(* Title : HyperNat.ML
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Description : Explicit construction of hypernaturals using
ultrafilters
*)
(*------------------------------------------------------------------------
Properties of hypnatrel
------------------------------------------------------------------------*)
(** Proving that hyprel is an equivalence relation **)
(** Natural deduction for hypnatrel - similar to hyprel! **)
Goalw [hypnatrel_def]
"((X,Y): hypnatrel) = ({n. X n = Y n}: FreeUltrafilterNat)";
by (Fast_tac 1);
qed "hypnatrel_iff";
Goalw [hypnatrel_def]
"{n. X n = Y n}: FreeUltrafilterNat ==> (X,Y): hypnatrel";
by (Fast_tac 1);
qed "hypnatrelI";
Goalw [hypnatrel_def]
"p: hypnatrel --> (EX X Y. \
\ p = (X,Y) & {n. X n = Y n} : FreeUltrafilterNat)";
by (Fast_tac 1);
qed "hypnatrelE_lemma";
val [major,minor] = Goal
"[| p: hypnatrel; \
\ !!X Y. [| p = (X,Y); {n. X n = Y n}: FreeUltrafilterNat |] \
\ ==> Q |] \
\ ==> Q";
by (cut_facts_tac [major RS (hypnatrelE_lemma RS mp)] 1);
by (REPEAT (eresolve_tac [asm_rl,exE,conjE,minor] 1));
qed "hypnatrelE";
AddSIs [hypnatrelI];
AddSEs [hypnatrelE];
Goalw [hypnatrel_def] "(x,x): hypnatrel";
by Auto_tac;
qed "hypnatrel_refl";
Goalw [hypnatrel_def] "(x,y): hypnatrel --> (y,x):hypnatrel";
by (auto_tac (claset() addIs [lemma_perm RS subst], simpset()));
qed_spec_mp "hypnatrel_sym";
Goalw [hypnatrel_def]
"(x,y): hypnatrel --> (y,z):hypnatrel --> (x,z):hypnatrel";
by Auto_tac;
by (Fuf_tac 1);
qed_spec_mp "hypnatrel_trans";
Goalw [equiv_def, refl_def, sym_def, trans_def]
"equiv UNIV hypnatrel";
by (auto_tac (claset() addSIs [hypnatrel_refl]
addSEs [hypnatrel_sym,hypnatrel_trans]
delrules [hypnatrelI,hypnatrelE],
simpset()));
qed "equiv_hypnatrel";
val equiv_hypnatrel_iff =
[UNIV_I, UNIV_I] MRS (equiv_hypnatrel RS eq_equiv_class_iff);
Goalw [hypnat_def,hypnatrel_def,quotient_def] "hypnatrel^^{x}:hypnat";
by (Blast_tac 1);
qed "hypnatrel_in_hypnat";
Goal "inj_on Abs_hypnat hypnat";
by (rtac inj_on_inverseI 1);
by (etac Abs_hypnat_inverse 1);
qed "inj_on_Abs_hypnat";
Addsimps [equiv_hypnatrel_iff,inj_on_Abs_hypnat RS inj_on_iff,
hypnatrel_iff, hypnatrel_in_hypnat, Abs_hypnat_inverse];
Addsimps [equiv_hypnatrel RS eq_equiv_class_iff];
val eq_hypnatrelD = equiv_hypnatrel RSN (2,eq_equiv_class);
Goal "inj(Rep_hypnat)";
by (rtac inj_inverseI 1);
by (rtac Rep_hypnat_inverse 1);
qed "inj_Rep_hypnat";
Goalw [hypnatrel_def] "x: hypnatrel ^^ {x}";
by (Step_tac 1);
by Auto_tac;
qed "lemma_hypnatrel_refl";
Addsimps [lemma_hypnatrel_refl];
Goalw [hypnat_def] "{} ~: hypnat";
by (auto_tac (claset() addSEs [quotientE],simpset()));
qed "hypnat_empty_not_mem";
Addsimps [hypnat_empty_not_mem];
Goal "Rep_hypnat x ~= {}";
by (cut_inst_tac [("x","x")] Rep_hypnat 1);
by Auto_tac;
qed "Rep_hypnat_nonempty";
Addsimps [Rep_hypnat_nonempty];
(*------------------------------------------------------------------------
hypnat_of_nat: the injection from nat to hypnat
------------------------------------------------------------------------*)
Goal "inj(hypnat_of_nat)";
by (rtac injI 1);
by (rewtac hypnat_of_nat_def);
by (dtac (inj_on_Abs_hypnat RS inj_onD) 1);
by (REPEAT (rtac hypnatrel_in_hypnat 1));
by (dtac eq_equiv_class 1);
by (rtac equiv_hypnatrel 1);
by (Fast_tac 1);
by (rtac ccontr 1 THEN rotate_tac 1 1);
by Auto_tac;
qed "inj_hypnat_of_nat";
val [prem] = Goal
"(!!x. z = Abs_hypnat(hypnatrel^^{x}) ==> P) ==> P";
by (res_inst_tac [("x1","z")]
(rewrite_rule [hypnat_def] Rep_hypnat RS quotientE) 1);
by (dres_inst_tac [("f","Abs_hypnat")] arg_cong 1);
by (res_inst_tac [("x","x")] prem 1);
by (asm_full_simp_tac (simpset() addsimps [Rep_hypnat_inverse]) 1);
qed "eq_Abs_hypnat";
(*-----------------------------------------------------------
Addition for hyper naturals: hypnat_add
-----------------------------------------------------------*)
Goalw [congruent2_def]
"congruent2 hypnatrel (%X Y. hypnatrel^^{%n. X n + Y n})";
by Safe_tac;
by (ALLGOALS(Fuf_tac));
qed "hypnat_add_congruent2";
Goalw [hypnat_add_def]
"Abs_hypnat(hypnatrel^^{%n. X n}) + Abs_hypnat(hypnatrel^^{%n. Y n}) = \
\ Abs_hypnat(hypnatrel^^{%n. X n + Y n})";
by (asm_simp_tac
(simpset() addsimps [[equiv_hypnatrel, hypnat_add_congruent2]
MRS UN_equiv_class2]) 1);
qed "hypnat_add";
Goal "(z::hypnat) + w = w + z";
by (res_inst_tac [("z","z")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","w")] eq_Abs_hypnat 1);
by (asm_simp_tac (simpset() addsimps (add_ac @ [hypnat_add])) 1);
qed "hypnat_add_commute";
Goal "((z1::hypnat) + z2) + z3 = z1 + (z2 + z3)";
by (res_inst_tac [("z","z1")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","z2")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","z3")] eq_Abs_hypnat 1);
by (asm_simp_tac (simpset() addsimps [hypnat_add,add_assoc]) 1);
qed "hypnat_add_assoc";
(*For AC rewriting*)
Goal "(x::hypnat)+(y+z)=y+(x+z)";
by (rtac (hypnat_add_commute RS trans) 1);
by (rtac (hypnat_add_assoc RS trans) 1);
by (rtac (hypnat_add_commute RS arg_cong) 1);
qed "hypnat_add_left_commute";
(* hypnat addition is an AC operator *)
val hypnat_add_ac = [hypnat_add_assoc,hypnat_add_commute,
hypnat_add_left_commute];
Goalw [hypnat_zero_def] "(0::hypnat) + z = z";
by (res_inst_tac [("z","z")] eq_Abs_hypnat 1);
by (asm_full_simp_tac (simpset() addsimps [hypnat_add]) 1);
qed "hypnat_add_zero_left";
Goal "z + (0::hypnat) = z";
by (simp_tac (simpset() addsimps
[hypnat_add_zero_left,hypnat_add_commute]) 1);
