(* Title : HOL/Real/Hyperreal/Hyper.ML ID : $Id: HyperDef.ML,v 1.4 1999/10/11 08:52:54 paulson Exp $ Author : Jacques D. Fleuriot Copyright : 1998 University of Cambridge Description : Ultrapower construction of hyperreals *) (*------------------------------------------------------------------------ Proof that the set of naturals is not finite ------------------------------------------------------------------------*) (*** based on James' proof that the set of naturals is not finite ***) Goal "finite (A::nat set) --> (? n. !m. Suc (n + m) ~: A)"; by (rtac impI 1); by (eres_inst_tac [("F","A")] finite_induct 1); by (Blast_tac 1 THEN etac exE 1); by (res_inst_tac [("x","n + x")] exI 1); by (rtac allI 1 THEN eres_inst_tac [("x","x + m")] allE 1); by (auto_tac (claset(), simpset() addsimps add_ac)); by (auto_tac (claset(), simpset() addsimps [add_assoc RS sym, less_add_Suc2 RS less_not_refl2])); qed_spec_mp "finite_exhausts"; Goal "finite (A :: nat set) --> (? n. n ~:A)"; by (rtac impI 1 THEN dtac finite_exhausts 1); by (Blast_tac 1); qed_spec_mp "finite_not_covers"; Goal "~ finite(UNIV:: nat set)"; by (fast_tac (claset() addSDs [finite_exhausts]) 1); qed "not_finite_nat"; (*------------------------------------------------------------------------ Existence of free ultrafilter over the naturals and proof of various properties of the FreeUltrafilterNat- an arbitrary free ultrafilter ------------------------------------------------------------------------*) Goal "EX U. U: FreeUltrafilter (UNIV::nat set)"; by (rtac (not_finite_nat RS FreeUltrafilter_Ex) 1); qed "FreeUltrafilterNat_Ex"; Goalw [FreeUltrafilterNat_def] "FreeUltrafilterNat: FreeUltrafilter(UNIV:: nat set)"; by (rtac (FreeUltrafilterNat_Ex RS exE) 1); by (rtac selectI2 1 THEN ALLGOALS(assume_tac)); qed "FreeUltrafilterNat_mem"; Addsimps [FreeUltrafilterNat_mem]; Goalw [FreeUltrafilterNat_def] "finite x ==> x ~: FreeUltrafilterNat"; by (rtac (FreeUltrafilterNat_Ex RS exE) 1); by (rtac selectI2 1 THEN assume_tac 1); by (blast_tac (claset() addDs [mem_FreeUltrafiltersetD1]) 1); qed "FreeUltrafilterNat_finite"; Goal "x: FreeUltrafilterNat ==> ~ finite x"; by (blast_tac (claset() addDs [FreeUltrafilterNat_finite]) 1); qed "FreeUltrafilterNat_not_finite"; Goalw [FreeUltrafilterNat_def] "{} ~: FreeUltrafilterNat"; by (rtac (FreeUltrafilterNat_Ex RS exE) 1); by (rtac selectI2 1 THEN assume_tac 1); by (blast_tac (claset() addDs [FreeUltrafilter_Ultrafilter, Ultrafilter_Filter,Filter_empty_not_mem]) 1); qed "FreeUltrafilterNat_empty"; Addsimps [FreeUltrafilterNat_empty]; Goal "[| X: FreeUltrafilterNat; Y: FreeUltrafilterNat |] \ \ ==> X Int Y : FreeUltrafilterNat"; by (cut_facts_tac [FreeUltrafilterNat_mem] 1); by (blast_tac (claset() addDs [FreeUltrafilter_Ultrafilter, Ultrafilter_Filter,mem_FiltersetD1]) 1); qed "FreeUltrafilterNat_Int"; Goal "[| X: FreeUltrafilterNat; X <= Y |] \ \ ==> Y : FreeUltrafilterNat"; by (cut_facts_tac [FreeUltrafilterNat_mem] 1); by (blast_tac (claset() addDs [FreeUltrafilter_Ultrafilter, Ultrafilter_Filter,mem_FiltersetD2]) 1); qed "FreeUltrafilterNat_subset"; Goal "X: FreeUltrafilterNat ==> -X ~: FreeUltrafilterNat"; by (Step_tac 1); by (dtac FreeUltrafilterNat_Int 1 THEN assume_tac 1); by Auto_tac; qed "FreeUltrafilterNat_Compl"; Goal "X~: FreeUltrafilterNat ==> -X : FreeUltrafilterNat"; by (cut_facts_tac [FreeUltrafilterNat_mem RS (FreeUltrafilter_iff RS iffD1)] 1); by (Step_tac 1 THEN dres_inst_tac [("x","X")] bspec 1); by (auto_tac (claset(),simpset() addsimps [UNIV_diff_Compl])); qed "FreeUltrafilterNat_Compl_mem"; Goal "(X ~: FreeUltrafilterNat) = (-X: FreeUltrafilterNat)"; by (blast_tac (claset() addDs [FreeUltrafilterNat_Compl, FreeUltrafilterNat_Compl_mem]) 1); qed "FreeUltrafilterNat_Compl_iff1"; Goal "(X: FreeUltrafilterNat) = (-X ~: FreeUltrafilterNat)"; by (auto_tac (claset(), simpset() addsimps [FreeUltrafilterNat_Compl_iff1 RS sym])); qed "FreeUltrafilterNat_Compl_iff2"; Goal "(UNIV::nat set) : FreeUltrafilterNat"; by (rtac (FreeUltrafilterNat_mem RS FreeUltrafilter_Ultrafilter RS Ultrafilter_Filter RS mem_FiltersetD4) 1); qed "FreeUltrafilterNat_UNIV"; Addsimps [FreeUltrafilterNat_UNIV]; Goal "{n::nat. True}: FreeUltrafilterNat"; by (subgoal_tac "{n::nat. True} = (UNIV::nat set)" 1); by Auto_tac; qed "FreeUltrafilterNat_Nat_set"; Addsimps [FreeUltrafilterNat_Nat_set]; Goal "{n. P(n) = P(n)} : FreeUltrafilterNat"; by (Simp_tac 1); qed "FreeUltrafilterNat_Nat_set_refl"; AddIs [FreeUltrafilterNat_Nat_set_refl]; Goal "{n::nat. P} : FreeUltrafilterNat ==> P"; by (rtac ccontr 1); by (rotate_tac 1 1); by (Asm_full_simp_tac 1); qed "FreeUltrafilterNat_P"; Goal "{n. P(n)} : FreeUltrafilterNat ==> EX n. P(n)"; by (rtac ccontr 1 THEN rotate_tac 1 1); by (Asm_full_simp_tac 1); qed "FreeUltrafilterNat_Ex_P"; Goal "ALL n. P(n) ==> {n. P(n)} : FreeUltrafilterNat"; by (auto_tac (claset() addIs [FreeUltrafilterNat_Nat_set],simpset())); qed "FreeUltrafilterNat_all"; (*----------------------------------------- Define and use Ultrafilter tactics -----------------------------------------*) use "fuf.ML"; (*------------------------------------------------------ Now prove one further property of our free ultrafilter -------------------------------------------------------*) Goal "X Un Y: FreeUltrafilterNat \ \ ==> X: FreeUltrafilterNat | Y: FreeUltrafilterNat"; by Auto_tac; by (Ultra_tac 1); qed "FreeUltrafilterNat_Un"; (*------------------------------------------------------------------------ Properties of hyprel ------------------------------------------------------------------------*) (** Proving that hyprel is an equivalence relation **) (** Natural deduction for hyprel **) Goalw [hyprel_def] "((X,Y): hyprel) = ({n. X n = Y n}: FreeUltrafilterNat)"; by (Fast_tac 1); qed "hyprel_iff"; Goalw [hyprel_def] "{n. X n = Y n}: FreeUltrafilterNat ==> (X,Y): hyprel"; by (Fast_tac 1); qed "hyprelI"; Goalw [hyprel_def] "p: hyprel --> (EX X Y. \ \ p = (X,Y) & {n. X n = Y n} : FreeUltrafilterNat)"; by (Fast_tac 1); qed "hyprelE_lemma"; val [major,minor] = goal thy "[| p: hyprel; \ \ !!X Y. [| p = (X,Y); {n. X n = Y n}: FreeUltrafilterNat\ \ |] ==> Q |] ==> Q"; by (cut_facts_tac [major RS (hyprelE_lemma RS mp)] 1); by (REPEAT (eresolve_tac [asm_rl,exE,conjE,minor] 1)); qed "hyprelE"; AddSIs [hyprelI]; AddSEs [hyprelE]; Goalw [hyprel_def] "(x,x): hyprel"; by (auto_tac (claset(),simpset() addsimps [FreeUltrafilterNat_Nat_set])); qed "hyprel_refl"; Goal "{n. X n = Y n} = {n. Y n = X n}"; by Auto_tac; qed "lemma_perm"; Goalw [hyprel_def] "(x,y): hyprel --> (y,x):hyprel"; by (auto_tac (claset() addIs [lemma_perm RS subst],simpset())); qed_spec_mp "hyprel_sym"; Goalw [hyprel_def] "(x,y): hyprel --> (y,z):hyprel --> (x,z):hyprel"; by Auto_tac; by (Ultra_tac 1); qed_spec_mp "hyprel_trans"; Goalw [equiv_def, refl_def, sym_def, trans_def] "equiv {x::nat=>real. True} hyprel"; by (auto_tac (claset() addSIs [hyprel_refl] addSEs [hyprel_sym,hyprel_trans] delrules [hyprelI,hyprelE], simpset() addsimps [FreeUltrafilterNat_Nat_set])); qed "equiv_hyprel"; val equiv_hyprel_iff = [TrueI, TrueI] MRS ([CollectI, CollectI] MRS (equiv_hyprel RS eq_equiv_class_iff)); Goalw [hypreal_def,hyprel_def,quotient_def] "hyprel^^{x}:hypreal"; by (Blast_tac 1); qed "hyprel_in_hypreal"; Goal "inj_on Abs_hypreal hypreal"; by (rtac inj_on_inverseI 1); by (etac Abs_hypreal_inverse 1); qed "inj_on_Abs_hypreal"; Addsimps [equiv_hyprel_iff,inj_on_Abs_hypreal RS inj_on_iff, hyprel_iff, hyprel_in_hypreal, Abs_hypreal_inverse]; Addsimps [equiv_hyprel RS eq_equiv_class_iff]; val eq_hyprelD = equiv_hyprel RSN (2,eq_equiv_class); Goal "inj(Rep_hypreal)"; by (rtac inj_inverseI 1); by (rtac Rep_hypreal_inverse 1); qed "inj_Rep_hypreal"; Goalw [hyprel_def] "x: hyprel ^^ {x}"; by (Step_tac 1); by (auto_tac (claset() addSIs [FreeUltrafilterNat_Nat_set],simpset())); qed "lemma_hyprel_refl"; Addsimps [lemma_hyprel_refl]; Goalw [hypreal_def] "{} ~: hypreal"; by (auto_tac (claset() addSEs [quotientE], simpset())); qed "hypreal_empty_not_mem"; Addsimps [hypreal_empty_not_mem]; Goal "Rep_hypreal x ~= {}"; by (cut_inst_tac [("x","x")] Rep_hypreal 1); by Auto_tac; qed "Rep_hypreal_nonempty"; Addsimps [Rep_hypreal_nonempty]; (*------------------------------------------------------------------------ hypreal_of_real: the injection from real to hypreal ------------------------------------------------------------------------*) Goal "inj(hypreal_of_real)"; by (rtac injI 1); by (rewtac hypreal_of_real_def); by (dtac (inj_on_Abs_hypreal RS inj_onD) 1); by (REPEAT (rtac hyprel_in_hypreal 1)); by (dtac eq_equiv_class 1); by (rtac equiv_hyprel 1); by (Fast_tac 1); by (rtac ccontr 1 THEN rotate_tac 1 1); by Auto_tac; qed "inj_hypreal_of_real"; val [prem] = goal thy "(!!x y. z = Abs_hypreal(hyprel^^{x}) ==> P) ==> P"; by (res_inst_tac [("x1","z")] (rewrite_rule [hypreal_def] Rep_hypreal RS quotientE) 1); by (dres_inst_tac [("f","Abs_hypreal")] arg_cong 1); by (res_inst_tac [("x","x")] prem 1); by (asm_full_simp_tac (simpset() addsimps [Rep_hypreal_inverse]) 1); qed "eq_Abs_hypreal"; (**** hypreal_minus: additive inverse on hypreal ****) Goalw [congruent_def] "congruent hyprel (%X. hyprel^^{%n. - (X n)})"; by Safe_tac; by (ALLGOALS Ultra_tac); qed "hypreal_minus_congruent"; (*Resolve th against the corresponding facts for hypreal_minus*) val hypreal_minus_ize = RSLIST [equiv_hyprel, hypreal_minus_congruent]; Goalw [hypreal_minus_def] "- (Abs_hypreal(hyprel^^{%n. X n})) = Abs_hypreal(hyprel ^^ {%n. -(X n)})"; by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1); by (simp_tac (simpset() addsimps [hyprel_in_hypreal RS Abs_hypreal_inverse,hypreal_minus_ize UN_equiv_class]) 1); qed "hypreal_minus"; Goal "- (- z) = (z::hypreal)"; by (res_inst_tac [("z","z")] eq_Abs_hypreal 1); by (asm_simp_tac (simpset() addsimps [hypreal_minus]) 1); qed "hypreal_minus_minus"; Addsimps [hypreal_minus_minus]; Goal "inj(%r::hypreal. -r)"; by (rtac injI 1); by (dres_inst_tac [("f","uminus")] arg_cong 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_minus_minus]) 1); qed "inj_hypreal_minus"; Goalw [hypreal_zero_def] "-0hr = 0hr"; by (simp_tac (simpset() addsimps [hypreal_minus]) 1); qed "hypreal_minus_zero"; Addsimps [hypreal_minus_zero]; Goal "(-x = 0hr) = (x = 0hr)"; by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); by (auto_tac (claset(),simpset() addsimps [hypreal_zero_def, hypreal_minus] @ real_add_ac)); qed "hypreal_minus_zero_iff"; Addsimps [hypreal_minus_zero_iff]; (**** hrinv: multiplicative inverse on hypreal ****) Goalw [congruent_def] "congruent hyprel (%X. hyprel^^{%n. if X n = 0r then 0r else rinv(X n)})"; by (Auto_tac THEN Ultra_tac 1); qed "hypreal_hrinv_congruent"; (* Resolve th against the corresponding facts for hrinv *) val hypreal_hrinv_ize = RSLIST [equiv_hyprel, hypreal_hrinv_congruent]; Goalw [hrinv_def] "hrinv (Abs_hypreal(hyprel^^{%n. X n})) = \ \ Abs_hypreal(hyprel ^^ {%n. if X n = 0r then 0r else rinv(X n)})"; by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1); by (simp_tac (simpset() addsimps [hyprel_in_hypreal RS Abs_hypreal_inverse,hypreal_hrinv_ize UN_equiv_class]) 1); qed "hypreal_hrinv"; Goal "z ~= 0hr ==> hrinv (hrinv z) = z"; by (res_inst_tac [("z","z")] eq_Abs_hypreal 1); by (rotate_tac 1 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_hrinv,hypreal_zero_def] setloop (split_tac [expand_if])) 1); by (ultra_tac (claset() addDs [rinv_not_zero,real_rinv_rinv],simpset()) 1); qed "hypreal_hrinv_hrinv"; Addsimps [hypreal_hrinv_hrinv]; Goalw [hypreal_one_def] "hrinv(1hr) = 1hr"; by (full_simp_tac (simpset() addsimps [hypreal_hrinv, real_zero_not_eq_one RS not_sym] setloop (split_tac [expand_if])) 1); qed "hypreal_hrinv_1"; Addsimps [hypreal_hrinv_1]; (**** hyperreal addition: hypreal_add ****) Goalw [congruent2_def] "congruent2 hyprel (%X Y. hyprel^^{%n. X n + Y n})"; by Safe_tac; by (ALLGOALS(Ultra_tac)); qed "hypreal_add_congruent2"; (*Resolve th against the corresponding facts for hyppreal_add*) val hypreal_add_ize = RSLIST [equiv_hyprel, hypreal_add_congruent2]; Goalw [hypreal_add_def] "Abs_hypreal(hyprel^^{%n. X n}) + Abs_hypreal(hyprel^^{%n. Y n}) = \ \ Abs_hypreal(hyprel^^{%n. X n + Y n})"; by (asm_simp_tac (simpset() addsimps [hypreal_add_ize UN_equiv_class2]) 1); qed "hypreal_add"; Goal "(z::hypreal) + w = w + z"; by (res_inst_tac [("z","z")] eq_Abs_hypreal 1); by (res_inst_tac [("z","w")] eq_Abs_hypreal 1); by (asm_simp_tac (simpset() addsimps (real_add_ac @ [hypreal_add])) 1); qed "hypreal_add_commute"; Goal "((z1::hypreal) + z2) + z3 = z1 + (z2 + z3)"; by (res_inst_tac [("z","z1")] eq_Abs_hypreal 1); by (res_inst_tac [("z","z2")] eq_Abs_hypreal 1); by (res_inst_tac [("z","z3")] eq_Abs_hypreal 1); by (asm_simp_tac (simpset() addsimps [hypreal_add, real_add_assoc]) 1); qed "hypreal_add_assoc"; (*For AC rewriting*) Goal "(x::hypreal)+(y+z)=y+(x+z)"; by (rtac (hypreal_add_commute RS trans) 1); by (rtac (hypreal_add_assoc RS trans) 1); by (rtac (hypreal_add_commute RS arg_cong) 1); qed "hypreal_add_left_commute"; (* hypreal addition is an AC operator *) val hypreal_add_ac = [hypreal_add_assoc,hypreal_add_commute, hypreal_add_left_commute]; Goalw [hypreal_zero_def] "0hr + z = z"; by (res_inst_tac [("z","z")] eq_Abs_hypreal 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_add]) 1); qed "hypreal_add_zero_left"; Goal "z + 0hr = z"; by (simp_tac (simpset() addsimps [hypreal_add_zero_left,hypreal_add_commute]) 1); qed "hypreal_add_zero_right"; Goalw [hypreal_zero_def] "z + -z = 0hr"; by (res_inst_tac [("z","z")] eq_Abs_hypreal 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_minus, hypreal_add]) 1); qed "hypreal_add_minus"; Goal "-z + z = 0hr"; by (simp_tac (simpset() addsimps [hypreal_add_commute,hypreal_add_minus]) 1); qed "hypreal_add_minus_left"; Addsimps [hypreal_add_minus,hypreal_add_minus_left, hypreal_add_zero_left,hypreal_add_zero_right]; Goal "? y. (x::hypreal) + y = 0hr"; by (fast_tac (claset() addIs [hypreal_add_minus]) 1); qed "hypreal_minus_ex"; Goal "?! y. (x::hypreal) + y = 0hr"; by (auto_tac (claset() addIs [hypreal_add_minus],simpset())); by (dres_inst_tac [("f","%x. ya+x")] arg_cong 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_add_assoc RS sym]) 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_add_commute]) 1); qed "hypreal_minus_ex1"; Goal "?! y. y + (x::hypreal) = 0hr"; by (auto_tac (claset() addIs [hypreal_add_minus_left],simpset())); by (dres_inst_tac [("f","%x. x+ya")] arg_cong 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_add_assoc]) 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_add_commute]) 1); qed "hypreal_minus_left_ex1"; Goal "x + y = 0hr ==> x = -y"; by (cut_inst_tac [("z","y")] hypreal_add_minus_left 1); by (res_inst_tac [("x1","y")] (hypreal_minus_left_ex1 RS ex1E) 1); by (Blast_tac 1); qed "hypreal_add_minus_eq_minus"; Goal "? y::hypreal. x = -y"; by (cut_inst_tac [("x","x")] hypreal_minus_ex 1); by (etac exE 1 THEN dtac hypreal_add_minus_eq_minus 1); by (Fast_tac 1); qed "hypreal_as_add_inverse_ex"; Goal "-(x + (y::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 [hypreal_minus, hypreal_add,real_minus_add_distrib])); qed "hypreal_minus_add_distrib"; Goal "-(y + -(x::hypreal)) = x + -y"; by (simp_tac (simpset() addsimps [hypreal_minus_add_distrib, hypreal_add_commute]) 1); qed "hypreal_minus_distrib1"; Goal "(x + - (y::hypreal)) + (y + - z) = x + -z"; by (res_inst_tac [("w1","y")] (hypreal_add_commute RS subst) 1); by (simp_tac (simpset() addsimps [hypreal_add_left_commute, hypreal_add_assoc]) 1); by (simp_tac (simpset() addsimps [hypreal_add_commute]) 1); qed "hypreal_add_minus_cancel1"; Goal "((x::hypreal) + 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 [hypreal_add_assoc RS sym]) 1); qed "hypreal_add_left_cancel"; Goal "z + (x + (y + -z)) = x + (y::hypreal)"; by (simp_tac (simpset() addsimps hypreal_add_ac) 1); qed "hypreal_add_minus_cancel2"; Addsimps [hypreal_add_minus_cancel2]; Goal "y + -(x + y) = -(x::hypreal)"; by (full_simp_tac (simpset() addsimps [hypreal_minus_add_distrib]) 1); by (rtac (hypreal_add_left_commute RS subst) 1); by (Full_simp_tac 1); qed "hypreal_add_minus_cancel"; Addsimps [hypreal_add_minus_cancel]; Goal "y + -(y + x) = -(x::hypreal)"; by (simp_tac (simpset() addsimps [hypreal_minus_add_distrib, hypreal_add_assoc RS sym]) 1); qed "hypreal_add_minus_cancelc"; Addsimps [hypreal_add_minus_cancelc]; Goal "(z + -x) + (y + -z) = (y + -(x::hypreal))"; by (full_simp_tac (simpset() addsimps [hypreal_minus_add_distrib RS sym, hypreal_add_left_cancel] @ hypreal_add_ac) 1); qed "hypreal_add_minus_cancel3"; Addsimps [hypreal_add_minus_cancel3]; Goal "(y + (x::hypreal)= z + x) = (y = z)"; by (simp_tac (simpset() addsimps [hypreal_add_commute, hypreal_add_left_cancel]) 1); qed "hypreal_add_right_cancel"; Goal "z + (y + -z) = (y::hypreal)"; by (simp_tac (simpset() addsimps hypreal_add_ac) 1); qed "hypreal_add_minus_cancel4"; Addsimps [hypreal_add_minus_cancel4]; Goal "z + (w + (x + (-z + y))) = w + x + (y::hypreal)"; by (simp_tac (simpset() addsimps hypreal_add_ac) 1); qed "hypreal_add_minus_cancel5"; Addsimps [hypreal_add_minus_cancel5]; (**** hyperreal multiplication: hypreal_mult ****) Goalw [congruent2_def] "congruent2 hyprel (%X Y. hyprel^^{%n. X n * Y n})"; by Safe_tac; by (ALLGOALS(Ultra_tac)); qed "hypreal_mult_congruent2"; (*Resolve th against the corresponding facts for hypreal_mult*) val hypreal_mult_ize = RSLIST [equiv_hyprel, hypreal_mult_congruent2]; Goalw [hypreal_mult_def] "Abs_hypreal(hyprel^^{%n. X n}) * Abs_hypreal(hyprel^^{%n. Y n}) = \ \ Abs_hypreal(hyprel^^{%n. X n * Y n})"; by (asm_simp_tac (simpset() addsimps [hypreal_mult_ize UN_equiv_class2]) 1); qed "hypreal_mult"; Goal "(z::hypreal) * w = w * z"; by (res_inst_tac [("z","z")] eq_Abs_hypreal 1); by (res_inst_tac [("z","w")] eq_Abs_hypreal 1); by (asm_simp_tac (simpset() addsimps ([hypreal_mult] @ real_mult_ac)) 1); qed "hypreal_mult_commute"; Goal "((z1::hypreal) * z2) * z3 = z1 * (z2 * z3)"; by (res_inst_tac [("z","z1")] eq_Abs_hypreal 1); by (res_inst_tac [("z","z2")] eq_Abs_hypreal 1); by (res_inst_tac [("z","z3")] eq_Abs_hypreal 1); by (asm_simp_tac (simpset() addsimps [hypreal_mult,real_mult_assoc]) 1); qed "hypreal_mult_assoc"; qed_goal "hypreal_mult_left_commute" thy "(z1::hypreal) * (z2 * z3) = z2 * (z1 * z3)" (fn _ => [rtac (hypreal_mult_commute RS trans) 1, rtac (hypreal_mult_assoc RS trans) 1, rtac (hypreal_mult_commute RS arg_cong) 1]); (* hypreal multiplication is an AC operator *) val hypreal_mult_ac = [hypreal_mult_assoc, hypreal_mult_commute, hypreal_mult_left_commute]; Goalw [hypreal_one_def] "1hr * z = z"; by (res_inst_tac [("z","z")] eq_Abs_hypreal 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_mult]) 1); qed "hypreal_mult_1"; Goal "z * 1hr = z"; by (simp_tac (simpset() addsimps [hypreal_mult_commute, hypreal_mult_1]) 1); qed "hypreal_mult_1_right"; Goalw [hypreal_zero_def] "0hr * z = 0hr"; by (res_inst_tac [("z","z")] eq_Abs_hypreal 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_mult,real_mult_0]) 1); qed "hypreal_mult_0"; Goal "z * 0hr = 0hr"; by (simp_tac (simpset() addsimps [hypreal_mult_commute, hypreal_mult_0]) 1); qed "hypreal_mult_0_right"; Addsimps [hypreal_mult_0,hypreal_mult_0_right]; Goal "-(x * y) = -x * (y::hypreal)"; 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 [hypreal_minus, hypreal_mult,real_minus_mult_eq1] @ real_mult_ac @ real_add_ac)); qed "hypreal_minus_mult_eq1"; Goal "-(x * y) = (x::hypreal) * -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 [hypreal_minus, hypreal_mult,real_minus_mult_eq2] @ real_mult_ac @ real_add_ac)); qed "hypreal_minus_mult_eq2"; Goal "-x*-y = x*(y::hypreal)"; by (full_simp_tac (simpset() addsimps [hypreal_minus_mult_eq2 RS sym, hypreal_minus_mult_eq1 RS sym]) 1); qed "hypreal_minus_mult_cancel"; Addsimps [hypreal_minus_mult_cancel]; Goal "-x*y = (x::hypreal)*-y"; by (full_simp_tac (simpset() addsimps [hypreal_minus_mult_eq2 RS sym, hypreal_minus_mult_eq1 RS sym]) 1); qed "hypreal_minus_mult_commute"; (*----------------------------------------------------------------------------- A few more theorems ----------------------------------------------------------------------------*) Goal "(z::hypreal) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"; by (asm_simp_tac (simpset() addsimps [hypreal_add_assoc RS sym]) 1); qed "hypreal_add_assoc_cong"; Goal "(z::hypreal) + (v + w) = v + (z + w)"; by (REPEAT (ares_tac [hypreal_add_commute RS hypreal_add_assoc_cong] 1)); qed "hypreal_add_assoc_swap"; Goal "((z1::hypreal) + z2) * w = (z1 * w) + (z2 * w)"; by (res_inst_tac [("z","z1")] eq_Abs_hypreal 1); by (res_inst_tac [("z","z2")] eq_Abs_hypreal 1); by (res_inst_tac [("z","w")] eq_Abs_hypreal 1); by (asm_simp_tac (simpset() addsimps [hypreal_mult,hypreal_add, real_add_mult_distrib]) 1); qed "hypreal_add_mult_distrib"; val hypreal_mult_commute'= read_instantiate [("z","w")] hypreal_mult_commute; Goal "(w::hypreal) * (z1 + z2) = (w * z1) + (w * z2)"; by (simp_tac (simpset() addsimps [hypreal_mult_commute',hypreal_add_mult_distrib]) 1); qed "hypreal_add_mult_distrib2"; val hypreal_mult_simps = [hypreal_mult_1, hypreal_mult_1_right]; Addsimps hypreal_mult_simps; (*** one and zero are distinct ***) Goalw [hypreal_zero_def,hypreal_one_def] "0hr ~= 1hr"; by (auto_tac (claset(),simpset() addsimps [real_zero_not_eq_one])); qed "hypreal_zero_not_eq_one"; (*** existence of inverse ***) Goalw [hypreal_one_def,hypreal_zero_def] "x ~= 0hr ==> x*hrinv(x) = 1hr"; by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); by (rotate_tac 1 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_hrinv, hypreal_mult] setloop (split_tac [expand_if])) 1); by (dtac FreeUltrafilterNat_Compl_mem 1); by (blast_tac (claset() addSIs [real_mult_inv_right, FreeUltrafilterNat_subset]) 1); qed "hypreal_mult_hrinv"; Goal "x ~= 0hr ==> hrinv(x)*x = 1hr"; by (asm_simp_tac (simpset() addsimps [hypreal_mult_hrinv, hypreal_mult_commute]) 1); qed "hypreal_mult_hrinv_left"; Goal "x ~= 0hr ==> ? y. (x::hypreal) * y = 1hr"; by (fast_tac (claset() addDs [hypreal_mult_hrinv]) 1); qed "hypreal_hrinv_ex"; Goal "x ~= 0hr ==> ? y. y * (x::hypreal) = 1hr"; by (fast_tac (claset() addDs [hypreal_mult_hrinv_left]) 1); qed "hypreal_hrinv_left_ex"; Goal "x ~= 0hr ==> ?! y. (x::hypreal) * y = 1hr"; by (auto_tac (claset() addIs [hypreal_mult_hrinv],simpset())); by (dres_inst_tac [("f","%x. ya*x")] arg_cong 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_mult_assoc RS sym]) 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_mult_commute]) 1); qed "hypreal_hrinv_ex1"; Goal "x ~= 0hr ==> ?! y. y * (x::hypreal) = 1hr"; by (auto_tac (claset() addIs [hypreal_mult_hrinv_left],simpset())); by (dres_inst_tac [("f","%x. x*ya")] arg_cong 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_mult_assoc]) 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_mult_commute]) 1); qed "hypreal_hrinv_left_ex1"; Goal "[| y~= 0hr; x * y = 1hr |] ==> x = hrinv y"; by (forw_inst_tac [("x","y")] hypreal_mult_hrinv_left 1); by (res_inst_tac [("x1","y")] (hypreal_hrinv_left_ex1 RS ex1E) 1); by (assume_tac 1); by (Blast_tac 1); qed "hypreal_mult_inv_hrinv"; Goal "x ~= 0hr ==> ? y. x = hrinv y"; by (forw_inst_tac [("x","x")] hypreal_hrinv_left_ex 1); by (etac exE 1 THEN forw_inst_tac [("x","y")] hypreal_mult_inv_hrinv 1); by (res_inst_tac [("x","y")] exI 2); by Auto_tac; qed "hypreal_as_inverse_ex"; Goal "(c::hypreal) ~= 0hr ==> (c*a=c*b) = (a=b)"; by Auto_tac; by (dres_inst_tac [("f","%x. x*hrinv c")] arg_cong 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_mult_hrinv] @ hypreal_mult_ac) 1); qed "hypreal_mult_left_cancel"; Goal "(c::hypreal) ~= 0hr ==> (a*c=b*c) = (a=b)"; by (Step_tac 1); by (dres_inst_tac [("f","%x. x*hrinv c")] arg_cong 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_mult_hrinv] @ hypreal_mult_ac) 1); qed "hypreal_mult_right_cancel"; Goalw [hypreal_zero_def] "x ~= 0hr ==> hrinv(x) ~= 0hr"; by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); by (rotate_tac 1 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_hrinv, hypreal_mult] setloop (split_tac [expand_if])) 1); by (dtac FreeUltrafilterNat_Compl_mem 1 THEN Clarify_tac 1); by (ultra_tac (claset() addIs [ccontr] addDs [rinv_not_zero],simpset()) 1); qed "hrinv_not_zero"; Addsimps [hypreal_mult_hrinv,hypreal_mult_hrinv_left]; Goal "[| x ~= 0hr; y ~= 0hr |] ==> x * y ~= 0hr"; by (Step_tac 1); by (dres_inst_tac [("f","%z. hrinv x*z")] arg_cong 