--- a/src/HOL/Real/Hyperreal/Lim.ML Sat Dec 30 22:03:46 2000 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2353 +0,0 @@
-(* Title : Lim.ML
- Author : Jacques D. Fleuriot
- Copyright : 1998 University of Cambridge
- Description : Theory of limits, continuity and
- differentiation of real=>real functions
-*)
-
-
-fun ARITH_PROVE str = prove_goal thy str
- (fn prems => [cut_facts_tac prems 1,arith_tac 1]);
-
-
-(*---------------------------------------------------------------
- Theory of limits, continuity and differentiation of
- real=>real functions
- ----------------------------------------------------------------*)
-
-Goalw [LIM_def] "(%x. k) -- x --> k";
-by Auto_tac;
-qed "LIM_const";
-Addsimps [LIM_const];
-
-(***-----------------------------------------------------------***)
-(*** Some Purely Standard Proofs - Can be used for comparison ***)
-(***-----------------------------------------------------------***)
-
-(*---------------
- LIM_add
- ---------------*)
-Goalw [LIM_def]
- "[| f -- x --> l; g -- x --> m |] \
-\ ==> (%x. f(x) + g(x)) -- x --> (l + m)";
-by (Step_tac 1);
-by (REPEAT(dres_inst_tac [("x","r/#2")] spec 1));
-by (dtac (rename_numerals (real_zero_less_two RS real_inverse_gt_zero
- RSN (2,real_mult_less_mono2))) 1);
-by (Asm_full_simp_tac 1);
-by (Clarify_tac 1);
-by (res_inst_tac [("R1.0","s"),("R2.0","sa")]
- real_linear_less2 1);
-by (res_inst_tac [("x","s")] exI 1);
-by (res_inst_tac [("x","sa")] exI 2);
-by (res_inst_tac [("x","sa")] exI 3);
-by (Step_tac 1);
-by (REPEAT(dres_inst_tac [("x","xa")] spec 1)
- THEN step_tac (claset() addSEs [real_less_trans]) 1);
-by (REPEAT(dres_inst_tac [("x","xa")] spec 2)
- THEN step_tac (claset() addSEs [real_less_trans]) 2);
-by (REPEAT(dres_inst_tac [("x","xa")] spec 3)
- THEN step_tac (claset() addSEs [real_less_trans]) 3);
-by (ALLGOALS(rtac (abs_sum_triangle_ineq RS real_le_less_trans)));
-by (ALLGOALS(rtac (real_sum_of_halves RS subst)));
-by (auto_tac (claset() addIs [real_add_less_mono],simpset()));
-qed "LIM_add";
-
-Goalw [LIM_def] "f -- a --> L ==> (%x. -f(x)) -- a --> -L";
-by (full_simp_tac (simpset() addsimps [real_minus_add_distrib RS sym]
- delsimps [real_minus_add_distrib, real_minus_minus]) 1);
-qed "LIM_minus";
-
-(*----------------------------------------------
- LIM_add_minus
- ----------------------------------------------*)
-Goal "[| f -- x --> l; g -- x --> m |] \
-\ ==> (%x. f(x) + -g(x)) -- x --> (l + -m)";
-by (blast_tac (claset() addDs [LIM_add,LIM_minus]) 1);
-qed "LIM_add_minus";
-
-(*----------------------------------------------
- LIM_zero
- ----------------------------------------------*)
-Goal "f -- a --> l ==> (%x. f(x) + -l) -- a --> #0";
-by (res_inst_tac [("z1","l")] (rename_numerals (real_add_minus RS subst)) 1);
-by (rtac LIM_add_minus 1 THEN Auto_tac);
-qed "LIM_zero";
-
-(*--------------------------
- Limit not zero
- --------------------------*)
-Goalw [LIM_def] "k ~= #0 ==> ~ ((%x. k) -- x --> #0)";
-by (res_inst_tac [("R1.0","k"),("R2.0","#0")] real_linear_less2 1);
-by (auto_tac (claset(), simpset() addsimps [real_abs_def]));
-by (res_inst_tac [("x","-k")] exI 1);
-by (res_inst_tac [("x","k")] exI 2);
-by Auto_tac;
-by (ALLGOALS(dres_inst_tac [("y","s")] real_dense));
-by Safe_tac;
-by (ALLGOALS(res_inst_tac [("x","r + x")] exI));
-by Auto_tac;
-qed "LIM_not_zero";
-
-(* [| k ~= #0; (%x. k) -- x --> #0 |] ==> R *)
-bind_thm("LIM_not_zeroE", LIM_not_zero RS notE);
-
-Goal "(%x. k) -- x --> L ==> k = L";
-by (rtac ccontr 1);
-by (dtac LIM_zero 1);
-by (rtac LIM_not_zeroE 1 THEN assume_tac 2);
-by (arith_tac 1);
-qed "LIM_const_eq";
-
-(*------------------------
- Limit is Unique
- ------------------------*)
-Goal "[| f -- x --> L; f -- x --> M |] ==> L = M";
-by (dtac LIM_minus 1);
-by (dtac LIM_add 1 THEN assume_tac 1);
-by (auto_tac (claset() addSDs [LIM_const_eq RS sym], simpset()));
-qed "LIM_unique";
-
-(*-------------
- LIM_mult_zero
- -------------*)
-Goalw [LIM_def] "[| f -- x --> #0; g -- x --> #0 |] \
-\ ==> (%x. f(x)*g(x)) -- x --> #0";
-by (Step_tac 1);
-by (dres_inst_tac [("x","#1")] spec 1);
-by (dres_inst_tac [("x","r")] spec 1);
-by (cut_facts_tac [real_zero_less_one] 1);
-by (asm_full_simp_tac (simpset() addsimps
- [abs_mult]) 1);
-by (Clarify_tac 1);
-by (res_inst_tac [("R1.0","s"),("R2.0","sa")]
- real_linear_less2 1);
-by (res_inst_tac [("x","s")] exI 1);
-by (res_inst_tac [("x","sa")] exI 2);
-by (res_inst_tac [("x","sa")] exI 3);
-by (Step_tac 1);
-by (REPEAT(dres_inst_tac [("x","xa")] spec 1)
- THEN step_tac (claset() addSEs [real_less_trans]) 1);
-by (REPEAT(dres_inst_tac [("x","xa")] spec 2)
- THEN step_tac (claset() addSEs [real_less_trans]) 2);
-by (REPEAT(dres_inst_tac [("x","xa")] spec 3)
- THEN step_tac (claset() addSEs [real_less_trans]) 3);
-by (ALLGOALS(res_inst_tac [("t","r")] (real_mult_1 RS subst)));
-by (ALLGOALS(rtac abs_mult_less2));
-by Auto_tac;
-qed "LIM_mult_zero";
-
-Goalw [LIM_def] "(%x. x) -- a --> a";
-by Auto_tac;
-qed "LIM_self";
-
-(*--------------------------------------------------------------
- Limits are equal for functions equal except at limit point
- --------------------------------------------------------------*)
-Goalw [LIM_def]
- "[| ALL x. x ~= a --> (f x = g x) |] \
-\ ==> (f -- a --> l) = (g -- a --> l)";
-by (auto_tac (claset(), simpset() addsimps [real_add_minus_iff]));
-qed "LIM_equal";
-
-Goal "[| (%x. f(x) + -g(x)) -- a --> #0; \
-\ g -- a --> l |] \
-\ ==> f -- a --> l";
-by (dtac LIM_add 1 THEN assume_tac 1);
-by (auto_tac (claset(), simpset() addsimps [real_add_assoc]));
-qed "LIM_trans";
-
-(***-------------------------------------------------------------***)
-(*** End of Purely Standard Proofs ***)
-(***-------------------------------------------------------------***)
-(*--------------------------------------------------------------
- Standard and NS definitions of Limit
- --------------------------------------------------------------*)
-Goalw [LIM_def,NSLIM_def,inf_close_def]
- "f -- x --> L ==> f -- x --NS> L";
-by (asm_full_simp_tac
- (simpset() addsimps [Infinitesimal_FreeUltrafilterNat_iff]) 1);
-by (Step_tac 1);
-by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),
- simpset() addsimps [real_add_minus_iff, starfun, hypreal_minus,
- hypreal_of_real_def, hypreal_add]));
-by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2 THEN Step_tac 1);
-by (dres_inst_tac [("x","u")] spec 1 THEN Clarify_tac 1);
-by (dres_inst_tac [("x","s")] spec 1 THEN Clarify_tac 1);
-by (subgoal_tac "ALL n::nat. (xa n) ~= x & \
-\ abs ((xa n) + - x) < s --> abs (f (xa n) + - L) < u" 1);
-by (Blast_tac 2);
-by (dtac FreeUltrafilterNat_all 1);
-by (Ultra_tac 1);
-qed "LIM_NSLIM";
-
-(*---------------------------------------------------------------------
- Limit: NS definition ==> standard definition
- ---------------------------------------------------------------------*)
-
-Goal "ALL s. #0 < s --> (EX xa. xa ~= x & \
-\ abs (xa + - x) < s & r <= abs (f xa + -L)) \
-\ ==> ALL n::nat. EX xa. xa ~= x & \
-\ abs(xa + -x) < inverse(real_of_posnat n) & r <= abs(f xa + -L)";
-by (Step_tac 1);
-by (cut_inst_tac [("n1","n")]
- (real_of_posnat_gt_zero RS real_inverse_gt_zero) 1);
-by Auto_tac;
-val lemma_LIM = result();
-
-Goal "ALL s. #0 < s --> (EX xa. xa ~= x & \
-\ abs (xa + - x) < s & r <= abs (f xa + -L)) \
-\ ==> EX X. ALL n::nat. X n ~= x & \
-\ abs(X n + -x) < inverse(real_of_posnat n) & r <= abs(f (X n) + -L)";
-by (dtac lemma_LIM 1);
-by (dtac choice 1);
-by (Blast_tac 1);
-val lemma_skolemize_LIM2 = result();
-
-Goal "ALL n. X n ~= x & \
-\ abs (X n + - x) < inverse (real_of_posnat n) & \
-\ r <= abs (f (X n) + - L) ==> \
-\ ALL n. abs (X n + - x) < inverse (real_of_posnat n)";
-by (Auto_tac );
-val lemma_simp = result();
-
-(*-------------------
- NSLIM => LIM
- -------------------*)
-
-Goalw [LIM_def,NSLIM_def,inf_close_def]
- "f -- x --NS> L ==> f -- x --> L";
-by (asm_full_simp_tac
- (simpset() addsimps [Infinitesimal_FreeUltrafilterNat_iff]) 1);
-by (EVERY1[Step_tac, rtac ccontr, Asm_full_simp_tac]);
-by (fold_tac [real_le_def]);
-by (dtac lemma_skolemize_LIM2 1);
-by (Step_tac 1);
-by (dres_inst_tac [("x","Abs_hypreal(hyprel^^{X})")] spec 1);
-by (asm_full_simp_tac
- (simpset() addsimps [starfun, hypreal_minus,
- hypreal_of_real_def,hypreal_add]) 1);
-by (Step_tac 1);
-by (dtac (lemma_simp RS real_seq_to_hypreal_Infinitesimal) 1);
-by (asm_full_simp_tac
- (simpset() addsimps
- [Infinitesimal_FreeUltrafilterNat_iff,hypreal_of_real_def,
- hypreal_minus, hypreal_add]) 1);
-by (Blast_tac 1);
-by (rotate_tac 2 1);
-by (dres_inst_tac [("x","r")] spec 1);
-by (Clarify_tac 1);
-by (dtac FreeUltrafilterNat_all 1);
-by (Ultra_tac 1);
-qed "NSLIM_LIM";
-
-
-(**** Key result ****)
-Goal "(f -- x --> L) = (f -- x --NS> L)";
-by (blast_tac (claset() addIs [LIM_NSLIM,NSLIM_LIM]) 1);
-qed "LIM_NSLIM_iff";
-
-(*-------------------------------------------------------------------*)
-(* Proving properties of limits using nonstandard definition and *)
-(* hence, the properties hold for standard limits as well *)
-(*-------------------------------------------------------------------*)
-(*------------------------------------------------
- NSLIM_mult and hence (trivially) LIM_mult
- ------------------------------------------------*)
-
-Goalw [NSLIM_def]
- "[| f -- x --NS> l; g -- x --NS> m |] \
-\ ==> (%x. f(x) * g(x)) -- x --NS> (l * m)";
-by (auto_tac (claset() addSIs [inf_close_mult_HFinite], simpset()));
-qed "NSLIM_mult";
-
-Goal "[| f -- x --> l; g -- x --> m |] \
-\ ==> (%x. f(x) * g(x)) -- x --> (l * m)";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, NSLIM_mult]) 1);
-qed "LIM_mult2";
-
-(*----------------------------------------------
- NSLIM_add and hence (trivially) LIM_add
- Note the much shorter proof
- ----------------------------------------------*)
-Goalw [NSLIM_def]
- "[| f -- x --NS> l; g -- x --NS> m |] \
-\ ==> (%x. f(x) + g(x)) -- x --NS> (l + m)";
-by (auto_tac (claset() addSIs [inf_close_add], simpset()));
-qed "NSLIM_add";
-
-Goal "[| f -- x --> l; g -- x --> m |] \
-\ ==> (%x. f(x) + g(x)) -- x --> (l + m)";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, NSLIM_add]) 1);
-qed "LIM_add2";
-
-(*----------------------------------------------
- NSLIM_const
- ----------------------------------------------*)
-Goalw [NSLIM_def] "(%x. k) -- x --NS> k";
-by Auto_tac;
-qed "NSLIM_const";
-
-Addsimps [NSLIM_const];
-
-Goal "(%x. k) -- x --> k";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff]) 1);
-qed "LIM_const2";
-
-(*----------------------------------------------
- NSLIM_minus
- ----------------------------------------------*)
-Goalw [NSLIM_def]
- "f -- a --NS> L ==> (%x. -f(x)) -- a --NS> -L";
-by Auto_tac;
-qed "NSLIM_minus";
-
-Goal "f -- a --> L ==> (%x. -f(x)) -- a --> -L";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, NSLIM_minus]) 1);
-qed "LIM_minus2";
-
-(*----------------------------------------------
- NSLIM_add_minus
- ----------------------------------------------*)
-Goal "[| f -- x --NS> l; g -- x --NS> m |] \
-\ ==> (%x. f(x) + -g(x)) -- x --NS> (l + -m)";
-by (blast_tac (claset() addDs [NSLIM_add,NSLIM_minus]) 1);
-qed "NSLIM_add_minus";
-
-Goal "[| f -- x --> l; g -- x --> m |] \
-\ ==> (%x. f(x) + -g(x)) -- x --> (l + -m)";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff,
- NSLIM_add_minus]) 1);
-qed "LIM_add_minus2";
-
-(*-----------------------------
- NSLIM_inverse
- -----------------------------*)
-Goalw [NSLIM_def]
- "[| f -- a --NS> L; L ~= #0 |] \
-\ ==> (%x. inverse(f(x))) -- a --NS> (inverse L)";