qed "hypnat_add_zero_right";
Addsimps [hypnat_add_zero_left,hypnat_add_zero_right];
(*-----------------------------------------------------------
Subtraction for hyper naturals: hypnat_minus
-----------------------------------------------------------*)
Goalw [congruent2_def]
"congruent2 hypnatrel (%X Y. hypnatrel^^{%n. X n - Y n})";
by Safe_tac;
by (ALLGOALS(Fuf_tac));
qed "hypnat_minus_congruent2";
Goalw [hypnat_minus_def]
"Abs_hypnat(hypnatrel^^{%n. X n}) - Abs_hypnat(hypnatrel^^{%n. Y n}) = \
\ Abs_hypnat(hypnatrel^^{%n. X n - Y n})";
by (asm_simp_tac
(simpset() addsimps [[equiv_hypnatrel, hypnat_minus_congruent2]
MRS UN_equiv_class2]) 1);
qed "hypnat_minus";
Goalw [hypnat_zero_def] "z - z = (0::hypnat)";
by (res_inst_tac [("z","z")] eq_Abs_hypnat 1);
by (asm_full_simp_tac (simpset() addsimps [hypnat_minus]) 1);
qed "hypnat_minus_zero";
Goalw [hypnat_zero_def] "(0::hypnat) - n = 0";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (asm_full_simp_tac (simpset() addsimps [hypnat_minus]) 1);
qed "hypnat_diff_0_eq_0";
Addsimps [hypnat_minus_zero,hypnat_diff_0_eq_0];
Goalw [hypnat_zero_def] "(m+n = (0::hypnat)) = (m=0 & n=0)";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1);
by (auto_tac (claset() addIs [FreeUltrafilterNat_subset]
addDs [FreeUltrafilterNat_Int],
simpset() addsimps [hypnat_add] ));
qed "hypnat_add_is_0";
AddIffs [hypnat_add_is_0];
Goal "!!i::hypnat. i-j-k = i - (j+k)";
by (res_inst_tac [("z","i")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","j")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","k")] eq_Abs_hypnat 1);
by (asm_full_simp_tac (simpset() addsimps
[hypnat_minus,hypnat_add,diff_diff_left]) 1);
qed "hypnat_diff_diff_left";
Goal "!!i::hypnat. i-j-k = i-k-j";
by (simp_tac (simpset() addsimps
[hypnat_diff_diff_left, hypnat_add_commute]) 1);
qed "hypnat_diff_commute";
Goal "!!n::hypnat. (n+m) - n = m";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1);
by (asm_full_simp_tac (simpset() addsimps [hypnat_minus,hypnat_add]) 1);
qed "hypnat_diff_add_inverse";
Addsimps [hypnat_diff_add_inverse];
Goal "!!n::hypnat.(m+n) - n = m";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1);
by (asm_full_simp_tac (simpset() addsimps [hypnat_minus,hypnat_add]) 1);
qed "hypnat_diff_add_inverse2";
Addsimps [hypnat_diff_add_inverse2];
Goal "!!k::hypnat. (k+m) - (k+n) = m - n";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","k")] eq_Abs_hypnat 1);
by (asm_full_simp_tac (simpset() addsimps [hypnat_minus,hypnat_add]) 1);
qed "hypnat_diff_cancel";
Addsimps [hypnat_diff_cancel];
Goal "!!m::hypnat. (m+k) - (n+k) = m - n";
val hypnat_add_commute_k = read_instantiate [("w","k")] hypnat_add_commute;
by (asm_simp_tac (simpset() addsimps ([hypnat_add_commute_k])) 1);
qed "hypnat_diff_cancel2";
Addsimps [hypnat_diff_cancel2];
Goalw [hypnat_zero_def] "!!n::hypnat. n - (n+m) = (0::hypnat)";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1);
by (asm_full_simp_tac (simpset() addsimps [hypnat_minus,hypnat_add]) 1);
qed "hypnat_diff_add_0";
Addsimps [hypnat_diff_add_0];
(*-----------------------------------------------------------
Multiplication for hyper naturals: hypnat_mult
-----------------------------------------------------------*)
Goalw [congruent2_def]
"congruent2 hypnatrel (%X Y. hypnatrel^^{%n. X n * Y n})";
by Safe_tac;
by (ALLGOALS(Fuf_tac));
qed "hypnat_mult_congruent2";
Goalw [hypnat_mult_def]
"Abs_hypnat(hypnatrel^^{%n. X n}) * Abs_hypnat(hypnatrel^^{%n. Y n}) = \
\ Abs_hypnat(hypnatrel^^{%n. X n * Y n})";
by (asm_simp_tac
(simpset() addsimps [[equiv_hypnatrel,hypnat_mult_congruent2] MRS
UN_equiv_class2]) 1);
qed "hypnat_mult";
Goal "(z::hypnat) * w = w * z";
by (res_inst_tac [("z","z")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","w")] eq_Abs_hypnat 1);
by (asm_simp_tac (simpset() addsimps ([hypnat_mult] @ mult_ac)) 1);
qed "hypnat_mult_commute";
Goal "((z1::hypnat) * z2) * z3 = z1 * (z2 * z3)";
by (res_inst_tac [("z","z1")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","z2")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","z3")] eq_Abs_hypnat 1);
by (asm_simp_tac (simpset() addsimps [hypnat_mult,mult_assoc]) 1);
qed "hypnat_mult_assoc";
Goal "(z1::hypnat) * (z2 * z3) = z2 * (z1 * z3)";
by (rtac (hypnat_mult_commute RS trans) 1);
by (rtac (hypnat_mult_assoc RS trans) 1);
by (rtac (hypnat_mult_commute RS arg_cong) 1);
qed "hypnat_mult_left_commute";
(* hypnat multiplication is an AC operator *)
val hypnat_mult_ac = [hypnat_mult_assoc, hypnat_mult_commute,
hypnat_mult_left_commute];
Goalw [hypnat_one_def] "1hn * z = z";
by (res_inst_tac [("z","z")] eq_Abs_hypnat 1);
by (asm_full_simp_tac (simpset() addsimps [hypnat_mult]) 1);
qed "hypnat_mult_1";
Addsimps [hypnat_mult_1];
Goal "z * 1hn = z";
by (simp_tac (simpset() addsimps [hypnat_mult_commute]) 1);
qed "hypnat_mult_1_right";
Addsimps [hypnat_mult_1_right];
Goalw [hypnat_zero_def] "(0::hypnat) * z = 0";
by (res_inst_tac [("z","z")] eq_Abs_hypnat 1);
by (asm_full_simp_tac (simpset() addsimps [hypnat_mult]) 1);
qed "hypnat_mult_0";
Addsimps [hypnat_mult_0];
Goal "z * (0::hypnat) = 0";
by (simp_tac (simpset() addsimps [hypnat_mult_commute]) 1);
qed "hypnat_mult_0_right";
Addsimps [hypnat_mult_0_right];
Goal "!!m::hypnat. (m - n) * k = (m * k) - (n * k)";
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","k")] eq_Abs_hypnat 1);
by (asm_simp_tac (simpset() addsimps [hypnat_mult,
hypnat_minus,diff_mult_distrib]) 1);
