--- a/src/HOL/Hyperreal/Lim.ML Fri Mar 19 10:50:06 2004 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2307 +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 (Clarify_tac 1);
-by (REPEAT(dres_inst_tac [("x","r/2")] spec 1));
-by (Asm_full_simp_tac 1);
-by (Clarify_tac 1);
-by (res_inst_tac [("x","s"),("y","sa")] linorder_cases 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 Safe_tac;
-by (REPEAT(dres_inst_tac [("x","xa")] spec 1)
- THEN step_tac (claset() addSEs [order_less_trans]) 1);
-by (REPEAT(dres_inst_tac [("x","xa")] spec 2)
- THEN step_tac (claset() addSEs [order_less_trans]) 2);
-by (REPEAT(dres_inst_tac [("x","xa")] spec 3)
- THEN step_tac (claset() addSEs [order_less_trans]) 3);
-by (ALLGOALS(rtac (abs_sum_triangle_ineq RS order_le_less_trans)));
-by (ALLGOALS(rtac (real_sum_of_halves RS subst)));
-by (auto_tac (claset() addIs [add_strict_mono],simpset()));
-qed "LIM_add";
-
-Goalw [LIM_def] "f -- a --> L ==> (%x. -f(x)) -- a --> -L";
-by (subgoal_tac "ALL x. abs(- f x + L) = abs(f x + - L)" 1);
-by (Asm_full_simp_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [real_abs_def]) 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 [("a1","l")] ((right_minus RS subst)) 1);
-by (rtac LIM_add_minus 1 THEN Auto_tac);
-qed "LIM_zero";
-
-(*--------------------------
- Limit not zero
- --------------------------*)
-Goalw [LIM_def] "k \\<noteq> 0 ==> ~ ((%x. k) -- x --> 0)";
-by (res_inst_tac [("x","k"),("y","0")] linorder_cases 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 \\<noteq> 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 Safe_tac;
-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 [("x","s"),("y","sa")]
- linorder_cases 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 Safe_tac;
-by (REPEAT(dres_inst_tac [("x","xa")] spec 1)
- THEN step_tac (claset() addSEs [order_less_trans]) 1);
-by (REPEAT(dres_inst_tac [("x","xa")] spec 2)
- THEN step_tac (claset() addSEs [order_less_trans]) 2);
-by (REPEAT(dres_inst_tac [("x","xa")] spec 3)
- THEN step_tac (claset() addSEs [order_less_trans]) 3);
-by (ALLGOALS(res_inst_tac [("t","r")] (real_mult_1 RS subst)));
-by (ALLGOALS(rtac abs_mult_less));
-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]
- "[| \\<forall>x. x \\<noteq> 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,approx_def]
- "f -- x --> L ==> f -- x --NS> L";
-by (asm_full_simp_tac
- (simpset() addsimps [Infinitesimal_FreeUltrafilterNat_iff]) 1);
-by Safe_tac;
-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 "\\<forall>n::nat. (xa n) \\<noteq> 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 "\\<forall>s. 0 < s --> (\\<exists>xa. xa \\<noteq> x & \
-\ abs (xa + - x) < s & r \\<le> abs (f xa + -L)) \
-\ ==> \\<forall>n::nat. \\<exists>xa. xa \\<noteq> x & \
-\ abs(xa + -x) < inverse(real(Suc n)) & r \\<le> abs(f xa + -L)";
-by (Clarify_tac 1);
-by (cut_inst_tac [("n1","n")]
- (real_of_nat_Suc_gt_zero RS positive_imp_inverse_positive) 1);
-by Auto_tac;
-qed "lemma_LIM";
-
-Goal "\\<forall>s. 0 < s --> (\\<exists>xa. xa \\<noteq> x & \
-\ abs (xa + - x) < s & r \\<le> abs (f xa + -L)) \
-\ ==> \\<exists>X. \\<forall>n::nat. X n \\<noteq> x & \
-\ abs(X n + -x) < inverse(real(Suc n)) & r \\<le> abs(f (X n) + -L)";
-by (dtac lemma_LIM 1);
-by (dtac choice 1);
-by (Blast_tac 1);
-qed "lemma_skolemize_LIM2";
-
-Goal "\\<forall>n. X n \\<noteq> x & \
-\ abs (X n + - x) < inverse (real(Suc n)) & \
-\ r \\<le> abs (f (X n) + - L) ==> \
-\ \\<forall>n. abs (X n + - x) < inverse (real(Suc n))";
-by (Auto_tac );
-qed "lemma_simp";
-
-(*-------------------
- NSLIM => LIM
- -------------------*)
-
-Goalw [LIM_def,NSLIM_def,approx_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 (asm_full_simp_tac (simpset() addsimps [linorder_not_less]) 1);
-by (dtac lemma_skolemize_LIM2 1);
-by Safe_tac;
-by (dres_inst_tac [("x","Abs_hypreal(hyprel``{X})")] spec 1);
-by (auto_tac
- (claset(),
- simpset() addsimps [starfun, hypreal_minus,
- hypreal_of_real_def,hypreal_add]));
-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 (dtac spec 1 THEN dtac mp 1 THEN assume_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 [approx_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 [approx_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 \\<noteq> 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_approx_inverse]));
-qed "NSLIM_inverse";
-
-Goal "[| f -- a --> L; \
-\ L \\<noteq> 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 [("a1","l")] ((right_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 \\<noteq> 0 ==> ~ ((%x. k) -- x --NS> 0)";
-by Auto_tac;
-by (res_inst_tac [("x","hypreal_of_real x + epsilon")] exI 1);
-by (auto_tac (claset() addIs [Infinitesimal_add_approx_self RS approx_sym],
- simpset() addsimps [hypreal_epsilon_not_zero]));
-qed "NSLIM_not_zero";
-
-(* [| k \\<noteq> 0; (%x. k) -- x --NS> 0 |] ==> R *)
-bind_thm("NSLIM_not_zeroE", NSLIM_not_zero RS notE);
-
-Goal "k \\<noteq> 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_approx],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 \\<approx> hypreal_of_real a |] \
-\ ==> ( *f* f) y \\<approx> 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;
-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 Safe_tac;
-by (Asm_full_simp_tac 1);
-by (rtac ((mem_infmal_iff RS iffD2) RS
- (Infinitesimal_add_approx_self RS approx_sym)) 1);
-by (rtac (approx_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, approx_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 [approx_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_approx],
- 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 \\<noteq> 0 |] ==> isCont (%x. inverse (f x)) x";
-by (blast_tac (claset() addIs [LIM_inverse]) 1);
-qed "isCont_inverse";
-
-Goal "[| isNSCont f x; f x \\<noteq> 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 [approx_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) \\<in> A}, of any open set A
- is always an open set
- ------------------------------------------------------------*%)
-Goal "[| isNSopen A; \\<forall>x. isNSCont f x |] \
-\ ==> isNSopen {x. f x \\<in> A}";
-by (auto_tac (claset(),simpset() addsimps [isNSopen_iff1]));
-by (dtac (mem_monad_approx RS approx_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")] approx_mem_monad2 1);
-by (blast_tac (claset() addIs [starfun_mem_starset]) 1);
-qed "isNSCont_isNSopen";
-
-Goalw [isNSCont_def]
- "\\<forall>A. isNSopen A --> isNSopen {x. f x \\<in> A} \
-\ ==> isNSCont f x";
-by (auto_tac (claset() addSIs [(mem_infmal_iff RS iffD1) RS
- (approx_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 [approx_sym RS approx_mem_monad],
- simpset() addsimps [hypreal_of_real_zero RS sym,STAR_starfun_rabs_add_minus]));
-qed "isNSopen_isNSCont";
-
-Goal "(\\<forall>x. isNSCont f x) = \
-\ (\\<forall>A. isNSopen A --> isNSopen {x. f(x) \\<in> A})";
-by (blast_tac (claset() addIs [isNSCont_isNSopen,
- isNSopen_isNSCont]) 1);
-qed "isNSCont_isNSopen_iff";
-
-(%*------- Standard version of same theorem --------*%)
-Goal "(\\<forall>x. isCont f x) = \
-\ (\\<forall>A. isopen A --> isopen {x. f(x) \\<in> 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 \\<approx> y|] ==> ( *f* f) x \\<approx> ( *f* f) y";
-by (Blast_tac 1);
-qed "isNSUContD";
-
-Goalw [isUCont_def,isCont_def,LIM_def]
- "isUCont f ==> isCont f x";
-by (Clarify_tac 1);
-by (dtac spec 1);
-by (Blast_tac 1);
-qed "isUCont_isCont";
-
-Goalw [isNSUCont_def,isUCont_def,approx_def]
- "isUCont f ==> isNSUCont f";
-by (asm_full_simp_tac (simpset() addsimps
- [Infinitesimal_FreeUltrafilterNat_iff]) 1);
-by Safe_tac;
-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 "\\<forall>n::nat. abs ((xa n) + - (xb n)) < s --> abs (f (xa n) + - f (xb n)) < u" 1);
-by (Blast_tac 2);
-by (thin_tac "\\<forall>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 "\\<forall>s. 0 < s --> (\\<exists>z y. abs (z + - y) < s & r \\<le> abs (f z + -f y)) \
-\ ==> \\<forall>n::nat. \\<exists>z y. \
-\ abs(z + -y) < inverse(real(Suc n)) & \
-\ r \\<le> abs(f z + -f y)";
-by (Clarify_tac 1);
-by (cut_inst_tac [("n1","n")]
- (real_of_nat_Suc_gt_zero RS positive_imp_inverse_positive) 1);
-by Auto_tac;
-qed "lemma_LIMu";
-
-Goal "\\<forall>s. 0 < s --> (\\<exists>z y. abs (z + - y) < s & r \\<le> abs (f z + -f y)) \
-\ ==> \\<exists>X Y. \\<forall>n::nat. \
-\ abs(X n + -(Y n)) < inverse(real(Suc n)) & \
-\ r \\<le> abs(f (X n) + -f (Y n))";
-by (dtac lemma_LIMu 1);
-by (dtac choice 1);
-by Safe_tac;
-by (dtac choice 1);
-by (Blast_tac 1);
-qed "lemma_skolemize_LIM2u";
-
-Goal "\\<forall>n. abs (X n + -Y n) < inverse (real(Suc n)) & \
-\ r \\<le> abs (f (X n) + - f(Y n)) ==> \
-\ \\<forall>n. abs (X n + - Y n) < inverse (real(Suc n))";
-by (Auto_tac );
-qed "lemma_simpu";
-
-Goalw [isNSUCont_def,isUCont_def,approx_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 (asm_full_simp_tac (simpset() addsimps [linorder_not_less]) 1);
-by (dtac lemma_skolemize_LIM2u 1);
-by Safe_tac;
-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" "epsilon" bspec]
- addSIs [inj_hypreal_of_real RS injD]
- addDs [approx_trans3],
- simpset()));
-qed "NSDeriv_unique";
-
-(*------------------------------------------------------------------------
- Differentiable
- ------------------------------------------------------------------------*)
-
-Goalw [differentiable_def]
- "f differentiable x ==> \\<exists>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 ==> \\<exists>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 Safe_tac;
-by (ALLGOALS(dtac spec));
-by Safe_tac;
-by (Blast_tac 1 THEN Blast_tac 2);
-by (ALLGOALS(res_inst_tac [("x","s")] exI));
-by Safe_tac;
-by (dres_inst_tac [("x","x + -a")] spec 1);
-by (dres_inst_tac [("x","x + a")] spec 2);
-by (auto_tac (claset(), simpset() addsimps 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;
-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) = \
-\ (\\<forall>xa. \
-\ xa \\<noteq> hypreal_of_real x & xa \\<approx> hypreal_of_real x --> \
-\ ( *f* (%z. (f z - f x) / (z - x))) xa \\<approx> hypreal_of_real D)";
-by (auto_tac (claset(), simpset() addsimps [NSDERIV_NSLIM_iff2, NSLIM_def]));
-qed "NSDERIV_iff2";
-
-
-Goal "(NSDERIV f x :> D) ==> \
-\ (\\<forall>u. \
-\ u \\<approx> hypreal_of_real x --> \
-\ ( *f* (%z. f z - f x)) u \\<approx> hypreal_of_real D * (u - hypreal_of_real x))";
-by (auto_tac (claset(), simpset() addsimps [NSDERIV_iff2]));
-by (case_tac "u = hypreal_of_real x" 1);
-by (auto_tac (claset(), simpset() addsimps [hypreal_diff_def]));
-by (dres_inst_tac [("x","u")] spec 1);
-by Auto_tac;
-by (dres_inst_tac [("c","u - hypreal_of_real x"),("b","hypreal_of_real D")]
- approx_mult1 1);