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_mult_assoc RS sym]) 1); qed "hypreal_mult_not_0"; bind_thm ("hypreal_mult_not_0E",hypreal_mult_not_0 RS notE); Goal "x ~= 0hr ==> x * x ~= 0hr"; by (blast_tac (claset() addDs [hypreal_mult_not_0]) 1); qed "hypreal_mult_self_not_zero"; Goal "[| x ~= 0hr; y ~= 0hr |] ==> hrinv(x*y) = hrinv(x)*hrinv(y)"; by (res_inst_tac [("c1","x")] (hypreal_mult_left_cancel RS iffD1) 1); by (auto_tac (claset(),simpset() addsimps [hypreal_mult_assoc RS sym, hypreal_mult_not_0])); by (res_inst_tac [("c1","y")] (hypreal_mult_right_cancel RS iffD1) 1); by (auto_tac (claset(),simpset() addsimps [hypreal_mult_not_0] @ hypreal_mult_ac)); by (auto_tac (claset(),simpset() addsimps [hypreal_mult_assoc RS sym,hypreal_mult_not_0])); qed "hrinv_mult_eq"; Goal "x ~= 0hr ==> hrinv(-x) = -hrinv(x)"; by (res_inst_tac [("c1","-x")] (hypreal_mult_right_cancel RS iffD1) 1); by Auto_tac; qed "hypreal_minus_hrinv"; Goal "[| x ~= 0hr; y ~= 0hr |] \ \ ==> hrinv(x*y) = hrinv(x)*hrinv(y)"; by (forw_inst_tac [("y","y")] hypreal_mult_not_0 1 THEN assume_tac 1); by (res_inst_tac [("c1","x")] (hypreal_mult_left_cancel RS iffD1) 1); by (auto_tac (claset(),simpset() addsimps [hypreal_mult_assoc RS sym])); by (res_inst_tac [("c1","y")] (hypreal_mult_left_cancel RS iffD1) 1); by (auto_tac (claset(),simpset() addsimps [hypreal_mult_left_commute])); by (asm_simp_tac (simpset() addsimps [hypreal_mult_assoc RS sym]) 1); qed "hypreal_hrinv_distrib"; (*------------------------------------------------------------------ Theorems for ordering ------------------------------------------------------------------*) (* prove introduction and elimination rules for hypreal_less *) Goalw [hypreal_less_def] "P < (Q::hypreal) = (EX X Y. X : Rep_hypreal(P) & \ \ Y : Rep_hypreal(Q) & \ \ {n. X n < Y n} : FreeUltrafilterNat)"; by (Fast_tac 1); qed "hypreal_less_iff"; Goalw [hypreal_less_def] "[| {n. X n < Y n} : FreeUltrafilterNat; \ \ X : Rep_hypreal(P); \ \ Y : Rep_hypreal(Q) |] ==> P < (Q::hypreal)"; by (Fast_tac 1); qed "hypreal_lessI"; Goalw [hypreal_less_def] "!! R1. [| R1 < (R2::hypreal); \ \ !!X Y. {n. X n < Y n} : FreeUltrafilterNat ==> P; \ \ !!X. X : Rep_hypreal(R1) ==> P; \ \ !!Y. Y : Rep_hypreal(R2) ==> P |] \ \ ==> P"; by Auto_tac; qed "hypreal_lessE"; Goalw [hypreal_less_def] "R1 < (R2::hypreal) ==> (EX X Y. {n. X n < Y n} : FreeUltrafilterNat & \ \ X : Rep_hypreal(R1) & \ \ Y : Rep_hypreal(R2))"; by (Fast_tac 1); qed "hypreal_lessD"; Goal "~ (R::hypreal) < R"; by (res_inst_tac [("z","R")] eq_Abs_hypreal 1); by (auto_tac (claset(),simpset() addsimps [hypreal_less_def])); by (Ultra_tac 1); qed "hypreal_less_not_refl"; (*** y < y ==> P ***) bind_thm("hypreal_less_irrefl",hypreal_less_not_refl RS notE); Goal "!!(x::hypreal). x < y ==> x ~= y"; by (auto_tac (claset(),simpset() addsimps [hypreal_less_not_refl])); qed "hypreal_not_refl2"; Goal "!!(R1::hypreal). [| R1 < R2; R2 < R3 |] ==> R1 < R3"; 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","R3")] eq_Abs_hypreal 1); by (auto_tac (claset() addSIs [exI],simpset() addsimps [hypreal_less_def])); by (ultra_tac (claset() addIs [real_less_trans],simpset()) 1); qed "hypreal_less_trans"; Goal "!! (R1::hypreal). [| R1 < R2; R2 < R1 |] ==> P"; by (dtac hypreal_less_trans 1 THEN assume_tac 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_less_not_refl]) 1); qed "hypreal_less_asym"; (*-------------------------------------------------------- TODO: The following theorem should have been proved first and then used througout the proofs as it probably makes many of them more straightforward. -------------------------------------------------------*) Goalw [hypreal_less_def] "(Abs_hypreal(hyprel^^{%n. X n}) < \ \ Abs_hypreal(hyprel^^{%n. Y n})) = \ \ ({n. X n < Y n} : FreeUltrafilterNat)"; by (auto_tac (claset() addSIs [lemma_hyprel_refl],simpset())); by (Ultra_tac 1); qed "hypreal_less"; (*--------------------------------------------------------------------------------- Hyperreals as a linearly ordered field ---------------------------------------------------------------------------------*) (*** sum order ***) Goalw [hypreal_zero_def] "[| 0hr < x; 0hr < y |] ==> 0hr < 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 [hypreal_less_def,hypreal_add])); by (auto_tac (claset() addSIs [exI],simpset() addsimps [hypreal_less_def,hypreal_add])); by (ultra_tac (claset() addIs [real_add_order],simpset()) 1); qed "hypreal_add_order"; (*** mult order ***) Goalw [hypreal_zero_def] "[| 0hr < x; 0hr < y |] ==> 0hr < 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() addSIs [exI],simpset() addsimps [hypreal_less_def,hypreal_mult])); by (ultra_tac (claset() addIs [real_mult_order],simpset()) 1); qed "hypreal_mult_order"; (*--------------------------------------------------------------------------------- Trichotomy of the hyperreals --------------------------------------------------------------------------------*) Goalw [hyprel_def] "? x. x: hyprel ^^ {%n. 0r}"; by (res_inst_tac [("x","%n. 0r")] exI 1); by (Step_tac 1); by (auto_tac (claset() addSIs [FreeUltrafilterNat_Nat_set],simpset())); qed "lemma_hyprel_0r_mem"; Goalw [hypreal_zero_def]"0hr < x | x = 0hr | x < 0hr"; by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); by (auto_tac (claset(),simpset() addsimps [hypreal_less_def])); by (cut_facts_tac [lemma_hyprel_0r_mem] 1 THEN etac exE 1); by (dres_inst_tac [("x","xa")] spec 1); by (dres_inst_tac [("x","x")] spec 1); by (cut_inst_tac [("x","x")] lemma_hyprel_refl 1); by Auto_tac; by (dres_inst_tac [("x","x")] spec 1); by (dres_inst_tac [("x","xa")] spec 1); by Auto_tac; by (Ultra_tac 1); by (auto_tac (claset() addIs [real_linear_less2],simpset())); qed "hypreal_trichotomy"; val prems = Goal "[| 0hr < x ==> P; \ \ x = 0hr ==> P; \ \ x < 0hr ==> P |] ==> P"; by (cut_inst_tac [("x","x")] hypreal_trichotomy 1); by (REPEAT (eresolve_tac (disjE::prems) 1)); qed "hypreal_trichotomyE"; (*---------------------------------------------------------------------------- More properties of < ----------------------------------------------------------------------------*) Goal "!!(A::hypreal). A < B ==> A + C < B + C"; by (res_inst_tac [("z","A")] eq_Abs_hypreal 1); by (res_inst_tac [("z","B")] eq_Abs_hypreal 1); by (res_inst_tac [("z","C")] eq_Abs_hypreal 1); by (auto_tac (claset() addSIs [exI],simpset() addsimps [hypreal_less_def,hypreal_add])); by (Ultra_tac 1); qed "hypreal_add_less_mono1"; Goal "!!(A::hypreal). A < B ==> C + A < C + B"; by (auto_tac (claset() addIs [hypreal_add_less_mono1], simpset() addsimps [hypreal_add_commute])); qed "hypreal_add_less_mono2"; Goal "((x::hypreal) < y) = (0hr < y + -x)"; by (Step_tac 1); by (dres_inst_tac [("C","-x")] hypreal_add_less_mono1 1); by (dres_inst_tac [("C","x")] hypreal_add_less_mono1 2); by (auto_tac (claset(),simpset() addsimps [hypreal_add_assoc])); qed "hypreal_less_minus_iff"; Goal "((x::hypreal) < y) = (x + -y< 0hr)"; by (Step_tac 1); by (dres_inst_tac [("C","-y")] hypreal_add_less_mono1 1); by (dres_inst_tac [("C","y")] hypreal_add_less_mono1 2); by (auto_tac (claset(),simpset() addsimps [hypreal_add_assoc])); qed "hypreal_less_minus_iff2"; Goal "!!(y1 :: hypreal). [| z1 < y1; z2 < y2 |] ==> z1 + z2 < y1 + y2"; by (dtac (hypreal_less_minus_iff RS iffD1) 1); by (dtac (hypreal_less_minus_iff RS iffD1) 1); by (dtac hypreal_add_order 1 THEN assume_tac 1); by (thin_tac "0hr < y2 + - z2" 1); by (dres_inst_tac [("C","z1 + z2")] hypreal_add_less_mono1 1); by (auto_tac (claset(),simpset() addsimps [hypreal_minus_add_distrib RS sym] @ hypreal_add_ac)); qed "hypreal_add_less_mono"; Goal "((x::hypreal) = y) = (0hr = x + - y)"; by Auto_tac; by (res_inst_tac [("x1","-y")] (hypreal_add_right_cancel RS iffD1) 1); by Auto_tac; qed "hypreal_eq_minus_iff"; Goal "((x::hypreal) = y) = (0hr = y + - x)"; by Auto_tac; by (res_inst_tac [("x1","-x")] (hypreal_add_right_cancel RS iffD1) 1); by Auto_tac; qed "hypreal_eq_minus_iff2"; Goal "(x = y + z) = (x + -z = (y::hypreal))"; by (auto_tac (claset(),simpset() addsimps [hypreal_add_assoc])); qed "hypreal_eq_minus_iff3"; Goal "(x = z + y) = (x + -z = (y::hypreal))"; by (auto_tac (claset(),simpset() addsimps hypreal_add_ac)); qed "hypreal_eq_minus_iff4"; Goal "(x ~= a) = (x + -a ~= 0hr)"; by (auto_tac (claset() addDs [sym RS (hypreal_eq_minus_iff RS iffD2)],simpset())); qed "hypreal_not_eq_minus_iff"; (*** linearity ***) Goal "(x::hypreal) < y | x = y | y < x"; by (stac hypreal_eq_minus_iff2 1); by (res_inst_tac [("x1","x")] (hypreal_less_minus_iff RS ssubst) 1); by (res_inst_tac [("x1","y")] (hypreal_less_minus_iff2 RS ssubst) 1); by (rtac hypreal_trichotomyE 1); by Auto_tac; qed "hypreal_linear"; Goal "!!