--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/CLim.ML Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,1191 @@
+(* Title : CLim.ML
+ Author : Jacques D. Fleuriot
+ Copyright : 2001 University of Edinburgh
+ Description : A first theory of limits, continuity and
+ differentiation for complex functions
+*)
+
+(*------------------------------------------------------------------------------------*)
+(* Limit of complex to complex function *)
+(*------------------------------------------------------------------------------------*)
+
+Goalw [NSCLIM_def,NSCRLIM_def]
+ "f -- a --NSC> L ==> (%x. Re(f x)) -- a --NSCR> Re(L)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [starfunC_approx_Re_Im_iff,
+ hRe_hcomplex_of_complex]));
+qed "NSCLIM_NSCRLIM_Re";
+
+Goalw [NSCLIM_def,NSCRLIM_def]
+ "f -- a --NSC> L ==> (%x. Im(f x)) -- a --NSCR> Im(L)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [starfunC_approx_Re_Im_iff,
+ hIm_hcomplex_of_complex]));
+qed "NSCLIM_NSCRLIM_Im";
+
+Goalw [CLIM_def,NSCLIM_def,capprox_def]
+ "f -- x --C> L ==> f -- x --NSC> L";
+by Auto_tac;
+by (res_inst_tac [("z","xa")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_complex_def,
+ starfunC,hcomplex_diff,CInfinitesimal_hcmod_iff,hcmod,
+ Infinitesimal_FreeUltrafilterNat_iff]));
+by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
+by (Step_tac 1);
+by (dres_inst_tac [("x","u")] spec 1 THEN Auto_tac);
+by (dres_inst_tac [("x","s")] spec 1 THEN Auto_tac);
+by (Ultra_tac 1);
+by (dtac sym 1 THEN Auto_tac);
+qed "CLIM_NSCLIM";
+
+Goal "(ALL t. P t) = (ALL X. P (Abs_hcomplex(hcomplexrel `` {X})))";
+by Auto_tac;
+by (res_inst_tac [("z","t")] eq_Abs_hcomplex 1);
+by Auto_tac;
+qed "eq_Abs_hcomplex_ALL";
+
+Goal "ALL s. 0 < s --> (EX xa. xa ~= x & \
+\ cmod (xa - x) < s & r <= cmod (f xa - L)) \
+\ ==> ALL (n::nat). EX xa. xa ~= x & \
+\ cmod(xa - x) < inverse(real(Suc n)) & r <= cmod(f xa - L)";
+by (Clarify_tac 1);
+by (cut_inst_tac [("n1","n")]
+ (real_of_nat_Suc_gt_zero RS real_inverse_gt_0) 1);
+by Auto_tac;
+val lemma_CLIM = result();
+
+(* not needed? *)
+Goal "ALL x z. EX y. Q x z y ==> EX f. ALL x z. Q x z (f x z)";
+by (rtac choice 1 THEN Step_tac 1);
+by (blast_tac (claset() addIs [choice]) 1);
+qed "choice2";
+
+Goal "ALL s. 0 < s --> (EX xa. xa ~= x & \
+\ cmod (xa - x) < s & r <= cmod (f xa - L)) \
+\ ==> EX X. ALL (n::nat). X n ~= x & \
+\ cmod(X n - x) < inverse(real(Suc n)) & r <= cmod(f (X n) - L)";
+by (dtac lemma_CLIM 1);
+by (dtac choice 1);
+by (Blast_tac 1);
+val lemma_skolemize_CLIM2 = result();
+
+Goal "ALL n. X n ~= x & \
+\ cmod (X n - x) < inverse (real(Suc n)) & \
+\ r <= cmod (f (X n) - L) ==> \
+\ ALL n. cmod (X n - x) < inverse (real(Suc n))";
+by (Auto_tac );
+val lemma_csimp = result();
+
+Goalw [CLIM_def,NSCLIM_def]
+ "f -- x --NSC> L ==> f -- x --C> L";
+by (auto_tac (claset(),simpset() addsimps [eq_Abs_hcomplex_ALL,
+ starfunC,CInfinitesimal_capprox_minus RS sym,hcomplex_diff,
+ CInfinitesimal_hcmod_iff,hcomplex_of_complex_def,
+ Infinitesimal_FreeUltrafilterNat_iff,hcmod]));
+by (EVERY1[rtac ccontr, Asm_full_simp_tac]);
+by (fold_tac [real_le_def]);
+by (dtac lemma_skolemize_CLIM2 1);
+by (Step_tac 1);
+by (dres_inst_tac [("x","X")] spec 1);
+by Auto_tac;
+by (dtac (lemma_csimp RS complex_seq_to_hcomplex_CInfinitesimal) 1);
+by (asm_full_simp_tac (simpset() addsimps [CInfinitesimal_hcmod_iff,
+ hcomplex_of_complex_def,Infinitesimal_FreeUltrafilterNat_iff,
+ hcomplex_diff,hcmod]) 1);
+by (Blast_tac 1);
+by (dres_inst_tac [("x","r")] spec 1);
+by (Clarify_tac 1);
+by (dtac FreeUltrafilterNat_all 1);
+by (Ultra_tac 1);
+by (arith_tac 1);
+qed "NSCLIM_CLIM";
+
+(**** First key result ****)
+
+Goal "(f -- x --C> L) = (f -- x --NSC> L)";
+by (blast_tac (claset() addIs [CLIM_NSCLIM,NSCLIM_CLIM]) 1);
+qed "CLIM_NSCLIM_iff";
+
+(*------------------------------------------------------------------------------------*)
+(* Limit of complex to real function *)
+(*------------------------------------------------------------------------------------*)
+
+Goalw [CRLIM_def,NSCRLIM_def,capprox_def]
+ "f -- x --CR> L ==> f -- x --NSCR> L";
+by Auto_tac;
+by (res_inst_tac [("z","xa")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_complex_def,
+ starfunCR,hcomplex_diff,CInfinitesimal_hcmod_iff,hcmod,hypreal_diff,
+ Infinitesimal_FreeUltrafilterNat_iff,Infinitesimal_approx_minus RS sym,
+ hypreal_of_real_def]));
+by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
+by (Step_tac 1);
+by (dres_inst_tac [("x","u")] spec 1 THEN Auto_tac);
+by (dres_inst_tac [("x","s")] spec 1 THEN Auto_tac);
+by (Ultra_tac 1);
+by (dtac sym 1 THEN Auto_tac);
+qed "CRLIM_NSCRLIM";
+
+Goal "ALL s. 0 < s --> (EX xa. xa ~= x & \
+\ cmod (xa - x) < s & r <= abs (f xa - L)) \
+\ ==> ALL (n::nat). EX xa. xa ~= x & \
+\ cmod(xa - x) < inverse(real(Suc n)) & r <= abs (f xa - L)";
+by (Clarify_tac 1);
+by (cut_inst_tac [("n1","n")]
+ (real_of_nat_Suc_gt_zero RS real_inverse_gt_0) 1);
+by Auto_tac;
+val lemma_CRLIM = result();
+
+Goal "ALL s. 0 < s --> (EX xa. xa ~= x & \
+\ cmod (xa - x) < s & r <= abs (f xa - L)) \
+\ ==> EX X. ALL (n::nat). X n ~= x & \
+\ cmod(X n - x) < inverse(real(Suc n)) & r <= abs (f (X n) - L)";
+by (dtac lemma_CRLIM 1);
+by (dtac choice 1);
+by (Blast_tac 1);
+val lemma_skolemize_CRLIM2 = result();
+
+Goal "ALL n. X n ~= x & \
+\ cmod (X n - x) < inverse (real(Suc n)) & \
+\ r <= abs (f (X n) - L) ==> \
+\ ALL n. cmod (X n - x) < inverse (real(Suc n))";
+by (Auto_tac );
+val lemma_crsimp = result();
+
+Goalw [CRLIM_def,NSCRLIM_def,capprox_def]
+ "f -- x --NSCR> L ==> f -- x --CR> L";
+by (auto_tac (claset(),simpset() addsimps [eq_Abs_hcomplex_ALL,
+ starfunCR,hcomplex_diff,hcomplex_of_complex_def,hypreal_diff,
+ CInfinitesimal_hcmod_iff,hcmod,Infinitesimal_approx_minus RS sym,
+ Infinitesimal_FreeUltrafilterNat_iff]));
+by (EVERY1[rtac ccontr, Asm_full_simp_tac]);
+by (fold_tac [real_le_def]);
+by (dtac lemma_skolemize_CRLIM2 1);
+by (Step_tac 1);
+by (dres_inst_tac [("x","X")] spec 1);
+by Auto_tac;
+by (dtac (lemma_crsimp RS complex_seq_to_hcomplex_CInfinitesimal) 1);
+by (asm_full_simp_tac (simpset() addsimps [CInfinitesimal_hcmod_iff,
+ hcomplex_of_complex_def,Infinitesimal_FreeUltrafilterNat_iff,
+ hcomplex_diff,hcmod]) 1);
+by (Blast_tac 1);
+by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_def,
+ hypreal_diff]));
+by (dres_inst_tac [("x","r")] spec 1);
+by (Clarify_tac 1);
+by (dtac FreeUltrafilterNat_all 1);
+by (Ultra_tac 1);
+qed "NSCRLIM_CRLIM";
+
+(** second key result **)
+Goal "(f -- x --CR> L) = (f -- x --NSCR> L)";
+by (blast_tac (claset() addIs [CRLIM_NSCRLIM,NSCRLIM_CRLIM]) 1);
+qed "CRLIM_NSCRLIM_iff";
+
+(** get this result easily now **)
+Goal "f -- a --C> L ==> (%x. Re(f x)) -- a --CR> Re(L)";
+by (auto_tac (claset() addDs [NSCLIM_NSCRLIM_Re],simpset()
+ addsimps [CLIM_NSCLIM_iff,CRLIM_NSCRLIM_iff RS sym]));
+qed "CLIM_CRLIM_Re";
+
+Goal "f -- a --C> L ==> (%x. Im(f x)) -- a --CR> Im(L)";
+by (auto_tac (claset() addDs [NSCLIM_NSCRLIM_Im],simpset()
+ addsimps [CLIM_NSCLIM_iff,CRLIM_NSCRLIM_iff RS sym]));
+qed "CLIM_CRLIM_Im";
+
+Goal "f -- a --C> L ==> (%x. cnj (f x)) -- a --C> cnj L";
+by (auto_tac (claset(),simpset() addsimps [CLIM_def,
+ complex_cnj_diff RS sym]));
+qed "CLIM_cnj";
+
+Goal "((%x. cnj (f x)) -- a --C> cnj L) = (f -- a --C> L)";
+by (auto_tac (claset(),simpset() addsimps [CLIM_def,
+ complex_cnj_diff RS sym]));
+qed "CLIM_cnj_iff";
+
+(*** NSLIM_add hence CLIM_add *)
+
+Goalw [NSCLIM_def]
+ "[| f -- x --NSC> l; g -- x --NSC> m |] \
+\ ==> (%x. f(x) + g(x)) -- x --NSC> (l + m)";
+by (auto_tac (claset() addSIs [capprox_add], simpset()));
+qed "NSCLIM_add";
+
+Goal "[| f -- x --C> l; g -- x --C> m |] \
+\ ==> (%x. f(x) + g(x)) -- x --C> (l + m)";
+by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_add]) 1);
+qed "CLIM_add";
+
+(*** NSLIM_mult hence CLIM_mult *)
+
+Goalw [NSCLIM_def]
+ "[| f -- x --NSC> l; g -- x --NSC> m |] \
+\ ==> (%x. f(x) * g(x)) -- x --NSC> (l * m)";
+by (auto_tac (claset() addSIs [capprox_mult_CFinite], simpset()));
+qed "NSCLIM_mult";
+
+Goal "[| f -- x --C> l; g -- x --C> m |] \
+\ ==> (%x. f(x) * g(x)) -- x --C> (l * m)";
+by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_mult]) 1);
+qed "CLIM_mult";
+
+(*** NSCLIM_const and CLIM_const ***)
+
+Goalw [NSCLIM_def] "(%x. k) -- x --NSC> k";
+by Auto_tac;
+qed "NSCLIM_const";
+Addsimps [NSCLIM_const];
+
+Goalw [CLIM_def] "(%x. k) -- x --C> k";
+by Auto_tac;
+qed "CLIM_const";
+Addsimps [CLIM_const];
+
+(*** NSCLIM_minus and CLIM_minus ***)
+
+Goalw [NSCLIM_def]
+ "f -- a --NSC> L ==> (%x. -f(x)) -- a --NSC> -L";
+by Auto_tac;
+qed "NSCLIM_minus";
+
+Goal "f -- a --C> L ==> (%x. -f(x)) -- a --C> -L";
+by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_minus]) 1);
+qed "CLIM_minus";
+
+(*** NSCLIM_diff hence CLIM_diff ***)
+
+Goalw [complex_diff_def]
+ "[| f -- x --NSC> l; g -- x --NSC> m |] \
+\ ==> (%x. f(x) - g(x)) -- x --NSC> (l - m)";
+by (auto_tac (claset(), simpset() addsimps [NSCLIM_add,NSCLIM_minus]));
+qed "NSCLIM_diff";
+
+Goal "[| f -- x --C> l; g -- x --C> m |] \
+\ ==> (%x. f(x) - g(x)) -- x --C> (l - m)";
+by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_diff]) 1);
+qed "CLIM_diff";
+
+(*** NSCLIM_inverse and hence CLIM_inverse *)
+
+Goalw [NSCLIM_def]
+ "[| f -- a --NSC> L; L ~= 0 |] \
+\ ==> (%x. inverse(f(x))) -- a --NSC> (inverse L)";
+by (Clarify_tac 1);
+by (dtac spec 1);
+by (auto_tac (claset(),
+ simpset() addsimps [hcomplex_of_complex_capprox_inverse]));
+qed "NSCLIM_inverse";
+
+Goal "[| f -- a --C> L; L ~= 0 |] \
+\ ==> (%x. inverse(f(x))) -- a --C> (inverse L)";
+by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_inverse]) 1);
+qed "CLIM_inverse";
+
+(*** NSCLIM_zero, CLIM_zero, etc. ***)
+
+Goal "f -- a --NSC> l ==> (%x. f(x) - l) -- a --NSC> 0";
+by (res_inst_tac [("z1","l")] (complex_add_minus_right_zero RS subst) 1);
+by (rewtac complex_diff_def);
+by (rtac NSCLIM_add 1 THEN Auto_tac);
+qed "NSCLIM_zero";
+
+Goal "f -- a --C> l ==> (%x. f(x) - l) -- a --C> 0";
+by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_zero]) 1);
+qed "CLIM_zero";
+
+Goal "(%x. f(x) - l) -- x --NSC> 0 ==> f -- x --NSC> l";
+by (dres_inst_tac [("g","%x. l"),("m","l")] NSCLIM_add 1);
+by Auto_tac;
+qed "NSCLIM_zero_cancel";
+
+Goal "(%x. f(x) - l) -- x --C> 0 ==> f -- x --C> l";
+by (dres_inst_tac [("g","%x. l"),("m","l")] CLIM_add 1);
+by Auto_tac;
+qed "CLIM_zero_cancel";
+
+(*** NSCLIM_not zero and hence CLIM_not_zero ***)
+
+(*not in simpset?*)
+Addsimps [hypreal_epsilon_not_zero];
+
+Goalw [NSCLIM_def] "k ~= 0 ==> ~ ((%x. k) -- x --NSC> 0)";
+by (auto_tac (claset(),simpset() delsimps [hcomplex_of_complex_zero]));
+by (res_inst_tac [("x","hcomplex_of_complex x + hcomplex_of_hypreal epsilon")] exI 1);
+by (auto_tac (claset() addIs [CInfinitesimal_add_capprox_self RS capprox_sym],simpset()
+ delsimps [hcomplex_of_complex_zero]));
+qed "NSCLIM_not_zero";
+
+(* [| k ~= 0; (%x. k) -- x --NSC> 0 |] ==> R *)
+bind_thm("NSCLIM_not_zeroE", NSCLIM_not_zero RS notE);
+
+Goal "k ~= 0 ==> ~ ((%x. k) -- x --C> 0)";
+by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_not_zero]) 1);
+qed "CLIM_not_zero";
+
+(*** NSCLIM_const hence CLIM_const ***)
+
+Goal "(%x. k) -- x --NSC> L ==> k = L";
+by (rtac ccontr 1);
+by (dtac NSCLIM_zero 1);
+by (rtac NSCLIM_not_zeroE 1 THEN assume_tac 2);
+by Auto_tac;
+qed "NSCLIM_const_eq";
+
+Goal "(%x. k) -- x --C> L ==> k = L";
+by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff,NSCLIM_const_eq]) 1);
+qed "CLIM_const_eq";
+
+(*** NSCLIM and hence CLIM are unique ***)
+
+Goal "[| f -- x --NSC> L; f -- x --NSC> M |] ==> L = M";
+by (dtac NSCLIM_minus 1);
+by (dtac NSCLIM_add 1 THEN assume_tac 1);
+by (auto_tac (claset() addSDs [NSCLIM_const_eq RS sym], simpset()));
+qed "NSCLIM_unique";
+
+Goal "[| f -- x --C> L; f -- x --C> M |] ==> L = M";
+by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_unique]) 1);
+qed "CLIM_unique";
+
+(*** NSCLIM_mult_zero and CLIM_mult_zero ***)
+
+Goal "[| f -- x --NSC> 0; g -- x --NSC> 0 |] \
+\ ==> (%x. f(x)*g(x)) -- x --NSC> 0";
+by (dtac NSCLIM_mult 1 THEN Auto_tac);
+qed "NSCLIM_mult_zero";
+
+Goal "[| f -- x --C> 0; g -- x --C> 0 |] \
+\ ==> (%x. f(x)*g(x)) -- x --C> 0";
+by (dtac CLIM_mult 1 THEN Auto_tac);
+qed "CLIM_mult_zero";
+
+(*** NSCLIM_self hence CLIM_self ***)
+
+Goalw [NSCLIM_def] "(%x. x) -- a --NSC> a";
+by (auto_tac (claset() addIs [starfunC_Idfun_capprox],simpset()));
+qed "NSCLIM_self";
+
+Goal "(%x. x) -- a --C> a";
+by (simp_tac (simpset() addsimps [CLIM_NSCLIM_iff,NSCLIM_self]) 1);
+qed "CLIM_self";
+
+(** another equivalence result **)
+Goalw [NSCLIM_def,NSCRLIM_def]
+ "(f -- x --NSC> L) = ((%y. cmod(f y - L)) -- x --NSCR> 0)";
+by (auto_tac (claset(),simpset() addsimps [CInfinitesimal_capprox_minus
+ RS sym,CInfinitesimal_hcmod_iff]));
+by (ALLGOALS(dtac spec) THEN Auto_tac);
+by (ALLGOALS(res_inst_tac [("z","xa")] eq_Abs_hcomplex));
+by (auto_tac (claset(),simpset() addsimps [hcomplex_diff,
+ starfunC,starfunCR,hcomplex_of_complex_def,hcmod,mem_infmal_iff]));
+qed "NSCLIM_NSCRLIM_iff";
+
+(** much, much easier standard proof **)
+Goalw [CLIM_def,CRLIM_def]
+ "(f -- x --C> L) = ((%y. cmod(f y - L)) -- x --CR> 0)";
+by Auto_tac;
+qed "CLIM_CRLIM_iff";
+
+(* so this is nicer nonstandard proof *)
+Goal "(f -- x --NSC> L) = ((%y. cmod(f y - L)) -- x --NSCR> 0)";
+by (auto_tac (claset(),simpset() addsimps [CRLIM_NSCRLIM_iff RS sym,
+ CLIM_CRLIM_iff,CLIM_NSCLIM_iff RS sym]));
+qed "NSCLIM_NSCRLIM_iff2";
+
+Goal "(f -- a --NSC> L) = ((%x. Re(f x)) -- a --NSCR> Re(L) & \
+\ (%x. Im(f x)) -- a --NSCR> Im(L))";
+by (auto_tac (claset() addIs [NSCLIM_NSCRLIM_Re,NSCLIM_NSCRLIM_Im],simpset()));
+by (auto_tac (claset(),simpset() addsimps [NSCLIM_def,NSCRLIM_def]));
+by (REPEAT(dtac spec 1) THEN Auto_tac);
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [capprox_approx_iff,starfunC,
+ hcomplex_of_complex_def,starfunCR,hypreal_of_real_def]));
+qed "NSCLIM_NSCRLIM_Re_Im_iff";
+
+Goal "(f -- a --C> L) = ((%x. Re(f x)) -- a --CR> Re(L) & \
+\ (%x. Im(f x)) -- a --CR> Im(L))";
+by (auto_tac (claset(),simpset() addsimps [CLIM_NSCLIM_iff,CRLIM_NSCRLIM_iff,
+ NSCLIM_NSCRLIM_Re_Im_iff]));
+qed "CLIM_CRLIM_Re_Im_iff";
+
+
+(*------------------------------------------------------------------------------------*)
+(* Continuity *)
+(*------------------------------------------------------------------------------------*)
+
+Goalw [isNSContc_def]
+ "[| isNSContc f a; y @c= hcomplex_of_complex a |] \
+\ ==> ( *fc* f) y @c= hcomplex_of_complex (f a)";
+by (Blast_tac 1);
+qed "isNSContcD";
+
+Goalw [isNSContc_def,NSCLIM_def]
+ "isNSContc f a ==> f -- a --NSC> (f a) ";
+by (Blast_tac 1);
+qed "isNSContc_NSCLIM";
+
+Goalw [isNSContc_def,NSCLIM_def]
+ "f -- a --NSC> (f a) ==> isNSContc f a";
+by Auto_tac;
+by (res_inst_tac [("Q","y = hcomplex_of_complex a")]
+ (excluded_middle RS disjE) 1);
+by Auto_tac;
+qed "NSCLIM_isNSContc";
+
+(*--------------------------------------------------*)
+(* NS continuity can be defined using NS Limit in *)
+(* similar fashion to standard def of continuity *)
+(* -------------------------------------------------*)
+
+Goal "(isNSContc f a) = (f -- a --NSC> (f a))";
+by (blast_tac (claset() addIs [isNSContc_NSCLIM,NSCLIM_isNSContc]) 1);
+qed "isNSContc_NSCLIM_iff";
+
+Goal "(isNSContc f a) = (f -- a --C> (f a))";
+by (asm_full_simp_tac (simpset() addsimps
+ [CLIM_NSCLIM_iff,isNSContc_NSCLIM_iff]) 1);
+qed "isNSContc_CLIM_iff";
+
+(*** key result for continuity ***)
+Goalw [isContc_def] "(isNSContc f a) = (isContc f a)";
+by (rtac isNSContc_CLIM_iff 1);
+qed "isNSContc_isContc_iff";
+
+Goal "isContc f a ==> isNSContc f a";
+by (etac (isNSContc_isContc_iff RS iffD2) 1);
+qed "isContc_isNSContc";
+
+Goal "isNSContc f a ==> isContc f a";
+by (etac (isNSContc_isContc_iff RS iffD1) 1);
+qed "isNSContc_isContc";
+
+(*--------------------------------------------------*)
+(* Alternative definition of continuity *)
+(* -------------------------------------------------*)
+
+Goalw [NSCLIM_def]
+ "(f -- a --NSC> L) = ((%h. f(a + h)) -- 0 --NSC> L)";
+by Auto_tac;
+by (dres_inst_tac [("x","hcomplex_of_complex a + x")] spec 1);
+by (dres_inst_tac [("x","- hcomplex_of_complex a + x")] spec 2);
+by (Step_tac 1);
+by (Asm_full_simp_tac 1);
+by (rtac ((mem_cinfmal_iff RS iffD2) RS
+ (CInfinitesimal_add_capprox_self RS capprox_sym)) 1);
+by (rtac (capprox_minus_iff2 RS iffD1) 4);
+by (asm_full_simp_tac (simpset() addsimps [hcomplex_add_commute]) 3);
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 2);
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 4);
+by (auto_tac (claset(),
+ simpset() addsimps [starfunC, hcomplex_of_complex_def,
+ hcomplex_minus, hcomplex_add]));
+qed "NSCLIM_h_iff";
+
+Goal "(f -- a --NSC> f a) = ((%h. f(a + h)) -- 0 --NSC> f a)";
+by (rtac NSCLIM_h_iff 1);
+qed "NSCLIM_isContc_iff";
+
+Goal "(f -- a --C> f a) = ((%h. f(a + h)) -- 0 --C> f(a))";
+by (simp_tac (simpset() addsimps [CLIM_NSCLIM_iff, NSCLIM_isContc_iff]) 1);
+qed "CLIM_isContc_iff";
+
+Goalw [isContc_def] "(isContc f x) = ((%h. f(x + h)) -- 0 --C> f(x))";
+by (simp_tac (simpset() addsimps [CLIM_isContc_iff]) 1);
+qed "isContc_iff";
+
+Goal "[| isContc f a; isContc g a |] ==> isContc (%x. f(x) + g(x)) a";
+by (auto_tac (claset() addIs [capprox_add],
+ simpset() addsimps [isNSContc_isContc_iff RS sym, isNSContc_def]));
+qed "isContc_add";
+
+Goal "[| isContc f a; isContc g a |] ==> isContc (%x. f(x) * g(x)) a";
+by (auto_tac (claset() addSIs [starfunC_mult_CFinite_capprox],
+ simpset() delsimps [starfunC_mult RS sym]
+ addsimps [isNSContc_isContc_iff RS sym, isNSContc_def]));
+qed "isContc_mult";
+
+(*** more theorems: note simple proofs ***)
+
+Goal "[| isContc f a; isContc g (f a) |] \
+\ ==> isContc (g o f) a";
+by (auto_tac (claset(),simpset() addsimps [isNSContc_isContc_iff RS sym,
+ isNSContc_def,starfunC_o RS sym]));
+qed "isContc_o";
+
+Goal "[| isContc f a; isContc g (f a) |] \
+\ ==> isContc (%x. g (f x)) a";
+by (auto_tac (claset() addDs [isContc_o],simpset() addsimps [o_def]));
+qed "isContc_o2";
+
+Goalw [isNSContc_def] "isNSContc f a ==> isNSContc (%x. - f x) a";
+by Auto_tac;
+qed "isNSContc_minus";
+
+Goal "isContc f a ==> isContc (%x. - f x) a";
+by (auto_tac (claset(),simpset() addsimps [isNSContc_isContc_iff RS sym,
+ isNSContc_minus]));
+qed "isContc_minus";
+
+Goalw [isContc_def]
+ "[| isContc f x; f x ~= 0 |] ==> isContc (%x. inverse (f x)) x";
+by (blast_tac (claset() addIs [CLIM_inverse]) 1);
+qed "isContc_inverse";
+
+Goal "[| isNSContc f x; f x ~= 0 |] ==> isNSContc (%x. inverse (f x)) x";
+by (auto_tac (claset() addIs [isContc_inverse],simpset() addsimps
+ [isNSContc_isContc_iff]));
+qed "isNSContc_inverse";
+
+Goalw [complex_diff_def]
+ "[| isContc f a; isContc g a |] ==> isContc (%x. f(x) - g(x)) a";
+by (auto_tac (claset() addIs [isContc_add,isContc_minus],simpset()));
+qed "isContc_diff";
+
+Goalw [isContc_def] "isContc (%x. k) a";
+by (Simp_tac 1);
+qed "isContc_const";
+Addsimps [isContc_const];
+
+Goalw [isNSContc_def] "isNSContc (%x. k) a";
+by (Simp_tac 1);
+qed "isNSContc_const";
+Addsimps [isNSContc_const];
+
+
+(*------------------------------------------------------------------------------------*)
+(* functions from complex to reals *)
+(* -----------------------------------------------------------------------------------*)
+
+Goalw [isNSContCR_def]
+ "[| isNSContCR f a; y @c= hcomplex_of_complex a |] \
+\ ==> ( *fcR* f) y @= hypreal_of_real (f a)";
+by (Blast_tac 1);
+qed "isNSContCRD";
+
+Goalw [isNSContCR_def,NSCRLIM_def]
+ "isNSContCR f a ==> f -- a --NSCR> (f a) ";
+by (Blast_tac 1);
+qed "isNSContCR_NSCRLIM";
+
+Goalw [isNSContCR_def,NSCRLIM_def]
+ "f -- a --NSCR> (f a) ==> isNSContCR f a";
+by Auto_tac;
+by (res_inst_tac [("Q","y = hcomplex_of_complex a")]
+ (excluded_middle RS disjE) 1);
+by Auto_tac;
+qed "NSCRLIM_isNSContCR";
+
+Goal "(isNSContCR f a) = (f -- a --NSCR> (f a))";
+by (blast_tac (claset() addIs [isNSContCR_NSCRLIM,NSCRLIM_isNSContCR]) 1);
+qed "isNSContCR_NSCRLIM_iff";
+
+Goal "(isNSContCR f a) = (f -- a --CR> (f a))";
+by (asm_full_simp_tac (simpset() addsimps
+ [CRLIM_NSCRLIM_iff,isNSContCR_NSCRLIM_iff]) 1);
+qed "isNSContCR_CRLIM_iff";
+
+(*** another key result for continuity ***)
+Goalw [isContCR_def] "(isNSContCR f a) = (isContCR f a)";
+by (rtac isNSContCR_CRLIM_iff 1);
+qed "isNSContCR_isContCR_iff";
+
+Goal "isContCR f a ==> isNSContCR f a";
+by (etac (isNSContCR_isContCR_iff RS iffD2) 1);
+qed "isContCR_isNSContCR";
+
+Goal "isNSContCR f a ==> isContCR f a";
+by (etac (isNSContCR_isContCR_iff RS iffD1) 1);
+qed "isNSContCR_isContCR";
+
+Goalw [isNSContCR_def] "isNSContCR cmod (a)";
+by (auto_tac (claset() addIs [capprox_hcmod_approx],
+ simpset() addsimps [starfunCR_cmod,hcmod_hcomplex_of_complex
+ RS sym]));
+qed "isNSContCR_cmod";
+Addsimps [isNSContCR_cmod];
+
+Goal "isContCR cmod (a)";
+by (auto_tac (claset(),simpset() addsimps [isNSContCR_isContCR_iff RS sym]));
+qed "isContCR_cmod";
+Addsimps [isContCR_cmod];
+
+Goalw [isContc_def,isContCR_def]
+ "isContc f a ==> isContCR (%x. Re (f x)) a";
+by (etac CLIM_CRLIM_Re 1);
+qed "isContc_isContCR_Re";
+
+Goalw [isContc_def,isContCR_def]
+ "isContc f a ==> isContCR (%x. Im (f x)) a";
+by (etac CLIM_CRLIM_Im 1);
+qed "isContc_isContCR_Im";
+
+(*------------------------------------------------------------------------------------*)
+(* Derivatives *)
+(*------------------------------------------------------------------------------------*)
+
+Goalw [cderiv_def]
+ "(CDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --C> D)";
+by (Blast_tac 1);
+qed "CDERIV_iff";
+
+Goalw [cderiv_def]
+ "(CDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NSC> D)";
+by (simp_tac (simpset() addsimps [CLIM_NSCLIM_iff]) 1);
+qed "CDERIV_NSC_iff";
+
+Goalw [cderiv_def]
+ "CDERIV f x :> D \
+\ ==> (%h. (f(x + h) - f(x))/h) -- 0 --C> D";
+by (Blast_tac 1);
+qed "CDERIVD";
+
+Goalw [cderiv_def]
+ "CDERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --NSC> D";
+by (asm_full_simp_tac (simpset() addsimps [CLIM_NSCLIM_iff]) 1);
+qed "NSC_DERIVD";
+
+(*** Uniqueness ***)
+
+Goalw [cderiv_def]
+ "[| CDERIV f x :> D; CDERIV f x :> E |] ==> D = E";
+by (blast_tac (claset() addIs [CLIM_unique]) 1);
+qed "CDERIV_unique";
+
+(*** uniqueness: a nonstandard proof ***)
+Goalw [nscderiv_def]
+ "[| NSCDERIV f x :> D; NSCDERIV f x :> E |] ==> D = E";
+by (auto_tac (claset() addSDs [inst "x" "hcomplex_of_hypreal epsilon" bspec]
+ addSIs [inj_hcomplex_of_complex RS injD]
+ addDs [capprox_trans3],
+ simpset()));
+qed "NSCDeriv_unique";
+
+
+(*------------------------------------------------------------------------------------*)
+(* Differentiability *)
+(*------------------------------------------------------------------------------------*)
+
+Goalw [cdifferentiable_def]
+ "f cdifferentiable x ==> EX D. CDERIV f x :> D";
+by (assume_tac 1);
+qed "cdifferentiableD";
+
+Goalw [cdifferentiable_def]
+ "CDERIV f x :> D ==> f cdifferentiable x";
+by (Blast_tac 1);
+qed "cdifferentiableI";
+
+Goalw [NSCdifferentiable_def]
+ "f NSCdifferentiable x ==> EX D. NSCDERIV f x :> D";
+by (assume_tac 1);
+qed "NSCdifferentiableD";
+
+Goalw [NSCdifferentiable_def]
+ "NSCDERIV f x :> D ==> f NSCdifferentiable x";
+by (Blast_tac 1);
+qed "NSCdifferentiableI";
+
+
+(*------------------------------------------------------------------------------------*)
+(* Alternative definition for differentiability *)
+(*------------------------------------------------------------------------------------*)
+
+Goalw [CLIM_def]
+ "((%h. (f(a + h) - f(a))/h) -- 0 --C> D) = \
+\ ((%x. (f(x) - f(a)) / (x - a)) -- a --C> D)";
+by (Step_tac 1);
+by (ALLGOALS(dtac spec));
+by (Step_tac 1);
+by (Blast_tac 1 THEN Blast_tac 2);
+by (ALLGOALS(res_inst_tac [("x","s")] exI));
+by (Step_tac 1);
+by (dres_inst_tac [("x","x - a")] spec 1);
+by (dres_inst_tac [("x","x + a")] spec 2);
+by (auto_tac (claset(), simpset() addsimps complex_add_ac));
+qed "CDERIV_CLIM_iff";
+
+Goalw [cderiv_def] "(CDERIV f x :> D) = \
+\ ((%z. (f(z) - f(x)) / (z - x)) -- x --C> D)";
+by (simp_tac (simpset() addsimps [CDERIV_CLIM_iff]) 1);
+qed "CDERIV_iff2";
+
+
+(*------------------------------------------------------------------------------------*)
+(* Equivalence of NS and standard defs of differentiation *)
+(*------------------------------------------------------------------------------------*)
+
+(*** first equivalence ***)
+Goalw [nscderiv_def,NSCLIM_def]
+ "(NSCDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NSC> D)";
+by Auto_tac;
+by (dres_inst_tac [("x","xa")] bspec 1);
+by (rtac ccontr 3);
+by (dres_inst_tac [("x","h")] spec 3);
+by (auto_tac (claset(),
+ simpset() addsimps [mem_cinfmal_iff, starfunC_lambda_cancel]));
+qed "NSCDERIV_NSCLIM_iff";
+
+(*** 2nd equivalence ***)
+Goal "(NSCDERIV f x :> D) = \
+\ ((%z. (f(z) - f(x)) / (z - x)) -- x --NSC> D)";
+by (full_simp_tac (simpset() addsimps
+ [NSCDERIV_NSCLIM_iff, CDERIV_CLIM_iff, CLIM_NSCLIM_iff RS sym]) 1);
+qed "NSCDERIV_NSCLIM_iff2";
+
+Goal "(NSCDERIV f x :> D) = \
+\ (ALL xa. xa ~= hcomplex_of_complex x & xa @c= hcomplex_of_complex x --> \
+\ ( *fc* (%z. (f z - f x) / (z - x))) xa @c= hcomplex_of_complex D)";
+by (auto_tac (claset(), simpset() addsimps [NSCDERIV_NSCLIM_iff2, NSCLIM_def]));
+qed "NSCDERIV_iff2";
+
+Goalw [cderiv_def] "(NSCDERIV f x :> D) = (CDERIV f x :> D)";
+by (simp_tac (simpset() addsimps [NSCDERIV_NSCLIM_iff,CLIM_NSCLIM_iff]) 1);
+qed "NSCDERIV_CDERIV_iff";
+
+Goalw [nscderiv_def]
+ "NSCDERIV f x :> D ==> isNSContc f x";
+by (auto_tac (claset(),simpset() addsimps [isNSContc_NSCLIM_iff,
+ NSCLIM_def,hcomplex_diff_def]));
+by (dtac (capprox_minus_iff RS iffD1) 1);
+by (dtac (CLAIM "x ~= a ==> x + - a ~= (0::hcomplex)") 1);
+by (dres_inst_tac [("x","- hcomplex_of_complex x + xa")] bspec 1);
+by (asm_full_simp_tac (simpset() addsimps [hcomplex_add_assoc RS sym]) 2);
+by (auto_tac (claset(),simpset() addsimps
+ [mem_cinfmal_iff RS sym,hcomplex_add_commute]));
+by (dres_inst_tac [("c","xa + - hcomplex_of_complex x")] capprox_mult1 1);
+by (auto_tac (claset() addIs [CInfinitesimal_subset_CFinite
+ RS subsetD],simpset() addsimps [hcomplex_mult_assoc]));
+by (dres_inst_tac [("x3","D")] (CFinite_hcomplex_of_complex RSN
+ (2,CInfinitesimal_CFinite_mult) RS (mem_cinfmal_iff RS iffD1)) 1);
+by (blast_tac (claset() addIs [capprox_trans,hcomplex_mult_commute RS subst,
+ (capprox_minus_iff RS iffD2)]) 1);
+qed "NSCDERIV_isNSContc";
+
+Goal "CDERIV f x :> D ==> isContc f x";
+by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_CDERIV_iff RS sym,
+ isNSContc_isContc_iff RS sym,NSCDERIV_isNSContc]) 1);
+qed "CDERIV_isContc";
+
+(*------------------------------------------------------------------------------------*)
+(* Differentiation rules for combinations of functions follow from clear, *)
+(* straightforard, algebraic manipulations *)
+(*------------------------------------------------------------------------------------*)
+
+(* use simple constant nslimit theorem *)
+Goal "(NSCDERIV (%x. k) x :> 0)";
+by (simp_tac (simpset() addsimps [NSCDERIV_NSCLIM_iff]) 1);
+qed "NSCDERIV_const";
+Addsimps [NSCDERIV_const];
+
+Goal "(CDERIV (%x. k) x :> 0)";
+by (simp_tac (simpset() addsimps [NSCDERIV_CDERIV_iff RS sym]) 1);
+qed "CDERIV_const";
+Addsimps [CDERIV_const];
+
+Goal "[| NSCDERIV f x :> Da; NSCDERIV g x :> Db |] \
+\ ==> NSCDERIV (%x. f x + g x) x :> Da + Db";
+by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_NSCLIM_iff,
+ NSCLIM_def]) 1 THEN REPEAT(Step_tac 1));
+by (auto_tac (claset(),
+ simpset() addsimps [hcomplex_add_divide_distrib,hcomplex_diff_def]));
+by (dres_inst_tac [("b","hcomplex_of_complex Da"),
+ ("d","hcomplex_of_complex Db")] capprox_add 1);
+by (auto_tac (claset(), simpset() addsimps hcomplex_add_ac));
+qed "NSCDERIV_add";
+
+Goal "[| CDERIV f x :> Da; CDERIV g x :> Db |] \
+\ ==> CDERIV (%x. f x + g x) x :> Da + Db";
+by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_add,
+ NSCDERIV_CDERIV_iff RS sym]) 1);
+qed "CDERIV_add";
+
+(*** lemmas for multiplication ***)
+
+Goal "((a::hcomplex)*b) - (c*d) = (b*(a - c)) + (c*(b - d))";
+by (simp_tac (simpset() addsimps [hcomplex_diff_mult_distrib2]) 1);
+val lemma_nscderiv1 = result();
+
+Goal "[| (x + y) / z = hcomplex_of_complex D + yb; z ~= 0; \
+\ z : CInfinitesimal; yb : CInfinitesimal |] \
+\ ==> x + y @c= 0";
+by (forw_inst_tac [("c1","z")] (hcomplex_mult_right_cancel RS iffD2) 1
+ THEN assume_tac 1);
+by (thin_tac "(x + y) / z = hcomplex_of_complex D + yb" 1);
+by (auto_tac (claset() addSIs [CInfinitesimal_CFinite_mult2, CFinite_add],
+ simpset() addsimps [mem_cinfmal_iff RS sym]));
+by (etac (CInfinitesimal_subset_CFinite RS subsetD) 1);
+val lemma_nscderiv2 = result();
+
+Goal "[| NSCDERIV f x :> Da; NSCDERIV g x :> Db |] \
+\ ==> NSCDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))";
+by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_NSCLIM_iff, NSCLIM_def]) 1
+ THEN REPEAT(Step_tac 1));
+by (auto_tac (claset(),
+ simpset() addsimps [starfunC_lambda_cancel, lemma_nscderiv1,
+ hcomplex_of_complex_zero]));
+by (simp_tac (simpset() addsimps [hcomplex_add_divide_distrib]) 1);
+by (REPEAT(dtac (bex_CInfinitesimal_iff2 RS iffD2) 1));
+by (auto_tac (claset(),
+ simpset() delsimps [hcomplex_times_divide1_eq]
+ addsimps [hcomplex_times_divide1_eq RS sym]));
+by (rewtac hcomplex_diff_def);
+by (dres_inst_tac [("D","Db")] lemma_nscderiv2 1);
+by (dtac (capprox_minus_iff RS iffD2 RS (bex_CInfinitesimal_iff2 RS iffD2)) 4);
+by (auto_tac (claset() addSIs [capprox_add_mono1],
+ simpset() addsimps [hcomplex_add_mult_distrib, hcomplex_add_mult_distrib2,
+ hcomplex_mult_commute, hcomplex_add_assoc]));
+by (res_inst_tac [("w1","hcomplex_of_complex Db * hcomplex_of_complex (f x)")]
+ (hcomplex_add_commute RS subst) 1);
+by (auto_tac (claset() addSIs [CInfinitesimal_add_capprox_self2 RS capprox_sym,
+ CInfinitesimal_add, CInfinitesimal_mult,
+ CInfinitesimal_hcomplex_of_complex_mult,
+ CInfinitesimal_hcomplex_of_complex_mult2],
+ simpset() addsimps [hcomplex_add_assoc RS sym]));
+qed "NSCDERIV_mult";
+
+Goal "[| CDERIV f x :> Da; CDERIV g x :> Db |] \
+\ ==> CDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))";
+by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_mult,
+ NSCDERIV_CDERIV_iff RS sym]) 1);
+qed "CDERIV_mult";
+
+Goal "NSCDERIV f x :> D ==> NSCDERIV (%x. c * f x) x :> c*D";
+by (asm_full_simp_tac
+ (simpset() addsimps [complex_times_divide1_eq RS sym, NSCDERIV_NSCLIM_iff,
+ complex_minus_mult_eq2, complex_add_mult_distrib2 RS sym,
+ complex_diff_def]
+ delsimps [complex_times_divide1_eq, complex_minus_mult_eq2 RS sym]) 1);
+by (etac (NSCLIM_const RS NSCLIM_mult) 1);
+qed "NSCDERIV_cmult";
+
+Goal "CDERIV f x :> D ==> CDERIV (%x. c * f x) x :> c*D";
+by (auto_tac (claset(),simpset() addsimps [NSCDERIV_cmult,NSCDERIV_CDERIV_iff
+ RS sym]));
+qed "CDERIV_cmult";
+
+Goal "NSCDERIV f x :> D ==> NSCDERIV (%x. -(f x)) x :> -D";
+by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_NSCLIM_iff,complex_diff_def]) 1);
+by (res_inst_tac [("t","f x")] (complex_minus_minus RS subst) 1);
+by (asm_simp_tac (simpset() addsimps [complex_minus_add_distrib RS sym]
+ delsimps [complex_minus_add_distrib, complex_minus_minus]) 1);
+by (etac NSCLIM_minus 1);
+qed "NSCDERIV_minus";
+
+Goal "CDERIV f x :> D ==> CDERIV (%x. -(f x)) x :> -D";
+by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_minus,NSCDERIV_CDERIV_iff RS sym]) 1);
+qed "CDERIV_minus";
+
+Goal "[| NSCDERIV f x :> Da; NSCDERIV g x :> Db |] \
+\ ==> NSCDERIV (%x. f x + -g x) x :> Da + -Db";
+by (blast_tac (claset() addDs [NSCDERIV_add,NSCDERIV_minus]) 1);
+qed "NSCDERIV_add_minus";
+
+Goal "[| CDERIV f x :> Da; CDERIV g x :> Db |] \
+\ ==> CDERIV (%x. f x + -g x) x :> Da + -Db";
+by (blast_tac (claset() addDs [CDERIV_add,CDERIV_minus]) 1);
+qed "CDERIV_add_minus";
+
+Goalw [complex_diff_def]
+ "[| NSCDERIV f x :> Da; NSCDERIV g x :> Db |] \
+\ ==> NSCDERIV (%x. f x - g x) x :> Da - Db";
+by (blast_tac (claset() addIs [NSCDERIV_add_minus]) 1);
+qed "NSCDERIV_diff";
+
+Goalw [complex_diff_def]
+ "[| CDERIV f x :> Da; CDERIV g x :> Db |] \
+\ ==> CDERIV (%x. f x - g x) x :> Da - Db";
+by (blast_tac (claset() addIs [CDERIV_add_minus]) 1);
+qed "CDERIV_diff";
+
+
+(*--------------------------------------------------*)
+(* Chain rule *)
+(*--------------------------------------------------*)
+
+(* lemmas *)
+Goalw [nscderiv_def]
+ "[| NSCDERIV g x :> D; \
+\ ( *fc* g) (hcomplex_of_complex(x) + xa) = hcomplex_of_complex(g x);\
+\ xa : CInfinitesimal; xa ~= 0 \
+\ |] ==> D = 0";
+by (dtac bspec 1);
+by Auto_tac;
+qed "NSCDERIV_zero";
+
+Goalw [nscderiv_def]
+ "[| NSCDERIV f x :> D; h: CInfinitesimal; h ~= 0 |] \
+\ ==> ( *fc* f) (hcomplex_of_complex(x) + h) - hcomplex_of_complex(f x) @c= 0";
+by (asm_full_simp_tac (simpset() addsimps [mem_cinfmal_iff RS sym]) 1);
+by (rtac CInfinitesimal_ratio 1);
+by (rtac capprox_hcomplex_of_complex_CFinite 3);
+by Auto_tac;
+qed "NSCDERIV_capprox";
+
+
+(*--------------------------------------------------*)
+(* from one version of differentiability *)
+(* *)
+(* f(x) - f(a) *)
+(* --------------- @= Db *)
+(* x - a *)
+(* -------------------------------------------------*)
+
+Goal "[| NSCDERIV f (g x) :> Da; \
+\ ( *fc* g) (hcomplex_of_complex(x) + xa) ~= hcomplex_of_complex (g x); \
+\ ( *fc* g) (hcomplex_of_complex(x) + xa) @c= hcomplex_of_complex (g x) \
+\ |] ==> (( *fc* f) (( *fc* g) (hcomplex_of_complex(x) + xa)) \
+\ - hcomplex_of_complex (f (g x))) \
+\ / (( *fc* g) (hcomplex_of_complex(x) + xa) - hcomplex_of_complex (g x)) \
+\ @c= hcomplex_of_complex (Da)";
+by (auto_tac (claset(),simpset() addsimps [NSCDERIV_NSCLIM_iff2, NSCLIM_def]));
+qed "NSCDERIVD1";
+
+(*--------------------------------------------------*)
+(* from other version of differentiability *)
+(* *)
+(* f(x + h) - f(x) *)
+(* ----------------- @= Db *)
+(* h *)
+(*--------------------------------------------------*)
+
+Goal "[| NSCDERIV g x :> Db; xa: CInfinitesimal; xa ~= 0 |] \
+\ ==> (( *fc* g) (hcomplex_of_complex(x) + xa) - hcomplex_of_complex(g x)) / xa \
+\ @c= hcomplex_of_complex (Db)";
+by (auto_tac (claset(),
+ simpset() addsimps [NSCDERIV_NSCLIM_iff, NSCLIM_def,
+ mem_cinfmal_iff, starfunC_lambda_cancel]));
+qed "NSCDERIVD2";
+
+Goal "(z::hcomplex) ~= 0 ==> x*y = (x*inverse(z))*(z*y)";
+by Auto_tac;
+qed "lemma_complex_chain";
+
+(*** chain rule ***)
+
+Goal "[| NSCDERIV f (g x) :> Da; NSCDERIV g x :> Db |] \
+\ ==> NSCDERIV (f o g) x :> Da * Db";
+by (asm_simp_tac (simpset() addsimps [NSCDERIV_NSCLIM_iff,
+ NSCLIM_def,mem_cinfmal_iff RS sym]) 1 THEN Step_tac 1);
+by (forw_inst_tac [("f","g")] NSCDERIV_capprox 1);
+by (auto_tac (claset(),
+ simpset() addsimps [starfunC_lambda_cancel2, starfunC_o RS sym]));
+by (case_tac "( *fc* g) (hcomplex_of_complex(x) + xa) = hcomplex_of_complex (g x)" 1);
+by (dres_inst_tac [("g","g")] NSCDERIV_zero 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_divide_def]));
+by (res_inst_tac [("z1","( *fc* g) (hcomplex_of_complex(x) + xa) - hcomplex_of_complex (g x)"),
+ ("y1","inverse xa")] (lemma_complex_chain RS ssubst) 1);
+by (Asm_simp_tac 1);
+by (rtac capprox_mult_hcomplex_of_complex 1);
+by (fold_tac [hcomplex_divide_def]);
+by (blast_tac (claset() addIs [NSCDERIVD2]) 2);
+by (auto_tac (claset() addSIs [NSCDERIVD1] addIs [capprox_minus_iff RS iffD2],
+ simpset() addsimps [symmetric hcomplex_diff_def]));
+qed "NSCDERIV_chain";
+
+(* standard version *)
+Goal "[| CDERIV f (g x) :> Da; CDERIV g x :> Db |] \
+\ ==> CDERIV (f o g) x :> Da * Db";
+by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_CDERIV_iff RS sym,
+ NSCDERIV_chain]) 1);
+qed "CDERIV_chain";
+
+Goal "[| CDERIV f (g x) :> Da; CDERIV g x :> Db |] \
+\ ==> CDERIV (%x. f (g x)) x :> Da * Db";
+by (auto_tac (claset() addDs [CDERIV_chain], simpset() addsimps [o_def]));
+qed "CDERIV_chain2";
+
+(*------------------------------------------------------------------------------------*)
+(* Differentiation of natural number powers *)
+(*------------------------------------------------------------------------------------*)
+
+Goal "NSCDERIV (%x. x) x :> 1";
+by (auto_tac (claset(),
+ simpset() addsimps [NSCDERIV_NSCLIM_iff,NSCLIM_def]));
+qed "NSCDERIV_Id";
+Addsimps [NSCDERIV_Id];
+
+Goal "CDERIV (%x. x) x :> 1";
+by (simp_tac (simpset() addsimps [NSCDERIV_CDERIV_iff RS sym]) 1);
+qed "CDERIV_Id";
+Addsimps [CDERIV_Id];
+
+bind_thm ("isContc_Id", CDERIV_Id RS CDERIV_isContc);
+
+(*derivative of linear multiplication*)
+Goal "CDERIV (op * c) x :> c";
+by (cut_inst_tac [("c","c"),("x","x")] (CDERIV_Id RS CDERIV_cmult) 1);
+by (Asm_full_simp_tac 1);
+qed "CDERIV_cmult_Id";
+Addsimps [CDERIV_cmult_Id];
+
+Goal "NSCDERIV (op * c) x :> c";
+by (simp_tac (simpset() addsimps [NSCDERIV_CDERIV_iff]) 1);
+qed "NSCDERIV_cmult_Id";
+Addsimps [NSCDERIV_cmult_Id];
+
+Goal "CDERIV (%x. x ^ n) x :> (complex_of_real (real n)) * (x ^ (n - 1))";
+by (induct_tac "n" 1);
+by (dtac (CDERIV_Id RS CDERIV_mult) 2);
+by (auto_tac (claset(),
+ simpset() addsimps [complex_of_real_add RS sym,
+ complex_add_mult_distrib,real_of_nat_Suc] delsimps [complex_of_real_add]));
+by (case_tac "n" 1);
+by (auto_tac (claset(),
+ simpset() addsimps [complex_mult_assoc, complex_add_commute]));
+by (auto_tac (claset(),simpset() addsimps [complex_mult_commute]));
+qed "CDERIV_pow";
+Addsimps [CDERIV_pow,simplify (simpset()) CDERIV_pow];
+
+(* NS version *)
+Goal "NSCDERIV (%x. x ^ n) x :> complex_of_real (real n) * (x ^ (n - 1))";
+by (simp_tac (simpset() addsimps [NSCDERIV_CDERIV_iff]) 1);
+qed "NSCDERIV_pow";
+
+Goal "\\<lbrakk> CDERIV f x :> D; D = E \\<rbrakk> \\<Longrightarrow> CDERIV f x :> E";
+by Auto_tac;
+qed "lemma_CDERIV_subst";
+
+(*used once, in NSCDERIV_inverse*)
+Goal "[| h: CInfinitesimal; x ~= 0 |] ==> hcomplex_of_complex x + h ~= 0";
+by Auto_tac;
+qed "CInfinitesimal_add_not_zero";
+
+(***
+Goal "[|(x::hcomplex) ~= 0; y ~= 0 |] \
+\ ==> inverse(x) + inverse(y) = (x + y)*inverse(x*y)";
+by (asm_full_simp_tac (simpset() addsimps [hcomplex_inverse_distrib,
+ hcomplex_add_mult_distrib,hcomplex_mult_assoc RS sym]) 1);
+qed "hcomplex_inverse_add";
+***)
+
+(*Can't get rid of x ~= 0 because it isn't continuous at zero*)
+
+Goalw [nscderiv_def]
+ "x ~= 0 ==> NSCDERIV (%x. inverse(x)) x :> (- (inverse x ^ 2))";
+by (rtac ballI 1 THEN Asm_full_simp_tac 1 THEN Step_tac 1);
+by (forward_tac [CInfinitesimal_add_not_zero] 1);
+by (asm_full_simp_tac (simpset() addsimps [hcomplex_add_commute,two_eq_Suc_Suc]) 2);
+by (auto_tac (claset(),
+ simpset() addsimps [starfunC_inverse_inverse,hcomplex_diff_def]
+ delsimps [hcomplex_minus_mult_eq1 RS sym,
+ hcomplex_minus_mult_eq2 RS sym]));
+by (asm_simp_tac
+ (simpset() addsimps [hcomplex_inverse_add,
+ hcomplex_inverse_distrib RS sym, hcomplex_minus_inverse RS sym]
+ @ hcomplex_add_ac @ hcomplex_mult_ac
+ delsimps [hcomplex_minus_mult_eq1 RS sym,
+ hcomplex_minus_mult_eq2 RS sym] ) 1);
+by (asm_simp_tac (simpset() addsimps [hcomplex_mult_assoc RS sym,
+ hcomplex_add_mult_distrib2]
+ delsimps [hcomplex_minus_mult_eq1 RS sym,
+ hcomplex_minus_mult_eq2 RS sym]) 1);
+by (res_inst_tac [("y"," inverse(- hcomplex_of_complex x * hcomplex_of_complex x)")]
+ capprox_trans 1);
+by (rtac inverse_add_CInfinitesimal_capprox2 1);
+by (auto_tac (claset() addSDs [hcomplex_of_complex_CFinite_diff_CInfinitesimal] addIs [CFinite_mult],
+ simpset() addsimps [hcomplex_minus_inverse RS sym]));
+by (rtac CInfinitesimal_CFinite_mult2 1);
+by Auto_tac;
+qed "NSCDERIV_inverse";
+
+Goal "x ~= 0 ==> CDERIV (%x. inverse(x)) x :> (-(inverse x ^ 2))";
+by (asm_simp_tac (simpset() addsimps [NSCDERIV_inverse,
+ NSCDERIV_CDERIV_iff RS sym] delsimps [complexpow_Suc]) 1);
+qed "CDERIV_inverse";
+
+
+(*------------------------------------------------------------------------------------*)
+(* Derivative of inverse *)
+(*------------------------------------------------------------------------------------*)
+
+Goal "[| CDERIV f x :> d; f(x) ~= 0 |] \
+\ ==> CDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ 2)))";
+by (rtac (complex_mult_commute RS subst) 1);
+by (asm_simp_tac (simpset() addsimps [complex_minus_mult_eq1,
+ complexpow_inverse] delsimps [complexpow_Suc,
+ complex_minus_mult_eq1 RS sym]) 1);
+by (fold_goals_tac [o_def]);
+by (blast_tac (claset() addSIs [CDERIV_chain,CDERIV_inverse]) 1);
+qed "CDERIV_inverse_fun";
+
+Goal "[| NSCDERIV f x :> d; f(x) ~= 0 |] \
+\ ==> NSCDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ 2)))";
+by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_CDERIV_iff,
+ CDERIV_inverse_fun] delsimps [complexpow_Suc]) 1);
+qed "NSCDERIV_inverse_fun";
+
+(*------------------------------------------------------------------------------------*)
+(* Derivative of quotient *)
+(*------------------------------------------------------------------------------------*)
+
+
+Goal "x ~= (0::complex) \\<Longrightarrow> (x * inverse(x) ^ 2) = inverse x";
+by (auto_tac (claset(),simpset() addsimps [two_eq_Suc_Suc]));
+qed "lemma_complex_mult_inverse_squared";
+Addsimps [lemma_complex_mult_inverse_squared];
+
+Goalw [complex_diff_def]
+ "[| CDERIV f x :> d; CDERIV g x :> e; g(x) ~= 0 |] \
+\ ==> CDERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ 2)";
+by (dres_inst_tac [("f","g")] CDERIV_inverse_fun 1);
+by (dtac CDERIV_mult 2);
+by (REPEAT(assume_tac 1));
+by (asm_full_simp_tac
+ (simpset() addsimps [complex_divide_def, complex_add_mult_distrib2,
+ complexpow_inverse,complex_minus_mult_eq1] @ complex_mult_ac
+ delsimps [complexpow_Suc, complex_minus_mult_eq1 RS sym,
+ complex_minus_mult_eq2 RS sym]) 1);
+qed "CDERIV_quotient";
+
+Goal "[| NSCDERIV f x :> d; NSCDERIV g x :> e; g(x) ~= 0 |] \
+\ ==> NSCDERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ 2)";
+by (asm_full_simp_tac (simpset() addsimps [NSCDERIV_CDERIV_iff,
+ CDERIV_quotient] delsimps [complexpow_Suc]) 1);
+qed "NSCDERIV_quotient";
+
+
+(*------------------------------------------------------------------------------------*)
+(* Caratheodory formulation of derivative at a point: standard proof *)
+(*------------------------------------------------------------------------------------*)
+
+
+Goalw [CLIM_def]
+ "[| ALL x. x ~= a --> (f x = g x) |] \
+\ ==> (f -- a --C> l) = (g -- a --C> l)";
+by (auto_tac (claset(), simpset() addsimps [complex_add_minus_iff]));
+qed "CLIM_equal";
+
+Goal "[| (%x. f(x) + -g(x)) -- a --C> 0; \
+\ g -- a --C> l |] \
+\ ==> f -- a --C> l";
+by (dtac CLIM_add 1 THEN assume_tac 1);
+by (auto_tac (claset(), simpset() addsimps [complex_add_assoc]));
+qed "CLIM_trans";
+
+Goal "(CDERIV f x :> l) = \
+\ (EX g. (ALL z. f z - f x = g z * (z - x)) & isContc g x & g x = l)";
+by (Step_tac 1);
+by (res_inst_tac
+ [("x","%z. if z = x then l else (f(z) - f(x)) / (z - x)")] exI 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mult_assoc,
+ CLAIM "z ~= x ==> z - x ~= (0::complex)"]));
+by (auto_tac (claset(),simpset() addsimps [isContc_iff,CDERIV_iff]));
+by (ALLGOALS(rtac (CLIM_equal RS iffD1)));
+by Auto_tac;
+qed "CARAT_CDERIV";
+
+Goal "NSCDERIV f x :> l ==> \
+\ EX g. (ALL z. f z - f x = g z * (z - x)) & isNSContc g x & g x = l";
+by (auto_tac (claset(),simpset() addsimps [NSCDERIV_CDERIV_iff,
+ isNSContc_isContc_iff,CARAT_CDERIV]));
+qed "CARAT_NSCDERIV";
+
+(* How about a NS proof? *)
+Goal "(ALL z. f z - f x = g z * (z - x)) & isNSContc g x & g x = l \
+\ ==> NSCDERIV f x :> l";
+by (auto_tac (claset(),
+ simpset() delsimprocs complex_cancel_factor
+ addsimps [NSCDERIV_iff2]));
+by (asm_full_simp_tac (simpset() addsimps [isNSContc_def]) 1);
+qed "CARAT_CDERIVD";
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/CLim.thy Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,82 @@
+(* Title : CLim.thy
+ Author : Jacques D. Fleuriot
+ Copyright : 2001 University of Edinburgh
+ Description : A first theory of limits, continuity and
+ differentiation for complex functions
+*)
+
+CLim = CSeries +
+
+constdefs
+
+ CLIM :: [complex=>complex,complex,complex] => bool
+ ("((_)/ -- (_)/ --C> (_))" [60, 0, 60] 60)
+ "f -- a --C> L ==
+ ALL r. 0 < r -->
+ (EX s. 0 < s & (ALL x. (x ~= a & (cmod(x - a) < s)
+ --> cmod(f x - L) < r)))"
+
+ NSCLIM :: [complex=>complex,complex,complex] => bool
+ ("((_)/ -- (_)/ --NSC> (_))" [60, 0, 60] 60)
+ "f -- a --NSC> L == (ALL x. (x ~= hcomplex_of_complex a &
+ x @c= hcomplex_of_complex a
+ --> ( *fc* f) x @c= hcomplex_of_complex L))"
+
+ (* f: C --> R *)
+ CRLIM :: [complex=>real,complex,real] => bool
+ ("((_)/ -- (_)/ --CR> (_))" [60, 0, 60] 60)
+ "f -- a --CR> L ==
+ ALL r. 0 < r -->
+ (EX s. 0 < s & (ALL x. (x ~= a & (cmod(x - a) < s)
+ --> abs(f x - L) < r)))"
+
+ NSCRLIM :: [complex=>real,complex,real] => bool
+ ("((_)/ -- (_)/ --NSCR> (_))" [60, 0, 60] 60)
+ "f -- a --NSCR> L == (ALL x. (x ~= hcomplex_of_complex a &
+ x @c= hcomplex_of_complex a
+ --> ( *fcR* f) x @= hypreal_of_real L))"
+
+
+ isContc :: [complex=>complex,complex] => bool
+ "isContc f a == (f -- a --C> (f a))"
+
+ (* NS definition dispenses with limit notions *)
+ isNSContc :: [complex=>complex,complex] => bool
+ "isNSContc f a == (ALL y. y @c= hcomplex_of_complex a -->
+ ( *fc* f) y @c= hcomplex_of_complex (f a))"
+
+ isContCR :: [complex=>real,complex] => bool
+ "isContCR f a == (f -- a --CR> (f a))"
+
+ (* NS definition dispenses with limit notions *)
+ isNSContCR :: [complex=>real,complex] => bool
+ "isNSContCR f a == (ALL y. y @c= hcomplex_of_complex a -->
+ ( *fcR* f) y @= hypreal_of_real (f a))"
+
+ (* differentiation: D is derivative of function f at x *)
+ cderiv:: [complex=>complex,complex,complex] => bool
+ ("(CDERIV (_)/ (_)/ :> (_))" [60, 0, 60] 60)
+ "CDERIV f x :> D == ((%h. (f(x + h) - f(x))/h) -- 0 --C> D)"
+
+ nscderiv :: [complex=>complex,complex,complex] => bool
+ ("(NSCDERIV (_)/ (_)/ :> (_))" [60, 0, 60] 60)
+ "NSCDERIV f x :> D == (ALL h: CInfinitesimal - {0}.
