(*  Title       : HyperPow.ML

    Author      : Jacques D. Fleuriot  

    Copyright   : 1998  University of Cambridge

    Description : Natural Powers of hyperreals theory

Exponentials on the hyperreals

*)

Goal "(0::hypreal) ^ (Suc n) = 0";

by Auto_tac;

qed "hrealpow_zero";

Addsimps [hrealpow_zero];

Goal "r ~= (0::hypreal) --> r ^ n ~= 0";

by (induct_tac "n" 1);

by Auto_tac;

qed_spec_mp "hrealpow_not_zero";

Goal "r ~= (0::hypreal) --> inverse(r ^ n) = (inverse r) ^ n";

by (induct_tac "n" 1);

by Auto_tac;

by (forw_inst_tac [("n","n")] hrealpow_not_zero 1);

by (auto_tac (claset(), simpset() addsimps [hypreal_inverse_distrib]));

qed_spec_mp "hrealpow_inverse";

Goal "abs (r::hypreal) ^ n = abs (r ^ n)";

by (induct_tac "n" 1);

by (auto_tac (claset(), simpset() addsimps [hrabs_mult]));

qed "hrealpow_hrabs";

Goal "(r::hypreal) ^ (n + m) = (r ^ n) * (r ^ m)";

by (induct_tac "n" 1);

by (auto_tac (claset(), simpset() addsimps hypreal_mult_ac));

qed "hrealpow_add";

Goal "(r::hypreal) ^ Suc 0 = r";

by (Simp_tac 1);

qed "hrealpow_one";

Addsimps [hrealpow_one];

Goal "(r::hypreal) ^ Suc (Suc 0) = r * r";

by (Simp_tac 1);

qed "hrealpow_two";

Goal "(0::hypreal) <= r --> 0 <= r ^ n";

by (induct_tac "n" 1);

by (auto_tac (claset(), simpset() addsimps [hypreal_0_le_mult_iff]));

qed_spec_mp "hrealpow_ge_zero";

Goal "(0::hypreal) < r --> 0 < r ^ n";

by (induct_tac "n" 1);

by (auto_tac (claset(), simpset() addsimps [hypreal_0_less_mult_iff]));

qed_spec_mp "hrealpow_gt_zero";

Goal "x <= y & (0::hypreal) < x --> x ^ n <= y ^ n";

by (induct_tac "n" 1);

by (auto_tac (claset() addSIs [hypreal_mult_le_mono], simpset()));

by (asm_simp_tac (simpset() addsimps [hrealpow_ge_zero]) 1);

qed_spec_mp "hrealpow_le";

Goal "x < y & (0::hypreal) < x & 0 < n --> x ^ n < y ^ n";

by (induct_tac "n" 1);

by (auto_tac (claset() addIs [hypreal_mult_less_mono,gr0I],

              simpset() addsimps [hrealpow_gt_zero]));

qed "hrealpow_less";

Goal "1 ^ n = (1::hypreal)";

by (induct_tac "n" 1);

by Auto_tac;

qed "hrealpow_eq_one";

Addsimps [hrealpow_eq_one];

Goal "abs(-(1 ^ n)) = (1::hypreal)";

by Auto_tac;  

qed "hrabs_minus_hrealpow_one";

Addsimps [hrabs_minus_hrealpow_one];

Goal "abs(-1 ^ n) = (1::hypreal)";

by (induct_tac "n" 1);

by Auto_tac;  

qed "hrabs_hrealpow_minus_one";

Addsimps [hrabs_hrealpow_minus_one];

Goal "((r::hypreal) * s) ^ n = (r ^ n) * (s ^ n)";

by (induct_tac "n" 1);

by (auto_tac (claset(), simpset() addsimps hypreal_mult_ac));

qed "hrealpow_mult";

Goal "(0::hypreal) <= r ^ Suc (Suc 0)";

by (auto_tac (claset(), simpset() addsimps [hypreal_0_le_mult_iff]));

qed "hrealpow_two_le";

Addsimps [hrealpow_two_le];

Goal "(0::hypreal) <= u ^ Suc (Suc 0) + v ^ Suc (Suc 0)";

by (simp_tac (HOL_ss addsimps [hrealpow_two_le, hypreal_le_add_order]) 1); 

qed "hrealpow_two_le_add_order";

Addsimps [hrealpow_two_le_add_order];