-by (ALLGOALS(dtac (hypreal_not_eq_minus_iff RS iffD1)));
-by (subgoal_tac "( *f* (%z. z - x)) u \\<noteq> (0::hypreal)" 2);
-by (auto_tac (claset(),
- simpset() addsimps [real_diff_def, hypreal_diff_def,
- (approx_minus_iff RS iffD1) RS (mem_infmal_iff RS iffD2),
- Infinitesimal_subset_HFinite RS subsetD]));
-qed "NSDERIVD5";
-
-Goal "(NSDERIV f x :> D) ==> \
-\ (\\<forall>h \\<in> Infinitesimal. \
-\ (( *f* f)(hypreal_of_real x + h) - \
-\ hypreal_of_real (f x))\\<approx> (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")] approx_mult1 2);
-by (auto_tac (claset() addIs [Infinitesimal_subset_HFinite RS subsetD],
- simpset() addsimps [hypreal_diff_def]));
-qed "NSDERIVD4";
-
-Goal "(NSDERIV f x :> D) ==> \
-\ (\\<forall>h \\<in> Infinitesimal - {0}. \
-\ (( *f* f)(hypreal_of_real x + h) - \
-\ hypreal_of_real (f x))\\<approx> (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")] approx_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 (approx_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")] approx_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 [approx_trans,
- hypreal_mult_commute RS subst,
- (approx_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 [add_divide_distrib]));
-by (dres_inst_tac [("b","hypreal_of_real Da"),
- ("d","hypreal_of_real Db")] approx_add 1);
-by (auto_tac (claset(), simpset() addsimps 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
- ----------------------------------------------------*)
-
-Goal "((a::hypreal)*b) + -(c*d) = (b*(a + -c)) + (c*(b + -d))";
-by (simp_tac (simpset() addsimps [right_distrib]) 1);
-qed "lemma_nsderiv1";
-
-Goal "[| (x + y) / z = hypreal_of_real D + yb; z \\<noteq> 0; \
-\ z \\<in> Infinitesimal; yb \\<in> Infinitesimal |] \
-\ ==> x + y \\<approx> 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);
-qed "lemma_nsderiv2";
-
-
-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);
-by (REPEAT (Step_tac 1));
-by (auto_tac (claset(),
- simpset() addsimps [starfun_lambda_cancel, lemma_nsderiv1]));
-by (simp_tac (simpset() addsimps [add_divide_distrib]) 1);
-by (REPEAT(dtac (bex_Infinitesimal_iff2 RS iffD2) 1));
-by (auto_tac (claset(),
- simpset() delsimps [times_divide_eq_right]
- addsimps [times_divide_eq_right RS sym]));
-by (dres_inst_tac [("D","Db")] lemma_nsderiv2 1);
-by (dtac (approx_minus_iff RS iffD2 RS (bex_Infinitesimal_iff2 RS iffD2)) 4);
-by (auto_tac (claset() addSIs [approx_add_mono1],
- simpset() addsimps [left_distrib, right_distrib,
- 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_approx_self2 RS approx_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
- (HOL_ss addsimps [times_divide_eq_right RS sym, NSDERIV_NSLIM_iff,
- minus_mult_right, right_distrib 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
- (HOL_ss addsimps [times_divide_eq_right RS sym, NSDERIV_NSLIM_iff,
- minus_mult_right, right_distrib 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 (dtac NSLIM_minus 1);
-by (subgoal_tac "ALL a::real. ALL b. - a + b = - (a + - b)" 1);
-by (asm_full_simp_tac (HOL_ss addsimps [thm"minus_divide_left" RS sym]) 1);
-by (Asm_full_simp_tac 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 \\<in> Infinitesimal; h \\<noteq> 0 |] \
-\ ==> \\<exists>e \\<in> 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 [times_divide_eq_right RS sym]
- delsimps [times_divide_eq_right]) 1);
-by (auto_tac (claset(),
- simpset() addsimps [left_distrib]));
-qed "increment_thm";
-
-Goal "[| NSDERIV f x :> D; h \\<approx> 0; h \\<noteq> 0 |] \
-\ ==> \\<exists>e \\<in> 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 \\<approx> 0; h \\<noteq> 0 |] \
-\ ==> increment f x h \\<approx> 0";
-by (dtac increment_thm2 1 THEN auto_tac (claset() addSIs
- [Infinitesimal_HFinite_mult2,HFinite_add],simpset() addsimps
- [left_distrib RS sym,mem_infmal_iff RS sym]));
-by (etac (Infinitesimal_subset_HFinite RS subsetD) 1);
-qed "increment_approx_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 \\<in> Infinitesimal;\
-\ xa \\<noteq> 0 \
-\ |] ==> D = 0";
-by (dtac bspec 1);
-by Auto_tac;
-qed "NSDERIV_zero";
-
-(* can be proved differently using NSLIM_isCont_iff *)
-Goalw [nsderiv_def]
- "[| NSDERIV f x :> D; h \\<in> Infinitesimal; h \\<noteq> 0 |] \
-\ ==> ( *f* f) (hypreal_of_real(x) + h) + -hypreal_of_real(f x) \\<approx> 0";
-by (asm_full_simp_tac (simpset() addsimps
- [mem_infmal_iff RS sym]) 1);
-by (rtac Infinitesimal_ratio 1);
-by (rtac approx_hypreal_of_real_HFinite 3);
-by Auto_tac;
-qed "NSDERIV_approx";
-
-(*---------------------------------------------------------------
- from one version of differentiability
-
- f(x) - f(a)
- --------------- \\<approx> Db
- x - a
- ---------------------------------------------------------------*)
-Goal "[| NSDERIV f (g x) :> Da; \
-\ ( *f* g) (hypreal_of_real(x) + xa) \\<noteq> hypreal_of_real (g x); \
-\ ( *f* g) (hypreal_of_real(x) + xa) \\<approx> 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)) \
-\ \\<approx> 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)
- ----------------- \\<approx> Db
- h
- --------------------------------------------------------------*)
-Goal "[| NSDERIV g x :> Db; xa \\<in> Infinitesimal; xa \\<noteq> 0 |] \
-\ ==> (( *f* g) (hypreal_of_real(x) + xa) + - hypreal_of_real(g x)) / xa \
-\ \\<approx> hypreal_of_real(Db)";
-by (auto_tac (claset(),
- simpset() addsimps [NSDERIV_NSLIM_iff, NSLIM_def,
- mem_infmal_iff, starfun_lambda_cancel]));
-qed "NSDERIVD2";
-
-Goal "(z::hypreal) \\<noteq> 0 ==> x*y = (x*inverse(z))*(z*y)";
-by Auto_tac;
-qed "lemma_chain";
-
-(*------------------------------------------------------
- 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,mem_infmal_iff RS sym]) 1 THEN Step_tac 1);
-by (forw_inst_tac [("f","g")] NSDERIV_approx 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]));
-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 approx_mult_hypreal_of_real 1);
-by (fold_tac [hypreal_divide_def]);
-by (blast_tac (claset() addIs [NSDERIVD1,
- approx_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]));
-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 n * (x ^ (n - Suc 0))";
-by (induct_tac "n" 1);
-by (dtac (DERIV_Id RS DERIV_mult) 2);
-by (auto_tac (claset(),
- simpset() addsimps [real_of_nat_Suc, left_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 n * (x ^ (n - Suc 0))";
-by (simp_tac (simpset() addsimps [NSDERIV_DERIV_iff, DERIV_pow]) 1);
-qed "NSDERIV_pow";
-
-(*---------------------------------------------------------------
- Power of -1
- ---------------------------------------------------------------*)
-
-(*Can't get rid of x \\<noteq> 0 because it isn't continuous at zero*)
-Goalw [nsderiv_def]
- "x \\<noteq> 0 ==> NSDERIV (%x. inverse(x)) x :> (- (inverse x ^ Suc (Suc 0)))";
-by (rtac ballI 1 THEN Asm_full_simp_tac 1 THEN Step_tac 1);
-by (ftac 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 [minus_mult_left RS sym,
- minus_mult_right RS sym]));
-by (asm_full_simp_tac
- (simpset() addsimps [hypreal_inverse_add,
- hypreal_inverse_distrib RS sym, hypreal_minus_inverse RS sym]
- @ add_ac @ mult_ac
- delsimps [inverse_mult_distrib,inverse_minus_eq,
- minus_mult_left RS sym,
- minus_mult_right RS sym] ) 1);
-by (asm_simp_tac (simpset() addsimps [hypreal_mult_assoc RS sym,
- right_distrib]
- delsimps [minus_mult_left RS sym,
- minus_mult_right RS sym]) 1);
-by (res_inst_tac [("y"," inverse(- hypreal_of_real x * hypreal_of_real x)")]
- approx_trans 1);
-by (rtac inverse_add_Infinitesimal_approx2 1);
-by (auto_tac (claset() addSDs [hypreal_of_real_HFinite_diff_Infinitesimal],
- simpset() addsimps [hypreal_minus_inverse RS sym,
- HFinite_minus_iff]));
-by (rtac Infinitesimal_HFinite_mult2 1);
-by Auto_tac;
-qed "NSDERIV_inverse";
-
-
-Goal "x \\<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))";
-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) \\<noteq> 0 |] \
-\ ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))";
-by (rtac (real_mult_commute RS subst) 1);
-by (asm_simp_tac (HOL_ss addsimps [minus_mult_left, power_inverse]) 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) \\<noteq> 0 |] \
-\ ==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))";
-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) \\<noteq> 0 |] \
-\ ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) + -(e*f(x))) / (g(x) ^ Suc (Suc 0))";
-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, right_distrib,
- power_inverse,minus_mult_left] @ mult_ac
- delsimps [realpow_Suc, minus_mult_right RS sym, minus_mult_left RS sym]) 1);
-qed "DERIV_quotient";
-
-Goal "[| NSDERIV f x :> d; DERIV g x :> e; g(x) \\<noteq> 0 |] \
-\ ==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x) \
-\ + -(e*f(x))) / (g(x) ^ Suc (Suc 0))";
-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) = \
-\ (\\<exists>g. (\\<forall>z. f z - f x = g z * (z - x)) & isCont g x & g x = l)";
-by Safe_tac;
-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 \\<noteq> x ==> z - x \\<noteq> (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 ==> \
-\ \\<exists>g. (\\<forall>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 "(\\<forall>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 field_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 "(\\<forall>n. (f::nat=>real) n \\<le> f (Suc n)) --> f m \\<le> f(m + no)";
-by (induct_tac "no" 1);
-by (auto_tac (claset() addIs [order_trans], simpset()));
-qed_spec_mp "lemma_f_mono_add";
-
-Goal "[| \\<forall>n. f(n) \\<le> f(Suc n); \
-\ \\<forall>n. g(Suc n) \\<le> g(n); \
-\ \\<forall>n. f(n) \\<le> 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 [order_trans], simpset()));
-by (res_inst_tac [("y","g(Suc na)")] order_trans 1);
-by (induct_tac "na" 2);
-by (auto_tac (claset() addIs [order_trans], simpset()));
-qed "f_inc_g_dec_Beq_f";
-
-Goal "[| \\<forall>n. f(n) \\<le> f(Suc n); \
-\ \\<forall>n. g(Suc n) \\<le> g(n); \
-\ \\<forall>n. f(n) \\<le> 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 "[| \\<forall>n. f n \\<le> f (Suc n); convergent f |] ==> f n \\<le> lim f";
-by (rtac (linorder_not_less RS iffD1) 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\\<le>na --> ?Q na"),("x","no + n")] spec 1);
-by Auto_tac;
-by (subgoal_tac "lim f \\<le> f(no + n)" 1);
-by (induct_tac "no" 2);
-by (auto_tac (claset() addIs [order_trans],
- simpset() addsimps [real_diff_def, real_abs_def]));
-by (dres_inst_tac [("x","f(no + n)"),("no1","no")]
- (lemma_f_mono_add RSN (2,order_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 "[| \\<forall>n. g(Suc n) \\<le> g(n); convergent g |] ==> lim g \\<le> g n";
-by (subgoal_tac "- (g n) \\<le> - (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 "[| \\<forall>n. f(n) \\<le> f(Suc n); \