+ (( *fc* f)(hcomplex_of_complex x + h)
+ - hcomplex_of_complex (f x))/h @c= hcomplex_of_complex D)"
+
+ cdifferentiable :: [complex=>complex,complex] => bool (infixl 60)
+ "f cdifferentiable x == (EX D. CDERIV f x :> D)"
+
+ NSCdifferentiable :: [complex=>complex,complex] => bool (infixl 60)
+ "f NSCdifferentiable x == (EX D. NSCDERIV f x :> D)"
+
+
+ isUContc :: (complex=>complex) => bool
+ "isUContc f == (ALL r. 0 < r -->
+ (EX s. 0 < s & (ALL x y. cmod(x - y) < s
+ --> cmod(f x - f y) < r)))"
+
+ isNSUContc :: (complex=>complex) => bool
+ "isNSUContc f == (ALL x y. x @c= y --> ( *fc* f) x @c= ( *fc* f) y)"
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/CSeries.ML Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,210 @@
+(* Title : CSeries.ML
+ Author : Jacques D. Fleuriot
+ Copyright : 2002 University of Edinburgh
+ Description : Finite summation and infinite series for complex numbers
+*)
+
+
+Goal "sumc (Suc n) n f = 0";
+by (induct_tac "n" 1);
+by (Auto_tac);
+qed "sumc_Suc_zero";
+Addsimps [sumc_Suc_zero];
+
+Goal "sumc m m f = 0";
+by (induct_tac "m" 1);
+by (Auto_tac);
+qed "sumc_eq_bounds";
+Addsimps [sumc_eq_bounds];
+
+Goal "sumc m (Suc m) f = f(m)";
+by (Auto_tac);
+qed "sumc_Suc_eq";
+Addsimps [sumc_Suc_eq];
+
+Goal "sumc (m+k) k f = 0";
+by (induct_tac "k" 1);
+by (Auto_tac);
+qed "sumc_add_lbound_zero";
+Addsimps [sumc_add_lbound_zero];
+
+Goal "sumc m n f + sumc m n g = sumc m n (%n. f n + g n)";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps real_add_ac));
+qed "sumc_add";
+
+Goal "r * sumc m n f = sumc m n (%n. r * f n)";
+by (induct_tac "n" 1);
+by (Auto_tac);
+by (auto_tac (claset(),simpset() addsimps
+ [complex_add_mult_distrib2]));
+qed "sumc_mult";
+
+Goal "n < p --> sumc 0 n f + sumc n p f = sumc 0 p f";
+by (induct_tac "p" 1);
+by (auto_tac (claset() addSDs [CLAIM "n < Suc na ==> n <= na",
+ leI] addDs [le_anti_sym], simpset()));
+qed_spec_mp "sumc_split_add";
+
+Goal "n < p ==> sumc 0 p f + \
+\ - sumc 0 n f = sumc n p f";
+by (dres_inst_tac [("f1","f")] (sumc_split_add RS sym) 1);
+by (asm_simp_tac (simpset() addsimps complex_add_ac) 1);
+qed "sumc_split_add_minus";
+
+Goal "cmod(sumc m n f) <= sumr m n (%i. cmod(f i))";
+by (induct_tac "n" 1);
+by (auto_tac (claset() addIs [complex_mod_triangle_ineq RS order_trans],
+ simpset()));
+qed "sumc_cmod";
+
+Goal "!!f g. (ALL r. m <= r & r < n --> f r = g r) \
+\ --> sumc m n f = sumc m n g";
+by (induct_tac "n" 1);
+by (Auto_tac);
+qed_spec_mp "sumc_fun_eq";
+
+Goal "sumc 0 n (%i. r) = complex_of_real (real n) * r";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),
+ simpset() addsimps [complex_add_mult_distrib,
+ complex_of_real_add RS sym,
+ real_of_nat_Suc]));
+qed "sumc_const";
+Addsimps [sumc_const];
+
+Goal "sumc 0 n f + -(complex_of_real(real n) * r) = sumc 0 n (%i. f i + -r)";
+by (full_simp_tac (simpset() addsimps [sumc_add RS sym]) 1);
+qed "sumc_add_mult_const";
+
+Goalw [complex_diff_def]
+ "sumc 0 n f - (complex_of_real(real n)*r) = sumc 0 n (%i. f i - r)";
+by (full_simp_tac (simpset() addsimps [sumc_add_mult_const]) 1);
+qed "sumc_diff_mult_const";
+
+Goal "n < m --> sumc m n f = 0";
+by (induct_tac "n" 1);
+by Auto_tac;
+qed_spec_mp "sumc_less_bounds_zero";
+Addsimps [sumc_less_bounds_zero];
+
+Goal "sumc m n (%i. - f i) = - sumc m n f";
+by (induct_tac "n" 1);
+by Auto_tac;
+qed "sumc_minus";
+
+Goal "sumc (m+k) (n+k) f = sumc m n (%i. f(i + k))";
+by (induct_tac "n" 1);
+by (Auto_tac);
+qed "sumc_shift_bounds";
+
+Goal "sumc 0 (2*n) (%i. (-1) ^ Suc i) = 0";
+by (induct_tac "n" 1);
+by (Auto_tac);
+qed "sumc_minus_one_complexpow_zero";
+Addsimps [sumc_minus_one_complexpow_zero];
+
+Goal "(ALL n. m <= Suc n --> f n = r) & m <= na \
+\ --> sumc m na f = (complex_of_real(real (na - m)) * r)";
+by (induct_tac "na" 1);
+by (auto_tac (claset(),simpset() addsimps [real_add_mult_distrib, Suc_diff_n,
+ real_of_nat_Suc,complex_of_real_add RS sym,
+ complex_add_mult_distrib]));
+qed_spec_mp "sumc_interval_const";
+
+Goal "(ALL n. m <= n --> f n = r) & m <= na \
+\ --> sumc m na f = (complex_of_real(real (na - m)) * r)";
+by (induct_tac "na" 1);
+by (auto_tac (claset(),simpset() addsimps [real_add_mult_distrib, Suc_diff_n,
+ real_of_nat_Suc,complex_of_real_add RS sym,
+ complex_add_mult_distrib]));
+qed_spec_mp "sumc_interval_const2";
+
+(***
+Goal "(ALL n. m <= n --> 0 <= cmod(f n)) & m < k --> cmod(sumc 0 m f) <= cmod(sumc 0 k f)";
+by (induct_tac "k" 1);
+by (Step_tac 1);
+by (ALLGOALS(asm_full_simp_tac (simpset() addsimps [less_Suc_eq_le])));
+by (ALLGOALS(dres_inst_tac [("x","n")] spec));
+by (Step_tac 1);
+by (dtac le_imp_less_or_eq 1 THEN Step_tac 1);
+by (dtac real_add_le_mono 2);
+by (dres_inst_tac [("i","sumr 0 m f")] (order_refl RS real_add_le_mono) 1);
+by (Auto_tac);
+qed_spec_mp "sumc_le";
+
+Goal "!!f g. (ALL r. m <= r & r < n --> f r <= g r) \
+\ --> sumc m n f <= sumc m n g";
+by (induct_tac "n" 1);
+by (auto_tac (claset() addIs [real_add_le_mono],
+ simpset() addsimps [le_def]));
+qed_spec_mp "sumc_le2";
+
+Goal "(ALL n. 0 <= f n) --> 0 <= sumc m n f";
+by (induct_tac "n" 1);
+by Auto_tac;
+by (dres_inst_tac [("x","n")] spec 1);
+by (arith_tac 1);
+qed_spec_mp "sumc_ge_zero";
+
+Goal "(ALL n. m <= n --> 0 <= f n) --> 0 <= sumc m n f";
+by (induct_tac "n" 1);
+by Auto_tac;
+by (dres_inst_tac [("x","n")] spec 1);
+by (arith_tac 1);
+qed_spec_mp "sumc_ge_zero2";
+***)
+
+Goal "0 <= sumr m n (%n. cmod (f n))";
+by (induct_tac "n" 1);
+by Auto_tac;
+by (res_inst_tac [("j","0")] real_le_trans 1);
+by Auto_tac;
+qed "sumr_cmod_ge_zero";
+Addsimps [sumr_cmod_ge_zero];
+AddSIs [sumr_cmod_ge_zero];
+
+Goal "abs (sumr m n (%n. cmod (f n))) = (sumr m n (%n. cmod (f n)))";
+by (rtac (abs_eqI1 RS ssubst) 1 THEN Auto_tac);
+qed "rabs_sumc_cmod_cancel";
+Addsimps [rabs_sumc_cmod_cancel];
+
+Goal "ALL n. N <= n --> f n = 0 \
+\ ==> ALL m n. N <= m --> sumc m n f = 0";
+by (Step_tac 1);
+by (induct_tac "n" 1);
+by (Auto_tac);
+qed "sumc_zero";
+
+Goal "ALL n. N <= n --> f (Suc n) = 0 \
+\ ==> ALL m n. Suc N <= m --> sumc m n f = 0";
+by (rtac sumc_zero 1 THEN Step_tac 1);
+by (dres_inst_tac [("x","n - 1")] spec 1);
+by Auto_tac;
+by (arith_tac 1);
+qed "fun_zero_sumc_zero";
+
+Goal "sumc 1 n (%n. f(n) * 0 ^ n) = 0";
+by (induct_tac "n" 1);
+by (case_tac "n" 2);
+by Auto_tac;
+qed "sumc_one_lb_complexpow_zero";
+Addsimps [sumc_one_lb_complexpow_zero];
+
+Goalw [complex_diff_def] "sumc m n f - sumc m n g = sumc m n (%n. f n - g n)";
+by (simp_tac (simpset() addsimps [sumc_add RS sym,sumc_minus]) 1);
+qed "sumc_diff";
+
+Goal "(ALL p. (m <= p & p < m + n --> (f p = g p))) --> sumc m n f = sumc m n g";
+by (induct_tac "n" 1);
+by (Auto_tac);
+qed_spec_mp "sumc_subst";
+
+Goal "sumc 0 n (%m. sumc (m * k) (m*k + k) f) = sumc 0 (n * k) f";
+by (subgoal_tac "k = 0 | 0 < k" 1);
+by (Auto_tac);
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps [sumc_split_add,add_commute]));
+qed "sumc_group";
+Addsimps [sumc_group];
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/CSeries.thy Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,30 @@
+(* Title : CSeries.thy
+ Author : Jacques D. Fleuriot
+ Copyright : 2002 University of Edinburgh
+ Description : Finite summation and infinite series for complex numbers
+*)
+
+CSeries = CStar +
+
+consts sumc :: "[nat,nat,(nat=>complex)] => complex"
+primrec
+ sumc_0 "sumc m 0 f = 0"
+ sumc_Suc "sumc m (Suc n) f = (if n < m then 0 else sumc m n f + f(n))"
+
+(*
+constdefs
+
+ needs convergence of complex sequences
+
+ csums :: [nat=>complex,complex] => bool (infixr 80)
+ "f sums s == (%n. sumr 0 n f) ----C> s"
+
+ csummable :: (nat=>complex) => bool
+ "csummable f == (EX s. f csums s)"
+
+ csuminf :: (nat=>complex) => complex
+ "csuminf f == (@s. f csums s)"
+*)
+
+end
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/CStar.ML Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,630 @@
+(* Title : CStar.ML
+ Author : Jacques D. Fleuriot
+ Copyright : 2001 University of Edinburgh
+ Description : defining *-transforms in NSA which extends sets of complex numbers,
+ and complex functions
+*)
+
+
+
+(*-----------------------------------------------------------------------------------*)
+(* Properties of the *-transform applied to sets of reals *)
+(* ----------------------------------------------------------------------------------*)
+
+Goalw [starsetC_def] "*sc*(UNIV::complex set) = (UNIV::hcomplex set)";
+by (Auto_tac);
+qed "STARC_complex_set";
+Addsimps [STARC_complex_set];
+
+Goalw [starsetC_def] "*sc* {} = {}";
+by (Auto_tac);
+qed "STARC_empty_set";
+Addsimps [STARC_empty_set];
+
+Goalw [starsetC_def] "*sc* (A Un B) = *sc* A Un *sc* B";
+by (Auto_tac);
+by (REPEAT(blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 2));
+by (dtac bspec 1 THEN assume_tac 1);
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (Auto_tac);
+by (Ultra_tac 1);
+qed "STARC_Un";
+
+Goalw [starsetC_n_def]
+ "*scn* (%n. (A n) Un (B n)) = *scn* A Un *scn* B";
+by Auto_tac;
+by (dres_inst_tac [("x","Xa")] bspec 1);
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 2);
+by (auto_tac (claset() addSDs [bspec], simpset()));
+by (TRYALL(Ultra_tac));
+qed "starsetC_n_Un";
+
+Goalw [InternalCSets_def]
+ "[| X : InternalCSets; Y : InternalCSets |] \
+\ ==> (X Un Y) : InternalCSets";
+by (auto_tac (claset(),
+ simpset() addsimps [starsetC_n_Un RS sym]));
+qed "InternalCSets_Un";
+
+Goalw [starsetC_def] "*sc* (A Int B) = *sc* A Int *sc* B";
+by (Auto_tac);
+by (blast_tac (claset() addIs [FreeUltrafilterNat_Int,
+ FreeUltrafilterNat_subset]) 3);
+by (REPEAT(blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 1));
+qed "STARC_Int";
+
+Goalw [starsetC_n_def]
+ "*scn* (%n. (A n) Int (B n)) = *scn* A Int *scn* B";
+by (Auto_tac);
+by (auto_tac (claset() addSDs [bspec],simpset()));
+by (TRYALL(Ultra_tac));
+qed "starsetC_n_Int";
+
+Goalw [InternalCSets_def]
+ "[| X : InternalCSets; Y : InternalCSets |] \
+\ ==> (X Int Y) : InternalCSets";
+by (auto_tac (claset(),
+ simpset() addsimps [starsetC_n_Int RS sym]));
+qed "InternalCSets_Int";
+
+Goalw [starsetC_def] "*sc* -A = -( *sc* A)";
+by (Auto_tac);
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 2);
+by (REPEAT(Step_tac 1) THEN Auto_tac);
+by (ALLGOALS(Ultra_tac));
+qed "STARC_Compl";
+
+Goalw [starsetC_n_def] "*scn* ((%n. - A n)) = -( *scn* A)";
+by (Auto_tac);
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 2);
+by (REPEAT(Step_tac 1) THEN Auto_tac);
+by (TRYALL(Ultra_tac));
+qed "starsetC_n_Compl";
+
+Goalw [InternalCSets_def]
+ "X :InternalCSets ==> -X : InternalCSets";
+by (auto_tac (claset(),
+ simpset() addsimps [starsetC_n_Compl RS sym]));
+qed "InternalCSets_Compl";
+
+Goal "x ~: *sc* F ==> x : *sc* (- F)";
+by (auto_tac (claset(),simpset() addsimps [STARC_Compl]));
+qed "STARC_mem_Compl";
+
+Goal "*sc* (A - B) = *sc* A - *sc* B";
+by (auto_tac (claset(),simpset() addsimps
+ [set_diff_iff2,STARC_Int,STARC_Compl]));
+qed "STARC_diff";
+
+Goalw [starsetC_n_def]
+ "*scn* (%n. (A n) - (B n)) = *scn* A - *scn* B";
+by (Auto_tac);
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 2);
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 3);
+by (auto_tac (claset() addSDs [bspec], simpset()));
+by (TRYALL(Ultra_tac));
+qed "starsetC_n_diff";
+
+Goalw [InternalCSets_def]
+ "[| X : InternalCSets; Y : InternalCSets |] \
+\ ==> (X - Y) : InternalCSets";
+by (auto_tac (claset(), simpset() addsimps [starsetC_n_diff RS sym]));
+qed "InternalCSets_diff";
+
+Goalw [starsetC_def] "A <= B ==> *sc* A <= *sc* B";
+by (REPEAT(blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 1));
+qed "STARC_subset";
+
+Goalw [starsetC_def,hcomplex_of_complex_def]
+ "a : A ==> hcomplex_of_complex a : *sc* A";
+by (auto_tac (claset() addIs [FreeUltrafilterNat_subset],
+ simpset()));
+qed "STARC_mem";
+
+Goalw [starsetC_def] "hcomplex_of_complex ` A <= *sc* A";
+by (auto_tac (claset(), simpset() addsimps [hcomplex_of_complex_def]));
+by (blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 1);
+qed "STARC_hcomplex_of_complex_image_subset";
+
+Goal "SComplex <= *sc* (UNIV:: complex set)";
+by Auto_tac;
+qed "STARC_SComplex_subset";
+
+Goalw [starsetC_def]
+ "*sc* X Int SComplex = hcomplex_of_complex ` X";
+by (auto_tac (claset(),
+ simpset() addsimps
+ [hcomplex_of_complex_def,SComplex_def]));
+by (fold_tac [hcomplex_of_complex_def]);
+by (rtac imageI 1 THEN rtac ccontr 1);
+by (dtac bspec 1);
+by (rtac lemma_hcomplexrel_refl 1);
+by (blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 2);
+by (Auto_tac);
+qed "STARC_hcomplex_of_complex_Int";
+
+Goal "x ~: hcomplex_of_complex ` A ==> ALL y: A. x ~= hcomplex_of_complex y";
+by (Auto_tac);
+qed "lemma_not_hcomplexA";
+
+Goalw [starsetC_n_def,starsetC_def] "*sc* X = *scn* (%n. X)";
+by Auto_tac;
+qed "starsetC_starsetC_n_eq";
+
+Goalw [InternalCSets_def] "( *sc* X) : InternalCSets";
+by (auto_tac (claset(),
+ simpset() addsimps [starsetC_starsetC_n_eq]));
+qed "InternalCSets_starsetC_n";
+Addsimps [InternalCSets_starsetC_n];
+
+Goal "X : InternalCSets ==> UNIV - X : InternalCSets";
+by (auto_tac (claset() addIs [InternalCSets_Compl], simpset()));
+qed "InternalCSets_UNIV_diff";
+
+(*-----------------------------------------------------------------------------------*)
+(* Nonstandard extension of a set (defined using a constant sequence) as a special *)
+(* case of an internal set *)
+(*-----------------------------------------------------------------------------------*)
+
+Goalw [starsetC_n_def,starsetC_def]
+ "ALL n. (As n = A) ==> *scn* As = *sc* A";
+by (Auto_tac);
+qed "starsetC_n_starsetC";
+
+(*-----------------------------------------------------------------------------------*)
+(* Theorems about nonstandard extensions of functions *)
+(*-----------------------------------------------------------------------------------*)
+
+Goalw [starfunC_n_def,starfunC_def]
+ "ALL n. (F n = f) ==> *fcn* F = *fc* f";
+by (Auto_tac);
+qed "starfunC_n_starfunC";
+
+Goalw [starfunRC_n_def,starfunRC_def]
+ "ALL n. (F n = f) ==> *fRcn* F = *fRc* f";
+by (Auto_tac);
+qed "starfunRC_n_starfunRC";
+
+Goalw [starfunCR_n_def,starfunCR_def]
+ "ALL n. (F n = f) ==> *fcRn* F = *fcR* f";
+by (Auto_tac);
+qed "starfunCR_n_starfunCR";
+
+Goalw [congruent_def]
+ "congruent hcomplexrel (%X. hcomplexrel``{%n. f (X n)})";
+by (safe_tac (claset()));
+by (ALLGOALS(Ultra_tac));
+qed "starfunC_congruent";
+
+(* f::complex => complex *)
+Goalw [starfunC_def]
+ "( *fc* f) (Abs_hcomplex(hcomplexrel``{%n. X n})) = \
+\ Abs_hcomplex(hcomplexrel `` {%n. f (X n)})";
+by (res_inst_tac [("f","Abs_hcomplex")] arg_cong 1);
+by Auto_tac;
+by (Ultra_tac 1);
+qed "starfunC";
+
+Goalw [starfunRC_def]
+ "( *fRc* f) (Abs_hypreal(hyprel``{%n. X n})) = \
+\ Abs_hcomplex(hcomplexrel `` {%n. f (X n)})";
+by (res_inst_tac [("f","Abs_hcomplex")] arg_cong 1);
+by Auto_tac;
+by (Ultra_tac 1);
+qed "starfunRC";
+
+Goalw [starfunCR_def]
+ "( *fcR* f) (Abs_hcomplex(hcomplexrel``{%n. X n})) = \
+\ Abs_hypreal(hyprel `` {%n. f (X n)})";
+by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);
+by Auto_tac;
+by (Ultra_tac 1);
+qed "starfunCR";
+
+(** multiplication: ( *f ) x ( *g ) = *(f x g) **)
+
+Goal "( *fc* f) z * ( *fc* g) z = ( *fc* (%x. f x * g x)) z";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(), simpset() addsimps [starfunC,hcomplex_mult]));
+qed "starfunC_mult";
+Addsimps [starfunC_mult RS sym];
+
+Goal "( *fRc* f) z * ( *fRc* g) z = ( *fRc* (%x. f x * g x)) z";
+by (res_inst_tac [("z","z")] eq_Abs_hypreal 1);
+by (auto_tac (claset(), simpset() addsimps [starfunRC,hcomplex_mult]));
+qed "starfunRC_mult";
+Addsimps [starfunRC_mult RS sym];
+
+Goal "( *fcR* f) z * ( *fcR* g) z = ( *fcR* (%x. f x * g x)) z";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(), simpset() addsimps [starfunCR,hypreal_mult]));
+qed "starfunCR_mult";
+Addsimps [starfunCR_mult RS sym];
+
+(** addition: ( *f ) + ( *g ) = *(f + g) **)
+
+Goal "( *fc* f) z + ( *fc* g) z = ( *fc* (%x. f x + g x)) z";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(), simpset() addsimps [starfunC,hcomplex_add]));
+qed "starfunC_add";
+Addsimps [starfunC_add RS sym];
+
+Goal "( *fRc* f) z + ( *fRc* g) z = ( *fRc* (%x. f x + g x)) z";
+by (res_inst_tac [("z","z")] eq_Abs_hypreal 1);
+by (auto_tac (claset(), simpset() addsimps [starfunRC,hcomplex_add]));
+qed "starfunRC_add";
+Addsimps [starfunRC_add RS sym];
+
+Goal "( *fcR* f) z + ( *fcR* g) z = ( *fcR* (%x. f x + g x)) z";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(), simpset() addsimps [starfunCR,hypreal_add]));
+qed "starfunCR_add";
+Addsimps [starfunCR_add RS sym];
+
+(** uminus **)
+Goal "( *fc* (%x. - f x)) x = - ( *fc* f) x";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [starfunC, hcomplex_minus]));
+qed "starfunC_minus";
+Addsimps [starfunC_minus];
+
+Goal "( *fRc* (%x. - f x)) x = - ( *fRc* f) x";
+by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [starfunRC, hcomplex_minus]));
+qed "starfunRC_minus";
+Addsimps [starfunRC_minus];
+
+Goal "( *fcR* (%x. - f x)) x = - ( *fcR* f) x";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [starfunCR, hypreal_minus]));
+qed "starfunCR_minus";
+Addsimps [starfunCR_minus];
+
+(** addition: ( *f ) - ( *g ) = *(f - g) **)
+
+Goalw [hcomplex_diff_def,complex_diff_def]
+ "( *fc* f) xa - ( *fc* g) xa = ( *fc* (%x. f x - g x)) xa";
+by (auto_tac (claset(),simpset() addsimps [starfunC_minus,starfunC_add RS sym]));
+qed "starfunC_diff";
+Addsimps [starfunC_diff RS sym];
+
+Goalw [hcomplex_diff_def,complex_diff_def]
+ "( *fRc* f) xa - ( *fRc* g) xa = ( *fRc* (%x. f x - g x)) xa";
+by (auto_tac (claset(),simpset() addsimps [starfunRC_minus,starfunRC_add RS sym]));
+qed "starfunRC_diff";
+Addsimps [starfunRC_diff RS sym];
+
+Goalw [hypreal_diff_def,real_diff_def]
+ "( *fcR* f) xa - ( *fcR* g) xa = ( *fcR* (%x. f x - g x)) xa";
+by (auto_tac (claset(),simpset() addsimps [starfunCR_minus,starfunCR_add RS sym]));
+qed "starfunCR_diff";
+Addsimps [starfunCR_diff RS sym];
+
+(** composition: ( *f ) o ( *g ) = *(f o g) **)
+
+Goal "(%x. ( *fc* f) (( *fc* g) x)) = *fc* (%x. f (g x))";
+by (rtac ext 1);
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [starfunC]));
+qed "starfunC_o2";
+
+Goalw [o_def] "( *fc* f) o ( *fc* g) = ( *fc* (f o g))";
+by (simp_tac (simpset() addsimps [starfunC_o2]) 1);
+qed "starfunC_o";
+
+Goal "(%x. ( *fc* f) (( *fRc* g) x)) = *fRc* (%x. f (g x))";
+by (rtac ext 1);
+by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [starfunRC,starfunC]));
+qed "starfunC_starfunRC_o2";
+
+Goal "(%x. ( *f* f) (( *fcR* g) x)) = *fcR* (%x. f (g x))";
+by (rtac ext 1);
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [starfunCR,starfun]));
+qed "starfun_starfunCR_o2";
+
+Goalw [o_def] "( *fc* f) o ( *fRc* g) = ( *fRc* (f o g))";
+by (simp_tac (simpset() addsimps [starfunC_starfunRC_o2]) 1);
+qed "starfunC_starfunRC_o";
+
+Goalw [o_def] "( *f* f) o ( *fcR* g) = ( *fcR* (f o g))";
+by (simp_tac (simpset() addsimps [starfun_starfunCR_o2]) 1);
+qed "starfun_starfunCR_o";
+
+Goal "( *fc* (%x. k)) z = hcomplex_of_complex k";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(), simpset() addsimps [starfunC, hcomplex_of_complex_def]));
+qed "starfunC_const_fun";
+Addsimps [starfunC_const_fun];
+
+Goal "( *fRc* (%x. k)) z = hcomplex_of_complex k";
+by (res_inst_tac [("z","z")] eq_Abs_hypreal 1);
+by (auto_tac (claset(), simpset() addsimps [starfunRC, hcomplex_of_complex_def]));
+qed "starfunRC_const_fun";
+Addsimps [starfunRC_const_fun];
+
+Goal "( *fcR* (%x. k)) z = hypreal_of_real k";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(), simpset() addsimps [starfunCR, hypreal_of_real_def]));
+qed "starfunCR_const_fun";
+Addsimps [starfunCR_const_fun];
+
+Goal "inverse (( *fc* f) x) = ( *fc* (%x. inverse (f x))) x";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(), simpset() addsimps [starfunC, hcomplex_inverse]));
+qed "starfunC_inverse";
+Addsimps [starfunC_inverse RS sym];
+
+Goal "inverse (( *fRc* f) x) = ( *fRc* (%x. inverse (f x))) x";
+by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
+by (auto_tac (claset(), simpset() addsimps [starfunRC, hcomplex_inverse]));
+qed "starfunRC_inverse";
+Addsimps [starfunRC_inverse RS sym];
+
+Goal "inverse (( *fcR* f) x) = ( *fcR* (%x. inverse (f x))) x";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(), simpset() addsimps [starfunCR, hypreal_inverse]));
+qed "starfunCR_inverse";
+Addsimps [starfunCR_inverse RS sym];
+
+Goal "( *fc* f) (hcomplex_of_complex a) = hcomplex_of_complex (f a)";
+by (auto_tac (claset(),
+ simpset() addsimps [starfunC,hcomplex_of_complex_def]));
+qed "starfunC_eq";
+Addsimps [starfunC_eq];
+
+Goal "( *fRc* f) (hypreal_of_real a) = hcomplex_of_complex (f a)";
+by (auto_tac (claset(),
+ simpset() addsimps [starfunRC,hcomplex_of_complex_def,hypreal_of_real_def]));
+qed "starfunRC_eq";
+Addsimps [starfunRC_eq];
+
+Goal "( *fcR* f) (hcomplex_of_complex a) = hypreal_of_real (f a)";
+by (auto_tac (claset(),
+ simpset() addsimps [starfunCR,hcomplex_of_complex_def,hypreal_of_real_def]));
+qed "starfunCR_eq";
+Addsimps [starfunCR_eq];
+
+Goal "( *fc* f) (hcomplex_of_complex a) @c= hcomplex_of_complex (f a)";
+by (Auto_tac);
+qed "starfunC_capprox";
+
+Goal "( *fRc* f) (hypreal_of_real a) @c= hcomplex_of_complex (f a)";
+by (Auto_tac);
+qed "starfunRC_capprox";
+
+Goal "( *fcR* f) (hcomplex_of_complex a) @= hypreal_of_real (f a)";
+by (Auto_tac);
+qed "starfunCR_approx";
+
+(*
+Goal "( *fcNat* (%n. z ^ n)) N = (hcomplex_of_complex z) hcpow N";
+*)
+
+Goalw [hypnat_of_nat_def]
+ "( *fc* (%z. z ^ n)) Z = Z hcpow hypnat_of_nat n";
+by (res_inst_tac [("z","Z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(), simpset() addsimps [hcpow,starfunC]));
+qed "starfunC_hcpow";
+
+Goal "( *fc* (%h. f (x + h))) xa = ( *fc* f) (hcomplex_of_complex x + xa)";
+by (res_inst_tac [("z","xa")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [starfunC,
+ hcomplex_of_complex_def,hcomplex_add]));
+qed "starfunC_lambda_cancel";
+
+Goal "( *fcR* (%h. f (x + h))) xa = ( *fcR* f) (hcomplex_of_complex x + xa)";
+by (res_inst_tac [("z","xa")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [starfunCR,
+ hcomplex_of_complex_def,hcomplex_add]));
+qed "starfunCR_lambda_cancel";
+
+Goal "( *fRc* (%h. f (x + h))) xa = ( *fRc* f) (hypreal_of_real x + xa)";
+by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [starfunRC,
+ hypreal_of_real_def,hypreal_add]));
+qed "starfunRC_lambda_cancel";
+
+Goal "( *fc* (%h. f(g(x + h)))) xa = ( *fc* (f o g)) (hcomplex_of_complex x + xa)";
+by (res_inst_tac [("z","xa")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [starfunC,
+ hcomplex_of_complex_def,hcomplex_add]));
+qed "starfunC_lambda_cancel2";
+
+Goal "( *fcR* (%h. f(g(x + h)))) xa = ( *fcR* (f o g)) (hcomplex_of_complex x + xa)";
+by (res_inst_tac [("z","xa")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [starfunCR,
+ hcomplex_of_complex_def,hcomplex_add]));
+qed "starfunCR_lambda_cancel2";
+
+Goal "( *fRc* (%h. f(g(x + h)))) xa = ( *fRc* (f o g)) (hypreal_of_real x + xa)";
+by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [starfunRC,
+ hypreal_of_real_def,hypreal_add]));
+qed "starfunRC_lambda_cancel2";
+
+Goal "[| ( *fc* f) xa @c= l; ( *fc* g) xa @c= m; \
+\ l: CFinite; m: CFinite \
+\ |] ==> ( *fc* (%x. f x * g x)) xa @c= l * m";
+by (dtac capprox_mult_CFinite 1);
+by (REPEAT(assume_tac 1));
+by (auto_tac (claset() addIs [capprox_sym RSN (2,capprox_CFinite)],
+ simpset()));
+qed "starfunC_mult_CFinite_capprox";
+
+Goal "[| ( *fcR* f) xa @= l; ( *fcR* g) xa @= m; \
+\ l: HFinite; m: HFinite \
+\ |] ==> ( *fcR* (%x. f x * g x)) xa @= l * m";
+by (dtac approx_mult_HFinite 1);
+by (REPEAT(assume_tac 1));
+by (auto_tac (claset() addIs [approx_sym RSN (2,approx_HFinite)],
+ simpset()));
+qed "starfunCR_mult_HFinite_capprox";
+
+Goal "[| ( *fRc* f) xa @c= l; ( *fRc* g) xa @c= m; \
+\ l: CFinite; m: CFinite \
+\ |] ==> ( *fRc* (%x. f x * g x)) xa @c= l * m";
+by (dtac capprox_mult_CFinite 1);
+by (REPEAT(assume_tac 1));
+by (auto_tac (claset() addIs [capprox_sym RSN (2,capprox_CFinite)],
+ simpset()));
+qed "starfunRC_mult_CFinite_capprox";
+
+Goal "[| ( *fc* f) xa @c= l; ( *fc* g) xa @c= m \
+\ |] ==> ( *fc* (%x. f x + g x)) xa @c= l + m";
+by (auto_tac (claset() addIs [capprox_add], simpset()));
+qed "starfunC_add_capprox";
+
+Goal "[| ( *fRc* f) xa @c= l; ( *fRc* g) xa @c= m \
+\ |] ==> ( *fRc* (%x. f x + g x)) xa @c= l + m";
+by (auto_tac (claset() addIs [capprox_add], simpset()));
+qed "starfunRC_add_capprox";
+
+Goal "[| ( *fcR* f) xa @= l; ( *fcR* g) xa @= m \
+\ |] ==> ( *fcR* (%x. f x + g x)) xa @= l + m";
+by (auto_tac (claset() addIs [approx_add], simpset()));
+qed "starfunCR_add_approx";
+
+Goal "*fcR* cmod = hcmod";
+by (rtac ext 1);
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [starfunCR,hcmod]));
+qed "starfunCR_cmod";
+
+Goal "( *fc* inverse) x = inverse(x)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [starfunC,hcomplex_inverse]));
+qed "starfunC_inverse_inverse";
+
+Goalw [hcomplex_divide_def,complex_divide_def]
+ "( *fc* f) xa / ( *fc* g) xa = ( *fc* (%x. f x / g x)) xa";
+by Auto_tac;
+qed "starfunC_divide";
+Addsimps [starfunC_divide RS sym];
+
+Goalw [hypreal_divide_def,real_divide_def]
+ "( *fcR* f) xa / ( *fcR* g) xa = ( *fcR* (%x. f x / g x)) xa";
+by Auto_tac;
+qed "starfunCR_divide";
+Addsimps [starfunCR_divide RS sym];
+
+Goalw [hcomplex_divide_def,complex_divide_def]
+ "( *fRc* f) xa / ( *fRc* g) xa = ( *fRc* (%x. f x / g x)) xa";
+by Auto_tac;
+qed "starfunRC_divide";
+Addsimps [starfunRC_divide RS sym];
+
+(*-----------------------------------------------------------------------------------*)
+(* Internal functions - some redundancy with *fc* now *)
+(*-----------------------------------------------------------------------------------*)
+
+Goalw [congruent_def]
+ "congruent hcomplexrel (%X. hcomplexrel``{%n. f n (X n)})";
+by (safe_tac (claset()));
+by (ALLGOALS(Fuf_tac));
+qed "starfunC_n_congruent";
+
+Goalw [starfunC_n_def]
+ "( *fcn* f) (Abs_hcomplex(hcomplexrel``{%n. X n})) = \
+\ Abs_hcomplex(hcomplexrel `` {%n. f n (X n)})";
+by (res_inst_tac [("f","Abs_hcomplex")] arg_cong 1);
+by Auto_tac;
+by (Ultra_tac 1);
+qed "starfunC_n";
+
+(** multiplication: ( *fn ) x ( *gn ) = *(fn x gn) **)
+
+Goal "( *fcn* f) z * ( *fcn* g) z = ( *fcn* (% i x. f i x * g i x)) z";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(), simpset() addsimps [starfunC_n,hcomplex_mult]));
+qed "starfunC_n_mult";
+
+(** addition: ( *fn ) + ( *gn ) = *(fn + gn) **)
+
+Goal "( *fcn* f) z + ( *fcn* g) z = ( *fcn* (%i x. f i x + g i x)) z";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(), simpset() addsimps [starfunC_n,hcomplex_add]));
+qed "starfunC_n_add";
+
+(** uminus **)
+
+Goal "- ( *fcn* g) z = ( *fcn* (%i x. - g i x)) z";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(), simpset() addsimps [starfunC_n,hcomplex_minus]));
+qed "starfunC_n_minus";
+
+(** subtraction: ( *fn ) - ( *gn ) = *(fn - gn) **)
+
+Goalw [hcomplex_diff_def,complex_diff_def]
+ "( *fcn* f) z - ( *fcn* g) z = ( *fcn* (%i x. f i x - g i x)) z";
+by (auto_tac (claset(),
+ simpset() addsimps [starfunC_n_add,starfunC_n_minus]));
+qed "starfunNat_n_diff";
+
+(** composition: ( *fn ) o ( *gn ) = *(fn o gn) **)
+
+Goal "( *fcn* (%i x. k)) z = hcomplex_of_complex k";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),
+ simpset() addsimps [starfunC_n, hcomplex_of_complex_def]));
+qed "starfunC_n_const_fun";
+Addsimps [starfunC_n_const_fun];
+
+Goal "( *fcn* f) (hcomplex_of_complex n) = Abs_hcomplex(hcomplexrel `` {%i. f i n})";
+by (auto_tac (claset(), simpset() addsimps [starfunC_n,hcomplex_of_complex_def]));
+qed "starfunC_n_eq";
+Addsimps [starfunC_n_eq];
+
+Goal "(( *fc* f) = ( *fc* g)) = (f = g)";
+by Auto_tac;
+by (rtac ext 1 THEN rtac ccontr 1);
+by (dres_inst_tac [("x","hcomplex_of_complex(x)")] fun_cong 1);
+by (auto_tac (claset(), simpset() addsimps [starfunC,hcomplex_of_complex_def]));
+qed "starfunC_eq_iff";
+
+Goal "(( *fRc* f) = ( *fRc* g)) = (f = g)";
+by Auto_tac;
+by (rtac ext 1 THEN rtac ccontr 1);
+by (dres_inst_tac [("x","hypreal_of_real(x)")] fun_cong 1);
+by Auto_tac;
+qed "starfunRC_eq_iff";
+
+Goal "(( *fcR* f) = ( *fcR* g)) = (f = g)";
+by Auto_tac;
+by (rtac ext 1 THEN rtac ccontr 1);
+by (dres_inst_tac [("x","hcomplex_of_complex(x)")] fun_cong 1);
+by Auto_tac;
+qed "starfunCR_eq_iff";
+
+(*** more theorems ***)
+
+Goal "(( *fc* f) x = z) = ((( *fcR* (%x. Re(f x))) x = hRe (z)) & \
+\ (( *fcR* (%x. Im(f x))) x = hIm (z)))";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [starfunCR,starfunC,
+ hIm,hRe,complex_Re_Im_cancel_iff]));
+by (ALLGOALS(Ultra_tac));
+qed "starfunC_eq_Re_Im_iff";
+
+Goal "(( *fc* f) x @c= z) = ((( *fcR* (%x. Re(f x))) x @= hRe (z)) & \
+\ (( *fcR* (%x. Im(f x))) x @= hIm (z)))";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [starfunCR,starfunC,
+ hIm,hRe,capprox_approx_iff]));
+qed "starfunC_approx_Re_Im_iff";
+
+Goal "x @c= hcomplex_of_complex a ==> ( *fc* (%x. x)) x @c= hcomplex_of_complex a";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [starfunC]));
+qed "starfunC_Idfun_capprox";
+
+Goal "( *fc* (%x. x)) x = x";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [starfunC]));
+qed "starfunC_Id";
+Addsimps [starfunC_Id];
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/CStar.thy Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,57 @@
+(* Title : CStar.thy
+ Author : Jacques D. Fleuriot
+ Copyright : 2001 University of Edinburgh
+ Description : defining *-transforms in NSA which extends sets of complex numbers,
+ and complex functions
+*)
+
+CStar = NSCA +
+
+constdefs
+
+ (* nonstandard extension of sets *)
+ starsetC :: complex set => hcomplex set ("*sc* _" [80] 80)
+ "*sc* A == {x. ALL X: Rep_hcomplex(x). {n::nat. X n : A}: FreeUltrafilterNat}"
+
+ (* internal sets *)
+ starsetC_n :: (nat => complex set) => hcomplex set ("*scn* _" [80] 80)
+ "*scn* As == {x. ALL X: Rep_hcomplex(x). {n::nat. X n : (As n)}: FreeUltrafilterNat}"
+
+ InternalCSets :: "hcomplex set set"
+ "InternalCSets == {X. EX As. X = *scn* As}"
+
+ (* star transform of functions f: Complex --> Complex *)
+
+ starfunC :: (complex => complex) => hcomplex => hcomplex ("*fc* _" [80] 80)
+ "*fc* f == (%x. Abs_hcomplex(UN X: Rep_hcomplex(x). hcomplexrel``{%n. f (X n)}))"
+
+ starfunC_n :: (nat => (complex => complex)) => hcomplex => hcomplex ("*fcn* _" [80] 80)
+ "*fcn* F == (%x. Abs_hcomplex(UN X: Rep_hcomplex(x). hcomplexrel``{%n. (F n)(X n)}))"
+
+ InternalCFuns :: (hcomplex => hcomplex) set
+ "InternalCFuns == {X. EX F. X = *fcn* F}"
+
+
+ (* star transform of functions f: Real --> Complex *)
+
+ starfunRC :: (real => complex) => hypreal => hcomplex ("*fRc* _" [80] 80)
+ "*fRc* f == (%x. Abs_hcomplex(UN X: Rep_hypreal(x). hcomplexrel``{%n. f (X n)}))"
+
+ starfunRC_n :: (nat => (real => complex)) => hypreal => hcomplex ("*fRcn* _" [80] 80)
+ "*fRcn* F == (%x. Abs_hcomplex(UN X: Rep_hypreal(x). hcomplexrel``{%n. (F n)(X n)}))"
+
+ InternalRCFuns :: (hypreal => hcomplex) set
+ "InternalRCFuns == {X. EX F. X = *fRcn* F}"
+
+ (* star transform of functions f: Complex --> Real; needed for Re and Im parts *)
+
+ starfunCR :: (complex => real) => hcomplex => hypreal ("*fcR* _" [80] 80)
+ "*fcR* f == (%x. Abs_hypreal(UN X: Rep_hcomplex(x). hyprel``{%n. f (X n)}))"
+
+ starfunCR_n :: (nat => (complex => real)) => hcomplex => hypreal ("*fcRn* _" [80] 80)
+ "*fcRn* F == (%x. Abs_hypreal(UN X: Rep_hcomplex(x). hyprel``{%n. (F n)(X n)}))"
+
+ InternalCRFuns :: (hcomplex => hypreal) set
+ "InternalCRFuns == {X. EX F. X = *fcRn* F}"
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/Complex.ML Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,1588 @@
+(* Title: Complex.ML
+ Author: Jacques D. Fleuriot
+ Copyright: 2001 University of Edinburgh
+ Description: Complex numbers
+*)
+
+Goal "inj Rep_complex";
+by (rtac inj_inverseI 1);
+by (rtac Rep_complex_inverse 1);
+qed "inj_Rep_complex";
+
+Goal "inj Abs_complex";
+by (rtac inj_inverseI 1);
+by (rtac Abs_complex_inverse 1);
+by (simp_tac (simpset() addsimps [complex_def]) 1);
+qed "inj_Abs_complex";
+Addsimps [inj_Abs_complex RS injD];
+
+Goal "(Abs_complex x = Abs_complex y) = (x = y)";
+by (auto_tac (claset() addDs [inj_Abs_complex RS injD],simpset()));
+qed "Abs_complex_cancel_iff";
+Addsimps [Abs_complex_cancel_iff];
+
+Goalw [complex_def] "(x,y) : complex";
+by (Auto_tac);
+qed "pair_mem_complex";
+Addsimps [pair_mem_complex];
+
+Goal "Rep_complex (Abs_complex (x,y)) = (x,y)";
+by (simp_tac (simpset() addsimps [Abs_complex_inverse]) 1);
+qed "Abs_complex_inverse2";
+Addsimps [Abs_complex_inverse2];
+
+val [prem] = goal Complex.thy
+ "(!!x y. z = Abs_complex(x,y) ==> P) ==> P";
+by (res_inst_tac [("p","Rep_complex z")] PairE 1);
+by (dres_inst_tac [("f","Abs_complex")] arg_cong 1);
+by (res_inst_tac [("x","x"),("y","y")] prem 1);
+by (asm_full_simp_tac (simpset() addsimps [Rep_complex_inverse]) 1);
+qed "eq_Abs_complex";
+
+Goalw [Re_def] "Re(Abs_complex(x,y)) = x";
+by (Simp_tac 1);
+qed "Re";
+Addsimps [Re];
+
+Goalw [Im_def] "Im(Abs_complex(x,y)) = y";
+by (Simp_tac 1);
+qed "Im";
+Addsimps [Im];
+
+Goal "Abs_complex(Re(z),Im(z)) = z";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (Asm_simp_tac 1);
+qed "Abs_complex_cancel";
+Addsimps [Abs_complex_cancel];
+
+Goal "(w=z) = (Re(w) = Re(z) & Im(w) = Im(z))";
+by (res_inst_tac [("z","w")] eq_Abs_complex 1);
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset() addDs [inj_Abs_complex RS injD],simpset()));
+qed "complex_Re_Im_cancel_iff";
+
+Goalw [complex_zero_def] "Re 0 = 0";
+by (Simp_tac 1);
+qed "complex_Re_zero";
+
+Goalw [complex_zero_def] "Im 0 = 0";
+by (Simp_tac 1);
+qed "complex_Im_zero";
+Addsimps [complex_Re_zero,complex_Im_zero];
+
+Goalw [complex_one_def] "Re 1 = 1";
+by (Simp_tac 1);
+qed "complex_Re_one";
+Addsimps [complex_Re_one];
+
+Goalw [complex_one_def] "Im 1 = 0";
+by (Simp_tac 1);
+qed "complex_Im_one";
+Addsimps [complex_Im_one];
+
+Goalw [i_def] "Re(ii) = 0";
+by Auto_tac;
+qed "complex_Re_i";
+Addsimps [complex_Re_i];
+
+Goalw [i_def] "Im(ii) = 1";
+by Auto_tac;
+qed "complex_Im_i";
+Addsimps [complex_Im_i];
+
+Goalw [complex_of_real_def] "Re(complex_of_real 0) = 0";
+by (Simp_tac 1);
+qed "Re_complex_of_real_zero";
+Addsimps [Re_complex_of_real_zero];
+
+Goalw [complex_of_real_def] "Im(complex_of_real 0) = 0";
+by (Simp_tac 1);
+qed "Im_complex_of_real_zero";
+Addsimps [Im_complex_of_real_zero];
+
+Goalw [complex_of_real_def] "Re(complex_of_real 1) = 1";
+by (Simp_tac 1);
+qed "Re_complex_of_real_one";
+Addsimps [Re_complex_of_real_one];
+
+Goalw [complex_of_real_def] "Im(complex_of_real 1) = 0";
+by (Simp_tac 1);
+qed "Im_complex_of_real_one";
+Addsimps [Im_complex_of_real_one];
+
+Goalw [complex_of_real_def] "Re(complex_of_real z) = z";
+by Auto_tac;
+qed "Re_complex_of_real";
+Addsimps [Re_complex_of_real];
+
+Goalw [complex_of_real_def] "Im(complex_of_real z) = 0";
+by Auto_tac;
+qed "Im_complex_of_real";
+Addsimps [Im_complex_of_real];
+
+(*** negation ***)
+
+Goalw [complex_minus_def] "- Abs_complex(x,y) = Abs_complex(-x,-y)";
+by (Simp_tac 1);
+qed "complex_minus";
+
+Goalw [Re_def] "Re (-z) = - Re z";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_minus]));
+qed "complex_Re_minus";
+
+Goalw [Im_def] "Im (-z) = - Im z";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_minus]));
+qed "complex_Im_minus";
+
+Goalw [complex_minus_def] "- (- z) = (z::complex)";
+by (Simp_tac 1);
+qed "complex_minus_minus";
+Addsimps [complex_minus_minus];
+
+Goal "inj(%r::complex. -r)";
+by (rtac injI 1);
+by (dres_inst_tac [("f","uminus")] arg_cong 1);
+by (Asm_full_simp_tac 1);
+qed "inj_complex_minus";
+
+Goalw [complex_zero_def] "-(0::complex) = 0";
+by (simp_tac (simpset() addsimps [complex_minus]) 1);
+qed "complex_minus_zero";
+Addsimps [complex_minus_zero];
+
+Goal "(-x = 0) = (x = (0::complex))";
+by (res_inst_tac [("z","x")] eq_Abs_complex 1);
+by (auto_tac (claset() addDs [inj_Abs_complex RS injD],simpset()
+ addsimps [complex_zero_def,complex_minus]));
+qed "complex_minus_zero_iff";
+Addsimps [complex_minus_zero_iff];
+
+Goal "(0 = -x) = (x = (0::real))";
+by (auto_tac (claset() addDs [sym],simpset()));
+qed "complex_minus_zero_iff2";
+Addsimps [complex_minus_zero_iff2];
+
+Goal "(-x ~= 0) = (x ~= (0::complex))";
+by Auto_tac;
+qed "complex_minus_not_zero_iff";
+
+(*** addition ***)
+
+Goalw [complex_add_def]
+ "Abs_complex(x1,y1) + Abs_complex(x2,y2) = Abs_complex(x1+x2,y1+y2)";
+by (Simp_tac 1);
+qed "complex_add";
+
+Goalw [Re_def] "Re(x + y) = Re(x) + Re(y)";
+by (res_inst_tac [("z","x")] eq_Abs_complex 1);
+by (res_inst_tac [("z","y")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_add]));
+qed "complex_Re_add";
+
+Goalw [Im_def] "Im(x + y) = Im(x) + Im(y)";
+by (res_inst_tac [("z","x")] eq_Abs_complex 1);
+by (res_inst_tac [("z","y")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_add]));
+qed "complex_Im_add";
+
+Goalw [complex_add_def] "(u::complex) + v = v + u";
+by (simp_tac (simpset() addsimps [real_add_commute]) 1);
+qed "complex_add_commute";
+
+Goalw [complex_add_def] "((u::complex) + v) + w = u + (v + w)";
+by (simp_tac (simpset() addsimps [real_add_assoc]) 1);
+qed "complex_add_assoc";
+
+Goalw [complex_add_def] "(x::complex) + (y + z) = y + (x + z)";
+by (simp_tac (simpset() addsimps [real_add_left_commute]) 1);
+qed "complex_add_left_commute";
+
+val complex_add_ac = [complex_add_assoc,complex_add_commute,
+ complex_add_left_commute];
+
+Goalw [complex_add_def,complex_zero_def] "(0::complex) + z = z";
+by (Simp_tac 1);
+qed "complex_add_zero_left";
+Addsimps [complex_add_zero_left];
+
+Goalw [complex_add_def,complex_zero_def] "z + (0::complex) = z";
+by (Simp_tac 1);
+qed "complex_add_zero_right";
+Addsimps [complex_add_zero_right];
+
+Goalw [complex_add_def,complex_minus_def,complex_zero_def]
+ "z + -z = (0::complex)";
+by (Simp_tac 1);
+qed "complex_add_minus_right_zero";
+Addsimps [complex_add_minus_right_zero];
+
+Goalw [complex_add_def,complex_minus_def,complex_zero_def]
+ "-z + z = (0::complex)";
+by (Simp_tac 1);
+qed "complex_add_minus_left_zero";
+Addsimps [complex_add_minus_left_zero];
+
+Goal "z + (- z + w) = (w::complex)";
+by (simp_tac (simpset() addsimps [complex_add_assoc RS sym]) 1);
+qed "complex_add_minus_cancel";
+
+Goal "(-z) + (z + w) = (w::complex)";
+by (simp_tac (simpset() addsimps [complex_add_assoc RS sym]) 1);
+qed "complex_minus_add_cancel";
+
+Addsimps [complex_add_minus_cancel, complex_minus_add_cancel];
+
+Goal "x + y = (0::complex) ==> x = -y";
+by (auto_tac (claset(),simpset() addsimps [complex_Re_Im_cancel_iff,
+ complex_Re_add,complex_Im_add,complex_Re_minus,complex_Im_minus]));
+qed "complex_add_minus_eq_minus";
+
+Goal "-(x + y) = -x + -(y::complex)";
+by (res_inst_tac [("z","x")] eq_Abs_complex 1);
+by (res_inst_tac [("z","y")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_minus,complex_add]));
+qed "complex_minus_add_distrib";
+Addsimps [complex_minus_add_distrib];
+
+Goal "((x::complex) + y = x + z) = (y = z)";
+by (Step_tac 1);
+by (dres_inst_tac [("f","%t.-x + t")] arg_cong 1);
+by (asm_full_simp_tac (simpset() addsimps [complex_add_assoc RS sym]) 1);
+qed "complex_add_left_cancel";
+AddIffs [complex_add_left_cancel];
+
+Goal "(y + (x::complex)= z + x) = (y = z)";
+by (simp_tac (simpset() addsimps [complex_add_commute]) 1);
+qed "complex_add_right_cancel";
+Addsimps [complex_add_right_cancel];
+
+Goal "((x::complex) = y) = (0 = x + - y)";
+by (Step_tac 1);
+by (res_inst_tac [("x1","-y")]
+ (complex_add_right_cancel RS iffD1) 2);
+by (Auto_tac);
+qed "complex_eq_minus_iff";
+
+Goal "((x::complex) = y) = (x + - y = 0)";
+by (Step_tac 1);
+by (res_inst_tac [("x1","-y")]
+ (complex_add_right_cancel RS iffD1) 2);
+by (Auto_tac);
+qed "complex_eq_minus_iff2";
+
+Goal "(0::complex) - x = -x";
+by (simp_tac (simpset() addsimps [complex_diff_def]) 1);
+qed "complex_diff_0";
+
+Goal "x - (0::complex) = x";
+by (simp_tac (simpset() addsimps [complex_diff_def]) 1);
+qed "complex_diff_0_right";
+
+Goal "x - x = (0::complex)";
+by (simp_tac (simpset() addsimps [complex_diff_def]) 1);
+qed "complex_diff_self";
+
+Addsimps [complex_diff_0, complex_diff_0_right, complex_diff_self];
+
+Goalw [complex_diff_def]
+ "Abs_complex(x1,y1) - Abs_complex(x2,y2) = Abs_complex(x1-x2,y1-y2)";
+by (simp_tac (simpset() addsimps [complex_add,complex_minus]) 1);
+qed "complex_diff";
+
+Goal "((x::complex) - y = z) = (x = z + y)";
+by (auto_tac (claset(),simpset() addsimps [complex_diff_def,complex_add_assoc]));
+qed "complex_diff_eq_eq";
+
+(*** complex multiplication ***)
+
+Goalw [complex_mult_def]
+ "Abs_complex(x1,y1) * Abs_complex(x2,y2) = \
+\ Abs_complex(x1*x2 - y1*y2,x1*y2 + y1*x2)";
+by (Simp_tac 1);
+qed "complex_mult";
+
+Goalw [complex_mult_def] "(w::complex) * z = z * w";
+by (simp_tac (simpset() addsimps [real_mult_commute,real_add_commute]) 1);
+qed "complex_mult_commute";
+
+Goalw [complex_mult_def] "((u::complex) * v) * w = u * (v * w)";
+by (simp_tac (simpset() addsimps [complex_Re_Im_cancel_iff,
+ real_mult_assoc,real_diff_mult_distrib2,
+ real_add_mult_distrib2,real_diff_mult_distrib,
+ real_add_mult_distrib,real_mult_left_commute]) 1);
+qed "complex_mult_assoc";
+
+Goalw [complex_mult_def] "(x::complex) * (y * z) = y * (x * z)";
+by (simp_tac (simpset() addsimps [complex_Re_Im_cancel_iff,
+ real_mult_left_commute,real_diff_mult_distrib2,
+ real_add_mult_distrib2]) 1);
+qed "complex_mult_left_commute";
+
+val complex_mult_ac = [complex_mult_assoc,complex_mult_commute,
+ complex_mult_left_commute];
+
+Goalw [complex_one_def] "(1::complex) * z = z";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (asm_simp_tac (simpset() addsimps [complex_mult]) 1);
+qed "complex_mult_one_left";
+Addsimps [complex_mult_one_left];
+
+Goal "z * (1::complex) = z";
+by (simp_tac (simpset() addsimps [complex_mult_commute]) 1);
+qed "complex_mult_one_right";
+Addsimps [complex_mult_one_right];
+
+Goalw [complex_zero_def] "(0::complex) * z = 0";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (asm_simp_tac (simpset() addsimps [complex_mult]) 1);
+qed "complex_mult_zero_left";
+Addsimps [complex_mult_zero_left];
+
+Goal "z * 0 = (0::complex)";
+by (simp_tac (simpset() addsimps [complex_mult_commute]) 1);
+qed "complex_mult_zero_right";
+Addsimps [complex_mult_zero_right];
+
+Goalw [complex_divide_def] "0 / z = (0::complex)";
+by Auto_tac;
+qed "complex_divide_zero";
+Addsimps [complex_divide_zero];
+
+Goal "-(x * y) = -x * (y::complex)";
+by (res_inst_tac [("z","x")] eq_Abs_complex 1);
+by (res_inst_tac [("z","y")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_minus,
+ real_diff_def]));
+qed "complex_minus_mult_eq1";
+
+Goal "-(x * y) = x * -(y::complex)";
+by (res_inst_tac [("z","x")] eq_Abs_complex 1);
+by (res_inst_tac [("z","y")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_minus,
+ real_diff_def]));
+qed "complex_minus_mult_eq2";
+
+Addsimps [ complex_minus_mult_eq1 RS sym, complex_minus_mult_eq2 RS sym];
+
+Goal "-(1::complex) * z = -z";
+by (Simp_tac 1);
+qed "complex_mult_minus_one";
+Addsimps [complex_mult_minus_one];
+
+Goal "z * -(1::complex) = -z";
+by (stac complex_mult_commute 1);
+by (Simp_tac 1);
+qed "complex_mult_minus_one_right";
+Addsimps [complex_mult_minus_one_right];
+
+Goal "-x * -y = x * (y::complex)";
+by (Simp_tac 1);
+qed "complex_minus_mult_cancel";
+Addsimps [complex_minus_mult_cancel];
+
+Goal "-x * y = x * -(y::complex)";
+by (Simp_tac 1);
+qed "complex_minus_mult_commute";
+
+qed_goal "complex_add_assoc_cong" thy
+ "!!z. (z::complex) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"
+ (fn _ => [(asm_simp_tac (simpset() addsimps [complex_add_assoc RS sym]) 1)]);
+
+qed_goal "complex_add_assoc_swap" thy "(z::complex) + (v + w) = v + (z + w)"
+ (fn _ => [(REPEAT (ares_tac [complex_add_commute RS complex_add_assoc_cong] 1))]);
+
+Goal "((z1::complex) + z2) * w = (z1 * w) + (z2 * w)";
+by (res_inst_tac [("z","z1")] eq_Abs_complex 1);
+by (res_inst_tac [("z","z2")] eq_Abs_complex 1);
+by (res_inst_tac [("z","w")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_add,
+ real_add_mult_distrib,real_diff_def] @ real_add_ac));