-by (Clarify_tac 1);
-by (dtac spec 1);
-by (auto_tac (claset(),
- simpset() addsimps [hypreal_of_real_inf_close_inverse]));
-qed "NSLIM_inverse";
-
-Goal "[| f -- a --> L; \
-\ L ~= #0 |] ==> (%x. inverse(f(x))) -- a --> (inverse L)";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, NSLIM_inverse]) 1);
-qed "LIM_inverse";
-
-(*------------------------------
- NSLIM_zero
- ------------------------------*)
-Goal "f -- a --NS> l ==> (%x. f(x) + -l) -- a --NS> #0";
-by (res_inst_tac [("z1","l")] (rename_numerals (real_add_minus RS subst)) 1);
-by (rtac NSLIM_add_minus 1 THEN Auto_tac);
-qed "NSLIM_zero";
-
-Goal "f -- a --> l ==> (%x. f(x) + -l) -- a --> #0";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, NSLIM_zero]) 1);
-qed "LIM_zero2";
-
-Goal "(%x. f(x) - l) -- x --NS> #0 ==> f -- x --NS> l";
-by (dres_inst_tac [("g","%x. l"),("m","l")] NSLIM_add 1);
-by (auto_tac (claset(),simpset() addsimps [real_diff_def, real_add_assoc]));
-qed "NSLIM_zero_cancel";
-
-Goal "(%x. f(x) - l) -- x --> #0 ==> f -- x --> l";
-by (dres_inst_tac [("g","%x. l"),("m","l")] LIM_add 1);
-by (auto_tac (claset(),simpset() addsimps [real_diff_def, real_add_assoc]));
-qed "LIM_zero_cancel";
-
-(*--------------------------
- NSLIM_not_zero
- --------------------------*)
-Goalw [NSLIM_def] "k ~= #0 ==> ~ ((%x. k) -- x --NS> #0)";
-by Auto_tac;
-by (res_inst_tac [("x","hypreal_of_real x + ehr")] exI 1);
-by (auto_tac (claset() addIs [Infinitesimal_add_inf_close_self
- RS inf_close_sym],simpset()));
-by (dres_inst_tac [("x1","-hypreal_of_real x")]
- (hypreal_add_left_cancel RS iffD2) 1);
-by (asm_full_simp_tac (simpset() addsimps [hypreal_add_assoc RS sym,
- hypreal_epsilon_not_zero]) 1);
-qed "NSLIM_not_zero";
-
-(* [| k ~= #0; (%x. k) -- x --NS> #0 |] ==> R *)
-bind_thm("NSLIM_not_zeroE", NSLIM_not_zero RS notE);
-
-Goal "k ~= #0 ==> ~ ((%x. k) -- x --> #0)";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, NSLIM_not_zero]) 1);
-qed "LIM_not_zero2";
-
-(*-------------------------------------
- NSLIM of constant function
- -------------------------------------*)
-Goal "(%x. k) -- x --NS> L ==> k = L";
-by (rtac ccontr 1);
-by (dtac NSLIM_zero 1);
-by (rtac NSLIM_not_zeroE 1 THEN assume_tac 2);
-by (arith_tac 1);
-qed "NSLIM_const_eq";
-
-Goal "(%x. k) -- x --> L ==> k = L";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff,
- NSLIM_const_eq]) 1);
-qed "LIM_const_eq2";
-
-(*------------------------
- NS Limit is Unique
- ------------------------*)
-(* can actually be proved more easily by unfolding def! *)
-Goal "[| f -- x --NS> L; f -- x --NS> M |] ==> L = M";
-by (dtac NSLIM_minus 1);
-by (dtac NSLIM_add 1 THEN assume_tac 1);
-by (auto_tac (claset() addSDs [NSLIM_const_eq RS sym], simpset()));
-qed "NSLIM_unique";
-
-Goal "[| f -- x --> L; f -- x --> M |] ==> L = M";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, NSLIM_unique]) 1);
-qed "LIM_unique2";
-
-(*--------------------
- NSLIM_mult_zero
- --------------------*)
-Goal "[| f -- x --NS> #0; g -- x --NS> #0 |] \
-\ ==> (%x. f(x)*g(x)) -- x --NS> #0";
-by (dtac NSLIM_mult 1 THEN Auto_tac);
-qed "NSLIM_mult_zero";
-
-(* we can use the corresponding thm LIM_mult2 *)
-(* for standard definition of limit *)
-
-Goal "[| f -- x --> #0; g -- x --> #0 |] \
-\ ==> (%x. f(x)*g(x)) -- x --> #0";
-by (dtac LIM_mult2 1 THEN Auto_tac);
-qed "LIM_mult_zero2";
-
-(*----------------------------
- NSLIM_self
- ----------------------------*)
-Goalw [NSLIM_def] "(%x. x) -- a --NS> a";
-by (auto_tac (claset() addIs [starfun_Idfun_inf_close],simpset()));
-qed "NSLIM_self";
-
-Goal "(%x. x) -- a --> a";
-by (simp_tac (simpset() addsimps [LIM_NSLIM_iff,NSLIM_self]) 1);
-qed "LIM_self2";
-
-(*-----------------------------------------------------------------------------
- Derivatives and Continuity - NS and Standard properties
- -----------------------------------------------------------------------------*)
-(*---------------
- Continuity
- ---------------*)
-
-Goalw [isNSCont_def]
- "[| isNSCont f a; y @= hypreal_of_real a |] \
-\ ==> (*f* f) y @= hypreal_of_real (f a)";
-by (Blast_tac 1);
-qed "isNSContD";
-
-Goalw [isNSCont_def,NSLIM_def]
- "isNSCont f a ==> f -- a --NS> (f a) ";
-by (Blast_tac 1);
-qed "isNSCont_NSLIM";
-
-Goalw [isNSCont_def,NSLIM_def]
- "f -- a --NS> (f a) ==> isNSCont f a";
-by Auto_tac;
-by (res_inst_tac [("Q","y = hypreal_of_real a")]
- (excluded_middle RS disjE) 1);
-by Auto_tac;
-qed "NSLIM_isNSCont";
-
-(*-----------------------------------------------------
- NS continuity can be defined using NS Limit in
- similar fashion to standard def of continuity
- -----------------------------------------------------*)
-Goal "(isNSCont f a) = (f -- a --NS> (f a))";
-by (blast_tac (claset() addIs [isNSCont_NSLIM,NSLIM_isNSCont]) 1);
-qed "isNSCont_NSLIM_iff";
-
-(*----------------------------------------------
- Hence, NS continuity can be given
- in terms of standard limit
- ---------------------------------------------*)
-Goal "(isNSCont f a) = (f -- a --> (f a))";
-by (asm_full_simp_tac (simpset() addsimps
- [LIM_NSLIM_iff,isNSCont_NSLIM_iff]) 1);
-qed "isNSCont_LIM_iff";
-
-(*-----------------------------------------------
- Moreover, it's trivial now that NS continuity
- is equivalent to standard continuity
- -----------------------------------------------*)
-Goalw [isCont_def] "(isNSCont f a) = (isCont f a)";
-by (rtac isNSCont_LIM_iff 1);
-qed "isNSCont_isCont_iff";
-
-(*----------------------------------------
- Standard continuity ==> NS continuity
- ----------------------------------------*)
-Goal "isCont f a ==> isNSCont f a";
-by (etac (isNSCont_isCont_iff RS iffD2) 1);
-qed "isCont_isNSCont";
-
-(*----------------------------------------
- NS continuity ==> Standard continuity
- ----------------------------------------*)
-Goal "isNSCont f a ==> isCont f a";
-by (etac (isNSCont_isCont_iff RS iffD1) 1);
-qed "isNSCont_isCont";
-
-(*--------------------------------------------------------------------------
- Alternative definition of continuity
- --------------------------------------------------------------------------*)
-(* Prove equivalence between NS limits - *)
-(* seems easier than using standard def *)
-Goalw [NSLIM_def] "(f -- a --NS> L) = ((%h. f(a + h)) -- #0 --NS> L)";
-by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_zero]));
-by (dres_inst_tac [("x","hypreal_of_real a + x")] spec 1);
-by (dres_inst_tac [("x","-hypreal_of_real a + x")] spec 2);
-by (Step_tac 1);
-by (Asm_full_simp_tac 1);
-by (rtac ((mem_infmal_iff RS iffD2) RS
- (Infinitesimal_add_inf_close_self RS inf_close_sym)) 1);
-by (rtac (inf_close_minus_iff2 RS iffD1) 4);
-by (asm_full_simp_tac (simpset() addsimps [hypreal_add_commute]) 3);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 2);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 4);
-by (auto_tac (claset(),
- simpset() addsimps [starfun, hypreal_of_real_def, hypreal_minus,
- hypreal_add, real_add_assoc, inf_close_refl, hypreal_zero_def]));
-qed "NSLIM_h_iff";
-
-Goal "(f -- a --NS> f a) = ((%h. f(a + h)) -- #0 --NS> f a)";
-by (rtac NSLIM_h_iff 1);
-qed "NSLIM_isCont_iff";
-
-Goal "(f -- a --> f a) = ((%h. f(a + h)) -- #0 --> f(a))";
-by (simp_tac (simpset() addsimps [LIM_NSLIM_iff, NSLIM_isCont_iff]) 1);
-qed "LIM_isCont_iff";
-
-Goalw [isCont_def] "(isCont f x) = ((%h. f(x + h)) -- #0 --> f(x))";
-by (simp_tac (simpset() addsimps [LIM_isCont_iff]) 1);
-qed "isCont_iff";
-
-(*--------------------------------------------------------------------------
- Immediate application of nonstandard criterion for continuity can offer
- very simple proofs of some standard property of continuous functions
- --------------------------------------------------------------------------*)
-(*------------------------
- sum continuous
- ------------------------*)
-Goal "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) + g(x)) a";
-by (auto_tac (claset() addIs [inf_close_add],
- simpset() addsimps [isNSCont_isCont_iff RS sym, isNSCont_def]));
-qed "isCont_add";
-
-(*------------------------
- mult continuous
- ------------------------*)
-Goal "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) * g(x)) a";
-by (auto_tac (claset() addSIs [starfun_mult_HFinite_inf_close],
- simpset() delsimps [starfun_mult RS sym]
- addsimps [isNSCont_isCont_iff RS sym, isNSCont_def]));
-qed "isCont_mult";
-
-(*-------------------------------------------
- composition of continuous functions
- Note very short straightforard proof!
- ------------------------------------------*)
-Goal "[| isCont f a; isCont g (f a) |] \
-\ ==> isCont (g o f) a";
-by (auto_tac (claset(),simpset() addsimps [isNSCont_isCont_iff RS sym,
- isNSCont_def,starfun_o RS sym]));
-qed "isCont_o";
-
-Goal "[| isCont f a; isCont g (f a) |] \
-\ ==> isCont (%x. g (f x)) a";
-by (auto_tac (claset() addDs [isCont_o],simpset() addsimps [o_def]));
-qed "isCont_o2";
-
-Goalw [isNSCont_def] "isNSCont f a ==> isNSCont (%x. - f x) a";
-by Auto_tac;
-qed "isNSCont_minus";
-
-Goal "isCont f a ==> isCont (%x. - f x) a";
-by (auto_tac (claset(),simpset() addsimps [isNSCont_isCont_iff RS sym,
- isNSCont_minus]));
-qed "isCont_minus";
-
-Goalw [isCont_def]
- "[| isCont f x; f x ~= #0 |] ==> isCont (%x. inverse (f x)) x";
-by (blast_tac (claset() addIs [LIM_inverse]) 1);
-qed "isCont_inverse";
-
-Goal "[| isNSCont f x; f x ~= #0 |] ==> isNSCont (%x. inverse (f x)) x";
-by (auto_tac (claset() addIs [isCont_inverse],simpset() addsimps
- [isNSCont_isCont_iff]));
-qed "isNSCont_inverse";
-
-Goalw [real_diff_def]
- "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) - g(x)) a";
-by (auto_tac (claset() addIs [isCont_add,isCont_minus],simpset()));
-qed "isCont_diff";
-
-Goalw [isCont_def] "isCont (%x. k) a";
-by (Simp_tac 1);
-qed "isCont_const";
-Addsimps [isCont_const];
-
-Goalw [isNSCont_def] "isNSCont (%x. k) a";
-by (Simp_tac 1);
-qed "isNSCont_const";
-Addsimps [isNSCont_const];
-
-Goalw [isNSCont_def] "isNSCont abs a";
-by (auto_tac (claset() addIs [inf_close_hrabs],
- simpset() addsimps [hypreal_of_real_hrabs RS sym,
- starfun_rabs_hrabs]));
-qed "isNSCont_rabs";
-Addsimps [isNSCont_rabs];
-
-Goal "isCont abs a";
-by (auto_tac (claset(), simpset() addsimps [isNSCont_isCont_iff RS sym]));
-qed "isCont_rabs";
-Addsimps [isCont_rabs];
-
-(****************************************************************
-(%* Leave as commented until I add topology theory or remove? *%)
-(%*------------------------------------------------------------
- Elementary topology proof for a characterisation of
- continuity now: a function f is continuous if and only
- if the inverse image, {x. f(x) : A}, of any open set A
- is always an open set
- ------------------------------------------------------------*%)
-Goal "[| isNSopen A; ALL x. isNSCont f x |] \
-\ ==> isNSopen {x. f x : A}";
-by (auto_tac (claset(),simpset() addsimps [isNSopen_iff1]));
-by (dtac (mem_monad_inf_close RS inf_close_sym) 1);
-by (dres_inst_tac [("x","a")] spec 1);
-by (dtac isNSContD 1 THEN assume_tac 1);
-by (dtac bspec 1 THEN assume_tac 1);
-by (dres_inst_tac [("x","( *f* f) x")] inf_close_mem_monad2 1);
-by (blast_tac (claset() addIs [starfun_mem_starset]) 1);
-qed "isNSCont_isNSopen";
-
-Goalw [isNSCont_def]
- "ALL A. isNSopen A --> isNSopen {x. f x : A} \
-\ ==> isNSCont f x";
-by (auto_tac (claset() addSIs [(mem_infmal_iff RS iffD1) RS
- (inf_close_minus_iff RS iffD2)],simpset() addsimps
- [Infinitesimal_def,SReal_iff]));
-by (dres_inst_tac [("x","{z. abs(z + -f(x)) < ya}")] spec 1);
-by (etac (isNSopen_open_interval RSN (2,impE)) 1);