-\ \\<forall>n. g(Suc n) \\<le> g(n); \
-\ \\<forall>n. f(n) \\<le> g(n) |] \
-\ ==> \\<exists>l m. l \\<le> m & ((\\<forall>n. f(n) \\<le> l) & f ----> l) & \
-\ ((\\<forall>n. m \\<le> 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 "[| \\<forall>n. f(n) \\<le> f(Suc n); \
-\ \\<forall>n. g(Suc n) \\<le> g(n); \
-\ \\<forall>n. f(n) \\<le> g(n); \
-\ (%n. f(n) - g(n)) ----> 0 |] \
-\ ==> \\<exists>l. ((\\<forall>n. f(n) \\<le> l) & f ----> l) & \
-\ ((\\<forall>n. l \\<le> 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 \\<le> b ==> \
-\ \\<forall>n. fst (Bolzano_bisect P a b n) \\<le> 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 \\<le> b ==> \
-\ \\<forall>n. fst(Bolzano_bisect P a b n) \\<le> 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 \\<le> b ==> \
-\ \\<forall>n. snd(Bolzano_bisect P a b (Suc n)) \\<le> 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 \\<le> 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, add_divide_distrib,
- Let_def, split_def]));
-by (auto_tac (claset(),
- simpset() addsimps (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 \\<le> b; b \\<le> c|] ==> P(a,c); \
-\ ~ P(a,b); a \\<le> 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 "[| \\<forall>a b c. P(a,b) & P(b,c) & a \\<le> b & b \\<le> c --> P(a,c); \
-\ ~ P(a,b); a \\<le> b |] ==> \
-\ \\<forall>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 "[| \\<forall>a b c. P(a,b) & P(b,c) & a \\<le> b & b \\<le> c --> P(a,c); \
-\ \\<forall>x. \\<exists>d::real. 0 < d & \
-\ (\\<forall>a b. a \\<le> x & x \\<le> b & (b - a) < d --> P(a,b)); \
-\ a \\<le> 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 [("y","abs(fst(Bolzano_bisect P a b(no + noa)) - l) + \
-\ abs(snd(Bolzano_bisect P a b(no + noa)) - l)")]
- order_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 add_strict_mono 1);
-by (ALLGOALS
- (asm_full_simp_tac (simpset() addsimps [symmetric real_diff_def])));
-qed "lemma_BOLZANO";
-
-
-Goal "((\\<forall>a b c. (a \\<le> b & b \\<le> c & P(a,b) & P(b,c)) --> P(a,c)) & \
-\ (\\<forall>x. \\<exists>d::real. 0 < d & \
-\ (\\<forall>a b. a \\<le> x & x \\<le> b & (b - a) < d --> P(a,b)))) \
-\ --> (\\<forall>a b. a \\<le> 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) \\<le> y & y \\<le> f(b); \
-\ a \\<le> b; \
-\ (\\<forall>x. a \\<le> x & x \\<le> b --> isCont f x) |] \
-\ ==> \\<exists>x. a \\<le> x & x \\<le> b & f(x) = y";
-by (rtac contrapos_pp 1);
-by (assume_tac 1);
-by (cut_inst_tac
- [("P","%(u,v). a \\<le> u & u \\<le> v & v \\<le> b --> ~(f(u) \\<le> y & y \\<le> f(v))")]
- lemma_BOLZANO2 1);
-by Safe_tac;
-by (ALLGOALS(Asm_full_simp_tac));
-by (asm_full_simp_tac (simpset() addsimps [isCont_iff,LIM_def]) 1);
-by (rtac ccontr 1);
-by (subgoal_tac "a \\<le> x & x \\<le> 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 [order_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 --> (\\<exists>s. 0<s & ?Q r s)"),
- ("x","abs(y - f x)")] spec 1);
-by Safe_tac;
-by (asm_full_simp_tac (simpset() addsimps []) 1);
-by (dres_inst_tac [("x","s")] spec 1);
-by (Clarify_tac 1);
-by (cut_inst_tac [("x","f x"),("y","y")] linorder_less_linear 1);
-by Safe_tac;
-by (dres_inst_tac [("x","ba - x")] spec 1);
-by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [thm"abs_if"])));
-by (dres_inst_tac [("x","aa - x")] spec 1);
-by (case_tac "x \\<le> 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) \\<le> y & y \\<le> f(a); \
-\ a \\<le> b; \
-\ (\\<forall>x. a \\<le> x & x \\<le> b --> isCont f x) \
-\ |] ==> \\<exists>x. a \\<le> x & x \\<le> b & f(x) = y";
-by (subgoal_tac "- f a \\<le> -y & -y \\<le> - f b" 1);
-by (thin_tac "f b \\<le> y & y \\<le> 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) \\<le> y & y \\<le> f(b) & a \\<le> b & \
-\ (\\<forall>x. a \\<le> x & x \\<le> b --> isCont f x)) \
-\ --> (\\<exists>x. a \\<le> x & x \\<le> b & f(x) = y)";
-by (blast_tac (claset() addIs [IVT]) 1);
-qed "IVT_objl";
-
-Goal "(f(b) \\<le> y & y \\<le> f(a) & a \\<le> b & \
-\ (\\<forall>x. a \\<le> x & x \\<le> b --> isCont f x)) \
-\ --> (\\<exists>x. a \\<le> x & x \\<le> 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 x) = real (x::nat)";
-by (auto_tac (claset() addIs [abs_eqI1], simpset()));
-qed "abs_real_of_nat_cancel";
-Addsimps [abs_real_of_nat_cancel];
-
-Goal "~ abs(x) + (1::real) < x";
-by (asm_full_simp_tac (simpset() addsimps [linorder_not_less]) 1);
-by (auto_tac (claset() addIs [abs_ge_self RS order_trans],simpset()));
-qed "abs_add_one_not_less_self";
-Addsimps [abs_add_one_not_less_self];
-
-
-Goal "[| a \\<le> b; \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x |]\
-\ ==> \\<exists>M. \\<forall>x. a \\<le> x & x \\<le> b --> f(x) \\<le> M";
-by (cut_inst_tac [("P","%(u,v). a \\<le> u & u \\<le> v & v \\<le> b --> \
-\ (\\<exists>M. \\<forall>x. u \\<le> x & x \\<le> v --> f x \\<le> M)")]
- lemma_BOLZANO2 1);
-by Safe_tac;
-by (ALLGOALS Asm_full_simp_tac);
-by (rename_tac "x xa ya M Ma" 1);
-by (cut_inst_tac [("x","M"),("y","Ma")] linorder_linear 1);
-by Safe_tac;
-by (res_inst_tac [("x","Ma")] exI 1);
-by (Clarify_tac 1);
-by (cut_inst_tac [("x","xb"),("y","xa")] linorder_linear 1);
-by (Force_tac 1);
-by (res_inst_tac [("x","M")] exI 1);
-by (Clarify_tac 1);
-by (cut_inst_tac [("x","xb"),("y","xa")] linorder_linear 1);
-by (Force_tac 1);
-by (case_tac "a \\<le> x & x \\<le> b" 1);
-by (res_inst_tac [("x","1")] exI 2);
-by (Force_tac 2);
-by (asm_full_simp_tac (simpset() addsimps [LIM_def,isCont_iff]) 1);
-by (dres_inst_tac [("x","x")] spec 1 THEN Auto_tac);
-by (thin_tac "\\<forall>M. \\<exists>x. a \\<le> x & x \\<le> b & ~ f x \\<le> M" 1);
-by (dres_inst_tac [("x","1")] spec 1);
-by Auto_tac;
-by (res_inst_tac [("x","s")] exI 1 THEN Clarify_tac 1);
-by (res_inst_tac [("x","abs(f x) + 1")] exI 1 THEN Clarify_tac 1);
-by (dres_inst_tac [("x","xa - x")] spec 1);
-by (auto_tac (claset(), simpset() addsimps [abs_ge_self]));
-by (REPEAT (arith_tac 1));
-qed "isCont_bounded";
-
-(*----------------------------------------------------------------------------*)
-(* Refine the above to existence of least upper bound *)
-(*----------------------------------------------------------------------------*)
-
-Goal "((\\<exists>x. x \\<in> S) & (\\<exists>y. isUb UNIV S (y::real))) --> \
-\ (\\<exists>t. isLub UNIV S t)";
-by (blast_tac (claset() addIs [reals_complete]) 1);
-qed "lemma_reals_complete";
-
-Goal "[| a \\<le> b; \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x |] \
-\ ==> \\<exists>M. (\\<forall>x. a \\<le> x & x \\<le> b --> f(x) \\<le> M) & \
-\ (\\<forall>N. N < M --> (\\<exists>x. a \\<le> x & x \\<le> b & N < f(x)))";
-by (cut_inst_tac [("S","Collect (%y. \\<exists>x. a \\<le> x & x \\<le> 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 [(linorder_not_less RS iffD1)],simpset()));
-qed "isCont_has_Ub";
-
-(*----------------------------------------------------------------------------*)
-(* Now show that it attains its upper bound *)
-(*----------------------------------------------------------------------------*)
-
-Goal "[| a \\<le> b; \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x |] \
-\ ==> \\<exists>M. (\\<forall>x. a \\<le> x & x \\<le> b --> f(x) \\<le> M) & \
-\ (\\<exists>x. a \\<le> x & x \\<le> 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 "\\<forall>x. a \\<le> x & x \\<le> b --> f x < M" 1 THEN Step_tac 1);
-by (rtac ccontr 2 THEN dtac (linorder_not_less RS iffD1) 2);
-by (dres_inst_tac [("z","M")] real_le_anti_sym 2);
-by (REPEAT(Blast_tac 2));
-by (subgoal_tac "\\<forall>x. a \\<le> x & x \\<le> b --> isCont (%x. inverse(M - f x)) x" 1);
-by Safe_tac;
-by (EVERY[rtac isCont_inverse 2, rtac isCont_diff 2, rtac notI 4]);
-by (ALLGOALS(asm_full_simp_tac (simpset() addsimps [diff_eq_eq])));
-by (Blast_tac 2);
-by (subgoal_tac
- "\\<exists>k. \\<forall>x. a \\<le> x & x \\<le> b --> (%x. inverse(M - (f x))) x \\<le> k" 1);
-by (rtac isCont_bounded 2);
-by Safe_tac;
-by (subgoal_tac "\\<forall>x. a \\<le> x & x \\<le> b --> 0 < inverse(M - f(x))" 1);
-by (Asm_full_simp_tac 1);
-by Safe_tac;
-by (asm_full_simp_tac (simpset() addsimps [less_diff_eq]) 2);
-by (subgoal_tac
- "\\<forall>x. a \\<le> x & x \\<le> b --> (%x. inverse(M - (f x))) x < (k + 1)" 1);
-by Safe_tac;
-by (res_inst_tac [("y","k")] order_le_less_trans 2);
-by (asm_full_simp_tac (simpset() addsimps [zero_less_one]) 3);
-by (Asm_full_simp_tac 2);
-by (subgoal_tac "\\<forall>x. a \\<le> x & x \\<le> b --> \
-\ inverse(k + 1) < inverse((%x. inverse(M - (f x))) x)" 1);
-by Safe_tac;
-by (rtac less_imp_inverse_less 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 (linorder_not_less RS iffD1) 1);
-by (dtac (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 [inverse_eq_divide, pos_divide_le_eq]) 1);
-by (cut_inst_tac [("a","k"),("b","M-f a")] zero_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 \\<le> b; \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x |] \
-\ ==> \\<exists>M. (\\<forall>x. a \\<le> x & x \\<le> b --> M \\<le> f(x)) & \
-\ (\\<exists>x. a \\<le> x & x \\<le> b & f(x) = M)";
-by (subgoal_tac "\\<forall>x. a \\<le> x & x \\<le> 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 Safe_tac;
-by Auto_tac;
-qed "isCont_eq_Lb";
-
-
-(* ------------------------------------------------------------------------- *)
-(* Another version. *)
-(* ------------------------------------------------------------------------- *)
-
-Goal "[|a \\<le> b; \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x |] \
-\ ==> \\<exists>L M. (\\<forall>x. a \\<le> x & x \\<le> b --> L \\<le> f(x) & f(x) \\<le> M) & \
-\ (\\<forall>y. L \\<le> y & y \\<le> M --> (\\<exists>x. a \\<le> x & x \\<le> b & (f(x) = y)))";
-by (ftac isCont_eq_Lb 1);
-by (ftac isCont_eq_Ub 2);
-by (REPEAT(assume_tac 1));
-by Safe_tac;
-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 Safe_tac;
-by (cut_inst_tac [("x","x"),("y","xa")] linorder_linear 1);
-by Safe_tac;
-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 Safe_tac;
-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 |] \
-\ ==> \\<exists>d. 0 < d & (\\<forall>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 [zero_less_mult_iff]) 2);
-by (dres_inst_tac [("x","h")] spec 1);
-by (asm_full_simp_tac
- (simpset() addsimps [real_abs_def, inverse_eq_divide,
- pos_le_divide_eq, pos_less_divide_eq]
- addsplits [split_if_asm]) 1);
-qed "DERIV_left_inc";
-
-val prems = goalw (the_context()) [deriv_def,LIM_def]
- "[| DERIV f x :> l; l < 0 |] ==> \
-\ \\<exists>d. 0 < d & (\\<forall>h. 0 < h & h < d --> f(x) < f(x - h))";
-by (cut_facts_tac prems 1); (*needed because arith removes the assumption l<0*)
-by (dres_inst_tac [("x","-l")] spec 1 THEN Auto_tac);