+qed "complex_add_mult_distrib";
+
+Goal "(w::complex) * (z1 + z2) = (w * z1) + (w * z2)";
+by (res_inst_tac [("z1","z1 + z2")] (complex_mult_commute RS ssubst) 1);
+by (simp_tac (simpset() addsimps [complex_add_mult_distrib]) 1);
+by (simp_tac (simpset() addsimps [complex_mult_commute]) 1);
+qed "complex_add_mult_distrib2";
+
+Goalw [complex_zero_def,complex_one_def] "(0::complex) ~= 1";
+by (simp_tac (simpset() addsimps [complex_Re_Im_cancel_iff]) 1);
+qed "complex_zero_not_eq_one";
+Addsimps [complex_zero_not_eq_one];
+Addsimps [complex_zero_not_eq_one RS not_sym];
+
+(*** inverse ***)
+Goalw [complex_inverse_def] "inverse (Abs_complex(x,y)) = \
+\ Abs_complex(x/(x ^ 2 + y ^ 2),-y/(x ^ 2 + y ^ 2))";
+by (Simp_tac 1);
+qed "complex_inverse";
+
+Goalw [complex_inverse_def,complex_zero_def] "inverse 0 = (0::complex)";
+by Auto_tac;
+qed "COMPLEX_INVERSE_ZERO";
+
+Goal "a / (0::complex) = 0";
+by (simp_tac (simpset() addsimps [complex_divide_def, COMPLEX_INVERSE_ZERO]) 1);
+qed "COMPLEX_DIVISION_BY_ZERO"; (*NOT for adding to default simpset*)
+
+fun complex_div_undefined_case_tac s i =
+ case_tac s i THEN
+ asm_simp_tac (simpset() addsimps [COMPLEX_DIVISION_BY_ZERO, COMPLEX_INVERSE_ZERO]) i;
+
+(*REMOVE?:
+lemmas:replace previous versions to accommodate new behaviour of simplification
+Goal "x ^ 2 + y ^ 2 = 0 ==> x = (0::real)";
+by (auto_tac (claset() addIs [real_sum_squares_cancel],
+ simpset() addsimps [CLAIM "2 = Suc(Suc 0)"]));
+qed "real_sum_squares_cancel";
+
+Goal "x ^ 2 + y ^ 2 = 0 ==> y = (0::real)";
+by (auto_tac (claset() addIs [real_sum_squares_cancel2],
+ simpset() addsimps [CLAIM "2 = Suc(Suc 0)"]));
+qed "real_sum_squares_cancel2";
+*)
+
+Goal "z ~= (0::complex) ==> inverse(z) * z = 1";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_inverse,
+ complex_one_def,complex_zero_def,real_add_divide_distrib RS sym,
+ realpow_two_eq_mult] @ real_mult_ac));
+by (dres_inst_tac [("y","y")] real_sum_squares_not_zero 1);
+by (dres_inst_tac [("x","x")] real_sum_squares_not_zero2 2);
+by Auto_tac;
+qed "complex_mult_inv_left";
+Addsimps [complex_mult_inv_left];
+
+Goal "z ~= (0::complex) ==> z * inverse(z) = 1";
+by (auto_tac (claset() addIs [complex_mult_commute RS subst],simpset()));
+qed "complex_mult_inv_right";
+Addsimps [complex_mult_inv_right];
+
+Goal "(c::complex) ~= 0 ==> (c*a=c*b) = (a=b)";
+by Auto_tac;
+by (dres_inst_tac [("f","%x. x*inverse c")] arg_cong 1);
+by (asm_full_simp_tac (simpset() addsimps complex_mult_ac) 1);
+qed "complex_mult_left_cancel";
+
+Goal "(c::complex) ~= 0 ==> (a*c=b*c) = (a=b)";
+by (Step_tac 1);
+by (dres_inst_tac [("f","%x. x*inverse c")] arg_cong 1);
+by (asm_full_simp_tac (simpset() addsimps complex_mult_ac) 1);
+qed "complex_mult_right_cancel";
+
+Goal "z ~= 0 ==> inverse(z::complex) ~= 0";
+by (Step_tac 1);
+by (ftac (complex_mult_right_cancel RS iffD2) 1);
+by (thin_tac "inverse z = 0" 2);
+by (assume_tac 1 THEN Auto_tac);
+qed "complex_inverse_not_zero";
+Addsimps [complex_inverse_not_zero];
+
+Goal "!!x. [| x ~= 0; y ~= (0::complex) |] ==> x * y ~= 0";
+by (Step_tac 1);
+by (dres_inst_tac [("f","%z. inverse x*z")] arg_cong 1);
+by (asm_full_simp_tac (simpset() addsimps [complex_mult_assoc RS sym]) 1);
+qed "complex_mult_not_zero";
+
+bind_thm ("complex_mult_not_zeroE",complex_mult_not_zero RS notE);
+
+Goal "inverse(inverse (x::complex)) = x";
+by (complex_div_undefined_case_tac "x = 0" 1);
+by (res_inst_tac [("c1","inverse x")] (complex_mult_right_cancel RS iffD1) 1);
+by (etac complex_inverse_not_zero 1);
+by (auto_tac (claset() addDs [complex_inverse_not_zero],simpset()));
+qed "complex_inverse_inverse";
+Addsimps [complex_inverse_inverse];
+
+Goalw [complex_one_def] "inverse(1::complex) = 1";
+by (simp_tac (simpset() addsimps [complex_inverse,realpow_num_two]) 1);
+qed "complex_inverse_one";
+Addsimps [complex_inverse_one];
+
+Goal "inverse(-x) = -inverse(x::complex)";
+by (complex_div_undefined_case_tac "x = 0" 1);
+by (res_inst_tac [("c1","-x")] (complex_mult_right_cancel RS iffD1) 1);
+by (stac complex_mult_inv_left 2);
+by Auto_tac;
+qed "complex_minus_inverse";
+
+Goal "inverse(x*y) = inverse x * inverse (y::complex)";
+by (complex_div_undefined_case_tac "x = 0" 1);
+by (complex_div_undefined_case_tac "y = 0" 1);
+by (res_inst_tac [("c1","x*y")] (complex_mult_left_cancel RS iffD1) 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mult_not_zero]
+ @ complex_mult_ac));
+by (auto_tac (claset(),simpset() addsimps [complex_mult_not_zero,
+ complex_mult_assoc RS sym]));
+qed "complex_inverse_distrib";
+
+
+(*** division ***)
+
+(*adding some of these theorems to simpset as for reals:
+ not 100% convinced for some*)
+
+Goal "(x::complex) * (y/z) = (x*y)/z";
+by (simp_tac (simpset() addsimps [complex_divide_def, complex_mult_assoc]) 1);
+qed "complex_times_divide1_eq";
+
+Goal "(y/z) * (x::complex) = (y*x)/z";
+by (simp_tac (simpset() addsimps [complex_divide_def]@complex_mult_ac) 1);
+qed "complex_times_divide2_eq";
+
+Addsimps [complex_times_divide1_eq, complex_times_divide2_eq];
+
+Goal "(x::complex) / (y/z) = (x*z)/y";
+by (simp_tac (simpset() addsimps [complex_divide_def, complex_inverse_distrib]@
+ complex_mult_ac) 1);
+qed "complex_divide_divide1_eq";
+
+Goal "((x::complex) / y) / z = x/(y*z)";
+by (simp_tac (simpset() addsimps [complex_divide_def, complex_inverse_distrib,
+ complex_mult_assoc]) 1);
+qed "complex_divide_divide2_eq";
+
+Addsimps [complex_divide_divide1_eq, complex_divide_divide2_eq];
+
+(** As with multiplication, pull minus signs OUT of the / operator **)
+
+Goal "(-x) / (y::complex) = - (x/y)";
+by (simp_tac (simpset() addsimps [complex_divide_def]) 1);
+qed "complex_minus_divide_eq";
+Addsimps [complex_minus_divide_eq];
+
+Goal "(x / -(y::complex)) = - (x/y)";
+by (simp_tac (simpset() addsimps [complex_divide_def, complex_minus_inverse]) 1);
+qed "complex_divide_minus_eq";
+Addsimps [complex_divide_minus_eq];
+
+Goal "(x+y)/(z::complex) = x/z + y/z";
+by (simp_tac (simpset() addsimps [complex_divide_def, complex_add_mult_distrib]) 1);
+qed "complex_add_divide_distrib";
+
+(*---------------------------------------------------------------------------*)
+(* Embedding properties for complex_of_real map *)
+(*---------------------------------------------------------------------------*)
+
+Goal "inj complex_of_real";
+by (rtac injI 1);
+by (auto_tac (claset() addDs [inj_Abs_complex RS injD],
+ simpset() addsimps [complex_of_real_def]));
+qed "inj_complex_of_real";
+
+Goalw [complex_one_def,complex_of_real_def]
+ "complex_of_real 1 = 1";
+by (rtac refl 1);
+qed "complex_of_real_one";
+Addsimps [complex_of_real_one];
+
+Goalw [complex_zero_def,complex_of_real_def]
+ "complex_of_real 0 = 0";
+by (rtac refl 1);
+qed "complex_of_real_zero";
+Addsimps [complex_of_real_zero];
+
+Goal "(complex_of_real x = complex_of_real y) = (x = y)";
+by (auto_tac (claset() addDs [inj_complex_of_real RS injD],simpset()));
+qed "complex_of_real_eq_iff";
+AddIffs [complex_of_real_eq_iff];
+
+Goal "complex_of_real(-x) = - complex_of_real x";
+by (simp_tac (simpset() addsimps [complex_of_real_def,complex_minus]) 1);
+qed "complex_of_real_minus";
+
+Goal "complex_of_real(inverse x) = inverse(complex_of_real x)";
+by (real_div_undefined_case_tac "x=0" 1);
+by (simp_tac (simpset() addsimps [DIVISION_BY_ZERO,COMPLEX_INVERSE_ZERO]) 1);
+by (auto_tac (claset(),simpset() addsimps [complex_inverse,
+ complex_of_real_def,realpow_num_two,real_divide_def,
+ real_inverse_distrib]));
+qed "complex_of_real_inverse";
+
+Goal "complex_of_real x + complex_of_real y = complex_of_real (x + y)";
+by (simp_tac (simpset() addsimps [complex_add,complex_of_real_def]) 1);
+qed "complex_of_real_add";
+
+Goal "complex_of_real x - complex_of_real y = complex_of_real (x - y)";
+by (simp_tac (simpset() addsimps [complex_of_real_minus RS sym,
+ complex_diff_def,complex_of_real_add]) 1);
+qed "complex_of_real_diff";
+
+Goal "complex_of_real x * complex_of_real y = complex_of_real (x * y)";
+by (simp_tac (simpset() addsimps [complex_mult,complex_of_real_def]) 1);
+qed "complex_of_real_mult";
+
+Goalw [complex_divide_def]
+ "complex_of_real x / complex_of_real y = complex_of_real(x/y)";
+by (real_div_undefined_case_tac "y=0" 1);
+by (simp_tac (simpset() addsimps [rename_numerals DIVISION_BY_ZERO,
+ COMPLEX_INVERSE_ZERO]) 1);
+by (asm_simp_tac (simpset() addsimps [complex_of_real_mult RS sym,
+ complex_of_real_inverse,real_divide_def]) 1);
+qed "complex_of_real_divide";
+
+Goal "complex_of_real (x ^ n) = (complex_of_real x) ^ n";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps [complex_of_real_mult RS sym]));
+qed "complex_of_real_pow";
+
+Goalw [cmod_def] "cmod (Abs_complex(x,y)) = sqrt(x ^ 2 + y ^ 2)";
+by (Simp_tac 1);
+qed "complex_mod";
+
+Goalw [cmod_def] "cmod(0) = 0";
+by (Simp_tac 1);
+qed "complex_mod_zero";
+Addsimps [complex_mod_zero];
+
+Goalw [cmod_def] "cmod(1) = 1";
+by (simp_tac (simpset() addsimps [realpow_num_two]) 1);
+qed "complex_mod_one";
+Addsimps [complex_mod_one];
+
+Goalw [complex_of_real_def] "cmod(complex_of_real x) = abs x";
+by (simp_tac (simpset() addsimps [complex_mod,realpow_num_two]) 1);
+qed "complex_mod_complex_of_real";
+Addsimps [complex_mod_complex_of_real];
+
+Goal "complex_of_real (abs x) = complex_of_real(cmod(complex_of_real x))";
+by (Simp_tac 1);
+qed "complex_of_real_abs";
+
+(*---------------------------------------------------------------------------*)
+(* conjugation is an automorphism *)
+(*---------------------------------------------------------------------------*)
+
+Goalw [cnj_def] "cnj (Abs_complex(x,y)) = Abs_complex(x,-y)";
+by (Simp_tac 1);
+qed "complex_cnj";
+
+Goal "inj cnj";
+by (rtac injI 1);
+by (auto_tac (claset(),simpset() addsimps [cnj_def,
+ Abs_complex_cancel_iff,complex_Re_Im_cancel_iff]));
+qed "inj_cnj";
+
+Goal "(cnj x = cnj y) = (x = y)";
+by (auto_tac (claset() addDs [inj_cnj RS injD],simpset()));
+qed "complex_cnj_cancel_iff";
+Addsimps [complex_cnj_cancel_iff];
+
+Goalw [cnj_def] "cnj (cnj z) = z";
+by (Simp_tac 1);
+qed "complex_cnj_cnj";
+Addsimps [complex_cnj_cnj];
+
+Goalw [complex_of_real_def] "cnj (complex_of_real x) = complex_of_real x";
+by (simp_tac (simpset() addsimps [complex_cnj]) 1);
+qed "complex_cnj_complex_of_real";
+Addsimps [complex_cnj_complex_of_real];
+
+Goal "cmod (cnj z) = cmod z";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (asm_simp_tac (simpset() addsimps [complex_cnj,complex_mod,realpow_num_two]) 1);
+qed "complex_mod_cnj";
+Addsimps [complex_mod_cnj];
+
+Goalw [cnj_def] "cnj (-z) = - cnj z";
+by (simp_tac (simpset() addsimps [complex_minus,
+ complex_Re_minus,complex_Im_minus]) 1);
+qed "complex_cnj_minus";
+
+Goal "cnj(inverse z) = inverse(cnj z)";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (asm_simp_tac (simpset() addsimps [complex_cnj,complex_inverse,
+ realpow_num_two]) 1);
+qed "complex_cnj_inverse";
+
+Goal "cnj(w + z) = cnj(w) + cnj(z)";
+by (res_inst_tac [("z","w")] eq_Abs_complex 1);
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (asm_simp_tac (simpset() addsimps [complex_cnj,complex_add]) 1);
+qed "complex_cnj_add";
+
+Goalw [complex_diff_def] "cnj(w - z) = cnj(w) - cnj(z)";
+by (simp_tac (simpset() addsimps [complex_cnj_add,complex_cnj_minus]) 1);
+qed "complex_cnj_diff";
+
+Goal "cnj(w * z) = cnj(w) * cnj(z)";
+by (res_inst_tac [("z","w")] eq_Abs_complex 1);
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (asm_simp_tac (simpset() addsimps [complex_cnj,complex_mult]) 1);
+qed "complex_cnj_mult";
+
+Goalw [complex_divide_def] "cnj(w / z) = (cnj w)/(cnj z)";
+by (simp_tac (simpset() addsimps [complex_cnj_mult,complex_cnj_inverse]) 1);
+qed "complex_cnj_divide";
+
+Goalw [cnj_def,complex_one_def] "cnj 1 = 1";
+by (Simp_tac 1);
+qed "complex_cnj_one";
+Addsimps [complex_cnj_one];
+
+Goal "cnj(z ^ n) = cnj(z) ^ n";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps [complex_cnj_mult]));
+qed "complex_cnj_pow";
+
+Goal "z + cnj z = complex_of_real (2 * Re(z))";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (asm_simp_tac (simpset() addsimps [complex_add,complex_cnj,
+ complex_of_real_def]) 1);
+qed "complex_add_cnj";
+
+Goal "z - cnj z = complex_of_real (2 * Im(z)) * ii";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (asm_simp_tac (simpset() addsimps [complex_add,complex_cnj,
+ complex_of_real_def,complex_diff_def,complex_minus,
+ i_def,complex_mult]) 1);
+qed "complex_diff_cnj";
+
+goalw Complex.thy [cnj_def,complex_zero_def]
+ "cnj 0 = 0";
+by Auto_tac;
+qed "complex_cnj_zero";
+Addsimps [complex_cnj_zero];
+
+goal Complex.thy "(cnj z = 0) = (z = 0)";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_zero_def,
+ complex_cnj]));
+qed "complex_cnj_zero_iff";
+AddIffs [complex_cnj_zero_iff];
+
+Goal "z * cnj z = complex_of_real (Re(z) ^ 2 + Im(z) ^ 2)";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_cnj,complex_mult,
+ complex_of_real_def,realpow_num_two]));
+qed "complex_mult_cnj";
+
+(*---------------------------------------------------------------------------*)
+(* algebra *)
+(*---------------------------------------------------------------------------*)
+
+Goal "(x*y = (0::complex)) = (x = 0 | y = 0)";
+by Auto_tac;
+by (auto_tac (claset() addIs [ccontr] addDs
+ [complex_mult_not_zero],simpset()));
+qed "complex_mult_zero_iff";
+AddIffs [complex_mult_zero_iff];
+
+Goalw [complex_zero_def] "(x + y = x) = (y = (0::complex))";
+by (res_inst_tac [("z","x")] eq_Abs_complex 1);
+by (res_inst_tac [("z","y")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_add]));
+qed "complex_add_left_cancel_zero";
+Addsimps [complex_add_left_cancel_zero];
+
+Goalw [complex_diff_def]
+ "((z1::complex) - z2) * w = (z1 * w) - (z2 * w)";
+by (simp_tac (simpset() addsimps [complex_add_mult_distrib]) 1);
+qed "complex_diff_mult_distrib";
+
+Goalw [complex_diff_def]
+ "(w::complex) * (z1 - z2) = (w * z1) - (w * z2)";
+by (simp_tac (simpset() addsimps [complex_add_mult_distrib2]) 1);
+qed "complex_diff_mult_distrib2";
+
+(*---------------------------------------------------------------------------*)
+(* modulus *)
+(*---------------------------------------------------------------------------*)
+
+(*
+Goal "[| sqrt(x) = 0; 0 <= x |] ==> x = 0";
+by (auto_tac (claset() addIs [real_sqrt_eq_zero_cancel],
+ simpset()));
+qed "real_sqrt_eq_zero_cancel2";
+*)
+
+Goal "(cmod x = 0) = (x = 0)";
+by (res_inst_tac [("z","x")] eq_Abs_complex 1);
+by (auto_tac (claset() addIs
+ [real_sum_squares_cancel,real_sum_squares_cancel2],
+ simpset() addsimps [complex_mod,complex_zero_def,
+ realpow_num_two]));
+qed "complex_mod_eq_zero_cancel";
+Addsimps [complex_mod_eq_zero_cancel];
+
+Goal "cmod (complex_of_real(real (n::nat))) = real n";
+by (Simp_tac 1);
+qed "complex_mod_complex_of_real_of_nat";
+Addsimps [complex_mod_complex_of_real_of_nat];
+
+Goal "cmod (-x) = cmod(x)";
+by (res_inst_tac [("z","x")] eq_Abs_complex 1);
+by (asm_simp_tac (simpset() addsimps [complex_mod,complex_minus,
+ realpow_num_two]) 1);
+qed "complex_mod_minus";
+Addsimps [complex_mod_minus];
+
+Goal "cmod(z * cnj(z)) = cmod(z) ^ 2";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (asm_simp_tac (simpset() addsimps [complex_mod,complex_cnj,
+ complex_mult, CLAIM "0 ^ 2 = (0::real)"]) 1);
+by (simp_tac (simpset() addsimps [realpow_two_eq_mult]) 1);
+qed "complex_mod_mult_cnj";
+
+Goalw [cmod_def] "cmod(Abs_complex(x,y)) ^ 2 = x ^ 2 + y ^ 2";
+by Auto_tac;
+qed "complex_mod_squared";
+
+Goalw [cmod_def] "0 <= cmod x";
+by (auto_tac (claset() addIs [real_sqrt_ge_zero],simpset()));
+qed "complex_mod_ge_zero";
+Addsimps [complex_mod_ge_zero];
+
+Goal "abs(cmod x) = cmod x";
+by (auto_tac (claset() addIs [abs_eqI1],simpset()));
+qed "abs_cmod_cancel";
+Addsimps [abs_cmod_cancel];
+
+Goal "cmod(x*y) = cmod(x) * cmod(y)";
+by (res_inst_tac [("z","x")] eq_Abs_complex 1);
+by (res_inst_tac [("z","y")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mult,
+ complex_mod,real_sqrt_mult_distrib2 RS sym] delsimps [realpow_Suc]));
+by (res_inst_tac [("n","1")] realpow_Suc_cancel_eq 1);
+by (auto_tac (claset(),simpset() addsimps [realpow_num_two RS sym]
+ delsimps [realpow_Suc]));
+by (auto_tac (claset(),simpset() addsimps [real_diff_def,realpow_num_two,
+ real_add_mult_distrib2,real_add_mult_distrib] @ real_add_ac @
+ real_mult_ac));
+qed "complex_mod_mult";
+
+Goal "cmod(x + y) ^ 2 = cmod(x) ^ 2 + cmod(y) ^ 2 + 2 * Re(x * cnj y)";
+by (res_inst_tac [("z","x")] eq_Abs_complex 1);
+by (res_inst_tac [("z","y")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_add,
+ complex_mod_squared,complex_mult,complex_cnj,real_diff_def]
+ delsimps [realpow_Suc]));
+by (auto_tac (claset(),simpset() addsimps [real_add_mult_distrib2,
+ real_add_mult_distrib,realpow_num_two] @ real_mult_ac @ real_add_ac));
+qed "complex_mod_add_squared_eq";
+
+Goal "Re(x * cnj y) <= cmod(x * cnj y)";
+by (res_inst_tac [("z","x")] eq_Abs_complex 1);
+by (res_inst_tac [("z","y")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mod,
+ complex_mult,complex_cnj,real_diff_def] delsimps [realpow_Suc]));
+qed "complex_Re_mult_cnj_le_cmod";
+Addsimps [complex_Re_mult_cnj_le_cmod];
+
+Goal "Re(x * cnj y) <= cmod(x * y)";
+by (cut_inst_tac [("x","x"),("y","y")] complex_Re_mult_cnj_le_cmod 1);
+by (asm_full_simp_tac (simpset() addsimps [complex_mod_mult]) 1);
+qed "complex_Re_mult_cnj_le_cmod2";
+Addsimps [complex_Re_mult_cnj_le_cmod2];
+
+Goal "((x::real) + y) ^ 2 = x ^ 2 + y ^ 2 + 2 * x * y";
+by (simp_tac (simpset() addsimps [real_add_mult_distrib,
+ real_add_mult_distrib2,realpow_num_two]) 1);
+qed "real_sum_squared_expand";
+
+Goal "cmod (x + y) ^ 2 <= (cmod(x) + cmod(y)) ^ 2";
+by (simp_tac (simpset() addsimps [real_sum_squared_expand,
+ complex_mod_add_squared_eq,real_mult_assoc,complex_mod_mult RS sym]) 1);
+qed "complex_mod_triangle_squared";
+Addsimps [complex_mod_triangle_squared];
+
+Goal "- cmod x <= cmod x";
+by (rtac (complex_mod_ge_zero RSN (2,real_le_trans)) 1);
+by (Simp_tac 1);
+qed "complex_mod_minus_le_complex_mod";
+Addsimps [complex_mod_minus_le_complex_mod];
+
+Goal "cmod (x + y) <= cmod(x) + cmod(y)";
+by (res_inst_tac [("n","1")] realpow_increasing 1);
+by (auto_tac (claset() addIs [(complex_mod_ge_zero RSN (2,real_le_trans))],
+ simpset() addsimps [realpow_num_two RS sym]));
+qed "complex_mod_triangle_ineq";
+Addsimps [complex_mod_triangle_ineq];
+
+Goal "cmod(b + a) - cmod b <= cmod a";
+by (cut_inst_tac [("x1","b"),("y1","a"),("z","-cmod b")]
+ (complex_mod_triangle_ineq RS real_add_le_mono1) 1);
+by (Simp_tac 1);
+qed "complex_mod_triangle_ineq2";
+Addsimps [complex_mod_triangle_ineq2];
+
+Goal "cmod (x - y) = cmod (y - x)";
+by (res_inst_tac [("z","x")] eq_Abs_complex 1);
+by (res_inst_tac [("z","y")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_diff,
+ complex_mod,real_diff_mult_distrib2,realpow_num_two,
+ real_diff_mult_distrib] @ real_add_ac @ real_mult_ac));
+qed "complex_mod_diff_commute";
+
+Goal "[| cmod x < r; cmod y < s |] ==> cmod (x + y) < r + s";
+by (auto_tac (claset() addIs [order_le_less_trans,
+ complex_mod_triangle_ineq],simpset()));
+qed "complex_mod_add_less";
+
+Goal "[| cmod x < r; cmod y < s |] ==> cmod (x * y) < r * s";
+by (auto_tac (claset() addIs [real_mult_less_mono'],simpset()
+ addsimps [complex_mod_mult]));
+qed "complex_mod_mult_less";
+
+goal Complex.thy "cmod(a) - cmod(b) <= cmod(a + b)";
+by (res_inst_tac [("R1.0","cmod(a)"),("R2.0","cmod(b)")]
+ real_linear_less2 1);
+by Auto_tac;
+by (dtac (ARITH_PROVE "a < b ==> a - (b::real) < 0") 1);
+by (rtac real_le_trans 1 THEN rtac order_less_imp_le 1);
+by Auto_tac;
+by (dtac (ARITH_PROVE "a < b ==> 0 < (b::real) - a") 1);
+by (rtac (ARITH_PROVE "a <= b + c ==> a - c <= (b::real)") 1);
+by (rtac (complex_mod_minus RS subst) 1);
+by (rtac real_le_trans 1);
+by (rtac complex_mod_triangle_ineq 2);
+by (auto_tac (claset(),simpset() addsimps complex_add_ac));
+qed "complex_mod_diff_ineq";
+Addsimps [complex_mod_diff_ineq];
+
+Goal "Re z <= cmod z";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mod]
+ delsimps [realpow_Suc]));
+qed "complex_Re_le_cmod";
+Addsimps [complex_Re_le_cmod];
+
+Goal "z ~= 0 ==> 0 < cmod z";
+by (cut_inst_tac [("x","z")] complex_mod_ge_zero 1);
+by (dtac order_le_imp_less_or_eq 1);
+by Auto_tac;
+qed "complex_mod_gt_zero";
+
+
+(*---------------------------------------------------------------------------*)
+(* a few more theorems *)
+(*---------------------------------------------------------------------------*)
+
+Goal "cmod(x ^ n) = cmod(x) ^ n";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mod_mult]));
+qed "complex_mod_complexpow";
+
+Goal "(-x::complex) ^ n = (if even n then (x ^ n) else -(x ^ n))";
+by (induct_tac "n" 1);
+by Auto_tac;
+qed "complexpow_minus";
+
+Goal "inverse (-x) = - inverse (x::complex)";
+by (res_inst_tac [("z","x")] eq_Abs_complex 1);
+by (asm_simp_tac (simpset() addsimps [complex_inverse,complex_minus,
+ realpow_num_two]) 1);
+qed "complex_inverse_minus";
+
+Goalw [complex_divide_def] "x / (1::complex) = x";
+by (Simp_tac 1);
+qed "complex_divide_one";
+Addsimps [complex_divide_one];
+
+Goal "cmod(inverse x) = inverse(cmod x)";
+by (complex_div_undefined_case_tac "x=0" 1);
+by (res_inst_tac [("c1","cmod x")] (real_mult_left_cancel RS iffD1) 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mod_mult RS sym]));
+qed "complex_mod_inverse";
+
+Goalw [complex_divide_def,real_divide_def]
+ "cmod(x/y) = cmod(x)/(cmod y)";
+by (auto_tac (claset(),simpset() addsimps [complex_mod_mult,
+ complex_mod_inverse]));
+qed "complex_mod_divide";
+
+Goalw [complex_divide_def]
+ "inverse(x/y) = y/(x::complex)";
+by (auto_tac (claset(),simpset() addsimps [complex_inverse_distrib,
+ complex_mult_commute]));
+qed "complex_inverse_divide";
+Addsimps [complex_inverse_divide];
+
+Goal "((r::complex) * s) ^ n = (r ^ n) * (s ^ n)";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps complex_mult_ac));
+qed "complexpow_mult";
+
+(*---------------------------------------------------------------------------*)
+(* More exponentiation *)
+(*---------------------------------------------------------------------------*)
+
+Goal "(0::complex) ^ (Suc n) = 0";
+by (Auto_tac);
+qed "complexpow_zero";
+Addsimps [complexpow_zero];
+
+Goal "r ~= (0::complex) --> r ^ n ~= 0";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mult_not_zero]));
+qed_spec_mp "complexpow_not_zero";
+Addsimps [complexpow_not_zero];
+AddIs [complexpow_not_zero];
+
+Goal "r ^ n = (0::complex) ==> r = 0";
+by (blast_tac (claset() addIs [ccontr]
+ addDs [complexpow_not_zero]) 1);
+qed "complexpow_zero_zero";
+
+Goalw [i_def] "ii ^ 2 = -(1::complex)";
+by (auto_tac (claset(),simpset() addsimps
+ [complex_mult,complex_one_def,complex_minus,realpow_num_two]));
+qed "complexpow_i_squared";
+Addsimps [complexpow_i_squared];
+
+Goalw [i_def,complex_zero_def] "ii ~= 0";
+by Auto_tac;
+qed "complex_i_not_zero";
+Addsimps [complex_i_not_zero];
+
+Goal "x * y ~= (0::complex) ==> x ~= 0";
+by Auto_tac;
+qed "complex_mult_eq_zero_cancel1";
+
+Goal "x * y ~= 0 ==> y ~= (0::complex)";
+by Auto_tac;
+qed "complex_mult_eq_zero_cancel2";
+
+Goal "(x * y ~= 0) = (x ~= 0 & y ~= (0::complex))";
+by Auto_tac;
+qed "complex_mult_not_eq_zero_iff";
+AddIffs [complex_mult_not_eq_zero_iff];
+
+Goal "inverse ((r::complex) ^ n) = (inverse r) ^ n";
+by (induct_tac "n" 1);
+by (auto_tac (claset(), simpset() addsimps [complex_inverse_distrib]));
+qed "complexpow_inverse";
+
+(*---------------------------------------------------------------------------*)
+(* sgn *)
+(*---------------------------------------------------------------------------*)
+
+Goalw [sgn_def] "sgn 0 = 0";
+by (Simp_tac 1);
+qed "sgn_zero";
+Addsimps[sgn_zero];
+
+Goalw [sgn_def] "sgn 1 = 1";
+by (Simp_tac 1);
+qed "sgn_one";
+Addsimps [sgn_one];
+
+Goalw [sgn_def] "sgn (-z) = - sgn(z)";
+by Auto_tac;
+qed "sgn_minus";
+
+Goalw [sgn_def]
+ "sgn z = z / complex_of_real (cmod z)";
+by (Simp_tac 1);
+qed "sgn_eq";
+
+Goal "EX x y. z = complex_of_real(x) + ii * complex_of_real(y)";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_of_real_def,
+ i_def,complex_mult,complex_add]));
+qed "complex_split";
+
+Goal "Re(complex_of_real(x) + ii * complex_of_real(y)) = x";
+by (auto_tac (claset(),simpset() addsimps [complex_of_real_def,
+ i_def,complex_mult,complex_add]));
+qed "Re_complex_i";
+Addsimps [Re_complex_i];
+
+Goal "Im(complex_of_real(x) + ii * complex_of_real(y)) = y";
+by (auto_tac (claset(),simpset() addsimps [complex_of_real_def,
+ i_def,complex_mult,complex_add]));
+qed "Im_complex_i";
+Addsimps [Im_complex_i];
+
+Goalw [i_def,complex_of_real_def] "ii * ii = complex_of_real (-1)";
+by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_add]));
+qed "i_mult_eq";
+
+Goalw [i_def,complex_one_def] "ii * ii = -(1::complex)";
+by (simp_tac (simpset() addsimps [complex_mult,complex_minus]) 1);
+qed "i_mult_eq2";
+Addsimps [i_mult_eq2];
+
+Goal "cmod (complex_of_real(x) + ii * complex_of_real(y)) = \
+\ sqrt (x ^ 2 + y ^ 2)";
+by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_add,
+ i_def,complex_of_real_def,cmod_def]));
+qed "cmod_i";
+
+Goalw [complex_of_real_def,i_def]
+ "complex_of_real xa + ii * complex_of_real ya = \
+\ complex_of_real xb + ii * complex_of_real yb \
+\ ==> xa = xb";
+by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_add]));
+qed "complex_eq_Re_eq";
+
+Goalw [complex_of_real_def,i_def]
+ "complex_of_real xa + ii * complex_of_real ya = \
+\ complex_of_real xb + ii * complex_of_real yb \
+\ ==> ya = yb";
+by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_add]));
+qed "complex_eq_Im_eq";
+
+Goal "(complex_of_real xa + ii * complex_of_real ya = \
+\ complex_of_real xb + ii * complex_of_real yb) = ((xa = xb) & (ya = yb))";
+by (auto_tac (claset() addIs [complex_eq_Im_eq,complex_eq_Re_eq],simpset()));
+qed "complex_eq_cancel_iff";
+AddIffs [complex_eq_cancel_iff];
+
+Goal "(complex_of_real xa + complex_of_real ya * ii = \
+\ complex_of_real xb + complex_of_real yb * ii ) = ((xa = xb) & (ya = yb))";
+by (auto_tac (claset(),simpset() addsimps [complex_mult_commute]));
+qed "complex_eq_cancel_iffA";
+AddIffs [complex_eq_cancel_iffA];
+
+Goal "(complex_of_real xa + complex_of_real ya * ii = \
+\ complex_of_real xb + ii * complex_of_real yb) = ((xa = xb) & (ya = yb))";
+by (auto_tac (claset(),simpset() addsimps [complex_mult_commute]));
+qed "complex_eq_cancel_iffB";
+AddIffs [complex_eq_cancel_iffB];
+
+Goal "(complex_of_real xa + ii * complex_of_real ya = \
+\ complex_of_real xb + complex_of_real yb * ii) = ((xa = xb) & (ya = yb))";
+by (auto_tac (claset(),simpset() addsimps [complex_mult_commute]));
+qed "complex_eq_cancel_iffC";
+AddIffs [complex_eq_cancel_iffC];
+
+Goal"(complex_of_real x + ii * complex_of_real y = \
+\ complex_of_real xa) = (x = xa & y = 0)";
+by (cut_inst_tac [("xa","x"),("ya","y"),("xb","xa"),("yb","0")]
+ complex_eq_cancel_iff 1);
+by (asm_full_simp_tac (simpset() delsimps [complex_eq_cancel_iff]) 1);
+qed "complex_eq_cancel_iff2";
+Addsimps [complex_eq_cancel_iff2];
+
+Goal"(complex_of_real x + complex_of_real y * ii = \
+\ complex_of_real xa) = (x = xa & y = 0)";
+by (auto_tac (claset(),simpset() addsimps [complex_mult_commute]));
+qed "complex_eq_cancel_iff2a";
+Addsimps [complex_eq_cancel_iff2a];
+
+Goal "(complex_of_real x + ii * complex_of_real y = \
+\ ii * complex_of_real ya) = (x = 0 & y = ya)";
+by (cut_inst_tac [("xa","x"),("ya","y"),("xb","0"),("yb","ya")]
+ complex_eq_cancel_iff 1);
+by (asm_full_simp_tac (simpset() delsimps [complex_eq_cancel_iff]) 1);
+qed "complex_eq_cancel_iff3";
+Addsimps [complex_eq_cancel_iff3];
+
+Goal "(complex_of_real x + complex_of_real y * ii = \
+\ ii * complex_of_real ya) = (x = 0 & y = ya)";
+by (auto_tac (claset(),simpset() addsimps [complex_mult_commute]));
+qed "complex_eq_cancel_iff3a";
+Addsimps [complex_eq_cancel_iff3a];
+
+Goalw [complex_of_real_def,i_def,complex_zero_def]
+ "complex_of_real x + ii * complex_of_real y = 0 \
+\ ==> x = 0";
+by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_add]));
+qed "complex_split_Re_zero";
+
+Goalw [complex_of_real_def,i_def,complex_zero_def]
+ "complex_of_real x + ii * complex_of_real y = 0 \
+\ ==> y = 0";
+by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_add]));
+qed "complex_split_Im_zero";
+
+Goalw [sgn_def,complex_divide_def]
+ "Re(sgn z) = Re(z)/cmod z";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_of_real_inverse RS sym]));
+by (auto_tac (claset(),simpset() addsimps [complex_of_real_def,
+ complex_mult,real_divide_def]));
+qed "Re_sgn";
+Addsimps [Re_sgn];
+
+Goalw [sgn_def,complex_divide_def]
+ "Im(sgn z) = Im(z)/cmod z";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_of_real_inverse RS sym]));
+by (auto_tac (claset(),simpset() addsimps [complex_of_real_def,
+ complex_mult,real_divide_def]));
+qed "Im_sgn";
+Addsimps [Im_sgn];
+
+Goalw [complex_of_real_def,i_def]
+ "inverse(complex_of_real x + ii * complex_of_real y) = \
+\ complex_of_real(x/(x ^ 2 + y ^ 2)) - \
+\ ii * complex_of_real(y/(x ^ 2 + y ^ 2))";
+by (auto_tac (claset(),simpset() addsimps [complex_mult,complex_add,
+ complex_diff_def,complex_minus,complex_inverse,real_divide_def]));
+qed "complex_inverse_complex_split";
+
+(*----------------------------------------------------------------------------*)
+(* Many of the theorems below need to be moved elsewhere e.g. Transc.ML. Also *)
+(* many of the theorems are not used - so should they be kept? *)
+(*----------------------------------------------------------------------------*)
+
+Goalw [i_def,complex_of_real_def]
+ "Re (ii * complex_of_real y) = 0";
+by (auto_tac (claset(),simpset() addsimps [complex_mult]));
+qed "Re_mult_i_eq";
+Addsimps [Re_mult_i_eq];
+
+Goalw [i_def,complex_of_real_def]
+ "Im (ii * complex_of_real y) = y";
+by (auto_tac (claset(),simpset() addsimps [complex_mult]));
+qed "Im_mult_i_eq";
+Addsimps [Im_mult_i_eq];
+
+Goalw [i_def,complex_of_real_def]
+ "cmod (ii * complex_of_real y) = abs y";
+by (auto_tac (claset(),simpset() addsimps [complex_mult,
+ complex_mod,realpow_num_two]));
+qed "complex_mod_mult_i";
+Addsimps [complex_mod_mult_i];
+
+Goalw [arg_def]
+ "0 < y ==> cos (arg(ii * complex_of_real y)) = 0";
+by (auto_tac (claset(),simpset() addsimps [abs_eqI2]));
+by (res_inst_tac [("a","pi/2")] someI2 1);
+by Auto_tac;
+by (res_inst_tac [("R2.0","0")] real_less_trans 1);
+by Auto_tac;
+qed "cos_arg_i_mult_zero";
+Addsimps [cos_arg_i_mult_zero];
+
+Goalw [arg_def]
+ "y < 0 ==> cos (arg(ii * complex_of_real y)) = 0";
+by (auto_tac (claset(),simpset() addsimps [abs_minus_eqI2]));
+by (res_inst_tac [("a","- pi/2")] someI2 1);
+by Auto_tac;
+by (res_inst_tac [("j","0")] real_le_trans 1);
+by Auto_tac;
+qed "cos_arg_i_mult_zero2";
+Addsimps [cos_arg_i_mult_zero2];
+
+Goalw [complex_zero_def,complex_of_real_def]
+ "(complex_of_real y ~= 0) = (y ~= 0)";
+by Auto_tac;
+qed "complex_of_real_not_zero_iff";
+Addsimps [complex_of_real_not_zero_iff];
+
+Goal "(complex_of_real y = 0) = (y = 0)";
+by Auto_tac;
+by (rtac ccontr 1 THEN dtac (complex_of_real_not_zero_iff RS iffD2) 1);
+by (Asm_full_simp_tac 1);
+qed "complex_of_real_zero_iff";
+Addsimps [complex_of_real_zero_iff];
+
+Goal "y ~= 0 ==> cos (arg(ii * complex_of_real y)) = 0";
+by (cut_inst_tac [("R1.0","y"),("R2.0","0")] real_linear 1);
+by Auto_tac;
+qed "cos_arg_i_mult_zero3";
+Addsimps [cos_arg_i_mult_zero3];
+
+(*---------------------------------------------------------------------------*)
+(* Finally! Polar form for complex numbers *)
+(*---------------------------------------------------------------------------*)
+
+Goal "EX r a. z = complex_of_real r * \
+\ (complex_of_real(cos a) + ii * complex_of_real(sin a))";
+by (cut_inst_tac [("z","z")] complex_split 1);
+by (auto_tac (claset(),simpset() addsimps [polar_Ex,
+ complex_add_mult_distrib2,complex_of_real_mult] @ complex_mult_ac));
+qed "complex_split_polar";
+
+Goalw [rcis_def,cis_def] "EX r a. z = rcis r a";
+by (rtac complex_split_polar 1);
+qed "rcis_Ex";
+
+Goal "Re(complex_of_real r * \
+\ (complex_of_real(cos a) + ii * complex_of_real(sin a))) = r * cos a";
+by (auto_tac (claset(),simpset() addsimps [complex_add_mult_distrib2,
+ complex_of_real_mult] @ complex_mult_ac));
+qed "Re_complex_polar";
+Addsimps [Re_complex_polar];
+
+Goalw [rcis_def,cis_def] "Re(rcis r a) = r * cos a";
+by Auto_tac;
+qed "Re_rcis";
+Addsimps [Re_rcis];
+
+Goal "Im(complex_of_real r * \
+\ (complex_of_real(cos a) + ii * complex_of_real(sin a))) = r * sin a";
+by (auto_tac (claset(),simpset() addsimps [complex_add_mult_distrib2,
+ complex_of_real_mult] @ complex_mult_ac));
+qed "Im_complex_polar";
+Addsimps [Im_complex_polar];
+
+Goalw [rcis_def,cis_def] "Im(rcis r a) = r * sin a";
+by Auto_tac;
+qed "Im_rcis";
+Addsimps [Im_rcis];
+
+Goal "cmod (complex_of_real r * \
+\ (complex_of_real(cos a) + ii * complex_of_real(sin a))) = abs r";
+by (auto_tac (claset(),simpset() addsimps [complex_add_mult_distrib2,
+ cmod_i,complex_of_real_mult,real_add_mult_distrib2
+ RS sym,realpow_mult] @ complex_mult_ac@ real_mult_ac
+ delsimps [realpow_Suc]));
+qed "complex_mod_complex_polar";
+Addsimps [complex_mod_complex_polar];
+
+Goalw [rcis_def,cis_def] "cmod(rcis r a) = abs r";
+by Auto_tac;
+qed "complex_mod_rcis";
+Addsimps [complex_mod_rcis];
+
+Goalw [cmod_def] "cmod z = sqrt (Re (z * cnj z))";
+by (rtac (real_sqrt_eq_iff RS iffD2) 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mult_cnj]));
+qed "complex_mod_sqrt_Re_mult_cnj";
+
+Goal "Re(cnj z) = Re z";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_cnj]));
+qed "complex_Re_cnj";
+Addsimps [complex_Re_cnj];
+
+Goal "Im(cnj z) = - Im z";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_cnj]));
+qed "complex_Im_cnj";
+Addsimps [complex_Im_cnj];
+
+Goal "Im (z * cnj z) = 0";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_cnj,complex_mult]));
+qed "complex_In_mult_cnj_zero";
+Addsimps [complex_In_mult_cnj_zero];
+
+Goal "[| Im w = 0; Im z = 0 |] ==> Re(w * z) = Re(w) * Re(z)";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (res_inst_tac [("z","w")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mult]));
+qed "complex_Re_mult";
+
+Goalw [complex_of_real_def] "Re (z * complex_of_real c) = Re(z) * c";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mult]));
+qed "complex_Re_mult_complex_of_real";
+Addsimps [complex_Re_mult_complex_of_real];
+
+Goalw [complex_of_real_def] "Im (z * complex_of_real c) = Im(z) * c";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mult]));
+qed "complex_Im_mult_complex_of_real";
+Addsimps [complex_Im_mult_complex_of_real];
+
+Goalw [complex_of_real_def] "Re (complex_of_real c * z) = c * Re(z)";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mult]));
+qed "complex_Re_mult_complex_of_real2";
+Addsimps [complex_Re_mult_complex_of_real2];
+
+Goalw [complex_of_real_def] "Im (complex_of_real c * z) = c * Im(z)";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mult]));
+qed "complex_Im_mult_complex_of_real2";
+Addsimps [complex_Im_mult_complex_of_real2];
+
+(*---------------------------------------------------------------------------*)
+(* (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b) *)
+(*---------------------------------------------------------------------------*)
+
+Goalw [rcis_def] "cis a = rcis 1 a";
+by (Simp_tac 1);
+qed "cis_rcis_eq";
+
+Goalw [rcis_def,cis_def]
+ "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)";
+by (auto_tac (claset(),simpset() addsimps [cos_add,sin_add,
+ complex_add_mult_distrib2,complex_add_mult_distrib]
+ @ complex_mult_ac @ complex_add_ac));
+by (auto_tac (claset(),simpset() addsimps [complex_add_mult_distrib2 RS sym,
+ complex_mult_assoc RS sym,complex_of_real_mult,complex_of_real_add,
+ complex_add_assoc RS sym,i_mult_eq] delsimps [i_mult_eq2]));
+by (auto_tac (claset(),simpset() addsimps complex_add_ac));
+by (auto_tac (claset(),simpset() addsimps [complex_add_assoc RS sym,
+ complex_of_real_add,real_add_mult_distrib2,
+ real_diff_def] @ real_mult_ac @ real_add_ac));
+qed "rcis_mult";
+
+Goal "cis a * cis b = cis (a + b)";
+by (simp_tac (simpset() addsimps [cis_rcis_eq,rcis_mult]) 1);
+qed "cis_mult";
+
+Goalw [cis_def] "cis 0 = 1";
+by Auto_tac;
+qed "cis_zero";
+Addsimps [cis_zero];
+
+Goalw [cis_def] "cis 0 = complex_of_real 1";
+by Auto_tac;
+qed "cis_zero2";
+Addsimps [cis_zero2];
+
+Goalw [rcis_def] "rcis 0 a = 0";
+by (Simp_tac 1);
+qed "rcis_zero_mod";
+Addsimps [rcis_zero_mod];
+
+Goalw [rcis_def] "rcis r 0 = complex_of_real r";
+by (Simp_tac 1);
+qed "rcis_zero_arg";
+Addsimps [rcis_zero_arg];
+
+Goalw [complex_of_real_def,complex_one_def]
+ "complex_of_real (-(1::real)) = -(1::complex)";
+by (simp_tac (simpset() addsimps [complex_minus]) 1);
+qed "complex_of_real_minus_one";
+
+Goal "ii * (ii * x) = - x";
+by (simp_tac (simpset() addsimps [complex_mult_assoc RS sym]) 1);
+qed "complex_i_mult_minus";
+Addsimps [complex_i_mult_minus];
+
+Goal "ii * ii * x = - x";
+by (Simp_tac 1);
+qed "complex_i_mult_minus2";
+Addsimps [complex_i_mult_minus2];
+
+Goalw [cis_def]
+ "cis (real (Suc n) * a) = cis a * cis (real n * a)";
+by (auto_tac (claset(),simpset() addsimps [real_of_nat_Suc,
+ real_add_mult_distrib,cos_add,sin_add,complex_add_mult_distrib,
+ complex_add_mult_distrib2,complex_of_real_add,complex_of_real_mult]
+ @ complex_mult_ac @ complex_add_ac));
+by (auto_tac (claset(),simpset() addsimps [complex_add_mult_distrib2 RS sym,
+ complex_mult_assoc RS sym,i_mult_eq,complex_of_real_mult,
+ complex_of_real_add,complex_add_assoc RS sym,complex_of_real_minus
+ RS sym,real_diff_def] @ real_mult_ac delsimps [i_mult_eq2]));
+qed "cis_real_of_nat_Suc_mult";
+
+Goal "(cis a) ^ n = cis (real n * a)";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps [cis_real_of_nat_Suc_mult]));
+qed "DeMoivre";
+
+Goalw [rcis_def]
+ "(rcis r a) ^ n = rcis (r ^ n) (real n * a)";
+by (auto_tac (claset(),simpset() addsimps [complexpow_mult,
+ DeMoivre,complex_of_real_pow]));
+qed "DeMoivre2";
+
+Goalw [cis_def] "inverse(cis a) = cis (-a)";
+by (auto_tac (claset(),simpset() addsimps [complex_inverse_complex_split,
+ complex_of_real_minus,complex_diff_def]));
+qed "cis_inverse";
+Addsimps [cis_inverse];
+
+Goal "inverse(rcis r a) = rcis (1/r) (-a)";
+by (real_div_undefined_case_tac "r=0" 1);
+by (simp_tac (simpset() addsimps [rename_numerals DIVISION_BY_ZERO,
+ COMPLEX_INVERSE_ZERO]) 1);
+by (auto_tac (claset(),simpset() addsimps [complex_inverse_complex_split,
+ complex_add_mult_distrib2,complex_of_real_mult,rcis_def,cis_def,
+ realpow_num_two] @ complex_mult_ac @ real_mult_ac));
+by (auto_tac (claset(),simpset() addsimps [real_add_mult_distrib2 RS sym,
+ complex_of_real_minus,complex_diff_def]));
+qed "rcis_inverse";
+
+Goalw [complex_divide_def] "cis a / cis b = cis (a - b)";
+by (auto_tac (claset(),simpset() addsimps [cis_mult,real_diff_def]));
+qed "cis_divide";
+
+Goalw [complex_divide_def]
+ "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)";
+by (real_div_undefined_case_tac "r2=0" 1);
+by (simp_tac (simpset() addsimps [rename_numerals DIVISION_BY_ZERO,
+ COMPLEX_INVERSE_ZERO]) 1);
+by (auto_tac (claset(),simpset() addsimps [rcis_inverse,rcis_mult,
+ real_diff_def]));
+qed "rcis_divide";
+
+Goalw [cis_def] "Re(cis a) = cos a";
+by Auto_tac;
+qed "Re_cis";
+Addsimps [Re_cis];
+
+Goalw [cis_def] "Im(cis a) = sin a";
+by Auto_tac;
+qed "Im_cis";
+Addsimps [Im_cis];
+
+Goal "cos (real n * a) = Re(cis a ^ n)";
+by (auto_tac (claset(),simpset() addsimps [DeMoivre]));
+qed "cos_n_Re_cis_pow_n";
+
+Goal "sin (real n * a) = Im(cis a ^ n)";
+by (auto_tac (claset(),simpset() addsimps [DeMoivre]));
+qed "sin_n_Im_cis_pow_n";
+
+Goalw [expi_def,cis_def]
+ "expi (ii * complex_of_real y) = \
+\ complex_of_real (cos y) + ii * complex_of_real (sin y)";
+by Auto_tac;
+qed "expi_Im_split";
+
+Goalw [expi_def]
+ "expi (ii * complex_of_real y) = cis y";
+by Auto_tac;
+qed "expi_Im_cis";
+
+Goalw [expi_def] "expi(a + b) = expi(a) * expi(b)";
+by (auto_tac (claset(),simpset() addsimps [complex_Re_add,exp_add,
+ complex_Im_add,cis_mult RS sym,complex_of_real_mult] @
+ complex_mult_ac));
+qed "expi_add";
+
+Goalw [expi_def]
+ "expi(complex_of_real x + ii * complex_of_real y) = \
+\ complex_of_real (exp(x)) * cis y";
+by Auto_tac;
+qed "expi_complex_split";
+
+Goalw [expi_def] "expi (0::complex) = 1";
+by Auto_tac;
+qed "expi_zero";
+Addsimps [expi_zero];
+
+goal Complex.thy
+ "Re (w * z) = Re w * Re z - Im w * Im z";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (res_inst_tac [("z","w")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mult]));
+qed "complex_Re_mult_eq";
+
+goal Complex.thy
+ "Im (w * z) = Re w * Im z + Im w * Re z";
+by (res_inst_tac [("z","z")] eq_Abs_complex 1);
+by (res_inst_tac [("z","w")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mult]));
+qed "complex_Im_mult_eq";
+
+goal Complex.thy
+ "EX a r. z = complex_of_real r * expi a";
+by (cut_inst_tac [("z","z")] rcis_Ex 1);
+by (auto_tac (claset(),simpset() addsimps [expi_def,rcis_def,
+ complex_mult_assoc RS sym,complex_of_real_mult]));
+by (res_inst_tac [("x","ii * complex_of_real a")] exI 1);
+by Auto_tac;
+qed "complex_expi_Ex";
+
+
+(****
+Goal "[| - pi < a; a <= pi |] ==> (-pi < a & a <= 0) | (0 <= a & a <= pi)";
+by Auto_tac;
+qed "lemma_split_interval";
+
+Goalw [arg_def]
+ "[| r ~= 0; - pi < a; a <= pi |] \
+\ ==> arg(complex_of_real r * \
+\ (complex_of_real(cos a) + ii * complex_of_real(sin a))) = a";
+by Auto_tac;
+by (cut_inst_tac [("R1.0","0"),("R2.0","r")] real_linear 1);
+by (auto_tac (claset(),simpset() addsimps (map (full_rename_numerals thy)
+ [rabs_eqI2,rabs_minus_eqI2,real_minus_rinv]) @ [real_divide_def,
+ real_minus_mult_eq2 RS sym] @ real_mult_ac));
+by (auto_tac (claset(),simpset() addsimps [real_mult_assoc RS sym]));
+by (dtac lemma_split_interval 1 THEN Step_tac 1);
+****)
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/Complex.thy Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,111 @@
+(* Title: Complex.thy
+ Author: Jacques D. Fleuriot
+ Copyright: 2001 University of Edinburgh
+ Description: Complex numbers
+*)
+
+Complex = HLog +
+
+typedef complex = "{p::(real*real). True}"
+
+instance
+ complex :: {ord,zero,one,plus,minus,times,power,inverse}
+
+consts
+ "ii" :: complex ("ii")
+
+constdefs
+
+ (*--- real and Imaginary parts ---*)
+
+ Re :: complex => real
+ "Re(z) == fst(Rep_complex z)"
+
+ Im :: complex => real
+ "Im(z) == snd(Rep_complex z)"
+
+ (*----------- modulus ------------*)
+
+ cmod :: complex => real
+ "cmod z == sqrt(Re(z) ^ 2 + Im(z) ^ 2)"
+
+ (*----- injection from reals -----*)
+
+ complex_of_real :: real => complex
+ "complex_of_real r == Abs_complex(r,0::real)"
+
+ (*------- complex conjugate ------*)
+
+ cnj :: complex => complex
+ "cnj z == Abs_complex(Re z, -Im z)"
+
+ (*------------ Argand -------------*)
+
+ sgn :: complex => complex
+ "sgn z == z / complex_of_real(cmod z)"
+
+ arg :: complex => real
+ "arg z == @a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a <= pi"
+
+defs
+
+ complex_zero_def
+ "0 == Abs_complex(0::real,0)"
+
+ complex_one_def
+ "1 == Abs_complex(1,0::real)"
+
+ (*------ imaginary unit ----------*)
+
+ i_def
+ "ii == Abs_complex(0::real,1)"
+
+ (*----------- negation -----------*)
+
+ complex_minus_def
+ "- (z::complex) == Abs_complex(-Re z, -Im z)"
+
+
+ (*----------- inverse -----------*)
+ complex_inverse_def
+ "inverse (z::complex) == Abs_complex(Re(z)/(Re(z) ^ 2 + Im(z) ^ 2),
+ -Im(z)/(Re(z) ^ 2 + Im(z) ^ 2))"
+
+ complex_add_def
+ "w + (z::complex) == Abs_complex(Re(w) + Re(z),Im(w) + Im(z))"
+
+ complex_diff_def
+ "w - (z::complex) == w + -(z::complex)"
+
+ complex_mult_def
+ "w * (z::complex) == Abs_complex(Re(w) * Re(z) - Im(w) * Im(z),
+ Re(w) * Im(z) + Im(w) * Re(z))"
+
+
+ (*----------- division ----------*)
+ complex_divide_def
+ "w / (z::complex) == w * inverse z"
+
+
+primrec
+ complexpow_0 "z ^ 0 = complex_of_real 1"
+ complexpow_Suc "z ^ (Suc n) = (z::complex) * (z ^ n)"
+
+
+constdefs
+
+ (* abbreviation for (cos a + i sin a) *)
+ cis :: real => complex
+ "cis a == complex_of_real(cos a) + ii * complex_of_real(sin a)"
+
+ (* abbreviation for r*(cos a + i sin a) *)
+ rcis :: [real, real] => complex
+ "rcis r a == complex_of_real r * cis a"
+
+ (* e ^ (x + iy) *)