qed "hypnat_diff_mult_distrib" ;
Goal "!!m::hypnat. k * (m - n) = (k * m) - (k * n)";
val hypnat_mult_commute_k = read_instantiate [("z","k")] hypnat_mult_commute;
by (simp_tac (simpset() addsimps [hypnat_diff_mult_distrib,
hypnat_mult_commute_k]) 1);
qed "hypnat_diff_mult_distrib2" ;
(*--------------------------
A Few more theorems
-------------------------*)
Goal "(z::hypnat) + v = z' + v' ==> z + (v + w) = z' + (v' + w)";
by (asm_simp_tac (simpset() addsimps [hypnat_add_assoc RS sym]) 1);
qed "hypnat_add_assoc_cong";
Goal "(z::hypnat) + (v + w) = v + (z + w)";
by (REPEAT (ares_tac [hypnat_add_commute RS hypnat_add_assoc_cong] 1));
qed "hypnat_add_assoc_swap";
Goal "((z1::hypnat) + z2) * w = (z1 * w) + (z2 * w)";
by (res_inst_tac [("z","z1")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","z2")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","w")] eq_Abs_hypnat 1);
by (asm_simp_tac (simpset() addsimps [hypnat_mult,hypnat_add,
add_mult_distrib]) 1);
qed "hypnat_add_mult_distrib";
Addsimps [hypnat_add_mult_distrib];
val hypnat_mult_commute'= read_instantiate [("z","w")] hypnat_mult_commute;
Goal "(w::hypnat) * (z1 + z2) = (w * z1) + (w * z2)";
by (simp_tac (simpset() addsimps [hypnat_mult_commute']) 1);
qed "hypnat_add_mult_distrib2";
Addsimps [hypnat_add_mult_distrib2];
(*** (hypnat) one and zero are distinct ***)
Goalw [hypnat_zero_def,hypnat_one_def] "(0::hypnat) ~= 1hn";
by Auto_tac;
qed "hypnat_zero_not_eq_one";
Addsimps [hypnat_zero_not_eq_one];
Addsimps [hypnat_zero_not_eq_one RS not_sym];
Goalw [hypnat_zero_def] "(m*n = (0::hypnat)) = (m=0 | n=0)";
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (auto_tac (claset() addSDs [FreeUltrafilterNat_Compl_mem],
simpset() addsimps [hypnat_mult]));
by (ALLGOALS(Fuf_tac));
qed "hypnat_mult_is_0";
Addsimps [hypnat_mult_is_0];
(*------------------------------------------------------------------
Theorems for ordering
------------------------------------------------------------------*)
(* prove introduction and elimination rules for hypnat_less *)
Goalw [hypnat_less_def]
"P < (Q::hypnat) = (EX X Y. X : Rep_hypnat(P) & \
\ Y : Rep_hypnat(Q) & \
\ {n. X n < Y n} : FreeUltrafilterNat)";
by (Fast_tac 1);
qed "hypnat_less_iff";
Goalw [hypnat_less_def]
"[| {n. X n < Y n} : FreeUltrafilterNat; \
\ X : Rep_hypnat(P); \
\ Y : Rep_hypnat(Q) |] ==> P < (Q::hypnat)";
by (Fast_tac 1);
qed "hypnat_lessI";
Goalw [hypnat_less_def]
"!! R1. [| R1 < (R2::hypnat); \
\ !!X Y. {n. X n < Y n} : FreeUltrafilterNat ==> P; \
\ !!X. X : Rep_hypnat(R1) ==> P; \
\ !!Y. Y : Rep_hypnat(R2) ==> P |] \
\ ==> P";
by Auto_tac;
qed "hypnat_lessE";
Goalw [hypnat_less_def]
"R1 < (R2::hypnat) ==> (EX X Y. {n. X n < Y n} : FreeUltrafilterNat & \
\ X : Rep_hypnat(R1) & \
\ Y : Rep_hypnat(R2))";
by (Fast_tac 1);
qed "hypnat_lessD";
Goal "~ (R::hypnat) < R";
by (res_inst_tac [("z","R")] eq_Abs_hypnat 1);
by (auto_tac (claset(),simpset() addsimps [hypnat_less_def]));
by (Fuf_empty_tac 1);
qed "hypnat_less_not_refl";
Addsimps [hypnat_less_not_refl];
bind_thm("hypnat_less_irrefl",hypnat_less_not_refl RS notE);
Goalw [hypnat_less_def,hypnat_zero_def] "~ n<(0::hypnat)";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by Auto_tac;
by (Fuf_empty_tac 1);
qed "hypnat_not_less0";
AddIffs [hypnat_not_less0];
(* n<(0::hypnat) ==> R *)
bind_thm ("hypnat_less_zeroE", hypnat_not_less0 RS notE);
Goalw [hypnat_less_def,hypnat_zero_def,hypnat_one_def]
"(n<1hn) = (n=0)";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (auto_tac (claset() addSIs [exI] addEs
[FreeUltrafilterNat_subset],simpset()));
by (Fuf_tac 1);
qed "hypnat_less_one";
AddIffs [hypnat_less_one];
Goal "!!(R1::hypnat). [| R1 < R2; R2 < R3 |] ==> R1 < R3";
by (res_inst_tac [("z","R1")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","R2")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","R3")] eq_Abs_hypnat 1);
by (auto_tac (claset(),simpset() addsimps [hypnat_less_def]));
by (res_inst_tac [("x","X")] exI 1);
by Auto_tac;
by (res_inst_tac [("x","Ya")] exI 1);
by Auto_tac;
by (Fuf_tac 1);
qed "hypnat_less_trans";
Goal "!! (R1::hypnat). [| R1 < R2; R2 < R1 |] ==> P";
by (dtac hypnat_less_trans 1 THEN assume_tac 1);
by (Asm_full_simp_tac 1);
qed "hypnat_less_asym";
(*----- hypnat less iff less a.e -----*)
(* See comments in HYPER for corresponding thm *)
Goalw [hypnat_less_def]
"(Abs_hypnat(hypnatrel^^{%n. X n}) < \
\ Abs_hypnat(hypnatrel^^{%n. Y n})) = \
\ ({n. X n < Y n} : FreeUltrafilterNat)";
by (auto_tac (claset() addSIs [lemma_hypnatrel_refl],simpset()));
by (Fuf_tac 1);
qed "hypnat_less";
Goal "~ m<n --> n+(m-n) = (m::hypnat)";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1);
by (auto_tac (claset(),simpset() addsimps
[hypnat_minus,hypnat_add,hypnat_less]));
by (dtac FreeUltrafilterNat_Compl_mem 1);
by (Fuf_tac 1);
qed_spec_mp "hypnat_add_diff_inverse";
Goal "n<=m ==> n+(m-n) = (m::hypnat)";
by (asm_full_simp_tac (simpset() addsimps
[hypnat_add_diff_inverse, hypnat_le_def]) 1);
qed "hypnat_le_add_diff_inverse";
Goal "n<=m ==> (m-n)+n = (m::hypnat)";
by (asm_simp_tac (simpset() addsimps [hypnat_le_add_diff_inverse,
hypnat_add_commute]) 1);
qed "hypnat_le_add_diff_inverse2";
(*---------------------------------------------------------------------------------
Hyper naturals as a linearly ordered field
---------------------------------------------------------------------------------*)
Goalw [hypnat_zero_def]