Goal "(0::hypreal) <= u ^ Suc (Suc 0) + v ^ Suc (Suc 0) + w ^ Suc (Suc 0)";

by (simp_tac (HOL_ss addsimps [hrealpow_two_le, hypreal_le_add_order]) 1); 

qed "hrealpow_two_le_add_order2";

Addsimps [hrealpow_two_le_add_order2];

Goal "[| 0 <= x; 0 <= y |] ==> (x+y = 0) = (x = 0 & y = (0::hypreal))";

by (auto_tac (claset() addIs [order_antisym], simpset()));

qed "hypreal_add_nonneg_eq_0_iff";

Goal "(x*y = 0) = (x = 0 | y = (0::hypreal))";

by Auto_tac;

qed "hypreal_mult_is_0";

Goal "(x*x + y*y + z*z = 0) = (x = 0 & y = 0 & z = (0::hypreal))";

by (simp_tac (HOL_ss addsimps [hypreal_le_square, hypreal_le_add_order, 

                         hypreal_add_nonneg_eq_0_iff, hypreal_mult_is_0]) 1);

qed "hypreal_three_squares_add_zero_iff";

Goal "(x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + z ^ Suc (Suc 0) = (0::hypreal)) = (x = 0 & y = 0 & z = 0)";

by (simp_tac (HOL_ss addsimps

               [hypreal_three_squares_add_zero_iff, hrealpow_two]) 1);

qed "hrealpow_three_squares_add_zero_iff";

Addsimps [hrealpow_three_squares_add_zero_iff];

Goal "abs(x ^ Suc (Suc 0)) = (x::hypreal) ^ Suc (Suc 0)";

by (auto_tac (claset(), 

              simpset() addsimps [hrabs_def, hypreal_0_le_mult_iff])); 

qed "hrabs_hrealpow_two";

Addsimps [hrabs_hrealpow_two];

Goal "abs(x) ^ Suc (Suc 0) = (x::hypreal) ^ Suc (Suc 0)";

by (simp_tac (simpset() addsimps [hrealpow_hrabs, hrabs_eqI1] 

                        delsimps [hpowr_Suc]) 1);

qed "hrealpow_two_hrabs";

Addsimps [hrealpow_two_hrabs];

Goal "(1::hypreal) < r ==> 1 < r ^ Suc (Suc 0)";

by (auto_tac (claset(), simpset() addsimps [hrealpow_two]));

by (res_inst_tac [("y","1*1")] order_le_less_trans 1); 

by (rtac hypreal_mult_less_mono 2); 

by Auto_tac;  

qed "hrealpow_two_gt_one";

Goal "(1::hypreal) <= r ==> 1 <= r ^ Suc (Suc 0)";

by (etac (order_le_imp_less_or_eq RS disjE) 1);

by (etac (hrealpow_two_gt_one RS order_less_imp_le) 1);

by Auto_tac;  

qed "hrealpow_two_ge_one";

Goal "(1::hypreal) <= 2 ^ n";

by (res_inst_tac [("y","1 ^ n")] order_trans 1);

by (rtac hrealpow_le 2);

by Auto_tac;

qed "two_hrealpow_ge_one";

Goal "hypreal_of_nat n < 2 ^ n";

by (induct_tac "n" 1);

by (auto_tac (claset(),

        simpset() addsimps [hypreal_of_nat_Suc, hypreal_add_mult_distrib]));

by (cut_inst_tac [("n","n")] two_hrealpow_ge_one 1);

by (arith_tac 1);

qed "two_hrealpow_gt";

Addsimps [two_hrealpow_gt,two_hrealpow_ge_one];

Goal "-1 ^ (2*n) = (1::hypreal)";

by (induct_tac "n" 1);

by Auto_tac;

qed "hrealpow_minus_one";

Goal "n+n = (2*n::nat)";

by Auto_tac; 

qed "double_lemma";

(*ugh: need to get rid fo the n+n*)

Goal "-1 ^ (n + n) = (1::hypreal)";

by (auto_tac (claset(), 

              simpset() addsimps [double_lemma, hrealpow_minus_one]));

qed "hrealpow_minus_one2";

Addsimps [hrealpow_minus_one2];

Goal "(-(x::hypreal)) ^ Suc (Suc 0) = x ^ Suc (Suc 0)";

by Auto_tac;

qed "hrealpow_minus_two";

Addsimps [hrealpow_minus_two];