+ expi :: complex => complex
+ "expi z == complex_of_real(exp (Re z)) * cis (Im z)"
+
+end
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/ComplexArith0.ML Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,321 @@
+(* Title: ComplexArith0.ML
+ Author: Jacques D. Fleuriot
+ Copyright: 2001 University of Edinburgh
+ Description: Assorted facts that need binary literals
+ Also, common factor cancellation (see e.g. HyperArith0)
+*)
+
+Goal "x - - y = x + (y::complex)";
+by (Simp_tac 1);
+qed "real_diff_minus_eq";
+Addsimps [real_diff_minus_eq];
+
+(** Division and inverse **)
+
+Goal "0/x = (0::complex)";
+by (simp_tac (simpset() addsimps [complex_divide_def]) 1);
+qed "complex_0_divide";
+Addsimps [complex_0_divide];
+
+Goalw [complex_divide_def] "x/(0::complex) = 0";
+by (stac COMPLEX_INVERSE_ZERO 1);
+by (Simp_tac 1);
+qed "COMPLEX_DIVIDE_ZERO";
+
+Goal "inverse (x::complex) = 1/x";
+by (simp_tac (simpset() addsimps [complex_divide_def]) 1);
+qed "complex_inverse_eq_divide";
+
+Goal "(inverse(x::complex) = 0) = (x = 0)";
+by (auto_tac (claset(),
+ simpset() addsimps [COMPLEX_INVERSE_ZERO]));
+by (rtac ccontr 1);
+by (blast_tac (claset() addDs [complex_inverse_not_zero]) 1);
+qed "complex_inverse_zero_iff";
+Addsimps [complex_inverse_zero_iff];
+
+Goal "(x/y = 0) = (x=0 | y=(0::complex))";
+by (auto_tac (claset(), simpset() addsimps [complex_divide_def]));
+qed "complex_divide_eq_0_iff";
+Addsimps [complex_divide_eq_0_iff];
+
+Goal "h ~= (0::complex) ==> h/h = 1";
+by (asm_simp_tac
+ (simpset() addsimps [complex_divide_def]) 1);
+qed "complex_divide_self_eq";
+Addsimps [complex_divide_self_eq];
+
+bind_thm ("complex_mult_minus_right", complex_minus_mult_eq2 RS sym);
+
+Goal "!!k::complex. (k*m = k*n) = (k = 0 | m=n)";
+by (case_tac "k=0" 1);
+by (auto_tac (claset(), simpset() addsimps [complex_mult_left_cancel]));
+qed "complex_mult_eq_cancel1";
+
+Goal "!!k::complex. (m*k = n*k) = (k = 0 | m=n)";
+by (case_tac "k=0" 1);
+by (auto_tac (claset(), simpset() addsimps [complex_mult_right_cancel]));
+qed "complex_mult_eq_cancel2";
+
+Goal "!!k::complex. k~=0 ==> (k*m) / (k*n) = (m/n)";
+by (asm_simp_tac
+ (simpset() addsimps [complex_divide_def, complex_inverse_distrib]) 1);
+by (subgoal_tac "k * m * (inverse k * inverse n) = \
+\ (k * inverse k) * (m * inverse n)" 1);
+by (Asm_full_simp_tac 1);
+by (asm_full_simp_tac (HOL_ss addsimps complex_mult_ac) 1);
+qed "complex_mult_div_cancel1";
+
+(*For ExtractCommonTerm*)
+Goal "(k*m) / (k*n) = (if k = (0::complex) then 0 else m/n)";
+by (simp_tac (simpset() addsimps [complex_mult_div_cancel1]) 1);
+qed "complex_mult_div_cancel_disj";
+
+
+local
+ open Complex_Numeral_Simprocs
+in
+
+val rel_complex_number_of = [eq_complex_number_of];
+
+
+structure CancelNumeralFactorCommon =
+ struct
+ val mk_coeff = mk_coeff
+ val dest_coeff = dest_coeff 1
+ val trans_tac = Real_Numeral_Simprocs.trans_tac
+ val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps complex_minus_from_mult_simps @ mult_1s))
+ THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@complex_mult_minus_simps))
+ THEN ALLGOALS (simp_tac (HOL_ss addsimps complex_mult_ac))
+ val numeral_simp_tac = ALLGOALS (simp_tac (HOL_ss addsimps rel_complex_number_of@bin_simps))
+ val simplify_meta_eq = simplify_meta_eq
+ end
+
+
+structure DivCancelNumeralFactor = CancelNumeralFactorFun
+ (open CancelNumeralFactorCommon
+ val prove_conv = Bin_Simprocs.prove_conv
+ val mk_bal = HOLogic.mk_binop "HOL.divide"
+ val dest_bal = HOLogic.dest_bin "HOL.divide" complexT
+ val cancel = complex_mult_div_cancel1 RS trans
+ val neg_exchanges = false
+)
+
+
+structure EqCancelNumeralFactor = CancelNumeralFactorFun
+ (open CancelNumeralFactorCommon
+ val prove_conv = Bin_Simprocs.prove_conv
+ val mk_bal = HOLogic.mk_eq
+ val dest_bal = HOLogic.dest_bin "op =" complexT
+ val cancel = complex_mult_eq_cancel1 RS trans
+ val neg_exchanges = false
+)
+
+val complex_cancel_numeral_factors_relations =
+ map prep_simproc
+ [("complexeq_cancel_numeral_factor",
+ ["(l::complex) * m = n", "(l::complex) = m * n"],
+ EqCancelNumeralFactor.proc)];
+
+val complex_cancel_numeral_factors_divide = prep_simproc
+ ("complexdiv_cancel_numeral_factor",
+ ["((l::complex) * m) / n", "(l::complex) / (m * n)",
+ "((number_of v)::complex) / (number_of w)"],
+ DivCancelNumeralFactor.proc);
+
+val complex_cancel_numeral_factors =
+ complex_cancel_numeral_factors_relations @
+ [complex_cancel_numeral_factors_divide];
+
+end;
+
+
+Addsimprocs complex_cancel_numeral_factors;
+
+
+(*examples:
+print_depth 22;
+set timing;
+set trace_simp;
+fun test s = (Goal s; by (Simp_tac 1));
+
+
+test "9*x = 12 * (y::complex)";
+test "(9*x) / (12 * (y::complex)) = z";
+
+test "-99*x = 132 * (y::complex)";
+
+test "999*x = -396 * (y::complex)";
+test "(999*x) / (-396 * (y::complex)) = z";
+
+test "-99*x = -81 * (y::complex)";
+test "(-99*x) / (-81 * (y::complex)) = z";
+
+test "-2 * x = -1 * (y::complex)";
+test "-2 * x = -(y::complex)";
+test "(-2 * x) / (-1 * (y::complex)) = z";
+
+*)
+
+
+(** Declarations for ExtractCommonTerm **)
+
+local
+ open Complex_Numeral_Simprocs
+in
+
+structure CancelFactorCommon =
+ struct
+ val mk_sum = long_mk_prod
+ val dest_sum = dest_prod
+ val mk_coeff = mk_coeff
+ val dest_coeff = dest_coeff
+ val find_first = find_first []
+ val trans_tac = Real_Numeral_Simprocs.trans_tac
+ val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps mult_1s@complex_mult_ac))
+ end;
+
+
+structure EqCancelFactor = ExtractCommonTermFun
+ (open CancelFactorCommon
+ val prove_conv = Bin_Simprocs.prove_conv
+ val mk_bal = HOLogic.mk_eq
+ val dest_bal = HOLogic.dest_bin "op =" complexT
+ val simplify_meta_eq = cancel_simplify_meta_eq complex_mult_eq_cancel1
+);
+
+
+structure DivideCancelFactor = ExtractCommonTermFun
+ (open CancelFactorCommon
+ val prove_conv = Bin_Simprocs.prove_conv
+ val mk_bal = HOLogic.mk_binop "HOL.divide"
+ val dest_bal = HOLogic.dest_bin "HOL.divide" complexT
+ val simplify_meta_eq = cancel_simplify_meta_eq complex_mult_div_cancel_disj
+);
+
+val complex_cancel_factor =
+ map prep_simproc
+ [("complex_eq_cancel_factor", ["(l::complex) * m = n", "(l::complex) = m * n"],
+ EqCancelFactor.proc),
+ ("complex_divide_cancel_factor", ["((l::complex) * m) / n", "(l::complex) / (m * n)"],
+ DivideCancelFactor.proc)];
+
+end;
+
+Addsimprocs complex_cancel_factor;
+
+
+(*examples:
+print_depth 22;
+set timing;
+set trace_simp;
+fun test s = (Goal s; by (Asm_simp_tac 1));
+
+test "x*k = k*(y::complex)";
+test "k = k*(y::complex)";
+test "a*(b*c) = (b::complex)";
+test "a*(b*c) = d*(b::complex)*(x*a)";
+
+
+test "(x*k) / (k*(y::complex)) = (uu::complex)";
+test "(k) / (k*(y::complex)) = (uu::complex)";
+test "(a*(b*c)) / ((b::complex)) = (uu::complex)";
+test "(a*(b*c)) / (d*(b::complex)*(x*a)) = (uu::complex)";
+
+(*FIXME: what do we do about this?*)
+test "a*(b*c)/(y*z) = d*(b::complex)*(x*a)/z";
+*)
+
+
+Goal "z~=0 ==> ((x::complex) = y/z) = (x*z = y)";
+by (subgoal_tac "(x*z = y) = (x*z = (y/z)*z)" 1);
+by (asm_simp_tac (simpset() addsimps [complex_divide_def, complex_mult_assoc]) 2);
+by (etac ssubst 1);
+by (stac complex_mult_eq_cancel2 1);
+by (Asm_simp_tac 1);
+qed "complex_eq_divide_eq";
+Addsimps [inst "z" "number_of ?w" complex_eq_divide_eq];
+
+Goal "z~=0 ==> (y/z = (x::complex)) = (y = x*z)";
+by (subgoal_tac "(y = x*z) = ((y/z)*z = x*z)" 1);
+by (asm_simp_tac (simpset() addsimps [complex_divide_def, complex_mult_assoc]) 2);
+by (etac ssubst 1);
+by (stac complex_mult_eq_cancel2 1);
+by (Asm_simp_tac 1);
+qed "complex_divide_eq_eq";
+Addsimps [inst "z" "number_of ?w" complex_divide_eq_eq];
+
+Goal "(m/k = n/k) = (k = 0 | m = (n::complex))";
+by (case_tac "k=0" 1);
+by (asm_simp_tac (simpset() addsimps [COMPLEX_DIVIDE_ZERO]) 1);
+by (asm_simp_tac (simpset() addsimps [complex_divide_eq_eq, complex_eq_divide_eq,
+ complex_mult_eq_cancel2]) 1);
+qed "complex_divide_eq_cancel2";
+
+Goal "(k/m = k/n) = (k = 0 | m = (n::complex))";
+by (case_tac "m=0 | n = 0" 1);
+by (auto_tac (claset(),
+ simpset() addsimps [COMPLEX_DIVIDE_ZERO, complex_divide_eq_eq,
+ complex_eq_divide_eq, complex_mult_eq_cancel1]));
+qed "complex_divide_eq_cancel1";
+
+(** Division by 1, -1 **)
+
+Goal "(x::complex)/1 = x";
+by (simp_tac (simpset() addsimps [complex_divide_def]) 1);
+qed "complex_divide_1";
+Addsimps [complex_divide_1];
+
+Goal "x/-1 = -(x::complex)";
+by (Simp_tac 1);
+qed "complex_divide_minus1";
+Addsimps [complex_divide_minus1];
+
+Goal "-1/(x::complex) = - (1/x)";
+by (simp_tac (simpset() addsimps [complex_divide_def, complex_minus_inverse]) 1);
+qed "complex_minus1_divide";
+Addsimps [complex_minus1_divide];
+
+
+Goal "(x = - y) = (y = - (x::complex))";
+by Auto_tac;
+qed "complex_equation_minus";
+
+Goal "(- x = y) = (- (y::complex) = x)";
+by Auto_tac;
+qed "complex_minus_equation";
+
+Goal "(x + - a = (0::complex)) = (x=a)";
+by (simp_tac (simpset() addsimps [complex_diff_eq_eq,symmetric complex_diff_def]) 1);
+qed "complex_add_minus_iff";
+Addsimps [complex_add_minus_iff];
+
+Goal "(-b = -a) = (b = (a::complex))";
+by Auto_tac;
+by (etac ( inj_complex_minus RS injD) 1);
+qed "complex_minus_eq_cancel";
+Addsimps [complex_minus_eq_cancel];
+
+(*Distributive laws for literals*)
+Addsimps (map (inst "w" "number_of ?v")
+ [complex_add_mult_distrib, complex_add_mult_distrib2,
+ complex_diff_mult_distrib, complex_diff_mult_distrib2]);
+
+Addsimps [inst "x" "number_of ?v" complex_equation_minus];
+
+Addsimps [inst "y" "number_of ?v" complex_minus_equation];
+
+Goal "(x+y = (0::complex)) = (y = -x)";
+by Auto_tac;
+by (dtac (sym RS (complex_diff_eq_eq RS iffD2)) 1);
+by Auto_tac;
+qed "complex_add_eq_0_iff";
+AddIffs [complex_add_eq_0_iff];
+
+Goalw [complex_diff_def]"-(x-y) = y - (x::complex)";
+by (auto_tac (claset(),simpset() addsimps [complex_add_commute]));
+qed "complex_minus_diff_eq";
+Addsimps [complex_minus_diff_eq];
+
+Addsimps [inst "x" "number_of ?w" complex_inverse_eq_divide];
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/ComplexArith0.thy Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,2 @@
+ComplexArith0 = ComplexBin
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/ComplexBin.ML Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,609 @@
+(* Title: ComplexBin.ML
+ Author: Jacques D. Fleuriot
+ Copyright: 2001 University of Edinburgh
+ Descrition: Binary arithmetic for the complex numbers
+*)
+
+(** complex_of_real (coercion from real to complex) **)
+
+Goal "complex_of_real (number_of w) = number_of w";
+by (simp_tac (simpset() addsimps [complex_number_of_def]) 1);
+qed "complex_number_of";
+Addsimps [complex_number_of];
+
+Goalw [complex_number_of_def] "Numeral0 = (0::complex)";
+by (Simp_tac 1);
+qed "complex_numeral_0_eq_0";
+
+Goalw [complex_number_of_def] "Numeral1 = (1::complex)";
+by (Simp_tac 1);
+qed "complex_numeral_1_eq_1";
+
+(** Addition **)
+
+Goal "(number_of v :: complex) + number_of v' = number_of (bin_add v v')";
+by (simp_tac
+ (HOL_ss addsimps [complex_number_of_def,
+ complex_of_real_add, add_real_number_of]) 1);
+qed "add_complex_number_of";
+Addsimps [add_complex_number_of];
+
+
+(** Subtraction **)
+
+Goalw [complex_number_of_def]
+ "- (number_of w :: complex) = number_of (bin_minus w)";
+by (simp_tac
+ (HOL_ss addsimps [minus_real_number_of, complex_of_real_minus RS sym]) 1);
+qed "minus_complex_number_of";
+Addsimps [minus_complex_number_of];
+
+Goalw [complex_number_of_def, complex_diff_def]
+ "(number_of v :: complex) - number_of w = number_of (bin_add v (bin_minus w))";
+by (Simp_tac 1);
+qed "diff_complex_number_of";
+Addsimps [diff_complex_number_of];
+
+
+(** Multiplication **)
+
+Goal "(number_of v :: complex) * number_of v' = number_of (bin_mult v v')";
+by (simp_tac
+ (HOL_ss addsimps [complex_number_of_def,
+ complex_of_real_mult, mult_real_number_of]) 1);
+qed "mult_complex_number_of";
+Addsimps [mult_complex_number_of];
+
+Goal "(2::complex) = 1 + 1";
+by (simp_tac (simpset() addsimps [complex_numeral_1_eq_1 RS sym]) 1);
+val lemma = result();
+
+(*For specialist use: NOT as default simprules*)
+Goal "2 * z = (z+z::complex)";
+by (simp_tac (simpset () addsimps [lemma, complex_add_mult_distrib]) 1);
+qed "complex_mult_2";
+
+Goal "z * 2 = (z+z::complex)";
+by (stac complex_mult_commute 1 THEN rtac complex_mult_2 1);
+qed "complex_mult_2_right";
+
+(** Equals (=) **)
+
+Goal "((number_of v :: complex) = number_of v') = \
+\ iszero (number_of (bin_add v (bin_minus v')))";
+by (simp_tac
+ (HOL_ss addsimps [complex_number_of_def,
+ complex_of_real_eq_iff, eq_real_number_of]) 1);
+qed "eq_complex_number_of";
+Addsimps [eq_complex_number_of];
+
+(*** New versions of existing theorems involving 0, 1 ***)
+
+Goal "- 1 = (-1::complex)";
+by (simp_tac (simpset() addsimps [complex_numeral_1_eq_1 RS sym]) 1);
+qed "complex_minus_1_eq_m1";
+
+Goal "-1 * z = -(z::complex)";
+by (simp_tac (simpset() addsimps [complex_minus_1_eq_m1 RS sym]) 1);
+qed "complex_mult_minus1";
+
+Goal "z * -1 = -(z::complex)";
+by (stac complex_mult_commute 1 THEN rtac complex_mult_minus1 1);
+qed "complex_mult_minus1_right";
+
+Addsimps [complex_mult_minus1,complex_mult_minus1_right];
+
+
+(*Maps 0 to Numeral0 and 1 to Numeral1 and -Numeral1 to -1*)
+val complex_numeral_ss =
+ hypreal_numeral_ss addsimps [complex_numeral_0_eq_0 RS sym, complex_numeral_1_eq_1 RS sym,
+ complex_minus_1_eq_m1];
+
+fun rename_numerals th =
+ asm_full_simplify complex_numeral_ss (Thm.transfer (the_context ()) th);
+
+(*Now insert some identities previously stated for 0 and 1c*)
+
+Addsimps [complex_numeral_0_eq_0,complex_numeral_1_eq_1];
+
+Goal "number_of v + (number_of w + z) = (number_of(bin_add v w) + z::complex)";
+by (auto_tac (claset(),simpset() addsimps [complex_add_assoc RS sym]));
+qed "complex_add_number_of_left";
+
+Goal "number_of v *(number_of w * z) = (number_of(bin_mult v w) * z::complex)";
+by (simp_tac (simpset() addsimps [complex_mult_assoc RS sym]) 1);
+qed "complex_mult_number_of_left";
+
+Goalw [complex_diff_def]
+ "number_of v + (number_of w - c) = number_of(bin_add v w) - (c::complex)";
+by (rtac complex_add_number_of_left 1);
+qed "complex_add_number_of_diff1";
+
+Goal "number_of v + (c - number_of w) = \
+\ number_of (bin_add v (bin_minus w)) + (c::complex)";
+by (auto_tac (claset(),simpset() addsimps [complex_diff_def]@ complex_add_ac));
+qed "complex_add_number_of_diff2";
+
+Addsimps [complex_add_number_of_left, complex_mult_number_of_left,
+ complex_add_number_of_diff1, complex_add_number_of_diff2];
+
+
+(**** Simprocs for numeric literals ****)
+
+(** Combining of literal coefficients in sums of products **)
+
+Goal "(x = y) = (x-y = (0::complex))";
+by (simp_tac (simpset() addsimps [complex_diff_eq_eq]) 1);
+qed "complex_eq_iff_diff_eq_0";
+
+(** For combine_numerals **)
+
+Goal "i*u + (j*u + k) = (i+j)*u + (k::complex)";
+by (asm_simp_tac (simpset() addsimps [complex_add_mult_distrib]
+ @ complex_add_ac) 1);
+qed "left_complex_add_mult_distrib";
+
+(** For cancel_numerals **)
+
+Goal "((x::complex) = u + v) = (x - (u + v) = 0)";
+by (auto_tac (claset(),simpset() addsimps [complex_diff_eq_eq]));
+qed "complex_eq_add_diff_eq_0";
+
+Goal "((x::complex) = n) = (x - n = 0)";
+by (auto_tac (claset(),simpset() addsimps [complex_diff_eq_eq]));
+qed "complex_eq_diff_eq_0";
+
+val complex_rel_iff_rel_0_rls = [complex_eq_diff_eq_0,complex_eq_add_diff_eq_0];
+
+Goal "!!i::complex. (i*u + m = j*u + n) = ((i-j)*u + m = n)";
+by (auto_tac (claset(), simpset() addsimps [complex_add_mult_distrib,
+ complex_diff_def] @ complex_add_ac));
+by (asm_simp_tac (simpset() addsimps [complex_add_assoc RS sym]) 1);
+by (simp_tac (simpset() addsimps [complex_add_assoc]) 1);
+qed "complex_eq_add_iff1";
+
+Goal "!!i::complex. (i*u + m = j*u + n) = (m = (j-i)*u + n)";
+by (simp_tac (simpset() addsimps [ complex_eq_add_iff1]) 1);
+by (auto_tac (claset(), simpset() addsimps [complex_diff_def,
+ complex_add_mult_distrib]@ complex_add_ac));
+qed "complex_eq_add_iff2";
+
+structure Complex_Numeral_Simprocs =
+struct
+
+(*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic in simprocs
+ isn't complicated by the abstract 0 and 1.*)
+val numeral_syms = [complex_numeral_0_eq_0 RS sym, complex_numeral_1_eq_1 RS sym];
+
+
+(*Utilities*)
+
+val complexT = Type("Complex.complex",[]);
+
+fun mk_numeral n = HOLogic.number_of_const complexT $ HOLogic.mk_bin n;
+
+val dest_numeral = Real_Numeral_Simprocs.dest_numeral;
+val find_first_numeral = Real_Numeral_Simprocs.find_first_numeral;
+
+val zero = mk_numeral 0;
+val mk_plus = HOLogic.mk_binop "op +";
+
+val uminus_const = Const ("uminus", complexT --> complexT);
+
+(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
+fun mk_sum [] = zero
+ | mk_sum [t,u] = mk_plus (t, u)
+ | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
+
+(*this version ALWAYS includes a trailing zero*)
+fun long_mk_sum [] = zero
+ | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
+
+val dest_plus = HOLogic.dest_bin "op +" complexT;
+
+(*decompose additions AND subtractions as a sum*)
+fun dest_summing (pos, Const ("op +", _) $ t $ u, ts) =
+ dest_summing (pos, t, dest_summing (pos, u, ts))
+ | dest_summing (pos, Const ("op -", _) $ t $ u, ts) =
+ dest_summing (pos, t, dest_summing (not pos, u, ts))
+ | dest_summing (pos, t, ts) =
+ if pos then t::ts else uminus_const$t :: ts;
+
+fun dest_sum t = dest_summing (true, t, []);
+
+val mk_diff = HOLogic.mk_binop "op -";
+val dest_diff = HOLogic.dest_bin "op -" complexT;
+
+val one = mk_numeral 1;
+val mk_times = HOLogic.mk_binop "op *";
+
+fun mk_prod [] = one
+ | mk_prod [t] = t
+ | mk_prod (t :: ts) = if t = one then mk_prod ts
+ else mk_times (t, mk_prod ts);
+
+val dest_times = HOLogic.dest_bin "op *" complexT;
+
+fun dest_prod t =
+ let val (t,u) = dest_times t
+ in dest_prod t @ dest_prod u end
+ handle TERM _ => [t];
+
+(*DON'T do the obvious simplifications; that would create special cases*)
+fun mk_coeff (k, ts) = mk_times (mk_numeral k, ts);
+
+(*Express t as a product of (possibly) a numeral with other sorted terms*)
+fun dest_coeff sign (Const ("uminus", _) $ t) = dest_coeff (~sign) t
+ | dest_coeff sign t =
+ let val ts = sort Term.term_ord (dest_prod t)
+ val (n, ts') = find_first_numeral [] ts
+ handle TERM _ => (1, ts)
+ in (sign*n, mk_prod ts') end;
+
+(*Find first coefficient-term THAT MATCHES u*)
+fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
+ | find_first_coeff past u (t::terms) =
+ let val (n,u') = dest_coeff 1 t
+ in if u aconv u' then (n, rev past @ terms)
+ else find_first_coeff (t::past) u terms
+ end
+ handle TERM _ => find_first_coeff (t::past) u terms;
+
+
+(*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1*)
+val add_0s = map rename_numerals [complex_add_zero_left, complex_add_zero_right];
+val mult_plus_1s = map rename_numerals [complex_mult_one_left, complex_mult_one_right];
+val mult_minus_1s = map rename_numerals
+ [complex_mult_minus1, complex_mult_minus1_right];
+val mult_1s = mult_plus_1s @ mult_minus_1s;
+
+(*To perform binary arithmetic*)
+val bin_simps =
+ [complex_numeral_0_eq_0 RS sym, complex_numeral_1_eq_1 RS sym,
+ add_complex_number_of, complex_add_number_of_left,
+ minus_complex_number_of, diff_complex_number_of, mult_complex_number_of,
+ complex_mult_number_of_left] @ bin_arith_simps @ bin_rel_simps;
+
+(*To evaluate binary negations of coefficients*)
+val complex_minus_simps = NCons_simps @
+ [complex_minus_1_eq_m1,minus_complex_number_of,
+ bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min,
+ bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min];
+
+(*To let us treat subtraction as addition*)
+val diff_simps = [complex_diff_def, complex_minus_add_distrib,
+ complex_minus_minus];
+
+(* push the unary minus down: - x * y = x * - y *)
+val complex_minus_mult_eq_1_to_2 =
+ [complex_minus_mult_eq1 RS sym, complex_minus_mult_eq2] MRS trans
+ |> standard;
+
+(*to extract again any uncancelled minuses*)
+val complex_minus_from_mult_simps =
+ [complex_minus_minus, complex_minus_mult_eq1 RS sym,
+ complex_minus_mult_eq2 RS sym];
+
+(*combine unary minus with numeric literals, however nested within a product*)
+val complex_mult_minus_simps =
+ [complex_mult_assoc, complex_minus_mult_eq1, complex_minus_mult_eq_1_to_2];
+
+(*Final simplification: cancel + and * *)
+val simplify_meta_eq =
+ Int_Numeral_Simprocs.simplify_meta_eq
+ [complex_add_zero_left, complex_add_zero_right,
+ complex_mult_zero_left, complex_mult_zero_right, complex_mult_one_left,
+ complex_mult_one_right];
+
+val prep_simproc = Real_Numeral_Simprocs.prep_simproc;
+
+
+structure CancelNumeralsCommon =
+ struct
+ val mk_sum = mk_sum
+ val dest_sum = dest_sum
+ val mk_coeff = mk_coeff
+ val dest_coeff = dest_coeff 1
+ val find_first_coeff = find_first_coeff []
+ val trans_tac = Real_Numeral_Simprocs.trans_tac
+ val norm_tac =
+ ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
+ complex_minus_simps@complex_add_ac))
+ THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@complex_mult_minus_simps))
+ THEN ALLGOALS
+ (simp_tac (HOL_ss addsimps complex_minus_from_mult_simps@
+ complex_add_ac@complex_mult_ac))
+ val numeral_simp_tac = ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps))
+ val simplify_meta_eq = simplify_meta_eq
+ end;
+
+
+structure EqCancelNumerals = CancelNumeralsFun
+ (open CancelNumeralsCommon
+ val prove_conv = Bin_Simprocs.prove_conv
+ val mk_bal = HOLogic.mk_eq
+ val dest_bal = HOLogic.dest_bin "op =" complexT
+ val bal_add1 = complex_eq_add_iff1 RS trans
+ val bal_add2 = complex_eq_add_iff2 RS trans
+);
+
+
+val cancel_numerals =
+ map prep_simproc
+ [("complexeq_cancel_numerals",
+ ["(l::complex) + m = n", "(l::complex) = m + n",
+ "(l::complex) - m = n", "(l::complex) = m - n",
+ "(l::complex) * m = n", "(l::complex) = m * n"],
+ EqCancelNumerals.proc)];
+
+structure CombineNumeralsData =
+ struct
+ val add = op + : int*int -> int
+ val mk_sum = long_mk_sum (*to work for e.g. #2*x + #3*x *)
+ val dest_sum = dest_sum
+ val mk_coeff = mk_coeff
+ val dest_coeff = dest_coeff 1
+ val left_distrib = left_complex_add_mult_distrib RS trans
+ val prove_conv = Bin_Simprocs.prove_conv_nohyps
+ val trans_tac = Real_Numeral_Simprocs.trans_tac
+ val norm_tac =
+ ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
+ complex_minus_simps@complex_add_ac))
+ THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@complex_mult_minus_simps))
+ THEN ALLGOALS (simp_tac (HOL_ss addsimps complex_minus_from_mult_simps@
+ complex_add_ac@complex_mult_ac))
+ val numeral_simp_tac = ALLGOALS
+ (simp_tac (HOL_ss addsimps add_0s@bin_simps))
+ val simplify_meta_eq = simplify_meta_eq
+ end;
+
+structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
+
+val combine_numerals =
+ prep_simproc ("complex_combine_numerals",
+ ["(i::complex) + j", "(i::complex) - j"],
+ CombineNumerals.proc);
+
+
+(** Declarations for ExtractCommonTerm **)
+
+(*this version ALWAYS includes a trailing one*)
+fun long_mk_prod [] = one
+ | long_mk_prod (t :: ts) = mk_times (t, mk_prod ts);
+
+(*Find first term that matches u*)
+fun find_first past u [] = raise TERM("find_first", [])
+ | find_first past u (t::terms) =
+ if u aconv t then (rev past @ terms)
+ else find_first (t::past) u terms
+ handle TERM _ => find_first (t::past) u terms;
+
+(*Final simplification: cancel + and * *)
+fun cancel_simplify_meta_eq cancel_th th =
+ Int_Numeral_Simprocs.simplify_meta_eq
+ [complex_mult_one_left, complex_mult_one_right]
+ (([th, cancel_th]) MRS trans);
+
+(*** Making constant folding work for 0 and 1 too ***)
+
+structure ComplexAbstractNumeralsData =
+ struct
+ val dest_eq = HOLogic.dest_eq o HOLogic.dest_Trueprop o concl_of
+ val is_numeral = Bin_Simprocs.is_numeral
+ val numeral_0_eq_0 = complex_numeral_0_eq_0
+ val numeral_1_eq_1 = complex_numeral_1_eq_1
+ val prove_conv = Bin_Simprocs.prove_conv_nohyps_novars
+ fun norm_tac simps = ALLGOALS (simp_tac (HOL_ss addsimps simps))
+ val simplify_meta_eq = Bin_Simprocs.simplify_meta_eq
+ end
+
+structure ComplexAbstractNumerals = AbstractNumeralsFun (ComplexAbstractNumeralsData)
+
+(*For addition, we already have rules for the operand 0.
+ Multiplication is omitted because there are already special rules for
+ both 0 and 1 as operands. Unary minus is trivial, just have - 1 = -1.
+ For the others, having three patterns is a compromise between just having
+ one (many spurious calls) and having nine (just too many!) *)
+val eval_numerals =
+ map prep_simproc
+ [("complex_add_eval_numerals",
+ ["(m::complex) + 1", "(m::complex) + number_of v"],
+ ComplexAbstractNumerals.proc add_complex_number_of),
+ ("complex_diff_eval_numerals",
+ ["(m::complex) - 1", "(m::complex) - number_of v"],
+ ComplexAbstractNumerals.proc diff_complex_number_of),
+ ("complex_eq_eval_numerals",
+ ["(m::complex) = 0", "(m::complex) = 1", "(m::complex) = number_of v"],
+ ComplexAbstractNumerals.proc eq_complex_number_of)]
+
+end;
+
+Addsimprocs Complex_Numeral_Simprocs.eval_numerals;
+Addsimprocs Complex_Numeral_Simprocs.cancel_numerals;
+Addsimprocs [Complex_Numeral_Simprocs.combine_numerals];
+
+(*The Abel_Cancel simprocs are now obsolete
+Delsimprocs [Complex_Cancel.sum_conv, Complex_Cancel.rel_conv];
+*)
+
+(*examples:
+print_depth 22;
+set timing;
+set trace_simp;
+fun test s = (Goal s, by (Simp_tac 1));
+
+test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::complex)";
+test " 2*u = (u::complex)";
+test "(i + j + 12 + (k::complex)) - 15 = y";
+test "(i + j + 12 + (k::complex)) - 5 = y";
+
+test "( 2*x - (u*v) + y) - v* 3*u = (w::complex)";
+test "( 2*x*u*v + (u*v)* 4 + y) - v*u* 4 = (w::complex)";
+test "( 2*x*u*v + (u*v)* 4 + y) - v*u = (w::complex)";
+test "u*v - (x*u*v + (u*v)* 4 + y) = (w::complex)";
+
+test "(i + j + 12 + (k::complex)) = u + 15 + y";
+test "(i + j* 2 + 12 + (k::complex)) = j + 5 + y";
+
+test " 2*y + 3*z + 6*w + 2*y + 3*z + 2*u = 2*y' + 3*z' + 6*w' + 2*y' + 3*z' + u + (vv::complex)";
+
+test "a + -(b+c) + b = (d::complex)";
+test "a + -(b+c) - b = (d::complex)";
+
+(*negative numerals*)
+test "(i + j + -2 + (k::complex)) - (u + 5 + y) = zz";
+
+test "(i + j + -12 + (k::complex)) - 15 = y";
+test "(i + j + 12 + (k::complex)) - -15 = y";
+test "(i + j + -12 + (k::complex)) - -15 = y";
+
+*)
+
+
+(** Constant folding for complex plus and times **)
+
+structure Complex_Times_Assoc_Data : ASSOC_FOLD_DATA =
+struct
+ val ss = HOL_ss
+ val eq_reflection = eq_reflection
+ val sg_ref = Sign.self_ref (Theory.sign_of (the_context ()))
+ val T = Complex_Numeral_Simprocs.complexT
+ val plus = Const ("op *", [T,T] ---> T)
+ val add_ac = complex_mult_ac
+end;
+
+structure Complex_Times_Assoc = Assoc_Fold (Complex_Times_Assoc_Data);
+
+Addsimprocs [Complex_Times_Assoc.conv];
+
+Addsimps [complex_of_real_zero_iff];
+
+(*Simplification of x-y = 0 *)
+
+AddIffs [complex_eq_iff_diff_eq_0 RS sym];
+
+(*** Real and imaginary stuff ***)
+
+Goalw [complex_number_of_def]
+ "((number_of xa :: complex) + ii * number_of ya = \
+\ number_of xb + ii * number_of yb) = \
+\ (((number_of xa :: complex) = number_of xb) & \
+\ ((number_of ya :: complex) = number_of yb))";
+by (auto_tac (claset(), HOL_ss addsimps [complex_eq_cancel_iff]));
+qed "complex_number_of_eq_cancel_iff";
+Addsimps [complex_number_of_eq_cancel_iff];
+
+Goalw [complex_number_of_def]
+ "((number_of xa :: complex) + number_of ya * ii = \
+\ number_of xb + number_of yb * ii) = \
+\ (((number_of xa :: complex) = number_of xb) & \
+\ ((number_of ya :: complex) = number_of yb))";
+by (auto_tac (claset(), HOL_ss addsimps [complex_eq_cancel_iffA]));
+qed "complex_number_of_eq_cancel_iffA";
+Addsimps [complex_number_of_eq_cancel_iffA];
+
+Goalw [complex_number_of_def]
+ "((number_of xa :: complex) + number_of ya * ii = \
+\ number_of xb + ii * number_of yb) = \
+\ (((number_of xa :: complex) = number_of xb) & \
+\ ((number_of ya :: complex) = number_of yb))";
+by (auto_tac (claset(), HOL_ss addsimps [complex_eq_cancel_iffB]));
+qed "complex_number_of_eq_cancel_iffB";
+Addsimps [complex_number_of_eq_cancel_iffB];
+
+Goalw [complex_number_of_def]
+ "((number_of xa :: complex) + ii * number_of ya = \
+\ number_of xb + number_of yb * ii) = \
+\ (((number_of xa :: complex) = number_of xb) & \
+\ ((number_of ya :: complex) = number_of yb))";
+by (auto_tac (claset(), HOL_ss addsimps [complex_eq_cancel_iffC]));
+qed "complex_number_of_eq_cancel_iffC";
+Addsimps [complex_number_of_eq_cancel_iffC];
+
+Goalw [complex_number_of_def]
+ "((number_of xa :: complex) + ii * number_of ya = \
+\ number_of xb) = \
+\ (((number_of xa :: complex) = number_of xb) & \
+\ ((number_of ya :: complex) = 0))";
+by (auto_tac (claset(), HOL_ss addsimps [complex_eq_cancel_iff2,
+ complex_of_real_zero_iff]));
+qed "complex_number_of_eq_cancel_iff2";
+Addsimps [complex_number_of_eq_cancel_iff2];
+
+Goalw [complex_number_of_def]
+ "((number_of xa :: complex) + number_of ya * ii = \
+\ number_of xb) = \
+\ (((number_of xa :: complex) = number_of xb) & \
+\ ((number_of ya :: complex) = 0))";
+by (auto_tac (claset(), HOL_ss addsimps [complex_eq_cancel_iff2a,
+ complex_of_real_zero_iff]));
+qed "complex_number_of_eq_cancel_iff2a";
+Addsimps [complex_number_of_eq_cancel_iff2a];
+
+Goalw [complex_number_of_def]
+ "((number_of xa :: complex) + ii * number_of ya = \
+\ ii * number_of yb) = \
+\ (((number_of xa :: complex) = 0) & \
+\ ((number_of ya :: complex) = number_of yb))";
+by (auto_tac (claset(), HOL_ss addsimps [complex_eq_cancel_iff3,
+ complex_of_real_zero_iff]));
+qed "complex_number_of_eq_cancel_iff3";
+Addsimps [complex_number_of_eq_cancel_iff3];
+
+Goalw [complex_number_of_def]
+ "((number_of xa :: complex) + number_of ya * ii= \
+\ ii * number_of yb) = \
+\ (((number_of xa :: complex) = 0) & \
+\ ((number_of ya :: complex) = number_of yb))";
+by (auto_tac (claset(), HOL_ss addsimps [complex_eq_cancel_iff3a,
+ complex_of_real_zero_iff]));
+qed "complex_number_of_eq_cancel_iff3a";
+Addsimps [complex_number_of_eq_cancel_iff3a];
+
+Goalw [complex_number_of_def] "cnj (number_of v :: complex) = number_of v";
+by (rtac complex_cnj_complex_of_real 1);
+qed "complex_number_of_cnj";
+Addsimps [complex_number_of_cnj];
+
+Goalw [complex_number_of_def]
+ "cmod(number_of v :: complex) = abs (number_of v :: real)";
+by (auto_tac (claset(), HOL_ss addsimps [complex_mod_complex_of_real]));
+qed "complex_number_of_cmod";
+Addsimps [complex_number_of_cmod];
+
+Goalw [complex_number_of_def]
+ "Re(number_of v :: complex) = number_of v";
+by (auto_tac (claset(), HOL_ss addsimps [Re_complex_of_real]));
+qed "complex_number_of_Re";
+Addsimps [complex_number_of_Re];
+
+Goalw [complex_number_of_def]
+ "Im(number_of v :: complex) = 0";
+by (auto_tac (claset(), HOL_ss addsimps [Im_complex_of_real]));
+qed "complex_number_of_Im";
+Addsimps [complex_number_of_Im];
+
+Goalw [expi_def]
+ "expi((2::complex) * complex_of_real pi * ii) = 1";
+by (auto_tac (claset(),simpset() addsimps [complex_Re_mult_eq,
+ complex_Im_mult_eq,cis_def]));
+qed "expi_two_pi_i";
+Addsimps [expi_two_pi_i];
+
+(*examples:
+print_depth 22;
+set timing;
+set trace_simp;
+fun test s = (Goal s, by (Simp_tac 1));
+
+test "23 * ii + 45 * ii= (x::complex)";
+
+test "5 * ii + 12 - 45 * ii= (x::complex)";
+test "5 * ii + 40 - 12 * ii + 9 = (x::complex) + 89 * ii";
+test "5 * ii + 40 - 12 * ii + 9 - 78 = (x::complex) + 89 * ii";
+
+test "l + 10 * ii + 90 + 3*l + 9 + 45 * ii= (x::complex)";
+test "87 + 10 * ii + 90 + 3*7 + 9 + 45 * ii= (x::complex)";
+
+
+*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/ComplexBin.thy Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,22 @@
+(* Title: ComplexBin.thy
+ Author: Jacques D. Fleuriot
+ Copyright: 2001 University of Edinburgh
+ Descrition: Binary arithmetic for the complex numbers
+ This case is reduced to that for the reals.
+*)
+
+ComplexBin = Complex +
+
+
+instance
+ complex :: number
+
+instance complex :: plus_ac0(complex_add_commute,complex_add_assoc,complex_add_zero_left)
+
+
+defs
+ complex_number_of_def
+ "number_of v == complex_of_real (number_of v)"
+ (*::bin=>complex ::bin=>complex*)
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/NSCA.ML Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,1451 @@
+(* Title : NSCA.ML
+ Author : Jacques D. Fleuriot
+ Copyright : 2001,2002 University of Edinburgh
+ Description : Infinite, infinitesimal complex number etc!
+*)
+
+(*--------------------------------------------------------------------------------------*)
+(* Closure laws for members of (embedded) set standard complex SComplex *)
+(* -------------------------------------------------------------------------------------*)
+
+Goalw [SComplex_def] "[| (x::hcomplex): SComplex; y: SComplex |] ==> x + y: SComplex";
+by (Step_tac 1);
+by (res_inst_tac [("x","r + ra")] exI 1);
+by (Simp_tac 1);
+qed "SComplex_add";
+
+Goalw [SComplex_def] "[| (x::hcomplex): SComplex; y: SComplex |] ==> x * y: SComplex";
+by (Step_tac 1);
+by (res_inst_tac [("x","r * ra")] exI 1);
+by (Simp_tac 1);
+qed "SComplex_mult";
+
+Goalw [SComplex_def] "x: SComplex ==> inverse x : SComplex";
+by (blast_tac (claset() addIs [hcomplex_of_complex_inverse RS sym]) 1);
+qed "SComplex_inverse";
+
+Goal "[| x: SComplex; y: SComplex |] ==> x/y: SComplex";
+by (asm_simp_tac (simpset() addsimps [SComplex_mult,SComplex_inverse,
+ hcomplex_divide_def]) 1);
+qed "SComplex_divide";
+
+Goalw [SComplex_def] "x: SComplex ==> -x : SComplex";
+by (blast_tac (claset() addIs [hcomplex_of_complex_minus RS sym]) 1);
+qed "SComplex_minus";
+
+Goal "(-x : SComplex) = (x: SComplex)";
+by Auto_tac;
+by (etac SComplex_minus 2);
+by (dtac SComplex_minus 1);
+by Auto_tac;
+qed "SComplex_minus_iff";
+Addsimps [SComplex_minus_iff];
+
+Goal "[| x + y : SComplex; y: SComplex |] ==> x: SComplex";
+by (dres_inst_tac [("x","y")] SComplex_minus 1);
+by (dtac SComplex_add 1);
+by (assume_tac 1);
+by Auto_tac;
+qed "SComplex_add_cancel";
+
+Goalw [hcomplex_of_complex_def]
+ "hcmod (hcomplex_of_complex r) : Reals";
+by (simp_tac (simpset() addsimps [hcmod,SReal_def,
+ hypreal_of_real_def]) 1);
+qed "SReal_hcmod_hcomplex_of_complex";
+Addsimps [SReal_hcmod_hcomplex_of_complex];
+
+Goalw [hcomplex_number_of_def]
+ "hcmod (number_of w ::hcomplex) : Reals";
+by (rtac SReal_hcmod_hcomplex_of_complex 1);
+qed "SReal_hcmod_number_of";
+Addsimps [SReal_hcmod_number_of];
+
+Goalw [SComplex_def] "x: SComplex ==> hcmod x : Reals";
+by Auto_tac;
+qed "SReal_hcmod_SComplex";
+
+Goalw [SComplex_def] "hcomplex_of_complex x: SComplex";
+by (Blast_tac 1);
+qed "SComplex_hcomplex_of_complex";
+Addsimps [SComplex_hcomplex_of_complex];
+
+Goalw [hcomplex_number_of_def] "(number_of w ::hcomplex) : SComplex";
+by (rtac SComplex_hcomplex_of_complex 1);
+qed "SComplex_number_of";
+Addsimps [SComplex_number_of];
+
+Goalw [hcomplex_divide_def] "r : SComplex ==> r/(number_of w::hcomplex) : SComplex";
+by (blast_tac (claset() addSIs [SComplex_number_of, SComplex_mult,
+ SComplex_inverse]) 1);
+qed "SComplex_divide_number_of";
+
+Goalw [SComplex_def] "{x. hcomplex_of_complex x : SComplex} = (UNIV::complex set)";
+by Auto_tac;
+qed "SComplex_UNIV_complex";
+
+Goalw [SComplex_def] "(x: SComplex) = (EX y. x = hcomplex_of_complex y)";
+by Auto_tac;
+qed "SComplex_iff";
+
+Goalw [SComplex_def] "hcomplex_of_complex `(UNIV::complex set) = SComplex";
+by Auto_tac;
+qed "hcomplex_of_complex_image";
+
+Goalw [SComplex_def] "inv hcomplex_of_complex `SComplex = (UNIV::complex set)";
+by Auto_tac;
+by (rtac (inj_hcomplex_of_complex RS inv_f_f RS subst) 1);
+by (Blast_tac 1);
+qed "inv_hcomplex_of_complex_image";
+
+Goalw [SComplex_def]
+ "[| EX x. x: P; P <= SComplex |] ==> EX Q. P = hcomplex_of_complex ` Q";
+by (Best_tac 1);
+qed "SComplex_hcomplex_of_complex_image";
+
+Goal "[| (x::hcomplex): SComplex; y: SComplex; hcmod x < hcmod y \
+\ |] ==> EX r: Reals. hcmod x< r & r < hcmod y";
+by (auto_tac (claset() addIs [SReal_dense], simpset()
+ addsimps [SReal_hcmod_SComplex]));
+qed "SComplex_SReal_dense";
+
+Goalw [SComplex_def,SReal_def]
+ "z : SComplex ==> hcmod z : Reals";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcmod,hypreal_of_real_def,
+ hcomplex_of_complex_def,cmod_def]));
+by (res_inst_tac [("x","cmod r")] exI 1);
+by (ultra_tac (claset(),simpset() addsimps [cmod_def]) 1);
+qed "SComplex_hcmod_SReal";
+
+Goal "0 : SComplex";
+by (auto_tac (claset(),simpset() addsimps [SComplex_def]));
+qed "SComplex_zero";
+Addsimps [SComplex_zero];
+
+Goal "1 : SComplex";
+by (auto_tac (claset(),simpset() addsimps [SComplex_def,hcomplex_of_complex_def,
+ hcomplex_one_def]));
+qed "SComplex_one";
+Addsimps [SComplex_one];
+
+(*
+Goalw [SComplex_def,SReal_def] "hcmod z : Reals ==> z : SComplex";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcmod,hypreal_of_real_def,
+ hcomplex_of_complex_def,cmod_def]));
+*)
+
+(*--------------------------------------------------------------------------------------------*)
+(* Set of finite elements is a subring of the extended complex numbers *)
+(* -------------------------------------------------------------------------------------------*)
+
+Goalw [CFinite_def] "[|x : CFinite; y : CFinite|] ==> (x+y) : CFinite";
+by (blast_tac (claset() addSIs [SReal_add,hcmod_add_less]) 1);
+qed "CFinite_add";
+
+Goalw [CFinite_def] "[|x : CFinite; y : CFinite|] ==> x*y : CFinite";
+by (blast_tac (claset() addSIs [SReal_mult,hcmod_mult_less]) 1);
+qed "CFinite_mult";
+
+Goalw [CFinite_def] "(-x : CFinite) = (x : CFinite)";
+by (Simp_tac 1);
+qed "CFinite_minus_iff";
+Addsimps [CFinite_minus_iff];
+
+Goalw [SComplex_def,CFinite_def] "SComplex <= CFinite";
+by Auto_tac;
+by (res_inst_tac [("x","1 + hcmod(hcomplex_of_complex r)")] bexI 1);
+by (auto_tac (claset() addIs [SReal_add],simpset()));
+qed "SComplex_subset_CFinite";
+Addsimps [ SComplex_subset_CFinite];
+
+Goal "hcmod (hcomplex_of_complex r) : HFinite";
+by (auto_tac (claset() addSIs [ SReal_subset_HFinite RS subsetD],simpset()));
+qed "HFinite_hcmod_hcomplex_of_complex";
+Addsimps [HFinite_hcmod_hcomplex_of_complex];
+
+Goal "hcomplex_of_complex x: CFinite";
+by (auto_tac (claset() addSIs [ SComplex_subset_CFinite RS subsetD],simpset()));
+qed "CFinite_hcomplex_of_complex";
+Addsimps [CFinite_hcomplex_of_complex];
+
+Goalw [CFinite_def] "x : CFinite ==> EX t: Reals. hcmod x < t";
+by Auto_tac;
+qed "CFiniteD";
+
+Goalw [CFinite_def] "(x : CFinite) = (hcmod x : HFinite)";
+by (auto_tac (claset(), simpset() addsimps [HFinite_def]));
+qed "CFinite_hcmod_iff";
+
+Goal "number_of w : CFinite";
+by (rtac (SComplex_number_of RS (SComplex_subset_CFinite RS subsetD)) 1);
+qed "CFinite_number_of";
+Addsimps [CFinite_number_of];
+
+Goal "[|x : CFinite; y <= hcmod x; 0 <= y |] ==> y: HFinite";
+by (auto_tac (claset() addIs [HFinite_bounded],simpset() addsimps
+ [CFinite_hcmod_iff]));
+qed "CFinite_bounded";
+
+(*--------------------------------------------------------------------------------------*)
+(* Set of complex infinitesimals is a subring of the nonstandard complex numbers *)
+(*--------------------------------------------------------------------------------------*)
+
+Goalw [CInfinitesimal_def]
+ "x : CInfinitesimal ==> ALL r: Reals. 0 < r --> hcmod x < r";
+by Auto_tac;
+qed "CInfinitesimalD";
+
+Goalw [CInfinitesimal_def] "0 : CInfinitesimal";
+by Auto_tac;
+qed "CInfinitesimal_zero";
+AddIffs [CInfinitesimal_zero];
+
+Goal "x/(2::hcomplex) + x/(2::hcomplex) = x";
+by Auto_tac;
+qed "hcomplex_sum_of_halves";
+
+Goalw [CInfinitesimal_def,Infinitesimal_def]
+ "(z : CInfinitesimal) = (hcmod z : Infinitesimal)";
+by Auto_tac;
+qed "CInfinitesimal_hcmod_iff";
+
+Goal "1 ~: CInfinitesimal";
+by (simp_tac (simpset() addsimps [CInfinitesimal_hcmod_iff]) 1);
+qed "one_not_CInfinitesimal";
+Addsimps [one_not_CInfinitesimal];
+
+Goal "[| x : CInfinitesimal; y : CInfinitesimal |] ==> (x+y) : CInfinitesimal";
+by (auto_tac (claset(),simpset() addsimps [CInfinitesimal_hcmod_iff]));
+by (rtac hrabs_le_Infinitesimal 1);
+by (res_inst_tac [("y","hcmod y")] Infinitesimal_add 1);
+by Auto_tac;
+qed "CInfinitesimal_add";
+
+Goalw [CInfinitesimal_def] "(-x:CInfinitesimal) = (x:CInfinitesimal)";
+by (Full_simp_tac 1);
+qed "CInfinitesimal_minus_iff";
+Addsimps [CInfinitesimal_minus_iff];
+
+Goal "[| x : CInfinitesimal; y : CInfinitesimal |] ==> x-y : CInfinitesimal";
+by (asm_simp_tac
+ (simpset() addsimps [hcomplex_diff_def, CInfinitesimal_add]) 1);
+qed "CInfinitesimal_diff";
+
+Goal "[| x : CInfinitesimal; y : CInfinitesimal |] ==> (x * y) : CInfinitesimal";
+by (auto_tac (claset() addIs [Infinitesimal_mult],simpset() addsimps
+ [CInfinitesimal_hcmod_iff,hcmod_mult]));
+qed "CInfinitesimal_mult";
+
+Goal "[| x : CInfinitesimal; y : CFinite |] ==> (x * y) : CInfinitesimal";
+by (auto_tac (claset() addIs [Infinitesimal_HFinite_mult],simpset()
+ addsimps [CInfinitesimal_hcmod_iff,CFinite_hcmod_iff,hcmod_mult]));
+qed "CInfinitesimal_CFinite_mult";
+
+Goal "[| x : CInfinitesimal; y : CFinite |] ==> (y * x) : CInfinitesimal";
+by (auto_tac (claset() addDs [CInfinitesimal_CFinite_mult],
+ simpset() addsimps [hcomplex_mult_commute]));
+qed "CInfinitesimal_CFinite_mult2";
+
+Goalw [CInfinite_def,HInfinite_def]
+ "(z : CInfinite) = (hcmod z : HInfinite)";
+by Auto_tac;
+qed "CInfinite_hcmod_iff";
+
+Goal "x: CInfinite ==> inverse x: CInfinitesimal";
+by (auto_tac (claset() addIs [HInfinite_inverse_Infinitesimal],
+ simpset() addsimps [CInfinitesimal_hcmod_iff,
+ CInfinite_hcmod_iff,hcmod_hcomplex_inverse]));
+qed "CInfinite_inverse_CInfinitesimal";
+
+Goal "[|x: CInfinite; y: CInfinite|] ==> (x*y): CInfinite";
+by (auto_tac (claset() addIs [HInfinite_mult],simpset() addsimps
+ [CInfinite_hcmod_iff,hcmod_mult]));
+qed "CInfinite_mult";
+
+Goalw [CInfinite_def] "(-x : CInfinite) = (x : CInfinite)";
+by (Simp_tac 1);
+qed "CInfinite_minus_iff";
+Addsimps [CInfinite_minus_iff];
+
+Goal "[|a: CFinite; b: CFinite; c: CFinite|] \
+\ ==> a*a + b*b + c*c : CFinite";
+by (auto_tac (claset() addIs [CFinite_mult,CFinite_add], simpset()));
+qed "CFinite_sum_squares";
+
+Goal "x ~: CInfinitesimal ==> x ~= 0";
+by Auto_tac;
+qed "not_CInfinitesimal_not_zero";
+
+Goal "x: CFinite - CInfinitesimal ==> x ~= 0";
+by Auto_tac;
+qed "not_CInfinitesimal_not_zero2";
+
+Goal "x : CFinite - CInfinitesimal ==> hcmod x : HFinite - Infinitesimal";
+by (auto_tac (claset(),simpset() addsimps [CFinite_hcmod_iff,CInfinitesimal_hcmod_iff]));
+qed "CFinite_diff_CInfinitesimal_hcmod";
+
+Goal "[| e : CInfinitesimal; hcmod x < hcmod e |] ==> x : CInfinitesimal";
+by (auto_tac (claset() addIs [hrabs_less_Infinitesimal],simpset()
+ addsimps [CInfinitesimal_hcmod_iff]));
+qed "hcmod_less_CInfinitesimal";
+
+Goal "[| e : CInfinitesimal; hcmod x <= hcmod e |] ==> x : CInfinitesimal";
+by (auto_tac (claset() addIs [hrabs_le_Infinitesimal],simpset()
+ addsimps [CInfinitesimal_hcmod_iff]));
+qed "hcmod_le_CInfinitesimal";
+
+Goal "[| e : CInfinitesimal; \
+\ e' : CInfinitesimal; \
+\ hcmod e' < hcmod x ; hcmod x < hcmod e \
+\ |] ==> x : CInfinitesimal";
+by (auto_tac (claset() addIs [Infinitesimal_interval],simpset()
+ addsimps [CInfinitesimal_hcmod_iff]));
+qed "CInfinitesimal_interval";
+
+Goal "[| e : CInfinitesimal; \
+\ e' : CInfinitesimal; \
+\ hcmod e' <= hcmod x ; hcmod x <= hcmod e \
+\ |] ==> x : CInfinitesimal";
+by (auto_tac (claset() addIs [Infinitesimal_interval2],simpset()
+ addsimps [CInfinitesimal_hcmod_iff]));
+qed "CInfinitesimal_interval2";
+
+Goal "[| x ~: CInfinitesimal; y ~: CInfinitesimal|] ==> (x*y) ~: CInfinitesimal";
+by (auto_tac (claset(),simpset() addsimps [CInfinitesimal_hcmod_iff,hcmod_mult]));
+by (dtac not_Infinitesimal_mult 1);
+by Auto_tac;
+qed "not_CInfinitesimal_mult";
+
+Goal "x*y : CInfinitesimal ==> x : CInfinitesimal | y : CInfinitesimal";
+by (auto_tac (claset() addDs [Infinitesimal_mult_disj],simpset() addsimps
+ [CInfinitesimal_hcmod_iff,hcmod_mult]));
+qed "CInfinitesimal_mult_disj";
+
+Goal "[| x : CFinite - CInfinitesimal; \
+\ y : CFinite - CInfinitesimal \
+\ |] ==> (x*y) : CFinite - CInfinitesimal";
+by (Clarify_tac 1);
+by (blast_tac (claset() addDs [CFinite_mult,not_CInfinitesimal_mult]) 1);
+qed "CFinite_CInfinitesimal_diff_mult";
+
+Goal "CInfinitesimal <= CFinite";
+by (auto_tac (claset() addIs [Infinitesimal_subset_HFinite RS subsetD],
