypreal_of_real_number_of";
    val hrealpow_HFinite = thm "hrealpow_HFinite";
    val hyperpow = thm "hyperpow";
    val hyperpow_zero = thm "hyperpow_zero";
    val hyperpow_not_zero = thm "hyperpow_not_zero";
    val hyperpow_inverse = thm "hyperpow_inverse";
    val hyperpow_hrabs = thm "hyperpow_hrabs";
    val hyperpow_add = thm "hyperpow_add";
    val hyperpow_one = thm "hyperpow_one";
    val hyperpow_two = thm "hyperpow_two";
    val hyperpow_gt_zero = thm "hyperpow_gt_zero";
    val hyperpow_ge_zero = thm "hyperpow_ge_zero";
    val hyperpow_le = thm "hyperpow_le";
    val hyperpow_eq_one = thm "hyperpow_eq_one";
    val hrabs_hyperpow_minus_one = thm "hrabs_hyperpow_minus_one";
    val hyperpow_mult = thm "hyperpow_mult";
    val hyperpow_two_le = thm "hyperpow_two_le";
    val hrabs_hyperpow_two = thm "hrabs_hyperpow_two";
    val hyperpow_two_hrabs = thm "hyperpow_two_hrabs";
    val hyperpow_two_gt_one = thm "hyperpow_two_gt_one";
    val hyperpow_two_ge_one = thm "hyperpow_two_ge_one";
    val two_hyperpow_ge_one = thm "two_hyperpow_ge_one";
    val hyperpow_minus_one2 = thm "hyperpow_minus_one2";
    val hyperpow_less_le = thm "hyperpow_less_le";
    val hyperpow_SHNat_le = thm "hyperpow_SHNat_le";
    val hyperpow_realpow = thm "hyperpow_realpow";
    val hyperpow_SReal = thm "hyperpow_SReal";
    val hyperpow_zero_HNatInfinite = thm "hyperpow_zero_HNatInfinite";
    val hyperpow_le_le = thm "hyperpow_le_le";
    val hyperpow_Suc_le_self2 = thm "hyperpow_Suc_le_self2";
    val lemma_Infinitesimal_hyperpow = thm "lemma_Infinitesimal_hyperpow";
    val Infinitesimal_hyperpow = thm "Infinitesimal_hyperpow";
    val hrealpow_hyperpow_Infinitesimal_iff = thm "hrealpow_hyperpow_Infinitesimal_iff";
    val Infinitesimal_hrealpow = thm "Infinitesimal_hrealpow";
    *}

    end

lemma hpowr_0:

      r ^ 0 = 1

lemma hpowr_Suc:

      r ^ Suc n = r * r ^ n

lemma hrealpow_two:

      r ^ Suc (Suc 0) = r * r

lemma hrealpow_two_le:

      0 ≤ r ^ Suc (Suc 0)

lemma hrealpow_two_le_add_order:

      0 ≤ u ^ Suc (Suc 0) + v ^ Suc (Suc 0)

lemma hrealpow_two_le_add_order2:

      0 ≤ u ^ Suc (Suc 0) + v ^ Suc (Suc 0) + w ^ Suc (Suc 0)

lemma hypreal_add_nonneg_eq_0_iff:

      [| 0 ≤ x; 0 ≤ y |] ==> (x + y = 0) = (x = 0 ∧ y = 0)

lemma hypreal_three_squares_add_zero_iff:

      (x * x + y * y + z * z = 0) = (x = 0 ∧ y = 0 ∧ z = 0)

lemma hrealpow_three_squares_add_zero_iff:

      (x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + z ^ Suc (Suc 0) = 0) =
      (x = 0 ∧ y = 0 ∧ z = 0)

lemma hrabs_hrealpow_two:

      ¦x ^ Suc (Suc 0)¦ = x ^ Suc (Suc 0)

lemma two_hrealpow_ge_one:

      1 ≤ 2 ^ n

lemma two_hrealpow_gt:

      hypreal_of_nat n < 2 ^ n

lemma hrealpow:

      star_n X ^ m = star_n (%n. X n ^ m)

lemma hrealpow_sum_square_expand:

      (x + y) ^ Suc (Suc 0) =
      x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + hypreal_of_nat (Suc (Suc 0)) * x * y

### Literal Arithmetic Involving Powers and Type @{typ hypreal}

lemma power_hypreal_of_real_number_of:

      number_of v ^ n = star_of (number_of v ^ n)

lemma hrealpow_HFinite:

      x ∈ HFinite ==> x ^ n ∈ HFinite

### Powers with Hypernatural Exponents

lemma hyperpow:

      star_n X pow star_n Y = star_n (%n. X n ^ Y n)

lemma hyperpow_zero:

      0 pow (n + 1) = 0

lemma hyperpow_not_zero:

      r ≠ 0 ==> r pow n ≠ 0

lemma hyperpow_inverse:

      r ≠ 0 ==> inverse (r pow n) = inverse r pow n

lemma hyperpow_hrabs:

      ¦r¦ pow n = ¦r pow n¦

lemma hyperpow_add:

      r pow (n + m) = r pow n * r pow m

lemma hyperpow_one:

      r pow 1 = r

lemma hyperpow_two:

      r pow (1 + 1) = r * r

lemma hyperpow_gt_zero:

      0 < r ==> 0 < r pow n

lemma hyperpow_ge_zero:

      0 ≤ r ==> 0 ≤ r pow n

lemma hyperpow_le:

      [| 0 < x; x ≤ y |] ==> x pow n ≤ y pow n

lemma hyperpow_eq_one:

      1 pow n = 1

lemma hrabs_hyperpow_minus_one:

      ¦-1 pow n¦ = 1

lemma hyperpow_mult:

      (r * s) pow n = r pow n * s pow n

lemma hyperpow_two_le:

      0 ≤ r pow (1 + 1)

lemma hrabs_hyperpow_two:

      ¦x pow (1 + 1)¦ = x pow (1 + 1)

lemma hyperpow_two_hrabs:

      ¦x¦ pow (1 + 1) = x pow (1 + 1)

lemma hypreal_mult_less_mono:

      [| u < v; x < y; 0 < v; 0 < x |] ==> u * x < v * y

lemma hyperpow_two_gt_one:

      1 < r ==> 1 < r pow (1 + 1)

lemma hyperpow_two_ge_one:

      1 ≤ r ==> 1 ≤ r pow (1 + 1)

lemma two_hyperpow_ge_one:

      1 ≤ 2 pow n

lemma hyperpow_minus_one2:

      -1 pow ((1 + 1) * n) = 1

lemma hyperpow_less_le:

      [| 0 ≤ r; r ≤ 1; n < N |] ==> r pow N ≤ r pow n

lemma hyperpow_SHNat_le:

      [| 0 ≤ r; r ≤ 1; N ∈ HNatInfinite |] ==> ∀n∈Nats. r pow N ≤ r pow n

lemma hyperpow_realpow:

      star_of r pow star_of n = star_of (r ^ n)

lemma hyperpow_SReal:

      star_of r pow star_of n ∈ Reals

lemma hyperpow_zero_HNatInfinite:

      N ∈ HNatInfinite ==> 0 pow N = 0

lemma hyperpow_le_le:

      [| 0 ≤ r; r ≤ 1; n ≤ N |] ==> r pow N ≤ r pow n

lemma hyperpow_Suc_le_self2:

      [| 0 ≤ r; r < 1 |] ==> r pow (n + 1) ≤ r

lemma lemma_Infinitesimal_hyperpow:

      [| x ∈ Infinitesimal; 0 < N |] ==> ¦x pow N¦ ≤ ¦x¦

lemma Infinitesimal_hyperpow:

      [| x ∈ Infinitesimal; 0 < N |] ==> x pow N ∈ Infinitesimal

lemma hrealpow_hyperpow_Infinitesimal_iff:

      (x ^ n ∈ Infinitesimal) = (x pow star_of n ∈ Infinitesimal)

lemma Infinitesimal_hrealpow:

      [| x ∈ Infinitesimal; 0 < n |] ==> x ^ n ∈ Infinitesimal