-by (auto_tac (claset(),simpset() addsimps [isNSopen_def,isNSnbhd_def]));
-by (dres_inst_tac [("x","x")] spec 1);
-by (auto_tac (claset() addDs [inf_close_sym RS inf_close_mem_monad],
- simpset() addsimps [hypreal_of_real_zero RS sym,STAR_starfun_rabs_add_minus]));
-qed "isNSopen_isNSCont";
-
-Goal "(ALL x. isNSCont f x) = \
-\ (ALL A. isNSopen A --> isNSopen {x. f(x) : A})";
-by (blast_tac (claset() addIs [isNSCont_isNSopen,
- isNSopen_isNSCont]) 1);
-qed "isNSCont_isNSopen_iff";
-
-(%*------- Standard version of same theorem --------*%)
-Goal "(ALL x. isCont f x) = \
-\ (ALL A. isopen A --> isopen {x. f(x) : A})";
-by (auto_tac (claset() addSIs [isNSCont_isNSopen_iff],
- simpset() addsimps [isNSopen_isopen_iff RS sym,
- isNSCont_isCont_iff RS sym]));
-qed "isCont_isopen_iff";
-*******************************************************************)
-
-(*-----------------------------------------------------------------
- Uniform continuity
- ------------------------------------------------------------------*)
-Goalw [isNSUCont_def]
- "[| isNSUCont f; x @= y|] ==> (*f* f) x @= (*f* f) y";
-by (Blast_tac 1);
-qed "isNSUContD";
-
-Goalw [isUCont_def,isCont_def,LIM_def]
- "isUCont f ==> EX x. isCont f x";
-by (Force_tac 1);
-qed "isUCont_isCont";
-
-Goalw [isNSUCont_def,isUCont_def,inf_close_def]
- "isUCont f ==> isNSUCont f";
-by (asm_full_simp_tac (simpset() addsimps
- [Infinitesimal_FreeUltrafilterNat_iff]) 1);
-by (Step_tac 1);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun,
- hypreal_minus, hypreal_add]));
-by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2 THEN Step_tac 1);
-by (dres_inst_tac [("x","u")] spec 1 THEN Clarify_tac 1);
-by (dres_inst_tac [("x","s")] spec 1 THEN Clarify_tac 1);
-by (subgoal_tac "ALL n::nat. abs ((xa n) + - (xb n)) < s --> abs (f (xa n) + - f (xb n)) < u" 1);
-by (Blast_tac 2);
-by (thin_tac "ALL x y. abs (x + - y) < s --> abs (f x + - f y) < u" 1);
-by (dtac FreeUltrafilterNat_all 1);
-by (Ultra_tac 1);
-qed "isUCont_isNSUCont";
-
-Goal "ALL s. #0 < s --> (EX z y. abs (z + - y) < s & r <= abs (f z + -f y)) \
-\ ==> ALL n::nat. EX z y. \
-\ abs(z + -y) < inverse(real_of_posnat n) & \
-\ r <= abs(f z + -f y)";
-by (Step_tac 1);
-by (cut_inst_tac [("n1","n")] (real_of_posnat_gt_zero RS real_inverse_gt_zero) 1);
-by Auto_tac;
-val lemma_LIMu = result();
-
-Goal "ALL s. #0 < s --> (EX z y. abs (z + - y) < s & r <= abs (f z + -f y)) \
-\ ==> EX X Y. ALL n::nat. \
-\ abs(X n + -(Y n)) < inverse(real_of_posnat n) & \
-\ r <= abs(f (X n) + -f (Y n))";
-by (dtac lemma_LIMu 1);
-by (dtac choice 1);
-by (Step_tac 1);
-by (dtac choice 1);
-by (Blast_tac 1);
-val lemma_skolemize_LIM2u = result();
-
-Goal "ALL n. abs (X n + -Y n) < inverse (real_of_posnat n) & \
-\ r <= abs (f (X n) + - f(Y n)) ==> \
-\ ALL n. abs (X n + - Y n) < inverse (real_of_posnat n)";
-by (Auto_tac );
-val lemma_simpu = result();
-
-Goalw [isNSUCont_def,isUCont_def,inf_close_def]
- "isNSUCont f ==> isUCont f";
-by (asm_full_simp_tac (simpset() addsimps
- [Infinitesimal_FreeUltrafilterNat_iff]) 1);
-by (EVERY1[Step_tac, rtac ccontr, Asm_full_simp_tac]);
-by (fold_tac [real_le_def]);
-by (dtac lemma_skolemize_LIM2u 1);
-by (Step_tac 1);
-by (dres_inst_tac [("x","Abs_hypreal(hyprel^^{X})")] spec 1);
-by (dres_inst_tac [("x","Abs_hypreal(hyprel^^{Y})")] spec 1);
-by (asm_full_simp_tac
- (simpset() addsimps [starfun, hypreal_minus,hypreal_add]) 1);
-by Auto_tac;
-by (dtac (lemma_simpu RS real_seq_to_hypreal_Infinitesimal2) 1);
-by (asm_full_simp_tac (simpset() addsimps
- [Infinitesimal_FreeUltrafilterNat_iff, hypreal_minus,hypreal_add]) 1);
-by (Blast_tac 1);
-by (rotate_tac 2 1);
-by (dres_inst_tac [("x","r")] spec 1);
-by (Clarify_tac 1);
-by (dtac FreeUltrafilterNat_all 1);
-by (Ultra_tac 1);
-qed "isNSUCont_isUCont";
-
-(*------------------------------------------------------------------
- Derivatives
- ------------------------------------------------------------------*)
-Goalw [deriv_def]
- "(DERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- #0 --> D)";
-by (Blast_tac 1);
-qed "DERIV_iff";
-
-Goalw [deriv_def]
- "(DERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- #0 --NS> D)";
-by (simp_tac (simpset() addsimps [LIM_NSLIM_iff]) 1);
-qed "DERIV_NS_iff";
-
-Goalw [deriv_def]
- "DERIV f x :> D \
-\ ==> (%h. (f(x + h) + - f(x))/h) -- #0 --> D";
-by (Blast_tac 1);
-qed "DERIVD";
-
-Goalw [deriv_def] "DERIV f x :> D ==> \
-\ (%h. (f(x + h) + - f(x))/h) -- #0 --NS> D";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff]) 1);
-qed "NS_DERIVD";
-
-(* Uniqueness *)
-Goalw [deriv_def]
- "[| DERIV f x :> D; DERIV f x :> E |] ==> D = E";
-by (blast_tac (claset() addIs [LIM_unique]) 1);
-qed "DERIV_unique";
-
-Goalw [nsderiv_def]
- "[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E";
-by (cut_facts_tac [Infinitesimal_epsilon, hypreal_epsilon_not_zero] 1);
-by (auto_tac (claset() addSDs [inst "x" "ehr" bspec]
- addSIs [inj_hypreal_of_real RS injD]
- addDs [inf_close_trans3],
- simpset()));
-qed "NSDeriv_unique";
-
-(*------------------------------------------------------------------------
- Differentiable
- ------------------------------------------------------------------------*)
-
-Goalw [differentiable_def]
- "f differentiable x ==> EX D. DERIV f x :> D";
-by (assume_tac 1);
-qed "differentiableD";
-
-Goalw [differentiable_def]
- "DERIV f x :> D ==> f differentiable x";
-by (Blast_tac 1);
-qed "differentiableI";
-
-Goalw [NSdifferentiable_def]
- "f NSdifferentiable x ==> EX D. NSDERIV f x :> D";
-by (assume_tac 1);
-qed "NSdifferentiableD";
-
-Goalw [NSdifferentiable_def]
- "NSDERIV f x :> D ==> f NSdifferentiable x";
-by (Blast_tac 1);
-qed "NSdifferentiableI";
-
-(*--------------------------------------------------------
- Alternative definition for differentiability
- -------------------------------------------------------*)
-
-Goalw [LIM_def]
- "((%h. (f(a + h) + - f(a))/h) -- #0 --> D) = \
-\ ((%x. (f(x) + -f(a)) / (x + -a)) -- a --> D)";
-by (Step_tac 1);
-by (ALLGOALS(dtac spec));
-by (Step_tac 1);
-by (Blast_tac 1 THEN Blast_tac 2);
-by (ALLGOALS(res_inst_tac [("x","s")] exI));
-by (Step_tac 1);
-by (dres_inst_tac [("x","x + -a")] spec 1);
-by (dres_inst_tac [("x","x + a")] spec 2);
-by (auto_tac (claset(), simpset() addsimps real_add_ac));
-qed "DERIV_LIM_iff";
-
-Goalw [deriv_def] "(DERIV f x :> D) = \
-\ ((%z. (f(z) + -f(x)) / (z + -x)) -- x --> D)";
-by (simp_tac (simpset() addsimps [DERIV_LIM_iff]) 1);
-qed "DERIV_iff2";
-
-(*--------------------------------------------------------
- Equivalence of NS and standard defs of differentiation
- -------------------------------------------------------*)
-(*-------------------------------------------
- First NSDERIV in terms of NSLIM
- -------------------------------------------*)
-
-(*--- first equivalence ---*)
-Goalw [nsderiv_def,NSLIM_def]
- "(NSDERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- #0 --NS> D)";
-by (auto_tac (claset(), simpset() addsimps [hypreal_of_real_zero]));
-by (dres_inst_tac [("x","xa")] bspec 1);
-by (rtac ccontr 3);
-by (dres_inst_tac [("x","h")] spec 3);
-by (auto_tac (claset(),
- simpset() addsimps [mem_infmal_iff, starfun_lambda_cancel]));
-qed "NSDERIV_NSLIM_iff";
-
-(*--- second equivalence ---*)
-Goal "(NSDERIV f x :> D) = \
-\ ((%z. (f(z) + -f(x)) / (z + -x)) -- x --NS> D)";
-by (full_simp_tac (simpset() addsimps
- [NSDERIV_NSLIM_iff, DERIV_LIM_iff, LIM_NSLIM_iff RS sym]) 1);
-qed "NSDERIV_NSLIM_iff2";
-
-(* while we're at it! *)
-Goalw [real_diff_def]
- "(NSDERIV f x :> D) = \
-\ (ALL xa. \
-\ xa ~= hypreal_of_real x & xa @= hypreal_of_real x --> \
-\ (*f* (%z. (f z - f x) / (z - x))) xa @= hypreal_of_real D)";
-by (auto_tac (claset(), simpset() addsimps [NSDERIV_NSLIM_iff2, NSLIM_def]));
-qed "NSDERIV_iff2";
-
-Addsimps [inf_close_refl];
-
-
-(*FIXME: replace by simprocs for cancellation of common factors*)
-Goal "h ~= (0::hypreal) ==> x*h/h = x";
-by (asm_simp_tac
- (simpset() addsimps [hypreal_divide_def, hypreal_mult_inverse_left,
- hypreal_mult_assoc]) 1);
-qed "hypreal_divide_mult_self_eq";
-Addsimps [hypreal_divide_mult_self_eq];
-
-(*FIXME: replace by simprocs for cancellation of common factors*)
-Goal "h ~= (0::hypreal) ==> (h*x)/h = x";
-by (asm_simp_tac
- (simpset() addsimps [inst "z" "h" hypreal_mult_commute]) 1);
-qed "hypreal_times_divide_self_eq";
-Addsimps [hypreal_times_divide_self_eq];
-
-(*FIXME: replace by simprocs for cancellation of common factors*)
-Goal "h ~= (0::hypreal) ==> h/h = 1hr";
-by (asm_simp_tac
- (simpset() addsimps [hypreal_divide_def, hypreal_mult_inverse_left]) 1);
-qed "hypreal_divide_self_eq";
-Addsimps [hypreal_divide_self_eq];
-
-
-Goal "(NSDERIV f x :> D) ==> \
-\ (ALL xa. \
-\ xa @= hypreal_of_real x --> \
-\ (*f* (%z. f z - f x)) xa @= hypreal_of_real D * (xa - hypreal_of_real x))";
-by (auto_tac (claset(), simpset() addsimps [NSDERIV_iff2]));
-by (case_tac "xa = hypreal_of_real x" 1);
-by (auto_tac (claset(),
- simpset() addsimps [hypreal_diff_def, hypreal_of_real_zero]));
-by (dres_inst_tac [("x","xa")] spec 1);
-by Auto_tac;
-by (dres_inst_tac [("c","xa - hypreal_of_real x"),("b","hypreal_of_real D")]
- inf_close_mult1 1);
-by (ALLGOALS(dtac (hypreal_not_eq_minus_iff RS iffD1)));
-by (subgoal_tac "(*f* (%z. z - x)) xa ~= (0::hypreal)" 2);
-by (rotate_tac ~1 2);
-by (auto_tac (claset(),
- simpset() addsimps [real_diff_def, hypreal_diff_def,
- (inf_close_minus_iff RS iffD1) RS (mem_infmal_iff RS iffD2),
- Infinitesimal_subset_HFinite RS subsetD]));
-qed "NSDERIVD5";
-
-Goal "(NSDERIV f x :> D) ==> \
-\ (ALL h: Infinitesimal. \
-\ ((*f* f)(hypreal_of_real x + h) - \
-\ hypreal_of_real (f x))@= (hypreal_of_real D) * h)";
-by (auto_tac (claset(),simpset() addsimps [nsderiv_def]));
-by (case_tac "h = (0::hypreal)" 1);
-by (auto_tac (claset(),simpset() addsimps [hypreal_diff_def]));
-by (dres_inst_tac [("x","h")] bspec 1);
-by (dres_inst_tac [("c","h")] inf_close_mult1 2);
-by (auto_tac (claset() addIs [Infinitesimal_subset_HFinite RS subsetD],
- simpset() addsimps [hypreal_diff_def]));
-qed "NSDERIVD4";
-
-Goal "(NSDERIV f x :> D) ==> \
-\ (ALL h: Infinitesimal - {0}. \
-\ ((*f* f)(hypreal_of_real x + h) - \
-\ hypreal_of_real (f x))@= (hypreal_of_real D) * h)";
-by (auto_tac (claset(),simpset() addsimps [nsderiv_def]));
-by (rtac ccontr 1 THEN dres_inst_tac [("x","h")] bspec 1);
-by (dres_inst_tac [("c","h")] inf_close_mult1 2);
-by (auto_tac (claset() addIs [Infinitesimal_subset_HFinite RS subsetD],
- simpset() addsimps [hypreal_mult_assoc, hypreal_diff_def]));
-qed "NSDERIVD3";
-
-(*--------------------------------------------------------------
- Now equivalence between NSDERIV and DERIV
- -------------------------------------------------------------*)
-Goalw [deriv_def] "(NSDERIV f x :> D) = (DERIV f x :> D)";
-by (simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff,LIM_NSLIM_iff]) 1);
-qed "NSDERIV_DERIV_iff";
-
-(*---------------------------------------------------
- Differentiability implies continuity
- nice and simple "algebraic" proof
- --------------------------------------------------*)
-Goalw [nsderiv_def]
- "NSDERIV f x :> D ==> isNSCont f x";
-by (auto_tac (claset(),simpset() addsimps
- [isNSCont_NSLIM_iff,NSLIM_def]));
-by (dtac (inf_close_minus_iff RS iffD1) 1);
-by (dtac (hypreal_not_eq_minus_iff RS iffD1) 1);
-by (dres_inst_tac [("x","-hypreal_of_real x + xa")] bspec 1);
-by (asm_full_simp_tac (simpset() addsimps
- [hypreal_add_assoc RS sym]) 2);
-by (auto_tac (claset(),simpset() addsimps
- [mem_infmal_iff RS sym,hypreal_add_commute]));
-by (dres_inst_tac [("c","xa + -hypreal_of_real x")] inf_close_mult1 1);