+ simpset() addsimps [CInfinitesimal_hcmod_iff,CFinite_hcmod_iff]));
+qed "CInfinitesimal_subset_CFinite";
+
+Goal "x: CInfinitesimal ==> x * hcomplex_of_complex r : CInfinitesimal";
+by (auto_tac (claset() addSIs [Infinitesimal_HFinite_mult],
+ simpset() addsimps [CInfinitesimal_hcmod_iff,hcmod_mult]));
+qed "CInfinitesimal_hcomplex_of_complex_mult";
+
+Goal "x: CInfinitesimal ==> hcomplex_of_complex r * x: CInfinitesimal";
+by (auto_tac (claset() addSIs [Infinitesimal_HFinite_mult2],
+ simpset() addsimps [CInfinitesimal_hcmod_iff,hcmod_mult]));
+qed "CInfinitesimal_hcomplex_of_complex_mult2";
+
+
+(*--------------------------------------------------------------------------------------*)
+(* Infinitely close relation @c= *)
+(* -------------------------------------------------------------------------------------*)
+
+(*
+Goalw [capprox_def,approx_def] "(z @c= w) = (hcmod z @= hcmod w)";
+by (auto_tac (claset(),simpset() addsimps [CInfinitesimal_hcmod_iff]));
+*)
+
+Goal "x:CInfinitesimal = (x @c= 0)";
+by (simp_tac (simpset() addsimps [CInfinitesimal_hcmod_iff,
+ capprox_def]) 1);
+qed "mem_cinfmal_iff";
+
+Goalw [capprox_def,hcomplex_diff_def]
+ " (x @c= y) = (x + -y @c= 0)";
+by (Simp_tac 1);
+qed "capprox_minus_iff";
+
+Goalw [capprox_def,hcomplex_diff_def]
+ " (x @c= y) = (-y + x @c= 0)";
+by (simp_tac (simpset() addsimps [hcomplex_add_commute]) 1);
+qed "capprox_minus_iff2";
+
+Goalw [capprox_def] "x @c= x";
+by (Simp_tac 1);
+qed "capprox_refl";
+Addsimps [capprox_refl];
+
+Goalw [capprox_def,CInfinitesimal_def]
+ "x @c= y ==> y @c= x";
+by (auto_tac (claset() addSDs [bspec],simpset() addsimps
+ [hcmod_diff_commute]));
+qed "capprox_sym";
+
+Goalw [capprox_def] "[| x @c= y; y @c= z |] ==> x @c= z";
+by (dtac CInfinitesimal_add 1);
+by (assume_tac 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_def]));
+qed "capprox_trans";
+
+Goal "[| r @c= x; s @c= x |] ==> r @c= s";
+by (blast_tac (claset() addIs [capprox_sym, capprox_trans]) 1);
+qed "capprox_trans2";
+
+Goal "[| x @c= r; x @c= s|] ==> r @c= s";
+by (blast_tac (claset() addIs [capprox_sym, capprox_trans]) 1);
+qed "capprox_trans3";
+
+Goal "(number_of w @c= x) = (x @c= number_of w)";
+by (blast_tac (claset() addIs [capprox_sym]) 1);
+qed "number_of_capprox_reorient";
+Addsimps [number_of_capprox_reorient];
+
+Goal "(x-y : CInfinitesimal) = (x @c= y)";
+by (auto_tac (claset(),
+ simpset() addsimps [hcomplex_diff_def, capprox_minus_iff RS sym,
+ mem_cinfmal_iff]));
+qed "CInfinitesimal_capprox_minus";
+
+Goalw [cmonad_def] "(x @c= y) = (cmonad(x)=cmonad(y))";
+by (auto_tac (claset() addDs [capprox_sym]
+ addSEs [capprox_trans,equalityCE],
+ simpset()));
+qed "capprox_monad_iff";
+
+Goal "[| x: CInfinitesimal; y: CInfinitesimal |] ==> x @c= y";
+by (asm_full_simp_tac (simpset() addsimps [mem_cinfmal_iff]) 1);
+by (blast_tac (claset() addIs [capprox_trans, capprox_sym]) 1);
+qed "Infinitesimal_capprox";
+
+val prem1::prem2::rest =
+goalw thy [capprox_def,hcomplex_diff_def]
+ "[| a @c= b; c @c= d |] ==> a+c @c= b+d";
+by (rtac (hcomplex_minus_add_distrib RS ssubst) 1);
+by (rtac (hcomplex_add_assoc RS ssubst) 1);
+by (res_inst_tac [("y1","c")] (hcomplex_add_left_commute RS subst) 1);
+by (rtac (hcomplex_add_assoc RS subst) 1);
+by (rtac ([prem1,prem2] MRS CInfinitesimal_add) 1);
+qed "capprox_add";
+
+Goal "a @c= b ==> -a @c= -b";
+by (rtac ((capprox_minus_iff RS iffD2) RS capprox_sym) 1);
+by (dtac (capprox_minus_iff RS iffD1) 1);
+by (simp_tac (simpset() addsimps [hcomplex_add_commute]) 1);
+qed "capprox_minus";
+
+Goal "-a @c= -b ==> a @c= b";
+by (auto_tac (claset() addDs [capprox_minus], simpset()));
+qed "capprox_minus2";
+
+Goal "(-a @c= -b) = (a @c= b)";
+by (blast_tac (claset() addIs [capprox_minus,capprox_minus2]) 1);
+qed "capprox_minus_cancel";
+Addsimps [capprox_minus_cancel];
+
+Goal "[| a @c= b; c @c= d |] ==> a + -c @c= b + -d";
+by (blast_tac (claset() addSIs [capprox_add,capprox_minus]) 1);
+qed "capprox_add_minus";
+
+Goalw [capprox_def,hcomplex_diff_def]
+ "[| a @c= b; c: CFinite|] ==> a*c @c= b*c";
+by (asm_full_simp_tac (simpset() addsimps [CInfinitesimal_CFinite_mult,
+ hcomplex_minus_mult_eq1,hcomplex_add_mult_distrib RS sym]
+ delsimps [hcomplex_minus_mult_eq1 RS sym]) 1);
+qed "capprox_mult1";
+
+Goal "[|a @c= b; c: CFinite|] ==> c*a @c= c*b";
+by (asm_simp_tac (simpset() addsimps [capprox_mult1,hcomplex_mult_commute]) 1);
+qed "capprox_mult2";
+
+Goal "[|u @c= v*x; x @c= y; v: CFinite|] ==> u @c= v*y";
+by (fast_tac (claset() addIs [capprox_mult2,capprox_trans]) 1);
+qed "capprox_mult_subst";
+
+Goal "[| u @c= x*v; x @c= y; v: CFinite |] ==> u @c= y*v";
+by (fast_tac (claset() addIs [capprox_mult1,capprox_trans]) 1);
+qed "capprox_mult_subst2";
+
+Goal "[| u @c= x*hcomplex_of_complex v; x @c= y |] ==> u @c= y*hcomplex_of_complex v";
+by (auto_tac (claset() addIs [capprox_mult_subst2], simpset()));
+qed "capprox_mult_subst_SComplex";
+
+Goalw [capprox_def] "a = b ==> a @c= b";
+by (Asm_simp_tac 1);
+qed "capprox_eq_imp";
+
+Goal "x: CInfinitesimal ==> -x @c= x";
+by (fast_tac (HOL_cs addIs [CInfinitesimal_minus_iff RS iffD2,
+ mem_cinfmal_iff RS iffD1,capprox_trans2]) 1);
+qed "CInfinitesimal_minus_capprox";
+
+Goalw [capprox_def]
+ "(EX y: CInfinitesimal. x - z = y) = (x @c= z)";
+by (Blast_tac 1);
+qed "bex_CInfinitesimal_iff";
+
+Goal "(EX y: CInfinitesimal. x = z + y) = (x @c= z)";
+by (asm_full_simp_tac (simpset() addsimps [bex_CInfinitesimal_iff RS sym]) 1);
+by (Force_tac 1);
+qed "bex_CInfinitesimal_iff2";
+
+Goal "[| y: CInfinitesimal; x + y = z |] ==> x @c= z";
+by (rtac (bex_CInfinitesimal_iff RS iffD1) 1);
+by (dtac (CInfinitesimal_minus_iff RS iffD2) 1);
+by (auto_tac (claset(), simpset() addsimps [hcomplex_add_assoc RS sym]));
+qed "CInfinitesimal_add_capprox";
+
+Goal "y: CInfinitesimal ==> x @c= x + y";
+by (rtac (bex_CInfinitesimal_iff RS iffD1) 1);
+by (dtac (CInfinitesimal_minus_iff RS iffD2) 1);
+by (auto_tac (claset(), simpset() addsimps [hcomplex_add_assoc RS sym]));
+qed "CInfinitesimal_add_capprox_self";
+
+Goal "y: CInfinitesimal ==> x @c= y + x";
+by (auto_tac (claset() addDs [CInfinitesimal_add_capprox_self],
+ simpset() addsimps [hcomplex_add_commute]));
+qed "CInfinitesimal_add_capprox_self2";
+
+Goal "y: CInfinitesimal ==> x @c= x + -y";
+by (blast_tac (claset() addSIs [CInfinitesimal_add_capprox_self,
+ CInfinitesimal_minus_iff RS iffD2]) 1);
+qed "CInfinitesimal_add_minus_capprox_self";
+
+Goal "[| y: CInfinitesimal; x+y @c= z|] ==> x @c= z";
+by (dres_inst_tac [("x","x")] (CInfinitesimal_add_capprox_self RS capprox_sym) 1);
+by (etac (capprox_trans3 RS capprox_sym) 1);
+by (assume_tac 1);
+qed "CInfinitesimal_add_cancel";
+
+Goal "[| y: CInfinitesimal; x @c= z + y|] ==> x @c= z";
+by (dres_inst_tac [("x","z")] (CInfinitesimal_add_capprox_self2 RS capprox_sym) 1);
+by (etac (capprox_trans3 RS capprox_sym) 1);
+by (asm_full_simp_tac (simpset() addsimps [hcomplex_add_commute]) 1);
+by (etac capprox_sym 1);
+qed "CInfinitesimal_add_right_cancel";
+
+Goal "d + b @c= d + c ==> b @c= c";
+by (dtac (capprox_minus_iff RS iffD1) 1);
+by (asm_full_simp_tac (simpset() addsimps
+ [hcomplex_minus_add_distrib,capprox_minus_iff RS sym]
+ @ hcomplex_add_ac) 1);
+qed "capprox_add_left_cancel";
+
+Goal "b + d @c= c + d ==> b @c= c";
+by (rtac capprox_add_left_cancel 1);
+by (asm_full_simp_tac (simpset() addsimps
+ [hcomplex_add_commute]) 1);
+qed "capprox_add_right_cancel";
+
+Goal "b @c= c ==> d + b @c= d + c";
+by (rtac (capprox_minus_iff RS iffD2) 1);
+by (asm_full_simp_tac (simpset() addsimps
+ [capprox_minus_iff RS sym] @ hcomplex_add_ac) 1);
+qed "capprox_add_mono1";
+
+Goal "b @c= c ==> b + a @c= c + a";
+by (asm_simp_tac (simpset() addsimps
+ [hcomplex_add_commute,capprox_add_mono1]) 1);
+qed "capprox_add_mono2";
+
+Goal "(a + b @c= a + c) = (b @c= c)";
+by (fast_tac (claset() addEs [capprox_add_left_cancel,
+ capprox_add_mono1]) 1);
+qed "capprox_add_left_iff";
+
+AddIffs [capprox_add_left_iff];
+
+
+Goal "(b + a @c= c + a) = (b @c= c)";
+by (simp_tac (simpset() addsimps [hcomplex_add_commute]) 1);
+qed "capprox_add_right_iff";
+
+AddIffs [capprox_add_right_iff];
+
+Goal "[| x: CFinite; x @c= y |] ==> y: CFinite";
+by (dtac (bex_CInfinitesimal_iff2 RS iffD2) 1);
+by (Step_tac 1);
+by (dtac (CInfinitesimal_subset_CFinite RS subsetD
+ RS (CFinite_minus_iff RS iffD2)) 1);
+by (dtac CFinite_add 1);
+by (assume_tac 1 THEN Auto_tac);
+qed "capprox_CFinite";
+
+Goal "x @c= hcomplex_of_complex D ==> x: CFinite";
+by (rtac (capprox_sym RSN (2,capprox_CFinite)) 1);
+by Auto_tac;
+qed "capprox_hcomplex_of_complex_CFinite";
+
+Goal "[|a @c= b; c @c= d; b: CFinite; d: CFinite|] ==> a*c @c= b*d";
+by (rtac capprox_trans 1);
+by (rtac capprox_mult2 2);
+by (rtac capprox_mult1 1);
+by (blast_tac (claset() addIs [capprox_CFinite, capprox_sym]) 2);
+by Auto_tac;
+qed "capprox_mult_CFinite";
+
+Goal "[|a @c= hcomplex_of_complex b; c @c= hcomplex_of_complex d |] \
+\ ==> a*c @c= hcomplex_of_complex b * hcomplex_of_complex d";
+by (blast_tac (claset() addSIs [capprox_mult_CFinite,
+ capprox_hcomplex_of_complex_CFinite,CFinite_hcomplex_of_complex]) 1);
+qed "capprox_mult_hcomplex_of_complex";
+
+Goal "[| a: SComplex; a ~= 0; a*x @c= 0 |] ==> x @c= 0";
+by (dtac (SComplex_inverse RS (SComplex_subset_CFinite RS subsetD)) 1);
+by (auto_tac (claset() addDs [capprox_mult2],
+ simpset() addsimps [hcomplex_mult_assoc RS sym]));
+qed "capprox_SComplex_mult_cancel_zero";
+
+Goal "[| a: SComplex; x @c= 0 |] ==> x*a @c= 0";
+by (auto_tac (claset() addDs [(SComplex_subset_CFinite RS subsetD),
+ capprox_mult1], simpset()));
+qed "capprox_mult_SComplex1";
+
+Goal "[| a: SComplex; x @c= 0 |] ==> a*x @c= 0";
+by (auto_tac (claset() addDs [(SComplex_subset_CFinite RS subsetD),
+ capprox_mult2], simpset()));
+qed "capprox_mult_SComplex2";
+
+Goal "[|a : SComplex; a ~= 0 |] ==> (a*x @c= 0) = (x @c= 0)";
+by (blast_tac (claset() addIs [capprox_SComplex_mult_cancel_zero,
+ capprox_mult_SComplex2]) 1);
+qed "capprox_mult_SComplex_zero_cancel_iff";
+Addsimps [capprox_mult_SComplex_zero_cancel_iff];
+
+Goal "[| a: SComplex; a ~= 0; a* w @c= a*z |] ==> w @c= z";
+by (dtac (SComplex_inverse RS (SComplex_subset_CFinite RS subsetD)) 1);
+by (auto_tac (claset() addDs [capprox_mult2],
+ simpset() addsimps [hcomplex_mult_assoc RS sym]));
+qed "capprox_SComplex_mult_cancel";
+
+Goal "[| a: SComplex; a ~= 0|] ==> (a* w @c= a*z) = (w @c= z)";
+by (auto_tac (claset() addSIs [capprox_mult2,SComplex_subset_CFinite RS subsetD]
+ addIs [capprox_SComplex_mult_cancel], simpset()));
+qed "capprox_SComplex_mult_cancel_iff1";
+Addsimps [capprox_SComplex_mult_cancel_iff1];
+
+Goal "(x @c= y) = (hcmod (y - x) @= 0)";
+by (rtac (capprox_minus_iff RS ssubst) 1);
+by (auto_tac (claset(),simpset() addsimps [capprox_def,
+ CInfinitesimal_hcmod_iff,mem_infmal_iff,symmetric hcomplex_diff_def,
+ hcmod_diff_commute]));
+qed "capprox_hcmod_approx_zero";
+
+Goal "(x @c= 0) = (hcmod x @= 0)";
+by (auto_tac (claset(),simpset() addsimps
+ [capprox_hcmod_approx_zero]));
+qed "capprox_approx_zero_iff";
+
+Goal "(-x @c= 0) = (x @c= 0)";
+by (auto_tac (claset(),simpset() addsimps
+ [capprox_hcmod_approx_zero]));
+qed "capprox_minus_zero_cancel_iff";
+Addsimps [capprox_minus_zero_cancel_iff];
+
+Goal "u @c= 0 ==> hcmod(x + u) - hcmod x : Infinitesimal";
+by (res_inst_tac [("e","hcmod u"),("e'","- hcmod u")] Infinitesimal_interval2 1);
+by (auto_tac (claset() addDs [capprox_approx_zero_iff RS iffD1],
+ simpset() addsimps [mem_infmal_iff RS sym,hypreal_diff_def]));
+by (res_inst_tac [("C","hcmod x")] hypreal_le_add_left_cancel 1);
+by (auto_tac (claset(),simpset() addsimps [symmetric hypreal_diff_def]));
+qed "Infinitesimal_hcmod_add_diff";
+
+Goal "u @c= 0 ==> hcmod(x + u) @= hcmod x";
+by (rtac (approx_minus_iff RS iffD2) 1);
+by (auto_tac (claset() addIs [Infinitesimal_hcmod_add_diff],
+ simpset() addsimps [mem_infmal_iff RS sym,symmetric hypreal_diff_def]));
+qed "approx_hcmod_add_hcmod";
+
+Goal "x @c= y ==> hcmod x @= hcmod y";
+by (auto_tac (claset() addIs [approx_hcmod_add_hcmod]
+ addSDs [bex_CInfinitesimal_iff2 RS iffD2],simpset() addsimps [mem_cinfmal_iff]));
+qed "capprox_hcmod_approx";
+
+(*--------------------------------------------------------------------------------------*)
+(* zero is the only complex number that is also infinitesimal *)
+(*--------------------------------------------------------------------------------------*)
+
+Goal "[| x: SComplex; y: CInfinitesimal; 0 < hcmod x |] ==> hcmod y < hcmod x";
+by (auto_tac (claset() addSIs [Infinitesimal_less_SReal,SComplex_hcmod_SReal],
+ simpset() addsimps [CInfinitesimal_hcmod_iff]));
+qed "CInfinitesimal_less_SComplex";
+
+Goal "y: CInfinitesimal ==> ALL r: SComplex. 0 < hcmod r --> hcmod y < hcmod r";
+by (blast_tac (claset() addIs [CInfinitesimal_less_SComplex]) 1);
+qed "CInfinitesimal_less_SComplex2";
+
+Goal "SComplex Int CInfinitesimal = {0}";
+by (auto_tac (claset(),simpset() addsimps [SComplex_def,CInfinitesimal_hcmod_iff]));
+by (cut_inst_tac [("r","r")] SReal_hcmod_hcomplex_of_complex 1);
+by (dres_inst_tac [("A","Reals")] IntI 1 THEN assume_tac 1);
+by (subgoal_tac "hcmod (hcomplex_of_complex r) = 0" 1);
+by (Asm_full_simp_tac 1);
+by (cut_facts_tac [SReal_Int_Infinitesimal_zero] 1);
+by (rotate_tac 2 1);
+by (Asm_full_simp_tac 1);
+qed "SComplex_Int_CInfinitesimal_zero";
+
+Goal "[| x: SComplex; x: CInfinitesimal|] ==> x = 0";
+by (cut_facts_tac [SComplex_Int_CInfinitesimal_zero] 1);
+by (Blast_tac 1);
+qed "SComplex_CInfinitesimal_zero";
+
+Goal "[| x : SComplex; x ~= 0 |] ==> x : CFinite - CInfinitesimal";
+by (auto_tac (claset() addDs [SComplex_CInfinitesimal_zero,
+ SComplex_subset_CFinite RS subsetD],
+ simpset()));
+qed "SComplex_CFinite_diff_CInfinitesimal";
+
+Goal "hcomplex_of_complex x ~= 0 ==> hcomplex_of_complex x : CFinite - CInfinitesimal";
+by (rtac SComplex_CFinite_diff_CInfinitesimal 1);
+by Auto_tac;
+qed "hcomplex_of_complex_CFinite_diff_CInfinitesimal";
+
+Goal "(hcomplex_of_complex x : CInfinitesimal) = (x=0)";
+by (auto_tac (claset(), simpset() addsimps [hcomplex_of_complex_zero]));
+by (rtac ccontr 1);
+by (rtac (hcomplex_of_complex_CFinite_diff_CInfinitesimal RS DiffD2) 1);
+by Auto_tac;
+qed "hcomplex_of_complex_CInfinitesimal_iff_0";
+AddIffs [hcomplex_of_complex_CInfinitesimal_iff_0];
+
+Goal "number_of w ~= (0::hcomplex) ==> number_of w ~: CInfinitesimal";
+by (fast_tac (claset() addDs [SComplex_number_of RS SComplex_CInfinitesimal_zero]) 1);
+qed "number_of_not_CInfinitesimal";
+Addsimps [number_of_not_CInfinitesimal];
+
+Goal "[| y: SComplex; x @c= y; y~= 0 |] ==> x ~= 0";
+by (auto_tac (claset() addDs [SComplex_CInfinitesimal_zero,
+ capprox_sym RS (mem_cinfmal_iff RS iffD2)],simpset()));
+qed "capprox_SComplex_not_zero";
+
+Goal "[| x @c= y; y : CFinite - CInfinitesimal |] \
+\ ==> x : CFinite - CInfinitesimal";
+by (auto_tac (claset() addIs [capprox_sym RSN (2,capprox_CFinite)],
+ simpset() addsimps [mem_cinfmal_iff]));
+by (dtac capprox_trans3 1 THEN assume_tac 1);
+by (blast_tac (claset() addDs [capprox_sym]) 1);
+qed "CFinite_diff_CInfinitesimal_capprox";
+
+Goal "[| y ~= 0; y: CInfinitesimal; x/y : CFinite |] ==> x : CInfinitesimal";
+by (dtac CInfinitesimal_CFinite_mult2 1);
+by (assume_tac 1);
+by (asm_full_simp_tac
+ (simpset() addsimps [hcomplex_divide_def, hcomplex_mult_assoc]) 1);
+qed "CInfinitesimal_ratio";
+
+Goal "[|x: SComplex; y: SComplex|] ==> (x @c= y) = (x = y)";
+by Auto_tac;
+by (rewrite_goals_tac [capprox_def]);
+by (dres_inst_tac [("x","y")] SComplex_minus 1);
+by (dtac SComplex_add 1 THEN assume_tac 1);
+by (rtac (CLAIM "x - y = 0 ==> x = (y::hcomplex)") 1);
+by (rtac SComplex_CInfinitesimal_zero 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_def]));
+qed "SComplex_capprox_iff";
+
+Goal "(number_of v @c= number_of w) = (number_of v = (number_of w :: hcomplex))";
+by (rtac SComplex_capprox_iff 1);
+by Auto_tac;
+qed "number_of_capprox_iff";
+Addsimps [number_of_capprox_iff];
+
+Goal "(number_of w : CInfinitesimal) = (number_of w = (0::hcomplex))";
+by (rtac iffI 1);
+by (fast_tac (claset() addDs [SComplex_number_of RS SComplex_CInfinitesimal_zero]) 1);
+by (Asm_simp_tac 1);
+qed "number_of_CInfinitesimal_iff";
+Addsimps [number_of_CInfinitesimal_iff];
+
+Goal "(hcomplex_of_complex k @c= hcomplex_of_complex m) = (k = m)";
+by Auto_tac;
+by (rtac (inj_hcomplex_of_complex RS injD) 1);
+by (rtac (SComplex_capprox_iff RS iffD1) 1);
+by Auto_tac;
+qed "hcomplex_of_complex_approx_iff";
+Addsimps [hcomplex_of_complex_approx_iff];
+
+Goal "(hcomplex_of_complex k @c= number_of w) = (k = number_of w)";
+by (stac (hcomplex_of_complex_approx_iff RS sym) 1);
+by Auto_tac;
+qed "hcomplex_of_complex_capprox_number_of_iff";
+Addsimps [hcomplex_of_complex_capprox_number_of_iff];
+
+Goal "[| r: SComplex; s: SComplex; r @c= x; s @c= x|] ==> r = s";
+by (blast_tac (claset() addIs [(SComplex_capprox_iff RS iffD1),
+ capprox_trans2]) 1);
+qed "capprox_unique_complex";
+
+Goal "Abs_hcomplex(hcomplexrel ``{%n. X n}) @c= Abs_hcomplex(hcomplexrel``{%n. Y n}) \
+\ ==> Abs_hypreal(hyprel `` {%n. Re(X n)}) @= \
+\ Abs_hypreal(hyprel `` {%n. Re(Y n)})";
+by (auto_tac (claset(),simpset() addsimps [approx_FreeUltrafilterNat_iff]));
+by (dtac (capprox_minus_iff RS iffD1) 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_minus,hcomplex_add,
+ mem_cinfmal_iff RS sym,CInfinitesimal_hcmod_iff,hcmod,
+ Infinitesimal_FreeUltrafilterNat_iff2]));
+by (dres_inst_tac [("x","m")] spec 1);
+by (Ultra_tac 1);
+by (res_inst_tac [("z","X x")] eq_Abs_complex 1);
+by (res_inst_tac [("z","Y x")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_minus,complex_add,
+ complex_mod] delsimps [realpow_Suc]));
+by (rtac order_le_less_trans 1 THEN assume_tac 2);
+by (dres_inst_tac [("t","Ya x")] sym 1);
+by (auto_tac (claset(),simpset() addsimps [abs_eqI1] delsimps [realpow_Suc]));
+qed "hcomplex_capproxD1";
+
+(* same proof *)
+Goal "Abs_hcomplex(hcomplexrel ``{%n. X n}) @c= Abs_hcomplex(hcomplexrel``{%n. Y n}) \
+\ ==> Abs_hypreal(hyprel `` {%n. Im(X n)}) @= \
+\ Abs_hypreal(hyprel `` {%n. Im(Y n)})";
+by (auto_tac (claset(),simpset() addsimps [approx_FreeUltrafilterNat_iff]));
+by (dtac (capprox_minus_iff RS iffD1) 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_minus,hcomplex_add,
+ mem_cinfmal_iff RS sym,CInfinitesimal_hcmod_iff,hcmod,
+ Infinitesimal_FreeUltrafilterNat_iff2]));
+by (dres_inst_tac [("x","m")] spec 1);
+by (Ultra_tac 1);
+by (res_inst_tac [("z","X x")] eq_Abs_complex 1);
+by (res_inst_tac [("z","Y x")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_minus,complex_add,
+ complex_mod] delsimps [realpow_Suc]));
+by (rtac order_le_less_trans 1 THEN assume_tac 2);
+by (dres_inst_tac [("t","Ya x")] sym 1);
+by (auto_tac (claset(),simpset() addsimps [abs_eqI1] delsimps [realpow_Suc]));
+qed "hcomplex_capproxD2";
+
+Goal "[| Abs_hypreal(hyprel `` {%n. Re(X n)}) @= \
+\ Abs_hypreal(hyprel `` {%n. Re(Y n)}); \
+\ Abs_hypreal(hyprel `` {%n. Im(X n)}) @= \
+\ Abs_hypreal(hyprel `` {%n. Im(Y n)}) \
+\ |] ==> Abs_hcomplex(hcomplexrel ``{%n. X n}) @c= Abs_hcomplex(hcomplexrel``{%n. Y n})";
+by (dtac (approx_minus_iff RS iffD1) 1);
+by (dtac (approx_minus_iff RS iffD1) 1);
+by (rtac (capprox_minus_iff RS iffD2) 1);
+by (auto_tac (claset(),simpset() addsimps [mem_cinfmal_iff RS sym,
+ mem_infmal_iff RS sym,hypreal_minus,hypreal_add,hcomplex_minus,
+ hcomplex_add,CInfinitesimal_hcmod_iff,hcmod,Infinitesimal_FreeUltrafilterNat_iff]));
+by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
+by Auto_tac;
+by (dres_inst_tac [("x","u/2")] spec 1);
+by (dres_inst_tac [("x","u/2")] spec 1);
+by (Step_tac 1);
+by (TRYALL(Force_tac));
+by (ultra_tac (claset(),HOL_ss) 1);
+by (dtac sym 1 THEN dtac sym 1);
+by (res_inst_tac [("z","X x")] eq_Abs_complex 1);
+by (res_inst_tac [("z","Y x")] eq_Abs_complex 1);
+by (auto_tac (claset(),HOL_ss addsimps [complex_minus,complex_add,
+ complex_mod,pair_mem_complex RS Abs_complex_inverse,snd_conv,
+ fst_conv,two_eq_Suc_Suc]));
+by (rtac (realpow_two_abs RS subst) 1);
+by (res_inst_tac [("x1","xa + - xb")] (realpow_two_abs RS subst) 1);
+by (simp_tac (simpset() addsimps [two_eq_Suc_Suc RS sym]) 1);
+by (rtac lemma_sqrt_hcomplex_capprox 1);
+by Auto_tac;
+qed "hcomplex_capproxI";
+
+Goal "(Abs_hcomplex(hcomplexrel ``{%n. X n}) @c= Abs_hcomplex(hcomplexrel``{%n. Y n})) =\
+\ (Abs_hypreal(hyprel `` {%n. Re(X n)}) @= Abs_hypreal(hyprel `` {%n. Re(Y n)}) & \
+\ Abs_hypreal(hyprel `` {%n. Im(X n)}) @= Abs_hypreal(hyprel `` {%n. Im(Y n)}))";
+by (blast_tac (claset() addIs [hcomplex_capproxI,hcomplex_capproxD1,hcomplex_capproxD2]) 1);
+qed "capprox_approx_iff";
+
+Goal "(hcomplex_of_hypreal x @c= hcomplex_of_hypreal z) = (x @= z)";
+by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","z")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,
+ capprox_approx_iff]));
+qed "hcomplex_of_hypreal_capprox_iff";
+Addsimps [hcomplex_of_hypreal_capprox_iff];
+
+Goal "Abs_hcomplex(hcomplexrel ``{%n. X n}) : CFinite \
+\ ==> Abs_hypreal(hyprel `` {%n. Re(X n)}) : HFinite";
+by (auto_tac (claset(),simpset() addsimps [CFinite_hcmod_iff,
+ hcmod,HFinite_FreeUltrafilterNat_iff]));
+by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
+by (res_inst_tac [("x","u")] exI 1 THEN Auto_tac);
+by (Ultra_tac 1);
+by (dtac sym 1 THEN res_inst_tac [("z","X x")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mod,two_eq_Suc_Suc] delsimps [realpow_Suc]));
+by (rtac ccontr 1 THEN dtac real_leI 1);
+by (dtac order_less_le_trans 1 THEN assume_tac 1);
+by (dtac (real_sqrt_ge_abs1 RSN (2,order_less_le_trans)) 1);
+by (auto_tac (claset(),simpset() addsimps [two_eq_Suc_Suc RS sym]));
+qed "CFinite_HFinite_Re";
+
+Goal "Abs_hcomplex(hcomplexrel ``{%n. X n}) : CFinite \
+\ ==> Abs_hypreal(hyprel `` {%n. Im(X n)}) : HFinite";
+by (auto_tac (claset(),simpset() addsimps [CFinite_hcmod_iff,
+ hcmod,HFinite_FreeUltrafilterNat_iff]));
+by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
+by (res_inst_tac [("x","u")] exI 1 THEN Auto_tac);
+by (Ultra_tac 1);
+by (dtac sym 1 THEN res_inst_tac [("z","X x")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mod] delsimps [realpow_Suc]));
+by (rtac ccontr 1 THEN dtac real_leI 1);
+by (dtac order_less_le_trans 1 THEN assume_tac 1);
+by (dtac (real_sqrt_ge_abs2 RSN (2,order_less_le_trans)) 1);
+by Auto_tac;
+qed "CFinite_HFinite_Im";
+
+Goal "[| Abs_hypreal(hyprel `` {%n. Re(X n)}) : HFinite; \
+\ Abs_hypreal(hyprel `` {%n. Im(X n)}) : HFinite \
+\ |] ==> Abs_hcomplex(hcomplexrel ``{%n. X n}) : CFinite";
+by (auto_tac (claset(),simpset() addsimps [CFinite_hcmod_iff,
+ hcmod,HFinite_FreeUltrafilterNat_iff]));
+by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
+by (res_inst_tac [("x","2*(u + ua)")] exI 1);
+by (Ultra_tac 1);
+by (dtac sym 1 THEN res_inst_tac [("z","X x")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_mod,two_eq_Suc_Suc] delsimps [realpow_Suc]));
+by (subgoal_tac "0 < u" 1 THEN arith_tac 2);
+by (subgoal_tac "0 < ua" 1 THEN arith_tac 2);
+by (rtac (realpow_two_abs RS subst) 1);
+by (res_inst_tac [("x1","Y x")] (realpow_two_abs RS subst) 1);
+by (simp_tac (simpset() addsimps [two_eq_Suc_Suc RS sym]) 1);
+by (rtac lemma_sqrt_hcomplex_capprox 1);
+by Auto_tac;
+qed "HFinite_Re_Im_CFinite";
+
+Goal "(Abs_hcomplex(hcomplexrel ``{%n. X n}) : CFinite) = \
+\ (Abs_hypreal(hyprel `` {%n. Re(X n)}) : HFinite & \
+\ Abs_hypreal(hyprel `` {%n. Im(X n)}) : HFinite)";
+by (blast_tac (claset() addIs [HFinite_Re_Im_CFinite,CFinite_HFinite_Im,
+ CFinite_HFinite_Re]) 1);
+qed "CFinite_HFinite_iff";
+
+Goal "Abs_hcomplex(hcomplexrel ``{%n. X n}) : SComplex \
+\ ==> Abs_hypreal(hyprel `` {%n. Re(X n)}) : Reals";
+by (auto_tac (claset(),simpset() addsimps [SComplex_def,
+ hcomplex_of_complex_def,SReal_def,hypreal_of_real_def]));
+by (res_inst_tac [("x","Re r")] exI 1);
+by (Ultra_tac 1);
+qed "SComplex_Re_SReal";
+
+Goal "Abs_hcomplex(hcomplexrel ``{%n. X n}) : SComplex \
+\ ==> Abs_hypreal(hyprel `` {%n. Im(X n)}) : Reals";
+by (auto_tac (claset(),simpset() addsimps [SComplex_def,
+ hcomplex_of_complex_def,SReal_def,hypreal_of_real_def]));
+by (res_inst_tac [("x","Im r")] exI 1);
+by (Ultra_tac 1);
+qed "SComplex_Im_SReal";
+
+Goal "[| Abs_hypreal(hyprel `` {%n. Re(X n)}) : Reals; \
+\ Abs_hypreal(hyprel `` {%n. Im(X n)}) : Reals \
+\ |] ==> Abs_hcomplex(hcomplexrel ``{%n. X n}) : SComplex";
+by (auto_tac (claset(),simpset() addsimps [SComplex_def,
+ hcomplex_of_complex_def,SReal_def,hypreal_of_real_def]));
+by (res_inst_tac [("x","complex_of_real r + ii * complex_of_real ra")] exI 1);
+by (Ultra_tac 1);
+by (res_inst_tac [("z","X x")] eq_Abs_complex 1);
+by (auto_tac (claset(),simpset() addsimps [complex_of_real_def,i_def,
+ complex_add,complex_mult]));
+qed "Reals_Re_Im_SComplex";
+
+Goal "(Abs_hcomplex(hcomplexrel ``{%n. X n}) : SComplex) = \
+\ (Abs_hypreal(hyprel `` {%n. Re(X n)}) : Reals & \
+\ Abs_hypreal(hyprel `` {%n. Im(X n)}) : Reals)";
+by (blast_tac (claset() addIs [SComplex_Re_SReal,SComplex_Im_SReal,
+ Reals_Re_Im_SComplex]) 1);
+qed "SComplex_SReal_iff";
+
+Goal "(Abs_hcomplex(hcomplexrel ``{%n. X n}) : CInfinitesimal) = \
+\ (Abs_hypreal(hyprel `` {%n. Re(X n)}) : Infinitesimal & \
+\ Abs_hypreal(hyprel `` {%n. Im(X n)}) : Infinitesimal)";
+by (auto_tac (claset(),simpset() addsimps [mem_cinfmal_iff,
+ mem_infmal_iff,hcomplex_zero_num,hypreal_zero_num,capprox_approx_iff]));
+qed "CInfinitesimal_Infinitesimal_iff";
+
+(*** more lemmas ****)
+Goal "(EX t. P t) = (EX X. P (Abs_hcomplex(hcomplexrel `` {X})))";
+by Auto_tac;
+by (res_inst_tac [("z","t")] eq_Abs_hcomplex 1);
+by Auto_tac;
+qed "eq_Abs_hcomplex_EX";
+
+Goal "(EX t : A. P t) = (EX X. (Abs_hcomplex(hcomplexrel `` {X})) : A & \
+\ P (Abs_hcomplex(hcomplexrel `` {X})))";
+by Auto_tac;
+by (res_inst_tac [("z","t")] eq_Abs_hcomplex 1);
+by Auto_tac;
+qed "eq_Abs_hcomplex_Bex";
+
+(* Here we go - easy proof now!! *)
+Goal "x:CFinite ==> EX t: SComplex. x @c= t";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [CFinite_HFinite_iff,
+ eq_Abs_hcomplex_Bex,SComplex_SReal_iff,capprox_approx_iff]));
+by (REPEAT(dtac st_part_Ex 1 THEN Step_tac 1));
+by (res_inst_tac [("z","t")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","ta")] eq_Abs_hypreal 1);
+by Auto_tac;
+by (res_inst_tac [("x","%n. complex_of_real (xa n) + ii * complex_of_real (xb n)")]
+ exI 1);
+by Auto_tac;
+qed "stc_part_Ex";
+
+Goal "x:CFinite ==> EX! t. t : SComplex & x @c= t";
+by (dtac stc_part_Ex 1 THEN Step_tac 1);
+by (dtac capprox_sym 2 THEN dtac capprox_sym 2
+ THEN dtac capprox_sym 2);
+by (auto_tac (claset() addSIs [capprox_unique_complex], simpset()));
+qed "stc_part_Ex1";
+
+Goalw [CFinite_def,CInfinite_def] "CFinite Int CInfinite = {}";
+by Auto_tac;
+qed "CFinite_Int_CInfinite_empty";
+Addsimps [CFinite_Int_CInfinite_empty];
+
+Goal "x: CFinite ==> x ~: CInfinite";
+by (EVERY1[Step_tac, dtac IntI, assume_tac]);
+by Auto_tac;
+qed "CFinite_not_CInfinite";
+
+Goal "x~: CFinite ==> x: CInfinite";
+by (auto_tac (claset() addIs [not_HFinite_HInfinite],
+ simpset() addsimps [CFinite_hcmod_iff,CInfinite_hcmod_iff]));
+qed "not_CFinite_CInfinite";
+
+Goal "x : CInfinite | x : CFinite";
+by (blast_tac (claset() addIs [not_CFinite_CInfinite]) 1);
+qed "CInfinite_CFinite_disj";
+
+Goal "(x : CInfinite) = (x ~: CFinite)";
+by (blast_tac (claset() addDs [CFinite_not_CInfinite,
+ not_CFinite_CInfinite]) 1);
+qed "CInfinite_CFinite_iff";
+
+Goal "(x : CFinite) = (x ~: CInfinite)";
+by (simp_tac (simpset() addsimps [CInfinite_CFinite_iff]) 1);
+qed "CFinite_CInfinite_iff";
+
+Goal "x ~: CInfinitesimal ==> x : CInfinite | x : CFinite - CInfinitesimal";
+by (fast_tac (claset() addIs [not_CFinite_CInfinite]) 1);
+qed "CInfinite_diff_CFinite_CInfinitesimal_disj";
+
+Goal "[| x : CFinite; x ~: CInfinitesimal |] ==> inverse x : CFinite";
+by (cut_inst_tac [("x","inverse x")] CInfinite_CFinite_disj 1);
+by (auto_tac (claset() addSDs [CInfinite_inverse_CInfinitesimal], simpset()));
+qed "CFinite_inverse";
+
+Goal "x : CFinite - CInfinitesimal ==> inverse x : CFinite";
+by (blast_tac (claset() addIs [CFinite_inverse]) 1);
+qed "CFinite_inverse2";
+
+Goal "x ~: CInfinitesimal ==> inverse(x) : CFinite";
+by (dtac CInfinite_diff_CFinite_CInfinitesimal_disj 1);
+by (blast_tac (claset() addIs [CFinite_inverse,
+ CInfinite_inverse_CInfinitesimal,
+ CInfinitesimal_subset_CFinite RS subsetD]) 1);
+qed "CInfinitesimal_inverse_CFinite";
+
+
+Goal "x : CFinite - CInfinitesimal ==> inverse x : CFinite - CInfinitesimal";
+by (auto_tac (claset() addIs [CInfinitesimal_inverse_CFinite], simpset()));
+by (dtac CInfinitesimal_CFinite_mult2 1);
+by (assume_tac 1);
+by (asm_full_simp_tac (simpset() addsimps [not_CInfinitesimal_not_zero]) 1);
+qed "CFinite_not_CInfinitesimal_inverse";
+
+Goal "[| x @c= y; y : CFinite - CInfinitesimal |] \
+\ ==> inverse x @c= inverse y";
+by (forward_tac [CFinite_diff_CInfinitesimal_capprox] 1);
+by (assume_tac 1);
+by (forward_tac [not_CInfinitesimal_not_zero2] 1);
+by (forw_inst_tac [("x","x")] not_CInfinitesimal_not_zero2 1);
+by (REPEAT(dtac CFinite_inverse2 1));
+by (dtac capprox_mult2 1 THEN assume_tac 1);
+by Auto_tac;
+by (dres_inst_tac [("c","inverse x")] capprox_mult1 1
+ THEN assume_tac 1);
+by (auto_tac (claset() addIs [capprox_sym],
+ simpset() addsimps [hcomplex_mult_assoc]));
+qed "capprox_inverse";
+
+bind_thm ("hcomplex_of_complex_capprox_inverse",
+ hcomplex_of_complex_CFinite_diff_CInfinitesimal RSN (2, capprox_inverse));
+
+Goal "[| x: CFinite - CInfinitesimal; \
+\ h : CInfinitesimal |] ==> inverse(x + h) @c= inverse x";
+by (auto_tac (claset() addIs [capprox_inverse, capprox_sym,
+ CInfinitesimal_add_capprox_self],
+ simpset()));
+qed "inverse_add_CInfinitesimal_capprox";
+
+Goal "[| x: CFinite - CInfinitesimal; \
+\ h : CInfinitesimal |] ==> inverse(h + x) @c= inverse x";
+by (rtac (hcomplex_add_commute RS subst) 1);
+by (blast_tac (claset() addIs [inverse_add_CInfinitesimal_capprox]) 1);
+qed "inverse_add_CInfinitesimal_capprox2";
+
+Goal "[| x : CFinite - CInfinitesimal; \
+\ h : CInfinitesimal |] ==> inverse(x + h) - inverse x @c= h";
+by (rtac capprox_trans2 1);
+by (auto_tac (claset() addIs [inverse_add_CInfinitesimal_capprox],
+ simpset() addsimps [mem_cinfmal_iff,hcomplex_diff_def,
+ capprox_minus_iff RS sym]));
+qed "inverse_add_CInfinitesimal_approx_CInfinitesimal";
+
+Goal "(x*x : CInfinitesimal) = (x : CInfinitesimal)";
+by (auto_tac (claset(), simpset() addsimps [CInfinitesimal_hcmod_iff,
+ hcmod_mult]));
+qed "CInfinitesimal_square_iff";
+AddIffs [CInfinitesimal_square_iff];
+
+Goal "[| a: CFinite-CInfinitesimal; a*w @c= a*z |] ==> w @c= z";
+by (Step_tac 1);
+by (ftac CFinite_inverse 1 THEN assume_tac 1);
+by (dtac not_CInfinitesimal_not_zero 1);
+by (auto_tac (claset() addDs [capprox_mult2],
+ simpset() addsimps [hcomplex_mult_assoc RS sym]));
+qed "capprox_CFinite_mult_cancel";
+
+Goal "a: CFinite-CInfinitesimal ==> (a * w @c= a * z) = (w @c= z)";
+by (auto_tac (claset() addIs [capprox_mult2,
+ capprox_CFinite_mult_cancel], simpset()));
+qed "capprox_CFinite_mult_cancel_iff1";
+
+
+(*---------------------------------------------------------------------------*)
+(* Theorems about monads *)
+(*---------------------------------------------------------------------------*)
+
+Goalw [cmonad_def] "(x @c= y) = (cmonad(x)=cmonad(y))";
+by (auto_tac (claset() addDs [capprox_sym]
+ addSEs [capprox_trans,equalityCE],
+ simpset()));
+qed "capprox_cmonad_iff";
+
+Goal "e : CInfinitesimal ==> cmonad (x+e) = cmonad x";
+by (fast_tac (claset() addSIs [CInfinitesimal_add_capprox_self RS capprox_sym,
+ capprox_cmonad_iff RS iffD1]) 1);
+qed "CInfinitesimal_cmonad_eq";
+
+Goalw [cmonad_def] "(u:cmonad x) = (-u:cmonad (-x))";
+by Auto_tac;
+qed "mem_cmonad_iff";
+
+Goalw [cmonad_def] "(x:CInfinitesimal) = (x:cmonad 0)";
+by (auto_tac (claset() addIs [capprox_sym],
+ simpset() addsimps [mem_cinfmal_iff]));
+qed "CInfinitesimal_cmonad_zero_iff";
+
+Goal "(x:cmonad 0) = (-x:cmonad 0)";
+by (simp_tac (simpset() addsimps [CInfinitesimal_cmonad_zero_iff RS sym]) 1);
+qed "cmonad_zero_minus_iff";
+
+Goal "(x:cmonad 0) = (hcmod x:monad 0)";
+by (auto_tac (claset(), simpset() addsimps
+ [CInfinitesimal_cmonad_zero_iff RS sym,
+ CInfinitesimal_hcmod_iff,Infinitesimal_monad_zero_iff RS sym]));
+qed "cmonad_zero_hcmod_iff";
+
+Goalw [cmonad_def] "x:cmonad x";
+by (Simp_tac 1);
+qed "mem_cmonad_self";
+Addsimps [mem_cmonad_self];
+
+(*---------------------------------------------------------------------------*)
+(* Theorems about standard part *)
+(*---------------------------------------------------------------------------*)
+Goalw [stc_def] "x: CFinite ==> stc x @c= x";
+by (forward_tac [stc_part_Ex] 1 THEN Step_tac 1);
+by (rtac someI2 1);
+by (auto_tac (claset() addIs [capprox_sym], simpset()));
+qed "stc_capprox_self";
+
+Goalw [stc_def] "x: CFinite ==> stc x: SComplex";
+by (forward_tac [stc_part_Ex] 1 THEN Step_tac 1);
+by (rtac someI2 1);
+by (auto_tac (claset() addIs [capprox_sym], simpset()));
+qed "stc_SComplex";
+
+Goal "x: CFinite ==> stc x: CFinite";
+by (etac (stc_SComplex RS (SComplex_subset_CFinite RS subsetD)) 1);
+qed "stc_CFinite";
+
+Goalw [stc_def] "x: SComplex ==> stc x = x";
+by (rtac some_equality 1);
+by (auto_tac (claset() addIs [(SComplex_subset_CFinite RS subsetD)],simpset()));
+by (blast_tac (claset() addDs [SComplex_capprox_iff RS iffD1]) 1);
+qed "stc_SComplex_eq";
+Addsimps [stc_SComplex_eq];
+
+Goal "stc (hcomplex_of_complex x) = hcomplex_of_complex x";
+by Auto_tac;
+qed "stc_hcomplex_of_complex";
+
+Goal "[| x: CFinite; y: CFinite; stc x = stc y |] ==> x @c= y";
+by (auto_tac (claset() addSDs [stc_capprox_self]
+ addSEs [capprox_trans3], simpset()));
+qed "stc_eq_capprox";
+
+Goal "[| x: CFinite; y: CFinite; x @c= y |] ==> stc x = stc y";
+by (EVERY1 [forward_tac [stc_capprox_self],
+ forw_inst_tac [("x","y")] stc_capprox_self,
+ dtac stc_SComplex,dtac stc_SComplex]);
+by (fast_tac (claset() addEs [capprox_trans,
+ capprox_trans2,SComplex_capprox_iff RS iffD1]) 1);
+qed "capprox_stc_eq";
+
+Goal "[| x: CFinite; y: CFinite|] ==> (x @c= y) = (stc x = stc y)";
+by (blast_tac (claset() addIs [capprox_stc_eq,stc_eq_capprox]) 1);
+qed "stc_eq_capprox_iff";
+
+Goal "[| x: SComplex; e: CInfinitesimal |] ==> stc(x + e) = x";
+by (forward_tac [stc_SComplex_eq RS subst] 1);
+by (assume_tac 2);
+by (forward_tac [SComplex_subset_CFinite RS subsetD] 1);
+by (forward_tac [CInfinitesimal_subset_CFinite RS subsetD] 1);
+by (dtac stc_SComplex_eq 1);
+by (rtac capprox_stc_eq 1);
+by (auto_tac (claset() addIs [CFinite_add],
+ simpset() addsimps [CInfinitesimal_add_capprox_self
+ RS capprox_sym]));
+qed "stc_CInfinitesimal_add_SComplex";
+
+Goal "[| x: SComplex; e: CInfinitesimal |] ==> stc(e + x) = x";
+by (rtac (hcomplex_add_commute RS subst) 1);
+by (blast_tac (claset() addSIs [stc_CInfinitesimal_add_SComplex]) 1);
+qed "stc_CInfinitesimal_add_SComplex2";
+
+Goal "x: CFinite ==> EX e: CInfinitesimal. x = stc(x) + e";
+by (blast_tac (claset() addSDs [(stc_capprox_self RS
+ capprox_sym),bex_CInfinitesimal_iff2 RS iffD2]) 1);
+qed "CFinite_stc_CInfinitesimal_add";
+
+Goal "[| x: CFinite; y: CFinite |] ==> stc (x + y) = stc(x) + stc(y)";
+by (forward_tac [CFinite_stc_CInfinitesimal_add] 1);
+by (forw_inst_tac [("x","y")] CFinite_stc_CInfinitesimal_add 1);
+by (Step_tac 1);
+by (subgoal_tac "stc (x + y) = stc ((stc x + e) + (stc y + ea))" 1);
+by (dtac sym 2 THEN dtac sym 2);
+by (Asm_full_simp_tac 2);
+by (asm_simp_tac (simpset() addsimps hcomplex_add_ac) 1);
+by (REPEAT(dtac stc_SComplex 1));
+by (dtac SComplex_add 1 THEN assume_tac 1);
+by (dtac CInfinitesimal_add 1 THEN assume_tac 1);
+by (rtac (hcomplex_add_assoc RS subst) 1);
+by (blast_tac (claset() addSIs [stc_CInfinitesimal_add_SComplex2]) 1);
+qed "stc_add";
+
+Goal "stc (number_of w) = number_of w";
+by (rtac (SComplex_number_of RS stc_SComplex_eq) 1);
+qed "stc_number_of";
+Addsimps [stc_number_of];
+
+Goal "stc 0 = 0";
+by (Simp_tac 1);
+qed "stc_zero";
+Addsimps [stc_zero];
+
+Goal "stc 1 = 1";
+by (Simp_tac 1);
+qed "stc_one";
+Addsimps [stc_one];
+
+Goal "y: CFinite ==> stc(-y) = -stc(y)";
+by (forward_tac [CFinite_minus_iff RS iffD2] 1);
+by (rtac hcomplex_add_minus_eq_minus 1);
+by (dtac (stc_add RS sym) 1 THEN assume_tac 1);
+by Auto_tac;
+qed "stc_minus";
+
+Goalw [hcomplex_diff_def]
+ "[| x: CFinite; y: CFinite |] ==> stc (x-y) = stc(x) - stc(y)";
+by (forw_inst_tac [("y1","y")] (stc_minus RS sym) 1);
+by (dres_inst_tac [("x1","y")] (CFinite_minus_iff RS iffD2) 1);
+by (auto_tac (claset() addIs [stc_add],simpset()));
+qed "stc_diff";
+
+Goal "[| x: CFinite; y: CFinite; \
+\ e: CInfinitesimal; \
+\ ea: CInfinitesimal |] \
+\ ==> e*y + x*ea + e*ea: CInfinitesimal";
+by (forw_inst_tac [("x","e"),("y","y")] CInfinitesimal_CFinite_mult 1);
+by (forw_inst_tac [("x","ea"),("y","x")] CInfinitesimal_CFinite_mult 2);
+by (dtac CInfinitesimal_mult 3);
+by (auto_tac (claset() addIs [CInfinitesimal_add],
+ simpset() addsimps hcomplex_add_ac @ hcomplex_mult_ac));
+qed "lemma_stc_mult";
+
+Goal "[| x: CFinite; y: CFinite |] \
+\ ==> stc (x * y) = stc(x) * stc(y)";
+by (forward_tac [CFinite_stc_CInfinitesimal_add] 1);
+by (forw_inst_tac [("x","y")] CFinite_stc_CInfinitesimal_add 1);
+by (Step_tac 1);
+by (subgoal_tac "stc (x * y) = stc ((stc x + e) * (stc y + ea))" 1);
+by (dtac sym 2 THEN dtac sym 2);
+by (Asm_full_simp_tac 2);
+by (thin_tac "x = stc x + e" 1);
+by (thin_tac "y = stc y + ea" 1);
+by (asm_full_simp_tac (simpset() addsimps
+ [hcomplex_add_mult_distrib,hcomplex_add_mult_distrib2]) 1);
+by (REPEAT(dtac stc_SComplex 1));
+by (full_simp_tac (simpset() addsimps [hcomplex_add_assoc]) 1);
+by (rtac stc_CInfinitesimal_add_SComplex 1);
+by (blast_tac (claset() addSIs [SComplex_mult]) 1);
+by (REPEAT(dtac (SComplex_subset_CFinite RS subsetD) 1));
+by (rtac (hcomplex_add_assoc RS subst) 1);
+by (blast_tac (claset() addSIs [lemma_stc_mult]) 1);
+qed "stc_mult";
+
+Goal "x: CInfinitesimal ==> stc x = 0";
+by (rtac (stc_zero RS subst) 1);
+by (rtac capprox_stc_eq 1);
+by (auto_tac (claset() addIs [CInfinitesimal_subset_CFinite RS subsetD],
+ simpset() addsimps [mem_cinfmal_iff RS sym]));
+qed "stc_CInfinitesimal";
+
+Goal "stc(x) ~= 0 ==> x ~: CInfinitesimal";
+by (fast_tac (claset() addIs [stc_CInfinitesimal]) 1);
+qed "stc_not_CInfinitesimal";
+
+Goal "[| x: CFinite; stc x ~= 0 |] \
+\ ==> stc(inverse x) = inverse (stc x)";
+by (res_inst_tac [("c1","stc x")] (hcomplex_mult_left_cancel RS iffD1) 1);
+by (auto_tac (claset(),
+ simpset() addsimps [stc_mult RS sym, stc_not_CInfinitesimal,
+ CFinite_inverse]));
+by (stac hcomplex_mult_inv_right 1);
+by Auto_tac;
+qed "stc_inverse";
+
+Goal "[| x: CFinite; y: CFinite; stc y ~= 0 |] \
+\ ==> stc(x/y) = (stc x) / (stc y)";
+by (auto_tac (claset(),
+ simpset() addsimps [hcomplex_divide_def, stc_mult, stc_not_CInfinitesimal,
+ CFinite_inverse, stc_inverse]));
+qed "stc_divide";
+Addsimps [stc_divide];
+
+Goal "x: CFinite ==> stc(stc(x)) = stc(x)";
+by (blast_tac (claset() addIs [stc_CFinite, stc_capprox_self,
+ capprox_stc_eq]) 1);
+qed "stc_idempotent";
+Addsimps [stc_idempotent];
+
+Goal "z : HFinite ==> hcomplex_of_hypreal z : CFinite";
+by (res_inst_tac [("z","z")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,
+ CFinite_HFinite_iff,symmetric hypreal_zero_def]));
+qed "CFinite_HFinite_hcomplex_of_hypreal";
+
+Goal "x : Reals ==> hcomplex_of_hypreal x : SComplex";
+by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,
+ SComplex_SReal_iff,symmetric hypreal_zero_def]));
+qed "SComplex_SReal_hcomplex_of_hypreal";
+
+Goalw [st_def,stc_def]
+ "z : HFinite ==> stc(hcomplex_of_hypreal z) = hcomplex_of_hypreal (st z)";
+by (ftac st_part_Ex 1 THEN Step_tac 1);
+by (rtac someI2 1);
+by (auto_tac (claset() addIs [approx_sym],simpset()));
+by (dtac CFinite_HFinite_hcomplex_of_hypreal 1);
+by (ftac stc_part_Ex 1 THEN Step_tac 1);
+by (rtac someI2 1);
+by (auto_tac (claset() addIs [capprox_sym] addSIs [capprox_unique_complex]
+ addDs [SComplex_SReal_hcomplex_of_hypreal],simpset()));
+qed "stc_hcomplex_of_hypreal";
+
+(*
+Goal "x: CFinite ==> hcmod(stc x) = st(hcmod x)";
+by (dtac stc_capprox_self 1);
+by (auto_tac (claset(),simpset() addsimps [bex_CInfinitesimal_iff2 RS sym]));
+
+
+approx_hcmod_add_hcmod
+*)
+
+(*---------------------------------------------------------------------------*)
+(* More nonstandard complex specific theorems *)
+(*---------------------------------------------------------------------------*)
+Goal "(hcnj z : CInfinitesimal) = (z : CInfinitesimal)";
+by (auto_tac (claset(),simpset() addsimps [CInfinitesimal_hcmod_iff]));
+qed "CInfinitesimal_hcnj_iff";
+Addsimps [CInfinitesimal_hcnj_iff];
+
+Goal "(Abs_hcomplex(hcomplexrel ``{%n. X n}) : CInfinite) = \
+\ (Abs_hypreal(hyprel `` {%n. Re(X n)}) : HInfinite | \
+\ Abs_hypreal(hyprel `` {%n. Im(X n)}) : HInfinite)";
+by (auto_tac (claset(),simpset() addsimps [CInfinite_CFinite_iff,
+ HInfinite_HFinite_iff,CFinite_HFinite_iff]));
+qed "CInfinite_HInfinite_iff";
+
+Goal "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y : CInfinitesimal) = \
+\ (x : Infinitesimal & y : Infinitesimal)";
+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 [iii_def,hcomplex_add,hcomplex_mult,
+ hcomplex_of_hypreal,CInfinitesimal_Infinitesimal_iff]));
+qed "hcomplex_split_CInfinitesimal_iff";
+
+Goal "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y : CFinite) = \
+\ (x : HFinite & y : HFinite)";
+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 [iii_def,hcomplex_add,hcomplex_mult,
+ hcomplex_of_hypreal,CFinite_HFinite_iff]));
+qed "hcomplex_split_CFinite_iff";
+
+Goal "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y : SComplex) = \
+\ (x : Reals & y : Reals)";
+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 [iii_def,hcomplex_add,hcomplex_mult,
+ hcomplex_of_hypreal,SComplex_SReal_iff]));
+qed "hcomplex_split_SComplex_iff";
+
+Goal "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y : CInfinite) = \
+\ (x : HInfinite | y : HInfinite)";
+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 [iii_def,hcomplex_add,hcomplex_mult,
+ hcomplex_of_hypreal,CInfinite_HInfinite_iff]));
+qed "hcomplex_split_CInfinite_iff";
+
+Goal "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y @c= \
+\ hcomplex_of_hypreal x' + iii * hcomplex_of_hypreal y') = \
+\ (x @= x' & y @= y')";
+by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","y")] eq_Abs_hypreal 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 [iii_def,hcomplex_add,hcomplex_mult,
+ hcomplex_of_hypreal,capprox_approx_iff]));
+qed "hcomplex_split_capprox_iff";
+
+(*** More theorems ***)
+
+Goal "ALL n. cmod (X n - x) < inverse (real (Suc n)) ==> \
+\ Abs_hcomplex(hcomplexrel``{X}) - hcomplex_of_complex x : CInfinitesimal";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_diff,
+ CInfinitesimal_hcmod_iff,hcomplex_of_complex_def,
+ Infinitesimal_FreeUltrafilterNat_iff,hcmod]));
+by (rtac bexI 1 THEN Auto_tac);
+by (auto_tac (claset() addDs [FreeUltrafilterNat_inverse_real_of_posnat,
+ FreeUltrafilterNat_all,FreeUltrafilterNat_Int]
+ addIs [order_less_trans, FreeUltrafilterNat_subset],
+ simpset()));
+qed "complex_seq_to_hcomplex_CInfinitesimal";
+
+Goal "hcomplex_of_hypreal epsilon : CInfinitesimal";
+by (auto_tac (claset(),simpset() addsimps [CInfinitesimal_hcmod_iff]));
+qed "CInfinitesimal_hcomplex_of_hypreal_epsilon";
+Addsimps [CInfinitesimal_hcomplex_of_hypreal_epsilon];
+
+Goal "(hcomplex_of_complex z @c= 0) = (z = 0)";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_complex_zero RS sym]
+ delsimps [hcomplex_of_complex_zero]));
+qed "hcomplex_of_complex_approx_zero_iff";
+
+Goal "(0 @c= hcomplex_of_complex z) = (z = 0)";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_complex_zero RS sym]
+ delsimps [hcomplex_of_complex_zero]));
+qed "hcomplex_of_complex_approx_zero_iff2";
+
+Addsimps [hcomplex_of_complex_approx_zero_iff,hcomplex_of_complex_approx_zero_iff2];
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/NSCA.thy Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,44 @@
+(* Title : NSCA.thy
+ Author : Jacques D. Fleuriot
+ Copyright : 2001,2002 University of Edinburgh
+ Description : Infinite, infinitesimal complex number etc!