-by (res_inst_tac [("x","s")] exI 1 THEN Auto_tac);
-by (dres_inst_tac [("x","-h")] spec 1);
-by (asm_full_simp_tac
- (simpset() addsimps [real_abs_def, inverse_eq_divide,
- pos_less_divide_eq,
- symmetric real_diff_def]
- addsplits [split_if_asm]) 1);
-by (subgoal_tac "0 < (f (x - h) - f x)/h" 1);
-by (asm_full_simp_tac (simpset() addsimps [pos_less_divide_eq]) 1);
-by (cut_facts_tac prems 1);
-by (arith_tac 1);
-qed "DERIV_left_dec";
-
-(*????previous proof, revealing arith problem:
-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 [mult_less_0_iff]) 2);
-by (dres_inst_tac [("x","-h")] spec 1);
-by (asm_full_simp_tac
- (simpset() addsimps [real_abs_def, inverse_eq_divide,
- pos_less_divide_eq,
- symmetric real_diff_def]
- addsplits [split_if_asm]
- delsimprocs [fast_real_arith_simproc]) 1);
-by (subgoal_tac "0 < (f (x - h) - f x)/h" 1);
-by (arith_tac 2);
-by (asm_full_simp_tac
- (simpset() addsimps [pos_less_divide_eq]) 1);
-qed "DERIV_left_dec";
-*)
-
-
-Goal "[| DERIV f x :> l; \
-\ \\<exists>d. 0 < d & (\\<forall>y. abs(x - y) < d --> f(y) \\<le> f(x)) |] \
-\ ==> l = 0";
-by (res_inst_tac [("x","l"),("y","0")] linorder_cases 1);
-by Safe_tac;
-by (dtac DERIV_left_dec 1);
-by (dtac DERIV_left_inc 3);
-by Safe_tac;
-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 Safe_tac;
-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; \
-\ \\<exists>d::real. 0 < d & (\\<forall>y. abs(x - y) < d --> f(x) \\<le> 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; \
-\ \\<exists>d. 0 < d & (\\<forall>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 |] ==> \
-\ \\<exists>d::real. 0 < d & (\\<forall>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")] linorder_linear 1);
-by Safe_tac;
-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 [less_diff_eq]));
-qed "lemma_interval_lt";
-
-Goal "[| a < x; x < b |] ==> \
-\ \\<exists>d::real. 0 < d & (\\<forall>y. abs(x - y) < d --> a \\<le> y & y \\<le> 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 \\<in> (a,b) such that f'(x0) = 0
- ----------------------------------------------------------------------*)
-
-Goal "[| a < b; f(a) = f(b); \
-\ \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x; \
-\ \\<forall>x. a < x & x < b --> f differentiable x \
-\ |] ==> \\<exists>z. a < z & z < b & DERIV f z :> 0";
-by (ftac (order_less_imp_le RS isCont_eq_Ub) 1);
-by (EVERY1[assume_tac,Step_tac]);
-by (ftac (order_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 "(\\<exists>l. DERIV f x :> l) & \
-\ (\\<exists>d. 0 < d & (\\<forall>y. abs(x - y) < d --> f(y) \\<le> 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 "(\\<exists>l. DERIV f xa :> l) & \
-\ (\\<exists>d. 0 < d & (\\<forall>y. abs(xa - y) < d --> f(xa) \\<le> 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 "\\<forall>x. a \\<le> x & x \\<le> 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")] (order_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")] (order_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_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 "(\\<exists>l. DERIV f r :> l) & \
-\ (\\<exists>d. 0 < d & (\\<forall>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 [order_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 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 [right_diff_distrib]));
-by (auto_tac (claset(), simpset() addsimps [left_diff_distrib]));
-qed "lemma_MVT";
-
-Goal "[| a < b; \
-\ \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x; \
-\ \\<forall>x. a < x & x < b --> f differentiable x |] \
-\ ==> \\<exists>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 Safe_tac;
-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 Safe_tac;
-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 Safe_tac;
-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; \
-\ \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x; \
-\ \\<forall>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 [diff_eq_eq]));
-qed "DERIV_isconst_end";
-
-Goal "[| a < b; \
-\ \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x; \
-\ \\<forall>x. a < x & x < b --> DERIV f x :> 0 |] \
-\ ==> \\<forall>x. a \\<le> x & x \\<le> b --> f x = f a";
-by Safe_tac;
-by (dres_inst_tac [("x","a")] order_le_imp_less_or_eq 1);
-by Safe_tac;
-by (dres_inst_tac [("b","x")] DERIV_isconst_end 1);
-by Auto_tac;
-qed "DERIV_isconst1";
-
-Goal "[| a < b; \
-\ \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x; \
-\ \\<forall>x. a < x & x < b --> DERIV f x :> 0; \
-\ a \\<le> x; x \\<le> b |] \
-\ ==> f x = f a";
-by (blast_tac (claset() addDs [DERIV_isconst1]) 1);
-qed "DERIV_isconst2";
-
-Goal "\\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)";
-by (res_inst_tac [("x","x"),("y","y")] linorder_cases 1);
-by (rtac sym 1);
-by (auto_tac (claset() addIs [DERIV_isCont,DERIV_isconst_end],simpset()));
-qed "DERIV_isconst_all";
-
-Goal "[|a \\<noteq> b; \\<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b - a) * k";
-by (res_inst_tac [("x","a"),("y","b")] linorder_cases 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 [left_distrib, real_diff_def]));
-qed "DERIV_const_ratio_const";
-
-Goal "[|a \\<noteq> b; \\<forall>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 \\<noteq> (b::real); \\<forall>x. DERIV v x :> k|] \
-\ ==> v((a + b)/2) = (v a + v b)/2";
-by (res_inst_tac [("x","a"),("y","b")] linorder_cases 1);
-by Safe_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 [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; \\<forall>z. abs(z - x) \\<le> d --> g(f z) = z; \
-\ \\<forall>z. abs(z - x) \\<le> d --> isCont f z |] \
-\ ==> ~(\\<forall>z. abs(z - x) \\<le> d --> f(z) \\<le> 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 Safe_tac;
-by (cut_inst_tac [("x","f(x - d)"),("y","f(x + d)")]
- (ARITH_PROVE "x \\<le> y | y \\<le> (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 Safe_tac;
-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 Safe_tac;
-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; \\<forall>z. abs(z - x) \\<le> d --> g(f z) = z; \
-\ \\<forall>z. abs(z - x) \\<le> d --> isCont f z |] \
-\ ==> ~(\\<forall>z. abs(z - x) \\<le> d --> f(x) \\<le> 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 *)
-(* ------------------------------------------------------------------------ *)
-
-val lemma_le = ARITH_PROVE "0 \\<le> (d::real) ==> -d \\<le> d";
-
-(* FIXME: awful proof - needs improvement *)
-Goal "[| 0 < d; \\<forall>z. abs(z - x) \\<le> d --> g(f z) = z; \
-\ \\<forall>z. abs(z - x) \\<le> d --> isCont f z |] \
-\ ==> \\<exists>e. 0 < e & \
-\ (\\<forall>y. \
-\ abs(y - f(x)) \\<le> e --> \
-\ (\\<exists>z. abs(z - x) \\<le> d & (f z = y)))";
-by (ftac order_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 Safe_tac;
-by (asm_full_simp_tac (simpset() addsimps [abs_le_interval_iff]) 1);
-by (subgoal_tac "L \\<le> f x & f x \\<le> 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 Safe_tac;
-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")]
- (real_lbound_gt_zero) 1);
-by Safe_tac;
-by (res_inst_tac [("x","e")] exI 1);
-by Safe_tac;
-by (asm_full_simp_tac (simpset() addsimps [abs_le_interval_iff]) 1);
-by (dres_inst_tac [("P","%v. ?PP v --> (\\<exists>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 \\<le> y; x \\<noteq> 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 Safe_tac;
-by (ALLGOALS(dres_inst_tac [("x","z")] spec));
-by (ALLGOALS(arith_tac));
-qed "isCont_inj_range";
-
-
-(* ------------------------------------------------------------------------ *)
-(* Continuity of inverse function *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "[| 0 < d; \\<forall>z. abs(z - x) \\<le> d --> g(f(z)) = z; \
-\ \\<forall>z. abs(z - x) \\<le> d --> isCont f z |] \
-\ ==> isCont g (f x)";
-by (simp_tac (simpset() addsimps [isCont_iff,LIM_def]) 1);
-by Safe_tac;
-by (dres_inst_tac [("d1.0","r")] (real_lbound_gt_zero) 1);
-by (assume_tac 1 THEN Step_tac 1);
-by (subgoal_tac "\\<forall>z. abs(z - x) \\<le> e --> (g(f z) = z)" 1);
-by (Force_tac 2);
-by (subgoal_tac "\\<forall>z. abs(z - x) \\<le> 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 Safe_tac;
-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_function";
-
--- a/src/HOL/Hyperreal/Lim.thy Fri Mar 19 10:50:06 2004 +0100
+++ b/src/HOL/Hyperreal/Lim.thy Fri Mar 19 10:51:03 2004 +0100
@@ -1,66 +1,67 @@
(* Title : Lim.thy
+ ID : $Id$
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
- Description : Theory of limits, continuity and
- differentiation of real=>real functions
+ Conversion to Isar and new proofs by Lawrence C Paulson, 2004
*)
-Lim = SEQ + RealDef +
+header{*Limits, Continuity and Differentiation*}
-(*-----------------------------------------------------------------------
- Limits, continuity and differentiation: standard and NS definitions
- -----------------------------------------------------------------------*)
+theory Lim = SEQ + RealDef:
+
+text{*Standard and Nonstandard Definitions*}
constdefs
- LIM :: [real=>real,real,real] => bool
+ LIM :: "[real=>real,real,real] => bool"
("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60)
"f -- a --> L ==
- ALL r. 0 < r -->
- (EX s. 0 < s & (ALL x. (x ~= a & (abs(x + -a) < s)
- --> abs(f x + -L) < r)))"
+ \<forall>r. 0 < r -->
+ (\<exists>s. 0 < s & (\<forall>x. (x \<noteq> a & (\<bar>x + -a\<bar> < s)
+ --> \<bar>f x + -L\<bar> < r)))"
- NSLIM :: [real=>real,real,real] => bool
+ NSLIM :: "[real=>real,real,real] => bool"
("((_)/ -- (_)/ --NS> (_))" [60, 0, 60] 60)
- "f -- a --NS> L == (ALL x. (x ~= hypreal_of_real a &
+ "f -- a --NS> L == (\<forall>x. (x \<noteq> hypreal_of_real a &
x @= hypreal_of_real a -->
- ( *f* f) x @= hypreal_of_real L))"
+ ( *f* f) x @= hypreal_of_real L))"
- isCont :: [real=>real,real] => bool
- "isCont f a == (f -- a --> (f a))"
+ isCont :: "[real=>real,real] => bool"
+ "isCont f a == (f -- a --> (f a))"
(* NS definition dispenses with limit notions *)
- isNSCont :: [real=>real,real] => bool
- "isNSCont f a == (ALL y. y @= hypreal_of_real a -->
+ isNSCont :: "[real=>real,real] => bool"
+ "isNSCont f a == (\<forall>y. y @= hypreal_of_real a -->
( *f* f) y @= hypreal_of_real (f a))"
(* differentiation: D is derivative of function f at x *)
- deriv:: [real=>real,real,real] => bool
+ deriv:: "[real=>real,real,real] => bool"
("(DERIV (_)/ (_)/ :> (_))" [60, 0, 60] 60)
- "DERIV f x :> D == ((%h. (f(x + h) + -f(x))/h) -- 0 --> D)"
+ "DERIV f x :> D == ((%h. (f(x + h) + -f x)/h) -- 0 --> D)"
- nsderiv :: [real=>real,real,real] => bool
+ nsderiv :: "[real=>real,real,real] => bool"
("(NSDERIV (_)/ (_)/ :> (_))" [60, 0, 60] 60)
- "NSDERIV f x :> D == (ALL h: Infinitesimal - {0}.