-by (auto_tac (claset() addIs [Infinitesimal_subset_HFinite
- RS subsetD],simpset() addsimps [hypreal_mult_assoc]));
-by (dres_inst_tac [("x3","D")] (HFinite_hypreal_of_real RSN
- (2,Infinitesimal_HFinite_mult) RS (mem_infmal_iff RS iffD1)) 1);
-by (blast_tac (claset() addIs [inf_close_trans,
- hypreal_mult_commute RS subst,
- (inf_close_minus_iff RS iffD2)]) 1);
-qed "NSDERIV_isNSCont";
-
-(* Now Sandard proof *)
-Goal "DERIV f x :> D ==> isCont f x";
-by (asm_full_simp_tac (simpset() addsimps
- [NSDERIV_DERIV_iff RS sym, isNSCont_isCont_iff RS sym,
- NSDERIV_isNSCont]) 1);
-qed "DERIV_isCont";
-
-(*----------------------------------------------------------------------------
- Differentiation rules for combinations of functions
- follow from clear, straightforard, algebraic
- manipulations
- ----------------------------------------------------------------------------*)
-(*-------------------------
- Constant function
- ------------------------*)
-
-(* use simple constant nslimit theorem *)
-Goal "(NSDERIV (%x. k) x :> #0)";
-by (simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff]) 1);
-qed "NSDERIV_const";
-Addsimps [NSDERIV_const];
-
-Goal "(DERIV (%x. k) x :> #0)";
-by (simp_tac (simpset() addsimps [NSDERIV_DERIV_iff RS sym]) 1);
-qed "DERIV_const";
-Addsimps [DERIV_const];
-
-(*-----------------------------------------------------
- Sum of functions- proved easily
- ----------------------------------------------------*)
-
-
-Goal "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] \
-\ ==> NSDERIV (%x. f x + g x) x :> Da + Db";
-by (asm_full_simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff,
- NSLIM_def]) 1 THEN REPEAT(Step_tac 1));
-by (auto_tac (claset(),
- simpset() addsimps [hypreal_add_divide_distrib]));
-by (thin_tac "xa @= hypreal_of_real #0" 1 THEN dtac inf_close_add 1);
-by (auto_tac (claset(),simpset() addsimps hypreal_add_ac));
-qed "NSDERIV_add";
-
-(* Standard theorem *)
-Goal "[| DERIV f x :> Da; DERIV g x :> Db |] \
-\ ==> DERIV (%x. f x + g x) x :> Da + Db";
-by (asm_full_simp_tac (simpset() addsimps [NSDERIV_add,
- NSDERIV_DERIV_iff RS sym]) 1);
-qed "DERIV_add";
-
-(*-----------------------------------------------------
- Product of functions - Proof is trivial but tedious
- and long due to rearrangement of terms
- ----------------------------------------------------*)
-(* lemma - terms manipulation tedious as usual*)
-
-Goal "((a::hypreal)*b) + -(c*d) = (b*(a + -c)) + \
-\ (c*(b + -d))";
-by (full_simp_tac (simpset() addsimps [hypreal_add_mult_distrib2,
- hypreal_minus_mult_eq2 RS sym,hypreal_mult_commute]) 1);
-val lemma_nsderiv1 = result();
-
-Goal "[| (x + y) / z = hypreal_of_real D + yb; z ~= 0; \
-\ z : Infinitesimal; yb : Infinitesimal |] \
-\ ==> x + y @= 0";
-by (forw_inst_tac [("c1","z")] (hypreal_mult_right_cancel RS iffD2) 1
- THEN assume_tac 1);
-by (thin_tac "(x + y) / z = hypreal_of_real D + yb" 1);
-by (auto_tac (claset() addSIs [Infinitesimal_HFinite_mult2, HFinite_add],
- simpset() addsimps [hypreal_mult_assoc, mem_infmal_iff RS sym]));
-by (etac (Infinitesimal_subset_HFinite RS subsetD) 1);
-val lemma_nsderiv2 = result();
-
-
-Goal "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] \
-\ ==> NSDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))";
-by (asm_full_simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff, NSLIM_def]) 1
- THEN REPEAT(Step_tac 1));
-by (auto_tac (claset(),
- simpset() addsimps [starfun_lambda_cancel, hypreal_of_real_zero,
- lemma_nsderiv1]));
-by (simp_tac (simpset() addsimps [hypreal_add_divide_distrib]) 1);
-by (REPEAT(dtac (bex_Infinitesimal_iff2 RS iffD2) 1));
-by (auto_tac (claset(),
- simpset() delsimps [hypreal_times_divide1_eq]
- addsimps [hypreal_times_divide1_eq RS sym]));
-by (dres_inst_tac [("D","Db")] lemma_nsderiv2 1);
-by (dtac (inf_close_minus_iff RS iffD2 RS (bex_Infinitesimal_iff2 RS iffD2)) 4);
-by (auto_tac (claset() addSIs [inf_close_add_mono1],
- simpset() addsimps [hypreal_add_mult_distrib, hypreal_add_mult_distrib2,
- hypreal_mult_commute, hypreal_add_assoc]));
-by (res_inst_tac [("w1","hypreal_of_real Db * hypreal_of_real (f x)")]
- (hypreal_add_commute RS subst) 1);
-by (auto_tac (claset() addSIs [Infinitesimal_add_inf_close_self2 RS inf_close_sym,
- Infinitesimal_add, Infinitesimal_mult,
- Infinitesimal_hypreal_of_real_mult,
- Infinitesimal_hypreal_of_real_mult2],
- simpset() addsimps [hypreal_add_assoc RS sym]));
-qed "NSDERIV_mult";
-
-Goal "[| DERIV f x :> Da; DERIV g x :> Db |] \
-\ ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))";
-by (asm_full_simp_tac (simpset() addsimps [NSDERIV_mult,
- NSDERIV_DERIV_iff RS sym]) 1);
-qed "DERIV_mult";
-
-(*----------------------------
- Multiplying by a constant
- ---------------------------*)
-Goal "NSDERIV f x :> D \
-\ ==> NSDERIV (%x. c * f x) x :> c*D";
-by (asm_full_simp_tac
- (simpset() addsimps [real_times_divide1_eq RS sym, NSDERIV_NSLIM_iff,
- real_minus_mult_eq2, real_add_mult_distrib2 RS sym]
- delsimps [real_times_divide1_eq, real_minus_mult_eq2 RS sym]) 1);
-by (etac (NSLIM_const RS NSLIM_mult) 1);
-qed "NSDERIV_cmult";
-
-(* let's do the standard proof though theorem *)
-(* LIM_mult2 follows from a NS proof *)
-
-Goalw [deriv_def]
- "DERIV f x :> D \
-\ ==> DERIV (%x. c * f x) x :> c*D";
-by (asm_full_simp_tac
- (simpset() addsimps [real_times_divide1_eq RS sym, NSDERIV_NSLIM_iff,
- real_minus_mult_eq2, real_add_mult_distrib2 RS sym]
- delsimps [real_times_divide1_eq, real_minus_mult_eq2 RS sym]) 1);
-by (etac (LIM_const RS LIM_mult2) 1);
-qed "DERIV_cmult";
-
-(*--------------------------------
- Negation of function
- -------------------------------*)
-Goal "NSDERIV f x :> D ==> NSDERIV (%x. -(f x)) x :> -D";
-by (asm_full_simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff]) 1);
-by (res_inst_tac [("t","f x")] (real_minus_minus RS subst) 1);
-by (asm_simp_tac (simpset() addsimps [real_minus_add_distrib RS sym,
- real_minus_mult_eq1 RS sym]
- delsimps [real_minus_add_distrib, real_minus_minus]) 1);
-by (etac NSLIM_minus 1);
-qed "NSDERIV_minus";
-
-Goal "DERIV f x :> D \
-\ ==> DERIV (%x. -(f x)) x :> -D";
-by (asm_full_simp_tac (simpset() addsimps
- [NSDERIV_minus,NSDERIV_DERIV_iff RS sym]) 1);
-qed "DERIV_minus";
-
-(*-------------------------------
- Subtraction
- ------------------------------*)
-Goal "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] \
-\ ==> NSDERIV (%x. f x + -g x) x :> Da + -Db";
-by (blast_tac (claset() addDs [NSDERIV_add,NSDERIV_minus]) 1);
-qed "NSDERIV_add_minus";
-
-Goal "[| DERIV f x :> Da; DERIV g x :> Db |] \
-\ ==> DERIV (%x. f x + -g x) x :> Da + -Db";
-by (blast_tac (claset() addDs [DERIV_add,DERIV_minus]) 1);
-qed "DERIV_add_minus";
-
-Goalw [real_diff_def]
- "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] \
-\ ==> NSDERIV (%x. f x - g x) x :> Da - Db";
-by (blast_tac (claset() addIs [NSDERIV_add_minus]) 1);
-qed "NSDERIV_diff";
-
-Goalw [real_diff_def]
- "[| DERIV f x :> Da; DERIV g x :> Db |] \
-\ ==> DERIV (%x. f x - g x) x :> Da - Db";
-by (blast_tac (claset() addIs [DERIV_add_minus]) 1);
-qed "DERIV_diff";
-
-(*---------------------------------------------------------------
- (NS) Increment
- ---------------------------------------------------------------*)
-Goalw [increment_def]
- "f NSdifferentiable x ==> \
-\ increment f x h = (*f* f) (hypreal_of_real(x) + h) + \
-\ -hypreal_of_real (f x)";
-by (Blast_tac 1);
-qed "incrementI";
-
-Goal "NSDERIV f x :> D ==> \
-\ increment f x h = (*f* f) (hypreal_of_real(x) + h) + \
-\ -hypreal_of_real (f x)";
-by (etac (NSdifferentiableI RS incrementI) 1);
-qed "incrementI2";
-
-(* The Increment theorem -- Keisler p. 65 *)
-Goal "[| NSDERIV f x :> D; h: Infinitesimal; h ~= 0 |] \
-\ ==> EX e: Infinitesimal. increment f x h = hypreal_of_real(D)*h + e*h";
-by (forw_inst_tac [("h","h")] incrementI2 1 THEN rewtac nsderiv_def);
-by (dtac bspec 1 THEN Auto_tac);
-by (dtac (bex_Infinitesimal_iff2 RS iffD2) 1 THEN Step_tac 1);
-by (forw_inst_tac [("b1","hypreal_of_real(D) + y")]
- (hypreal_mult_right_cancel RS iffD2) 1);
-by (thin_tac "((*f* f) (hypreal_of_real(x) + h) + \
-\ - hypreal_of_real (f x)) / h = hypreal_of_real(D) + y" 2);
-by (assume_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [hypreal_times_divide1_eq RS sym]
- delsimps [hypreal_times_divide1_eq]) 1);
-by (auto_tac (claset(),
- simpset() addsimps [hypreal_add_mult_distrib]));
-qed "increment_thm";
-
-Goal "[| NSDERIV f x :> D; h @= 0; h ~= 0 |] \
-\ ==> EX e: Infinitesimal. increment f x h = \
-\ hypreal_of_real(D)*h + e*h";
-by (blast_tac (claset() addSDs [mem_infmal_iff RS iffD2]
- addSIs [increment_thm]) 1);
-qed "increment_thm2";
-
-Goal "[| NSDERIV f x :> D; h @= 0; h ~= 0 |] \
-\ ==> increment f x h @= 0";
-by (dtac increment_thm2 1 THEN auto_tac (claset() addSIs
- [Infinitesimal_HFinite_mult2,HFinite_add],simpset() addsimps
- [hypreal_add_mult_distrib RS sym,mem_infmal_iff RS sym]));
-by (etac (Infinitesimal_subset_HFinite RS subsetD) 1);
-qed "increment_inf_close_zero";
-
-(*---------------------------------------------------------------
- Similarly to the above, the chain rule admits an entirely
- straightforward derivation. Compare this with Harrison's
- HOL proof of the chain rule, which proved to be trickier and
- required an alternative characterisation of differentiability-
- the so-called Carathedory derivative. Our main problem is
- manipulation of terms.
- --------------------------------------------------------------*)
-
-(* lemmas *)
-Goalw [nsderiv_def]
- "[| NSDERIV g x :> D; \
-\ (*f* g) (hypreal_of_real(x) + xa) = hypreal_of_real(g x);\
-\ xa : Infinitesimal;\
-\ xa ~= 0 \
-\ |] ==> D = #0";
-by (dtac bspec 1);
-by Auto_tac;
-by (asm_full_simp_tac (simpset() addsimps [hypreal_of_real_zero RS sym]) 1);
-qed "NSDERIV_zero";
-
-(* can be proved differently using NSLIM_isCont_iff *)
-Goalw [nsderiv_def]
- "[| NSDERIV f x :> D; \
-\ h: Infinitesimal; h ~= 0 \
-\ |] ==> (*f* f) (hypreal_of_real(x) + h) + -hypreal_of_real(f x) @= 0";
-by (asm_full_simp_tac (simpset() addsimps
- [mem_infmal_iff RS sym]) 1);
-by (rtac Infinitesimal_ratio 1);
-by (rtac inf_close_hypreal_of_real_HFinite 3);
-by Auto_tac;
-qed "NSDERIV_inf_close";
-
-(*---------------------------------------------------------------
- from one version of differentiability
-
- f(x) - f(a)
- --------------- @= Db
- x - a
- ---------------------------------------------------------------*)
-Goal "[| NSDERIV f (g x) :> Da; \
-\ (*f* g) (hypreal_of_real(x) + xa) ~= hypreal_of_real (g x); \
-\ (*f* g) (hypreal_of_real(x) + xa) @= hypreal_of_real (g x) \
-\ |] ==> ((*f* f) ((*f* g) (hypreal_of_real(x) + xa)) \
-\ + - hypreal_of_real (f (g x))) \
-\ / ((*f* g) (hypreal_of_real(x) + xa) + - hypreal_of_real (g x)) \
-\ @= hypreal_of_real(Da)";
-by (auto_tac (claset(),
- simpset() addsimps [NSDERIV_NSLIM_iff2, NSLIM_def]));
-qed "NSDERIVD1";
-
-(*--------------------------------------------------------------
- from other version of differentiability
-
- f(x + h) - f(x)
- ----------------- @= Db
- h
- --------------------------------------------------------------*)
-Goal "[| NSDERIV g x :> Db; xa: Infinitesimal; xa ~= 0 |] \
-\ ==> ((*f* g) (hypreal_of_real(x) + xa) + - hypreal_of_real(g x)) / xa \
-\ @= hypreal_of_real(Db)";
-by (auto_tac (claset(),
- simpset() addsimps [NSDERIV_NSLIM_iff, NSLIM_def,
- hypreal_of_real_zero, mem_infmal_iff, starfun_lambda_cancel]));
-qed "NSDERIVD2";
-
-(*------------------------------------------------------
- This proof uses both definitions of differentiability.