"[| (0::hypnat) < x; 0 < y |] ==> 0 < x + y";
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","y")] eq_Abs_hypnat 1);
by (auto_tac (claset(),simpset() addsimps
[hypnat_less_def,hypnat_add]));
by (REPEAT(Step_tac 1));
by (Fuf_tac 1);
qed "hypnat_add_order";
Goalw [hypnat_zero_def]
"!!(x::hypnat). [| (0::hypnat) < x; 0 < y |] ==> 0 < x * y";
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","y")] eq_Abs_hypnat 1);
by (auto_tac (claset(),simpset() addsimps
[hypnat_less_def,hypnat_mult]));
by (REPEAT(Step_tac 1));
by (Fuf_tac 1);
qed "hypnat_mult_order";
(*---------------------------------------------------------------------------------
Trichotomy of the hyper naturals
--------------------------------------------------------------------------------*)
Goalw [hypnatrel_def] "EX x. x: hypnatrel ^^ {%n. 0}";
by (res_inst_tac [("x","%n. 0")] exI 1);
by (Step_tac 1);
by Auto_tac;
qed "lemma_hypnatrel_0_mem";
(* linearity is actually proved further down! *)
Goalw [hypnat_zero_def, hypnat_less_def]
"(0::hypnat) < x | x = 0 | x < 0";
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1);
by (Auto_tac );
by (REPEAT(Step_tac 1));
by (REPEAT(dtac FreeUltrafilterNat_Compl_mem 1));
by (Fuf_tac 1);
qed "hypnat_trichotomy";
Goal "!!P. [| (0::hypnat) < x ==> P; \
\ x = 0 ==> P; \
\ x < 0 ==> P |] ==> P";
by (cut_inst_tac [("x","x")] hypnat_trichotomy 1);
by Auto_tac;
qed "hypnat_trichotomyE";
(*----------------------------------------------------------------------------
More properties of <
----------------------------------------------------------------------------*)
Goal "!!(A::hypnat). A < B ==> A + C < B + C";
by (res_inst_tac [("z","A")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","B")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","C")] eq_Abs_hypnat 1);
by (auto_tac (claset(), simpset() addsimps [hypnat_less_def,hypnat_add]));
by (REPEAT(Step_tac 1));
by (Fuf_tac 1);
qed "hypnat_add_less_mono1";
Goal "!!(A::hypnat). A < B ==> C + A < C + B";
by (auto_tac (claset() addIs [hypnat_add_less_mono1],
simpset() addsimps [hypnat_add_commute]));
qed "hypnat_add_less_mono2";
Goal "!!k l::hypnat. [|i<j; k<l |] ==> i + k < j + l";
by (etac (hypnat_add_less_mono1 RS hypnat_less_trans) 1);
by (simp_tac (simpset() addsimps [hypnat_add_commute]) 1);
(*j moves to the end because it is free while k, l are bound*)
by (etac hypnat_add_less_mono1 1);
qed "hypnat_add_less_mono";
(*---------------------------------------
hypnat_minus_less
---------------------------------------*)
Goalw [hypnat_less_def,hypnat_zero_def]
"((x::hypnat) < y) = ((0::hypnat) < y - x)";
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","y")] eq_Abs_hypnat 1);
by (auto_tac (claset(),simpset() addsimps
[hypnat_minus]));
by (REPEAT(Step_tac 1));
by (REPEAT(Step_tac 2));
by (ALLGOALS(fuf_tac (claset() addDs [sym],simpset())));
(*** linearity ***)
Goalw [hypnat_less_def] "(x::hypnat) < y | x = y | y < x";
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","y")] eq_Abs_hypnat 1);
by (Auto_tac );
by (REPEAT(Step_tac 1));
by (REPEAT(dtac FreeUltrafilterNat_Compl_mem 1));
by (Fuf_tac 1);
qed "hypnat_linear";
Goal "!!(x::hypnat). [| x < y ==> P; x = y ==> P; \
\ y < x ==> P |] ==> P";
by (cut_inst_tac [("x","x"),("y","y")] hypnat_linear 1);
by Auto_tac;
qed "hypnat_linear_less2";
Goal "((w::hypnat) ~= z) = (w<z | z<w)";
by (cut_facts_tac [hypnat_linear] 1);
by Auto_tac;
qed "hypnat_neq_iff";
(* Axiom 'order_less_le' of class 'order': *)
Goal "(w::hypnat) < z = (w <= z & w ~= z)";
by (simp_tac (simpset() addsimps [hypnat_le_def, hypnat_neq_iff]) 1);
by (blast_tac (claset() addIs [hypnat_less_asym]) 1);
qed "hypnat_less_le";
(*----------------------------------------------------------------------------
Properties of <=
----------------------------------------------------------------------------*)
(*------ hypnat le iff nat le a.e ------*)
Goalw [hypnat_le_def,le_def]
"(Abs_hypnat(hypnatrel^^{%n. X n}) <= \
\ Abs_hypnat(hypnatrel^^{%n. Y n})) = \
\ ({n. X n <= Y n} : FreeUltrafilterNat)";
by (auto_tac (claset() addSDs [FreeUltrafilterNat_Compl_mem],
simpset() addsimps [hypnat_less]));
by (Fuf_tac 1 THEN Fuf_empty_tac 1);
qed "hypnat_le";
(*---------------------------------------------------------*)
(*---------------------------------------------------------*)
Goalw [hypnat_le_def] "z < w ==> z <= (w::hypnat)";
by (fast_tac (claset() addEs [hypnat_less_asym]) 1);
qed "hypnat_less_imp_le";
Goalw [hypnat_le_def] "!!(x::hypnat). x <= y ==> x < y | x = y";
by (cut_facts_tac [hypnat_linear] 1);
by (fast_tac (claset() addEs [hypnat_less_irrefl,hypnat_less_asym]) 1);
qed "hypnat_le_imp_less_or_eq";
Goalw [hypnat_le_def] "z<w | z=w ==> z <=(w::hypnat)";
by (cut_facts_tac [hypnat_linear] 1);
by (blast_tac (claset() addDs [hypnat_less_irrefl,hypnat_less_asym]) 1);
qed "hypnat_less_or_eq_imp_le";
Goal "(x <= (y::hypnat)) = (x < y | x=y)";
by (REPEAT(ares_tac [iffI, hypnat_less_or_eq_imp_le,
hypnat_le_imp_less_or_eq] 1));
qed "hypnat_le_less";
(* Axiom 'linorder_linear' of class 'linorder': *)
Goal "(z::hypnat) <= w | w <= z";
by (simp_tac (simpset() addsimps [hypnat_le_less]) 1);
by (cut_facts_tac [hypnat_linear] 1);
by (Blast_tac 1);
qed "hypnat_le_linear";
Goal "w <= (w::hypnat)";
by (simp_tac (simpset() addsimps [hypnat_le_less]) 1);
qed "hypnat_le_refl";
Addsimps [hypnat_le_refl];
Goal "[| i <= j; j <= k |] ==> i <= (k::hypnat)";