(x::hypreal). [| x < y ==> P; x = y ==> P; \ \ y < x ==> P |] ==> P"; by (cut_inst_tac [("x","x"),("y","y")] hypreal_linear 1); by Auto_tac; qed "hypreal_linear_less2"; (*------------------------------------------------------------------------------ Properties of <= ------------------------------------------------------------------------------*) (*------ hypreal le iff reals le a.e ------*) Goalw [hypreal_le_def,real_le_def] "(Abs_hypreal(hyprel^^{%n. X n}) <= \ \ Abs_hypreal(hyprel^^{%n. Y n})) = \ \ ({n. X n <= Y n} : FreeUltrafilterNat)"; by (auto_tac (claset(),simpset() addsimps [hypreal_less])); by (ALLGOALS(Ultra_tac)); qed "hypreal_le"; (*---------------------------------------------------------*) (*---------------------------------------------------------*) Goalw [hypreal_le_def] "~(w < z) ==> z <= (w::hypreal)"; by (assume_tac 1); qed "hypreal_leI"; Goalw [hypreal_le_def] "z<=w ==> ~(w<(z::hypreal))"; by (assume_tac 1); qed "hypreal_leD"; val hypreal_leE = make_elim hypreal_leD; Goal "(~(w < z)) = (z <= (w::hypreal))"; by (fast_tac (claset() addSIs [hypreal_leI,hypreal_leD]) 1); qed "hypreal_less_le_iff"; Goalw [hypreal_le_def] "~ z <= w ==> w<(z::hypreal)"; by (Fast_tac 1); qed "not_hypreal_leE"; Goalw [hypreal_le_def] "z < w ==> z <= (w::hypreal)"; by (fast_tac (claset() addEs [hypreal_less_asym]) 1); qed "hypreal_less_imp_le"; Goalw [hypreal_le_def] "!!(x::hypreal). x <= y ==> x < y | x = y"; by (cut_facts_tac [hypreal_linear] 1); by (fast_tac (claset() addEs [hypreal_less_irrefl,hypreal_less_asym]) 1); qed "hypreal_le_imp_less_or_eq"; Goalw [hypreal_le_def] "z z <=(w::hypreal)"; by (cut_facts_tac [hypreal_linear] 1); by (fast_tac (claset() addEs [hypreal_less_irrefl,hypreal_less_asym]) 1); qed "hypreal_less_or_eq_imp_le"; Goal "(x <= (y::hypreal)) = (x < y | x=y)"; by (REPEAT(ares_tac [iffI, hypreal_less_or_eq_imp_le, hypreal_le_imp_less_or_eq] 1)); qed "hypreal_le_eq_less_or_eq"; Goal "w <= (w::hypreal)"; by (simp_tac (simpset() addsimps [hypreal_le_eq_less_or_eq]) 1); qed "hypreal_le_refl"; Addsimps [hypreal_le_refl]; Goal "[| i <= j; j < k |] ==> i < (k::hypreal)"; by (dtac hypreal_le_imp_less_or_eq 1); by (fast_tac (claset() addIs [hypreal_less_trans]) 1); qed "hypreal_le_less_trans"; Goal "!! (i::hypreal). [| i < j; j <= k |] ==> i < k"; by (dtac hypreal_le_imp_less_or_eq 1); by (fast_tac (claset() addIs [hypreal_less_trans]) 1); qed "hypreal_less_le_trans"; Goal "[| i <= j; j <= k |] ==> i <= (k::hypreal)"; by (EVERY1 [dtac hypreal_le_imp_less_or_eq, dtac hypreal_le_imp_less_or_eq, rtac hypreal_less_or_eq_imp_le, fast_tac (claset() addIs [hypreal_less_trans])]); qed "hypreal_le_trans"; Goal "[| z <= w; w <= z |] ==> z = (w::hypreal)"; by (EVERY1 [dtac hypreal_le_imp_less_or_eq, dtac hypreal_le_imp_less_or_eq, fast_tac (claset() addEs [hypreal_less_irrefl,hypreal_less_asym])]); qed "hypreal_le_anti_sym"; Goal "[| 0hr < x; 0hr <= y |] ==> 0hr < x + y"; by (auto_tac (claset() addDs [sym,hypreal_le_imp_less_or_eq] addIs [hypreal_add_order],simpset())); qed "hypreal_add_order_le"; (*------------------------------------------------------------------------ ------------------------------------------------------------------------*) Goal "[| ~ y < x; y ~= x |] ==> x < (y::hypreal)"; by (rtac not_hypreal_leE 1); by (fast_tac (claset() addDs [hypreal_le_imp_less_or_eq]) 1); qed "not_less_not_eq_hypreal_less"; Goal "(0hr < -R) = (R < 0hr)"; by (Step_tac 1); by (dres_inst_tac [("C","R")] hypreal_add_less_mono1 1); by (dres_inst_tac [("C","-R")] hypreal_add_less_mono1 2); by (auto_tac (claset(),simpset() addsimps [hypreal_add_assoc])); qed "hypreal_minus_zero_less_iff"; Goal "(-R < 0hr) = (0hr < R)"; by (Step_tac 1); by (dres_inst_tac [("C","R")] hypreal_add_less_mono1 1); by (dres_inst_tac [("C","-R")] hypreal_add_less_mono1 2); by (auto_tac (claset(),simpset() addsimps [hypreal_add_assoc])); qed "hypreal_minus_zero_less_iff2"; Goal "((x::hypreal) < y) = (-y < -x)"; by (stac hypreal_less_minus_iff 1); by (res_inst_tac [("x1","x")] (hypreal_less_minus_iff RS ssubst) 1); by (simp_tac (simpset() addsimps [hypreal_add_commute]) 1); qed "hypreal_less_swap_iff"; Goal "(0hr < x) = (-x < x)"; by (Step_tac 1); by (rtac ccontr 2 THEN forward_tac [hypreal_leI RS hypreal_le_imp_less_or_eq] 2); by (Step_tac 2); by (dtac (hypreal_minus_zero_less_iff RS iffD2) 2); by (dres_inst_tac [("R2.0","-x")] hypreal_less_trans 2); by (Auto_tac ); by (ftac hypreal_add_order 1 THEN assume_tac 1); by (dres_inst_tac [("C","-x"),("B","x + x")] hypreal_add_less_mono1 1); by (auto_tac (claset(),simpset() addsimps [hypreal_add_assoc])); qed "hypreal_gt_zero_iff"; Goal "(x < 0hr) = (x < -x)"; by (rtac (hypreal_minus_zero_less_iff RS subst) 1); by (stac hypreal_gt_zero_iff 1); by (Full_simp_tac 1); qed "hypreal_lt_zero_iff"; Goalw [hypreal_le_def] "(0hr <= x) = (-x <= x)"; by (auto_tac (claset(),simpset() addsimps [hypreal_lt_zero_iff RS sym])); qed "hypreal_ge_zero_iff"; Goalw [hypreal_le_def] "(x <= 0hr) = (x <= -x)"; by (auto_tac (claset(),simpset() addsimps [hypreal_gt_zero_iff RS sym])); qed "hypreal_le_zero_iff"; Goal "[| x < 0hr; y < 0hr |] ==> 0hr < x * y"; by (REPEAT(dtac (hypreal_minus_zero_less_iff RS iffD2) 1)); by (dtac hypreal_mult_order 1 THEN assume_tac 1); by (Asm_full_simp_tac 1); qed "hypreal_mult_less_zero1"; Goal "[| 0hr <= x; 0hr <= y |] ==> 0hr <= x * y"; by (REPEAT(dtac hypreal_le_imp_less_or_eq 1)); by (auto_tac (claset() addIs [hypreal_mult_order, hypreal_less_imp_le],simpset())); qed "hypreal_le_mult_order"; Goal "[| x <= 0hr; y <= 0hr |] ==> 0hr <= x * y"; by (rtac hypreal_less_or_eq_imp_le 1); by (dtac hypreal_le_imp_less_or_eq 1 THEN etac disjE 1); by Auto_tac; by (dtac hypreal_le_imp_less_or_eq 1); by (auto_tac (claset() addDs [hypreal_mult_less_zero1],simpset())); qed "real_mult_le_zero1"; Goal "[| 0hr <= x; y < 0hr |] ==> x * y <= 0hr"; by (rtac hypreal_less_or_eq_imp_le 1); by (dtac hypreal_le_imp_less_or_eq 1 THEN etac disjE 1); by Auto_tac; by (dtac (hypreal_minus_zero_less_iff RS iffD2) 1); by (rtac (hypreal_minus_zero_less_iff RS subst) 1); by (blast_tac (claset() addDs [hypreal_mult_order] addIs [hypreal_minus_mult_eq2 RS ssubst]) 1); qed "hypreal_mult_le_zero"; Goal "[| 0hr < x; y < 0hr |] ==> x*y < 0hr"; by (dtac (hypreal_minus_zero_less_iff RS iffD2) 1); by (dtac hypreal_mult_order 1 THEN assume_tac 1); by (rtac (hypreal_minus_zero_less_iff RS iffD1) 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_minus_mult_eq2]) 1); qed "hypreal_mult_less_zero"; Goalw [hypreal_one_def,hypreal_zero_def,hypreal_less_def] "0hr < 1hr"; by (res_inst_tac [("x","%n. 0r")] exI 1); by (res_inst_tac [("x","%n. 1r")] exI 1); by (auto_tac (claset(),simpset() addsimps [real_zero_less_one, FreeUltrafilterNat_Nat_set])); qed "hypreal_zero_less_one"; Goal "[| 0hr <= x; 0hr <= y |] ==> 0hr <= x + y"; by (REPEAT(dtac hypreal_le_imp_less_or_eq 1)); by (auto_tac (claset() addIs [hypreal_add_order, hypreal_less_imp_le],simpset())); qed "hypreal_le_add_order"; Goal "!!(q1::hypreal). q1 <= q2 ==> x + q1 <= x + q2"; by (dtac hypreal_le_imp_less_or_eq 1); by (Step_tac 1); by (auto_tac (claset() addSIs [hypreal_le_refl, hypreal_less_imp_le,hypreal_add_less_mono1], simpset() addsimps [hypreal_add_commute])); qed "hypreal_add_left_le_mono1"; Goal "!!