Goal "(0::hypreal) < r & r < 1 --> r ^ Suc n < r ^ n";

by (induct_tac "n" 1);

by (auto_tac (claset(),

              simpset() addsimps [hypreal_mult_less_mono2]));

qed_spec_mp "hrealpow_Suc_less";

Goal "(0::hypreal) <= r & r < 1 --> r ^ Suc n <= r ^ n";

by (induct_tac "n" 1);

by (auto_tac (claset() addIs [order_less_imp_le]

                       addSDs [order_le_imp_less_or_eq,hrealpow_Suc_less],

              simpset() addsimps [hypreal_mult_less_mono2]));

qed_spec_mp "hrealpow_Suc_le";

Goal "Abs_hypreal(hyprel``{%n. X n}) ^ m = Abs_hypreal(hyprel``{%n. (X n) ^ m})";

by (induct_tac "m" 1);

by (auto_tac (claset(),

              simpset() addsimps [hypreal_one_def, hypreal_mult]));

qed "hrealpow";

Goal "(x + (y::hypreal)) ^ Suc (Suc 0) = \

\     x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + (hypreal_of_nat (Suc (Suc 0)))*x*y";

by (simp_tac (simpset() addsimps

              [hypreal_add_mult_distrib2, hypreal_add_mult_distrib, 

               hypreal_of_nat_zero, hypreal_of_nat_Suc]) 1);

qed "hrealpow_sum_square_expand";

(*** Literal arithmetic involving powers, type hypreal ***)

Goal "hypreal_of_real (x ^ n) = hypreal_of_real x ^ n";

by (induct_tac "n" 1); 

by (ALLGOALS (asm_simp_tac (simpset() addsimps [nat_mult_distrib])));

qed "hypreal_of_real_power";

Addsimps [hypreal_of_real_power];

Goal "(number_of v :: hypreal) ^ n = hypreal_of_real ((number_of v) ^ n)";

by (simp_tac (HOL_ss addsimps [hypreal_number_of_def, 

                               hypreal_of_real_power]) 1);

qed "power_hypreal_of_real_number_of";

Addsimps [inst "n" "number_of ?w" power_hypreal_of_real_number_of];

(*---------------------------------------------------------------

   we'll prove the following theorem by going down to the

   level of the ultrafilter and relying on the analogous

   property for the real rather than prove it directly 

   using induction: proof is much simpler this way!

 ---------------------------------------------------------------*)

Goal "[|(0::hypreal) <= x; 0 <= y;x ^ Suc n <= y ^ Suc n |] ==> x <= y";

by (full_simp_tac (simpset() addsimps [hypreal_zero_def]) 1);

by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);

by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);

by (auto_tac (claset(),

              simpset() addsimps [hrealpow,hypreal_le,hypreal_mult]));

by (ultra_tac (claset() addIs [realpow_increasing], simpset()) 1);

qed "hrealpow_increasing";

(*By antisymmetry with the above we conclude x=y, replacing the deleted 

  theorem hrealpow_Suc_cancel_eq*)

Goal "x : HFinite --> x ^ n : HFinite";

by (induct_tac "n" 1);

by (auto_tac (claset() addIs [HFinite_mult], simpset()));

qed_spec_mp "hrealpow_HFinite";

(*---------------------------------------------------------------

                  Hypernaturals Powers

 --------------------------------------------------------------*)

Goalw [congruent_def]

     "congruent hyprel \

\    (%X Y. hyprel``{%n. ((X::nat=>real) n ^ (Y::nat=>nat) n)})";

by (safe_tac (claset() addSIs [ext]));

by (ALLGOALS(Fuf_tac));

qed "hyperpow_congruent";

Goalw [hyperpow_def]

  "Abs_hypreal(hyprel``{%n. X n}) pow Abs_hypnat(hypnatrel``{%n. Y n}) = \

\  Abs_hypreal(hyprel``{%n. X n ^ Y n})";

by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);

by (auto_tac (claset() addSIs [lemma_hyprel_refl,bexI],

    simpset() addsimps [hyprel_in_hypreal RS 

    Abs_hypreal_inverse,equiv_hyprel,hyperpow_congruent]));

by (Fuf_tac 1);

qed "hyperpow";

Goalw [hypnat_one_def] "(0::hypreal) pow (n + (1::hypnat)) = 0";

by (simp_tac (simpset() addsimps [hypreal_zero_def]) 1);

by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);

by (auto_tac (claset(), simpset() addsimps [hyperpow,hypnat_add]));

qed "hyperpow_zero";