by (EVERY1 [dtac hypnat_le_imp_less_or_eq, dtac hypnat_le_imp_less_or_eq,
rtac hypnat_less_or_eq_imp_le,
fast_tac (claset() addIs [hypnat_less_trans])]);
qed "hypnat_le_trans";
Goal "[| z <= w; w <= z |] ==> z = (w::hypnat)";
by (EVERY1 [dtac hypnat_le_imp_less_or_eq, dtac hypnat_le_imp_less_or_eq,
fast_tac (claset() addEs [hypnat_less_irrefl,hypnat_less_asym])]);
qed "hypnat_le_anti_sym";
Goal "[| (0::hypnat) <= x; 0 <= y |] ==> 0 <= x * y";
by (REPEAT(dtac hypnat_le_imp_less_or_eq 1));
by (auto_tac (claset() addIs [hypnat_mult_order, hypnat_less_imp_le],
simpset() addsimps [hypnat_le_refl]));
qed "hypnat_le_mult_order";
Goalw [hypnat_one_def,hypnat_zero_def,hypnat_less_def]
"(0::hypnat) < 1hn";
by (res_inst_tac [("x","%n. 0")] exI 1);
by (res_inst_tac [("x","%n. 1")] exI 1);
by Auto_tac;
qed "hypnat_zero_less_one";
Goal "[| (0::hypnat) <= x; 0 <= y |] ==> 0 <= x + y";
by (REPEAT(dtac hypnat_le_imp_less_or_eq 1));
by (auto_tac (claset() addIs [hypnat_add_order,
hypnat_less_imp_le],simpset() addsimps [hypnat_le_refl]));
qed "hypnat_le_add_order";
Goal "!!(q1::hypnat). q1 <= q2 ==> x + q1 <= x + q2";
by (dtac hypnat_le_imp_less_or_eq 1);
by (Step_tac 1);
by (auto_tac (claset() addSIs [hypnat_le_refl,
hypnat_less_imp_le,hypnat_add_less_mono1],
simpset() addsimps [hypnat_add_commute]));
qed "hypnat_add_le_mono2";
Goal "!!(q1::hypnat). q1 <= q2 ==> q1 + x <= q2 + x";
by (auto_tac (claset() addDs [hypnat_add_le_mono2],
simpset() addsimps [hypnat_add_commute]));
qed "hypnat_add_le_mono1";
Goal "!!k l::hypnat. [|i<=j; k<=l |] ==> i + k <= j + l";
by (etac (hypnat_add_le_mono1 RS hypnat_le_trans) 1);
by (simp_tac (simpset() addsimps [hypnat_add_commute]) 1);
(*j moves to the end because it is free while k, l are bound*)
by (etac hypnat_add_le_mono1 1);
qed "hypnat_add_le_mono";
Goalw [hypnat_zero_def]
"!!x::hypnat. [| (0::hypnat) < z; x < y |] ==> x * z < y * z";
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","y")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","z")] eq_Abs_hypnat 1);
by (auto_tac (claset(),simpset() addsimps
[hypnat_less,hypnat_mult]));
by (Fuf_tac 1);
qed "hypnat_mult_less_mono1";
Goal "!!x::hypnat. [| 0 < z; x < y |] ==> z * x < z * y";
by (auto_tac (claset() addIs [hypnat_mult_less_mono1],
simpset() addsimps [hypnat_mult_commute]));
qed "hypnat_mult_less_mono2";
Addsimps [hypnat_mult_less_mono2,hypnat_mult_less_mono1];
Goal "(x::hypnat) <= n + x";
by (res_inst_tac [("x","n")] hypnat_trichotomyE 1);
by (auto_tac (claset() addDs [(hypnat_less_imp_le RS
hypnat_add_le_mono1)],simpset() addsimps [hypnat_le_refl]));
qed "hypnat_add_self_le";
Addsimps [hypnat_add_self_le];
Goal "(x::hypnat) < x + 1hn";
by (cut_facts_tac [hypnat_zero_less_one
RS hypnat_add_less_mono2] 1);
by Auto_tac;
qed "hypnat_add_one_self_less";
Addsimps [hypnat_add_one_self_less];
Goalw [hypnat_le_def] "~ x + 1hn <= x";
by (Simp_tac 1);
qed "not_hypnat_add_one_le_self";
Addsimps [not_hypnat_add_one_le_self];
Goal "((0::hypnat) < n) = (1hn <= n)";
by (res_inst_tac [("x","n")] hypnat_trichotomyE 1);
by (auto_tac (claset(),simpset() addsimps [hypnat_le_def]));
qed "hypnat_gt_zero_iff";
Addsimps [hypnat_le_add_diff_inverse, hypnat_le_add_diff_inverse2,
hypnat_less_imp_le RS hypnat_le_add_diff_inverse2];
Goal "(0 < n) = (EX m. n = m + 1hn)";
by (Step_tac 1);
by (res_inst_tac [("x","m")] hypnat_trichotomyE 2);
by (rtac hypnat_less_trans 2);
by (res_inst_tac [("x","n - 1hn")] exI 1);
by (auto_tac (claset(),simpset() addsimps
[hypnat_gt_zero_iff,hypnat_add_commute]));
qed "hypnat_gt_zero_iff2";
Goalw [hypnat_zero_def] "(0::hypnat) <= n";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (asm_simp_tac (simpset() addsimps [hypnat_le]) 1);
qed "hypnat_le_zero";
Addsimps [hypnat_le_zero];
(*------------------------------------------------------------------
hypnat_of_nat: properties embedding of naturals in hypernaturals
-----------------------------------------------------------------*)
(** hypnat_of_nat preserves field and order properties **)
Goalw [hypnat_of_nat_def]
"hypnat_of_nat ((z1::nat) + z2) = \
\ hypnat_of_nat z1 + hypnat_of_nat z2";
by (asm_simp_tac (simpset() addsimps [hypnat_add]) 1);
qed "hypnat_of_nat_add";
Goalw [hypnat_of_nat_def]
"hypnat_of_nat ((z1::nat) - z2) = \
\ hypnat_of_nat z1 - hypnat_of_nat z2";
by (asm_simp_tac (simpset() addsimps [hypnat_minus]) 1);
qed "hypnat_of_nat_minus";
Goalw [hypnat_of_nat_def]
"hypnat_of_nat (z1 * z2) = hypnat_of_nat z1 * hypnat_of_nat z2";
by (full_simp_tac (simpset() addsimps [hypnat_mult]) 1);
qed "hypnat_of_nat_mult";
Goalw [hypnat_less_def,hypnat_of_nat_def]
"(z1 < z2) = (hypnat_of_nat z1 < hypnat_of_nat z2)";
by (auto_tac (claset() addSIs [exI] addIs
[FreeUltrafilterNat_all],simpset()));
by (rtac FreeUltrafilterNat_P 1 THEN Fuf_tac 1);
qed "hypnat_of_nat_less_iff";
Addsimps [hypnat_of_nat_less_iff RS sym];
Goalw [hypnat_le_def,le_def]
"(z1 <= z2) = (hypnat_of_nat z1 <= hypnat_of_nat z2)";
by Auto_tac;
qed "hypnat_of_nat_le_iff";
Goalw [hypnat_of_nat_def,hypnat_one_def] "hypnat_of_nat 1 = 1hn";
by (Simp_tac 1);
qed "hypnat_of_nat_one";
Goalw [hypnat_of_nat_def,hypnat_zero_def] "hypnat_of_nat 0 = 0";
by (Simp_tac 1);
qed "hypnat_of_nat_zero";
Goal "(hypnat_of_nat n = 0) = (n = 0)";
by (auto_tac (claset() addIs [FreeUltrafilterNat_P],
simpset() addsimps [hypnat_of_nat_def,
hypnat_zero_def]));
qed "hypnat_of_nat_zero_iff";
Goal "(hypnat_of_nat n ~= 0) = (n ~= 0)";
by (full_simp_tac (simpset() addsimps [hypnat_of_nat_zero_iff]) 1);