- (( *f* f)(hypreal_of_real x + h) +
+ "NSDERIV f x :> D == (\<forall>h \<in> Infinitesimal - {0}.
+ (( *f* f)(hypreal_of_real x + h) +
- hypreal_of_real (f x))/h @= hypreal_of_real D)"
- differentiable :: [real=>real,real] => bool (infixl 60)
- "f differentiable x == (EX D. DERIV f x :> D)"
+ differentiable :: "[real=>real,real] => bool" (infixl "differentiable" 60)
+ "f differentiable x == (\<exists>D. DERIV f x :> D)"
- NSdifferentiable :: [real=>real,real] => bool (infixl 60)
- "f NSdifferentiable x == (EX D. NSDERIV f x :> D)"
+ NSdifferentiable :: "[real=>real,real] => bool"
+ (infixl "NSdifferentiable" 60)
+ "f NSdifferentiable x == (\<exists>D. NSDERIV f x :> D)"
- increment :: [real=>real,real,hypreal] => hypreal
- "increment f x h == (@inc. f NSdifferentiable x &
+ increment :: "[real=>real,real,hypreal] => hypreal"
+ "increment f x h == (@inc. f NSdifferentiable x &
inc = ( *f* f)(hypreal_of_real x + h) + -hypreal_of_real (f x))"
- isUCont :: (real=>real) => bool
- "isUCont f == (ALL r. 0 < r -->
- (EX s. 0 < s & (ALL x y. abs(x + -y) < s
- --> abs(f x + -f y) < r)))"
+ isUCont :: "(real=>real) => bool"
+ "isUCont f == (\<forall>r. 0 < r -->
+ (\<exists>s. 0 < s & (\<forall>x y. \<bar>x + -y\<bar> < s
+ --> \<bar>f x + -f y\<bar> < r)))"
- isNSUCont :: (real=>real) => bool
- "isNSUCont f == (ALL x y. x @= y --> ( *f* f) x @= ( *f* f) y)"
+ isNSUCont :: "(real=>real) => bool"
+ "isNSUCont f == (\<forall>x y. x @= y --> ( *f* f) x @= ( *f* f) y)"
(*Used in the proof of the Bolzano theorem*)
@@ -72,8 +73,2258 @@
"Bolzano_bisect P a b (Suc n) =
(let (x,y) = Bolzano_bisect P a b n
in if P(x, (x+y)/2) then ((x+y)/2, y)
- else (x, (x+y)/2) )"
-
+ else (x, (x+y)/2))"
+
+
+
+section{*Some Purely Standard Proofs*}
+
+lemma LIM_eq:
+ "f -- a --> L =
+ (\<forall>r. 0<r --> (\<exists>s. 0 < s & (\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> \<bar>f x - L\<bar> < r)))"
+by (simp add: LIM_def diff_minus)
+
+lemma LIM_D:
+ "[| f -- a --> L; 0<r |]
+ ==> \<exists>s. 0 < s & (\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> \<bar>f x - L\<bar> < r)"
+by (simp add: LIM_eq)
+
+lemma LIM_const: "(%x. k) -- x --> k"
+by (simp add: LIM_def)
+declare LIM_const [simp]
+
+lemma LIM_add:
+ assumes f: "f -- a --> L" and g: "g -- a --> M"
+ shows "(%x. f x + g(x)) -- a --> (L + M)"
+proof (simp add: LIM_eq, clarify)
+ fix r :: real
+ assume r: "0<r"
+ from LIM_D [OF f half_gt_zero [OF r]]
+ obtain fs
+ where fs: "0 < fs"
+ and fs_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < fs --> \<bar>f x - L\<bar> < r/2"
+ by blast
+ from LIM_D [OF g half_gt_zero [OF r]]
+ obtain gs
+ where gs: "0 < gs"
+ and gs_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < gs --> \<bar>g x - M\<bar> < r/2"
+ by blast
+ show "\<exists>s. 0 < s \<and>
+ (\<forall>x. x \<noteq> a \<and> \<bar>x-a\<bar> < s \<longrightarrow> \<bar>f x + g x - (L + M)\<bar> < r)"
+ proof (intro exI conjI strip)
+ show "0 < min fs gs" by (simp add: fs gs)
+ fix x :: real
+ assume "x \<noteq> a \<and> \<bar>x-a\<bar> < min fs gs"
+ with fs_lt gs_lt
+ have "\<bar>f x - L\<bar> < r/2" and "\<bar>g x - M\<bar> < r/2" by (auto simp add: fs_lt)
+ hence "\<bar>f x - L\<bar> + \<bar>g x - M\<bar> < r" by arith
+ thus "\<bar>f x + g x - (L + M)\<bar> < r"
+ by (blast intro: abs_diff_triangle_ineq order_le_less_trans)
+ qed
+qed
+
+lemma LIM_minus: "f -- a --> L ==> (%x. -f(x)) -- a --> -L"
+apply (simp add: LIM_eq)
+apply (subgoal_tac "\<forall>x. \<bar>- f x + L\<bar> = \<bar>f x - L\<bar>")
+apply (simp_all add: abs_if)
+done
+
+lemma LIM_add_minus:
+ "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
+by (blast dest: LIM_add LIM_minus)
+
+lemma LIM_diff:
+ "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) - g(x)) -- x --> l-m"
+by (simp add: diff_minus LIM_add_minus)
+
+
+lemma LIM_const_not_eq: "k \<noteq> L ==> ~ ((%x. k) -- a --> L)"
+proof (simp add: linorder_neq_iff LIM_eq, elim disjE)
+ assume k: "k < L"
+ show "\<exists>r. 0 < r \<and>
+ (\<forall>s. 0 < s \<longrightarrow> (\<exists>x. (x < a \<or> a < x) \<and> \<bar>x-a\<bar> < s) \<and> \<not> \<bar>k-L\<bar> < r)"
+ proof (intro exI conjI strip)
+ show "0 < L-k" by (simp add: k)
+ fix s :: real
+ assume s: "0<s"
+ { from s show "s/2 + a < a \<or> a < s/2 + a" by arith
+ next
+ from s show "\<bar>s / 2 + a - a\<bar> < s" by (simp add: abs_if)
+ next
+ from s show "~ \<bar>k-L\<bar> < L-k" by (simp add: abs_if) }
+ qed
+next
+ assume k: "L < k"
+ show "\<exists>r. 0 < r \<and>
+ (\<forall>s. 0 < s \<longrightarrow> (\<exists>x. (x < a \<or> a < x) \<and> \<bar>x-a\<bar> < s) \<and> \<not> \<bar>k-L\<bar> < r)"
+ proof (intro exI conjI strip)
+ show "0 < k-L" by (simp add: k)
+ fix s :: real
+ assume s: "0<s"
+ { from s show "s/2 + a < a \<or> a < s/2 + a" by arith
+ next
+ from s show "\<bar>s / 2 + a - a\<bar> < s" by (simp add: abs_if)
+ next
+ from s show "~ \<bar>k-L\<bar> < k-L" by (simp add: abs_if) }
+ qed
+qed
+
+lemma LIM_const_eq: "(%x. k) -- x --> L ==> k = L"
+apply (rule ccontr)
+apply (blast dest: LIM_const_not_eq)
+done
+
+lemma LIM_unique: "[| f -- a --> L; f -- a --> M |] ==> L = M"
+apply (drule LIM_diff, assumption)
+apply (auto dest!: LIM_const_eq)
+done
+
+lemma LIM_mult_zero:
+ assumes f: "f -- a --> 0" and g: "g -- a --> 0"
+ shows "(%x. f(x) * g(x)) -- a --> 0"
+proof (simp add: LIM_eq, clarify)
+ fix r :: real
+ assume r: "0<r"
+ from LIM_D [OF f zero_less_one]
+ obtain fs
+ where fs: "0 < fs"
+ and fs_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < fs --> \<bar>f x\<bar> < 1"
+ by auto
+ from LIM_D [OF g r]
+ obtain gs
+ where gs: "0 < gs"
+ and gs_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < gs --> \<bar>g x\<bar> < r"
+ by auto
+ show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> a \<and> \<bar>x-a\<bar> < s \<longrightarrow> \<bar>f x\<bar> * \<bar>g x\<bar> < r)"
+ proof (intro exI conjI strip)
+ show "0 < min fs gs" by (simp add: fs gs)
+ fix x :: real
+ assume "x \<noteq> a \<and> \<bar>x-a\<bar> < min fs gs"
+ with fs_lt gs_lt
+ have "\<bar>f x\<bar> < 1" and "\<bar>g x\<bar> < r" by (auto simp add: fs_lt)
+ hence "\<bar>f x\<bar> * \<bar>g x\<bar> < 1*r" by (rule abs_mult_less)
+ thus "\<bar>f x\<bar> * \<bar>g x\<bar> < r" by simp
+ qed
+qed
+
+lemma LIM_self: "(%x. x) -- a --> a"
+by (auto simp add: LIM_def)
+
+text{*Limits are equal for functions equal except at limit point*}
+lemma LIM_equal:
+ "[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
+by (simp add: LIM_def)
+
+text{*Two uses in Hyperreal/Transcendental.ML*}
+lemma LIM_trans:
+ "[| (%x. f(x) + -g(x)) -- a --> 0; g -- a --> l |] ==> f -- a --> l"
+apply (drule LIM_add, assumption)
+apply (auto simp add: add_assoc)
+done
+
+
+subsection{*Relationships Between Standard and Nonstandard Concepts*}
+
+text{*Standard and NS definitions of Limit*} (*NEEDS STRUCTURING*)
+lemma LIM_NSLIM:
+ "f -- x --> L ==> f -- x --NS> L"
+apply (simp add: LIM_def NSLIM_def approx_def)
+apply (simp add: Infinitesimal_FreeUltrafilterNat_iff, safe)
+apply (rule_tac z = xa in eq_Abs_hypreal)
+apply (auto simp add: real_add_minus_iff starfun hypreal_minus hypreal_of_real_def hypreal_add)
+apply (rule bexI, rule_tac [2] lemma_hyprel_refl, clarify)
+apply (drule_tac x = u in spec, clarify)
+apply (drule_tac x = s in spec, clarify)
+apply (subgoal_tac "\<forall>n::nat. (xa n) \<noteq> x & abs ((xa n) + - x) < s --> abs (f (xa n) + - L) < u")
+prefer 2 apply blast
+apply (drule FreeUltrafilterNat_all, ultra)
+done
+
+(*---------------------------------------------------------------------
+ Limit: NS definition ==> standard definition
+ ---------------------------------------------------------------------*)
+
+lemma lemma_LIM: "\<forall>s. 0 < s --> (\<exists>xa. xa \<noteq> x &
+ \<bar>xa + - x\<bar> < s & r \<le> \<bar>f xa + -L\<bar>)
+ ==> \<forall>n::nat. \<exists>xa. xa \<noteq> x &
+ \<bar>xa + -x\<bar> < inverse(real(Suc n)) & r \<le> \<bar>f xa + -L\<bar>"
+apply clarify
+apply (cut_tac n1 = n in real_of_nat_Suc_gt_zero [THEN positive_imp_inverse_positive], auto)
+done
+
+lemma lemma_skolemize_LIM2:
+ "\<forall>s. 0 < s --> (\<exists>xa. xa \<noteq> x &
+ \<bar>xa + - x\<bar> < s & r \<le> \<bar>f xa + -L\<bar>)
+ ==> \<exists>X. \<forall>n::nat. X n \<noteq> x &
+ \<bar>X n + -x\<bar> < inverse(real(Suc n)) & r \<le> abs(f (X n) + -L)"
+apply (drule lemma_LIM)
+apply (drule choice, blast)
+done
+
+lemma lemma_simp: "\<forall>n. X n \<noteq> x &
+ \<bar>X n + - x\<bar> < inverse (real(Suc n)) &
+ r \<le> abs (f (X n) + - L) ==>
+ \<forall>n. \<bar>X n + - x\<bar> < inverse (real(Suc n))"
+by auto
+
+
+(*-------------------
+ NSLIM => LIM
+ -------------------*)
+
+lemma NSLIM_LIM: "f -- x --NS> L ==> f -- x --> L"
+apply (simp add: LIM_def NSLIM_def approx_def)
+apply (simp add: Infinitesimal_FreeUltrafilterNat_iff, clarify)
+apply (rule ccontr, simp)
+apply (simp add: linorder_not_less)
+apply (drule lemma_skolemize_LIM2, safe)
+apply (drule_tac x = "Abs_hypreal (hyprel``{X}) " in spec)