(q1::hypreal). q1 <= q2 ==> q1 + x <= q2 + x"; by (auto_tac (claset() addDs [hypreal_add_left_le_mono1], simpset() addsimps [hypreal_add_commute])); qed "hypreal_add_le_mono1"; Goal "!!k l::hypreal. [|i<=j; k<=l |] ==> i + k <= j + l"; by (etac (hypreal_add_le_mono1 RS hypreal_le_trans) 1); by (simp_tac (simpset() addsimps [hypreal_add_commute]) 1); (*j moves to the end because it is free while k, l are bound*) by (etac hypreal_add_le_mono1 1); qed "hypreal_add_le_mono"; Goal "!!k l::hypreal. [|i i + k < j + l"; by (auto_tac (claset() addSDs [hypreal_le_imp_less_or_eq] addIs [hypreal_add_less_mono1,hypreal_add_less_mono],simpset())); qed "hypreal_add_less_le_mono"; Goal "!!k l::hypreal. [|i<=j; k i + k < j + l"; by (auto_tac (claset() addSDs [hypreal_le_imp_less_or_eq] addIs [hypreal_add_less_mono2,hypreal_add_less_mono],simpset())); qed "hypreal_add_le_less_mono"; Goal "(0hr*x x*z < y*z"; by (rotate_tac 1 1); by (dtac (hypreal_less_minus_iff RS iffD1) 1); by (rtac (hypreal_less_minus_iff RS iffD2) 1); by (dtac hypreal_mult_order 1 THEN assume_tac 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_add_mult_distrib2, hypreal_minus_mult_eq2 RS sym, hypreal_mult_commute ]) 1); qed "hypreal_mult_less_mono1"; Goal "[| 0hr z*x x*z<=y*z"; by (EVERY1 [rtac hypreal_less_or_eq_imp_le, dtac hypreal_le_imp_less_or_eq]); by (auto_tac (claset() addIs [hypreal_mult_less_mono1],simpset())); qed "hypreal_mult_le_less_mono1"; Goal "[| 0hr<=z; x z*x<=z*y"; by (asm_simp_tac (simpset() addsimps [hypreal_mult_commute, hypreal_mult_le_less_mono1]) 1); qed "hypreal_mult_le_less_mono2"; Goal "[| 0hr<=z; x<=y |] ==> z*x<=z*y"; by (dres_inst_tac [("x","x")] hypreal_le_imp_less_or_eq 1); by (auto_tac (claset() addIs [hypreal_mult_le_less_mono2,hypreal_le_refl],simpset())); qed "hypreal_mult_le_le_mono1"; val prem1::prem2::prem3::rest = goal thy "[| 0hr y*x y*x y*x < t*s"; by (dres_inst_tac [("x","x")] hypreal_le_imp_less_or_eq 1); by (fast_tac (claset() addIs [hypreal_mult_le_less_trans]) 1); qed "hypreal_mult_le_le_trans"; Goal "[| 0hr < r1; r1 r1 * x < r2 * y"; by (dres_inst_tac [("x","x")] hypreal_mult_less_mono2 1); by (dres_inst_tac [("R1.0","0hr")] hypreal_less_trans 2); by (dres_inst_tac [("x","r1")] hypreal_mult_less_mono1 3); by Auto_tac; by (blast_tac (claset() addIs [hypreal_less_trans]) 1); qed "hypreal_mult_less_mono"; Goal "[| 0hr < r1; r1 0hr < r2 * y"; by (dres_inst_tac [("R1.0","0hr")] hypreal_less_trans 1); by (assume_tac 1); by (blast_tac (claset() addIs [hypreal_mult_order]) 1); qed "hypreal_mult_order_trans"; Goal "[| 0hr < r1; r1 <= r2; 0hr <= x; x <= y |] \ \ ==> r1 * x <= r2 * y"; by (rtac hypreal_less_or_eq_imp_le 1); by (REPEAT(dtac hypreal_le_imp_less_or_eq 1)); by (auto_tac (claset() addIs [hypreal_mult_less_mono, hypreal_mult_less_mono1,hypreal_mult_less_mono2, hypreal_mult_order_trans,hypreal_mult_order],simpset())); qed "hypreal_mult_le_mono"; (*---------------------------------------------------------- hypreal_of_real preserves field and order properties -----------------------------------------------------------*) Goalw [hypreal_of_real_def] "hypreal_of_real ((z1::real) + z2) = \ \ hypreal_of_real z1 + hypreal_of_real z2"; by (asm_simp_tac (simpset() addsimps [hypreal_add, hypreal_add_mult_distrib]) 1); qed "hypreal_of_real_add"; Goalw [hypreal_of_real_def] "hypreal_of_real ((z1::real) * z2) = hypreal_of_real z1 * hypreal_of_real z2"; by (full_simp_tac (simpset() addsimps [hypreal_mult, hypreal_add_mult_distrib2]) 1); qed "hypreal_of_real_mult"; Goalw [hypreal_less_def,hypreal_of_real_def] "(z1 < z2) = (hypreal_of_real z1 < hypreal_of_real z2)"; by Auto_tac; by (res_inst_tac [("x","%n. z1")] exI 1); by (Step_tac 1); by (res_inst_tac [("x","%n. z2")] exI 2); by Auto_tac; by (rtac FreeUltrafilterNat_P 1); by (Ultra_tac 1); qed "hypreal_of_real_less_iff"; Addsimps [hypreal_of_real_less_iff RS sym]; Goalw [hypreal_le_def,real_le_def] "(z1 <= z2) = (hypreal_of_real z1 <= hypreal_of_real z2)"; by Auto_tac; qed "hypreal_of_real_le_iff"; Goalw [hypreal_of_real_def] "hypreal_of_real (-r) = - hypreal_of_real r"; by (auto_tac (claset(),simpset() addsimps [hypreal_minus])); qed "hypreal_of_real_minus"; Goal "0hr < x ==> 0hr < hrinv x"; by (EVERY1[rtac ccontr, dtac hypreal_leI]); by (forward_tac [hypreal_minus_zero_less_iff2 RS iffD2] 1); by (forward_tac [hypreal_not_refl2 RS not_sym] 1); by (dtac (hypreal_not_refl2 RS not_sym RS hrinv_not_zero) 1); by (EVERY1[dtac hypreal_le_imp_less_or_eq, Step_tac]); by (dtac hypreal_mult_less_zero1 1 THEN assume_tac 1); by (auto_tac (claset() addIs [hypreal_zero_less_one RS hypreal_less_asym], simpset() addsimps [hypreal_minus_mult_eq1 RS sym, hypreal_minus_zero_less_iff])); qed "hypreal_hrinv_gt_zero"; Goal "x < 0hr ==> hrinv x < 0hr"; by (ftac hypreal_not_refl2 1); by (dtac (hypreal_minus_zero_less_iff RS iffD2) 1); by (rtac (hypreal_minus_zero_less_iff RS iffD1) 1); by (dtac (hypreal_minus_hrinv RS sym) 1); by (auto_tac (claset() addIs [hypreal_hrinv_gt_zero], simpset())); qed "hypreal_hrinv_less_zero"; Goalw [hypreal_of_real_def,hypreal_one_def] "hypreal_of_real 1r = 1hr"; by (Step_tac 1); qed "hypreal_of_real_one"; Goalw [hypreal_of_real_def,hypreal_zero_def] "hypreal_of_real 0r = 0hr"; by (Step_tac 1); qed "hypreal_of_real_zero"; Goal "(hypreal_of_real r = 0hr) = (r = 0r)"; by (auto_tac (claset() addIs [FreeUltrafilterNat_P], simpset() addsimps [hypreal_of_real_def, hypreal_zero_def,FreeUltrafilterNat_Nat_set])); qed "hypreal_of_real_zero_iff"; Goal "(hypreal_of_real r ~= 0hr) = (r ~= 0r)"; by (full_simp_tac (simpset() addsimps [hypreal_of_real_zero_iff]) 1); qed "hypreal_of_real_not_zero_iff"; Goal "r ~= 0r ==> hrinv (hypreal_of_real r) = \ \ hypreal_of_real (rinv r)"; by (res_inst_tac [("c1","hypreal_of_real r")] (hypreal_mult_left_cancel RS iffD1) 1); by (etac (hypreal_of_real_not_zero_iff RS iffD2) 1); by (forward_tac [hypreal_of_real_not_zero_iff RS iffD2] 1); by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_mult RS sym,hypreal_of_real_one])); qed "hypreal_of_real_hrinv"; Goal "hypreal_of_real r ~= 0hr ==> hrinv (hypreal_of_real r) = \ \ hypreal_of_real (rinv r)"; by (etac (hypreal_of_real_not_zero_iff RS iffD1 RS hypreal_of_real_hrinv) 1); qed "hypreal_of_real_hrinv2"; Goal "x+x=x*(1hr+1hr)"; by (simp_tac (simpset() addsimps [hypreal_add_mult_distrib2]) 1); qed "hypreal_add_self"; Goal "1hr < 1hr + 1hr"; by (rtac (hypreal_less_minus_iff RS iffD2) 1); by (full_simp_tac (simpset() addsimps [hypreal_zero_less_one, hypreal_add_assoc]) 1); qed "hypreal_one_less_two"; Goal "0hr < 1hr + 1hr"; by (rtac ([hypreal_zero_less_one, hypreal_one_less_two] MRS hypreal_less_trans) 1); qed "hypreal_zero_less_two"; Goal "1hr + 1hr ~= 0hr"; by (rtac (hypreal_zero_less_two RS hypreal_not_refl2 RS not_sym) 1); qed "hypreal_two_not_zero"; Addsimps [hypreal_two_not_zero]; Goal "x*hrinv(1hr + 1hr) + x*hrinv(1hr + 1hr) = x"; by (stac hypreal_add_self 1); by (full_simp_tac (simpset() addsimps [hypreal_mult_assoc]) 1); qed "hypreal_sum_of_halves"; Goal "z ~= 0hr ==> x*y = (x*hrinv(z))*(z*y)"; by (asm_simp_tac (simpset() addsimps hypreal_mult_ac) 1); qed "lemma_chain"; Goal "0hr < r ==> 0hr < r*hrinv(1hr+1hr)"; by (dtac (hypreal_zero_less_two RS hypreal_hrinv_gt_zero RS hypreal_mult_less_mono1) 1); by Auto_tac; qed "hypreal_half_gt_zero"; (* TODO: remove redundant 0hr < x *) Goal "[| 0hr < r; 0hr < x; r < x |] ==> hrinv x < hrinv r"; by (ftac hypreal_hrinv_gt_zero 1); by (forw_inst_tac [("x","x")] hypreal_hrinv_gt_zero 1); by (forw_inst_tac [("x","r"),("z","hrinv r")] hypreal_mult_less_mono1 1); by (assume_tac 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_not_refl2 RS not_sym RS hypreal_mult_hrinv]) 1); by (ftac hypreal_hrinv_gt_zero 1); by (forw_inst_tac [("x","1hr"),("z","hrinv x")] hypreal_mult_less_mono2 1); by (assume_tac 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_not_refl2 RS not_sym RS hypreal_mult_hrinv_left,hypreal_mult_assoc RS sym]) 1); qed "hypreal_hrinv_less_swap"; Goal "[| 0hr < r; 0hr < x|] ==> (r < x) = (hrinv x < hrinv r)"; by (auto_tac (claset() addIs [hypreal_hrinv_less_swap],simpset())); by (res_inst_tac [("t","r")] (hypreal_hrinv_hrinv RS subst) 1); by (etac (hypreal_not_refl2 RS not_sym) 1); by (res_inst_tac [("t","x")] (hypreal_hrinv_hrinv RS subst) 1); by (etac (hypreal_not_refl2 RS not_sym) 1); by (auto_tac (claset() addIs [hypreal_hrinv_less_swap], simpset() addsimps [hypreal_hrinv_gt_zero])); qed "hypreal_hrinv_less_iff"; Goal "[| 0hr < z; x < y |] ==> x*hrinv(z) < y*hrinv(z)"; by (blast_tac (claset() addSIs [hypreal_mult_less_mono1, hypreal_hrinv_gt_zero]) 1); qed "hypreal_mult_hrinv_less_mono1"; Goal "[| 0hr < z; x < y |] ==> hrinv(z)*x < hrinv(z)*y"; by (blast_tac (claset() addSIs [hypreal_mult_less_mono2, hypreal_hrinv_gt_zero]) 1); qed "hypreal_mult_hrinv_less_mono2"; Goal "[| 0hr < z; x*z < y*z |] ==> x < y"; by (forw_inst_tac [("x","x*z")] hypreal_mult_hrinv_less_mono1 1); by (dtac (hypreal_not_refl2 RS not_sym) 2); by (auto_tac (claset() addSDs [hypreal_mult_hrinv], simpset() addsimps hypreal_mult_ac)); qed "hypreal_less_mult_right_cancel"; Goal "[| 0hr < z; z*x < z*y |] ==> x < y"; by (auto_tac (claset() addIs [hypreal_less_mult_right_cancel], simpset() addsimps [hypreal_mult_commute])); qed "hypreal_less_mult_left_cancel"; Goal "[| 0hr < r; 0hr < ra; \ \ r < x; ra < y |] \ \ ==> r*ra < x*y"; by (forw_inst_tac [("R2.0","r")] hypreal_less_trans 1); by (dres_inst_tac [("z","ra"),("x","r")] hypreal_mult_less_mono1 2); by (dres_inst_tac [("z","x"),("x","ra")] hypreal_mult_less_mono2 3); by (auto_tac (claset() addIs [hypreal_less_trans],simpset())); qed "hypreal_mult_less_gt_zero"; Goal "[| 0hr < r; 0hr < ra; \ \ r <= x; ra <= y |] \ \ ==> r*ra <= x*y"; by (REPEAT(dtac hypreal_le_imp_less_or_eq 1)); by (rtac hypreal_less_or_eq_imp_le 1); by (auto_tac (claset() addIs [hypreal_mult_less_mono1, hypreal_mult_less_mono2,hypreal_mult_less_gt_zero], simpset())); qed "hypreal_mult_le_ge_zero"; Goal "? (x::hypreal). x < y"; by (rtac (hypreal_add_zero_right RS subst) 1); by (res_inst_tac [("x","y + -1hr")] exI 1); by (auto_tac (claset() addSIs [hypreal_add_less_mono2], simpset() addsimps [hypreal_minus_zero_less_iff2, hypreal_zero_less_one] delsimps [hypreal_add_zero_right])); qed "hypreal_less_Ex"; Goal "!!(A::hypreal). A + C < B + C ==> A < B"; by (dres_inst_tac [("C","-C")] hypreal_add_less_mono1 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_add_assoc]) 1); qed "hypreal_less_add_right_cancel"; Goal "!!(A::hypreal). C + A < C + B ==> A < B"; by (dres_inst_tac [("C","-C")] hypreal_add_less_mono2 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_add_assoc RS sym]) 1); qed "hypreal_less_add_left_cancel"; Goal "0hr <= x*x"; by (res_inst_tac [("x","0hr"),("y","x")] hypreal_linear_less2 1); by (auto_tac (claset() addIs [hypreal_mult_order, hypreal_mult_less_zero1,hypreal_less_imp_le], simpset())); qed "hypreal_le_square"; Addsimps [hypreal_le_square]; Goalw [hypreal_le_def] "- (x*x) <= 0hr"; by (auto_tac (claset() addSDs [(hypreal_le_square RS hypreal_le_less_trans)],simpset() addsimps [hypreal_minus_zero_less_iff,hypreal_less_not_refl])); qed "hypreal_less_minus_square"; Addsimps [hypreal_less_minus_square]; Goal "[|x ~= 0hr; y ~= 0hr |] ==> \ \ hrinv(x) + hrinv(y) = (x + y)*hrinv(x*y)"; by (asm_full_simp_tac (simpset() addsimps [hypreal_hrinv_distrib, hypreal_add_mult_distrib,hypreal_mult_assoc RS sym]) 1); by (stac hypreal_mult_assoc 1); by (rtac (hypreal_mult_left_commute RS subst) 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_add_commute]) 1); qed "hypreal_hrinv_add"; Goal "x = -x ==> x = 0hr"; by (dtac (hypreal_eq_minus_iff RS iffD1 RS sym) 1); by (Asm_full_simp_tac 1); by (dtac (hypreal_add_self RS subst) 1); by (rtac ccontr 1); by (blast_tac (claset() addDs [hypreal_two_not_zero RSN (2,hypreal_mult_not_0)]) 1); qed "hypreal_self_eq_minus_self_zero"; Goal "(x + x = 0hr) = (x = 0hr)"; by Auto_tac; by (dtac (hypreal_add_self RS subst) 1); by (rtac ccontr 1 THEN rtac hypreal_mult_not_0E 1); by Auto_tac; qed "hypreal_add_self_zero_cancel"; Addsimps [hypreal_add_self_zero_cancel]; Goal "(x + x + y = y) = (x = 0hr)"; by Auto_tac; by (dtac (hypreal_eq_minus_iff RS iffD1) 1 THEN dtac sym 1); by (auto_tac (claset(),simpset() addsimps [hypreal_add_assoc])); qed "hypreal_add_self_zero_cancel2"; Addsimps [hypreal_add_self_zero_cancel2]; Goal "(x + (x + y) = y) = (x = 0hr)"; by (simp_tac (simpset() addsimps [hypreal_add_assoc RS sym]) 1); qed "hypreal_add_self_zero_cancel2a"; Addsimps [hypreal_add_self_zero_cancel2a]; Goal "(b = -a) = (-b = (a::hypreal))"; by Auto_tac; qed "hypreal_minus_eq_swap"; Goal "(-b = -a) = (b = (a::hypreal))"; by (asm_full_simp_tac (simpset() addsimps [hypreal_minus_eq_swap]) 1); qed "hypreal_minus_eq_cancel"; Addsimps [hypreal_minus_eq_cancel]; Goal "x < x + 1hr"; by (cut_inst_tac [("C","x")] (hypreal_zero_less_one RS hypreal_add_less_mono2) 1); by (Asm_full_simp_tac 1); qed "hypreal_less_self_add_one"; Addsimps [hypreal_less_self_add_one]; Goal "((x::hypreal) + x = y + y) = (x = y)"; by (auto_tac (claset() addIs [hypreal_two_not_zero RS hypreal_mult_left_cancel RS iffD1],simpset() addsimps [hypreal_add_mult_distrib])); qed "hypreal_add_self_cancel"; Addsimps [hypreal_add_self_cancel]; Goal "(y = x + - y + x) = (y = (x::hypreal))"; by Auto_tac; by (dres_inst_tac [("x1","y")] (hypreal_add_right_cancel RS iffD2) 1); by (auto_tac (claset(),simpset() addsimps hypreal_add_ac)); qed "hypreal_add_self_minus_cancel"; Addsimps [hypreal_add_self_minus_cancel]; Goal "(y = x + (- y + x)) = (y = (x::hypreal))"; by (asm_full_simp_tac (simpset() addsimps [hypreal_add_assoc RS sym])1); qed "hypreal_add_self_minus_cancel2"; Addsimps [hypreal_add_self_minus_cancel2]; Goal "z + -x = y + (y + (-x + -z)) = (y = (z::hypreal))"; by Auto_tac; by (dres_inst_tac [("x1","z")] (hypreal_add_right_cancel RS iffD2) 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_minus_add_distrib RS sym] @ hypreal_add_ac) 1); by (asm_full_simp_tac (simpset() addsimps [hypreal_add_assoc RS sym,hypreal_add_right_cancel]) 1); qed "hypreal_add_self_minus_cancel3"; Addsimps [hypreal_add_self_minus_cancel3]; (* check why this does not work without 2nd substiution anymore! *) Goal "x < y ==> x < (x + y)*hrinv(1hr + 1hr)"; by (dres_inst_tac [("C","x")] hypreal_add_less_mono2 1); by (dtac (hypreal_add_self RS subst) 1); by (dtac (hypreal_zero_less_two RS hypreal_hrinv_gt_zero RS hypreal_mult_less_mono1) 1); by (auto_tac (claset() addDs [hypreal_two_not_zero RS (hypreal_mult_hrinv RS subst)],simpset() addsimps [hypreal_mult_assoc])); qed "hypreal_less_half_sum"; (* check why this does not work without 2nd substiution anymore! *) Goal "x < y ==> (x + y)*hrinv(1hr + 1hr) < y"; by (dres_inst_tac [("C","y")] hypreal_add_less_mono1 1); by (dtac (hypreal_add_self RS subst) 1); by (dtac (hypreal_zero_less_two RS hypreal_hrinv_gt_zero RS hypreal_mult_less_mono1) 1); by (auto_tac (claset() addDs [hypreal_two_not_zero RS (hypreal_mult_hrinv RS subst)],simpset() addsimps [hypreal_mult_assoc])); qed "hypreal_gt_half_sum"; Goal "!!(x::hypreal). x < y ==> EX r. x < r & r < y"; by (blast_tac (claset() addSIs [hypreal_less_half_sum, hypreal_gt_half_sum]) 1); qed "hypreal_dense"; Goal "(x * x = 0hr) = (x = 0hr)"; by Auto_tac; by (blast_tac (claset() addIs [hypreal_mult_not_0E]) 1); qed "hypreal_mult_self_eq_zero_iff"; Addsimps [hypreal_mult_self_eq_zero_iff]; Goal "(0hr = x * x) = (x = 0hr)"; by (auto_tac (claset() addDs [sym],simpset())); qed "hypreal_mult_self_eq_zero_iff2"; Addsimps [hypreal_mult_self_eq_zero_iff2]; Goal "(x*x + y*y = 0hr) = (x = 0hr & y = 0hr)"; by Auto_tac; by (dtac (sym RS (hypreal_eq_minus_iff3 RS iffD1)) 1); by (dtac (sym RS (hypreal_eq_minus_iff4 RS iffD1)) 2); by (ALLGOALS(rtac ccontr)); by (ALLGOALS(dtac hypreal_mult_self_not_zero)); by (cut_inst_tac [("x1","x")] (hypreal_le_square RS hypreal_le_imp_less_or_eq) 1); by (cut_inst_tac [("x1","y")] (hypreal_le_square RS hypreal_le_imp_less_or_eq) 2); by (auto_tac (claset() addDs [sym],simpset())); by (dres_inst_tac [("x1","y")] (hypreal_less_minus_square RS hypreal_le_less_trans) 1); by (dres_inst_tac [("x1","x")] (hypreal_less_minus_square RS hypreal_le_less_trans) 2); by (auto_tac (claset(),simpset() addsimps [hypreal_less_not_refl])); qed "hypreal_squares_add_zero_iff"; Addsimps [hypreal_squares_add_zero_iff]; Goal "x * x ~= 0hr ==> 0hr < x* x + y*y + z*z"; by (cut_inst_tac [("x1","x")] (hypreal_le_square RS hypreal_le_imp_less_or_eq) 1); by (auto_tac (claset() addSIs [hypreal_add_order_le],simpset())); qed "hypreal_sum_squares3_gt_zero"; Goal "x * x ~= 0hr ==> 0hr < y*y + x*x + z*z"; by (dtac hypreal_sum_squares3_gt_zero 