qed "hypnat_of_nat_not_zero_iff";
Goalw [hypnat_of_nat_def,hypnat_one_def]
"hypnat_of_nat (Suc n) = hypnat_of_nat n + 1hn";
by (auto_tac (claset(),simpset() addsimps [hypnat_add]));
qed "hypnat_of_nat_Suc";
(*---------------------------------------------------------------------------------
Existence of infinite hypernatural number
---------------------------------------------------------------------------------*)
Goal "hypnatrel^^{%n::nat. n} : hypnat";
by Auto_tac;
qed "hypnat_omega";
Goalw [hypnat_omega_def] "Rep_hypnat(whn) : hypnat";
by (rtac Rep_hypnat 1);
qed "Rep_hypnat_omega";
(* See Hyper.thy for similar argument*)
(* existence of infinite number not corresponding to any natural number *)
(* use assumption that member FreeUltrafilterNat is not finite *)
(* a few lemmas first *)
Goalw [hypnat_omega_def,hypnat_of_nat_def]
"~ (EX x. hypnat_of_nat x = whn)";
by (auto_tac (claset() addDs [FreeUltrafilterNat_not_finite],
simpset()));
qed "not_ex_hypnat_of_nat_eq_omega";
Goal "hypnat_of_nat x ~= whn";
by (cut_facts_tac [not_ex_hypnat_of_nat_eq_omega] 1);
by Auto_tac;
qed "hypnat_of_nat_not_eq_omega";
Addsimps [hypnat_of_nat_not_eq_omega RS not_sym];
(*-----------------------------------------------------------
Properties of the set SNat of embedded natural numbers
(cf. set SReal in NSA.thy/NSA.ML)
----------------------------------------------------------*)
(* Infinite hypernatural not in embedded SNat *)
Goalw [SHNat_def] "whn ~: SNat";
by Auto_tac;
qed "SHNAT_omega_not_mem";
Addsimps [SHNAT_omega_not_mem];
(*-----------------------------------------------------------------------
Closure laws for members of (embedded) set standard naturals SNat
-----------------------------------------------------------------------*)
Goalw [SHNat_def]
"!!x::hypnat. [| x: SNat; y: SNat |] ==> x + y: SNat";
by (Step_tac 1);
by (res_inst_tac [("x","N + Na")] exI 1);
by (simp_tac (simpset() addsimps [hypnat_of_nat_add]) 1);
qed "SHNat_add";
Goalw [SHNat_def]
"!!x::hypnat. [| x: SNat; y: SNat |] ==> x - y: SNat";
by (Step_tac 1);
by (res_inst_tac [("x","N - Na")] exI 1);
by (simp_tac (simpset() addsimps [hypnat_of_nat_minus]) 1);
qed "SHNat_minus";
Goalw [SHNat_def]
"!!x::hypnat. [| x: SNat; y: SNat |] ==> x * y: SNat";
by (Step_tac 1);
by (res_inst_tac [("x","N * Na")] exI 1);
by (simp_tac (simpset() addsimps [hypnat_of_nat_mult]) 1);
qed "SHNat_mult";
Goal"!!x::hypnat. [| x + y : SNat; y: SNat |] ==> x: SNat";
by (dres_inst_tac [("x","x+y")] SHNat_minus 1);
by Auto_tac;
qed "SHNat_add_cancel";
Goalw [SHNat_def] "hypnat_of_nat x : SNat";
by (Blast_tac 1);
qed "SHNat_hypnat_of_nat";
Addsimps [SHNat_hypnat_of_nat];
Goal "hypnat_of_nat 1 : SNat";
by (Simp_tac 1);
qed "SHNat_hypnat_of_nat_one";
Goal "hypnat_of_nat 0 : SNat";
by (Simp_tac 1);
qed "SHNat_hypnat_of_nat_zero";
Goal "1hn : SNat";
by (simp_tac (simpset() addsimps [SHNat_hypnat_of_nat_one,
hypnat_of_nat_one RS sym]) 1);
qed "SHNat_one";
Goal "(0::hypnat) : SNat";
by (simp_tac (simpset() addsimps [SHNat_hypnat_of_nat_zero,
hypnat_of_nat_zero RS sym]) 1);
qed "SHNat_zero";
Addsimps [SHNat_hypnat_of_nat_one,SHNat_hypnat_of_nat_zero,
SHNat_one,SHNat_zero];
Goal "1hn + 1hn : SNat";
by (rtac ([SHNat_one,SHNat_one] MRS SHNat_add) 1);
qed "SHNat_two";
Goalw [SHNat_def] "{x. hypnat_of_nat x : SNat} = (UNIV::nat set)";
by Auto_tac;
qed "SHNat_UNIV_nat";
Goalw [SHNat_def] "(x: SNat) = (EX y. x = hypnat_of_nat y)";
by Auto_tac;
qed "SHNat_iff";
Goalw [SHNat_def] "hypnat_of_nat ``(UNIV::nat set) = SNat";
by Auto_tac;
qed "hypnat_of_nat_image";
Goalw [SHNat_def] "inv hypnat_of_nat ``SNat = (UNIV::nat set)";
by Auto_tac;
by (rtac (inj_hypnat_of_nat RS inv_f_f RS subst) 1);
by (Blast_tac 1);
qed "inv_hypnat_of_nat_image";
Goalw [SHNat_def]
"[| EX x. x: P; P <= SNat |] ==> EX Q. P = hypnat_of_nat `` Q";
by (Best_tac 1);
qed "SHNat_hypnat_of_nat_image";
Goalw [SHNat_def]
"SNat = hypnat_of_nat `` (UNIV::nat set)";
by Auto_tac;
qed "SHNat_hypnat_of_nat_iff";
Goalw [SHNat_def] "SNat <= (UNIV::hypnat set)";
by Auto_tac;
qed "SHNat_subset_UNIV";
Goal "{n. n <= Suc m} = {n. n <= m} Un {n. n = Suc m}";
by (auto_tac (claset(),simpset() addsimps [le_Suc_eq]));
qed "leSuc_Un_eq";
Goal "finite {n::nat. n <= m}";
by (nat_ind_tac "m" 1);
by (auto_tac (claset(),simpset() addsimps [leSuc_Un_eq]));
qed "finite_nat_le_segment";
Goal "{n::nat. m < n} : FreeUltrafilterNat";
by (cut_inst_tac [("m2","m")] (finite_nat_le_segment RS
FreeUltrafilterNat_finite RS FreeUltrafilterNat_Compl_mem) 1);
by (Fuf_tac 1);
qed "lemma_unbounded_set";
Addsimps [lemma_unbounded_set];
Goalw [SHNat_def,hypnat_of_nat_def, hypnat_less_def,hypnat_omega_def]
"ALL n: SNat. n < whn";
by (Clarify_tac 1);
by (auto_tac (claset() addSIs [exI],simpset()));
qed "hypnat_omega_gt_SHNat";
Goal "hypnat_of_nat n < whn";
by (cut_facts_tac [hypnat_omega_gt_SHNat] 1);
by (dres_inst_tac [("x","hypnat_of_nat n")] bspec 1);
by Auto_tac;
qed "hypnat_of_nat_less_whn";
Addsimps [hypnat_of_nat_less_whn];
Goal "hypnat_of_nat n <= whn";
by (rtac (hypnat_of_nat_less_whn RS hypnat_less_imp_le) 1);
qed "hypnat_of_nat_le_whn";
Addsimps [hypnat_of_nat_le_whn];
Goal "0 < whn";
by (rtac (hypnat_omega_gt_SHNat RS ballE) 1);
by Auto_tac;
qed "hypnat_zero_less_hypnat_omega";
Addsimps [hypnat_zero_less_hypnat_omega];
Goal "1hn < whn";
by (rtac (hypnat_omega_gt_SHNat RS ballE) 1);
by Auto_tac;
qed "hypnat_one_less_hypnat_omega";