Addsimps [hyperpow_zero];

Goal "r ~= (0::hypreal) --> r pow n ~= 0";

by (simp_tac (simpset() addsimps [hypreal_zero_def]) 1);

by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);

by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);

by (auto_tac (claset(), simpset() addsimps [hyperpow]));

by (dtac FreeUltrafilterNat_Compl_mem 1);

by (fuf_empty_tac (claset() addIs [realpow_not_zero RS notE],

    simpset()) 1);

qed_spec_mp "hyperpow_not_zero";

Goal "r ~= (0::hypreal) --> inverse(r pow n) = (inverse r) pow n";

by (simp_tac (simpset() addsimps [hypreal_zero_def]) 1);

by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);

by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);

by (auto_tac (claset() addSDs [FreeUltrafilterNat_Compl_mem],

              simpset() addsimps [hypreal_inverse,hyperpow]));

by (rtac FreeUltrafilterNat_subset 1);

by (auto_tac (claset() addDs [realpow_not_zero] 

                       addIs [realpow_inverse], 

              simpset()));

qed "hyperpow_inverse";

Goal "abs r pow n = abs (r pow n)";

by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);

by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);

by (auto_tac (claset(),

              simpset() addsimps [hypreal_hrabs, hyperpow,realpow_abs RS sym]));

qed "hyperpow_hrabs";

Goal "r pow (n + m) = (r pow n) * (r pow m)";

by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);

by (res_inst_tac [("z","m")] eq_Abs_hypnat 1);

by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);

by (auto_tac (claset(),

          simpset() addsimps [hyperpow,hypnat_add, hypreal_mult,realpow_add]));

qed "hyperpow_add";

Goalw [hypnat_one_def] "r pow (1::hypnat) = r";

by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);

by (auto_tac (claset(), simpset() addsimps [hyperpow]));

qed "hyperpow_one";

Addsimps [hyperpow_one];

Goalw [hypnat_one_def] 

     "r pow ((1::hypnat) + (1::hypnat)) = r * r";

by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);

by (auto_tac (claset(),

              simpset() addsimps [hyperpow,hypnat_add, hypreal_mult]));

qed "hyperpow_two";

Goal "(0::hypreal) < r --> 0 < r pow n";

by (simp_tac (simpset() addsimps [hypreal_zero_def]) 1);

by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);

by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);

by (auto_tac (claset() addSEs [FreeUltrafilterNat_subset, realpow_gt_zero],

              simpset() addsimps [hyperpow,hypreal_less, hypreal_le]));

qed_spec_mp "hyperpow_gt_zero";

Goal "(0::hypreal) <= r --> 0 <= r pow n";

by (simp_tac (simpset() addsimps [hypreal_zero_def]) 1);

by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);

by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);

by (auto_tac (claset() addSEs [FreeUltrafilterNat_subset, realpow_ge_zero],

              simpset() addsimps [hyperpow,hypreal_le]));

qed "hyperpow_ge_zero";

Goal "(0::hypreal) < x & x <= y --> x pow n <= y pow n";

by (full_simp_tac (simpset() addsimps [hypreal_zero_def]) 1);

by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);

by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);

by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);

by (auto_tac (claset(),

              simpset() addsimps [hyperpow,hypreal_le,hypreal_less]));

by (etac (FreeUltrafilterNat_Int RS FreeUltrafilterNat_subset) 1

    THEN assume_tac 1);

by (auto_tac (claset() addIs [realpow_le], simpset()));

qed_spec_mp "hyperpow_le";

Goal "1 pow n = (1::hypreal)";

by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);

by (auto_tac (claset(), simpset() addsimps [hypreal_one_def, hyperpow]));

qed "hyperpow_eq_one";

Addsimps [hyperpow_eq_one];

Goal "abs(-(1 pow n)) = (1::hypreal)";

by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);

by (auto_tac (claset(), 

              simpset() addsimps [hyperpow, hypreal_hrabs, hypreal_one_def]));

qed "hrabs_minus_hyperpow_one";

Addsimps [hrabs_minus_hyperpow_one];

Goal "abs(-1 pow n) = (1::hypreal)";

by (subgoal_tac "abs((- (1::hypreal)) pow n) = (1::hypreal)" 1);

by (Asm_full_simp_tac 1); 

by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);

by (auto_tac (claset(),

              simpset() addsimps [hypreal_one_def, hyperpow,hypreal_minus,

                                  hypreal_hrabs]));

qed "hrabs_hyperpow_minus_one";