- ------------------------------------------------------*)
-Goal "[| NSDERIV f (g x) :> Da; NSDERIV g x :> Db |] \
-\ ==> NSDERIV (f o g) x :> Da * Db";
-by (asm_simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff,
- NSLIM_def,hypreal_of_real_zero,mem_infmal_iff RS sym]) 1 THEN Step_tac 1);
-by (forw_inst_tac [("f","g")] NSDERIV_inf_close 1);
-by (auto_tac (claset(),
- simpset() addsimps [starfun_lambda_cancel2, starfun_o RS sym]));
-by (case_tac "(*f* g) (hypreal_of_real(x) + xa) = hypreal_of_real (g x)" 1);
-by (dres_inst_tac [("g","g")] NSDERIV_zero 1);
-by (auto_tac (claset(),
- simpset() addsimps [hypreal_divide_def, hypreal_of_real_zero]));
-by (res_inst_tac [("z1","(*f* g) (hypreal_of_real(x) + xa) + -hypreal_of_real (g x)"),
- ("y1","inverse xa")] (lemma_chain RS ssubst) 1);
-by (etac (hypreal_not_eq_minus_iff RS iffD1) 1);
-by (rtac inf_close_mult_hypreal_of_real 1);
-by (fold_tac [hypreal_divide_def]);
-by (blast_tac (claset() addIs [NSDERIVD1,
- inf_close_minus_iff RS iffD2]) 1);
-by (blast_tac (claset() addIs [NSDERIVD2]) 1);
-qed "NSDERIV_chain";
-
-(* standard version *)
-Goal "[| DERIV f (g x) :> Da; \
-\ DERIV g x :> Db \
-\ |] ==> DERIV (f o g) x :> Da * Db";
-by (asm_full_simp_tac (simpset() addsimps [NSDERIV_DERIV_iff RS sym,
- NSDERIV_chain]) 1);
-qed "DERIV_chain";
-
-Goal "[| DERIV f (g x) :> Da; DERIV g x :> Db |] \
-\ ==> DERIV (%x. f (g x)) x :> Da * Db";
-by (auto_tac (claset() addDs [DERIV_chain], simpset() addsimps [o_def]));
-qed "DERIV_chain2";
-
-(*------------------------------------------------------------------
- Differentiation of natural number powers
- ------------------------------------------------------------------*)
-Goal "NSDERIV (%x. x) x :> #1";
-by (auto_tac (claset(),
- simpset() addsimps [NSDERIV_NSLIM_iff,
- NSLIM_def ,starfun_Id, hypreal_of_real_zero,
- hypreal_of_real_one]));
-qed "NSDERIV_Id";
-Addsimps [NSDERIV_Id];
-
-(*derivative of the identity function*)
-Goal "DERIV (%x. x) x :> #1";
-by (simp_tac (simpset() addsimps [NSDERIV_DERIV_iff RS sym]) 1);
-qed "DERIV_Id";
-Addsimps [DERIV_Id];
-
-bind_thm ("isCont_Id", DERIV_Id RS DERIV_isCont);
-
-(*derivative of linear multiplication*)
-Goal "DERIV (op * c) x :> c";
-by (cut_inst_tac [("c","c"),("x","x")] (DERIV_Id RS DERIV_cmult) 1);
-by (Asm_full_simp_tac 1);
-qed "DERIV_cmult_Id";
-Addsimps [DERIV_cmult_Id];
-
-Goal "NSDERIV (op * c) x :> c";
-by (simp_tac (simpset() addsimps [NSDERIV_DERIV_iff]) 1);
-qed "NSDERIV_cmult_Id";
-Addsimps [NSDERIV_cmult_Id];
-
-Goal "DERIV (%x. x ^ n) x :> real_of_nat n * (x ^ (n - 1))";
-by (induct_tac "n" 1);
-by (dtac (DERIV_Id RS DERIV_mult) 2);
-by (auto_tac (claset(),simpset() addsimps
- [real_add_mult_distrib]));
-by (case_tac "0 < n" 1);
-by (dres_inst_tac [("x","x")] realpow_minus_mult 1);
-by (auto_tac (claset(),simpset() addsimps
- [real_mult_assoc,real_add_commute]));
-qed "DERIV_pow";
-
-(* NS version *)
-Goal "NSDERIV (%x. x ^ n) x :> real_of_nat n * (x ^ (n - 1))";
-by (simp_tac (simpset() addsimps [NSDERIV_DERIV_iff,DERIV_pow]) 1);
-qed "NSDERIV_pow";
-
-(*---------------------------------------------------------------
- Power of -1
- ---------------------------------------------------------------*)
-
-(*Can't get rid of x ~= #0 because it isn't continuous at zero*)
-Goalw [nsderiv_def]
- "x ~= #0 ==> NSDERIV (%x. inverse(x)) x :> (- (inverse x ^ 2))";
-by (rtac ballI 1 THEN Asm_full_simp_tac 1 THEN Step_tac 1);
-by (forward_tac [Infinitesimal_add_not_zero] 1);
-by (asm_full_simp_tac (simpset() addsimps [hypreal_add_commute]) 2);
-by (auto_tac (claset(),
- simpset() addsimps [starfun_inverse_inverse, realpow_two]
- delsimps [hypreal_minus_mult_eq1 RS sym,
- hypreal_minus_mult_eq2 RS sym]));
-by (asm_full_simp_tac
- (simpset() addsimps [hypreal_inverse_add,
- hypreal_inverse_distrib RS sym, hypreal_minus_inverse RS sym]
- @ hypreal_add_ac @ hypreal_mult_ac
- delsimps [hypreal_minus_mult_eq1 RS sym,
- hypreal_minus_mult_eq2 RS sym] ) 1);
-by (asm_simp_tac (simpset() addsimps [hypreal_mult_assoc RS sym,
- hypreal_add_mult_distrib2]
- delsimps [hypreal_minus_mult_eq1 RS sym,
- hypreal_minus_mult_eq2 RS sym]) 1);
-by (res_inst_tac [("y"," inverse(- hypreal_of_real x * hypreal_of_real x)")]
- inf_close_trans 1);
-by (rtac inverse_add_Infinitesimal_inf_close2 1);
-by (auto_tac (claset() addSDs [hypreal_of_real_HFinite_diff_Infinitesimal],
- simpset() addsimps [hypreal_minus_inverse RS sym,
- HFinite_minus_iff RS sym,
- Infinitesimal_minus_iff RS sym]));
-by (rtac Infinitesimal_HFinite_mult2 1);
-by Auto_tac;
-qed "NSDERIV_inverse";
-
-
-Goal "x ~= #0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ 2))";
-by (asm_simp_tac (simpset() addsimps [NSDERIV_inverse,
- NSDERIV_DERIV_iff RS sym] delsimps [realpow_Suc]) 1);
-qed "DERIV_inverse";
-
-(*--------------------------------------------------------------
- Derivative of inverse
- -------------------------------------------------------------*)
-Goal "[| DERIV f x :> d; f(x) ~= #0 |] \
-\ ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ 2)))";
-by (rtac (real_mult_commute RS subst) 1);
-by (asm_simp_tac (simpset() addsimps [real_minus_mult_eq1,
- realpow_inverse] delsimps [realpow_Suc,
- real_minus_mult_eq1 RS sym]) 1);
-by (fold_goals_tac [o_def]);
-by (blast_tac (claset() addSIs [DERIV_chain,DERIV_inverse]) 1);
-qed "DERIV_inverse_fun";
-
-Goal "[| NSDERIV f x :> d; f(x) ~= #0 |] \
-\ ==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ 2)))";
-by (asm_full_simp_tac (simpset() addsimps [NSDERIV_DERIV_iff,
- DERIV_inverse_fun] delsimps [realpow_Suc]) 1);
-qed "NSDERIV_inverse_fun";
-
-(*--------------------------------------------------------------
- Derivative of quotient
- -------------------------------------------------------------*)
-Goal "[| DERIV f x :> d; DERIV g x :> e; g(x) ~= #0 |] \
-\ ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) + -(e*f(x))) / (g(x) ^ 2)";
-by (dres_inst_tac [("f","g")] DERIV_inverse_fun 1);
-by (dtac DERIV_mult 2);
-by (REPEAT(assume_tac 1));
-by (asm_full_simp_tac
- (simpset() addsimps [real_divide_def, real_add_mult_distrib2,
- realpow_inverse,real_minus_mult_eq1] @ real_mult_ac
- delsimps [realpow_Suc, real_minus_mult_eq1 RS sym,
- real_minus_mult_eq2 RS sym]) 1);
-qed "DERIV_quotient";
-
-Goal "[| NSDERIV f x :> d; DERIV g x :> e; g(x) ~= #0 |] \
-\ ==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x) \
-\ + -(e*f(x))) / (g(x) ^ 2)";
-by (asm_full_simp_tac (simpset() addsimps [NSDERIV_DERIV_iff,
- DERIV_quotient] delsimps [realpow_Suc]) 1);
-qed "NSDERIV_quotient";
-
-(* ------------------------------------------------------------------------ *)
-(* Caratheodory formulation of derivative at a point: standard proof *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "(DERIV f x :> l) = \
-\ (EX g. (ALL z. f z - f x = g z * (z - x)) & isCont g x & g x = l)";
-by (Step_tac 1);
-by (res_inst_tac
- [("x","%z. if z = x then l else (f(z) - f(x)) / (z - x)")] exI 1);
-by (auto_tac (claset(),simpset() addsimps [real_mult_assoc,
- ARITH_PROVE "z ~= x ==> z - x ~= (#0::real)"]));
-by (auto_tac (claset(),simpset() addsimps [isCont_iff,DERIV_iff]));
-by (ALLGOALS(rtac (LIM_equal RS iffD1)));
-by (auto_tac (claset(),simpset() addsimps [real_diff_def,real_mult_assoc]));
-qed "CARAT_DERIV";
-
-Goal "NSDERIV f x :> l ==> \
-\ EX g. (ALL z. f z - f x = g z * (z - x)) & isNSCont g x & g x = l";
-by (auto_tac (claset(),simpset() addsimps [NSDERIV_DERIV_iff,
- isNSCont_isCont_iff,CARAT_DERIV]));
-qed "CARAT_NSDERIV";
-
-(* How about a NS proof? *)
-Goal "(ALL z. f z - f x = g z * (z - x)) & isNSCont g x & g x = l \
-\ ==> NSDERIV f x :> l";
-by (auto_tac (claset(),
- simpset() delsimprocs real_cancel_factor
- addsimps [NSDERIV_iff2]));
-by (auto_tac (claset(),
- simpset() addsimps [hypreal_mult_assoc]));
-by (asm_full_simp_tac (simpset() addsimps [hypreal_eq_minus_iff3 RS sym,
- hypreal_diff_def]) 1);
-by (asm_full_simp_tac (simpset() addsimps [isNSCont_def]) 1);
-qed "CARAT_DERIVD";
-
-
-
-(*--------------------------------------------------------------------------*)
-(* Lemmas about nested intervals and proof by bisection (cf.Harrison) *)
-(* All considerably tidied by lcp *)
-(*--------------------------------------------------------------------------*)
-
-Goal "(ALL n. (f::nat=>real) n <= f (Suc n)) --> f m <= f(m + no)";
-by (induct_tac "no" 1);
-by (auto_tac (claset() addIs [real_le_trans],simpset()));
-qed_spec_mp "lemma_f_mono_add";
-
-Goal "[| ALL n. f(n) <= f(Suc n); \
-\ ALL n. g(Suc n) <= g(n); \
-\ ALL n. f(n) <= g(n) |] \
-\ ==> Bseq f";
-by (res_inst_tac [("k","f 0"),("K","g 0")] BseqI2 1 THEN rtac allI 1);
-by (induct_tac "n" 1);
-by (auto_tac (claset() addIs [real_le_trans], simpset()));
-by (res_inst_tac [("j","g(Suc na)")] real_le_trans 1);
-by (induct_tac "na" 2);
-by (auto_tac (claset() addIs [real_le_trans], simpset()));
-qed "f_inc_g_dec_Beq_f";
-
-Goal "[| ALL n. f(n) <= f(Suc n); \
-\ ALL n. g(Suc n) <= g(n); \
-\ ALL n. f(n) <= g(n) |] \
-\ ==> Bseq g";
-by (stac (Bseq_minus_iff RS sym) 1);
-by (res_inst_tac [("g","%x. -(f x)")] f_inc_g_dec_Beq_f 1);
-by Auto_tac;
-qed "f_inc_g_dec_Beq_g";
-
-Goal "[| ALL n. f n <= f (Suc n); convergent f |] ==> f n <= lim f";
-by (rtac real_leI 1);
-by (auto_tac (claset(),
- simpset() addsimps [convergent_LIMSEQ_iff, LIMSEQ_iff, monoseq_Suc]));
-by (dtac real_less_sum_gt_zero 1);
-by (dres_inst_tac [("x","f n + - lim f")] spec 1);
-by Safe_tac;
-by (dres_inst_tac [("P","%na. no<=na --> ?Q na"),("x","no + n")] spec 2);
-by Auto_tac;
-by (subgoal_tac "lim f <= f(no + n)" 1);
-by (induct_tac "no" 2);
-by (auto_tac (claset() addIs [real_le_trans],
- simpset() addsimps [real_diff_def, real_abs_def]));
-by (dres_inst_tac [("i","f(no + n)"),("no1","no")]
- (lemma_f_mono_add RSN (2,real_less_le_trans)) 1);
-by (auto_tac (claset(), simpset() addsimps [add_commute]));
-qed "f_inc_imp_le_lim";
-
-Goal "convergent g ==> lim (%x. - g x) = - (lim g)";
-by (rtac (LIMSEQ_minus RS limI) 1);
-by (asm_full_simp_tac (simpset() addsimps [convergent_LIMSEQ_iff]) 1);
-qed "lim_uminus";
-
-Goal "[| ALL n. g(Suc n) <= g(n); convergent g |] ==> lim g <= g n";
-by (subgoal_tac "- (g n) <= - (lim g)" 1);
-by (cut_inst_tac [("f", "%x. - (g x)")] f_inc_imp_le_lim 2);
-by (auto_tac (claset(),
- simpset() addsimps [lim_uminus, convergent_minus_iff RS sym]));
-qed "g_dec_imp_lim_le";
-
-Goal "[| ALL n. f(n) <= f(Suc n); \
-\ ALL n. g(Suc n) <= g(n); \
-\ ALL n. f(n) <= g(n) |] \
-\ ==> EX l m. l <= m & ((ALL n. f(n) <= l) & f ----> l) & \
-\ ((ALL n. m <= g(n)) & g ----> m)";
-by (subgoal_tac "monoseq f & monoseq g" 1);
-by (force_tac (claset(), simpset() addsimps [LIMSEQ_iff,monoseq_Suc]) 2);
-by (subgoal_tac "Bseq f & Bseq g" 1);