Addsimps [hypnat_one_less_hypnat_omega];
(*--------------------------------------------------------------------------
Theorems about infinite hypernatural numbers -- HNatInfinite
-------------------------------------------------------------------------*)
Goalw [HNatInfinite_def,SHNat_def] "whn : HNatInfinite";
by Auto_tac;
qed "HNatInfinite_whn";
Addsimps [HNatInfinite_whn];
Goalw [HNatInfinite_def] "x: SNat ==> x ~: HNatInfinite";
by (Simp_tac 1);
qed "SHNat_not_HNatInfinite";
Goalw [HNatInfinite_def] "x ~: HNatInfinite ==> x: SNat";
by (Asm_full_simp_tac 1);
qed "not_HNatInfinite_SHNat";
Goalw [HNatInfinite_def] "x ~: SNat ==> x: HNatInfinite";
by (Simp_tac 1);
qed "not_SHNat_HNatInfinite";
Goalw [HNatInfinite_def] "x: HNatInfinite ==> x ~: SNat";
by (Asm_full_simp_tac 1);
qed "HNatInfinite_not_SHNat";
Goal "(x: SNat) = (x ~: HNatInfinite)";
by (blast_tac (claset() addSIs [SHNat_not_HNatInfinite,
not_HNatInfinite_SHNat]) 1);
qed "SHNat_not_HNatInfinite_iff";
Goal "(x ~: SNat) = (x: HNatInfinite)";
by (blast_tac (claset() addSIs [not_SHNat_HNatInfinite,
HNatInfinite_not_SHNat]) 1);
qed "not_SHNat_HNatInfinite_iff";
Goal "x : SNat | x : HNatInfinite";
by (simp_tac (simpset() addsimps [SHNat_not_HNatInfinite_iff]) 1);
qed "SHNat_HNatInfinite_disj";
(*-------------------------------------------------------------------
Proof of alternative definition for set of Infinite hypernatural
numbers --- HNatInfinite = {N. ALL n: SNat. n < N}
-------------------------------------------------------------------*)
Goal "!!N (xa::nat=>nat). \
\ (ALL N. {n. xa n ~= N} : FreeUltrafilterNat) ==> \
\ ({n. N < xa n} : FreeUltrafilterNat)";
by (nat_ind_tac "N" 1);
by (dres_inst_tac [("x","0")] spec 1);
by (rtac ccontr 1 THEN dtac FreeUltrafilterNat_Compl_mem 1
THEN dtac FreeUltrafilterNat_Int 1 THEN assume_tac 1);
by (Asm_full_simp_tac 1);
by (dres_inst_tac [("x","Suc N")] spec 1);
by (fuf_tac (claset() addSDs [Suc_leI],simpset() addsimps
[le_eq_less_or_eq]) 1);
qed "HNatInfinite_FreeUltrafilterNat_lemma";
(*** alternative definition ***)
Goalw [HNatInfinite_def,SHNat_def,hypnat_of_nat_def]
"HNatInfinite = {N. ALL n:SNat. n < N}";
by (Step_tac 1);
by (dres_inst_tac [("x","Abs_hypnat \
\ (hypnatrel ^^ {%n. N})")] bspec 2);
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (auto_tac (claset(),simpset() addsimps [hypnat_less_iff]));
by (auto_tac (claset() addSIs [exI] addEs
[HNatInfinite_FreeUltrafilterNat_lemma],
simpset() addsimps [FreeUltrafilterNat_Compl_iff1,
CLAIM "- {n. xa n = N} = {n. xa n ~= N}"]));
qed "HNatInfinite_iff";
(*--------------------------------------------------------------------
Alternative definition for HNatInfinite using Free ultrafilter
--------------------------------------------------------------------*)
Goal "x : HNatInfinite ==> EX X: Rep_hypnat x. \
\ ALL u. {n. u < X n}: FreeUltrafilterNat";
by (auto_tac (claset(),simpset() addsimps [hypnat_less_def,
HNatInfinite_iff,SHNat_iff,hypnat_of_nat_def]));
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1);
by (EVERY[Auto_tac, rtac bexI 1,
rtac lemma_hypnatrel_refl 2, Step_tac 1]);
by (dres_inst_tac [("x","hypnat_of_nat u")] bspec 1);
by (Simp_tac 1);
by (auto_tac (claset(),
simpset() addsimps [hypnat_of_nat_def]));
by (Fuf_tac 1);
qed "HNatInfinite_FreeUltrafilterNat";
Goal "EX X: Rep_hypnat x. ALL u. {n. u < X n}: FreeUltrafilterNat \
\ ==> x: HNatInfinite";
by (auto_tac (claset(),simpset() addsimps [hypnat_less_def,
HNatInfinite_iff,SHNat_iff,hypnat_of_nat_def]));
by (rtac exI 1 THEN Auto_tac);
qed "FreeUltrafilterNat_HNatInfinite";
Goal "(x : HNatInfinite) = (EX X: Rep_hypnat x. \
\ ALL u. {n. u < X n}: FreeUltrafilterNat)";
by (blast_tac (claset() addIs [HNatInfinite_FreeUltrafilterNat,
FreeUltrafilterNat_HNatInfinite]) 1);
qed "HNatInfinite_FreeUltrafilterNat_iff";
Goal "x : HNatInfinite ==> 1hn < x";
by (auto_tac (claset(),simpset() addsimps [HNatInfinite_iff]));
qed "HNatInfinite_gt_one";
Addsimps [HNatInfinite_gt_one];
Goal "0 ~: HNatInfinite";
by (auto_tac (claset(),simpset()
addsimps [HNatInfinite_iff]));
by (dres_inst_tac [("a","1hn")] equals0D 1);
by (Asm_full_simp_tac 1);
qed "zero_not_mem_HNatInfinite";
Addsimps [zero_not_mem_HNatInfinite];
Goal "x : HNatInfinite ==> x ~= 0";
by Auto_tac;
qed "HNatInfinite_not_eq_zero";
Goal "x : HNatInfinite ==> 1hn <= x";
by (blast_tac (claset() addIs [hypnat_less_imp_le,
HNatInfinite_gt_one]) 1);
qed "HNatInfinite_ge_one";
Addsimps [HNatInfinite_ge_one];
(*--------------------------------------------------
Closure Rules
--------------------------------------------------*)
Goal "[| x: HNatInfinite; y: HNatInfinite |] \
\ ==> x + y: HNatInfinite";
by (auto_tac (claset(),simpset() addsimps [HNatInfinite_iff]));
by (dtac bspec 1 THEN assume_tac 1);
by (dtac (SHNat_zero RSN (2,bspec)) 1);
by (dtac hypnat_add_less_mono 1 THEN assume_tac 1);
by (Asm_full_simp_tac 1);
qed "HNatInfinite_add";
Goal "[| x: HNatInfinite; y: SNat |] ==> x + y: HNatInfinite";
by (rtac ccontr 1 THEN dtac not_HNatInfinite_SHNat 1);
by (dres_inst_tac [("x","x + y")] SHNat_minus 1);
by (auto_tac (claset(),simpset() addsimps
[SHNat_not_HNatInfinite_iff]));
qed "HNatInfinite_SHNat_add";
Goal "[| x: HNatInfinite; y: SNat |] ==> x - y: HNatInfinite";
by (rtac ccontr 1 THEN dtac not_HNatInfinite_SHNat 1);
by (dres_inst_tac [("x","x - y")] SHNat_add 1);
by (subgoal_tac "y <= x" 2);
by (auto_tac (claset() addSDs [hypnat_le_add_diff_inverse2],