Addsimps [hrabs_hyperpow_minus_one];

Goal "(r * s) pow n = (r pow n) * (s pow n)";

by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);

by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);

by (res_inst_tac [("z","s")] eq_Abs_hypreal 1);

by (auto_tac (claset(),

       simpset() addsimps [hyperpow, hypreal_mult,realpow_mult]));

qed "hyperpow_mult";

Goal "(0::hypreal) <= r pow ((1::hypnat) + (1::hypnat))";

by (auto_tac (claset(), 

              simpset() addsimps [hyperpow_two, hypreal_0_le_mult_iff]));

qed "hyperpow_two_le";

Addsimps [hyperpow_two_le];

Goal "abs(x pow ((1::hypnat) + (1::hypnat))) = x pow ((1::hypnat) + (1::hypnat))";

by (simp_tac (simpset() addsimps [hrabs_eqI1]) 1);

qed "hrabs_hyperpow_two";

Addsimps [hrabs_hyperpow_two];

Goal "abs(x) pow ((1::hypnat) + (1::hypnat))  = x pow ((1::hypnat) + (1::hypnat))";

by (simp_tac (simpset() addsimps [hyperpow_hrabs,hrabs_eqI1]) 1);

qed "hyperpow_two_hrabs";

Addsimps [hyperpow_two_hrabs];

(*? very similar to hrealpow_two_gt_one *)

Goal "(1::hypreal) < r ==> 1 < r pow ((1::hypnat) + (1::hypnat))";

by (auto_tac (claset(), simpset() addsimps [hyperpow_two]));

by (res_inst_tac [("y","1*1")] order_le_less_trans 1); 

by (rtac hypreal_mult_less_mono 2); 

by Auto_tac;  

qed "hyperpow_two_gt_one";

Goal "(1::hypreal) <= r ==> 1 <= r pow ((1::hypnat) + (1::hypnat))";

by (auto_tac (claset() addSDs [order_le_imp_less_or_eq] 

                       addIs [hyperpow_two_gt_one,order_less_imp_le],

              simpset()));

qed "hyperpow_two_ge_one";

Goal "(1::hypreal) <= 2 pow n";

by (res_inst_tac [("y","1 pow n")] order_trans 1);

by (rtac hyperpow_le 2);

by Auto_tac;

qed "two_hyperpow_ge_one";

Addsimps [two_hyperpow_ge_one];

Addsimps [simplify (simpset()) realpow_minus_one];

Goal "-1 pow (((1::hypnat) + (1::hypnat))*n) = (1::hypreal)";

by (subgoal_tac "(-((1::hypreal))) pow (((1::hypnat) + (1::hypnat))*n) = (1::hypreal)" 1);

by (Asm_full_simp_tac 1); 

by (simp_tac (HOL_ss addsimps [hypreal_one_def]) 1);

by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);

by (auto_tac (claset(),

              simpset() addsimps [double_lemma, hyperpow, hypnat_add,

                                  hypreal_minus]));

qed "hyperpow_minus_one2";

Addsimps [hyperpow_minus_one2];

Goalw [hypnat_one_def]

     "(0::hypreal) < r & r < 1 --> r pow (n + (1::hypnat)) < r pow n";

by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);

by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);

by (auto_tac (claset() addSDs [conjI RS realpow_Suc_less] 

                  addEs [FreeUltrafilterNat_Int RS FreeUltrafilterNat_subset],

              simpset() addsimps [hypreal_zero_def, hypreal_one_def, 

                                  hyperpow, hypreal_less, hypnat_add]));

qed_spec_mp "hyperpow_Suc_less";

Goalw [hypnat_one_def]

     "0 <= r & r < (1::hypreal) --> r pow (n + (1::hypnat)) <= r pow n";

by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);

by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);

by (auto_tac (claset() addSDs [conjI RS realpow_Suc_le] 

                 addEs [FreeUltrafilterNat_Int RS FreeUltrafilterNat_subset],

              simpset() addsimps [hypreal_zero_def, hypreal_one_def, hyperpow,

                                  hypreal_le,hypnat_add, hypreal_less]));

qed_spec_mp "hyperpow_Suc_le";

Goalw [hypnat_one_def]