-by (blast_tac (claset() addIs [f_inc_g_dec_Beq_f, f_inc_g_dec_Beq_g]) 2);
-by (auto_tac (claset() addSDs [Bseq_monoseq_convergent],
- simpset() addsimps [convergent_LIMSEQ_iff]));
-by (res_inst_tac [("x","lim f")] exI 1);
-by (res_inst_tac [("x","lim g")] exI 1);
-by (auto_tac (claset() addIs [LIMSEQ_le], simpset()));
-by (auto_tac (claset(),
- simpset() addsimps [f_inc_imp_le_lim, g_dec_imp_lim_le,
- convergent_LIMSEQ_iff]));
-qed "lemma_nest";
-
-Goal "[| ALL n. f(n) <= f(Suc n); \
-\ ALL n. g(Suc n) <= g(n); \
-\ ALL n. f(n) <= g(n); \
-\ (%n. f(n) - g(n)) ----> 0 |] \
-\ ==> EX l. ((ALL n. f(n) <= l) & f ----> l) & \
-\ ((ALL n. l <= g(n)) & g ----> l)";
-by (dtac lemma_nest 1 THEN Auto_tac);
-by (subgoal_tac "l = m" 1);
-by (dres_inst_tac [("X","f")] LIMSEQ_diff 2);
-by (auto_tac (claset() addIs [LIMSEQ_unique], simpset()));
-qed "lemma_nest_unique";
-
-
-Goal "a <= b ==> \
-\ ALL n. fst (Bolzano_bisect P a b n) <= snd (Bolzano_bisect P a b n)";
-by (rtac allI 1);
-by (induct_tac "n" 1);
-by (auto_tac (claset(), simpset() addsimps [Let_def, split_def]));
-qed "Bolzano_bisect_le";
-
-Goal "a <= b ==> \
-\ ALL n. fst(Bolzano_bisect P a b n) <= fst (Bolzano_bisect P a b (Suc n))";
-by (rtac allI 1);
-by (induct_tac "n" 1);
-by (auto_tac (claset(),
- simpset() addsimps [Bolzano_bisect_le, Let_def, split_def]));
-qed "Bolzano_bisect_fst_le_Suc";
-
-Goal "a <= b ==> \
-\ ALL n. snd(Bolzano_bisect P a b (Suc n)) <= snd (Bolzano_bisect P a b n)";
-by (rtac allI 1);
-by (induct_tac "n" 1);
-by (auto_tac (claset(),
- simpset() addsimps [Bolzano_bisect_le, Let_def, split_def]));
-qed "Bolzano_bisect_Suc_le_snd";
-
-Goal "((x::real) = y / (#2 * z)) = (#2 * x = y/z)";
-by Auto_tac;
-by (dres_inst_tac [("f","%u. (#1/#2)*u")] arg_cong 1);
-by Auto_tac;
-qed "eq_divide_2_times_iff";
-
-Goal "a <= b ==> \
-\ snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) = \
-\ (b-a) / (#2 ^ n)";
-by (induct_tac "n" 1);
-by (auto_tac (claset(),
- simpset() addsimps [eq_divide_2_times_iff, real_add_divide_distrib,
- Let_def, split_def]));
-by (auto_tac (claset(),
- simpset() addsimps (real_add_ac@[Bolzano_bisect_le, real_diff_def])));
-qed "Bolzano_bisect_diff";
-
-val Bolzano_nest_unique =
- [Bolzano_bisect_fst_le_Suc, Bolzano_bisect_Suc_le_snd, Bolzano_bisect_le]
- MRS lemma_nest_unique;
-
-(*P_prem is a looping simprule, so it works better if it isn't an assumption*)
-val P_prem::notP_prem::rest =
-Goal "[| !!a b c. [| P(a,b); P(b,c); a <= b; b <= c|] ==> P(a,c); \
-\ ~ P(a,b); a <= b |] ==> \
-\ ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))";
-by (cut_facts_tac rest 1);
-by (induct_tac "n" 1);
-by (auto_tac (claset(),
- simpset() delsimps [surjective_pairing RS sym]
- addsimps [notP_prem, Let_def, split_def]));
-by (swap_res_tac [P_prem] 1);
-by (assume_tac 1);
-by (auto_tac (claset(), simpset() addsimps [Bolzano_bisect_le]));
-qed "not_P_Bolzano_bisect";
-
-(*Now we re-package P_prem as a formula*)
-Goal "[| ALL a b c. P(a,b) & P(b,c) & a <= b & b <= c --> P(a,c); \
-\ ~ P(a,b); a <= b |] ==> \
-\ ALL n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))";
-by (blast_tac (claset() addSEs [not_P_Bolzano_bisect RSN (2,rev_notE)]) 1);
-qed "not_P_Bolzano_bisect'";
-
-
-Goal "[| ALL a b c. P(a,b) & P(b,c) & a <= b & b <= c --> P(a,c); \
-\ ALL x. EX d::real. 0 < d & \
-\ (ALL a b. a <= x & x <= b & (b - a) < d --> P(a,b)); \
-\ a <= b |] \
-\ ==> P(a,b)";
-by (rtac (inst "P1" "P" Bolzano_nest_unique RS exE) 1);
-by (REPEAT (assume_tac 1));
-by (rtac LIMSEQ_minus_cancel 1);
-by (asm_simp_tac (simpset() addsimps [Bolzano_bisect_diff,
- LIMSEQ_divide_realpow_zero]) 1);
-by (rtac ccontr 1);
-by (dtac not_P_Bolzano_bisect' 1);
-by (REPEAT (assume_tac 1));
-by (rename_tac "l" 1);
-by (dres_inst_tac [("x","l")] spec 1 THEN Clarify_tac 1);
-by (rewtac LIMSEQ_def);
-by (dres_inst_tac [("P", "%r. #0<r --> ?Q r"), ("x","d/#2")] spec 1);
-by (dres_inst_tac [("P", "%r. #0<r --> ?Q r"), ("x","d/#2")] spec 1);
-by (dtac real_less_half_sum 1);
-by Safe_tac;
-(*linear arithmetic bug if we just use Asm_simp_tac*)
-by (ALLGOALS Asm_full_simp_tac);
-by (dres_inst_tac [("x","fst(Bolzano_bisect P a b (no + noa))")] spec 1);
-by (dres_inst_tac [("x","snd(Bolzano_bisect P a b (no + noa))")] spec 1);
-by Safe_tac;
-by (ALLGOALS Asm_simp_tac);
-by (res_inst_tac [("j","abs(fst(Bolzano_bisect P a b(no + noa)) - l) + \
-\ abs(snd(Bolzano_bisect P a b(no + noa)) - l)")]
- real_le_less_trans 1);
-by (asm_simp_tac (simpset() addsimps [real_abs_def]) 1);
-by (rtac (real_sum_of_halves RS subst) 1);
-by (rtac real_add_less_mono 1);
-by (ALLGOALS
- (asm_full_simp_tac (simpset() addsimps [symmetric real_diff_def])));
-qed "lemma_BOLZANO";
-
-
-Goal "((ALL a b c. (a <= b & b <= c & P(a,b) & P(b,c)) --> P(a,c)) & \
-\ (ALL x. EX d::real. 0 < d & \
-\ (ALL a b. a <= x & x <= b & (b - a) < d --> P(a,b)))) \
-\ --> (ALL a b. a <= b --> P(a,b))";
-by (Clarify_tac 1);
-by (blast_tac (claset() addIs [lemma_BOLZANO]) 1);
-qed "lemma_BOLZANO2";
-
-
-(*----------------------------------------------------------------------------*)
-(* Intermediate Value Theorem (prove contrapositive by bisection) *)
-(*----------------------------------------------------------------------------*)
-
-Goal "[| f(a) <= y & y <= f(b); \
-\ a <= b; \
-\ (ALL x. a <= x & x <= b --> isCont f x) |] \
-\ ==> EX x. a <= x & x <= b & f(x) = y";
-by (rtac contrapos_pp 1);
-by (assume_tac 1);
-by (cut_inst_tac
- [("P","%(u,v). a <= u & u <= v & v <= b --> ~(f(u) <= y & y <= f(v))")]
- lemma_BOLZANO2 1);
-by (Step_tac 1);
-by (ALLGOALS(Asm_full_simp_tac));
-by (Blast_tac 2);
-by (asm_full_simp_tac (simpset() addsimps [isCont_iff,LIM_def]) 1);
-by (rtac ccontr 1);
-by (subgoal_tac "a <= x & x <= b" 1);
-by (Asm_full_simp_tac 2);
-by (dres_inst_tac [("P", "%d. #0<d --> ?P d"),("x","#1")] spec 2);
-by (Step_tac 2);
-by (Asm_full_simp_tac 2);
-by (Asm_full_simp_tac 2);
-by (REPEAT(blast_tac (claset() addIs [real_le_trans]) 2));
-by (REPEAT(dres_inst_tac [("x","x")] spec 1));
-by (Asm_full_simp_tac 1);
-by (dres_inst_tac [("P", "%r. ?P r --> (EX s. #0<s & ?Q r s)"),
- ("x","abs(y - f x)")] spec 1);
-by (Step_tac 1);
-by (asm_full_simp_tac (simpset() addsimps []) 1);
-by (dres_inst_tac [("x","s")] spec 1);
-by (Clarify_tac 1);
-by (cut_inst_tac [("R1.0","f x"),("R2.0","y")] real_linear 1);
-by Safe_tac;
-by (dres_inst_tac [("x","ba - x")] spec 1);
-by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [abs_iff])));
-by (dres_inst_tac [("x","aa - x")] spec 1);
-by (case_tac "x <= aa" 1);
-by (ALLGOALS Asm_full_simp_tac);
-by (dres_inst_tac [("z","x"),("w","aa")] real_le_anti_sym 1);
-by (assume_tac 1 THEN Asm_full_simp_tac 1);
-qed "IVT";
-
-
-Goal "[| f(b) <= y & y <= f(a); \
-\ a <= b; \
-\ (ALL x. a <= x & x <= b --> isCont f x) \
-\ |] ==> EX x. a <= x & x <= b & f(x) = y";
-by (subgoal_tac "- f a <= -y & -y <= - f b" 1);
-by (thin_tac "f b <= y & y <= f a" 1);
-by (dres_inst_tac [("f","%x. - f x")] IVT 1);
-by (auto_tac (claset() addIs [isCont_minus],simpset()));
-qed "IVT2";
-
-
-(*HOL style here: object-level formulations*)
-Goal "(f(a) <= y & y <= f(b) & a <= b & \
-\ (ALL x. a <= x & x <= b --> isCont f x)) \
-\ --> (EX x. a <= x & x <= b & f(x) = y)";
-by (blast_tac (claset() addIs [IVT]) 1);
-qed "IVT_objl";
-
-Goal "(f(b) <= y & y <= f(a) & a <= b & \
-\ (ALL x. a <= x & x <= b --> isCont f x)) \
-\ --> (EX x. a <= x & x <= b & f(x) = y)";
-by (blast_tac (claset() addIs [IVT2]) 1);
-qed "IVT2_objl";
-
-(*---------------------------------------------------------------------------*)
-(* By bisection, function continuous on closed interval is bounded above *)
-(*---------------------------------------------------------------------------*)
-
-Goal "abs (real_of_nat x) = real_of_nat x";
-by (auto_tac (claset() addIs [abs_eqI1],simpset()));
-qed "abs_real_of_nat_cancel";
-Addsimps [abs_real_of_nat_cancel];
-
-Goal "~ abs(x) + 1r < x";
-by (rtac real_leD 1);
-by (auto_tac (claset() addIs [abs_ge_self RS real_le_trans],simpset()));
-qed "abs_add_one_not_less_self";
-Addsimps [abs_add_one_not_less_self];
-
-
-Goal "[| a <= b; ALL x. a <= x & x <= b --> isCont f x |]\
-\ ==> EX M. ALL x. a <= x & x <= b --> f(x) <= M";
-by (cut_inst_tac [("P","%(u,v). a <= u & u <= v & v <= b --> \
-\ (EX M. ALL x. u <= x & x <= v --> f x <= M)")]
- lemma_BOLZANO2 1);
-by (Step_tac 1);
-by (ALLGOALS(Asm_full_simp_tac));
-by (cut_inst_tac [("x","M"),("y","Ma")]
- (CLAIM "x <= y | y <= (x::real)") 1);
-by (Step_tac 1);
-by (res_inst_tac [("x","Ma")] exI 1);
-by (Step_tac 1);
-by (cut_inst_tac [("x","xb"),("y","xa")]
- (CLAIM "x <= y | y <= (x::real)") 1);
-by (Step_tac 1);
-by (rtac real_le_trans 1 THEN assume_tac 2);
-by (Asm_full_simp_tac 1);
-by (Asm_full_simp_tac 1);
-by (res_inst_tac [("x","M")] exI 1);
-by (Step_tac 1);
-by (cut_inst_tac [("x","xb"),("y","xa")]
- (CLAIM "x <= y | y <= (x::real)") 1);
-by (Step_tac 1);
-by (Asm_full_simp_tac 1);
-by (rtac real_le_trans 1 THEN assume_tac 2);
-by (Asm_full_simp_tac 1);
-by (case_tac "a <= x & x <= b" 1);
-by (res_inst_tac [("x","#1")] exI 2);
-by (auto_tac (claset(),
- simpset() addsimps [LIM_def,isCont_iff]));
-by (dres_inst_tac [("x","x")] spec 1 THEN Auto_tac);
-by (thin_tac "ALL M. EX x. a <= x & x <= b & ~ f x <= M" 1);
-by (dres_inst_tac [("x","#1")] spec 1);
-by Auto_tac;
-by (res_inst_tac [("x","s")] exI 1 THEN Auto_tac);
-by (res_inst_tac [("x","abs(f x) + #1")] exI 1 THEN Step_tac 1);
-by (dres_inst_tac [("x","xa - x")] spec 1 THEN Auto_tac);
-by (res_inst_tac [("j","abs(f xa)")] real_le_trans 3);
-by (res_inst_tac [("j","abs(f x) + abs(f(xa) - f(x))")] real_le_trans 4);
-by (auto_tac (claset() addIs [abs_triangle_ineq RSN (2, real_le_trans)],
- simpset() addsimps [real_diff_def,abs_ge_self]));
-by (auto_tac (claset(),
- simpset() addsimps [real_abs_def] addsplits [split_if_asm]));
-qed "isCont_bounded";
-
-(*----------------------------------------------------------------------------*)
-(* Refine the above to existence of least upper bound *)
-(*----------------------------------------------------------------------------*)
-
-Goal "((EX x. x : S) & (EX y. isUb UNIV S (y::real))) --> \
-\ (EX t. isLub UNIV S t)";
-by (blast_tac (claset() addIs [reals_complete]) 1);
-qed "lemma_reals_complete";
-
-Goal "[| a <= b; ALL x. a <= x & x <= b --> isCont f x |] \
-\ ==> EX M. (ALL x. a <= x & x <= b --> f(x) <= M) & \
-\ (ALL N. N < M --> (EX x. a <= x & x <= b & N < f(x)))";
-by (cut_inst_tac [("S","Collect (%y. EX x. a <= x & x <= b & y = f x)")]
- lemma_reals_complete 1);
-by Auto_tac;
-by (dtac isCont_bounded 1 THEN assume_tac 1);
-by (auto_tac (claset(),simpset() addsimps [isUb_def,leastP_def,
- isLub_def,setge_def,setle_def]));
-by (rtac exI 1 THEN Auto_tac);