simpset() addsimps [not_SHNat_HNatInfinite_iff RS sym]));
by (auto_tac (claset() addSIs [hypnat_less_imp_le],
simpset() addsimps [not_SHNat_HNatInfinite_iff,HNatInfinite_iff]));
qed "HNatInfinite_SHNat_diff";
Goal "x: HNatInfinite ==> x + 1hn: HNatInfinite";
by (auto_tac (claset() addIs [HNatInfinite_SHNat_add],
simpset()));
qed "HNatInfinite_add_one";
Goal "x: HNatInfinite ==> x - 1hn: HNatInfinite";
by (rtac ccontr 1 THEN dtac not_HNatInfinite_SHNat 1);
by (dres_inst_tac [("x","x - 1hn"),("y","1hn")] SHNat_add 1);
by (auto_tac (claset(),simpset() addsimps
[not_SHNat_HNatInfinite_iff RS sym]));
qed "HNatInfinite_minus_one";
Goal "x : HNatInfinite ==> EX y. x = y + 1hn";
by (res_inst_tac [("x","x - 1hn")] exI 1);
by Auto_tac;
qed "HNatInfinite_is_Suc";
(*---------------------------------------------------------------
HNat : the hypernaturals embedded in the hyperreals
Obtained using the NS extension of the naturals
--------------------------------------------------------------*)
Goalw [HNat_def,starset_def, hypreal_of_nat_def,hypreal_of_real_def]
"hypreal_of_nat N : HNat";
by Auto_tac;
by (Ultra_tac 1);
by (res_inst_tac [("x","N")] exI 1);
by Auto_tac;
qed "HNat_hypreal_of_nat";
Addsimps [HNat_hypreal_of_nat];
Goalw [HNat_def,starset_def]
"[| x: HNat; y: HNat |] ==> x + y: HNat";
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
by (auto_tac (claset() addSDs [bspec] addIs [lemma_hyprel_refl],
simpset() addsimps [hypreal_add]));
by (Ultra_tac 1);
by (dres_inst_tac [("t","Y xb")] sym 1);
by (auto_tac (claset(),simpset() addsimps [real_of_nat_add RS sym]));
qed "HNat_add";
Goalw [HNat_def,starset_def]
"[| x: HNat; y: HNat |] ==> x * y: HNat";
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
by (auto_tac (claset() addSDs [bspec] addIs [lemma_hyprel_refl],
simpset() addsimps [hypreal_mult]));
by (Ultra_tac 1);
by (dres_inst_tac [("t","Y xb")] sym 1);
by (auto_tac (claset(),simpset() addsimps [real_of_nat_mult RS sym]));
qed "HNat_mult";
(*---------------------------------------------------------------
Embedding of the hypernaturals into the hyperreal
--------------------------------------------------------------*)
Goal "(Ya : hyprel ^^{%n. f(n)}) = \
\ ({n. f n = Ya n} : FreeUltrafilterNat)";
by Auto_tac;
qed "lemma_hyprel_FUFN";
Goalw [hypreal_of_hypnat_def]
"hypreal_of_hypnat (Abs_hypnat(hypnatrel^^{%n. X n})) = \
\ Abs_hypreal(hyprel ^^ {%n. real_of_nat (X n)})";
by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);
by (auto_tac (claset() addEs [FreeUltrafilterNat_Int RS
FreeUltrafilterNat_subset],simpset() addsimps
[lemma_hyprel_FUFN]));
qed "hypreal_of_hypnat";
Goal "inj(hypreal_of_hypnat)";
by (rtac injI 1);
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","y")] eq_Abs_hypnat 1);
by (auto_tac (claset(), simpset() addsimps [hypreal_of_hypnat]));
qed "inj_hypreal_of_hypnat";
Goal "(hypreal_of_hypnat n = hypreal_of_hypnat m) = (n = m)";
by (auto_tac (claset(),simpset() addsimps [inj_hypreal_of_hypnat RS injD]));
qed "hypreal_of_hypnat_eq_cancel";
Addsimps [hypreal_of_hypnat_eq_cancel];
Goal "(hypnat_of_nat n = hypnat_of_nat m) = (n = m)";
by (auto_tac (claset() addDs [inj_hypnat_of_nat RS injD],
simpset()));
qed "hypnat_of_nat_eq_cancel";
Addsimps [hypnat_of_nat_eq_cancel];
Goalw [hypnat_zero_def]
"hypreal_of_hypnat 0 = #0";
by (simp_tac (HOL_ss addsimps
[zero_eq_numeral_0 RS sym, hypreal_zero_def]) 1);
by (simp_tac (simpset() addsimps [hypreal_of_hypnat, real_of_nat_zero]) 1);
qed "hypreal_of_hypnat_zero";
Goalw [hypnat_one_def]
"hypreal_of_hypnat 1hn = #1";
by (simp_tac (HOL_ss addsimps
[one_eq_numeral_1 RS sym, hypreal_one_def]) 1);
by (simp_tac (simpset() addsimps [hypreal_of_hypnat, real_of_nat_one]) 1);
qed "hypreal_of_hypnat_one";
Goal "hypreal_of_hypnat n1 + hypreal_of_hypnat n2 = hypreal_of_hypnat (n1 + n2)";
by (res_inst_tac [("z","n1")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","n2")] eq_Abs_hypnat 1);
by (asm_simp_tac (simpset() addsimps [hypreal_of_hypnat,
hypreal_add,hypnat_add,real_of_nat_add]) 1);
qed "hypreal_of_hypnat_add";
Goal "hypreal_of_hypnat n1 * hypreal_of_hypnat n2 = hypreal_of_hypnat (n1 * n2)";
by (res_inst_tac [("z","n1")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","n2")] eq_Abs_hypnat 1);
by (asm_simp_tac (simpset() addsimps [hypreal_of_hypnat,
hypreal_mult,hypnat_mult,real_of_nat_mult]) 1);
qed "hypreal_of_hypnat_mult";
Goal "(hypreal_of_hypnat n < hypreal_of_hypnat m) = (n < m)";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1);
by (asm_simp_tac (simpset() addsimps
[hypreal_of_hypnat,hypreal_less,hypnat_less]) 1);
qed "hypreal_of_hypnat_less_iff";
Addsimps [hypreal_of_hypnat_less_iff];
Goal "(hypreal_of_hypnat N = #0) = (N = 0)";
by (simp_tac (simpset() addsimps [hypreal_of_hypnat_zero RS sym]) 1);
qed "hypreal_of_hypnat_eq_zero_iff";
Addsimps [hypreal_of_hypnat_eq_zero_iff];
Goal "ALL n. N <= n ==> N = (0::hypnat)";
by (dres_inst_tac [("x","0")] spec 1);
by (res_inst_tac [("z","N")] eq_Abs_hypnat 1);
by (auto_tac (claset(),simpset() addsimps [hypnat_le,hypnat_zero_def]));
qed "hypnat_eq_zero";
Addsimps [hypnat_eq_zero];
Goal "~ (ALL n. n = (0::hypnat))";
by Auto_tac;
by (res_inst_tac [("x","1hn")] exI 1);
by (Simp_tac 1);
qed "hypnat_not_all_eq_zero";
Addsimps [hypnat_not_all_eq_zero];
Goal "n ~= 0 ==> (n <= 1hn) = (n = 1hn)";
by (auto_tac (claset(), simpset() addsimps [hypnat_le_less]));
qed "hypnat_le_one_eq_one";
Addsimps [hypnat_le_one_eq_one];