     "(0::hypreal) <= r & r < 1 & n < N --> r pow N <= r pow n";

by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);

by (res_inst_tac [("z","N")] eq_Abs_hypnat 1);

by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);

by (auto_tac (claset(),

              simpset() addsimps [hyperpow, hypreal_le,hypreal_less,

                           hypnat_less, hypreal_zero_def, hypreal_one_def]));

by (etac (FreeUltrafilterNat_Int RS FreeUltrafilterNat_subset) 1);

by (etac FreeUltrafilterNat_Int 1);

by (auto_tac (claset() addSDs [conjI RS realpow_less_le], simpset()));

qed_spec_mp "hyperpow_less_le";

Goal "[| (0::hypreal) <= r;  r < 1 |]  \

\     ==> ALL N n. n < N --> r pow N <= r pow n";

by (blast_tac (claset() addSIs [hyperpow_less_le]) 1);

qed "hyperpow_less_le2";

Goal "[| 0 <= r;  r < (1::hypreal);  N : HNatInfinite |]  \

\     ==> ALL n: Nats. r pow N <= r pow n";

by (auto_tac (claset() addSIs [hyperpow_less_le],

              simpset() addsimps [HNatInfinite_iff]));

qed "hyperpow_SHNat_le";

Goalw [hypreal_of_real_def,hypnat_of_nat_def] 

      "(hypreal_of_real r) pow (hypnat_of_nat n) = hypreal_of_real (r ^ n)";

by (auto_tac (claset(), simpset() addsimps [hyperpow]));

qed "hyperpow_realpow";

Goalw [SReal_def] "(hypreal_of_real r) pow (hypnat_of_nat n) : Reals";

by (simp_tac (simpset() delsimps [hypreal_of_real_power]

			addsimps [hyperpow_realpow]) 1); 

qed "hyperpow_SReal";

Addsimps [hyperpow_SReal];

Goal "N : HNatInfinite ==> (0::hypreal) pow N = 0";

by (dtac HNatInfinite_is_Suc 1);

by Auto_tac;

qed "hyperpow_zero_HNatInfinite";

Addsimps [hyperpow_zero_HNatInfinite];

Goal "[| (0::hypreal) <= r; r < 1; n <= N |] ==> r pow N <= r pow n";

by (dres_inst_tac [("y","N")] hypnat_le_imp_less_or_eq 1);

by (auto_tac (claset() addIs [hyperpow_less_le], simpset()));

qed "hyperpow_le_le";

Goal "[| (0::hypreal) < r; r < 1 |] ==> r pow (n + (1::hypnat)) <= r";

by (dres_inst_tac [("n","(1::hypnat)")] (order_less_imp_le RS hyperpow_le_le) 1);

by Auto_tac;

qed "hyperpow_Suc_le_self";

Goal "[| (0::hypreal) <= r; r < 1 |] ==> r pow (n + (1::hypnat)) <= r";

by (dres_inst_tac [("n","(1::hypnat)")] hyperpow_le_le 1);

by Auto_tac;

qed "hyperpow_Suc_le_self2";

Goalw [Infinitesimal_def]

     "[| x : Infinitesimal; 0 < N |] ==> abs (x pow N) <= abs x";

by (auto_tac (claset() addSIs [hyperpow_Suc_le_self2],

              simpset() addsimps [hyperpow_hrabs RS sym,

                                  hypnat_gt_zero_iff2, hrabs_ge_zero]));

qed "lemma_Infinitesimal_hyperpow";

Goal "[| x : Infinitesimal; 0 < N |] ==> x pow N : Infinitesimal";

by (rtac hrabs_le_Infinitesimal 1);

by (rtac lemma_Infinitesimal_hyperpow 2); 

by Auto_tac;  

qed "Infinitesimal_hyperpow";

Goalw [hypnat_of_nat_def] 

     "(x ^ n : Infinitesimal) = (x pow (hypnat_of_nat n) : Infinitesimal)";

by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);

by (auto_tac (claset(), simpset() addsimps [hrealpow, hyperpow]));

qed "hrealpow_hyperpow_Infinitesimal_iff";

Goal "[| x : Infinitesimal; 0 < n |] ==> x ^ n : Infinitesimal";

by (auto_tac (claset() addSIs [Infinitesimal_hyperpow],

    simpset() addsimps [hrealpow_hyperpow_Infinitesimal_iff,

                        hypnat_of_nat_less_iff,hypnat_of_nat_zero] 

              delsimps [hypnat_of_nat_less_iff RS sym]));

qed "Infinitesimal_hrealpow";