+*)
+
+NSCA = NSComplexArith0 +
+
+consts
+
+ (* infinitely close *)
+ "@c=" :: [hcomplex,hcomplex] => bool (infixl 50)
+
+
+constdefs
+ (* standard complex numbers reagarded as an embedded subset of NS complex *)
+ SComplex :: "hcomplex set"
+ "SComplex == {x. EX r. x = hcomplex_of_complex r}"
+
+ CInfinitesimal :: "hcomplex set"
+ "CInfinitesimal == {x. ALL r: Reals. 0 < r --> hcmod x < r}"
+
+ CFinite :: "hcomplex set"
+ "CFinite == {x. EX r: Reals. hcmod x < r}"
+
+ CInfinite :: "hcomplex set"
+ "CInfinite == {x. ALL r: Reals. r < hcmod x}"
+
+ (* standard part map *)
+ stc :: hcomplex => hcomplex
+ "stc x == (@r. x : CFinite & r:SComplex & r @c= x)"
+
+ cmonad :: hcomplex => hcomplex set
+ "cmonad x == {y. x @c= y}"
+
+ cgalaxy :: hcomplex => hcomplex set
+ "cgalaxy x == {y. (x - y) : CFinite}"
+
+
+defs
+
+ capprox_def "x @c= y == (x - y) : CInfinitesimal"
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/NSComplex.ML Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,1895 @@
+(* Title: NSComplex.ML
+ Author: Jacques D. Fleuriot
+ Copyhright: 2001 University of Edinburgh
+ Description: Nonstandard Complex numbers
+*)
+
+Goalw [hcomplexrel_def]
+ "((X,Y): hcomplexrel) = ({n. X n = Y n}: FreeUltrafilterNat)";
+by (Fast_tac 1);
+qed "hcomplexrel_iff";
+
+Goalw [hcomplexrel_def]
+ "!!X. {n. X n = Y n}: FreeUltrafilterNat \
+\ ==> (X,Y): hcomplexrel";
+by (Fast_tac 1);
+qed "hcomplexrelI";
+
+Goalw [hcomplexrel_def]
+ "p: hcomplexrel --> (EX X Y. \
+\ p = (X,Y) & {n. X n = Y n} : FreeUltrafilterNat)";
+by (Fast_tac 1);
+qed "hcomplexrelE_lemma";
+
+val [major,minor] = goal thy
+ "[| p: hcomplexrel; \
+\ !!X Y. [| p = (X,Y); {n. X n = Y n}: FreeUltrafilterNat\
+\ |] ==> Q |] ==> Q";
+by (cut_facts_tac [major RS (hcomplexrelE_lemma RS mp)] 1);
+by (REPEAT (eresolve_tac [asm_rl,exE,conjE,minor] 1));
+qed "hcomplexrelE";
+
+AddSIs [hcomplexrelI];
+AddSEs [hcomplexrelE];
+
+Goalw [hcomplexrel_def] "(x,x): hcomplexrel";
+by (Auto_tac);
+qed "hcomplexrel_refl";
+
+Goalw [hcomplexrel_def] "(x,y): hcomplexrel --> (y,x):hcomplexrel";
+by (auto_tac (claset() addIs [lemma_perm RS subst],simpset()));
+qed_spec_mp "hcomplexrel_sym";
+
+Goalw [hcomplexrel_def]
+ "(x,y): hcomplexrel --> (y,z):hcomplexrel --> (x,z):hcomplexrel";
+by (Auto_tac);
+by (Ultra_tac 1);
+qed_spec_mp "hcomplexrel_trans";
+
+Goalw [equiv_def, refl_def, sym_def, trans_def]
+ "equiv {x::nat=>complex. True} hcomplexrel";
+by (auto_tac (claset() addSIs [hcomplexrel_refl] addSEs
+ [hcomplexrel_sym,hcomplexrel_trans] delrules [hcomplexrelI,hcomplexrelE],
+ simpset()));
+qed "equiv_hcomplexrel";
+
+val equiv_hcomplexrel_iff =
+ [TrueI, TrueI] MRS
+ ([CollectI, CollectI] MRS
+ (equiv_hcomplexrel RS eq_equiv_class_iff));
+
+Goalw [hcomplex_def,hcomplexrel_def,quotient_def] "hcomplexrel``{x}:hcomplex";
+by (Blast_tac 1);
+qed "hcomplexrel_in_hcomplex";
+
+Goal "inj_on Abs_hcomplex hcomplex";
+by (rtac inj_on_inverseI 1);
+by (etac Abs_hcomplex_inverse 1);
+qed "inj_on_Abs_hcomplex";
+
+Addsimps [equiv_hcomplexrel_iff,inj_on_Abs_hcomplex RS inj_on_iff,
+ hcomplexrel_iff, hcomplexrel_in_hcomplex, Abs_hcomplex_inverse];
+
+Addsimps [equiv_hcomplexrel RS eq_equiv_class_iff];
+val eq_hcomplexrelD = equiv_hcomplexrel RSN (2,eq_equiv_class);
+
+Goal "inj(Rep_hcomplex)";
+by (rtac inj_inverseI 1);
+by (rtac Rep_hcomplex_inverse 1);
+qed "inj_Rep_hcomplex";
+
+Goalw [hcomplexrel_def] "x: hcomplexrel `` {x}";
+by (Step_tac 1);
+by (Auto_tac);
+qed "lemma_hcomplexrel_refl";
+Addsimps [lemma_hcomplexrel_refl];
+
+Goalw [hcomplex_def] "{} ~: hcomplex";
+by (auto_tac (claset() addSEs [quotientE],simpset()));
+qed "hcomplex_empty_not_mem";
+Addsimps [hcomplex_empty_not_mem];
+
+Goal "Rep_hcomplex x ~= {}";
+by (cut_inst_tac [("x","x")] Rep_hcomplex 1);
+by (Auto_tac);
+qed "Rep_hcomplex_nonempty";
+Addsimps [Rep_hcomplex_nonempty];
+
+val [prem] = goal thy
+ "(!!x. z = Abs_hcomplex(hcomplexrel `` {x}) ==> P) ==> P";
+by (res_inst_tac [("x1","z")]
+ (rewrite_rule [hcomplex_def] Rep_hcomplex RS quotientE) 1);
+by (dres_inst_tac [("f","Abs_hcomplex")] arg_cong 1);
+by (res_inst_tac [("x","x")] prem 1);
+by (asm_full_simp_tac (simpset() addsimps [Rep_hcomplex_inverse]) 1);
+qed "eq_Abs_hcomplex";
+
+(*-----------------------------------------------------------------------*)
+(* Properties of nonstandard real and imaginary parts *)
+(*-----------------------------------------------------------------------*)
+
+Goalw [hRe_def]
+ "hRe(Abs_hcomplex (hcomplexrel `` {X})) = \
+\ Abs_hypreal(hyprel `` {%n. Re(X n)})";
+by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);
+by (Auto_tac THEN Ultra_tac 1);
+qed "hRe";
+
+Goalw [hIm_def]
+ "hIm(Abs_hcomplex (hcomplexrel `` {X})) = \
+\ Abs_hypreal(hyprel `` {%n. Im(X n)})";
+by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);
+by (Auto_tac THEN Ultra_tac 1);
+qed "hIm";
+
+Goal "(w=z) = (hRe(w) = hRe(z) & hIm(w) = hIm(z))";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","w")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hRe,hIm,
+ complex_Re_Im_cancel_iff]));
+by (ALLGOALS(Ultra_tac));
+qed "hcomplex_hRe_hIm_cancel_iff";
+
+Goalw [hcomplex_zero_def] "hRe 0 = 0";
+by (simp_tac (simpset() addsimps [hRe,hypreal_zero_num]) 1);
+qed "hcomplex_hRe_zero";
+Addsimps [hcomplex_hRe_zero];
+
+Goalw [hcomplex_zero_def] "hIm 0 = 0";
+by (simp_tac (simpset() addsimps [hIm,hypreal_zero_num]) 1);
+qed "hcomplex_hIm_zero";
+Addsimps [hcomplex_hIm_zero];
+
+Goalw [hcomplex_one_def] "hRe 1 = 1";
+by (simp_tac (simpset() addsimps [hRe,hypreal_one_num]) 1);
+qed "hcomplex_hRe_one";
+Addsimps [hcomplex_hRe_one];
+
+Goalw [hcomplex_one_def] "hIm 1 = 0";
+by (simp_tac (simpset() addsimps [hIm,hypreal_one_def,hypreal_zero_num]) 1);
+qed "hcomplex_hIm_one";
+Addsimps [hcomplex_hIm_one];
+
+(*-----------------------------------------------------------------------*)
+(* hcomplex_of_complex: the injection from complex to hcomplex *)
+(* ----------------------------------------------------------------------*)
+
+Goal "inj(hcomplex_of_complex)";
+by (rtac injI 1 THEN rtac ccontr 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_complex_def]));
+qed "inj_hcomplex_of_complex";
+
+Goalw [iii_def,hcomplex_of_complex_def] "iii = hcomplex_of_complex ii";
+by (Simp_tac 1);
+qed "hcomplex_of_complex_i";
+
+(*-----------------------------------------------------------------------*)
+(* Addition for nonstandard complex numbers: hcomplex_add *)
+(* ----------------------------------------------------------------------*)
+
+Goalw [congruent2_def]
+ "congruent2 hcomplexrel (%X Y. hcomplexrel `` {%n. X n + Y n})";
+by (safe_tac (claset()));
+by (ALLGOALS(Ultra_tac));
+qed "hcomplex_add_congruent2";
+
+Goalw [hcomplex_add_def]
+ "Abs_hcomplex(hcomplexrel``{%n. X n}) + Abs_hcomplex(hcomplexrel``{%n. Y n}) = \
+\ Abs_hcomplex(hcomplexrel``{%n. X n + Y n})";
+by (res_inst_tac [("f","Abs_hcomplex")] arg_cong 1);
+by Auto_tac;
+by (Ultra_tac 1);
+qed "hcomplex_add";
+
+Goal "(z::hcomplex) + w = w + z";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","w")] eq_Abs_hcomplex 1);
+by (asm_simp_tac (simpset() addsimps (complex_add_ac @ [hcomplex_add])) 1);
+qed "hcomplex_add_commute";
+
+Goal "((z1::hcomplex) + z2) + z3 = z1 + (z2 + z3)";
+by (res_inst_tac [("z","z1")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","z2")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","z3")] eq_Abs_hcomplex 1);
+by (asm_simp_tac (simpset() addsimps [hcomplex_add,complex_add_assoc]) 1);
+qed "hcomplex_add_assoc";
+
+(*For AC rewriting*)
+Goal "(x::hcomplex)+(y+z)=y+(x+z)";
+by (rtac (hcomplex_add_commute RS trans) 1);
+by (rtac (hcomplex_add_assoc RS trans) 1);
+by (rtac (hcomplex_add_commute RS arg_cong) 1);
+qed "hcomplex_add_left_commute";
+
+(* hcomplex addition is an AC operator *)
+val hcomplex_add_ac = [hcomplex_add_assoc,hcomplex_add_commute,
+ hcomplex_add_left_commute];
+
+Goalw [hcomplex_zero_def] "(0::hcomplex) + z = z";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (asm_full_simp_tac (simpset() addsimps
+ [hcomplex_add]) 1);
+qed "hcomplex_add_zero_left";
+
+Goal "z + (0::hcomplex) = z";
+by (simp_tac (simpset() addsimps
+ [hcomplex_add_zero_left,hcomplex_add_commute]) 1);
+qed "hcomplex_add_zero_right";
+Addsimps [hcomplex_add_zero_left,hcomplex_add_zero_right];
+
+Goal "hRe(x + y) = hRe(x) + hRe(y)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hRe,hcomplex_add,
+ hypreal_add,complex_Re_add]));
+qed "hRe_add";
+
+Goal "hIm(x + y) = hIm(x) + hIm(y)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hIm,hcomplex_add,
+ hypreal_add,complex_Im_add]));
+qed "hIm_add";
+
+(*-----------------------------------------------------------------------*)
+(* hypreal_minus: additive inverse on nonstandard complex *)
+(* ----------------------------------------------------------------------*)
+
+Goalw [congruent_def]
+ "congruent hcomplexrel (%X. hcomplexrel `` {%n. - (X n)})";
+by (safe_tac (claset()));
+by (ALLGOALS(Ultra_tac));
+qed "hcomplex_minus_congruent";
+
+Goalw [hcomplex_minus_def]
+ "- (Abs_hcomplex(hcomplexrel `` {%n. X n})) = \
+\ Abs_hcomplex(hcomplexrel `` {%n. -(X n)})";
+by (res_inst_tac [("f","Abs_hcomplex")] arg_cong 1);
+by (Auto_tac THEN Ultra_tac 1);
+qed "hcomplex_minus";
+
+Goal "- (- z) = (z::hcomplex)";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (asm_simp_tac (simpset() addsimps [hcomplex_minus]) 1);
+qed "hcomplex_minus_minus";
+Addsimps [hcomplex_minus_minus];
+
+Goal "inj(%z::hcomplex. -z)";
+by (rtac injI 1);
+by (dres_inst_tac [("f","uminus")] arg_cong 1);
+by (Asm_full_simp_tac 1);
+qed "inj_hcomplex_minus";
+
+Goalw [hcomplex_zero_def] "- 0 = (0::hcomplex)";
+by (simp_tac (simpset() addsimps [hcomplex_minus]) 1);
+qed "hcomplex_minus_zero";
+Addsimps [hcomplex_minus_zero];
+
+Goal "(-x = 0) = (x = (0::hcomplex))";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_zero_def,
+ hcomplex_minus] @ complex_add_ac));
+qed "hcomplex_minus_zero_iff";
+Addsimps [hcomplex_minus_zero_iff];
+
+Goal "(0 = -x) = (x = (0::hcomplex))";
+by (auto_tac (claset() addDs [sym],simpset()));
+qed "hcomplex_minus_zero_iff2";
+Addsimps [hcomplex_minus_zero_iff2];
+
+Goal "(-x ~= 0) = (x ~= (0::hcomplex))";
+by Auto_tac;
+qed "hcomplex_minus_not_zero_iff";
+
+Goal "z + - z = (0::hcomplex)";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_add,hcomplex_minus,
+ hcomplex_zero_def]));
+qed "hcomplex_add_minus_right";
+Addsimps [hcomplex_add_minus_right];
+
+Goal "-z + z = (0::hcomplex)";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_add,hcomplex_minus,
+ hcomplex_zero_def]));
+qed "hcomplex_add_minus_left";
+Addsimps [hcomplex_add_minus_left];
+
+Goal "z + (- z + w) = (w::hcomplex)";
+by (simp_tac (simpset() addsimps [hcomplex_add_assoc RS sym]) 1);
+qed "hcomplex_add_minus_cancel";
+
+Goal "(-z) + (z + w) = (w::hcomplex)";
+by (simp_tac (simpset() addsimps [hcomplex_add_assoc RS sym]) 1);
+qed "hcomplex_minus_add_cancel";
+
+Goal "hRe(-z) = - hRe(z)";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hRe,hcomplex_minus,
+ hypreal_minus,complex_Re_minus]));
+qed "hRe_minus";
+
+Goal "hIm(-z) = - hIm(z)";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hIm,hcomplex_minus,
+ hypreal_minus,complex_Im_minus]));
+qed "hIm_minus";
+
+Goalw [hcomplex_zero_def]
+ "x + y = (0::hcomplex) ==> x = -y";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_add,hcomplex_minus]));
+by (ultra_tac (claset() addIs [complex_add_minus_eq_minus],simpset()) 1);
+qed "hcomplex_add_minus_eq_minus";
+
+Goal "-(x + y) = -x + -(y::hcomplex)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_add,hcomplex_minus]));
+qed "hcomplex_minus_add_distrib";
+Addsimps [hcomplex_minus_add_distrib];
+
+Goal "((x::hcomplex) + y = x + z) = (y = z)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_add]));
+qed "hcomplex_add_left_cancel";
+AddIffs [hcomplex_add_left_cancel];
+
+Goal "(y + (x::hcomplex)= z + x) = (y = z)";
+by (simp_tac (simpset() addsimps [hcomplex_add_commute]) 1);
+qed "hcomplex_add_right_cancel";
+AddIffs [hcomplex_add_right_cancel];
+
+Goal "((x::hcomplex) = y) = ((0::hcomplex) = x + - y)";
+by (Step_tac 1);
+by (res_inst_tac [("x1","-y")]
+ (hcomplex_add_right_cancel RS iffD1) 2);
+by (Auto_tac);
+qed "hcomplex_eq_minus_iff";
+
+Goal "((x::hcomplex) = y) = (x + - y = (0::hcomplex))";
+by (Step_tac 1);
+by (res_inst_tac [("x1","-y")]
+ (hcomplex_add_right_cancel RS iffD1) 2);
+by (Auto_tac);
+qed "hcomplex_eq_minus_iff2";
+
+(*-----------------------------------------------------------------------*)
+(* Subraction for nonstandard complex numbers: hcomplex_diff *)
+(* ----------------------------------------------------------------------*)
+
+Goalw [hcomplex_diff_def]
+ "Abs_hcomplex(hcomplexrel``{%n. X n}) - Abs_hcomplex(hcomplexrel``{%n. Y n}) = \
+\ Abs_hcomplex(hcomplexrel``{%n. X n - Y n})";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_minus,hcomplex_add,
+ complex_diff_def]));
+qed "hcomplex_diff";
+
+Goalw [hcomplex_diff_def] "(z::hcomplex) - z = (0::hcomplex)";
+by (Simp_tac 1);
+qed "hcomplex_diff_zero";
+Addsimps [hcomplex_diff_zero];
+
+Goal "(0::hcomplex) - x = -x";
+by (simp_tac (simpset() addsimps [hcomplex_diff_def]) 1);
+qed "hcomplex_diff_0";
+
+Goal "x - (0::hcomplex) = x";
+by (simp_tac (simpset() addsimps [hcomplex_diff_def]) 1);
+qed "hcomplex_diff_0_right";
+
+Goal "x - x = (0::hcomplex)";
+by (simp_tac (simpset() addsimps [hcomplex_diff_def]) 1);
+qed "hcomplex_diff_self";
+
+Addsimps [hcomplex_diff_0, hcomplex_diff_0_right, hcomplex_diff_self];
+
+Goal "((x::hcomplex) - y = z) = (x = z + y)";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_def,hcomplex_add_assoc]));
+qed "hcomplex_diff_eq_eq";
+
+(*-----------------------------------------------------------------------*)
+(* Multiplication for nonstandard complex numbers: hcomplex_mult *)
+(* ----------------------------------------------------------------------*)
+
+Goalw [hcomplex_mult_def]
+ "Abs_hcomplex(hcomplexrel``{%n. X n}) * Abs_hcomplex(hcomplexrel``{%n. Y n}) = \
+\ Abs_hcomplex(hcomplexrel``{%n. X n * Y n})";
+by (res_inst_tac [("f","Abs_hcomplex")] arg_cong 1);
+by (Auto_tac THEN Ultra_tac 1);
+qed "hcomplex_mult";
+
+Goal "(w::hcomplex) * z = z * w";
+by (res_inst_tac [("z","w")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult,
+ complex_mult_commute]));
+qed "hcomplex_mult_commute";
+
+Goal "((u::hcomplex) * v) * w = u * (v * w)";
+by (res_inst_tac [("z","u")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","v")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","w")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult,
+ complex_mult_assoc]));
+qed "hcomplex_mult_assoc";
+
+Goal "(x::hcomplex) * (y * z) = y * (x * z)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult,
+ complex_mult_left_commute]));
+qed "hcomplex_mult_left_commute";
+
+val hcomplex_mult_ac = [hcomplex_mult_assoc,hcomplex_mult_commute,
+ hcomplex_mult_left_commute];
+
+Goalw [hcomplex_one_def] "(1::hcomplex) * z = z";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult]));
+qed "hcomplex_mult_one_left";
+Addsimps [hcomplex_mult_one_left];
+
+Goal "z * (1::hcomplex) = z";
+by (simp_tac (simpset() addsimps [hcomplex_mult_commute]) 1);
+qed "hcomplex_mult_one_right";
+Addsimps [hcomplex_mult_one_right];
+
+Goalw [hcomplex_zero_def] "(0::hcomplex) * z = 0";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult]));
+qed "hcomplex_mult_zero_left";
+Addsimps [hcomplex_mult_zero_left];
+
+Goal "z * (0::hcomplex) = 0";
+by (simp_tac (simpset() addsimps [hcomplex_mult_commute]) 1);
+qed "hcomplex_mult_zero_right";
+Addsimps [hcomplex_mult_zero_right];
+
+Goal "-(x * y) = -x * (y::hcomplex)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult,
+ hcomplex_minus]));
+qed "hcomplex_minus_mult_eq1";
+
+Goal "-(x * y) = x * -(y::hcomplex)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult,
+ hcomplex_minus]));
+qed "hcomplex_minus_mult_eq2";
+
+Addsimps [hcomplex_minus_mult_eq1 RS sym,hcomplex_minus_mult_eq2 RS sym];
+
+Goal "- 1 * (z::hcomplex) = -z";
+by (Simp_tac 1);
+qed "hcomplex_mult_minus_one";
+Addsimps [hcomplex_mult_minus_one];
+
+Goal "(z::hcomplex) * - 1 = -z";
+by (stac hcomplex_mult_commute 1);
+by (Simp_tac 1);
+qed "hcomplex_mult_minus_one_right";
+Addsimps [hcomplex_mult_minus_one_right];
+
+Goal "-x * -y = x * (y::hcomplex)";
+by Auto_tac;
+qed "hcomplex_minus_mult_cancel";
+Addsimps [hcomplex_minus_mult_cancel];
+
+Goal "-x * y = x * -(y::hcomplex)";
+by Auto_tac;
+qed "hcomplex_minus_mult_commute";
+
+qed_goal "hcomplex_add_assoc_cong" thy
+ "!!z. (z::hcomplex) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"
+ (fn _ => [(asm_simp_tac (simpset() addsimps [hcomplex_add_assoc RS sym]) 1)]);
+
+qed_goal "hcomplex_add_assoc_swap" thy "(z::hcomplex) + (v + w) = v + (z + w)"
+ (fn _ => [(REPEAT (ares_tac [hcomplex_add_commute RS hcomplex_add_assoc_cong] 1))]);
+
+Goal "((z1::hcomplex) + z2) * w = (z1 * w) + (z2 * w)";
+by (res_inst_tac [("z","z1")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","z2")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","w")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult,hcomplex_add,
+ complex_add_mult_distrib]));
+qed "hcomplex_add_mult_distrib";
+
+Goal "(w::hcomplex) * (z1 + z2) = (w * z1) + (w * z2)";
+by (res_inst_tac [("z1","z1 + z2")] (hcomplex_mult_commute RS ssubst) 1);
+by (simp_tac (simpset() addsimps [hcomplex_add_mult_distrib]) 1);
+by (simp_tac (simpset() addsimps [hcomplex_mult_commute]) 1);
+qed "hcomplex_add_mult_distrib2";
+
+Goalw [hcomplex_zero_def,hcomplex_one_def] "(0::hcomplex) ~= (1::hcomplex)";
+by Auto_tac;
+qed "hcomplex_zero_not_eq_one";
+Addsimps [hcomplex_zero_not_eq_one];
+Addsimps [hcomplex_zero_not_eq_one RS not_sym];
+
+(*-----------------------------------------------------------------------*)
+(* Inverse of nonstandard complex number *)
+(*-----------------------------------------------------------------------*)
+
+Goalw [hcinv_def]
+ "inverse (Abs_hcomplex(hcomplexrel `` {%n. X n})) = \
+\ Abs_hcomplex(hcomplexrel `` {%n. inverse (X n)})";
+by (res_inst_tac [("f","Abs_hcomplex")] arg_cong 1);
+by (Auto_tac THEN Ultra_tac 1);
+qed "hcomplex_inverse";
+
+Goalw [hcomplex_zero_def] "inverse (0::hcomplex) = 0";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_inverse]));
+qed "HCOMPLEX_INVERSE_ZERO";
+
+Goal "a / (0::hcomplex) = 0";
+by (simp_tac (simpset() addsimps [hcomplex_divide_def, HCOMPLEX_INVERSE_ZERO]) 1);
+qed "HCOMPLEX_DIVISION_BY_ZERO"; (*NOT for adding to default simpset*)
+
+fun hcomplex_div_undefined_case_tac s i =
+ case_tac s i THEN
+ asm_simp_tac (simpset() addsimps [HCOMPLEX_DIVISION_BY_ZERO, HCOMPLEX_INVERSE_ZERO]) i;
+
+Goalw [hcomplex_zero_def,hcomplex_one_def]
+ "z ~= (0::hcomplex) ==> inverse(z) * z = (1::hcomplex)";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_inverse,hcomplex_mult]));
+by (Ultra_tac 1);
+by (rtac ccontr 1 THEN dtac (complex_mult_inv_left) 1);
+by Auto_tac;
+qed "hcomplex_mult_inv_left";
+Addsimps [hcomplex_mult_inv_left];
+
+Goal "z ~= (0::hcomplex) ==> z * inverse(z) = (1::hcomplex)";
+by (auto_tac (claset() addIs [hcomplex_mult_commute RS subst],simpset()));
+qed "hcomplex_mult_inv_right";
+Addsimps [hcomplex_mult_inv_right];
+
+Goal "(c::hcomplex) ~= (0::hcomplex) ==> (c*a=c*b) = (a=b)";
+by Auto_tac;
+by (dres_inst_tac [("f","%x. x*inverse c")] arg_cong 1);
+by (asm_full_simp_tac (simpset() addsimps hcomplex_mult_ac) 1);
+qed "hcomplex_mult_left_cancel";
+
+Goal "(c::hcomplex) ~= (0::hcomplex) ==> (a*c=b*c) = (a=b)";
+by (Step_tac 1);
+by (dres_inst_tac [("f","%x. x*inverse c")] arg_cong 1);
+by (asm_full_simp_tac (simpset() addsimps hcomplex_mult_ac) 1);
+qed "hcomplex_mult_right_cancel";
+
+Goal "z ~= (0::hcomplex) ==> inverse(z) ~= 0";
+by (Step_tac 1);
+by (ftac (hcomplex_mult_right_cancel RS iffD2) 1);
+by (thin_tac "inverse z = 0" 2);
+by (assume_tac 1 THEN Auto_tac);
+qed "hcomplex_inverse_not_zero";
+Addsimps [hcomplex_inverse_not_zero];
+
+Goal "[| x ~= (0::hcomplex); y ~= 0 |] ==> x * y ~= 0";
+by (Step_tac 1);
+by (dres_inst_tac [("f","%z. inverse x*z")] arg_cong 1);
+by (asm_full_simp_tac (simpset() addsimps [hcomplex_mult_assoc RS sym]) 1);
+qed "hcomplex_mult_not_zero";
+
+bind_thm ("hcomplex_mult_not_zeroE",hcomplex_mult_not_zero RS notE);
+
+Goal "inverse(inverse x) = (x::hcomplex)";
+by (hcomplex_div_undefined_case_tac "x = 0" 1);
+by (res_inst_tac [("c1","inverse x")] (hcomplex_mult_right_cancel RS iffD1) 1);
+by (etac hcomplex_inverse_not_zero 1);
+by (auto_tac (claset() addDs [hcomplex_inverse_not_zero],simpset()));
+qed "hcomplex_inverse_inverse";
+Addsimps [hcomplex_inverse_inverse];
+
+Goalw [hcomplex_one_def] "inverse((1::hcomplex)) = 1";
+by (simp_tac (simpset() addsimps [hcomplex_inverse]) 1);
+qed "hcomplex_inverse_one";
+Addsimps [hcomplex_inverse_one];
+
+Goal "inverse(-x) = -inverse(x::hcomplex)";
+by (hcomplex_div_undefined_case_tac "x = 0" 1);
+by (res_inst_tac [("c1","-x")] (hcomplex_mult_right_cancel RS iffD1) 1);
+by (stac hcomplex_mult_inv_left 2);
+by Auto_tac;
+qed "hcomplex_minus_inverse";
+
+Goal "inverse(x*y) = inverse x * inverse (y::hcomplex)";
+by (hcomplex_div_undefined_case_tac "x = 0" 1);
+by (hcomplex_div_undefined_case_tac "y = 0" 1);
+by (res_inst_tac [("c1","x*y")] (hcomplex_mult_left_cancel RS iffD1) 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult_not_zero]
+ @ hcomplex_mult_ac));
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult_not_zero,
+ hcomplex_mult_assoc RS sym]));
+qed "hcomplex_inverse_distrib";
+
+(*** division ***)
+
+(* adding some of these theorems to simpset as for reals: not 100% convinced for some*)
+
+Goal "(x::hcomplex) * (y/z) = (x*y)/z";
+by (simp_tac (simpset() addsimps [hcomplex_divide_def, hcomplex_mult_assoc]) 1);
+qed "hcomplex_times_divide1_eq";
+
+Goal "(y/z) * (x::hcomplex) = (y*x)/z";
+by (simp_tac (simpset() addsimps [hcomplex_divide_def] @ hcomplex_mult_ac) 1);
+qed "hcomplex_times_divide2_eq";
+
+Addsimps [hcomplex_times_divide1_eq, hcomplex_times_divide2_eq];
+
+Goal "(x::hcomplex) / (y/z) = (x*z)/y";
+by (simp_tac (simpset() addsimps [hcomplex_divide_def, hcomplex_inverse_distrib]@
+ hcomplex_mult_ac) 1);
+qed "hcomplex_divide_divide1_eq";
+
+Goal "((x::hcomplex) / y) / z = x/(y*z)";
+by (simp_tac (simpset() addsimps [hcomplex_divide_def, hcomplex_inverse_distrib,
+ hcomplex_mult_assoc]) 1);
+qed "hcomplex_divide_divide2_eq";
+
+Addsimps [hcomplex_divide_divide1_eq, hcomplex_divide_divide2_eq];
+
+(** As with multiplication, pull minus signs OUT of the / operator **)
+
+Goal "(-x) / (y::hcomplex) = - (x/y)";
+by (simp_tac (simpset() addsimps [hcomplex_divide_def]) 1);
+qed "hcomplex_minus_divide_eq";
+Addsimps [hcomplex_minus_divide_eq];
+
+Goal "(x / -(y::hcomplex)) = - (x/y)";
+by (simp_tac (simpset() addsimps [hcomplex_divide_def, hcomplex_minus_inverse]) 1);
+qed "hcomplex_divide_minus_eq";
+Addsimps [hcomplex_divide_minus_eq];
+
+Goal "(x+y)/(z::hcomplex) = x/z + y/z";
+by (simp_tac (simpset() addsimps [hcomplex_divide_def, hcomplex_add_mult_distrib]) 1);
+qed "hcomplex_add_divide_distrib";
+
+(*---------------------------------------------------------------------------*)
+(* Embedding properties for hcomplex_of_hypreal map *)
+(*---------------------------------------------------------------------------*)
+
+Goalw [hcomplex_of_hypreal_def]
+ "hcomplex_of_hypreal (Abs_hypreal(hyprel `` {%n. X n})) = \
+\ Abs_hcomplex(hcomplexrel `` {%n. complex_of_real (X n)})";
+by (res_inst_tac [("f","Abs_hcomplex")] arg_cong 1);
+by (Auto_tac THEN Ultra_tac 1);
+qed "hcomplex_of_hypreal";
+
+Goal "inj hcomplex_of_hypreal";
+by (rtac injI 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 [hcomplex_of_hypreal]));
+qed "inj_hcomplex_of_hypreal";
+
+Goal "(hcomplex_of_hypreal x = hcomplex_of_hypreal y) = (x = y)";
+by (auto_tac (claset() addDs [inj_hcomplex_of_hypreal RS injD],simpset()));
+qed "hcomplex_of_hypreal_cancel_iff";
+AddIffs [hcomplex_of_hypreal_cancel_iff];
+
+Goal "hcomplex_of_hypreal(-x) = - hcomplex_of_hypreal x";
+by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,
+ hcomplex_minus,hypreal_minus,complex_of_real_minus]));
+qed "hcomplex_of_hypreal_minus";
+
+Goal "hcomplex_of_hypreal(inverse x) = inverse(hcomplex_of_hypreal x)";
+by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,
+ hypreal_inverse,hcomplex_inverse,complex_of_real_inverse]));
+qed "hcomplex_of_hypreal_inverse";
+
+Goal "hcomplex_of_hypreal x + hcomplex_of_hypreal y = \
+\ hcomplex_of_hypreal (x + y)";
+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 [hcomplex_of_hypreal,
+ hypreal_add,hcomplex_add,complex_of_real_add]));
+qed "hcomplex_of_hypreal_add";
+
+Goalw [hcomplex_diff_def]
+ "hcomplex_of_hypreal x - hcomplex_of_hypreal y = \
+\ hcomplex_of_hypreal (x - y)";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal_minus
+ RS sym,hcomplex_of_hypreal_add,hypreal_diff_def]));
+qed "hcomplex_of_hypreal_diff";
+
+Goal "hcomplex_of_hypreal x * hcomplex_of_hypreal y = \
+\ hcomplex_of_hypreal (x * y)";
+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 [hcomplex_of_hypreal,
+ hypreal_mult,hcomplex_mult,complex_of_real_mult]));
+qed "hcomplex_of_hypreal_mult";
+
+Goalw [hcomplex_divide_def]
+ "hcomplex_of_hypreal x / hcomplex_of_hypreal y = hcomplex_of_hypreal(x/y)";
+by (hypreal_div_undefined_case_tac "y=0" 1);
+by (simp_tac (simpset() addsimps [rename_numerals HYPREAL_DIVISION_BY_ZERO,
+ HCOMPLEX_INVERSE_ZERO]) 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal_mult,
+ hcomplex_of_hypreal_inverse RS sym]));
+by (simp_tac (simpset() addsimps [hypreal_divide_def]) 1);
+qed "hcomplex_of_hypreal_divide";
+
+Goalw [hcomplex_one_def]
+ "hcomplex_of_hypreal 1 = 1";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,hypreal_one_num]));
+qed "hcomplex_of_hypreal_one";
+
+Goalw [hcomplex_zero_def,hypreal_zero_def]
+ "hcomplex_of_hypreal 0 = 0";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal]));
+qed "hcomplex_of_hypreal_zero";
+
+Addsimps [hcomplex_of_hypreal_one,hcomplex_of_hypreal_zero,
+ rename_numerals hcomplex_of_hypreal_zero];
+
+Goal "hcomplex_of_hypreal (x ^ n) = (hcomplex_of_hypreal x) ^ n";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal_mult RS sym]));
+qed "hcomplex_of_hypreal_pow";
+
+Goal "hRe(hcomplex_of_hypreal z) = z";
+by (res_inst_tac [("z","z")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,hRe]));
+qed "hRe_hcomplex_of_hypreal";
+Addsimps [hRe_hcomplex_of_hypreal];
+
+Goal "hIm(hcomplex_of_hypreal z) = 0";
+by (res_inst_tac [("z","z")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,hIm,
+ hypreal_zero_num]));
+qed "hIm_hcomplex_of_hypreal";
+Addsimps [hIm_hcomplex_of_hypreal];
+
+Goal "hcomplex_of_hypreal epsilon ~= 0";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,
+ epsilon_def,hcomplex_zero_def]));
+qed "hcomplex_of_hypreal_epsilon_not_zero";
+Addsimps [hcomplex_of_hypreal_epsilon_not_zero];
+
+(*---------------------------------------------------------------------------*)
+(* Modulus (absolute value) of nonstandard complex number *)
+(*---------------------------------------------------------------------------*)
+
+Goalw [hcmod_def]
+ "hcmod (Abs_hcomplex(hcomplexrel `` {%n. X n})) = \
+\ Abs_hypreal(hyprel `` {%n. cmod (X n)})";
+by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);
+by (Auto_tac THEN Ultra_tac 1);
+qed "hcmod";
+
+Goalw [hcomplex_zero_def,hypreal_zero_def]
+ "hcmod(0) = 0";
+by (auto_tac (claset(),simpset() addsimps [hcmod]));
+qed "hcmod_zero";
+Addsimps [hcmod_zero,rename_numerals hcmod_zero];
+
+Goalw [hcomplex_one_def]
+ "hcmod(1) = 1";
+by (auto_tac (claset(),simpset() addsimps [hcmod,hypreal_one_num]));
+qed "hcmod_one";
+Addsimps [hcmod_one];
+
+Goal "hcmod(hcomplex_of_hypreal x) = abs x";
+by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcmod,hcomplex_of_hypreal,
+ hypreal_hrabs]));
+qed "hcmod_hcomplex_of_hypreal";
+Addsimps [hcmod_hcomplex_of_hypreal];
+
+Goal "hcomplex_of_hypreal (abs x) = \
+\ hcomplex_of_hypreal(hcmod(hcomplex_of_hypreal x))";
+by (Simp_tac 1);
+qed "hcomplex_of_hypreal_abs";
+
+(*---------------------------------------------------------------------------*)
+(* conjugation *)
+(*---------------------------------------------------------------------------*)
+
+Goalw [hcnj_def]
+ "hcnj (Abs_hcomplex(hcomplexrel `` {%n. X n})) = \
+\ Abs_hcomplex(hcomplexrel `` {%n. cnj(X n)})";
+by (res_inst_tac [("f","Abs_hcomplex")] arg_cong 1);
+by (Auto_tac THEN Ultra_tac 1);
+qed "hcnj";
+
+Goal "inj hcnj";
+by (rtac injI 1);
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcnj]));
+qed "inj_hcnj";
+
+Goal "(hcnj x = hcnj y) = (x = y)";
+by (auto_tac (claset() addDs [inj_hcnj RS injD],simpset()));
+qed "hcomplex_hcnj_cancel_iff";
+Addsimps [hcomplex_hcnj_cancel_iff];
+
+Goal "hcnj (hcnj z) = z";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcnj]));
+qed "hcomplex_hcnj_hcnj";
+Addsimps [hcomplex_hcnj_hcnj];
+
+Goal "hcnj (hcomplex_of_hypreal x) = hcomplex_of_hypreal x";
+by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcnj,hcomplex_of_hypreal]));
+qed "hcomplex_hcnj_hcomplex_of_hypreal";
+Addsimps [hcomplex_hcnj_hcomplex_of_hypreal];
+
+Goal "hcmod (hcnj z) = hcmod z";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcnj,hcmod]));
+qed "hcomplex_hmod_hcnj";
+Addsimps [hcomplex_hmod_hcnj];
+
+Goal "hcnj (-z) = - hcnj z";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcnj,hcomplex_minus,
+ complex_cnj_minus]));
+qed "hcomplex_hcnj_minus";
+
+Goal "hcnj(inverse z) = inverse(hcnj z)";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcnj,hcomplex_inverse,
+ complex_cnj_inverse]));
+qed "hcomplex_hcnj_inverse";
+
+Goal "hcnj(w + z) = hcnj(w) + hcnj(z)";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","w")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcnj,hcomplex_add,
+ complex_cnj_add]));
+qed "hcomplex_hcnj_add";
+
+Goal "hcnj(w - z) = hcnj(w) - hcnj(z)";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","w")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcnj,hcomplex_diff,
+ complex_cnj_diff]));
+qed "hcomplex_hcnj_diff";
+
+Goal "hcnj(w * z) = hcnj(w) * hcnj(z)";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","w")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcnj,hcomplex_mult,
+ complex_cnj_mult]));
+qed "hcomplex_hcnj_mult";
+
+Goalw [hcomplex_divide_def] "hcnj(w / z) = (hcnj w)/(hcnj z)";
+by (simp_tac (simpset() addsimps [hcomplex_hcnj_mult,hcomplex_hcnj_inverse]) 1);
+qed "hcomplex_hcnj_divide";
+
+Goalw [hcomplex_one_def] "hcnj 1 = 1";
+by (simp_tac (simpset() addsimps [hcnj]) 1);
+qed "hcnj_one";
+Addsimps [hcnj_one];
+
+Goal "hcnj(z ^ n) = hcnj(z) ^ n";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_hcnj_mult]));
+qed "hcomplex_hcnj_pow";
+
+(* MOVE to NSComplexBin
+Goal "z + hcnj z = \
+\ hcomplex_of_hypreal (2 * hRe(z))";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),HOL_ss addsimps [hRe,hcnj,hcomplex_add,
+ hypreal_mult,hcomplex_of_hypreal,complex_add_cnj]));
+qed "hcomplex_add_hcnj";
+
+Goal "z - hcnj z = \
+\ hcomplex_of_hypreal (hypreal_of_real 2 * hIm(z)) * iii";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hIm,hcnj,hcomplex_diff,
+ hypreal_of_real_def,hypreal_mult,hcomplex_of_hypreal,
+ complex_diff_cnj,iii_def,hcomplex_mult]));
+qed "hcomplex_diff_hcnj";
+*)
+
+Goalw [hcomplex_zero_def]
+ "hcnj 0 = 0";
+by (auto_tac (claset(),simpset() addsimps [hcnj]));
+qed "hcomplex_hcnj_zero";
+Addsimps [hcomplex_hcnj_zero];
+
+Goal "(hcnj z = 0) = (z = 0)";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_zero_def,
+ hcnj]));
+qed "hcomplex_hcnj_zero_iff";
+AddIffs [hcomplex_hcnj_zero_iff];
+
+Goal "z * hcnj z = hcomplex_of_hypreal (hRe(z) ^ 2 + hIm(z) ^ 2)";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcnj,hcomplex_mult,
+ hcomplex_of_hypreal,hRe,hIm,hypreal_add,hypreal_mult,
+ complex_mult_cnj,two_eq_Suc_Suc]));
+qed "hcomplex_mult_hcnj";
+
+
+(*---------------------------------------------------------------------------*)
+(* some algebra etc. *)
+(*---------------------------------------------------------------------------*)
+
+Goal "(x*y = (0::hcomplex)) = (x = 0 | y = 0)";
+by Auto_tac;
+by (auto_tac (claset() addIs [ccontr] addDs
+ [hcomplex_mult_not_zero],simpset()));
+qed "hcomplex_mult_zero_iff";
+Addsimps [hcomplex_mult_zero_iff];
+
+Goal "(x + y = x) = (y = (0::hcomplex))";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_add,
+ hcomplex_zero_def]));
+qed "hcomplex_add_left_cancel_zero";
+Addsimps [hcomplex_add_left_cancel_zero];
+
+Goalw [hcomplex_diff_def]
+ "((z1::hcomplex) - z2) * w = (z1 * w) - (z2 * w)";
+by (simp_tac (simpset() addsimps [hcomplex_add_mult_distrib]) 1);
+qed "hcomplex_diff_mult_distrib";
+
+Goalw [hcomplex_diff_def]
+ "(w::hcomplex) * (z1 - z2) = (w * z1) - (w * z2)";
+by (simp_tac (simpset() addsimps [hcomplex_add_mult_distrib2]) 1);
+qed "hcomplex_diff_mult_distrib2";
+
+(*---------------------------------------------------------------------------*)
+(* More theorems about hcmod *)
+(*---------------------------------------------------------------------------*)
+
+Goal "(hcmod x = 0) = (x = 0)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcmod,hcomplex_zero_def,
+ hypreal_zero_num]));
+qed "hcomplex_hcmod_eq_zero_cancel";
+Addsimps [hcomplex_hcmod_eq_zero_cancel];
+
+(* not proved already? strange! *)
+Goalw [hypreal_le_def]
+ "(hypreal_of_nat n <= hypreal_of_nat m) = (n <= m)";
+by Auto_tac;
+qed "hypreal_of_nat_le_iff";
+Addsimps [hypreal_of_nat_le_iff];
+
+Goal "0 <= hypreal_of_nat n";
+by (simp_tac (simpset() addsimps [hypreal_of_nat_zero RS sym]
+ delsimps [hypreal_of_nat_zero]) 1);
+qed "hypreal_of_nat_ge_zero";
+Addsimps [hypreal_of_nat_ge_zero];
+
+Addsimps [hypreal_of_nat_ge_zero RS hrabs_eqI1];
+
+Goal "0 <= hypreal_of_hypnat n";
+by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
+by (asm_simp_tac (simpset() addsimps [hypreal_of_hypnat,
+ hypreal_zero_num,hypreal_le]) 1);
+qed "hypreal_of_hypnat_ge_zero";
+Addsimps [hypreal_of_hypnat_ge_zero];
+
+Addsimps [hypreal_of_hypnat_ge_zero RS hrabs_eqI1];
+
+Goal "hcmod (hcomplex_of_hypreal(hypreal_of_nat n)) = hypreal_of_nat n";
+by Auto_tac;
+qed "hcmod_hcomplex_of_hypreal_of_nat";
+Addsimps [hcmod_hcomplex_of_hypreal_of_nat];
+
+Goal "hcmod (hcomplex_of_hypreal(hypreal_of_hypnat n)) = hypreal_of_hypnat n";
+by Auto_tac;
+qed "hcmod_hcomplex_of_hypreal_of_hypnat";
+Addsimps [hcmod_hcomplex_of_hypreal_of_hypnat];
+
+Goal "hcmod (-x) = hcmod(x)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcmod,hcomplex_minus]));
+qed "hcmod_minus";
+Addsimps [hcmod_minus];
+
+Goal "hcmod(z * hcnj(z)) = hcmod(z) ^ 2";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcmod,hcomplex_mult,
+ hcnj,hypreal_mult,complex_mod_mult_cnj,two_eq_Suc_Suc]));
+qed "hcmod_mult_hcnj";
+
+Goal "(0::hypreal) <= hcmod x";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcmod,
+ hypreal_zero_num,hypreal_le]));
+qed "hcmod_ge_zero";
+Addsimps [hcmod_ge_zero];
+
+Goal "abs(hcmod x) = hcmod x";
+by (auto_tac (claset() addIs [hrabs_eqI1],simpset()));
+qed "hrabs_hcmod_cancel";
+Addsimps [hrabs_hcmod_cancel];
+
+Goal "hcmod(x*y) = hcmod(x) * hcmod(y)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcmod,hcomplex_mult,
+ hypreal_mult,complex_mod_mult]));
+qed "hcmod_mult";
+
+Goal "hcmod(x + y) ^ 2 = hcmod(x) ^ 2 + hcmod(y) ^ 2 + 2 * hRe(x * hcnj y)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcmod,hcomplex_add,
+ hypreal_mult,hRe,hcnj,hcomplex_mult,two_eq_Suc_Suc,
+ realpow_two RS sym] delsimps [realpow_Suc]));
+by (auto_tac (claset(),simpset() addsimps [two_eq_Suc_Suc RS sym,
+ complex_mod_add_squared_eq,hypreal_add RS sym,hypreal_mult RS sym,
+ symmetric hypreal_of_real_def]));
+qed "hcmod_add_squared_eq";
+
+Goal "hRe(x * hcnj y) <= hcmod(x * hcnj y)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcmod,hcnj,
+ hcomplex_mult,hRe,hypreal_le]));
+qed "hcomplex_hRe_mult_hcnj_le_hcmod";
+Addsimps [hcomplex_hRe_mult_hcnj_le_hcmod];
+
+Goal "hRe(x * hcnj y) <= hcmod(x * y)";
+by (cut_inst_tac [("x","x"),("y","y")] hcomplex_hRe_mult_hcnj_le_hcmod 1);
+by (asm_full_simp_tac (simpset() addsimps [hcmod_mult]) 1);
+qed "hcomplex_hRe_mult_hcnj_le_hcmod2";
+Addsimps [hcomplex_hRe_mult_hcnj_le_hcmod2];
+
+Goal "hcmod (x + y) ^ 2 <= (hcmod(x) + hcmod(y)) ^ 2";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcmod,hcnj,
+ hcomplex_add,hypreal_mult,hypreal_add,hypreal_le,
+ realpow_two RS sym,two_eq_Suc_Suc] delsimps [realpow_Suc]));
+by (simp_tac (simpset() addsimps [two_eq_Suc_Suc RS sym]) 1);
+qed "hcmod_triangle_squared";
+Addsimps [hcmod_triangle_squared];
+
+Goal "hcmod (x + y) <= hcmod(x) + hcmod(y)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcmod,
+ hcomplex_add,hypreal_add,hypreal_le]));
+qed "hcmod_triangle_ineq";
+Addsimps [hcmod_triangle_ineq];
+
+Goal "hcmod(b + a) - hcmod b <= hcmod a";
+by (cut_inst_tac [("x1","b"),("y1","a"),("x","-hcmod b")]
+ (hcmod_triangle_ineq RS hypreal_add_le_mono1) 1);
+by (asm_full_simp_tac (simpset() addsimps hypreal_add_ac) 1);
+qed "hcmod_triangle_ineq2";
+Addsimps [hcmod_triangle_ineq2];
+
+Goal "hcmod (x - y) = hcmod (y - x)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcmod,
+ hcomplex_diff,complex_mod_diff_commute]));
+qed "hcmod_diff_commute";
+
+Goal "[| hcmod x < r; hcmod y < s |] ==> hcmod (x + y) < r + s";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 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 [hcmod,hcomplex_add,
+ hypreal_add,hypreal_less]));
+by (ultra_tac (claset() addIs [complex_mod_add_less],simpset()) 1);
+qed "hcmod_add_less";
+
+Goal "[| hcmod x < r; hcmod y < s |] ==> hcmod (x * y) < r * s";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","y")] eq_Abs_hcomplex 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 [hcmod,hypreal_mult,
+ hypreal_less,hcomplex_mult]));
+by (ultra_tac (claset() addIs [complex_mod_mult_less],simpset()) 1);
+qed "hcmod_mult_less";
+
+goal NSComplex.thy "hcmod(a) - hcmod(b) <= hcmod(a + b)";
+by (res_inst_tac [("z","a")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","b")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcmod,hcomplex_add,
+ hypreal_diff,hypreal_le]));
+qed "hcmod_diff_ineq";
+Addsimps [hcmod_diff_ineq];
+
+
+(*---------------------------------------------------------------------------*)
+(* a few nonlinear theorems *)
+(*---------------------------------------------------------------------------*)
+
+Goalw [hcpow_def]
+ "Abs_hcomplex(hcomplexrel``{%n. X n}) hcpow \
+\ Abs_hypnat(hypnatrel``{%n. Y n}) = \
+\ Abs_hcomplex(hcomplexrel``{%n. X n ^ Y n})";
+by (res_inst_tac [("f","Abs_hcomplex")] arg_cong 1);
+by (Auto_tac THEN Ultra_tac 1);
+qed "hcpow";
+
+Goal "hcomplex_of_hypreal (x pow n) = (hcomplex_of_hypreal x) hcpow n";
+by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,
+ hyperpow,hcpow,complex_of_real_pow]));
+qed "hcomplex_of_hypreal_hyperpow";
+
+Goal "hcmod(x ^ n) = hcmod(x) ^ n";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps [hcmod_mult]));
+qed "hcmod_hcomplexpow";
+
+Goal "hcmod(x hcpow n) = hcmod(x) pow n";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
+by (auto_tac (claset(),simpset() addsimps [hcpow,hyperpow,
+ hcmod,complex_mod_complexpow]));
+qed "hcmod_hcpow";
+
+Goal "(-x::hcomplex) ^ n = (if even n then (x ^ n) else -(x ^ n))";
+by (induct_tac "n" 1);
+by Auto_tac;
+qed "hcomplexpow_minus";
+
+Goal "(-x::hcomplex) hcpow n = \
+\ (if ( *pNat* even) n then (x hcpow n) else -(x hcpow n))";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
+by (auto_tac (claset(),simpset() addsimps [hcpow,hyperpow,starPNat,
+ hcomplex_minus]));
+by (ALLGOALS(ultra_tac (claset(),simpset() addsimps [complexpow_minus])));
+qed "hcpow_minus";
+
+Goal "inverse(-x) = - inverse (x::hcomplex)";
+by (res_inst_tac [("z","x")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_inverse,hcomplex_minus,
+ complex_inverse_minus]));
+qed "hccomplex_inverse_minus";
+
+Goalw [hcomplex_divide_def] "x / (1::hcomplex) = x";
+by (Simp_tac 1);
+qed "hcomplex_div_one";
+Addsimps [hcomplex_div_one];
+
+Goal "hcmod(inverse x) = inverse(hcmod x)";
+by (hcomplex_div_undefined_case_tac "x = 0" 1);
+by (res_inst_tac [("c1","hcmod x")] (hypreal_mult_left_cancel RS iffD1) 1);
+by (auto_tac (claset(),simpset() addsimps [hcmod_mult RS sym]));
+qed "hcmod_hcomplex_inverse";
+
+Goalw [hcomplex_divide_def,hypreal_divide_def]
+ "hcmod(x/y) = hcmod(x)/(hcmod y)";
+by (auto_tac (claset(),simpset() addsimps [hcmod_mult,
+ hcmod_hcomplex_inverse]));
+qed "hcmod_divide";
+
+Goalw [hcomplex_divide_def]
+ "inverse(x/y) = y/(x::hcomplex)";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_inverse_distrib,
+ hcomplex_mult_commute]));
+qed "hcomplex_inverse_divide";
+Addsimps [hcomplex_inverse_divide];
+
+Goal "((r::hcomplex) * s) ^ n = (r ^ n) * (s ^ n)";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps hcomplex_mult_ac));
+qed "hcomplexpow_mult";
+
+Goal "((r::hcomplex) * s) hcpow n = (r hcpow n) * (s hcpow n)";
+by (res_inst_tac [("z","r")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","s")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
+by (auto_tac (claset(),simpset() addsimps [hcpow,hypreal_mult,
+ hcomplex_mult,complexpow_mult]));
+qed "hcpow_mult";
+
+Goal "(0::hcomplex) ^ (Suc n) = 0";
+by (Auto_tac);
+qed "hcomplexpow_zero";
+Addsimps [hcomplexpow_zero];
+
+Goalw [hcomplex_zero_def,hypnat_one_def]
+ "0 hcpow (n + 1) = 0";
+by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
+by (auto_tac (claset(),simpset() addsimps [hcpow,hypnat_add]));
+qed "hcpow_zero";
+Addsimps [hcpow_zero];
+
+Goalw [hSuc_def]
+ "0 hcpow (hSuc n) = 0";
+by (Simp_tac 1);
+qed "hcpow_zero2";
+Addsimps [hcpow_zero2];
+
+Goal "r ~= (0::hcomplex) --> r ^ n ~= 0";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps
+ [hcomplex_mult_not_zero]));
+qed_spec_mp "hcomplexpow_not_zero";
+Addsimps [hcomplexpow_not_zero];
+AddIs [hcomplexpow_not_zero];
+
+Goal "r ~= 0 ==> r hcpow n ~= (0::hcomplex)";
+by (res_inst_tac [("z","r")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
+by (auto_tac (claset(),simpset() addsimps [hcpow,
+ hcomplex_zero_def]));
+by (ultra_tac (claset() addDs [complexpow_zero_zero],simpset()) 1);
+qed "hcpow_not_zero";
+Addsimps [hcpow_not_zero];
+AddIs [hcpow_not_zero];
+
+Goal "r ^ n = (0::hcomplex) ==> r = 0";
+by (blast_tac (claset() addIs [ccontr]
+ addDs [hcomplexpow_not_zero]) 1);
+qed "hcomplexpow_zero_zero";
+
+Goal "r hcpow n = (0::hcomplex) ==> r = 0";
+by (blast_tac (claset() addIs [ccontr]
+ addDs [hcpow_not_zero]) 1);
+qed "hcpow_zero_zero";
+
+Goalw [iii_def] "iii * iii = - 1";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult,
+ hcomplex_one_def,hcomplex_minus]));
+qed "hcomplex_i_mult_eq";
+Addsimps [hcomplex_i_mult_eq];
+
+Goal "iii ^ 2 = - 1";
+by (simp_tac (simpset() addsimps [two_eq_Suc_Suc]) 1);
+qed "hcomplexpow_i_squared";
+Addsimps [hcomplexpow_i_squared];
+
+Goalw [iii_def,hcomplex_zero_def] "iii ~= 0";
+by Auto_tac;
+qed "hcomplex_i_not_zero";
+Addsimps [hcomplex_i_not_zero];
+
+Goal "x * y ~= (0::hcomplex) ==> x ~= 0";
+by Auto_tac;
+qed "hcomplex_mult_eq_zero_cancel1";
+
+Goal "x * y ~= (0::hcomplex) ==> y ~= 0";
+by Auto_tac;
+qed "hcomplex_mult_eq_zero_cancel2";
+
+Goal "(x * y ~= (0::hcomplex)) = (x ~= 0 & y ~= 0)";
+by Auto_tac;
+qed "hcomplex_mult_not_eq_zero_iff";
+AddIffs [hcomplex_mult_not_eq_zero_iff];
+
+Goalw [hcomplex_divide_def,complex_divide_def]
+ "Abs_hcomplex(hcomplexrel``{%n. X n}) / Abs_hcomplex(hcomplexrel``{%n. Y n}) = \
+\ Abs_hcomplex(hcomplexrel``{%n. X n / Y n})";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_inverse,hcomplex_mult]));
+qed "hcomplex_divide";
+
+(*---------------------------------------------------------------------------*)
+(* hsgn *)
+(*---------------------------------------------------------------------------*)
+
+Goalw [hsgn_def]
+ "hsgn (Abs_hcomplex(hcomplexrel `` {%n. X n})) = \
+\ Abs_hcomplex(hcomplexrel `` {%n. sgn (X n)})";
+by (res_inst_tac [("f","Abs_hcomplex")] arg_cong 1);
+by (Auto_tac THEN Ultra_tac 1);
+qed "hsgn";
+
+Addsimps [rename_numerals sgn_zero];
+Goalw [hcomplex_zero_def] "hsgn 0 = 0";
+by (simp_tac (simpset() addsimps [hsgn]) 1);
+qed "hsgn_zero";
+Addsimps[hsgn_zero];
+
+Addsimps [rename_numerals sgn_one];
+
+Goalw [hcomplex_one_def] "hsgn 1 = 1";
+by (simp_tac (simpset() addsimps [hsgn]) 1);
+qed "hsgn_one";
+Addsimps[hsgn_one];
+
+Goal "hsgn (-z) = - hsgn(z)";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hsgn,hcomplex_minus,
+ sgn_minus]));
+qed "hsgn_minus";
+
+Goal "hsgn z = z / hcomplex_of_hypreal (hcmod z)";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hsgn,hcomplex_divide,
+ hcomplex_of_hypreal,hcmod,sgn_eq]));
+qed "hsgn_eq";
+
+Goal "(EX (x::hypreal) y. P x y) = \
+\ (EX f g. P (Abs_hypreal(hyprel `` {f})) (Abs_hypreal(hyprel `` {g})))";
+by Auto_tac;
+by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
+by Auto_tac;
+qed "lemma_hypreal_P_EX2";
+
+Goal "ALL (n::nat). EX x y. (z n) = complex_of_real(x) + ii * complex_of_real(y)";
+by (blast_tac (claset() addIs [complex_split]) 1);
+qed "complex_split2";
+
+(* Interesting proof! *)
+Goal "EX x y. z = hcomplex_of_hypreal(x) + iii * hcomplex_of_hypreal(y)";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [lemma_hypreal_P_EX2,
+ hcomplex_of_hypreal,iii_def,hcomplex_add,hcomplex_mult]));
+by (cut_inst_tac [("z","x")] complex_split2 1);
+by (REPEAT(dtac choice 1 THEN Step_tac 1));
+by (res_inst_tac [("x","f")] exI 1);
+by (res_inst_tac [("x","fa")] exI 1);
+by Auto_tac;
+qed "hcomplex_split";
+
+Goal "hRe(hcomplex_of_hypreal(x) + iii * hcomplex_of_hypreal(y)) = x";
+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 [hRe,iii_def,hcomplex_add,
+ hcomplex_mult,hcomplex_of_hypreal]));
+qed "hRe_hcomplex_i";
+Addsimps [hRe_hcomplex_i];
+
+Goal "hIm(hcomplex_of_hypreal(x) + iii * hcomplex_of_hypreal(y)) = y";
+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 [hIm,iii_def,hcomplex_add,
+ hcomplex_mult,hcomplex_of_hypreal]));
+qed "hIm_hcomplex_i";
+Addsimps [hIm_hcomplex_i];
+
+Goal "hcmod (hcomplex_of_hypreal(x) + iii * hcomplex_of_hypreal(y)) = \
+\ ( *f* sqrt) (x ^ 2 + y ^ 2)";
+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 [hcomplex_of_hypreal,
+ iii_def,hcomplex_add,hcomplex_mult,starfun,hypreal_mult,
+ hypreal_add,hcmod,cmod_i,two_eq_Suc_Suc]));
+qed "hcmod_i";
+
+Goalw [iii_def]
+ "hcomplex_of_hypreal xa + iii * hcomplex_of_hypreal ya = \
+\ hcomplex_of_hypreal xb + iii * hcomplex_of_hypreal yb \
+\ ==> xa = xb";
+by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","ya")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","xb")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","yb")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult,hcomplex_add,
+ hcomplex_of_hypreal]));
+by (Ultra_tac 1);
+qed "hcomplex_eq_hRe_eq";
+
+Goalw [iii_def]
+ "hcomplex_of_hypreal xa + iii * hcomplex_of_hypreal ya = \
+\ hcomplex_of_hypreal xb + iii * hcomplex_of_hypreal yb \
+\ ==> ya = yb";
+by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","ya")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","xb")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","yb")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult,hcomplex_add,
+ hcomplex_of_hypreal]));
+by (Ultra_tac 1);
+qed "hcomplex_eq_hIm_eq";
+
+Goal "(hcomplex_of_hypreal xa + iii * hcomplex_of_hypreal ya = \
+\ hcomplex_of_hypreal xb + iii * hcomplex_of_hypreal yb) = \
+\ ((xa = xb) & (ya = yb))";
+by (auto_tac (claset() addIs [hcomplex_eq_hIm_eq,hcomplex_eq_hRe_eq],simpset()));
+qed "hcomplex_eq_cancel_iff";
+Addsimps [hcomplex_eq_cancel_iff];
+
+Goal "(hcomplex_of_hypreal xa + hcomplex_of_hypreal ya * iii = \
+\ hcomplex_of_hypreal xb + hcomplex_of_hypreal yb * iii ) = ((xa = xb) & (ya = yb))";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult_commute]));