-by (REPEAT(dtac spec 1) THEN Auto_tac);
-by (dres_inst_tac [("x","x")] spec 1);
-by (auto_tac (claset() addSIs [real_leI],simpset()));
-qed "isCont_has_Ub";
-
-(*----------------------------------------------------------------------------*)
-(* Now show that it attains its upper bound *)
-(*----------------------------------------------------------------------------*)
-
-Goal "[| a <= b; ALL x. a <= x & x <= b --> isCont f x |] \
-\ ==> EX M. (ALL x. a <= x & x <= b --> f(x) <= M) & \
-\ (EX x. a <= x & x <= b & f(x) = M)";
-by (ftac isCont_has_Ub 1 THEN assume_tac 1);
-by (Clarify_tac 1);
-by (res_inst_tac [("x","M")] exI 1);
-by (Asm_full_simp_tac 1);
-by (rtac ccontr 1);
-by (subgoal_tac "ALL x. a <= x & x <= b --> f x < M" 1 THEN Step_tac 1);
-by (rtac ccontr 2 THEN dtac real_leI 2);
-by (dres_inst_tac [("z","M")] real_le_anti_sym 2);
-by (REPEAT(Blast_tac 2));
-by (subgoal_tac "ALL x. a <= x & x <= b --> isCont (%x. inverse(M - f x)) x" 1);
-by (Step_tac 1);
-by (EVERY[rtac isCont_inverse 2, rtac isCont_diff 2, rtac notI 4]);
-by (ALLGOALS(asm_full_simp_tac (simpset() addsimps [real_diff_eq_eq])));
-by (Blast_tac 2);
-by (subgoal_tac
- "EX k. ALL x. a <= x & x <= b --> (%x. inverse(M - (f x))) x <= k" 1);
-by (rtac isCont_bounded 2);
-by (Step_tac 1);
-by (subgoal_tac "ALL x. a <= x & x <= b --> #0 < inverse(M - f(x))" 1);
-by (Asm_full_simp_tac 1);
-by (Step_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [real_less_diff_eq]) 2);
-by (subgoal_tac
- "ALL x. a <= x & x <= b --> (%x. inverse(M - (f x))) x < (k + #1)" 1);
-by (Step_tac 1);
-by (res_inst_tac [("j","k")] real_le_less_trans 2);
-by (asm_full_simp_tac (simpset() addsimps [real_zero_less_one]) 3);
-by (Asm_full_simp_tac 2);
-by (subgoal_tac "ALL x. a <= x & x <= b --> \
-\ inverse(k + #1) < inverse((%x. inverse(M - (f x))) x)" 1);
-by (Step_tac 1);
-by (rtac real_inverse_less_swap 2);
-by (ALLGOALS Asm_full_simp_tac);
-by (dres_inst_tac [("P", "%N. N<M --> ?Q N"),
- ("x","M - inverse(k + #1)")] spec 1);
-by (Step_tac 1 THEN dtac real_leI 1);
-by (dtac (real_le_diff_eq RS iffD1) 1);
-by (REPEAT(dres_inst_tac [("x","a")] spec 1));
-by (Asm_full_simp_tac 1);
-by (asm_full_simp_tac
- (simpset() addsimps [real_inverse_eq_divide, pos_real_divide_le_eq]) 1);
-by (cut_inst_tac [("x","k"),("y","M-f a")] real_0_less_mult_iff 1);
-by (Asm_full_simp_tac 1);
-(*last one*)
-by (REPEAT(dres_inst_tac [("x","x")] spec 1));
-by (Asm_full_simp_tac 1);
-qed "isCont_eq_Ub";
-
-
-(*----------------------------------------------------------------------------*)
-(* Same theorem for lower bound *)
-(*----------------------------------------------------------------------------*)
-
-Goal "[| a <= b; ALL x. a <= x & x <= b --> isCont f x |] \
-\ ==> EX M. (ALL x. a <= x & x <= b --> M <= f(x)) & \
-\ (EX x. a <= x & x <= b & f(x) = M)";
-by (subgoal_tac "ALL x. a <= x & x <= b --> isCont (%x. -(f x)) x" 1);
-by (blast_tac (claset() addIs [isCont_minus]) 2);
-by (dres_inst_tac [("f","(%x. -(f x))")] isCont_eq_Ub 1);
-by (Step_tac 1);
-by Auto_tac;
-qed "isCont_eq_Lb";
-
-
-(* ------------------------------------------------------------------------- *)
-(* Another version. *)
-(* ------------------------------------------------------------------------- *)
-
-Goal "[|a <= b; ALL x. a <= x & x <= b --> isCont f x |] \
-\ ==> EX L M. (ALL x. a <= x & x <= b --> L <= f(x) & f(x) <= M) & \
-\ (ALL y. L <= y & y <= M --> (EX x. a <= x & x <= b & (f(x) = y)))";
-by (ftac isCont_eq_Lb 1);
-by (ftac isCont_eq_Ub 2);
-by (REPEAT(assume_tac 1));
-by (Step_tac 1);
-by (res_inst_tac [("x","f x")] exI 1);
-by (res_inst_tac [("x","f xa")] exI 1);
-by (Asm_full_simp_tac 1);
-by (Step_tac 1);
-by (cut_inst_tac [("x","x"),("y","xa")] (CLAIM "x <= y | y <= (x::real)") 1);
-by (Step_tac 1);
-by (cut_inst_tac [("f","f"),("a","x"),("b","xa"),("y","y")] IVT_objl 1);
-by (cut_inst_tac [("f","f"),("a","xa"),("b","x"),("y","y")] IVT2_objl 2);
-by (Step_tac 1);
-by (res_inst_tac [("x","xb")] exI 2);
-by (res_inst_tac [("x","xb")] exI 4);
-by (ALLGOALS(Asm_full_simp_tac));
-qed "isCont_Lb_Ub";
-
-(*----------------------------------------------------------------------------*)
-(* If f'(x) > 0 then x is locally strictly increasing at the right *)
-(*----------------------------------------------------------------------------*)
-
-Goalw [deriv_def,LIM_def]
- "[| DERIV f x :> l; #0 < l |] ==> \
-\ EX d. #0 < d & (ALL h. #0 < h & h < d --> f(x) < f(x + h))";
-by (dtac spec 1 THEN Auto_tac);
-by (res_inst_tac [("x","s")] exI 1 THEN Auto_tac);
-by (subgoal_tac "#0 < l*h" 1);
-by (asm_full_simp_tac (simpset() addsimps [real_0_less_mult_iff]) 2);
-by (dres_inst_tac [("x","h")] spec 1);
-by (asm_full_simp_tac
- (simpset() addsimps [real_abs_def, real_inverse_eq_divide,
- pos_real_le_divide_eq, pos_real_less_divide_eq]
- addsplits [split_if_asm]) 1);
-qed "DERIV_left_inc";
-
-Goalw [deriv_def,LIM_def]
- "[| DERIV f x :> l; l < #0 |] ==> \
-\ EX d. #0 < d & (ALL h. #0 < h & h < d --> f(x) < f(x - h))";
-by (dres_inst_tac [("x","-l")] spec 1 THEN Auto_tac);
-by (res_inst_tac [("x","s")] exI 1 THEN Auto_tac);
-by (subgoal_tac "l*h < #0" 1);
-by (asm_full_simp_tac (simpset() addsimps [real_mult_less_0_iff]) 2);
-by (dres_inst_tac [("x","-h")] spec 1);
-by (asm_full_simp_tac
- (simpset() addsimps [real_abs_def, real_inverse_eq_divide,
- pos_real_less_divide_eq,
- symmetric real_diff_def]
- addsplits [split_if_asm]) 1);
-by (subgoal_tac "#0 < (f (x - h) - f x)/h" 1);
-by (arith_tac 2);
-by (asm_full_simp_tac
- (simpset() addsimps [pos_real_less_divide_eq]) 1);
-qed "DERIV_left_dec";
-
-
-Goal "[| DERIV f x :> l; \
-\ EX d. #0 < d & (ALL y. abs(x - y) < d --> f(y) <= f(x)) |] \
-\ ==> l = #0";
-by (res_inst_tac [("R1.0","l"),("R2.0","#0")] real_linear_less2 1);
-by (Step_tac 1);
-by (dtac DERIV_left_dec 1);
-by (dtac DERIV_left_inc 3);
-by (Step_tac 1);
-by (dres_inst_tac [("d1.0","d"),("d2.0","da")] real_lbound_gt_zero 1);
-by (dres_inst_tac [("d1.0","d"),("d2.0","da")] real_lbound_gt_zero 3);
-by (Step_tac 1);
-by (dres_inst_tac [("x","x - e")] spec 1);
-by (dres_inst_tac [("x","x + e")] spec 2);
-by (auto_tac (claset(), simpset() addsimps [real_abs_def]));
-qed "DERIV_local_max";
-
-(*----------------------------------------------------------------------------*)
-(* Similar theorem for a local minimum *)
-(*----------------------------------------------------------------------------*)
-
-Goal "[| DERIV f x :> l; \
-\ EX d::real. #0 < d & (ALL y. abs(x - y) < d --> f(x) <= f(y)) |] \
-\ ==> l = #0";
-by (dtac (DERIV_minus RS DERIV_local_max) 1);
-by Auto_tac;
-qed "DERIV_local_min";
-
-(*----------------------------------------------------------------------------*)
-(* In particular if a function is locally flat *)
-(*----------------------------------------------------------------------------*)
-
-Goal "[| DERIV f x :> l; \
-\ EX d. #0 < d & (ALL y. abs(x - y) < d --> f(x) = f(y)) |] \
-\ ==> l = #0";
-by (auto_tac (claset() addSDs [DERIV_local_max],simpset()));
-qed "DERIV_local_const";
-
-(*----------------------------------------------------------------------------*)
-(* Lemma about introducing open ball in open interval *)
-(*----------------------------------------------------------------------------*)
-
-Goal "[| a < x; x < b |] ==> \
-\ EX d::real. #0 < d & (ALL y. abs(x - y) < d --> a < y & y < b)";
-by (simp_tac (simpset() addsimps [abs_interval_iff]) 1);
-by (cut_inst_tac [("x","x - a"),("y","b - x")]
- (CLAIM "x <= y | y <= (x::real)") 1);
-by (Step_tac 1);
-by (res_inst_tac [("x","x - a")] exI 1);
-by (res_inst_tac [("x","b - x")] exI 2);
-by Auto_tac;
-by (auto_tac (claset(),simpset() addsimps [real_less_diff_eq]));
-qed "lemma_interval_lt";
-
-Goal "[| a < x; x < b |] ==> \
-\ EX d::real. #0 < d & (ALL y. abs(x - y) < d --> a <= y & y <= b)";
-by (dtac lemma_interval_lt 1);
-by Auto_tac;
-by (auto_tac (claset() addSIs [exI] ,simpset()));
-qed "lemma_interval";
-
-(*-----------------------------------------------------------------------
- Rolle's Theorem
- If f is defined and continuous on the finite closed interval [a,b]
- and differentiable a least on the open interval (a,b), and f(a) = f(b),
- then x0 : (a,b) such that f'(x0) = #0
- ----------------------------------------------------------------------*)
-
-Goal "[| a < b; f(a) = f(b); \
-\ ALL x. a <= x & x <= b --> isCont f x; \
-\ ALL x. a < x & x < b --> f differentiable x \
-\ |] ==> EX z. a < z & z < b & DERIV f z :> #0";
-by (ftac (real_less_imp_le RS isCont_eq_Ub) 1);
-by (EVERY1[assume_tac,Step_tac]);
-by (ftac (real_less_imp_le RS isCont_eq_Lb) 1);
-by (EVERY1[assume_tac,Step_tac]);
-by (case_tac "a < x & x < b" 1 THEN etac conjE 1);
-by (Asm_full_simp_tac 2);
-by (forw_inst_tac [("a","a"),("x","x")] lemma_interval 1);
-by (EVERY1[assume_tac,etac exE]);
-by (res_inst_tac [("x","x")] exI 1 THEN Asm_full_simp_tac 1);
-by (subgoal_tac "(EX l. DERIV f x :> l) & \
-\ (EX d. #0 < d & (ALL y. abs(x - y) < d --> f(y) <= f(x)))" 1);
-by (Clarify_tac 1 THEN rtac conjI 2);
-by (blast_tac (claset() addIs [differentiableD]) 2);
-by (Blast_tac 2);
-by (ftac DERIV_local_max 1);
-by (EVERY1[Blast_tac,Blast_tac]);
-by (case_tac "a < xa & xa < b" 1 THEN etac conjE 1);
-by (Asm_full_simp_tac 2);
-by (forw_inst_tac [("a","a"),("x","xa")] lemma_interval 1);
-by (EVERY1[assume_tac,etac exE]);
-by (res_inst_tac [("x","xa")] exI 1 THEN Asm_full_simp_tac 1);
-by (subgoal_tac "(EX l. DERIV f xa :> l) & \
-\ (EX d. #0 < d & (ALL y. abs(xa - y) < d --> f(xa) <= f(y)))" 1);
-by (Clarify_tac 1 THEN rtac conjI 2);
-by (blast_tac (claset() addIs [differentiableD]) 2);
-by (Blast_tac 2);
-by (ftac DERIV_local_min 1);
-by (EVERY1[Blast_tac,Blast_tac]);
-by (subgoal_tac "ALL x. a <= x & x <= b --> f(x) = f(b)" 1);
-by (Clarify_tac 2);
-by (rtac real_le_anti_sym 2);
-by (subgoal_tac "f b = f x" 2);
-by (Asm_full_simp_tac 2);
-by (res_inst_tac [("x1","a"),("y1","x")] (real_le_imp_less_or_eq RS disjE) 2);
-by (assume_tac 2);
-by (dres_inst_tac [("z","x"),("w","b")] real_le_anti_sym 2);
-by (subgoal_tac "f b = f xa" 5);
-by (Asm_full_simp_tac 5);
-by (res_inst_tac [("x1","a"),("y1","xa")] (real_le_imp_less_or_eq RS disjE) 5);
-by (assume_tac 5);
-by (dres_inst_tac [("z","xa"),("w","b")] real_le_anti_sym 5);
-by (REPEAT(Asm_full_simp_tac 2));
-by (dtac real_dense 1 THEN etac exE 1);
-by (res_inst_tac [("x","r")] exI 1 THEN Asm_full_simp_tac 1);
-by (etac conjE 1);
-by (forw_inst_tac [("a","a"),("x","r")] lemma_interval 1);
-by (EVERY1[assume_tac, etac exE]);
-by (subgoal_tac "(EX l. DERIV f r :> l) & \
-\ (EX d. #0 < d & (ALL y. abs(r - y) < d --> f(r) = f(y)))" 1);
-by (Clarify_tac 1 THEN rtac conjI 2);
-by (blast_tac (claset() addIs [differentiableD]) 2);