1); by (auto_tac (claset(),simpset() addsimps hypreal_add_ac)); qed "hypreal_sum_squares3_gt_zero2"; Goal "x * x ~= 0hr ==> 0hr < y*y + z*z + x*x"; by (dtac hypreal_sum_squares3_gt_zero 1); by (auto_tac (claset(),simpset() addsimps hypreal_add_ac)); qed "hypreal_sum_squares3_gt_zero3"; Goal "(x*x + y*y + z*z = 0hr) = \ \ (x = 0hr & y = 0hr & z = 0hr)"; by Auto_tac; by (ALLGOALS(rtac ccontr)); by (ALLGOALS(dtac hypreal_mult_self_not_zero)); by (auto_tac (claset() addDs [hypreal_not_refl2 RS not_sym, hypreal_sum_squares3_gt_zero3,hypreal_sum_squares3_gt_zero, hypreal_sum_squares3_gt_zero2],simpset() delsimps [hypreal_mult_self_eq_zero_iff])); qed "hypreal_three_squares_add_zero_iff"; Addsimps [hypreal_three_squares_add_zero_iff]; Goal "(x::hypreal)*x <= 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 (auto_tac (claset(),simpset() addsimps [hypreal_mult,hypreal_add,hypreal_le])); qed "hypreal_self_le_add_pos"; Addsimps [hypreal_self_le_add_pos]; Goal "(x::hypreal)*x <= x*x + y*y + z*z"; 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","z")] eq_Abs_hypreal 1); by (auto_tac (claset(),simpset() addsimps [hypreal_mult,hypreal_add,hypreal_le, real_le_add_order])); qed "hypreal_self_le_add_pos2"; Addsimps [hypreal_self_le_add_pos2]; (*--------------------------------------------------------------------------------- Embedding of the naturals in the hyperreals ---------------------------------------------------------------------------------*) Goalw [hypreal_of_posnat_def] "hypreal_of_posnat 0 = 1hr"; by (full_simp_tac (simpset() addsimps [pnat_one_iff RS sym,real_of_preal_def]) 1); by (fold_tac [real_one_def]); by (rtac hypreal_of_real_one 1); qed "hypreal_of_posnat_one"; Goalw [hypreal_of_posnat_def] "hypreal_of_posnat 1 = 1hr + 1hr"; by (full_simp_tac (simpset() addsimps [real_of_preal_def,real_one_def, hypreal_one_def,hypreal_add,hypreal_of_real_def,pnat_two_eq, real_add,prat_of_pnat_add RS sym,preal_of_prat_add RS sym] @ pnat_add_ac) 1); qed "hypreal_of_posnat_two"; Goalw [hypreal_of_posnat_def] "hypreal_of_posnat n1 + hypreal_of_posnat n2 = \ \ hypreal_of_posnat (n1 + n2) + 1hr"; by (full_simp_tac (simpset() addsimps [hypreal_of_posnat_one RS sym, hypreal_of_real_add RS sym,hypreal_of_posnat_def,real_of_preal_add RS sym, preal_of_prat_add RS sym,prat_of_pnat_add RS sym,pnat_of_nat_add]) 1); qed "hypreal_of_posnat_add"; Goal "hypreal_of_posnat (n + 1) = hypreal_of_posnat n + 1hr"; by (res_inst_tac [("x1","1hr")] (hypreal_add_right_cancel RS iffD1) 1); by (rtac (hypreal_of_posnat_add RS subst) 1); by (full_simp_tac (simpset() addsimps [hypreal_of_posnat_two,hypreal_add_assoc]) 1); qed "hypreal_of_posnat_add_one"; Goalw [real_of_posnat_def,hypreal_of_posnat_def] "hypreal_of_posnat n = hypreal_of_real (real_of_posnat n)"; by (rtac refl 1); qed "hypreal_of_real_of_posnat"; Goalw [hypreal_of_posnat_def] "(n < m) = (hypreal_of_posnat n < hypreal_of_posnat m)"; by Auto_tac; qed "hypreal_of_posnat_less_iff"; Addsimps [hypreal_of_posnat_less_iff RS sym]; (*--------------------------------------------------------------------------------- Existence of infinite hyperreal number ---------------------------------------------------------------------------------*) Goal "hyprel^^{%n::nat. real_of_posnat n} : hypreal"; by Auto_tac; qed "hypreal_omega"; Goalw [omega_def] "Rep_hypreal(whr) : hypreal"; by (rtac Rep_hypreal 1); qed "Rep_hypreal_omega"; (* existence of infinite number not corresponding to any real number *) (* use assumption that member FreeUltrafilterNat is not finite *) (* a few lemmas first *) Goal "{n::nat. x = real_of_posnat n} = {} | \ \ (? y. {n::nat. x = real_of_posnat n} = {y})"; by (auto_tac (claset() addDs [inj_real_of_posnat RS injD],simpset())); qed "lemma_omega_empty_singleton_disj"; Goal "finite {n::nat. x = real_of_posnat n}"; by (cut_inst_tac [("x","x")] lemma_omega_empty_singleton_disj 1); by Auto_tac; qed "lemma_finite_omega_set"; Goalw [omega_def,hypreal_of_real_def] "~ (? x. hypreal_of_real x = whr)"; by (auto_tac (claset(),simpset() addsimps [lemma_finite_omega_set RS FreeUltrafilterNat_finite])); qed "not_ex_hypreal_of_real_eq_omega"; Goal "hypreal_of_real x ~= whr"; by (cut_facts_tac [not_ex_hypreal_of_real_eq_omega] 1); by Auto_tac; qed "hypreal_of_real_not_eq_omega"; (* existence of infinitesimal number also not *) (* corresponding to any real number *) Goal "{n::nat. x = rinv(real_of_posnat n)} = {} | \ \ (? y. {n::nat. x = rinv(real_of_posnat n)} = {y})"; by (Step_tac 1 THEN Step_tac 1); by (auto_tac (claset() addIs [real_of_posnat_rinv_inj],simpset())); qed "lemma_epsilon_empty_singleton_disj"; Goal "finite {n::nat. x = rinv(real_of_posnat n)}"; by (cut_inst_tac [("x","x")] lemma_epsilon_empty_singleton_disj 1); by Auto_tac; qed "lemma_finite_epsilon_set"; Goalw [epsilon_def,hypreal_of_real_def] "~ (? x. hypreal_of_real x = ehr)"; by (auto_tac (claset(),simpset() addsimps [lemma_finite_epsilon_set RS FreeUltrafilterNat_finite])); qed "not_ex_hypreal_of_real_eq_epsilon"; Goal "hypreal_of_real x ~= ehr"; by (cut_facts_tac [not_ex_hypreal_of_real_eq_epsilon] 1); by Auto_tac; qed "hypreal_of_real_not_eq_epsilon"; Goalw [epsilon_def,hypreal_zero_def] "ehr ~= 0hr"; by (auto_tac (claset(),simpset() addsimps [real_of_posnat_rinv_not_zero])); qed "hypreal_epsilon_not_zero"; Goalw [omega_def,hypreal_zero_def] "whr ~= 0hr"; by (Simp_tac 1); qed "hypreal_omega_not_zero"; Goal "ehr = hrinv(whr)"; by (asm_full_simp_tac (simpset() addsimps [hypreal_hrinv,omega_def,epsilon_def] setloop (split_tac [expand_if])) 1); qed "hypreal_epsilon_hrinv_omega"; (*---------------------------------------------------------------- Another embedding of the naturals in the hyperreals (see hypreal_of_posnat) ----------------------------------------------------------------*) Goalw [hypreal_of_nat_def] "hypreal_of_nat 0 = 0hr"; by (full_simp_tac (simpset() addsimps [hypreal_of_posnat_one]) 1); qed "hypreal_of_nat_zero"; Goalw [hypreal_of_nat_def] "hypreal_of_nat 1 = 1hr"; by (full_simp_tac (simpset() addsimps [hypreal_of_posnat_two, hypreal_add_assoc]) 1); qed "hypreal_of_nat_one"; Goalw [hypreal_of_nat_def] "hypreal_of_nat n1 + hypreal_of_nat n2 = \ \ hypreal_of_nat (n1 + n2)"; by (full_simp_tac (simpset() addsimps hypreal_add_ac) 1); by (simp_tac (simpset() addsimps [hypreal_of_posnat_add, hypreal_add_assoc RS sym]) 1); by (rtac (hypreal_add_commute RS subst) 1); by (simp_tac (simpset() addsimps [hypreal_add_left_cancel, hypreal_add_assoc]) 1); qed "hypreal_of_nat_add"; Goal "hypreal_of_nat 2 = 1hr + 1hr"; by (simp_tac (simpset() addsimps [hypreal_of_nat_one RS sym,hypreal_of_nat_add]) 1); qed "hypreal_of_nat_two"; Goalw [hypreal_of_nat_def] "(n < m) = (hypreal_of_nat n < hypreal_of_nat m)"; by (auto_tac (claset() addIs [hypreal_add_less_mono1],simpset())); by (dres_inst_tac [("C","1hr")] hypreal_add_less_mono1 1); by (full_simp_tac (simpset() addsimps [hypreal_add_assoc]) 1); qed "hypreal_of_nat_less_iff"; Addsimps [hypreal_of_nat_less_iff RS sym]; (* naturals embedded in hyperreals is an hyperreal *) Goalw [hypreal_of_nat_def,real_of_nat_def] "hypreal_of_nat m = Abs_hypreal(hyprel^^{%n. real_of_nat m})"; by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_def, hypreal_of_real_of_posnat,hypreal_minus,hypreal_one_def,hypreal_add])); qed "hypreal_of_nat_iff"; Goal "inj hypreal_of_nat"; by (rtac injI 1); by (auto_tac (claset() addSDs [FreeUltrafilterNat_P], simpset() addsimps [split_if_mem1, hypreal_of_nat_iff, real_add_right_cancel,inj_real_of_nat RS injD])); qed "inj_hypreal_of_nat"; Goalw [hypreal_of_nat_def,hypreal_of_real_def,hypreal_of_posnat_def, real_of_posnat_def,hypreal_one_def,real_of_nat_def] "hypreal_of_nat n = hypreal_of_real (real_of_nat n)"; by (simp_tac (simpset() addsimps [hypreal_add,hypreal_minus]) 1); qed "hypreal_of_nat_real_of_nat"; Goal "hypreal_of_posnat (Suc n) = hypreal_of_posnat n + 1hr"; by (stac (hypreal_of_posnat_add_one RS sym) 1); by (Simp_tac 1); qed "hypreal_of_posnat_Suc"; Goalw [hypreal_of_nat_def] "hypreal_of_nat (Suc n) = hypreal_of_nat n + 1hr"; by (simp_tac (simpset() addsimps [hypreal_of_posnat_Suc] @ hypreal_add_ac) 1); qed "hypreal_of_nat_Suc";