+qed "hcomplex_eq_cancel_iffA";
+AddIffs [hcomplex_eq_cancel_iffA];
+
+Goal "(hcomplex_of_hypreal xa + hcomplex_of_hypreal ya * iii = \
+\ hcomplex_of_hypreal xb + iii * hcomplex_of_hypreal yb) = ((xa = xb) & (ya = yb))";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult_commute]));
+qed "hcomplex_eq_cancel_iffB";
+AddIffs [hcomplex_eq_cancel_iffB];
+
+Goal "(hcomplex_of_hypreal xa + iii * hcomplex_of_hypreal ya = \
+\ hcomplex_of_hypreal xb + hcomplex_of_hypreal yb * iii) = ((xa = xb) & (ya = yb))";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult_commute]));
+qed "hcomplex_eq_cancel_iffC";
+AddIffs [hcomplex_eq_cancel_iffC];
+
+Goal"(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y = \
+\ hcomplex_of_hypreal xa) = (x = xa & y = 0)";
+by (cut_inst_tac [("xa","x"),("ya","y"),("xb","xa"),("yb","0")]
+ hcomplex_eq_cancel_iff 1);
+by (asm_full_simp_tac (simpset() delsimps [hcomplex_eq_cancel_iff]) 1);
+qed "hcomplex_eq_cancel_iff2";
+Addsimps [hcomplex_eq_cancel_iff2];
+
+Goal"(hcomplex_of_hypreal x + hcomplex_of_hypreal y * iii = \
+\ hcomplex_of_hypreal xa) = (x = xa & y = 0)";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult_commute]));
+qed "hcomplex_eq_cancel_iff2a";
+Addsimps [hcomplex_eq_cancel_iff2a];
+
+Goal "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y = \
+\ iii * hcomplex_of_hypreal ya) = (x = 0 & y = ya)";
+by (cut_inst_tac [("xa","x"),("ya","y"),("xb","0"),("yb","ya")]
+ hcomplex_eq_cancel_iff 1);
+by (asm_full_simp_tac (simpset() delsimps [hcomplex_eq_cancel_iff]) 1);
+qed "hcomplex_eq_cancel_iff3";
+Addsimps [hcomplex_eq_cancel_iff3];
+
+Goal "(hcomplex_of_hypreal x + hcomplex_of_hypreal y * iii = \
+\ iii * hcomplex_of_hypreal ya) = (x = 0 & y = ya)";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult_commute]));
+qed "hcomplex_eq_cancel_iff3a";
+Addsimps [hcomplex_eq_cancel_iff3a];
+
+Goalw [iii_def]
+ "hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y = 0 \
+\ ==> x = 0";
+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 [hcomplex_of_hypreal,
+ hcomplex_add,hcomplex_mult,hcomplex_zero_def,hypreal_zero_num]));
+by (ultra_tac (claset(),simpset() addsimps [complex_split_Re_zero]) 1);
+qed "hcomplex_split_hRe_zero";
+
+Goalw [iii_def]
+ "hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y = 0 \
+\ ==> y = 0";
+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 [hcomplex_of_hypreal,
+ hcomplex_add,hcomplex_mult,hcomplex_zero_def,hypreal_zero_num]));
+by (ultra_tac (claset(),simpset() addsimps [complex_split_Im_zero]) 1);
+qed "hcomplex_split_hIm_zero";
+
+Goal "hRe(hsgn z) = hRe(z)/hcmod z";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hsgn,hcmod,hRe,hypreal_divide]));
+qed "hRe_hsgn";
+Addsimps [hRe_hsgn];
+
+Goal "hIm(hsgn z) = hIm(z)/hcmod z";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hsgn,hcmod,hIm,hypreal_divide]));
+qed "hIm_hsgn";
+Addsimps [hIm_hsgn];
+
+Goal "(x*x + y*y = 0) = ((x::real) = 0 & y = 0)";
+by (auto_tac (claset() addIs [real_sum_squares_cancel],simpset()));
+qed "real_two_squares_add_zero_iff";
+Addsimps [real_two_squares_add_zero_iff];
+
+Goal "inverse(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y) = \
+\ hcomplex_of_hypreal(x/(x ^ 2 + y ^ 2)) - \
+\ iii * hcomplex_of_hypreal(y/(x ^ 2 + y ^ 2))";
+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 [hcomplex_of_hypreal,
+ hcomplex_mult,hcomplex_add,iii_def,starfun,hypreal_mult,
+ hypreal_add,hcomplex_inverse,hypreal_divide,hcomplex_diff,
+ complex_inverse_complex_split,two_eq_Suc_Suc]));
+qed "hcomplex_inverse_complex_split";
+
+Goalw [iii_def]
+ "hRe (iii * hcomplex_of_hypreal y) = 0";
+by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,
+ hcomplex_mult,hRe,hypreal_zero_num]));
+qed "hRe_mult_i_eq";
+Addsimps [hRe_mult_i_eq];
+
+Goalw [iii_def]
+ "hIm (iii * hcomplex_of_hypreal y) = y";
+by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,
+ hcomplex_mult,hIm,hypreal_zero_num]));
+qed "hIm_mult_i_eq";
+Addsimps [hIm_mult_i_eq];
+
+Goal "hcmod (iii * hcomplex_of_hypreal y) = abs y";
+by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,
+ hcmod,hypreal_hrabs,iii_def,hcomplex_mult]));
+qed "hcmod_mult_i";
+Addsimps [hcmod_mult_i];
+
+Goal "hcmod (hcomplex_of_hypreal y * iii) = abs y";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_mult_commute]));
+qed "hcmod_mult_i2";
+Addsimps [hcmod_mult_i2];
+
+(*---------------------------------------------------------------------------*)
+(* harg *)
+(*---------------------------------------------------------------------------*)
+
+Goalw [harg_def]
+ "harg (Abs_hcomplex(hcomplexrel `` {%n. X n})) = \
+\ Abs_hypreal(hyprel `` {%n. arg (X n)})";
+by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);
+by (Auto_tac THEN Ultra_tac 1);
+qed "harg";
+
+Goal "0 < y ==> ( *f* cos) (harg(iii * hcomplex_of_hypreal y)) = 0";
+by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,
+ iii_def,hcomplex_mult,hypreal_zero_num,hypreal_less,starfun,
+ harg]));
+by (Ultra_tac 1);
+qed "cos_harg_i_mult_zero";
+Addsimps [cos_harg_i_mult_zero];
+
+Goal "y < 0 ==> ( *f* cos) (harg(iii * hcomplex_of_hypreal y)) = 0";
+by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,
+ iii_def,hcomplex_mult,hypreal_zero_num,hypreal_less,starfun,
+ harg]));
+by (Ultra_tac 1);
+qed "cos_harg_i_mult_zero2";
+Addsimps [cos_harg_i_mult_zero2];
+
+Goal "(hcomplex_of_hypreal y ~= 0) = (y ~= 0)";
+by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,
+ hypreal_zero_num,hcomplex_zero_def]));
+qed "hcomplex_of_hypreal_not_zero_iff";
+Addsimps [hcomplex_of_hypreal_not_zero_iff];
+
+Goal "(hcomplex_of_hypreal y = 0) = (y = 0)";
+by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,
+ hypreal_zero_num,hcomplex_zero_def]));
+qed "hcomplex_of_hypreal_zero_iff";
+Addsimps [hcomplex_of_hypreal_zero_iff];
+
+Goal "y ~= 0 ==> ( *f* cos) (harg(iii * hcomplex_of_hypreal y)) = 0";
+by (cut_inst_tac [("x","y"),("y","0")] hypreal_linear 1);
+by Auto_tac;
+qed "cos_harg_i_mult_zero3";
+Addsimps [cos_harg_i_mult_zero3];
+
+(*---------------------------------------------------------------------------*)
+(* Polar form for nonstandard complex numbers *)
+(*---------------------------------------------------------------------------*)
+
+Goal "ALL n. EX r a. (z n) = complex_of_real r * \
+\ (complex_of_real(cos a) + ii * complex_of_real(sin a))";
+by (blast_tac (claset() addIs [complex_split_polar]) 1);
+qed "complex_split_polar2";
+
+Goal
+ "EX r a. z = hcomplex_of_hypreal r * \
+\ (hcomplex_of_hypreal(( *f* cos) a) + iii * hcomplex_of_hypreal(( *f* sin) a))";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [lemma_hypreal_P_EX2,
+ hcomplex_of_hypreal,iii_def,starfun,hcomplex_add,hcomplex_mult]));
+by (cut_inst_tac [("z","x")] complex_split_polar2 1);
+by (REPEAT(dtac choice 1 THEN Step_tac 1));
+by (res_inst_tac [("x","f")] exI 1);
+by (res_inst_tac [("x","fa")] exI 1);
+by Auto_tac;
+qed "hcomplex_split_polar";
+
+Goalw [hcis_def]
+ "hcis (Abs_hypreal(hyprel `` {%n. X n})) = \
+\ Abs_hcomplex(hcomplexrel `` {%n. cis (X n)})";
+by (res_inst_tac [("f","Abs_hcomplex")] arg_cong 1);
+by (Auto_tac THEN Ultra_tac 1);
+qed "hcis";
+
+Goal
+ "hcis a = \
+\ (hcomplex_of_hypreal(( *f* cos) a) + \
+\ iii * hcomplex_of_hypreal(( *f* sin) a))";
+by (res_inst_tac [("z","a")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [starfun, hcis,
+ hcomplex_of_hypreal,iii_def,hcomplex_mult,hcomplex_add,
+ cis_def]));
+qed "hcis_eq";
+
+Goalw [hrcis_def]
+ "hrcis (Abs_hypreal(hyprel `` {%n. X n})) (Abs_hypreal(hyprel `` {%n. Y n})) = \
+\ Abs_hcomplex(hcomplexrel `` {%n. rcis (X n) (Y n)})";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_of_hypreal,hcomplex_mult,
+ hcis,rcis_def]));
+qed "hrcis";
+
+Goal "EX r a. z = hrcis r a";
+by (simp_tac (simpset() addsimps [hrcis_def,hcis_eq]) 1);
+by (rtac hcomplex_split_polar 1);
+qed "hrcis_Ex";
+
+Goal "hRe(hcomplex_of_hypreal r * \
+\ (hcomplex_of_hypreal(( *f* cos) a) + \
+\ iii * hcomplex_of_hypreal(( *f* sin) a))) = r * ( *f* cos) a";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_add_mult_distrib2,
+ hcomplex_of_hypreal_mult] @ hcomplex_mult_ac));
+qed "hRe_hcomplex_polar";
+Addsimps [hRe_hcomplex_polar];
+
+Goalw [hrcis_def] "hRe(hrcis r a) = r * ( *f* cos) a";
+by (auto_tac (claset(),simpset() addsimps [hcis_eq]));
+qed "hRe_hrcis";
+Addsimps [hRe_hrcis];
+
+Goal "hIm(hcomplex_of_hypreal r * \
+\ (hcomplex_of_hypreal(( *f* cos) a) + \
+\ iii * hcomplex_of_hypreal(( *f* sin) a))) = r * ( *f* sin) a";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_add_mult_distrib2,
+ hcomplex_of_hypreal_mult] @ hcomplex_mult_ac));
+qed "hIm_hcomplex_polar";
+Addsimps [hIm_hcomplex_polar];
+
+Goalw [hrcis_def] "hIm(hrcis r a) = r * ( *f* sin) a";
+by (auto_tac (claset(),simpset() addsimps [hcis_eq]));
+qed "hIm_hrcis";
+Addsimps [hIm_hrcis];
+
+Goal "hcmod (hcomplex_of_hypreal r * \
+\ (hcomplex_of_hypreal(( *f* cos) a) + \
+\ iii * hcomplex_of_hypreal(( *f* sin) a))) = abs r";
+by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","a")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [iii_def,starfun,
+ hcomplex_of_hypreal,hcomplex_mult,hcmod,hcomplex_add,
+ hypreal_hrabs]));
+qed "hcmod_complex_polar";
+Addsimps [hcmod_complex_polar];
+
+Goalw [hrcis_def] "hcmod(hrcis r a) = abs r";
+by (auto_tac (claset(),simpset() addsimps [hcis_eq]));
+qed "hcmod_hrcis";
+Addsimps [hcmod_hrcis];
+
+(*---------------------------------------------------------------------------*)
+(* (r1 * hrcis a) * (r2 * hrcis b) = r1 * r2 * hrcis (a + b) *)
+(*---------------------------------------------------------------------------*)
+
+Goalw [hrcis_def] "hcis a = hrcis 1 a";
+by (Simp_tac 1);
+qed "hcis_hrcis_eq";
+Addsimps [hcis_hrcis_eq RS sym];
+
+Goalw [hrcis_def]
+ "hrcis r1 a * hrcis r2 b = hrcis (r1*r2) (a + b)";
+by (res_inst_tac [("z","r1")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","r2")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","a")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","b")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hrcis,hcis,
+ hypreal_add,hypreal_mult,hcomplex_of_hypreal,
+ hcomplex_mult,cis_mult RS sym,complex_of_real_mult
+ RS sym] addsimps complex_mult_ac));
+qed "hrcis_mult";
+
+Goal "hcis a * hcis b = hcis (a + b)";
+by (res_inst_tac [("z","a")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","b")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcis,hcomplex_mult,
+ hypreal_add,cis_mult]));
+qed "hcis_mult";
+
+Goalw [hcomplex_one_def]
+ "hcis 0 = 1";
+by (auto_tac (claset(),simpset() addsimps [hcis,hypreal_zero_num]));
+qed "hcis_zero";
+Addsimps [hcis_zero];
+
+Goalw [hcomplex_zero_def]
+ "hrcis 0 a = 0";
+by (res_inst_tac [("z","a")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hrcis,hypreal_zero_num]));
+qed "hrcis_zero_mod";
+Addsimps [hrcis_zero_mod];
+
+Goal "hrcis r 0 = hcomplex_of_hypreal r";
+by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hrcis,
+ hypreal_zero_num,hcomplex_of_hypreal]));
+qed "hrcis_zero_arg";
+Addsimps [hrcis_zero_arg];
+
+Goal "iii * (iii * x) = - x";
+by (simp_tac (simpset() addsimps [hcomplex_mult_assoc RS sym]) 1);
+qed "hcomplex_i_mult_minus";
+Addsimps [hcomplex_i_mult_minus];
+
+Goal "iii * iii * x = - x";
+by (Simp_tac 1);
+qed "hcomplex_i_mult_minus2";
+Addsimps [hcomplex_i_mult_minus2];
+
+(* Move to one of the hyperreal theories *)
+Goal "hypreal_of_nat m = Abs_hypreal(hyprel `` {%n. real m})";
+by (induct_tac "m" 1);
+by (auto_tac (claset(),simpset() addsimps [hypreal_zero_def,
+ hypreal_of_nat_Suc,hypreal_zero_num,
+ hypreal_one_num,hypreal_add,real_of_nat_Suc]));
+qed "hypreal_of_nat";
+
+Goal
+ "hcis (hypreal_of_nat (Suc n) * a) = hcis a * hcis (hypreal_of_nat n * a)";
+by (res_inst_tac [("z","a")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hypreal_of_nat,hcis,
+ hypreal_mult,hcomplex_mult,cis_real_of_nat_Suc_mult]));
+qed "hcis_hypreal_of_nat_Suc_mult";
+
+Goal "(hcis a) ^ n = hcis (hypreal_of_nat n * a)";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps [hcis_hypreal_of_nat_Suc_mult]));
+qed "NSDeMoivre";
+
+Goal "hcis (hypreal_of_hypnat (n + 1) * a) = \
+\ hcis a * hcis (hypreal_of_hypnat n * a)";
+by (res_inst_tac [("z","a")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
+by (auto_tac (claset(),simpset() addsimps [hcis,hypreal_of_hypnat,
+ hypnat_add,hypnat_one_def,hypreal_mult,hcomplex_mult,
+ cis_real_of_nat_Suc_mult]));
+qed "hcis_hypreal_of_hypnat_Suc_mult";
+
+Goal "(hcis a) hcpow n = hcis (hypreal_of_hypnat n * a)";
+by (res_inst_tac [("z","a")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
+by (auto_tac (claset(),simpset() addsimps [hcis,hypreal_of_hypnat,
+ hypreal_mult,hcpow,DeMoivre]));
+qed "NSDeMoivre_ext";
+
+Goalw [hrcis_def]
+ "(hrcis r a) ^ n = hrcis (r ^ n) (hypreal_of_nat n * a)";
+by (auto_tac (claset(),simpset() addsimps [hcomplexpow_mult,
+ NSDeMoivre,hcomplex_of_hypreal_pow]));
+qed "DeMoivre2";
+
+Goalw [hrcis_def]
+ "(hrcis r a) hcpow n = hrcis (r pow n) (hypreal_of_hypnat n * a)";
+by (auto_tac (claset(),simpset() addsimps [hcpow_mult,
+ NSDeMoivre_ext,hcomplex_of_hypreal_hyperpow]));
+qed "DeMoivre2_ext";
+
+Goal "inverse(hcis a) = hcis (-a)";
+by (res_inst_tac [("z","a")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_inverse,hcis,hypreal_minus]));
+qed "hcis_inverse";
+Addsimps [hcis_inverse];
+
+Goal "inverse(hrcis r a) = hrcis (inverse r) (-a)";
+by (res_inst_tac [("z","a")] eq_Abs_hypreal 1);
+by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcomplex_inverse,hrcis,hypreal_minus,
+ hypreal_inverse,rcis_inverse]));
+by (Ultra_tac 1);
+by (rewtac real_divide_def);
+by (auto_tac (claset(),simpset() addsimps [INVERSE_ZERO]));
+qed "hrcis_inverse";
+
+Goal "hRe(hcis a) = ( *f* cos) a";
+by (res_inst_tac [("z","a")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcis,starfun,hRe]));
+qed "hRe_hcis";
+Addsimps [hRe_hcis];
+
+Goal "hIm(hcis a) = ( *f* sin) a";
+by (res_inst_tac [("z","a")] eq_Abs_hypreal 1);
+by (auto_tac (claset(),simpset() addsimps [hcis,starfun,hIm]));
+qed "hIm_hcis";
+Addsimps [hIm_hcis];
+
+Goal "( *f* cos) (hypreal_of_nat n * a) = hRe(hcis a ^ n)";
+by (auto_tac (claset(),simpset() addsimps [NSDeMoivre]));
+qed "cos_n_hRe_hcis_pow_n";
+
+Goal "( *f* sin) (hypreal_of_nat n * a) = hIm(hcis a ^ n)";
+by (auto_tac (claset(),simpset() addsimps [NSDeMoivre]));
+qed "sin_n_hIm_hcis_pow_n";
+
+Goal "( *f* cos) (hypreal_of_hypnat n * a) = hRe(hcis a hcpow n)";
+by (auto_tac (claset(),simpset() addsimps [NSDeMoivre_ext]));
+qed "cos_n_hRe_hcis_hcpow_n";
+
+Goal "( *f* sin) (hypreal_of_hypnat n * a) = hIm(hcis a hcpow n)";
+by (auto_tac (claset(),simpset() addsimps [NSDeMoivre_ext]));
+qed "sin_n_hIm_hcis_hcpow_n";
+
+Goalw [hexpi_def] "hexpi(a + b) = hexpi(a) * hexpi(b)";
+by (res_inst_tac [("z","a")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","b")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hcis,hRe,hIm,
+ hcomplex_add,hcomplex_mult,hypreal_mult,starfun,
+ hcomplex_of_hypreal,cis_mult RS sym,complex_Im_add,
+ complex_Re_add,exp_add,complex_of_real_mult]));
+qed "hexpi_add";
+
+(*----------------------------------------------------------------------------------*)
+(* hcomplex_of_complex preserves field and order properties *)
+(*----------------------------------------------------------------------------------*)
+
+Goalw [hcomplex_of_complex_def]
+ "hcomplex_of_complex (z1 + z2) = hcomplex_of_complex z1 + hcomplex_of_complex z2";
+by (simp_tac (simpset() addsimps [hcomplex_add]) 1);
+qed "hcomplex_of_complex_add";
+Addsimps [hcomplex_of_complex_add];
+
+Goalw [hcomplex_of_complex_def]
+ "hcomplex_of_complex (z1 * z2) = hcomplex_of_complex z1 * hcomplex_of_complex z2";
+by (simp_tac (simpset() addsimps [hcomplex_mult]) 1);
+qed "hcomplex_of_complex_mult";
+Addsimps [hcomplex_of_complex_mult];
+
+Goalw [hcomplex_of_complex_def]
+ "(hcomplex_of_complex z1 = hcomplex_of_complex z2) = (z1 = z2)";
+by Auto_tac;
+qed "hcomplex_of_complex_eq_iff";
+Addsimps [hcomplex_of_complex_eq_iff];
+
+Goalw [hcomplex_of_complex_def] "hcomplex_of_complex (-r) = - hcomplex_of_complex r";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_minus]));
+qed "hcomplex_of_complex_minus";
+Addsimps [hcomplex_of_complex_minus];
+
+Goalw [hcomplex_of_complex_def,hcomplex_one_def]
+ "hcomplex_of_complex 1 = 1";
+by Auto_tac;
+qed "hcomplex_of_complex_one";
+
+Goalw [hcomplex_of_complex_def,hcomplex_zero_def]
+ "hcomplex_of_complex 0 = 0";
+by (Simp_tac 1);
+qed "hcomplex_of_complex_zero";
+
+Goal "(hcomplex_of_complex r = 0) = (r = 0)";
+by (auto_tac (claset() addIs [FreeUltrafilterNat_P],
+ simpset() addsimps [hcomplex_of_complex_def,
+ hcomplex_zero_def]));
+qed "hcomplex_of_complex_zero_iff";
+
+Goal "hcomplex_of_complex (inverse r) = inverse (hcomplex_of_complex r)";
+by (case_tac "r=0" 1);
+by (asm_simp_tac (simpset() addsimps [COMPLEX_INVERSE_ZERO,
+ HCOMPLEX_INVERSE_ZERO, hcomplex_of_complex_zero,
+ COMPLEX_DIVIDE_ZERO]) 1);
+by (res_inst_tac [("c1","hcomplex_of_complex r")]
+ (hcomplex_mult_left_cancel RS iffD1) 1);
+by (stac (hcomplex_of_complex_mult RS sym) 2);
+by (auto_tac (claset(),
+ simpset() addsimps [hcomplex_of_complex_one, hcomplex_of_complex_zero_iff]));
+qed "hcomplex_of_complex_inverse";
+Addsimps [hcomplex_of_complex_inverse];
+
+Goal "hcomplex_of_complex (z1 / z2) = hcomplex_of_complex z1 / hcomplex_of_complex z2";
+by (simp_tac (simpset() addsimps [hcomplex_divide_def, complex_divide_def]) 1);
+qed "hcomplex_of_complex_divide";
+Addsimps [hcomplex_of_complex_divide];
+
+Goalw [hcomplex_of_complex_def,hypreal_of_real_def]
+ "hRe (hcomplex_of_complex z) = hypreal_of_real (Re z)";
+by (auto_tac (claset(),simpset() addsimps [hRe]));
+qed "hRe_hcomplex_of_complex";
+
+Goalw [hcomplex_of_complex_def,hypreal_of_real_def]
+ "hIm (hcomplex_of_complex z) = hypreal_of_real (Im z)";
+by (auto_tac (claset(),simpset() addsimps [hIm]));
+qed "hIm_hcomplex_of_complex";
+
+Goalw [hypreal_of_real_def,hcomplex_of_complex_def]
+ "hcmod (hcomplex_of_complex x) = hypreal_of_real (cmod x)";
+by (auto_tac (claset(),simpset() addsimps [hcmod]));
+qed "hcmod_hcomplex_of_complex";
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/NSComplex.thy Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,127 @@
+(* Title: NSComplex.thy
+ Author: Jacques D. Fleuriot
+ Copyright: 2001 University of Edinburgh
+ Description: Nonstandard Complex numbers
+*)
+
+NSComplex = NSInduct +
+
+constdefs
+ hcomplexrel :: "((nat=>complex)*(nat=>complex)) set"
+ "hcomplexrel == {p. EX X Y. p = ((X::nat=>complex),Y) &
+ {n::nat. X(n) = Y(n)}: FreeUltrafilterNat}"
+
+typedef hcomplex = "{x::nat=>complex. True}//hcomplexrel" (quotient_def)
+
+instance
+ hcomplex :: {zero,one,plus,times,minus,power,inverse}
+
+defs
+ hcomplex_zero_def
+ "0 == Abs_hcomplex(hcomplexrel `` {%n. (0::complex)})"
+
+ hcomplex_one_def
+ "1 == Abs_hcomplex(hcomplexrel `` {%n. (1::complex)})"
+
+
+ hcomplex_minus_def
+ "- z == Abs_hcomplex(UN X: Rep_hcomplex(z). hcomplexrel `` {%n::nat. - (X n)})"
+
+ hcomplex_diff_def
+ "w - z == w + -(z::hcomplex)"
+
+constdefs
+
+ hcomplex_of_complex :: complex => hcomplex
+ "hcomplex_of_complex z == Abs_hcomplex(hcomplexrel `` {%n. z})"
+
+ hcinv :: hcomplex => hcomplex
+ "inverse(P) == Abs_hcomplex(UN X: Rep_hcomplex(P).
+ hcomplexrel `` {%n. inverse(X n)})"
+
+ (*--- real and Imaginary parts ---*)
+
+ hRe :: hcomplex => hypreal
+ "hRe(z) == Abs_hypreal(UN X:Rep_hcomplex(z). hyprel `` {%n. Re (X n)})"
+
+ hIm :: hcomplex => hypreal
+ "hIm(z) == Abs_hypreal(UN X:Rep_hcomplex(z). hyprel `` {%n. Im (X n)})"
+
+
+ (*----------- modulus ------------*)
+
+ hcmod :: hcomplex => hypreal
+ "hcmod z == Abs_hypreal(UN X: Rep_hcomplex(z).
+ hyprel `` {%n. cmod (X n)})"
+
+ (*------ imaginary unit ----------*)
+
+ iii :: hcomplex
+ "iii == Abs_hcomplex(hcomplexrel `` {%n. ii})"
+
+ (*------- complex conjugate ------*)
+
+ hcnj :: hcomplex => hcomplex
+ "hcnj z == Abs_hcomplex(UN X:Rep_hcomplex(z). hcomplexrel `` {%n. cnj (X n)})"
+
+ (*------------ Argand -------------*)
+
+ hsgn :: hcomplex => hcomplex
+ "hsgn z == Abs_hcomplex(UN X:Rep_hcomplex(z). hcomplexrel `` {%n. sgn(X n)})"
+
+ harg :: hcomplex => hypreal
+ "harg z == Abs_hypreal(UN X:Rep_hcomplex(z). hyprel `` {%n. arg(X n)})"
+
+ (* abbreviation for (cos a + i sin a) *)
+ hcis :: hypreal => hcomplex
+ "hcis a == Abs_hcomplex(UN X:Rep_hypreal(a). hcomplexrel `` {%n. cis (X n)})"
+
+ (* abbreviation for r*(cos a + i sin a) *)
+ hrcis :: [hypreal, hypreal] => hcomplex
+ "hrcis r a == hcomplex_of_hypreal r * hcis a"
+
+ (*----- injection from hyperreals -----*)
+
+ hcomplex_of_hypreal :: hypreal => hcomplex
+ "hcomplex_of_hypreal r == Abs_hcomplex(UN X:Rep_hypreal(r).
+ hcomplexrel `` {%n. complex_of_real (X n)})"
+
+ (*------------ e ^ (x + iy) ------------*)
+
+ hexpi :: hcomplex => hcomplex
+ "hexpi z == hcomplex_of_hypreal(( *f* exp) (hRe z)) * hcis (hIm z)"
+
+
+defs
+
+
+ (*----------- division ----------*)
+
+ hcomplex_divide_def
+ "w / (z::hcomplex) == w * inverse z"
+
+ hcomplex_add_def
+ "w + z == Abs_hcomplex(UN X:Rep_hcomplex(w). UN Y:Rep_hcomplex(z).
+ hcomplexrel `` {%n. X n + Y n})"
+
+ hcomplex_mult_def
+ "w * z == Abs_hcomplex(UN X:Rep_hcomplex(w). UN Y:Rep_hcomplex(z).
+ hcomplexrel `` {%n. X n * Y n})"
+
+
+primrec
+ hcomplexpow_0 "z ^ 0 = 1"
+ hcomplexpow_Suc "z ^ (Suc n) = (z::hcomplex) * (z ^ n)"
+
+
+consts
+ "hcpow" :: [hcomplex,hypnat] => hcomplex (infixr 80)
+
+defs
+ (* hypernatural powers of nonstandard complex numbers *)
+ hcpow_def
+ "(z::hcomplex) hcpow (n::hypnat)
+ == Abs_hcomplex(UN X:Rep_hcomplex(z). UN Y: Rep_hypnat(n).
+ hcomplexrel `` {%n. (X n) ^ (Y n)})"
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/NSComplexArith0.ML Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,331 @@
+(* Title: NSComplexArith0.ML
+ Author: Jacques D. Fleuriot
+ Copyright: 2001 University of Edinburgh
+ Description: Assorted facts that need binary literals
+ Also, common factor cancellation (see e.g. HyperArith0)
+*)
+
+(****
+Goal "((x * y = #0) = (x = #0 | y = (#0::hcomplex)))";
+by (auto_tac (claset(),simpset() addsimps [rename_numerals hcomplex_mult_zero_iff]));
+qed "hcomplex_mult_is_0";
+AddIffs [hcomplex_mult_is_0];
+****)
+
+(** Division and inverse **)
+
+Goal "0/x = (0::hcomplex)";
+by (simp_tac (simpset() addsimps [hcomplex_divide_def]) 1);
+qed "hcomplex_0_divide";
+Addsimps [hcomplex_0_divide];
+
+Goalw [hcomplex_divide_def] "x/(0::hcomplex) = 0";
+by (stac HCOMPLEX_INVERSE_ZERO 1);
+by (Simp_tac 1);
+qed "HCOMPLEX_DIVIDE_ZERO";
+
+Goal "inverse (x::hcomplex) = 1/x";
+by (simp_tac (simpset() addsimps [hcomplex_divide_def]) 1);
+qed "hcomplex_inverse_eq_divide";
+
+Goal "(inverse(x::hcomplex) = 0) = (x = 0)";
+by (auto_tac (claset(),
+ simpset() addsimps [HCOMPLEX_INVERSE_ZERO]));
+by (blast_tac (claset() addIs [ccontr] addDs [hcomplex_inverse_not_zero]) 1);
+qed "hcomplex_inverse_zero_iff";
+Addsimps [hcomplex_inverse_zero_iff];
+
+Goal "(x/y = 0) = (x=0 | y=(0::hcomplex))";
+by (auto_tac (claset(), simpset() addsimps [hcomplex_divide_def]));
+qed "hcomplex_divide_eq_0_iff";
+Addsimps [hcomplex_divide_eq_0_iff];
+
+Goal "h ~= (0::hcomplex) ==> h/h = 1";
+by (asm_simp_tac
+ (simpset() addsimps [hcomplex_divide_def]) 1);
+qed "hcomplex_divide_self_eq";
+Addsimps [hcomplex_divide_self_eq];
+
+bind_thm ("hcomplex_mult_minus_right", hcomplex_minus_mult_eq2 RS sym);
+
+Goal "!!k::hcomplex. (k*m = k*n) = (k = 0 | m=n)";
+by (case_tac "k=0" 1);
+by (auto_tac (claset(), simpset() addsimps [hcomplex_mult_left_cancel]));
+qed "hcomplex_mult_eq_cancel1";
+
+Goal "!!k::hcomplex. (m*k = n*k) = (k = 0 | m=n)";
+by (case_tac "k=0" 1);
+by (auto_tac (claset(), simpset() addsimps [hcomplex_mult_right_cancel]));
+qed "hcomplex_mult_eq_cancel2";
+
+Goal "!!k::hcomplex. k~=0 ==> (k*m) / (k*n) = (m/n)";
+by (asm_simp_tac
+ (simpset() addsimps [hcomplex_divide_def, hcomplex_inverse_distrib]) 1);
+by (subgoal_tac "k * m * (inverse k * inverse n) = \
+\ (k * inverse k) * (m * inverse n)" 1);
+by (Asm_full_simp_tac 1);
+by (asm_full_simp_tac (HOL_ss addsimps hcomplex_mult_ac) 1);
+qed "hcomplex_mult_div_cancel1";
+
+(*For ExtractCommonTerm*)
+Goal "(k*m) / (k*n) = (if k = (0::hcomplex) then 0 else m/n)";
+by (simp_tac (simpset() addsimps [hcomplex_mult_div_cancel1]) 1);
+qed "hcomplex_mult_div_cancel_disj";
+
+
+local
+ open HComplex_Numeral_Simprocs
+in
+
+val rel_hcomplex_number_of = [eq_hcomplex_number_of];
+
+
+structure CancelNumeralFactorCommon =
+ struct
+ val mk_coeff = mk_coeff
+ val dest_coeff = dest_coeff 1
+ val trans_tac = Real_Numeral_Simprocs.trans_tac
+ val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps hcomplex_minus_from_mult_simps @ mult_1s))
+ THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@hcomplex_mult_minus_simps))
+ THEN ALLGOALS (simp_tac (HOL_ss addsimps hcomplex_mult_ac))
+ val numeral_simp_tac =
+ ALLGOALS (simp_tac (HOL_ss addsimps rel_hcomplex_number_of@bin_simps))
+ val simplify_meta_eq = simplify_meta_eq
+ end
+
+
+structure DivCancelNumeralFactor = CancelNumeralFactorFun
+ (open CancelNumeralFactorCommon
+ val prove_conv = Bin_Simprocs.prove_conv
+ val mk_bal = HOLogic.mk_binop "HOL.divide"
+ val dest_bal = HOLogic.dest_bin "HOL.divide" hcomplexT
+ val cancel = hcomplex_mult_div_cancel1 RS trans
+ val neg_exchanges = false
+)
+
+
+structure EqCancelNumeralFactor = CancelNumeralFactorFun
+ (open CancelNumeralFactorCommon
+ val prove_conv = Bin_Simprocs.prove_conv
+ val mk_bal = HOLogic.mk_eq
+ val dest_bal = HOLogic.dest_bin "op =" hcomplexT
+ val cancel = hcomplex_mult_eq_cancel1 RS trans
+ val neg_exchanges = false
+)
+
+
+val hcomplex_cancel_numeral_factors_relations =
+ map prep_simproc
+ [("hcomplexeq_cancel_numeral_factor",
+ ["(l::hcomplex) * m = n", "(l::hcomplex) = m * n"],
+ EqCancelNumeralFactor.proc)];
+
+val hcomplex_cancel_numeral_factors_divide = prep_simproc
+ ("hcomplexdiv_cancel_numeral_factor",
+ ["((l::hcomplex) * m) / n", "(l::hcomplex) / (m * n)",
+ "((number_of v)::hcomplex) / (number_of w)"],
+ DivCancelNumeralFactor.proc);
+
+val hcomplex_cancel_numeral_factors =
+ hcomplex_cancel_numeral_factors_relations @
+ [hcomplex_cancel_numeral_factors_divide];
+
+end;
+
+
+Addsimprocs hcomplex_cancel_numeral_factors;
+
+
+(*examples:
+print_depth 22;
+set timing;
+set trace_simp;
+fun test s = (Goal s; by (Simp_tac 1));
+
+
+test "#9*x = #12 * (y::hcomplex)";
+test "(#9*x) / (#12 * (y::hcomplex)) = z";
+
+test "#-99*x = #132 * (y::hcomplex)";
+
+test "#999*x = #-396 * (y::hcomplex)";
+test "(#999*x) / (#-396 * (y::hcomplex)) = z";
+
+test "#-99*x = #-81 * (y::hcomplex)";
+test "(#-99*x) / (#-81 * (y::hcomplex)) = z";
+
+test "#-2 * x = #-1 * (y::hcomplex)";
+test "#-2 * x = -(y::hcomplex)";
+test "(#-2 * x) / (#-1 * (y::hcomplex)) = z";
+
+*)
+
+
+(** Declarations for ExtractCommonTerm **)
+
+local
+ open HComplex_Numeral_Simprocs
+in
+
+structure CancelFactorCommon =
+ struct
+ val mk_sum = long_mk_prod
+ val dest_sum = dest_prod
+ val mk_coeff = mk_coeff
+ val dest_coeff = dest_coeff
+ val find_first = find_first []
+ val trans_tac = Real_Numeral_Simprocs.trans_tac
+ val norm_tac = ALLGOALS (simp_tac (HOL_ss addsimps mult_1s@hcomplex_mult_ac))
+ end;
+
+
+structure EqCancelFactor = ExtractCommonTermFun
+ (open CancelFactorCommon
+ val prove_conv = Bin_Simprocs.prove_conv
+ val mk_bal = HOLogic.mk_eq
+ val dest_bal = HOLogic.dest_bin "op =" hcomplexT
+ val simplify_meta_eq = cancel_simplify_meta_eq hcomplex_mult_eq_cancel1
+);
+
+
+structure DivideCancelFactor = ExtractCommonTermFun
+ (open CancelFactorCommon
+ val prove_conv = Bin_Simprocs.prove_conv
+ val mk_bal = HOLogic.mk_binop "HOL.divide"
+ val dest_bal = HOLogic.dest_bin "HOL.divide" hcomplexT
+ val simplify_meta_eq = cancel_simplify_meta_eq hcomplex_mult_div_cancel_disj
+);
+
+val hcomplex_cancel_factor =
+ map prep_simproc
+ [("hcomplex_eq_cancel_factor", ["(l::hcomplex) * m = n", "(l::hcomplex) = m * n"],
+ EqCancelFactor.proc),
+ ("hcomplex_divide_cancel_factor", ["((l::hcomplex) * m) / n", "(l::hcomplex) / (m * n)"],
+ DivideCancelFactor.proc)];
+
+end;
+
+Addsimprocs hcomplex_cancel_factor;
+
+
+(*examples:
+print_depth 22;
+set timing;
+set trace_simp;
+fun test s = (Goal s; by (Asm_simp_tac 1));
+
+test "x*k = k*(y::hcomplex)";
+test "k = k*(y::hcomplex)";
+test "a*(b*c) = (b::hcomplex)";
+test "a*(b*c) = d*(b::hcomplex)*(x*a)";
+
+
+test "(x*k) / (k*(y::hcomplex)) = (uu::hcomplex)";
+test "(k) / (k*(y::hcomplex)) = (uu::hcomplex)";
+test "(a*(b*c)) / ((b::hcomplex)) = (uu::hcomplex)";
+test "(a*(b*c)) / (d*(b::hcomplex)*(x*a)) = (uu::hcomplex)";
+
+(*FIXME: what do we do about this?*)
+test "a*(b*c)/(y*z) = d*(b::hcomplex)*(x*a)/z";
+*)
+
+
+Goal "z~=0 ==> ((x::hcomplex) = y/z) = (x*z = y)";
+by (subgoal_tac "(x*z = y) = (x*z = (y/z)*z)" 1);
+by (asm_simp_tac (simpset() addsimps [hcomplex_divide_def, hcomplex_mult_assoc]) 2);
+by (etac ssubst 1);
+by (stac hcomplex_mult_eq_cancel2 1);
+by (Asm_simp_tac 1);
+qed "hcomplex_eq_divide_eq";
+Addsimps [inst "z" "number_of ?w" hcomplex_eq_divide_eq];
+
+Goal "z~=0 ==> (y/z = (x::hcomplex)) = (y = x*z)";
+by (subgoal_tac "(y = x*z) = ((y/z)*z = x*z)" 1);
+by (asm_simp_tac (simpset() addsimps [hcomplex_divide_def, hcomplex_mult_assoc]) 2);
+by (etac ssubst 1);
+by (stac hcomplex_mult_eq_cancel2 1);
+by (Asm_simp_tac 1);
+qed "hcomplex_divide_eq_eq";
+Addsimps [inst "z" "number_of ?w" hcomplex_divide_eq_eq];
+
+Goal "(m/k = n/k) = (k = 0 | m = (n::hcomplex))";
+by (case_tac "k=0" 1);
+by (asm_simp_tac (simpset() addsimps [HCOMPLEX_DIVIDE_ZERO]) 1);
+by (asm_simp_tac (simpset() addsimps [hcomplex_divide_eq_eq, hcomplex_eq_divide_eq,
+ hcomplex_mult_eq_cancel2]) 1);
+qed "hcomplex_divide_eq_cancel2";
+
+Goal "(k/m = k/n) = (k = 0 | m = (n::hcomplex))";
+by (case_tac "m=0 | n = 0" 1);
+by (auto_tac (claset(),
+ simpset() addsimps [HCOMPLEX_DIVIDE_ZERO, hcomplex_divide_eq_eq,
+ hcomplex_eq_divide_eq, hcomplex_mult_eq_cancel1]));
+qed "hcomplex_divide_eq_cancel1";
+
+(** Division by 1, -1 **)
+
+Goal "(x::hcomplex)/1 = x";
+by (simp_tac (simpset() addsimps [hcomplex_divide_def]) 1);
+qed "hcomplex_divide_1";
+Addsimps [hcomplex_divide_1];
+
+Goal "x/-1 = -(x::hcomplex)";
+by (Simp_tac 1);
+qed "hcomplex_divide_minus1";
+Addsimps [hcomplex_divide_minus1];
+
+Goal "-1/(x::hcomplex) = - (1/x)";
+by (simp_tac (simpset() addsimps [hcomplex_divide_def, hcomplex_minus_inverse]) 1);
+qed "hcomplex_minus1_divide";
+Addsimps [hcomplex_minus1_divide];
+
+
+Goal "(x = - y) = (y = - (x::hcomplex))";
+by Auto_tac;
+qed "hcomplex_equation_minus";
+
+Goal "(- x = y) = (- (y::hcomplex) = x)";
+by Auto_tac;
+qed "hcomplex_minus_equation";
+
+Goal "(x + - a = (0::hcomplex)) = (x=a)";
+by (simp_tac (simpset() addsimps [hcomplex_diff_eq_eq,symmetric hcomplex_diff_def]) 1);
+qed "hcomplex_add_minus_iff";
+Addsimps [hcomplex_add_minus_iff];
+
+Goal "(-b = -a) = (b = (a::hcomplex))";
+by Auto_tac;
+by (etac ( inj_hcomplex_minus RS injD) 1);
+qed "hcomplex_minus_eq_cancel";
+Addsimps [hcomplex_minus_eq_cancel];
+
+(*Distributive laws for literals*)
+Addsimps (map (inst "w" "number_of ?v")
+ [hcomplex_add_mult_distrib, hcomplex_add_mult_distrib2,
+ hcomplex_diff_mult_distrib, hcomplex_diff_mult_distrib2]);
+
+Addsimps [inst "x" "number_of ?v" hcomplex_equation_minus];
+
+Addsimps [inst "y" "number_of ?v" hcomplex_minus_equation];
+
+Goal "(x+y = (0::hcomplex)) = (y = -x)";
+by Auto_tac;
+by (dtac (sym RS (hcomplex_diff_eq_eq RS iffD2)) 1);
+by Auto_tac;
+qed "hcomplex_add_eq_0_iff";
+AddIffs [hcomplex_add_eq_0_iff];
+
+Goalw [hcomplex_diff_def]"-(x-y) = y - (x::hcomplex)";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_add_commute]));
+qed "hcomplex_minus_diff_eq";
+Addsimps [hcomplex_minus_diff_eq];
+
+Addsimps [inst "x" "number_of ?w" hcomplex_inverse_eq_divide];
+
+Goal "[|(x::hcomplex) ~= 0; y ~= 0 |] \
+\ ==> inverse(x) + inverse(y) = (x + y)*inverse(x*y)";
+by (asm_full_simp_tac (simpset() addsimps [hcomplex_inverse_distrib,
+ hcomplex_add_mult_distrib,hcomplex_mult_assoc RS sym]) 1);
+qed "hcomplex_inverse_add";
+
+Addsimps [hcomplex_of_complex_zero,hcomplex_of_complex_one];
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/NSComplexArith0.thy Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,4 @@
+NSComplexArith0 = NSComplexBin
+
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/NSComplexBin.ML Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,637 @@
+(* Title: NSComplexBin.ML
+ Author: Jacques D. Fleuriot
+ Copyright: 2001 University of Edinburgh
+ Descrition: Binary arithmetic for the nonstandard complex numbers
+*)
+
+(** hcomplex_of_complex (coercion from complex to nonstandard complex) **)
+
+Goal "hcomplex_of_complex (number_of w) = number_of w";
+by (simp_tac (simpset() addsimps [hcomplex_number_of_def]) 1);
+qed "hcomplex_number_of";
+Addsimps [hcomplex_number_of];
+
+Goalw [hypreal_of_real_def]
+ "hcomplex_of_hypreal (hypreal_of_real x) = \
+\ hcomplex_of_complex(complex_of_real x)";
+by (simp_tac (simpset() addsimps [hcomplex_of_hypreal,
+ hcomplex_of_complex_def,complex_of_real_def]) 1);
+qed "hcomplex_of_hypreal_eq_hcomplex_of_complex";
+
+Goalw [complex_number_of_def,hypreal_number_of_def]
+ "hcomplex_of_complex (number_of w) = hcomplex_of_hypreal(number_of w)";
+by (rtac (hcomplex_of_hypreal_eq_hcomplex_of_complex RS sym) 1);
+qed "hcomplex_hypreal_number_of";
+
+Goalw [hcomplex_number_of_def] "Numeral0 = (0::hcomplex)";
+by (simp_tac (simpset() addsimps [hcomplex_of_complex_zero RS sym]) 1);
+qed "hcomplex_numeral_0_eq_0";
+
+Goalw [hcomplex_number_of_def] "Numeral1 = (1::hcomplex)";
+by (simp_tac (simpset() addsimps [hcomplex_of_complex_one RS sym]) 1);
+qed "hcomplex_numeral_1_eq_1";
+
+(*
+Goal "z + hcnj z = \
+\ hcomplex_of_hypreal (2 * hRe(z))";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),HOL_ss addsimps [hRe,hcnj,hcomplex_add,
+ hypreal_mult,hcomplex_of_hypreal,complex_add_cnj]));
+qed "hcomplex_add_hcnj";
+
+Goal "z - hcnj z = \
+\ hcomplex_of_hypreal (hypreal_of_real #2 * hIm(z)) * iii";
+by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(),simpset() addsimps [hIm,hcnj,hcomplex_diff,
+ hypreal_of_real_def,hypreal_mult,hcomplex_of_hypreal,
+ complex_diff_cnj,iii_def,hcomplex_mult]));
+qed "hcomplex_diff_hcnj";
+*)
+
+(** Addition **)
+
+Goal "(number_of v :: hcomplex) + number_of v' = number_of (bin_add v v')";
+by (simp_tac
+ (HOL_ss addsimps [hcomplex_number_of_def,
+ hcomplex_of_complex_add RS sym, add_complex_number_of]) 1);
+qed "add_hcomplex_number_of";
+Addsimps [add_hcomplex_number_of];
+
+
+(** Subtraction **)
+
+Goalw [hcomplex_number_of_def]
+ "- (number_of w :: hcomplex) = number_of (bin_minus w)";
+by (simp_tac
+ (HOL_ss addsimps [minus_complex_number_of, hcomplex_of_complex_minus RS sym]) 1);
+qed "minus_hcomplex_number_of";
+Addsimps [minus_hcomplex_number_of];
+
+Goalw [hcomplex_number_of_def, hcomplex_diff_def]
+ "(number_of v :: hcomplex) - number_of w = \
+\ number_of (bin_add v (bin_minus w))";
+by (Simp_tac 1);
+qed "diff_hcomplex_number_of";
+Addsimps [diff_hcomplex_number_of];
+
+
+(** Multiplication **)
+
+Goal "(number_of v :: hcomplex) * number_of v' = number_of (bin_mult v v')";
+by (simp_tac
+ (HOL_ss addsimps [hcomplex_number_of_def,
+ hcomplex_of_complex_mult RS sym, mult_complex_number_of]) 1);
+qed "mult_hcomplex_number_of";
+Addsimps [mult_hcomplex_number_of];
+
+Goal "(2::hcomplex) = 1 + 1";
+by (simp_tac (simpset() addsimps [hcomplex_numeral_1_eq_1 RS sym]) 1);
+val lemma = result();
+
+(*For specialist use: NOT as default simprules*)
+Goal "2 * z = (z+z::hcomplex)";
+by (simp_tac (simpset() addsimps [lemma, hcomplex_add_mult_distrib]) 1);
+qed "hcomplex_mult_2";
+
+Goal "z * 2 = (z+z::hcomplex)";
+by (stac hcomplex_mult_commute 1 THEN rtac hcomplex_mult_2 1);
+qed "hcomplex_mult_2_right";
+
+(** Equals (=) **)
+
+Goal "((number_of v :: hcomplex) = number_of v') = \
+\ iszero (number_of (bin_add v (bin_minus v')))";
+by (simp_tac
+ (HOL_ss addsimps [hcomplex_number_of_def,
+ hcomplex_of_complex_eq_iff, eq_complex_number_of]) 1);
+qed "eq_hcomplex_number_of";
+Addsimps [eq_hcomplex_number_of];
+
+(*** New versions of existing theorems involving 0, 1hc ***)
+
+Goal "- 1 = (-1::hcomplex)";
+by (simp_tac (simpset() addsimps [hcomplex_numeral_1_eq_1 RS sym]) 1);
+qed "hcomplex_minus_1_eq_m1";
+
+Goal "-1 * z = -(z::hcomplex)";
+by (simp_tac (simpset() addsimps [hcomplex_minus_1_eq_m1 RS sym]) 1);
+qed "hcomplex_mult_minus1";
+
+Goal "z * -1 = -(z::hcomplex)";
+by (stac hcomplex_mult_commute 1 THEN rtac hcomplex_mult_minus1 1);
+qed "hcomplex_mult_minus1_right";
+
+Addsimps [hcomplex_mult_minus1,hcomplex_mult_minus1_right];
+
+(*Maps 0 to Numeral0 and 1 to Numeral1 and -Numeral1 to -1*)
+val hcomplex_numeral_ss =
+ complex_numeral_ss addsimps [hcomplex_numeral_0_eq_0 RS sym, hcomplex_numeral_1_eq_1 RS sym,
+ hcomplex_minus_1_eq_m1];
+
+fun rename_numerals th =
+ asm_full_simplify hcomplex_numeral_ss (Thm.transfer (the_context ()) th);
+
+
+(*Now insert some identities previously stated for 0 and 1hc*)
+
+
+Addsimps [hcomplex_numeral_0_eq_0,hcomplex_numeral_1_eq_1];
+
+Goal "number_of v + (number_of w + z) = (number_of(bin_add v w) + z::hcomplex)";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_add_assoc RS sym]));
+qed "hcomplex_add_number_of_left";
+
+Goal "number_of v *(number_of w * z) = (number_of(bin_mult v w) * z::hcomplex)";
+by (simp_tac (simpset() addsimps [hcomplex_mult_assoc RS sym]) 1);
+qed "hcomplex_mult_number_of_left";
+
+Goalw [hcomplex_diff_def]
+ "number_of v + (number_of w - c) = number_of(bin_add v w) - (c::hcomplex)";
+by (rtac hcomplex_add_number_of_left 1);
+qed "hcomplex_add_number_of_diff1";
+
+Goal "number_of v + (c - number_of w) = \
+\ number_of (bin_add v (bin_minus w)) + (c::hcomplex)";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_def]@ hcomplex_add_ac));
+qed "hcomplex_add_number_of_diff2";
+
+Addsimps [hcomplex_add_number_of_left, hcomplex_mult_number_of_left,
+ hcomplex_add_number_of_diff1, hcomplex_add_number_of_diff2];
+
+
+(**** Simprocs for numeric literals ****)
+
+(** Combining of literal coefficients in sums of products **)
+
+Goal "(x = y) = (x-y = (0::hcomplex))";
+by (simp_tac (simpset() addsimps [hcomplex_diff_eq_eq]) 1);
+qed "hcomplex_eq_iff_diff_eq_0";
+
+(** For combine_numerals **)
+
+Goal "i*u + (j*u + k) = (i+j)*u + (k::hcomplex)";
+by (asm_simp_tac (simpset() addsimps [hcomplex_add_mult_distrib]
+ @ hcomplex_add_ac) 1);
+qed "left_hcomplex_add_mult_distrib";
+
+(** For cancel_numerals **)
+
+Goal "((x::hcomplex) = u + v) = (x - (u + v) = 0)";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_eq_eq]));
+qed "hcomplex_eq_add_diff_eq_0";
+
+Goal "((x::hcomplex) = n) = (x - n = 0)";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_eq_eq]));
+qed "hcomplex_eq_diff_eq_0";
+
+val hcomplex_rel_iff_rel_0_rls = [hcomplex_eq_diff_eq_0,hcomplex_eq_add_diff_eq_0];
+
+Goal "!!i::hcomplex. (i*u + m = j*u + n) = ((i-j)*u + m = n)";
+by (auto_tac (claset(), simpset() addsimps [hcomplex_add_mult_distrib,
+ hcomplex_diff_def] @ hcomplex_add_ac));
+by (asm_simp_tac (simpset() addsimps [hcomplex_add_assoc RS sym]) 1);
+by (simp_tac (simpset() addsimps [hcomplex_add_assoc]) 1);
+qed "hcomplex_eq_add_iff1";
+
+Goal "!!i::hcomplex. (i*u + m = j*u + n) = (m = (j-i)*u + n)";
+by (res_inst_tac [("z","i")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","j")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","u")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","m")] eq_Abs_hcomplex 1);
+by (res_inst_tac [("z","n")] eq_Abs_hcomplex 1);
+by (auto_tac (claset(), simpset() addsimps [hcomplex_diff,hcomplex_add,
+ hcomplex_mult,complex_eq_add_iff2]));
+qed "hcomplex_eq_add_iff2";
+
+
+structure HComplex_Numeral_Simprocs =
+struct
+
+(*Utilities*)
+
+val hcomplexT = Type("NSComplex.hcomplex",[]);
+
+fun mk_numeral n = HOLogic.number_of_const hcomplexT $ HOLogic.mk_bin n;
+
+val dest_numeral = Complex_Numeral_Simprocs.dest_numeral;
+
+val find_first_numeral = Complex_Numeral_Simprocs.find_first_numeral;
+
+val zero = mk_numeral 0;
+val mk_plus = HOLogic.mk_binop "op +";
+
+val uminus_const = Const ("uminus", hcomplexT --> hcomplexT);
+
+(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
+fun mk_sum [] = zero
+ | mk_sum [t,u] = mk_plus (t, u)
+ | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
+
+(*this version ALWAYS includes a trailing zero*)
+fun long_mk_sum [] = zero
+ | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
+
+val dest_plus = HOLogic.dest_bin "op +" hcomplexT;
+
+(*decompose additions AND subtractions as a sum*)
+fun dest_summing (pos, Const ("op +", _) $ t $ u, ts) =
+ dest_summing (pos, t, dest_summing (pos, u, ts))
+ | dest_summing (pos, Const ("op -", _) $ t $ u, ts) =
+ dest_summing (pos, t, dest_summing (not pos, u, ts))
+ | dest_summing (pos, t, ts) =
+ if pos then t::ts else uminus_const$t :: ts;
+
+fun dest_sum t = dest_summing (true, t, []);
+
+val mk_diff = HOLogic.mk_binop "op -";
+val dest_diff = HOLogic.dest_bin "op -" hcomplexT;
+
+val one = mk_numeral 1;
+val mk_times = HOLogic.mk_binop "op *";
+
+fun mk_prod [] = one
+ | mk_prod [t] = t
+ | mk_prod (t :: ts) = if t = one then mk_prod ts
+ else mk_times (t, mk_prod ts);
+
+val dest_times = HOLogic.dest_bin "op *" hcomplexT;
+
+fun dest_prod t =
+ let val (t,u) = dest_times t
+ in dest_prod t @ dest_prod u end
+ handle TERM _ => [t];
+
+(*DON'T do the obvious simplifications; that would create special cases*)
+fun mk_coeff (k, ts) = mk_times (mk_numeral k, ts);
+
+(*Express t as a product of (possibly) a numeral with other sorted terms*)
+fun dest_coeff sign (Const ("uminus", _) $ t) = dest_coeff (~sign) t
+ | dest_coeff sign t =
+ let val ts = sort Term.term_ord (dest_prod t)
+ val (n, ts') = find_first_numeral [] ts
+ handle TERM _ => (1, ts)
+ in (sign*n, mk_prod ts') end;
+
+(*Find first coefficient-term THAT MATCHES u*)
+fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
+ | find_first_coeff past u (t::terms) =
+ let val (n,u') = dest_coeff 1 t
+ in if u aconv u' then (n, rev past @ terms)
+ else find_first_coeff (t::past) u terms
+ end
+ handle TERM _ => find_first_coeff (t::past) u terms;
+
+
+
+(*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1*)
+val add_0s = map rename_numerals [hcomplex_add_zero_left, hcomplex_add_zero_right];
+val mult_plus_1s = map rename_numerals [hcomplex_mult_one_left, hcomplex_mult_one_right];
+val mult_minus_1s = map rename_numerals [hcomplex_mult_minus1, hcomplex_mult_minus1_right];
+val mult_1s = mult_plus_1s @ mult_minus_1s;
+
+(*To perform binary arithmetic*)
+val bin_simps =
+ [hcomplex_numeral_0_eq_0 RS sym, hcomplex_numeral_1_eq_1 RS sym,
+ add_hcomplex_number_of, hcomplex_add_number_of_left,
+ minus_hcomplex_number_of, diff_hcomplex_number_of, mult_hcomplex_number_of,
+ hcomplex_mult_number_of_left] @ bin_arith_simps @ bin_rel_simps;
+
+(*To evaluate binary negations of coefficients*)
+val hcomplex_minus_simps = NCons_simps @
+ [hcomplex_minus_1_eq_m1,minus_hcomplex_number_of,
+ bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min,
+ bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min];
+
+
+(*To let us treat subtraction as addition*)
+val diff_simps = [hcomplex_diff_def, hcomplex_minus_add_distrib,
+ hcomplex_minus_minus];
+
+(*push the unary minus down: - x * y = x * - y *)
+val hcomplex_minus_mult_eq_1_to_2 =
+ [hcomplex_minus_mult_eq1 RS sym, hcomplex_minus_mult_eq2] MRS trans
+ |> standard;
+
+(*to extract again any uncancelled minuses*)
+val hcomplex_minus_from_mult_simps =
+ [hcomplex_minus_minus, hcomplex_minus_mult_eq1 RS sym,
+ hcomplex_minus_mult_eq2 RS sym];
+
+(*combine unary minus with numeric literals, however nested within a product*)
+val hcomplex_mult_minus_simps =
+ [hcomplex_mult_assoc, hcomplex_minus_mult_eq1, hcomplex_minus_mult_eq_1_to_2];
+
+(*Final simplification: cancel + and * *)
+val simplify_meta_eq =
+ Int_Numeral_Simprocs.simplify_meta_eq
+ [hcomplex_add_zero_left, hcomplex_add_zero_right,
+ hcomplex_mult_zero_left, hcomplex_mult_zero_right, hcomplex_mult_one_left,
+ hcomplex_mult_one_right];
+
+val prep_simproc = Complex_Numeral_Simprocs.prep_simproc;
+
+
+structure CancelNumeralsCommon =
+ struct
+ val mk_sum = mk_sum
+ val dest_sum = dest_sum
+ val mk_coeff = mk_coeff
+ val dest_coeff = dest_coeff 1
+ val find_first_coeff = find_first_coeff []
+ val trans_tac = Real_Numeral_Simprocs.trans_tac
+ val norm_tac =
+ ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
+ hcomplex_minus_simps@hcomplex_add_ac))
+ THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@hcomplex_mult_minus_simps))
+ THEN ALLGOALS
+ (simp_tac (HOL_ss addsimps hcomplex_minus_from_mult_simps@
+ hcomplex_add_ac@hcomplex_mult_ac))
+ val numeral_simp_tac = ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps))
+ val simplify_meta_eq = simplify_meta_eq
+ end;
+
+
+structure EqCancelNumerals = CancelNumeralsFun
+ (open CancelNumeralsCommon
+ val prove_conv = Bin_Simprocs.prove_conv
+ val mk_bal = HOLogic.mk_eq
+ val dest_bal = HOLogic.dest_bin "op =" hcomplexT
+ val bal_add1 = hcomplex_eq_add_iff1 RS trans
+ val bal_add2 = hcomplex_eq_add_iff2 RS trans
+);
+
+
+val cancel_numerals =
+ map prep_simproc
+ [("hcomplexeq_cancel_numerals",
+ ["(l::hcomplex) + m = n", "(l::hcomplex) = m + n",
+ "(l::hcomplex) - m = n", "(l::hcomplex) = m - n",
+ "(l::hcomplex) * m = n", "(l::hcomplex) = m * n"],
+ EqCancelNumerals.proc)];
+
+structure CombineNumeralsData =
+ struct
+ val add = op + : int*int -> int
+ val mk_sum = long_mk_sum (*to work for e.g. #2*x + #3*x *)
+ val dest_sum = dest_sum
+ val mk_coeff = mk_coeff
+ val dest_coeff = dest_coeff 1
+ val left_distrib = left_hcomplex_add_mult_distrib RS trans
+ val prove_conv = Bin_Simprocs.prove_conv_nohyps
+ val trans_tac = Real_Numeral_Simprocs.trans_tac
+ val norm_tac =
+ ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
+ hcomplex_minus_simps@hcomplex_add_ac))
+ THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@hcomplex_mult_minus_simps))
+ THEN ALLGOALS (simp_tac (HOL_ss addsimps hcomplex_minus_from_mult_simps@
+ hcomplex_add_ac@hcomplex_mult_ac))
+ val numeral_simp_tac = ALLGOALS
+ (simp_tac (HOL_ss addsimps add_0s@bin_simps))
+ val simplify_meta_eq = simplify_meta_eq
+ end;
+
+structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
+
+val combine_numerals =
+ prep_simproc ("hcomplex_combine_numerals",
+ ["(i::hcomplex) + j", "(i::hcomplex) - j"],
+ CombineNumerals.proc);
+
+(** Declarations for ExtractCommonTerm **)
+
+(*this version ALWAYS includes a trailing one*)
+fun long_mk_prod [] = one
+ | long_mk_prod (t :: ts) = mk_times (t, mk_prod ts);
+
+(*Find first term that matches u*)
+fun find_first past u [] = raise TERM("find_first", [])
+ | find_first past u (t::terms) =
+ if u aconv t then (rev past @ terms)
+ else find_first (t::past) u terms
+ handle TERM _ => find_first (t::past) u terms;
+
+(*Final simplification: cancel + and * *)
+fun cancel_simplify_meta_eq cancel_th th =
+ Int_Numeral_Simprocs.simplify_meta_eq
+ [hcomplex_mult_one_left, hcomplex_mult_one_right]
+ (([th, cancel_th]) MRS trans);
+
+(*** Making constant folding work for 0 and 1 too ***)
+
+structure HComplexAbstractNumeralsData =
+ struct
+ val dest_eq = HOLogic.dest_eq o HOLogic.dest_Trueprop o concl_of
+ val is_numeral = Bin_Simprocs.is_numeral
+ val numeral_0_eq_0 = hcomplex_numeral_0_eq_0
+ val numeral_1_eq_1 = hcomplex_numeral_1_eq_1
+ val prove_conv = Bin_Simprocs.prove_conv_nohyps_novars
+ fun norm_tac simps = ALLGOALS (simp_tac (HOL_ss addsimps simps))
+ val simplify_meta_eq = Bin_Simprocs.simplify_meta_eq
+ end
+
+structure HComplexAbstractNumerals = AbstractNumeralsFun (HComplexAbstractNumeralsData)
+
+(*For addition, we already have rules for the operand 0.