+apply (auto simp add: starfun hypreal_minus hypreal_of_real_def hypreal_add)
+apply (drule lemma_simp [THEN real_seq_to_hypreal_Infinitesimal])
+apply (simp add: Infinitesimal_FreeUltrafilterNat_iff hypreal_of_real_def hypreal_minus hypreal_add, blast)
+apply (drule spec, drule mp, assumption)
+apply (drule FreeUltrafilterNat_all, ultra)
+done
+
+
+(**** Key result ****)
+lemma LIM_NSLIM_iff: "(f -- x --> L) = (f -- x --NS> L)"
+by (blast intro: LIM_NSLIM NSLIM_LIM)
+
+(*-------------------------------------------------------------------*)
+(* Proving properties of limits using nonstandard definition and *)
+(* hence, the properties hold for standard limits as well *)
+(*-------------------------------------------------------------------*)
+(*------------------------------------------------
+ NSLIM_mult and hence (trivially) LIM_mult
+ ------------------------------------------------*)
+
+lemma NSLIM_mult:
+ "[| f -- x --NS> l; g -- x --NS> m |]
+ ==> (%x. f(x) * g(x)) -- x --NS> (l * m)"
+apply (simp add: NSLIM_def)
+apply (auto intro!: approx_mult_HFinite)
+done
+
+lemma LIM_mult2: "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) * g(x)) -- x --> (l * m)"
+by (simp add: LIM_NSLIM_iff NSLIM_mult)
+
+(*----------------------------------------------
+ NSLIM_add and hence (trivially) LIM_add
+ Note the much shorter proof
+ ----------------------------------------------*)
+lemma NSLIM_add:
+ "[| f -- x --NS> l; g -- x --NS> m |]
+ ==> (%x. f(x) + g(x)) -- x --NS> (l + m)"
+apply (simp add: NSLIM_def)
+apply (auto intro!: approx_add)
+done
+
+lemma LIM_add2: "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + g(x)) -- x --> (l + m)"
+by (simp add: LIM_NSLIM_iff NSLIM_add)
+
+
+lemma NSLIM_const: "(%x. k) -- x --NS> k"
+by (simp add: NSLIM_def)
+
+declare NSLIM_const [simp]
+
+lemma LIM_const2: "(%x. k) -- x --> k"
+by (simp add: LIM_NSLIM_iff)
+
+
+lemma NSLIM_minus: "f -- a --NS> L ==> (%x. -f(x)) -- a --NS> -L"
+by (simp add: NSLIM_def)
+
+lemma LIM_minus2: "f -- a --> L ==> (%x. -f(x)) -- a --> -L"
+by (simp add: LIM_NSLIM_iff NSLIM_minus)
+
+
+lemma NSLIM_add_minus: "[| f -- x --NS> l; g -- x --NS> m |] ==> (%x. f(x) + -g(x)) -- x --NS> (l + -m)"
+by (blast dest: NSLIM_add NSLIM_minus)
+
+lemma LIM_add_minus2: "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
+by (simp add: LIM_NSLIM_iff NSLIM_add_minus)
+
+
+lemma NSLIM_inverse:
+ "[| f -- a --NS> L; L \<noteq> 0 |]
+ ==> (%x. inverse(f(x))) -- a --NS> (inverse L)"
+apply (simp add: NSLIM_def, clarify)
+apply (drule spec)
+apply (auto simp add: hypreal_of_real_approx_inverse)
+done
+
+lemma LIM_inverse: "[| f -- a --> L; L \<noteq> 0 |] ==> (%x. inverse(f(x))) -- a --> (inverse L)"
+by (simp add: LIM_NSLIM_iff NSLIM_inverse)
+
+
+lemma NSLIM_zero:
+ assumes f: "f -- a --NS> l" shows "(%x. f(x) + -l) -- a --NS> 0"
+proof -;
+ have "(\<lambda>x. f x + - l) -- a --NS> l + -l"
+ by (rule NSLIM_add_minus [OF f NSLIM_const])
+ thus ?thesis by simp
+qed
+
+lemma LIM_zero2: "f -- a --> l ==> (%x. f(x) + -l) -- a --> 0"
+by (simp add: LIM_NSLIM_iff NSLIM_zero)
+
+lemma NSLIM_zero_cancel: "(%x. f(x) - l) -- x --NS> 0 ==> f -- x --NS> l"
+apply (drule_tac g = "%x. l" and m = l in NSLIM_add)
+apply (auto simp add: diff_minus add_assoc)
+done
+
+lemma LIM_zero_cancel: "(%x. f(x) - l) -- x --> 0 ==> f -- x --> l"
+apply (drule_tac g = "%x. l" and M = l in LIM_add)
+apply (auto simp add: diff_minus add_assoc)
+done
+
+
+
+lemma NSLIM_not_zero: "k \<noteq> 0 ==> ~ ((%x. k) -- x --NS> 0)"
+apply (simp add: NSLIM_def)
+apply (rule_tac x = "hypreal_of_real x + epsilon" in exI)
+apply (auto intro: Infinitesimal_add_approx_self [THEN approx_sym]
+ simp add: hypreal_epsilon_not_zero)
+done
+
+lemma NSLIM_const_not_eq: "k \<noteq> L ==> ~ ((%x. k) -- x --NS> L)"
+apply (simp add: NSLIM_def)
+apply (rule_tac x = "hypreal_of_real x + epsilon" in exI)
+apply (auto intro: Infinitesimal_add_approx_self [THEN approx_sym]
+ simp add: hypreal_epsilon_not_zero)
+done
+
+lemma NSLIM_const_eq: "(%x. k) -- x --NS> L ==> k = L"
+apply (rule ccontr)
+apply (blast dest: NSLIM_const_not_eq)
+done
+
+(* can actually be proved more easily by unfolding def! *)
+lemma NSLIM_unique: "[| f -- x --NS> L; f -- x --NS> M |] ==> L = M"
+apply (drule NSLIM_minus)
+apply (drule NSLIM_add, assumption)
+apply (auto dest!: NSLIM_const_eq [symmetric])
+done
+
+lemma LIM_unique2: "[| f -- x --> L; f -- x --> M |] ==> L = M"
+by (simp add: LIM_NSLIM_iff NSLIM_unique)
+
+
+lemma NSLIM_mult_zero: "[| f -- x --NS> 0; g -- x --NS> 0 |] ==> (%x. f(x)*g(x)) -- x --NS> 0"
+by (drule NSLIM_mult, auto)
+
+(* we can use the corresponding thm LIM_mult2 *)
+(* for standard definition of limit *)
+
+lemma LIM_mult_zero2: "[| f -- x --> 0; g -- x --> 0 |] ==> (%x. f(x)*g(x)) -- x --> 0"
+by (drule LIM_mult2, auto)
+
+
+lemma NSLIM_self: "(%x. x) -- a --NS> a"
+by (simp add: NSLIM_def)
+
+
+(*-----------------------------------------------------------------------------
+ Derivatives and Continuity - NS and Standard properties
+ -----------------------------------------------------------------------------*)
+text{*Continuity*}
+
+lemma isNSContD: "[| isNSCont f a; y \<approx> hypreal_of_real a |] ==> ( *f* f) y \<approx> hypreal_of_real (f a)"
+by (simp add: isNSCont_def)
+
+lemma isNSCont_NSLIM: "isNSCont f a ==> f -- a --NS> (f a) "
+by (simp add: isNSCont_def NSLIM_def)
+
+lemma NSLIM_isNSCont: "f -- a --NS> (f a) ==> isNSCont f a"
+apply (simp add: isNSCont_def NSLIM_def, auto)
+apply (rule_tac Q = "y = hypreal_of_real a" in excluded_middle [THEN disjE], auto)
+done
+
+(*-----------------------------------------------------
+ NS continuity can be defined using NS Limit in
+ similar fashion to standard def of continuity
+ -----------------------------------------------------*)
+lemma isNSCont_NSLIM_iff: "(isNSCont f a) = (f -- a --NS> (f a))"
+by (blast intro: isNSCont_NSLIM NSLIM_isNSCont)
+
+(*----------------------------------------------
+ Hence, NS continuity can be given
+ in terms of standard limit
+ ---------------------------------------------*)
+lemma isNSCont_LIM_iff: "(isNSCont f a) = (f -- a --> (f a))"
+by (simp add: LIM_NSLIM_iff isNSCont_NSLIM_iff)
+
+(*-----------------------------------------------
+ Moreover, it's trivial now that NS continuity
+ is equivalent to standard continuity
+ -----------------------------------------------*)
+lemma isNSCont_isCont_iff: "(isNSCont f a) = (isCont f a)"
+apply (simp add: isCont_def)
+apply (rule isNSCont_LIM_iff)
+done
+
+(*----------------------------------------
+ Standard continuity ==> NS continuity
+ ----------------------------------------*)
+lemma isCont_isNSCont: "isCont f a ==> isNSCont f a"
+by (erule isNSCont_isCont_iff [THEN iffD2])
+
+(*----------------------------------------
+ NS continuity ==> Standard continuity
+ ----------------------------------------*)
+lemma isNSCont_isCont: "isNSCont f a ==> isCont f a"
+by (erule isNSCont_isCont_iff [THEN iffD1])
+
+text{*Alternative definition of continuity*}
+(* Prove equivalence between NS limits - *)
+(* seems easier than using standard def *)
+lemma NSLIM_h_iff: "(f -- a --NS> L) = ((%h. f(a + h)) -- 0 --NS> L)"
+apply (simp add: NSLIM_def, auto)
+apply (drule_tac x = "hypreal_of_real a + x" in spec)
+apply (drule_tac [2] x = "-hypreal_of_real a + x" in spec, safe, simp)
+apply (rule mem_infmal_iff [THEN iffD2, THEN Infinitesimal_add_approx_self [THEN approx_sym]])
+apply (rule_tac [4] approx_minus_iff2 [THEN iffD1])
+ prefer 3 apply (simp add: add_commute)
+apply (rule_tac [2] z = x in eq_Abs_hypreal)
+apply (rule_tac [4] z = x in eq_Abs_hypreal)
+apply (auto simp add: starfun hypreal_of_real_def hypreal_minus hypreal_add add_assoc approx_refl hypreal_zero_def)
+done
+
+lemma NSLIM_isCont_iff: "(f -- a --NS> f a) = ((%h. f(a + h)) -- 0 --NS> f a)"
+by (rule NSLIM_h_iff)
+
+lemma LIM_isCont_iff: "(f -- a --> f a) = ((%h. f(a + h)) -- 0 --> f(a))"
+by (simp add: LIM_NSLIM_iff NSLIM_isCont_iff)
+
+lemma isCont_iff: "(isCont f x) = ((%h. f(x + h)) -- 0 --> f(x))"
+by (simp add: isCont_def LIM_isCont_iff)
+
+(*--------------------------------------------------------------------------
+ Immediate application of nonstandard criterion for continuity can offer
+ very simple proofs of some standard property of continuous functions
+ --------------------------------------------------------------------------*)
+text{*sum continuous*}
+lemma isCont_add: "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) + g(x)) a"
+by (auto intro: approx_add simp add: isNSCont_isCont_iff [symmetric] isNSCont_def)
+
+text{*mult continuous*}
+lemma isCont_mult: "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) * g(x)) a"
+by (auto intro!: starfun_mult_HFinite_approx
+ simp del: starfun_mult [symmetric]
+ simp add: isNSCont_isCont_iff [symmetric] isNSCont_def)
+
+(*-------------------------------------------
+ composition of continuous functions
+ Note very short straightforard proof!