-by (EVERY1[ftac DERIV_local_const, Blast_tac, Blast_tac]);
-by (res_inst_tac [("x","d")] exI 1);
-by (EVERY1[rtac conjI, Blast_tac, rtac allI, rtac impI]);
-by (res_inst_tac [("s","f b")] trans 1);
-by (blast_tac (claset() addSDs [real_less_imp_le]) 1);
-by (rtac sym 1 THEN Blast_tac 1);
-qed "Rolle";
-
-(*----------------------------------------------------------------------------*)
-(* Mean value theorem *)
-(*----------------------------------------------------------------------------*)
-
-Goal "f a - (f b - f a)/(b - a) * a = \
-\ f b - (f b - f a)/(b - a) * (b::real)";
-by (case_tac "a = b" 1);
-by (asm_full_simp_tac (simpset() addsimps [REAL_DIVIDE_ZERO]) 1);
-by (res_inst_tac [("c1","b - a")] (real_mult_left_cancel RS iffD1) 1);
-by (arith_tac 1);
-by (auto_tac (claset(),
- simpset() addsimps [real_diff_mult_distrib2]));
-by (auto_tac (claset(),
- simpset() addsimps [real_diff_mult_distrib]));
-qed "lemma_MVT";
-
-Goal "[| a < b; \
-\ ALL x. a <= x & x <= b --> isCont f x; \
-\ ALL x. a < x & x < b --> f differentiable x |] \
-\ ==> EX l z. a < z & z < b & DERIV f z :> l & \
-\ (f(b) - f(a) = (b - a) * l)";
-by (dres_inst_tac [("f","%x. f(x) - (((f(b) - f(a)) / (b - a)) * x)")]
- Rolle 1);
-by (rtac lemma_MVT 1);
-by (Step_tac 1);
-by (rtac isCont_diff 1 THEN Blast_tac 1);
-by (rtac (isCont_const RS isCont_mult) 1);
-by (rtac isCont_Id 1);
-by (dres_inst_tac [("P", "%x. ?Pre x --> f differentiable x"),
- ("x","x")] spec 1);
-by (asm_full_simp_tac (simpset() addsimps [differentiable_def]) 1);
-by (Step_tac 1);
-by (res_inst_tac [("x","xa - ((f(b) - f(a)) / (b - a))")] exI 1);
-by (rtac DERIV_diff 1 THEN assume_tac 1);
-(*derivative of a linear function is the constant...*)
-by (subgoal_tac "(%x. (f b - f a) * x / (b - a)) = \
-\ op * ((f b - f a) / (b - a))" 1);
-by (rtac ext 2 THEN Simp_tac 2);
-by (Asm_full_simp_tac 1);
-(*final case*)
-by (res_inst_tac [("x","((f(b) - f(a)) / (b - a))")] exI 1);
-by (res_inst_tac [("x","z")] exI 1);
-by (Step_tac 1);
-by (Asm_full_simp_tac 2);
-by (subgoal_tac "DERIV (%x. ((f(b) - f(a)) / (b - a)) * x) z :> \
-\ ((f(b) - f(a)) / (b - a))" 1);
-by (rtac DERIV_cmult_Id 2);
-by (dtac DERIV_add 1 THEN assume_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [real_add_assoc, real_diff_def]) 1);
-qed "MVT";
-
-(*----------------------------------------------------------------------------*)
-(* Theorem that function is constant if its derivative is 0 over an interval. *)
-(*----------------------------------------------------------------------------*)
-
-Goal "[| a < b; \
-\ ALL x. a <= x & x <= b --> isCont f x; \
-\ ALL x. a < x & x < b --> DERIV f x :> #0 |] \
-\ ==> (f b = f a)";
-by (dtac MVT 1 THEN assume_tac 1);
-by (blast_tac (claset() addIs [differentiableI]) 1);
-by (auto_tac (claset() addSDs [DERIV_unique],simpset()
- addsimps [real_diff_eq_eq]));
-qed "DERIV_isconst_end";
-
-Goal "[| a < b; \
-\ ALL x. a <= x & x <= b --> isCont f x; \
-\ ALL x. a < x & x < b --> DERIV f x :> #0 |] \
-\ ==> ALL x. a <= x & x <= b --> f x = f a";
-by (Step_tac 1);
-by (dres_inst_tac [("x","a")] real_le_imp_less_or_eq 1);
-by (Step_tac 1);
-by (dres_inst_tac [("b","x")] DERIV_isconst_end 1);
-by Auto_tac;
-qed "DERIV_isconst1";
-
-Goal "[| a < b; \
-\ ALL x. a <= x & x <= b --> isCont f x; \
-\ ALL x. a < x & x < b --> DERIV f x :> #0; \
-\ a <= x; x <= b |] \
-\ ==> f x = f a";
-by (blast_tac (claset() addDs [DERIV_isconst1]) 1);
-qed "DERIV_isconst2";
-
-Goal "ALL x. DERIV f x :> #0 ==> f(x) = f(y)";
-by (res_inst_tac [("R1.0","x"),("R2.0","y")] real_linear_less2 1);
-by (rtac sym 1);
-by (auto_tac (claset() addIs [DERIV_isCont,DERIV_isconst_end],simpset()));
-qed "DERIV_isconst_all";
-
-Goal "[|a ~= b; ALL x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b - a) * k";
-by (res_inst_tac [("R1.0","a"),("R2.0","b")] real_linear_less2 1);
-by Auto_tac;
-by (ALLGOALS(dres_inst_tac [("f","f")] MVT));
-by (auto_tac (claset() addDs [DERIV_isCont,DERIV_unique],simpset() addsimps
- [differentiable_def]));
-by (auto_tac (claset() addDs [DERIV_unique],
- simpset() addsimps [real_add_mult_distrib, real_diff_def,
- real_minus_mult_eq1 RS sym]));
-qed "DERIV_const_ratio_const";
-
-Goal "[|a ~= b; ALL x. DERIV f x :> k |] ==> (f(b) - f(a))/(b - a) = k";
-by (res_inst_tac [("c1","b - a")] (real_mult_right_cancel RS iffD1) 1);
-by (auto_tac (claset() addSDs [DERIV_const_ratio_const],
- simpset() addsimps [real_mult_assoc]));
-qed "DERIV_const_ratio_const2";
-
-Goal "((a + b) /#2 - a) = (b - a)/(#2::real)";
-by Auto_tac;
-qed "real_average_minus_first";
-Addsimps [real_average_minus_first];
-
-Goal "((b + a)/#2 - a) = (b - a)/(#2::real)";
-by Auto_tac;
-qed "real_average_minus_second";
-Addsimps [real_average_minus_second];
-
-
-(* Gallileo's "trick": average velocity = av. of end velocities *)
-Goal "[|a ~= (b::real); ALL x. DERIV v x :> k|] \
-\ ==> v((a + b)/#2) = (v a + v b)/#2";
-by (res_inst_tac [("R1.0","a"),("R2.0","b")] real_linear_less2 1);
-by Auto_tac;
-by (ftac DERIV_const_ratio_const2 1 THEN assume_tac 1);
-by (ftac DERIV_const_ratio_const2 2 THEN assume_tac 2);
-by (dtac real_less_half_sum 1);
-by (dtac real_gt_half_sum 2);
-by (ftac (real_not_refl2 RS DERIV_const_ratio_const2) 1 THEN assume_tac 1);
-by (dtac ((real_not_refl2 RS not_sym) RS DERIV_const_ratio_const2) 2
- THEN assume_tac 2);
-by (ALLGOALS (dres_inst_tac [("f","%u. (b-a)*u")] arg_cong));
-by (auto_tac (claset(), simpset() addsimps [real_inverse_eq_divide]));
-by (asm_full_simp_tac (simpset() addsimps [real_add_commute, eq_commute]) 1);
-qed "DERIV_const_average";
-
-
-(* ------------------------------------------------------------------------ *)
-(* Dull lemma that an continuous injection on an interval must have a strict*)
-(* maximum at an end point, not in the middle. *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "[|#0 < d; ALL z. abs(z - x) <= d --> g(f z) = z; \
-\ ALL z. abs(z - x) <= d --> isCont f z |] \
-\ ==> ~(ALL z. abs(z - x) <= d --> f(z) <= f(x))";
-by (rtac notI 1);
-by (rotate_tac 3 1);
-by (forw_inst_tac [("x","x - d")] spec 1);
-by (forw_inst_tac [("x","x + d")] spec 1);
-by (Step_tac 1);
-by (cut_inst_tac [("x","f(x - d)"),("y","f(x + d)")]
- (ARITH_PROVE "x <= y | y <= (x::real)") 4);
-by (etac disjE 4);
-by (REPEAT(arith_tac 1));
-by (cut_inst_tac [("f","f"),("a","x - d"),("b","x"),("y","f(x + d)")]
- IVT_objl 1);
-by (Step_tac 1);
-by (arith_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [abs_le_interval_iff]) 1);
-by (dres_inst_tac [("f","g")] arg_cong 1);
-by (rotate_tac 2 1);
-by (forw_inst_tac [("x","xa")] spec 1);
-by (dres_inst_tac [("x","x + d")] spec 1);
-by (asm_full_simp_tac (simpset() addsimps [abs_le_interval_iff]) 1);
-(* 2nd case: similar *)
-by (cut_inst_tac [("f","f"),("a","x"),("b","x + d"),("y","f(x - d)")]
- IVT2_objl 1);
-by (Step_tac 1);
-by (arith_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [abs_le_interval_iff]) 1);
-by (dres_inst_tac [("f","g")] arg_cong 1);
-by (rotate_tac 2 1);
-by (forw_inst_tac [("x","xa")] spec 1);
-by (dres_inst_tac [("x","x - d")] spec 1);
-by (asm_full_simp_tac (simpset() addsimps [abs_le_interval_iff]) 1);
-qed "lemma_isCont_inj";
-
-(* ------------------------------------------------------------------------ *)
-(* Similar version for lower bound *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "[|#0 < d; ALL z. abs(z - x) <= d --> g(f z) = z; \
-\ ALL z. abs(z - x) <= d --> isCont f z |] \
-\ ==> ~(ALL z. abs(z - x) <= d --> f(x) <= f(z))";
-by (auto_tac (claset() addSDs [(asm_full_simplify (simpset())
- (read_instantiate [("f","%x. - f x"),("g","%y. g(-y)"),("x","x"),("d","d")]
- lemma_isCont_inj))],simpset() addsimps [isCont_minus]));
-qed "lemma_isCont_inj2";
-
-(* ------------------------------------------------------------------------ *)
-(* Show there's an interval surrounding f(x) in f[[x - d, x + d]] *)
-(* Also from John's theory *)
-(* ------------------------------------------------------------------------ *)
-
-Addsimps [zero_eq_numeral_0,one_eq_numeral_1];
-
-val lemma_le = ARITH_PROVE "#0 <= (d::real) ==> -d <= d";
-
-(* FIXME: awful proof - needs improvement *)
-Goal "[| #0 < d; ALL z. abs(z - x) <= d --> g(f z) = z; \
-\ ALL z. abs(z - x) <= d --> isCont f z |] \
-\ ==> EX e. #0 < e & \
-\ (ALL y. \
-\ abs(y - f(x)) <= e --> \
-\ (EX z. abs(z - x) <= d & (f z = y)))";
-by (ftac real_less_imp_le 1);
-by (dtac (lemma_le RS (asm_full_simplify (simpset()) (read_instantiate
- [("f","f"),("a","x - d"),("b","x + d")] isCont_Lb_Ub))) 1);
-by (Step_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [abs_le_interval_iff]) 1);
-by (subgoal_tac "L <= f x & f x <= M" 1);
-by (dres_inst_tac [("P", "%v. ?P v --> ?Q v & ?R v"), ("x","x")] spec 2);
-by (Asm_full_simp_tac 2);
-by (subgoal_tac "L < f x & f x < M" 1);
-by (Step_tac 1);
-by (dres_inst_tac [("x","L")] (ARITH_PROVE "x < y ==> #0 < y - (x::real)") 1);
-by (dres_inst_tac [("x","f x")] (ARITH_PROVE "x < y ==> #0 < y - (x::real)") 1);
-by (dres_inst_tac [("d1.0","f x - L"),("d2.0","M - f x")]
- (rename_numerals real_lbound_gt_zero) 1);
-by (Step_tac 1);
-by (res_inst_tac [("x","e")] exI 1);
-by (Step_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [abs_le_interval_iff]) 1);
-by (dres_inst_tac [("P","%v. ?PP v --> (EX xa. ?Q v xa)"),("x","y")] spec 1);
-by (Step_tac 1 THEN REPEAT(arith_tac 1));
-by (res_inst_tac [("x","xa")] exI 1);
-by (arith_tac 1);
-by (ALLGOALS(etac (ARITH_PROVE "[|x <= y; x ~= y |] ==> x < (y::real)")));
-by (ALLGOALS(rotate_tac 3));
-by (dtac lemma_isCont_inj2 1);
-by (assume_tac 2);
-by (dtac lemma_isCont_inj 3);
-by (assume_tac 4);
-by (TRYALL(assume_tac));
-by (Step_tac 1);
-by (ALLGOALS(dres_inst_tac [("x","z")] spec));
-by (ALLGOALS(arith_tac));
-qed "isCont_inj_range";
-
-
-(* ------------------------------------------------------------------------ *)
-(* Continuity of inverse function *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "[| #0 < d; ALL z. abs(z - x) <= d --> g(f(z)) = z; \
-\ ALL z. abs(z - x) <= d --> isCont f z |] \
-\ ==> isCont g (f x)";
-by (simp_tac (simpset() addsimps [isCont_iff,LIM_def]) 1);
-by (Step_tac 1);
-by (dres_inst_tac [("d1.0","r")] (rename_numerals real_lbound_gt_zero) 1);
-by (assume_tac 1 THEN Step_tac 1);
-by (subgoal_tac "ALL z. abs(z - x) <= e --> (g(f z) = z)" 1);
-by (Force_tac 2);
-by (subgoal_tac "ALL z. abs(z - x) <= e --> isCont f z" 1);
-by (Force_tac 2);
-by (dres_inst_tac [("d","e")] isCont_inj_range 1);
-by (assume_tac 2 THEN assume_tac 1);
-by (Step_tac 1);
-by (res_inst_tac [("x","ea")] exI 1);
-by Auto_tac;
-by (rotate_tac 4 1);
-by (dres_inst_tac [("x","f(x) + xa")] spec 1);
-by Auto_tac;
-by (dtac sym 1 THEN Auto_tac);
-by (arith_tac 1);
-qed "isCont_inverse";
-