+ Multiplication is omitted because there are already special rules for
+ both 0 and 1 as operands. Unary minus is trivial, just have - 1 = -1.
+ For the others, having three patterns is a compromise between just having
+ one (many spurious calls) and having nine (just too many!) *)
+val eval_numerals =
+ map prep_simproc
+ [("hcomplex_add_eval_numerals",
+ ["(m::hcomplex) + 1", "(m::hcomplex) + number_of v"],
+ HComplexAbstractNumerals.proc add_hcomplex_number_of),
+ ("hcomplex_diff_eval_numerals",
+ ["(m::hcomplex) - 1", "(m::hcomplex) - number_of v"],
+ HComplexAbstractNumerals.proc diff_hcomplex_number_of),
+ ("hcomplex_eq_eval_numerals",
+ ["(m::hcomplex) = 0", "(m::hcomplex) = 1", "(m::hcomplex) = number_of v"],
+ HComplexAbstractNumerals.proc eq_hcomplex_number_of)]
+
+end;
+
+Addsimprocs HComplex_Numeral_Simprocs.eval_numerals;
+Addsimprocs HComplex_Numeral_Simprocs.cancel_numerals;
+Addsimprocs [HComplex_Numeral_Simprocs.combine_numerals];
+
+
+(*examples:
+print_depth 22;
+set timing;
+set trace_simp;
+fun test s = (Goal s, by (Simp_tac 1));
+
+test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::hcomplex)";
+test " 2*u = (u::hcomplex)";
+test "(i + j + 12 + (k::hcomplex)) - 15 = y";
+test "(i + j + 12 + (k::hcomplex)) - 5 = y";
+
+test "( 2*x - (u*v) + y) - v* 3*u = (w::hcomplex)";
+test "( 2*x*u*v + (u*v)* 4 + y) - v*u* 4 = (w::hcomplex)";
+test "( 2*x*u*v + (u*v)* 4 + y) - v*u = (w::hcomplex)";
+test "u*v - (x*u*v + (u*v)* 4 + y) = (w::hcomplex)";
+
+test "(i + j + 12 + (k::hcomplex)) = u + 15 + y";
+test "(i + j* 2 + 12 + (k::hcomplex)) = j + 5 + y";
+
+test " 2*y + 3*z + 6*w + 2*y + 3*z + 2*u = 2*y' + 3*z' + 6*w' + 2*y' + 3*z' + u + (vv::hcomplex)";
+
+test "a + -(b+c) + b = (d::hcomplex)";
+test "a + -(b+c) - b = (d::hcomplex)";
+
+(*negative numerals*)
+test "(i + j + -2 + (k::hcomplex)) - (u + 5 + y) = zz";
+
+test "(i + j + -12 + (k::hcomplex)) - 15 = y";
+test "(i + j + 12 + (k::hcomplex)) - -15 = y";
+test "(i + j + -12 + (k::hcomplex)) - -15 = y";
+*)
+
+(** Constant folding for hcomplex plus and times **)
+
+structure HComplex_Times_Assoc_Data : ASSOC_FOLD_DATA =
+struct
+ val ss = HOL_ss
+ val eq_reflection = eq_reflection
+ val sg_ref = Sign.self_ref (Theory.sign_of (the_context ()))
+ val T = HComplex_Numeral_Simprocs.hcomplexT
+ val plus = Const ("op *", [T,T] ---> T)
+ val add_ac = hcomplex_mult_ac
+end;
+
+structure HComplex_Times_Assoc = Assoc_Fold (HComplex_Times_Assoc_Data);
+
+Addsimprocs [HComplex_Times_Assoc.conv];
+
+Addsimps [hcomplex_of_complex_zero_iff];
+
+(*Simplification of x-y = 0 *)
+
+AddIffs [hcomplex_eq_iff_diff_eq_0 RS sym];
+
+(** extra thms **)
+
+Goal "(hcnj z = 0) = (z = 0)";
+by (auto_tac (claset(),simpset() addsimps [hcomplex_hcnj_zero_iff]));
+qed "hcomplex_hcnj_num_zero_iff";
+Addsimps [hcomplex_hcnj_num_zero_iff];
+
+Goal "0 = Abs_hcomplex (hcomplexrel `` {%n. 0})";
+by (simp_tac (simpset() addsimps [hcomplex_zero_def RS meta_eq_to_obj_eq RS sym]) 1);
+qed "hcomplex_zero_num";
+
+Goal "1 = Abs_hcomplex (hcomplexrel `` {%n. 1})";
+by (simp_tac (simpset() addsimps [hcomplex_one_def RS meta_eq_to_obj_eq RS sym]) 1);
+qed "hcomplex_one_num";
+
+(*** Real and imaginary stuff ***)
+
+Goalw [hcomplex_number_of_def]
+ "((number_of xa :: hcomplex) + iii * number_of ya = \
+\ number_of xb + iii * number_of yb) = \
+\ (((number_of xa :: hcomplex) = number_of xb) & \
+\ ((number_of ya :: hcomplex) = number_of yb))";
+by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff,
+ hcomplex_hypreal_number_of]));
+qed "hcomplex_number_of_eq_cancel_iff";
+Addsimps [hcomplex_number_of_eq_cancel_iff];
+
+Goalw [hcomplex_number_of_def]
+ "((number_of xa :: hcomplex) + number_of ya * iii = \
+\ number_of xb + number_of yb * iii) = \
+\ (((number_of xa :: hcomplex) = number_of xb) & \
+\ ((number_of ya :: hcomplex) = number_of yb))";
+by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffA,
+ hcomplex_hypreal_number_of]));
+qed "hcomplex_number_of_eq_cancel_iffA";
+Addsimps [hcomplex_number_of_eq_cancel_iffA];
+
+Goalw [hcomplex_number_of_def]
+ "((number_of xa :: hcomplex) + number_of ya * iii = \
+\ number_of xb + iii * number_of yb) = \
+\ (((number_of xa :: hcomplex) = number_of xb) & \
+\ ((number_of ya :: hcomplex) = number_of yb))";
+by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffB,
+ hcomplex_hypreal_number_of]));
+qed "hcomplex_number_of_eq_cancel_iffB";
+Addsimps [hcomplex_number_of_eq_cancel_iffB];
+
+Goalw [hcomplex_number_of_def]
+ "((number_of xa :: hcomplex) + iii * number_of ya = \
+\ number_of xb + number_of yb * iii) = \
+\ (((number_of xa :: hcomplex) = number_of xb) & \
+\ ((number_of ya :: hcomplex) = number_of yb))";
+by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffC,
+ hcomplex_hypreal_number_of]));
+qed "hcomplex_number_of_eq_cancel_iffC";
+Addsimps [hcomplex_number_of_eq_cancel_iffC];
+
+Goalw [hcomplex_number_of_def]
+ "((number_of xa :: hcomplex) + iii * number_of ya = \
+\ number_of xb) = \
+\ (((number_of xa :: hcomplex) = number_of xb) & \
+\ ((number_of ya :: hcomplex) = 0))";
+by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff2,
+ hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
+qed "hcomplex_number_of_eq_cancel_iff2";
+Addsimps [hcomplex_number_of_eq_cancel_iff2];
+
+Goalw [hcomplex_number_of_def]
+ "((number_of xa :: hcomplex) + number_of ya * iii = \
+\ number_of xb) = \
+\ (((number_of xa :: hcomplex) = number_of xb) & \
+\ ((number_of ya :: hcomplex) = 0))";
+by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff2a,
+ hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
+qed "hcomplex_number_of_eq_cancel_iff2a";
+Addsimps [hcomplex_number_of_eq_cancel_iff2a];
+
+Goalw [hcomplex_number_of_def]
+ "((number_of xa :: hcomplex) + iii * number_of ya = \
+\ iii * number_of yb) = \
+\ (((number_of xa :: hcomplex) = 0) & \
+\ ((number_of ya :: hcomplex) = number_of yb))";
+by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff3,
+ hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
+qed "hcomplex_number_of_eq_cancel_iff3";
+Addsimps [hcomplex_number_of_eq_cancel_iff3];
+
+Goalw [hcomplex_number_of_def]
+ "((number_of xa :: hcomplex) + number_of ya * iii= \
+\ iii * number_of yb) = \
+\ (((number_of xa :: hcomplex) = 0) & \
+\ ((number_of ya :: hcomplex) = number_of yb))";
+by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff3a,
+ hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
+qed "hcomplex_number_of_eq_cancel_iff3a";
+Addsimps [hcomplex_number_of_eq_cancel_iff3a];
+
+Goalw [hcomplex_number_of_def] "hcnj (number_of v :: hcomplex) = number_of v";
+by (rtac (hcomplex_hypreal_number_of RS ssubst) 1);
+by (rtac hcomplex_hcnj_hcomplex_of_hypreal 1);
+qed "hcomplex_number_of_hcnj";
+Addsimps [hcomplex_number_of_hcnj];
+
+Goalw [hcomplex_number_of_def]
+ "hcmod(number_of v :: hcomplex) = abs (number_of v :: hypreal)";
+by (rtac (hcomplex_hypreal_number_of RS ssubst) 1);
+by (auto_tac (claset(), HOL_ss addsimps [hcmod_hcomplex_of_hypreal]));
+qed "hcomplex_number_of_hcmod";
+Addsimps [hcomplex_number_of_hcmod];
+
+Goalw [hcomplex_number_of_def]
+ "hRe(number_of v :: hcomplex) = number_of v";
+by (rtac (hcomplex_hypreal_number_of RS ssubst) 1);
+by (auto_tac (claset(), HOL_ss addsimps [hRe_hcomplex_of_hypreal]));
+qed "hcomplex_number_of_hRe";
+Addsimps [hcomplex_number_of_hRe];
+
+Goalw [hcomplex_number_of_def]
+ "hIm(number_of v :: hcomplex) = 0";
+by (rtac (hcomplex_hypreal_number_of RS ssubst) 1);
+by (auto_tac (claset(), HOL_ss addsimps [hIm_hcomplex_of_hypreal]));
+qed "hcomplex_number_of_hIm";
+Addsimps [hcomplex_number_of_hIm];
+
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/NSComplexBin.thy Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,19 @@
+(* Title: NSComplexBin.thy
+ Author: Jacques D. Fleuriot
+ Copyright: 2001 University of Edinburgh
+ Descrition: Binary arithmetic for the nonstandard complex numbers
+ This case is reduced to that for the complexes (hence reals).
+*)
+
+NSComplexBin = NSComplex +
+
+
+instance
+ hcomplex :: number
+
+defs
+ hcomplex_number_of_def
+ "number_of v == hcomplex_of_complex (number_of v)"
+ (*::bin=>complex ::bin=>complex*)
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/NSInduct.ML Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,138 @@
+(* Title: NSInduct.ML
+ Author: Jacques D. Fleuriot
+ Copyright: 2001 University of Edinburgh
+ Description: Nonstandard characterization of induction etc.
+*)
+
+Goalw [starPNat_def]
+"(( *pNat* P) (Abs_hypnat(hypnatrel``{%n. X n}))) = \
+\ ({n. P (X n)} : FreeUltrafilterNat)";
+by (Auto_tac);
+by (Ultra_tac 1);
+qed "starPNat";
+
+Goalw [hypnat_of_nat_def] "( *pNat* P) (hypnat_of_nat n) = P n";
+by (auto_tac (claset(),simpset() addsimps [starPNat]));
+qed "starPNat_hypnat_of_nat";
+Addsimps [starPNat_hypnat_of_nat];
+
+Goalw [hypnat_zero_def,hypnat_one_def]
+ "!!P. (( *pNat* P) 0 & \
+\ (ALL n. ( *pNat* P)(n) --> ( *pNat* P)(n + 1))) \
+\ --> ( *pNat* P)(n)";
+by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
+by (auto_tac (claset(),simpset() addsimps [starPNat]));
+by (Ultra_tac 1);
+by (etac nat_induct 1);
+by (dres_inst_tac [("x","hypnat_of_nat n")] spec 1);
+by (rtac ccontr 1);
+by (auto_tac (claset(),simpset() addsimps [starPNat,
+ hypnat_of_nat_def,hypnat_add]));
+qed "hypnat_induct_obj";
+
+Goal
+ "!!P. [| ( *pNat* P) 0; \
+\ !!n. ( *pNat* P)(n) ==> ( *pNat* P)(n + 1)|] \
+\ ==> ( *pNat* P)(n)";
+by (cut_inst_tac [("P","P"),("n","n")] hypnat_induct_obj 1);
+by (Auto_tac);
+qed "hypnat_induct";
+
+fun hypnat_ind_tac a i =
+ res_inst_tac [("n",a)] hypnat_induct i THEN rename_last_tac a [""] (i+1);
+
+Goalw [starPNat2_def]
+"(( *pNat2* P) (Abs_hypnat(hypnatrel``{%n. X n})) \
+\ (Abs_hypnat(hypnatrel``{%n. Y n}))) = \
+\ ({n. P (X n) (Y n)} : FreeUltrafilterNat)";
+by (Auto_tac);
+by (Ultra_tac 1);
+qed "starPNat2";
+
+Goalw [starPNat2_def] "( *pNat2* (op =)) = (op =)";
+by (rtac ext 1);
+by (rtac ext 1);
+by (res_inst_tac [("z","x")] eq_Abs_hypnat 1);
+by (res_inst_tac [("z","y")] eq_Abs_hypnat 1);
+by (Auto_tac THEN Ultra_tac 1);
+qed "starPNat2_eq_iff";
+
+Goal "( *pNat2* (%x y. x = y)) X Y = (X = Y)";
+by (simp_tac (simpset() addsimps [starPNat2_eq_iff]) 1);
+qed "starPNat2_eq_iff2";
+
+Goal "(EX h. P(h::hypnat)) = (EX x. P(Abs_hypnat(hypnatrel `` {x})))";
+by (Auto_tac);
+by (res_inst_tac [("z","h")] eq_Abs_hypnat 1);
+by (Auto_tac);
+val lemma_hyp = result();
+
+Goal "(EX (v::nat=>nat). ALL (x::nat). P (g v) (f x)) = \
+\ (EX (v::nat=>nat). ALL x. EX f': Rep_hypnat(f x). P (g v) (Abs_hypnat(hypnatrel `` {f'})))";
+by (Auto_tac);
+by (ALLGOALS(res_inst_tac [("x","v")] exI));
+by (Step_tac 1);
+by (ALLGOALS(res_inst_tac [("z","f x")] eq_Abs_hypnat));
+by (Auto_tac);
+by (rtac bexI 1);
+by (dres_inst_tac [("x","x")] spec 3);
+by (dtac sym 1);
+by (Auto_tac);
+by (subgoal_tac
+ "Abs_hypnat (hypnatrel `` {X}) = Abs_hypnat (hypnatrel `` {Y})" 1);
+by (Asm_simp_tac 1);
+by (Auto_tac);
+val lemma_hyp2 = result();
+
+Goalw [hSuc_def] "hSuc m ~= 0";
+by Auto_tac;
+qed "hSuc_not_zero";
+
+bind_thm ("zero_not_hSuc", hSuc_not_zero RS not_sym);
+
+Goalw [hSuc_def,hypnat_one_def]
+ "(hSuc m = hSuc n) = (m = n)";
+by (res_inst_tac [("z","m")] eq_Abs_hypnat 1);
+by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
+by (auto_tac (claset(),simpset() addsimps [hypnat_add]));
+qed "hSuc_hSuc_eq";
+
+AddIffs [hSuc_not_zero,zero_not_hSuc,hSuc_hSuc_eq];
+
+Goalw [hypnat_le_def] "m <= (n::hypnat) | n <= m";
+by (auto_tac (claset() addDs [hypnat_less_trans],simpset()));
+qed "hypnat_le_linear";
+
+val hypnat_less_le = hypnat_less_imp_le;
+
+Goal "c : (S :: nat set) ==> (LEAST n. n : S) : S";
+by (etac LeastI 1);
+qed "nonempty_nat_set_Least_mem";
+
+Goalw [InternalNatSets_def,starsetNat_n_def]
+ "[| S : InternalNatSets; S ~= {} |] ==> EX n: S. ALL m: S. n <= m";
+by (auto_tac (claset(),simpset() addsimps [lemma_hyp]));
+by (res_inst_tac [("z","x")] eq_Abs_hypnat 1);
+by (auto_tac (claset() addSDs [bspec],simpset() addsimps [hypnat_le]));
+by (dres_inst_tac [("x","%m. LEAST n. n : As m")] spec 1);
+by Auto_tac;
+by (ultra_tac (claset() addDs [nonempty_nat_set_Least_mem],simpset()) 1);
+by (dres_inst_tac [("x","x")] bspec 1 THEN Auto_tac);
+by (ultra_tac (claset() addIs [Least_le],simpset()) 1);
+qed "nonempty_InternalNatSet_has_least";
+
+(* Goldblatt p 129 Thm 11.3.2 *)
+Goal "[| X : InternalNatSets; 0 : X; ALL n. n : X --> n + 1 : X |] \
+\ ==> X = (UNIV:: hypnat set)";
+by (rtac ccontr 1);
+by (ftac InternalNatSets_Compl 1);
+by (dres_inst_tac [("S","- X")] nonempty_InternalNatSet_has_least 1);
+by Auto_tac;
+by (subgoal_tac "1 <= n" 1);
+by (dres_inst_tac [("x","n - 1")] bspec 1);
+by (Step_tac 1);
+by (dres_inst_tac [("x","n - 1")] spec 1);
+by (dres_inst_tac [("x","1"),("q1.0","n")] hypnat_add_le_mono1 2);
+by (auto_tac (claset(),simpset() addsimps [hypnat_le_def]));
+qed "internal_induct";
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/NSInduct.thy Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,23 @@
+(* Title: NSInduct.thy
+ Author: Jacques D. Fleuriot
+ Copyright: 2001 University of Edinburgh
+ Description: Nonstandard characterization of induction etc.
+*)
+
+NSInduct = ComplexArith0 +
+
+constdefs
+
+ starPNat :: (nat => bool) => hypnat => bool ("*pNat* _" [80] 80)
+ "*pNat* P == (%x. EX X. (X: Rep_hypnat(x) &
+ {n. P(X n)} : FreeUltrafilterNat))"
+
+
+ starPNat2 :: (nat => nat => bool) => hypnat =>hypnat => bool ("*pNat2* _" [80] 80)
+ "*pNat2* P == (%x y. EX X. EX Y. (X: Rep_hypnat(x) & Y: Rep_hypnat(y) &
+ {n. P(X n) (Y n)} : FreeUltrafilterNat))"
+
+ hSuc :: hypnat => hypnat
+ "hSuc n == n + 1"
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/ROOT.ML Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,9 @@
+(* Title: HOL/Complex/ROOT.ML
+ ID: $Id$
+ Author: Jacques Fleuriot
+
+The Complex Numbers
+*)
+
+with_path "../Hyperreal" use_thy "Hyperreal";
+time_use_thy "CLim";
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/ex/NSPrimes.ML Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,492 @@
+(* Title : NSPrimes.ML
+ Author : Jacques D. Fleuriot
+ Copyright : 2002 University of Edinburgh
+ Description : The nonstandard primes as an extension of
+ the prime numbers
+*)
+
+(*--------------------------------------------------------------*)
+(* A "choice" theorem for ultrafilters cf. almost everywhere *)
+(* quantification *)
+(*--------------------------------------------------------------*)
+
+Goal "{n. EX m. Q n m} : FreeUltrafilterNat \
+\ ==> EX f. {n. Q n (f n)} : FreeUltrafilterNat";
+by (res_inst_tac [("x","%n. (@x. Q n x)")] exI 1);
+by (Ultra_tac 1 THEN rtac someI 1 THEN Auto_tac);
+qed "UF_choice";
+
+Goal "({n. P n} : FreeUltrafilterNat --> {n. Q n} : FreeUltrafilterNat) = \
+\ ({n. P n --> Q n} : FreeUltrafilterNat)";
+by Auto_tac;
+by (ALLGOALS(Ultra_tac));
+qed "UF_if";
+
+Goal "({n. P n} : FreeUltrafilterNat & {n. Q n} : FreeUltrafilterNat) = \
+\ ({n. P n & Q n} : FreeUltrafilterNat)";
+by Auto_tac;
+by (ALLGOALS(Ultra_tac));
+qed "UF_conj";
+
+Goal "(ALL f. {n. Q n (f n)} : FreeUltrafilterNat) = \
+\ ({n. ALL m. Q n m} : FreeUltrafilterNat)";
+by Auto_tac;
+by (Ultra_tac 2);
+by (rtac ccontr 1);
+by (rtac swap 1 THEN assume_tac 2);
+by (simp_tac (simpset() addsimps [FreeUltrafilterNat_Compl_iff1,
+ CLAIM "-{n. Q n} = {n. ~ Q n}"]) 1);
+by (rtac UF_choice 1);
+by (Ultra_tac 1 THEN Auto_tac);
+qed "UF_choice_ccontr";
+
+Goal "ALL M. EX N. 0 < N & (ALL m. 0 < m & (m::nat) <= M --> m dvd N)";
+by (rtac allI 1);
+by (induct_tac "M" 1);
+by Auto_tac;
+by (res_inst_tac [("x","N * (Suc n)")] exI 1);
+by (Step_tac 1 THEN Force_tac 1);
+by (dtac le_imp_less_or_eq 1 THEN etac disjE 1);
+by (force_tac (claset() addSIs [dvd_mult2],simpset()) 1);
+by (force_tac (claset() addSIs [dvd_mult],simpset()) 1);
+qed "dvd_by_all";
+
+bind_thm ("dvd_by_all2",dvd_by_all RS spec);
+
+Goal "(EX (x::hypnat). P x) = (EX f. P (Abs_hypnat(hypnatrel `` {f})))";
+by Auto_tac;
+by (res_inst_tac [("z","x")] eq_Abs_hypnat 1);
+by Auto_tac;
+qed "lemma_hypnat_P_EX";
+
+Goal "(ALL (x::hypnat). P x) = (ALL f. P (Abs_hypnat(hypnatrel `` {f})))";
+by Auto_tac;
+by (res_inst_tac [("z","x")] eq_Abs_hypnat 1);
+by Auto_tac;
+qed "lemma_hypnat_P_ALL";
+
+Goalw [hdvd_def]
+ "(Abs_hypnat(hypnatrel``{%n. X n}) hdvd \
+\ Abs_hypnat(hypnatrel``{%n. Y n})) = \
+\ ({n. X n dvd Y n} : FreeUltrafilterNat)";
+by (Auto_tac THEN Ultra_tac 1);
+qed "hdvd";
+
+Goal "(hypnat_of_nat n <= 0) = (n = 0)";
+by (stac (hypnat_of_nat_zero RS sym) 1);
+by (auto_tac (claset(),simpset() addsimps [hypnat_of_nat_le_iff RS sym]));
+qed "hypnat_of_nat_le_zero_iff";
+Addsimps [hypnat_of_nat_le_zero_iff];
+
+
+(* Goldblatt: Exercise 5.11(2) - p. 57 *)
+Goal "ALL M. EX N. 0 < N & (ALL m. 0 < m & (m::hypnat) <= M --> m hdvd N)";
+by (Step_tac 1);
+by (res_inst_tac [("z","M")] eq_Abs_hypnat 1);
+by (auto_tac (claset(),simpset() addsimps [lemma_hypnat_P_EX,
+ lemma_hypnat_P_ALL,hypnat_zero_def,hypnat_le,hypnat_less,hdvd]));
+by (cut_facts_tac [dvd_by_all] 1);
+by (dtac (CLAIM "(ALL (M::nat). EX N. 0 < N & (ALL m. 0 < m & m <= M \
+\ --> m dvd N)) \
+\ ==> ALL (n::nat). EX N. 0 < N & (ALL m. 0 < (m::nat) & m <= (x n) \
+\ --> m dvd N)") 1);
+by (dtac choice 1);
+by (Step_tac 1);
+by (res_inst_tac [("x","f")] exI 1);
+by Auto_tac;
+by (ALLGOALS(Ultra_tac));
+by Auto_tac;
+qed "hdvd_by_all";
+
+bind_thm ("hdvd_by_all2",hdvd_by_all RS spec);
+
+(* Goldblatt: Exercise 5.11(2) - p. 57 *)
+Goal "EX (N::hypnat). 0 < N & (ALL n : - {0::nat}. hypnat_of_nat(n) hdvd N)";
+by (cut_facts_tac [hdvd_by_all] 1);
+by (dres_inst_tac [("x","whn")] spec 1);
+by Auto_tac;
+by (rtac exI 1 THEN Auto_tac);
+by (dres_inst_tac [("x","hypnat_of_nat n")] spec 1);
+by (auto_tac (claset(),simpset() addsimps [symmetric hypnat_le_def,
+ hypnat_of_nat_zero_iff]));
+qed "hypnat_dvd_all_hypnat_of_nat";
+
+
+(*--------------------------------------------------------------*)
+(* the nonstandard extension of the set prime numbers consists *)
+(* of precisely those hypernaturals > 1 that have no nontrivial *)
+(* factors *)
+(*--------------------------------------------------------------*)
+
+(* Goldblatt: Exercise 5.11(3a) - p 57 *)
+Goalw [starprime_def,thm "prime_def"]
+ "starprime = {p. 1 < p & (ALL m. m hdvd p --> m = 1 | m = p)}";
+by (auto_tac (claset(),simpset() addsimps
+ [CLAIM "{p. Q(p) & R(p)} = {p. Q(p)} Int {p. R(p)}",NatStar_Int]));
+by (ALLGOALS(res_inst_tac [("z","x")] eq_Abs_hypnat));
+by (res_inst_tac [("z","m")] eq_Abs_hypnat 2);
+by (auto_tac (claset(),simpset() addsimps [hdvd,hypnat_one_def,hypnat_less,
+ lemma_hypnat_P_ALL,starsetNat_def]));
+by (dtac bspec 1 THEN dtac bspec 2 THEN Auto_tac);
+by (Ultra_tac 1 THEN Ultra_tac 1);
+by (rtac ccontr 1);
+by (dtac FreeUltrafilterNat_Compl_mem 1);
+by (auto_tac (claset(),simpset() addsimps [CLAIM "- {p. Q p} = {p. ~ Q p}"]));
+by (dtac UF_choice 1 THEN Auto_tac);
+by (dres_inst_tac [("x","f")] spec 1 THEN Auto_tac);
+by (ALLGOALS(Ultra_tac));
+qed "starprime";
+
+Goalw [thm "prime_def"] "2 : prime";
+by Auto_tac;
+by (ftac dvd_imp_le 1);
+by (auto_tac (claset() addDs [dvd_0_left],simpset() addsimps
+ [ARITH_PROVE "(m <= (2::nat)) = (m = 0 | m = 1 | m = 2)"]));
+qed "prime_two";
+Addsimps [prime_two];
+
+(* proof uses course-of-value induction *)
+Goal "Suc 0 < n --> (EX k : prime. k dvd n)";
+by (res_inst_tac [("n","n")] nat_less_induct 1);
+by Auto_tac;
+by (case_tac "n : prime" 1);
+by (res_inst_tac [("x","n")] bexI 1 THEN Auto_tac);
+by (dtac (conjI RS (thm "not_prime_ex_mk")) 1 THEN Auto_tac);
+by (dres_inst_tac [("x","m")] spec 1 THEN Auto_tac);
+by (res_inst_tac [("x","ka")] bexI 1);
+by (auto_tac (claset() addIs [dvd_mult2],simpset()));
+qed_spec_mp "prime_factor_exists";
+
+(* Goldblatt Exercise 5.11(3b) - p 57 *)
+Goal "1 < n ==> (EX k : starprime. k hdvd n)";
+by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
+by (auto_tac (claset(),simpset() addsimps [hypnat_one_def,
+ hypnat_less,starprime_def,lemma_hypnat_P_EX,lemma_hypnat_P_ALL,
+ hdvd,starsetNat_def,CLAIM "(ALL X: S. P X) = (ALL X. X : S --> P X)",
+ UF_if]));
+by (res_inst_tac [("x","%n. @y. y : prime & y dvd x n")] exI 1);
+by Auto_tac;
+by (ALLGOALS (Ultra_tac));
+by (dtac sym 1 THEN Asm_simp_tac 1);
+by (ALLGOALS(rtac someI2_ex));
+by (auto_tac (claset() addSDs [prime_factor_exists],simpset()));
+qed_spec_mp "hyperprime_factor_exists";
+
+(* behaves as expected! *)
+Goal "( *sNat* insert x A) = insert (hypnat_of_nat x) ( *sNat* A)";
+by (auto_tac (claset(),simpset() addsimps [starsetNat_def,
+ hypnat_of_nat_def]));
+by (ALLGOALS(res_inst_tac [("z","xa")] eq_Abs_hypnat));
+by Auto_tac;
+by (TRYALL(dtac bspec));
+by Auto_tac;
+by (ALLGOALS(Ultra_tac));
+qed "NatStar_insert";
+
+(* Goldblatt Exercise 3.10(1) - p. 29 *)
+Goal "finite A ==> *sNat* A = hypnat_of_nat ` A";
+by (res_inst_tac [("P","%x. *sNat* x = hypnat_of_nat ` x")] finite_induct 1);
+by (auto_tac (claset(),simpset() addsimps [NatStar_insert]));
+qed "NatStar_hypnat_of_nat";
+
+(* proved elsewhere? *)
+Goal "{x} ~: FreeUltrafilterNat";
+by (auto_tac (claset() addSIs [FreeUltrafilterNat_finite],simpset()));
+qed "FreeUltrafilterNat_singleton_not_mem";
+Addsimps [FreeUltrafilterNat_singleton_not_mem];
+
+(*-------------------------------------------------------------------------------*)
+(* Another characterization of infinite set of natural numbers *)
+(*-------------------------------------------------------------------------------*)
+
+Goal "finite N ==> EX n. (ALL i:N. i<(n::nat))";
+by (eres_inst_tac [("F","N")] finite_induct 1);
+by Auto_tac;
+by (res_inst_tac [("x","Suc n + x")] exI 1);
+by Auto_tac;
+qed "finite_nat_set_bounded";
+
+Goal "finite N = (EX n. (ALL i:N. i<(n::nat)))";
+by (blast_tac (claset() addIs [finite_nat_set_bounded,
+ bounded_nat_set_is_finite]) 1);
+qed "finite_nat_set_bounded_iff";
+
+Goal "(~ finite N) = (ALL n. EX i:N. n <= (i::nat))";
+by (auto_tac (claset(),simpset() addsimps [finite_nat_set_bounded_iff,
+ le_def]));
+qed "not_finite_nat_set_iff";
+
+Goal "(ALL i:N. i<=(n::nat)) ==> finite N";
+by (rtac finite_subset 1);
+ by (rtac finite_atMost 2);
+by Auto_tac;
+qed "bounded_nat_set_is_finite2";
+
+Goal "finite N ==> EX n. (ALL i:N. i<=(n::nat))";
+by (eres_inst_tac [("F","N")] finite_induct 1);
+by Auto_tac;
+by (res_inst_tac [("x","n + x")] exI 1);
+by Auto_tac;
+qed "finite_nat_set_bounded2";
+
+Goal "finite N = (EX n. (ALL i:N. i<=(n::nat)))";
+by (blast_tac (claset() addIs [finite_nat_set_bounded2,
+ bounded_nat_set_is_finite2]) 1);
+qed "finite_nat_set_bounded_iff2";
+
+Goal "(~ finite N) = (ALL n. EX i:N. n < (i::nat))";
+by (auto_tac (claset(),simpset() addsimps [finite_nat_set_bounded_iff2,
+ le_def]));
+qed "not_finite_nat_set_iff2";
+
+(*-------------------------------------------------------------------------------*)
+(* An injective function cannot define an embedded natural number *)
+(*-------------------------------------------------------------------------------*)
+
+Goal "ALL m n. m ~= n --> f n ~= f m \
+\ ==> {n. f n = N} = {} | (EX m. {n. f n = N} = {m})";
+by Auto_tac;
+by (dres_inst_tac [("x","x")] spec 1 THEN Auto_tac);
+by (subgoal_tac "ALL n. (f n = f x) = (x = n)" 1);
+by Auto_tac;
+qed "lemma_infinite_set_singleton";
+
+Goalw [SHNat_def,hypnat_of_nat_def]
+ "inj f ==> Abs_hypnat(hypnatrel `` {f}) ~: Nats";
+by Auto_tac;
+by (subgoal_tac "ALL m n. m ~= n --> f n ~= f m" 1);
+by Auto_tac;
+by (dtac injD 2);
+by (assume_tac 2 THEN Force_tac 2);
+by (dres_inst_tac[("N","N")] lemma_infinite_set_singleton 1);
+by Auto_tac;
+qed "inj_fun_not_hypnat_in_SHNat";
+
+Goalw [starsetNat_def]
+ "range f <= A ==> Abs_hypnat(hypnatrel `` {f}) : *sNat* A";
+by Auto_tac;
+by (Ultra_tac 1);
+by (dres_inst_tac [("c","f x")] subsetD 1);
+by (rtac rangeI 1);
+by Auto_tac;
+qed "range_subset_mem_starsetNat";
+
+(*--------------------------------------------------------------------------------*)
+(* Gleason Proposition 11-5.5. pg 149, pg 155 (ex. 3) and pg. 360 *)
+(* Let E be a nonvoid ordered set with no maximal elements (note: effectively an *)
+(* infinite set if we take E = N (Nats)). Then there exists an order-preserving *)
+(* injection from N to E. Of course, (as some doofus will undoubtedly point out! *)
+(* :-)) can use notion of least element in proof (i.e. no need for choice) if *)
+(* dealing with nats as we have well-ordering property *)
+(*--------------------------------------------------------------------------------*)
+
+Goal "E ~= {} ==> EX x. EX X : (Pow E - {{}}). x: X";
+by Auto_tac;
+val lemmaPow3 = result();
+
+Goalw [choicefun_def] "E ~= {} ==> choicefun E : E";
+by (rtac (lemmaPow3 RS someI2_ex) 1);
+by Auto_tac;
+qed "choicefun_mem_set";
+Addsimps [choicefun_mem_set];
+
+Goal "[| E ~={}; ALL x. EX y: E. x < y |] ==> injf_max n E : E";
+by (induct_tac "n" 1);
+by (Force_tac 1);
+by (simp_tac (simpset() addsimps [choicefun_def]) 1);
+by (rtac (lemmaPow3 RS someI2_ex) 1);
+by Auto_tac;
+qed "injf_max_mem_set";
+
+Goal "ALL x. EX y: E. x < y ==> injf_max n E < injf_max (Suc n) E";
+by (simp_tac (simpset() addsimps [choicefun_def]) 1);
+by (rtac (lemmaPow3 RS someI2_ex) 1);
+by Auto_tac;
+qed "injf_max_order_preserving";
+
+Goal "ALL x. EX y: E. x < y \
+\ ==> ALL n m. m < n --> injf_max m E < injf_max n E";
+by (rtac allI 1);
+by (induct_tac "n" 1);
+by Auto_tac;
+by (simp_tac (simpset() addsimps [choicefun_def]) 1);
+by (rtac (lemmaPow3 RS someI2_ex) 1);
+by Auto_tac;
+by (dtac (CLAIM "m < Suc n ==> m = n | m < n") 1);
+by Auto_tac;
+by (dres_inst_tac [("x","m")] spec 1);
+by Auto_tac;
+by (dtac subsetD 1 THEN Auto_tac);
+by (dres_inst_tac [("x","injf_max m E")] order_less_trans 1);
+by Auto_tac;
+qed "injf_max_order_preserving2";
+
+Goal "ALL x. EX y: E. x < y ==> inj (%n. injf_max n E)";
+by (rtac injI 1);
+by (rtac ccontr 1);
+by Auto_tac;
+by (dtac injf_max_order_preserving2 1);
+by (cut_inst_tac [("m","x"),("n","y")] less_linear 1);
+by Auto_tac;
+by (auto_tac (claset() addSDs [spec],simpset()));
+qed "inj_injf_max";
+
+Goal "[| (E::('a::{order} set)) ~= {}; ALL x. EX y: E. x < y |] \
+\ ==> EX f. range f <= E & inj (f::nat => 'a) & (ALL m. f m < f(Suc m))";
+by (res_inst_tac [("x","%n. injf_max n E")] exI 1);
+by (Step_tac 1);
+by (rtac injf_max_mem_set 1);
+by (rtac inj_injf_max 3);
+by (rtac injf_max_order_preserving 4);
+by Auto_tac;
+qed "infinite_set_has_order_preserving_inj";
+
+(*-------------------------------------------------------------------------------*)
+(* Only need fact that we can have an injective function from N to A for proof *)
+(*-------------------------------------------------------------------------------*)
+
+Goal "~ finite A ==> hypnat_of_nat ` A < ( *sNat* A)";
+by Auto_tac;
+by (rtac subsetD 1);
+by (rtac NatStar_hypreal_of_real_image_subset 1);
+by Auto_tac;
+by (subgoal_tac "A ~= {}" 1);
+by (Force_tac 2);
+by (dtac infinite_set_has_order_preserving_inj 1);
+by (etac (not_finite_nat_set_iff2 RS iffD1) 1);
+by Auto_tac;
+by (dtac inj_fun_not_hypnat_in_SHNat 1);
+by (dtac range_subset_mem_starsetNat 1);
+by (auto_tac (claset(),simpset() addsimps [SHNat_def]));
+qed "hypnat_infinite_has_nonstandard";
+
+Goal "*sNat* A = hypnat_of_nat ` A ==> finite A";
+by (rtac ccontr 1);
+by (auto_tac (claset() addDs [hypnat_infinite_has_nonstandard],
+ simpset()));
+qed "starsetNat_eq_hypnat_of_nat_image_finite";
+
+Goal "( *sNat* A = hypnat_of_nat ` A) = (finite A)";
+by (blast_tac (claset() addSIs [starsetNat_eq_hypnat_of_nat_image_finite,
+ NatStar_hypnat_of_nat]) 1);
+qed "finite_starsetNat_iff";
+
+Goal "(~ finite A) = (hypnat_of_nat ` A < *sNat* A)";
+by (rtac iffI 1);
+by (blast_tac (claset() addSIs [hypnat_infinite_has_nonstandard]) 1);
+by (auto_tac (claset(),simpset() addsimps [finite_starsetNat_iff RS sym]));
+qed "hypnat_infinite_has_nonstandard_iff";
+
+
+(*-------------------------------------------------------------------------------*)
+(* Existence of infinitely many primes: a nonstandard proof *)
+(*-------------------------------------------------------------------------------*)
+
+Goal "~ (ALL n:- {0}. hypnat_of_nat n hdvd 1)";
+by Auto_tac;
+by (res_inst_tac [("x","2")] bexI 1);
+by (auto_tac (claset(),simpset() addsimps [hypnat_of_nat_def,
+ hypnat_one_def,hdvd,dvd_def]));
+val lemma_not_dvd_hypnat_one = result();
+Addsimps [lemma_not_dvd_hypnat_one];
+
+Goal "EX n:- {0}. ~ hypnat_of_nat n hdvd 1";
+by (cut_facts_tac [lemma_not_dvd_hypnat_one] 1);
+by (auto_tac (claset(),simpset() delsimps [lemma_not_dvd_hypnat_one]));
+val lemma_not_dvd_hypnat_one2 = result();
+Addsimps [lemma_not_dvd_hypnat_one2];
+
+(* not needed here *)
+Goalw [hypnat_zero_def,hypnat_one_def]
+ "[| 0 < (N::hypnat); N ~= 1 |] ==> 1 < N";
+by (res_inst_tac [("z","N")] eq_Abs_hypnat 1);
+by (auto_tac (claset(),simpset() addsimps [hypnat_less]));
+by (Ultra_tac 1);
+qed "hypnat_gt_zero_gt_one";
+
+Goalw [hypnat_zero_def,hypnat_one_def]
+ "0 < N ==> 1 < (N::hypnat) + 1";
+by (res_inst_tac [("z","N")] eq_Abs_hypnat 1);
+by (auto_tac (claset(),simpset() addsimps [hypnat_less,hypnat_add]));
+qed "hypnat_add_one_gt_one";
+
+Goal "0 ~: prime";
+by (Step_tac 1);
+by (dtac (thm "prime_g_zero") 1);
+by Auto_tac;
+qed "zero_not_prime";
+Addsimps [zero_not_prime];
+
+Goalw [starprime_def,starsetNat_def,hypnat_of_nat_def]
+ "hypnat_of_nat 0 ~: starprime";
+by (auto_tac (claset() addSIs [bexI],simpset()));
+qed "hypnat_of_nat_zero_not_prime";
+Addsimps [hypnat_of_nat_zero_not_prime];
+
+Goalw [starprime_def,starsetNat_def,hypnat_zero_def]
+ "0 ~: starprime";
+by (auto_tac (claset() addSIs [bexI],simpset()));
+qed "hypnat_zero_not_prime";
+Addsimps [hypnat_zero_not_prime];
+
+Goal "1 ~: prime";
+by (Step_tac 1);
+by (dtac (thm "prime_g_one") 1);
+by Auto_tac;
+qed "one_not_prime";
+Addsimps [one_not_prime];
+
+Goal "Suc 0 ~: prime";
+by (Step_tac 1);
+by (dtac (thm "prime_g_one") 1);
+by Auto_tac;
+qed "one_not_prime2";
+Addsimps [one_not_prime2];
+
+Goalw [starprime_def,starsetNat_def,hypnat_of_nat_def]
+ "hypnat_of_nat 1 ~: starprime";
+by (auto_tac (claset() addSIs [bexI],simpset()));
+qed "hypnat_of_nat_one_not_prime";
+Addsimps [hypnat_of_nat_one_not_prime];
+
+Goalw [starprime_def,starsetNat_def,hypnat_one_def]
+ "1 ~: starprime";
+by (auto_tac (claset() addSIs [bexI],simpset()));
+qed "hypnat_one_not_prime";
+Addsimps [hypnat_one_not_prime];
+
+Goal "[| k hdvd m; k hdvd n |] ==> k hdvd (m - n)";
+by (res_inst_tac [("z","k")] eq_Abs_hypnat 1);
+by (res_inst_tac [("z","m")] eq_Abs_hypnat 1);
+by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
+by (auto_tac (claset(),simpset() addsimps [hdvd,hypnat_minus]));
+by (Ultra_tac 1);
+by (blast_tac (claset() addIs [dvd_diff]) 1);
+qed "hdvd_diff";
+
+Goalw [dvd_def] "x dvd (1::nat) ==> x = 1";
+by Auto_tac;
+qed "dvd_one_eq_one";
+
+Goalw [hypnat_one_def] "x hdvd 1 ==> x = 1";
+by (res_inst_tac [("z","x")] eq_Abs_hypnat 1);
+by (auto_tac (claset(),simpset() addsimps [hdvd]));
+qed "hdvd_one_eq_one";
+
+Goal "~ finite prime";
+by (rtac (hypnat_infinite_has_nonstandard_iff RS iffD2) 1);
+by (cut_facts_tac [hypnat_dvd_all_hypnat_of_nat] 1);
+by (etac exE 1);
+by (etac conjE 1);
+by (subgoal_tac "1 < N + 1" 1);
+by (blast_tac (claset() addIs [hypnat_add_one_gt_one]) 2);
+by (dtac hyperprime_factor_exists 1);
+by (auto_tac (claset() addIs [NatStar_mem],simpset()));
+by (subgoal_tac "k ~: hypnat_of_nat ` prime" 1);
+by (force_tac (claset(),simpset() addsimps [starprime_def]) 1);
+by (Step_tac 1);
+by (dres_inst_tac [("x","x")] bspec 1);
+by (rtac ccontr 1 THEN Asm_full_simp_tac 1);
+by (dtac hdvd_diff 1 THEN assume_tac 1);
+by (auto_tac (claset() addDs [hdvd_one_eq_one],simpset()));
+qed "not_finite_prime";
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/ex/NSPrimes.thy Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,36 @@
+(* Title : NSPrimes.thy
+ Author : Jacques D. Fleuriot
+ Copyright : 2002 University of Edinburgh
+ Description : The nonstandard primes as an extension of
+ the prime numbers
+
+These can be used to derive an alternative proof of the infinitude of primes by
+considering a property of nonstandard sets.
+*)
+
+NSPrimes = Factorization + HSeries +
+
+consts
+ hdvd :: [hypnat, hypnat] => bool (infixl 50)
+
+defs
+ hdvd_def "(M::hypnat) hdvd N ==
+ EX X Y. X: Rep_hypnat(M) & Y: Rep_hypnat(N) &
+ {n::nat. X n dvd Y n} : FreeUltrafilterNat"
+
+constdefs
+ starprime :: hypnat set
+ "starprime == ( *sNat* prime)"
+
+constdefs
+ choicefun :: 'a set => 'a
+ "choicefun E == (@x. EX X: Pow(E) -{{}}. x : X)"
+
+consts injf_max :: "nat => ('a::{order} set) => 'a"
+primrec
+ injf_max_zero "injf_max 0 E = choicefun E"
+ injf_max_Suc "injf_max (Suc n) E = choicefun({e. e : E & injf_max n E < e})"
+
+end
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/ex/ROOT.ML Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,12 @@
+(* Title: HOL/Hyperreal/ex/ROOT.ML
+ ID: $Id$
+
+Miscellaneous HOL-Hyperreal Examples.
+*)
+
+no_document (with_path "../../NumberTheory" use_thy) "Factorization";
+(*These two examples just need the theory Primes.*)
+use_thy "Sqrt";
+use_thy "Sqrt_Script";
+(*This example also needs the theory Factorization.*)
+use_thy "NSPrimes";
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/ex/Sqrt.thy Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,147 @@
+(* Title: HOL/Hyperreal/ex/Sqrt.thy
+ ID: $Id$
+ Author: Markus Wenzel, TU Muenchen
+ License: GPL (GNU GENERAL PUBLIC LICENSE)
+*)
+
+header {* Square roots of primes are irrational *}
+
+theory Sqrt = Primes + Hyperreal:
+
+subsection {* Preliminaries *}
+
+text {*
+ The set of rational numbers, including the key representation
+ theorem.
+*}
+
+constdefs
+ rationals :: "real set" ("\<rat>")
+ "\<rat> \<equiv> {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}"
+
+theorem rationals_rep: "x \<in> \<rat> \<Longrightarrow>
+ \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real m / real n \<and> gcd (m, n) = 1"
+proof -
+ assume "x \<in> \<rat>"
+ then obtain m n :: nat where
+ n: "n \<noteq> 0" and x_rat: "\<bar>x\<bar> = real m / real n"
+ by (unfold rationals_def) blast
+ let ?gcd = "gcd (m, n)"
+ from n have gcd: "?gcd \<noteq> 0" by (simp add: gcd_zero)
+ let ?k = "m div ?gcd"
+ let ?l = "n div ?gcd"
+ let ?gcd' = "gcd (?k, ?l)"
+ have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"
+ by (rule dvd_mult_div_cancel)
+ have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"
+ by (rule dvd_mult_div_cancel)
+
+ from n and gcd_l have "?l \<noteq> 0"
+ by (auto iff del: neq0_conv)
+ moreover
+ have "\<bar>x\<bar> = real ?k / real ?l"
+ proof -
+ from gcd have "real ?k / real ?l =
+ real (?gcd * ?k) / real (?gcd * ?l)"
+ by (simp add: real_mult_div_cancel1)
+ also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
+ also from x_rat have "\<dots> = \<bar>x\<bar>" ..
+ finally show ?thesis ..
+ qed
+ moreover
+ have "?gcd' = 1"
+ proof -
+ have "?gcd * ?gcd' = gcd (?gcd * ?k, ?gcd * ?l)"
+ by (rule gcd_mult_distrib2)
+ with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
+ with gcd show ?thesis by simp
+ qed
+ ultimately show ?thesis by blast
+qed
+
+lemma [elim?]: "r \<in> \<rat> \<Longrightarrow>
+ (\<And>m n. n \<noteq> 0 \<Longrightarrow> \<bar>r\<bar> = real m / real n \<Longrightarrow> gcd (m, n) = 1 \<Longrightarrow> C)
+ \<Longrightarrow> C"
+ using rationals_rep by blast
+
+
+subsection {* Main theorem *}
+
+text {*
+ The square root of any prime number (including @{text 2}) is
+ irrational.
+*}
+
+theorem sqrt_prime_irrational: "p \<in> prime \<Longrightarrow> sqrt (real p) \<notin> \<rat>"
+proof
+ assume p_prime: "p \<in> prime"
+ then have p: "1 < p" by (simp add: prime_def)
+ assume "sqrt (real p) \<in> \<rat>"
+ then obtain m n where
+ n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n"
+ and gcd: "gcd (m, n) = 1" ..
+ have eq: "m\<twosuperior> = p * n\<twosuperior>"
+ proof -
+ from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
+ then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)"
+ by (auto simp add: power_two real_power_two)
+ also have "(sqrt (real p))\<twosuperior> = real p" by simp
+ also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp
+ finally show ?thesis ..
+ qed
+ have "p dvd m \<and> p dvd n"
+ proof
+ from eq have "p dvd m\<twosuperior>" ..
+ with p_prime show "p dvd m" by (rule prime_dvd_power_two)
+ then obtain k where "m = p * k" ..
+ with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power_two mult_ac)
+ with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power_two)
+ then have "p dvd n\<twosuperior>" ..
+ with p_prime show "p dvd n" by (rule prime_dvd_power_two)
+ qed
+ then have "p dvd gcd (m, n)" ..
+ with gcd have "p dvd 1" by simp
+ then have "p \<le> 1" by (simp add: dvd_imp_le)
+ with p show False by simp
+qed
+
+corollary "sqrt (real (2::nat)) \<notin> \<rat>"
+ by (rule sqrt_prime_irrational) (rule two_is_prime)
+
+
+subsection {* Variations *}
+
+text {*
+ Here is an alternative version of the main proof, using mostly
+ linear forward-reasoning. While this results in less top-down
+ structure, it is probably closer to proofs seen in mathematics.
+*}
+
+theorem "p \<in> prime \<Longrightarrow> sqrt (real p) \<notin> \<rat>"
+proof
+ assume p_prime: "p \<in> prime"
+ then have p: "1 < p" by (simp add: prime_def)
+ assume "sqrt (real p) \<in> \<rat>"
+ then obtain m n where
+ n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n"
+ and gcd: "gcd (m, n) = 1" ..
+ from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
+ then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)"
+ by (auto simp add: power_two real_power_two)
+ also have "(sqrt (real p))\<twosuperior> = real p" by simp
+ also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp
+ finally have eq: "m\<twosuperior> = p * n\<twosuperior>" ..
+ then have "p dvd m\<twosuperior>" ..
+ with p_prime have dvd_m: "p dvd m" by (rule prime_dvd_power_two)
+ then obtain k where "m = p * k" ..
+ with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power_two mult_ac)
+ with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power_two)
+ then have "p dvd n\<twosuperior>" ..
+ with p_prime have "p dvd n" by (rule prime_dvd_power_two)
+ with dvd_m have "p dvd gcd (m, n)" by (rule gcd_greatest)
+ with gcd have "p dvd 1" by simp
+ then have "p \<le> 1" by (simp add: dvd_imp_le)
+ with p show False by simp
+qed
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/ex/Sqrt_Script.thy Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,78 @@
+(* Title: HOL/Hyperreal/ex/Sqrt_Script.thy
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 2001 University of Cambridge
+*)
+
+header {* Square roots of primes are irrational (script version) *}
+
+theory Sqrt_Script = Primes + Hyperreal:
+
+text {*
+ \medskip Contrast this linear Isabelle/Isar script with Markus
+ Wenzel's more mathematical version.
+*}
+
+subsection {* Preliminaries *}
+
+lemma prime_nonzero: "p \<in> prime \<Longrightarrow> p \<noteq> 0"
+ by (force simp add: prime_def)
+
+lemma prime_dvd_other_side:
+ "n * n = p * (k * k) \<Longrightarrow> p \<in> prime \<Longrightarrow> p dvd n"
+ apply (subgoal_tac "p dvd n * n", blast dest: prime_dvd_mult)
+ apply (rule_tac j = "k * k" in dvd_mult_left, simp)
+ done
+
+lemma reduction: "p \<in> prime \<Longrightarrow>
+ 0 < k \<Longrightarrow> k * k = p * (j * j) \<Longrightarrow> k < p * j \<and> 0 < j"
+ apply (rule ccontr)
+ apply (simp add: linorder_not_less)
+ apply (erule disjE)
+ apply (frule mult_le_mono, assumption)
+ apply auto
+ apply (force simp add: prime_def)
+ done
+
+lemma rearrange: "(j::nat) * (p * j) = k * k \<Longrightarrow> k * k = p * (j * j)"
+ by (simp add: mult_ac)
+
+lemma prime_not_square:
+ "p \<in> prime \<Longrightarrow> (\<And>k. 0 < k \<Longrightarrow> m * m \<noteq> p * (k * k))"
+ apply (induct m rule: nat_less_induct)
+ apply clarify
+ apply (frule prime_dvd_other_side, assumption)
+ apply (erule dvdE)
+ apply (simp add: nat_mult_eq_cancel_disj prime_nonzero)
+ apply (blast dest: rearrange reduction)
+ done
+
+
+subsection {* The set of rational numbers *}
+
+constdefs
+ rationals :: "real set" ("\<rat>")
+ "\<rat> \<equiv> {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}"
+
+
+subsection {* Main theorem *}
+
+text {*
+ The square root of any prime number (including @{text 2}) is
+ irrational.
+*}
+
+theorem prime_sqrt_irrational:
+ "p \<in> prime \<Longrightarrow> x * x = real p \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<notin> \<rat>"
+ apply (simp add: rationals_def real_abs_def)
+ apply clarify
+ apply (erule_tac P = "real m / real n * ?x = ?y" in rev_mp)
+ apply (simp del: real_of_nat_mult
+ add: real_divide_eq_eq prime_not_square
+ real_of_nat_mult [symmetric])
+ done
+
+lemmas two_sqrt_irrational =
+ prime_sqrt_irrational [OF two_is_prime]
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/ex/document/root.tex Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,20 @@
+
+\documentclass[11pt,a4paper]{article}
+\usepackage[latin1]{inputenc}
+\usepackage{isabelle,isabellesym}
+\usepackage{pdfsetup}
+
+\urlstyle{rm}
+\isabellestyle{it}
+
+\begin{document}
+
+\title{Miscellaneous HOL-Hyperreal Examples}
+\maketitle
+
+\tableofcontents
+
+\parindent 0pt\parskip 0.5ex
+\input{session}
+
+\end{document}