+ ------------------------------------------*)
+lemma isCont_o: "[| isCont f a; isCont g (f a) |] ==> isCont (g o f) a"
+by (auto simp add: isNSCont_isCont_iff [symmetric] isNSCont_def starfun_o [symmetric])
+
+lemma isCont_o2: "[| isCont f a; isCont g (f a) |] ==> isCont (%x. g (f x)) a"
+by (auto dest: isCont_o simp add: o_def)
+
+lemma isNSCont_minus: "isNSCont f a ==> isNSCont (%x. - f x) a"
+by (simp add: isNSCont_def)
+
+lemma isCont_minus: "isCont f a ==> isCont (%x. - f x) a"
+by (auto simp add: isNSCont_isCont_iff [symmetric] isNSCont_minus)
+
+lemma isCont_inverse:
+ "[| isCont f x; f x \<noteq> 0 |] ==> isCont (%x. inverse (f x)) x"
+apply (simp add: isCont_def)
+apply (blast intro: LIM_inverse)
+done
+
+lemma isNSCont_inverse: "[| isNSCont f x; f x \<noteq> 0 |] ==> isNSCont (%x. inverse (f x)) x"
+by (auto intro: isCont_inverse simp add: isNSCont_isCont_iff)
+
+lemma isCont_diff:
+ "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) - g(x)) a"
+apply (simp add: diff_minus)
+apply (auto intro: isCont_add isCont_minus)
+done
+
+lemma isCont_const: "isCont (%x. k) a"
+by (simp add: isCont_def)
+declare isCont_const [simp]
+
+lemma isNSCont_const: "isNSCont (%x. k) a"
+by (simp add: isNSCont_def)
+declare isNSCont_const [simp]
+
+lemma isNSCont_rabs: "isNSCont abs a"
+apply (simp add: isNSCont_def)
+apply (auto intro: approx_hrabs simp add: hypreal_of_real_hrabs [symmetric] starfun_rabs_hrabs)
+done
+declare isNSCont_rabs [simp]
+
+lemma isCont_rabs: "isCont abs a"
+by (auto simp add: isNSCont_isCont_iff [symmetric])
+declare isCont_rabs [simp]
+
+(****************************************************************
+(%* 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) \<in> A}, of any open set A
+ is always an open set
+ ------------------------------------------------------------*%)
+Goal "[| isNSopen A; \<forall>x. isNSCont f x |]
+ ==> isNSopen {x. f x \<in> A}"
+by (auto_tac (claset(),simpset() addsimps [isNSopen_iff1]));
+by (dtac (mem_monad_approx RS approx_sym);
+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")] approx_mem_monad2 1);
+by (blast_tac (claset() addIs [starfun_mem_starset]);
+qed "isNSCont_isNSopen";
+
+Goalw [isNSCont_def]
+ "\<forall>A. isNSopen A --> isNSopen {x. f x \<in> A} \
+\ ==> isNSCont f x";
+by (auto_tac (claset() addSIs [(mem_infmal_iff RS iffD1) RS
+ (approx_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));
+by (auto_tac (claset(),simpset() addsimps [isNSopen_def,isNSnbhd_def]));
+by (dres_inst_tac [("x","x")] spec 1);
+by (auto_tac (claset() addDs [approx_sym RS approx_mem_monad],
+ simpset() addsimps [hypreal_of_real_zero RS sym,STAR_starfun_rabs_add_minus]));
+qed "isNSopen_isNSCont";
+
+Goal "(\<forall>x. isNSCont f x) = \
+\ (\<forall>A. isNSopen A --> isNSopen {x. f(x) \<in> A})";
+by (blast_tac (claset() addIs [isNSCont_isNSopen,
+ isNSopen_isNSCont]);
+qed "isNSCont_isNSopen_iff";
+
+(%*------- Standard version of same theorem --------*%)
+Goal "(\<forall>x. isCont f x) = \
+\ (\<forall>A. isopen A --> isopen {x. f(x) \<in> 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";
+*******************************************************************)
+
+text{*Uniform continuity*}
+lemma isNSUContD: "[| isNSUCont f; x \<approx> y|] ==> ( *f* f) x \<approx> ( *f* f) y"
+by (simp add: isNSUCont_def)
+
+lemma isUCont_isCont: "isUCont f ==> isCont f x"
+by (simp add: isUCont_def isCont_def LIM_def, meson)
+
+lemma isUCont_isNSUCont: "isUCont f ==> isNSUCont f"
+apply (simp add: isNSUCont_def isUCont_def approx_def)
+apply (simp add: Infinitesimal_FreeUltrafilterNat_iff, safe)
+apply (rule_tac z = x in eq_Abs_hypreal)
+apply (rule_tac z = y in eq_Abs_hypreal)
+apply (auto simp add: starfun hypreal_minus hypreal_add)
+apply (rule bexI, rule_tac [2] lemma_hyprel_refl, safe)
+apply (drule_tac x = u in spec, clarify)
+apply (drule_tac x = s in spec, clarify)
+apply (subgoal_tac "\<forall>n::nat. abs ((xa n) + - (xb n)) < s --> abs (f (xa n) + - f (xb n)) < u")
+prefer 2 apply blast
+apply (erule_tac V = "\<forall>x y. \<bar>x + - y\<bar> < s --> \<bar>f x + - f y\<bar> < u" in thin_rl)
+apply (drule FreeUltrafilterNat_all, ultra)
+done
+
+lemma lemma_LIMu: "\<forall>s. 0 < s --> (\<exists>z y. \<bar>z + - y\<bar> < s & r \<le> \<bar>f z + -f y\<bar>)
+ ==> \<forall>n::nat. \<exists>z y.
+ \<bar>z + -y\<bar> < inverse(real(Suc n)) &
+ r \<le> \<bar>f z + -f y\<bar>"
+apply clarify
+apply (cut_tac n1 = n in real_of_nat_Suc_gt_zero [THEN positive_imp_inverse_positive], auto)
+done
+
+lemma lemma_skolemize_LIM2u: "\<forall>s. 0 < s --> (\<exists>z y. \<bar>z + - y\<bar> < s & r \<le> \<bar>f z + -f y\<bar>)
+ ==> \<exists>X Y. \<forall>n::nat.
+ abs(X n + -(Y n)) < inverse(real(Suc n)) &
+ r \<le> abs(f (X n) + -f (Y n))"
+apply (drule lemma_LIMu)
+apply (drule choice, safe)
+apply (drule choice, blast)
+done
+
+lemma lemma_simpu: "\<forall>n. \<bar>X n + -Y n\<bar> < inverse (real(Suc n)) &
+ r \<le> abs (f (X n) + - f(Y n)) ==>
+ \<forall>n. \<bar>X n + - Y n\<bar> < inverse (real(Suc n))"
+apply auto
+done
+
+lemma isNSUCont_isUCont:
+ "isNSUCont f ==> isUCont f"
+apply (simp add: isNSUCont_def isUCont_def approx_def)
+apply (simp add: Infinitesimal_FreeUltrafilterNat_iff, safe)
+apply (rule ccontr, simp)
+apply (simp add: linorder_not_less)
+apply (drule lemma_skolemize_LIM2u, safe)
+apply (drule_tac x = "Abs_hypreal (hyprel``{X}) " in spec)
+apply (drule_tac x = "Abs_hypreal (hyprel``{Y}) " in spec)
+apply (simp add: starfun hypreal_minus hypreal_add, auto)
+apply (drule lemma_simpu [THEN real_seq_to_hypreal_Infinitesimal2])
+apply (simp add: Infinitesimal_FreeUltrafilterNat_iff hypreal_minus hypreal_add, blast)
+apply (rotate_tac 2)
+apply (drule_tac x = r in spec, clarify)
+apply (drule FreeUltrafilterNat_all, ultra)
+done
+
+text{*Derivatives*}
+lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- 0 --> D)"
+by (simp add: deriv_def)
+
+lemma DERIV_NS_iff:
+ "(DERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- 0 --NS> D)"
+by (simp add: deriv_def LIM_NSLIM_iff)
+
+lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) + - f(x))/h) -- 0 --> D"
+by (simp add: deriv_def)
+
+lemma NS_DERIV_D: "DERIV f x :> D ==>
+ (%h. (f(x + h) + - f(x))/h) -- 0 --NS> D"
+by (simp add: deriv_def LIM_NSLIM_iff)
+
+subsubsection{*Uniqueness*}
+
+lemma DERIV_unique:
+ "[| DERIV f x :> D; DERIV f x :> E |] ==> D = E"
+apply (simp add: deriv_def)
+apply (blast intro: LIM_unique)
+done
+
+lemma NSDeriv_unique:
+ "[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E"
+apply (simp add: nsderiv_def)
+apply (cut_tac Infinitesimal_epsilon hypreal_epsilon_not_zero)
+apply (auto dest!: bspec [where x=epsilon]
+ intro!: inj_hypreal_of_real [THEN injD]
+ dest: approx_trans3)
+done
+
+subsubsection{*Differentiable*}
+
+lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
+by (simp add: differentiable_def)
+
+lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
+by (force simp add: differentiable_def)
+
+lemma NSdifferentiableD: "f NSdifferentiable x ==> \<exists>D. NSDERIV f x :> D"
+by (simp add: NSdifferentiable_def)
+
+lemma NSdifferentiableI: "NSDERIV f x :> D ==> f NSdifferentiable x"
+by (force simp add: NSdifferentiable_def)
+
+subsubsection{*Alternative definition for differentiability*}
+
+lemma LIM_I:
+ "(!!r. 0<r ==> (\<exists>s. 0 < s & (\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> \<bar>f x - L\<bar> < r)))
+ ==> f -- a --> L"
+by (simp add: LIM_eq)
+
+lemma DERIV_LIM_iff:
+ "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
+ ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
+proof (intro iffI LIM_I)
+ fix r::real
+ assume r: "0<r"
+ assume "(\<lambda>h. (f (a + h) - f a) / h) -- 0 --> D"
+ from LIM_D [OF this r]
+ obtain s
+ where s: "0 < s"
+ and s_lt: "\<forall>x. x \<noteq> 0 & \<bar>x\<bar> < s --> \<bar>(f (a + x) - f a) / x - D\<bar> < r"
+ by auto
+ show "\<exists>s. 0 < s \<and>
+ (\<forall>x. x \<noteq> a \<and> \<bar>x-a\<bar> < s \<longrightarrow> \<bar>(f x - f a) / (x-a) - D\<bar> < r)"
+ proof (intro exI conjI strip)
+ show "0 < s" by (rule s)
+ next
+ fix x::real
+ assume "x \<noteq> a \<and> \<bar>x-a\<bar> < s"
+ with s_lt [THEN spec [where x="x-a"]]
+ show "\<bar>(f x - f a) / (x-a) - D\<bar> < r" by auto
+ qed
+next
+ fix r::real
+ assume r: "0<r"
+ assume "(\<lambda>x. (f x - f a) / (x-a)) -- a --> D"
+ from LIM_D [OF this r]
+ obtain s
+ where s: "0 < s"
+ and s_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> \<bar>(f x - f a)/(x-a) - D\<bar> < r"
+ by auto
+ show "\<exists>s. 0 < s \<and>
+ (\<forall>x. x \<noteq> 0 & \<bar>x - 0\<bar> < s --> \<bar>(f (a + x) - f a) / x - D\<bar> < r)"
+ proof (intro exI conjI strip)
+ show "0 < s" by (rule s)
+ next
+ fix x::real
+ assume "x \<noteq> 0 \<and> \<bar>x - 0\<bar> < s"
+ with s_lt [THEN spec [where x="x+a"]]
+ show "\<bar>(f (a + x) - f a) / x - D\<bar> < r" by (auto simp add: add_ac)
+ qed
+qed
+
+lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --> D)"
+by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)
+
+
+subsection{*Equivalence of NS and standard definitions of differentiation*}
+
+text{*First NSDERIV in terms of NSLIM*}
+
+(*--- first equivalence ---*)
+lemma NSDERIV_NSLIM_iff:
+ "(NSDERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- 0 --NS> D)"
+apply (simp add: nsderiv_def NSLIM_def, auto)
+apply (drule_tac x = xa in bspec)
+apply (rule_tac [3] ccontr)
+apply (drule_tac [3] x = h in spec)
+apply (auto simp add: mem_infmal_iff starfun_lambda_cancel)
+done
+
+(*--- second equivalence ---*)
+lemma NSDERIV_NSLIM_iff2:
+ "(NSDERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --NS> D)"
+by (simp add: NSDERIV_NSLIM_iff DERIV_LIM_iff diff_minus [symmetric]
+ LIM_NSLIM_iff [symmetric])
+
+(* while we're at it! *)
+lemma NSDERIV_iff2:
+ "(NSDERIV f x :> D) =
+ (\<forall>w.
+ w \<noteq> hypreal_of_real x & w \<approx> hypreal_of_real x -->
+ ( *f* (%z. (f z - f x) / (z-x))) w \<approx> hypreal_of_real D)"
+by (simp add: NSDERIV_NSLIM_iff2 NSLIM_def)
+
+(*FIXME DELETE*)
+lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x + -a \<noteq> (0::hypreal))"
+by (auto dest: hypreal_eq_minus_iff [THEN iffD2])
+
+lemma NSDERIVD5:
+ "(NSDERIV f x :> D) ==>
+ (\<forall>u. u \<approx> hypreal_of_real x -->
+ ( *f* (%z. f z - f x)) u \<approx> hypreal_of_real D * (u - hypreal_of_real x))"
+apply (auto simp add: NSDERIV_iff2)
+apply (case_tac "u = hypreal_of_real x", auto)
+apply (drule_tac x = u in spec, auto)
+apply (drule_tac c = "u - hypreal_of_real x" and b = "hypreal_of_real D" in approx_mult1)
+apply (drule_tac [!] hypreal_not_eq_minus_iff [THEN iffD1])
+apply (subgoal_tac [2] "( *f* (%z. z-x)) u \<noteq> (0::hypreal) ")
+apply (auto simp add: diff_minus
+ approx_minus_iff [THEN iffD1, THEN mem_infmal_iff [THEN iffD2]]
+ Infinitesimal_subset_HFinite [THEN subsetD])
+done
+
+lemma NSDERIVD4:
+ "(NSDERIV f x :> D) ==>
+ (\<forall>h \<in> Infinitesimal.
+ (( *f* f)(hypreal_of_real x + h) -
+ hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
+apply (auto simp add: nsderiv_def)
+apply (case_tac "h = (0::hypreal) ")
+apply (auto simp add: diff_minus)
+apply (drule_tac x = h in bspec)
+apply (drule_tac [2] c = h in approx_mult1)
+apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
+ simp add: diff_minus)
+done
+
+lemma NSDERIVD3:
+ "(NSDERIV f x :> D) ==>
+ (\<forall>h \<in> Infinitesimal - {0}.
+ (( *f* f)(hypreal_of_real x + h) -
+ hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
+apply (auto simp add: nsderiv_def)
+apply (rule ccontr, drule_tac x = h in bspec)
+apply (drule_tac [2] c = h in approx_mult1)
+apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
+ simp add: mult_assoc diff_minus)
+done
+
+text{*Now equivalence between NSDERIV and DERIV*}
+lemma NSDERIV_DERIV_iff: "(NSDERIV f x :> D) = (DERIV f x :> D)"
+by (simp add: deriv_def NSDERIV_NSLIM_iff LIM_NSLIM_iff)
+
+(*---------------------------------------------------
+ Differentiability implies continuity
+ nice and simple "algebraic" proof
+ --------------------------------------------------*)
+lemma NSDERIV_isNSCont: "NSDERIV f x :> D ==> isNSCont f x"
+apply (auto simp add: nsderiv_def isNSCont_NSLIM_iff NSLIM_def)
+apply (drule approx_minus_iff [THEN iffD1])
+apply (drule hypreal_not_eq_minus_iff [THEN iffD1])
+apply (drule_tac x = "-hypreal_of_real x + xa" in bspec)
+ prefer 2 apply (simp add: add_assoc [symmetric])
+apply (auto simp add: mem_infmal_iff [symmetric] hypreal_add_commute)
+apply (drule_tac c = "xa + -hypreal_of_real x" in approx_mult1)
+apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
+ simp add: mult_assoc)
+apply (drule_tac x3=D in
+ HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult,
+ THEN mem_infmal_iff [THEN iffD1]])
+apply (auto simp add: mult_commute
+ intro: approx_trans approx_minus_iff [THEN iffD2])
+done
+
+text{*Now Sandard proof*}
+lemma DERIV_isCont: "DERIV f x :> D ==> isCont f x"
+by (simp add: NSDERIV_DERIV_iff [symmetric] isNSCont_isCont_iff [symmetric]
+ NSDERIV_isNSCont)
+
+
+(*----------------------------------------------------------------------------
+ Differentiation rules for combinations of functions
+ follow from clear, straightforard, algebraic
+ manipulations
+ ----------------------------------------------------------------------------*)
+text{*Constant function*}
+
+(* use simple constant nslimit theorem *)
+lemma NSDERIV_const: "(NSDERIV (%x. k) x :> 0)"
+by (simp add: NSDERIV_NSLIM_iff)
+declare NSDERIV_const [simp]
+
+lemma DERIV_const: "(DERIV (%x. k) x :> 0)"
+by (simp add: NSDERIV_DERIV_iff [symmetric])
+declare DERIV_const [simp]
+
+(*-----------------------------------------------------
+ Sum of functions- proved easily
+ ----------------------------------------------------*)
+
+
+lemma NSDERIV_add: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
+ ==> NSDERIV (%x. f x + g x) x :> Da + Db"
+apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
+apply (auto simp add: add_divide_distrib dest!: spec)
+apply (drule_tac b = "hypreal_of_real Da" and d = "hypreal_of_real Db" in approx_add)
+apply (auto simp add: add_ac)
+done
+
+(* Standard theorem *)
+lemma DERIV_add: "[| DERIV f x :> Da; DERIV g x :> Db |]
+ ==> DERIV (%x. f x + g x) x :> Da + Db"
+apply (simp add: NSDERIV_add NSDERIV_DERIV_iff [symmetric])
+done
+
+(*-----------------------------------------------------
+ Product of functions - Proof is trivial but tedious
+ and long due to rearrangement of terms
+ ----------------------------------------------------*)
+
+lemma lemma_nsderiv1: "((a::hypreal)*b) + -(c*d) = (b*(a + -c)) + (c*(b + -d))"
+by (simp add: right_distrib)
+
+lemma lemma_nsderiv2: "[| (x + y) / z = hypreal_of_real D + yb; z \<noteq> 0;
+ z \<in> Infinitesimal; yb \<in> Infinitesimal |]
+ ==> x + y \<approx> 0"
+apply (frule_tac c1 = z in hypreal_mult_right_cancel [THEN iffD2], assumption)
+apply (erule_tac V = " (x + y) / z = hypreal_of_real D + yb" in thin_rl)
+apply (auto intro!: Infinitesimal_HFinite_mult2 HFinite_add
+ simp add: hypreal_mult_assoc mem_infmal_iff [symmetric])
+apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
+done
+
+
+lemma NSDERIV_mult: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
+ ==> NSDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
+apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
+apply (auto dest!: spec
+ simp add: starfun_lambda_cancel lemma_nsderiv1)
+apply (simp (no_asm) add: add_divide_distrib)
+apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
+apply (auto simp del: times_divide_eq_right simp add: times_divide_eq_right [symmetric])
+apply (drule_tac D = Db in lemma_nsderiv2)
+apply (drule_tac [4]
+ approx_minus_iff [THEN iffD2, THEN bex_Infinitesimal_iff2 [THEN iffD2]])
+apply (auto intro!: approx_add_mono1
+ simp add: left_distrib right_distrib mult_commute add_assoc)
+apply (rule_tac b1 = "hypreal_of_real Db * hypreal_of_real (f x)"
+ in add_commute [THEN subst])
+apply (auto intro!: Infinitesimal_add_approx_self2 [THEN approx_sym]
+ Infinitesimal_add Infinitesimal_mult
+ Infinitesimal_hypreal_of_real_mult
+ Infinitesimal_hypreal_of_real_mult2
+ simp add: add_assoc [symmetric])
+done
+
+lemma DERIV_mult:
+ "[| DERIV f x :> Da; DERIV g x :> Db |]
+ ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
+by (simp add: NSDERIV_mult NSDERIV_DERIV_iff [symmetric])
+
+text{*Multiplying by a constant*}
+lemma NSDERIV_cmult: "NSDERIV f x :> D
+ ==> NSDERIV (%x. c * f x) x :> c*D"
+apply (simp only: times_divide_eq_right [symmetric] NSDERIV_NSLIM_iff
+ minus_mult_right right_distrib [symmetric])
+apply (erule NSLIM_const [THEN NSLIM_mult])
+done
+
+(* let's do the standard proof though theorem *)
+(* LIM_mult2 follows from a NS proof *)
+
+lemma DERIV_cmult:
+ "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
+apply (simp only: deriv_def times_divide_eq_right [symmetric]
+ NSDERIV_NSLIM_iff minus_mult_right right_distrib [symmetric])
+apply (erule LIM_const [THEN LIM_mult2])
+done
+
+text{*Negation of function*}
+lemma NSDERIV_minus: "NSDERIV f x :> D ==> NSDERIV (%x. -(f x)) x :> -D"
+proof (simp add: NSDERIV_NSLIM_iff)
+ assume "(\<lambda>h. (f (x + h) + - f x) / h) -- 0 --NS> D"
+ hence deriv: "(\<lambda>h. - ((f(x+h) + - f x) / h)) -- 0 --NS> - D"
+ by (rule NSLIM_minus)
+ have "\<forall>h. - ((f (x + h) + - f x) / h) = (- f (x + h) + f x) / h"
+ by (simp add: minus_divide_left)
+ with deriv
+ show "(\<lambda>h. (- f (x + h) + f x) / h) -- 0 --NS> - D" by simp
+qed
+
+
+lemma DERIV_minus: "DERIV f x :> D ==> DERIV (%x. -(f x)) x :> -D"
+by (simp add: NSDERIV_minus NSDERIV_DERIV_iff [symmetric])
+
+text{*Subtraction*}
+lemma NSDERIV_add_minus: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] ==> NSDERIV (%x. f x + -g x) x :> Da + -Db"
+by (blast dest: NSDERIV_add NSDERIV_minus)
+
+lemma DERIV_add_minus: "[| DERIV f x :> Da; DERIV g x :> Db |] ==> DERIV (%x. f x + -g x) x :> Da + -Db"
+by (blast dest: DERIV_add DERIV_minus)
+
+lemma NSDERIV_diff:
+ "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
+ ==> NSDERIV (%x. f x - g x) x :> Da-Db"
+apply (simp add: diff_minus)
+apply (blast intro: NSDERIV_add_minus)
+done
+
+lemma DERIV_diff:
+ "[| DERIV f x :> Da; DERIV g x :> Db |]
+ ==> DERIV (%x. f x - g x) x :> Da-Db"
+apply (simp add: diff_minus)
+apply (blast intro: DERIV_add_minus)
+done
+
+(*---------------------------------------------------------------
+ (NS) Increment
+ ---------------------------------------------------------------*)
+lemma incrementI:
+ "f NSdifferentiable x ==>
+ increment f x h = ( *f* f) (hypreal_of_real(x) + h) +
+ -hypreal_of_real (f x)"
+by (simp add: increment_def)
+
+lemma incrementI2: "NSDERIV f x :> D ==>
+ increment f x h = ( *f* f) (hypreal_of_real(x) + h) +
+ -hypreal_of_real (f x)"
+apply (erule NSdifferentiableI [THEN incrementI])
+done
+
+(* The Increment theorem -- Keisler p. 65 *)
+lemma increment_thm: "[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |]
+ ==> \<exists>e \<in> Infinitesimal. increment f x h = hypreal_of_real(D)*h + e*h"
+apply (frule_tac h = h in incrementI2, simp add: nsderiv_def)
+apply (drule bspec, auto)
+apply (drule bex_Infinitesimal_iff2 [THEN iffD2], clarify)
+apply (frule_tac b1 = "hypreal_of_real (D) + y"
+ in hypreal_mult_right_cancel [THEN iffD2])
+apply (erule_tac [2] V = "(( *f* f) (hypreal_of_real (x) + h) + - hypreal_of_real (f x)) / h = hypreal_of_real (D) + y" in thin_rl)
+apply assumption
+apply (simp add: times_divide_eq_right [symmetric] del: times_divide_eq_right)
+apply (auto simp add: left_distrib)
+done
+
+lemma increment_thm2:
+ "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
+ ==> \<exists>e \<in> Infinitesimal. increment f x h =
+ hypreal_of_real(D)*h + e*h"
+by (blast dest!: mem_infmal_iff [THEN iffD2] intro!: increment_thm)
+
+
+lemma increment_approx_zero: "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
+ ==> increment f x h \<approx> 0"
+apply (drule increment_thm2,
+ auto intro!: Infinitesimal_HFinite_mult2 HFinite_add simp add: left_distrib [symmetric] mem_infmal_iff [symmetric])
+apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
+done
+
+text{* 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 *)
+lemma NSDERIV_zero:
+ "[| NSDERIV g x :> D;
+ ( *f* g) (hypreal_of_real(x) + xa) = hypreal_of_real(g x);
+ xa \<in> Infinitesimal;
+ xa \<noteq> 0
+ |] ==> D = 0"
+apply (simp add: nsderiv_def)
+apply (drule bspec, auto)
+done
+
+(* can be proved differently using NSLIM_isCont_iff *)
+lemma NSDERIV_approx:
+ "[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |]
+ ==> ( *f* f) (hypreal_of_real(x) + h) + -hypreal_of_real(f x) \<approx> 0"
+apply (simp add: nsderiv_def)
+apply (simp add: mem_infmal_iff [symmetric])
+apply (rule Infinitesimal_ratio)
+apply (rule_tac [3] approx_hypreal_of_real_HFinite, auto)
+done
+
+(*---------------------------------------------------------------
+ from one version of differentiability
+
+ f(x) - f(a)
+ --------------- \<approx> Db
+ x - a
+ ---------------------------------------------------------------*)
+lemma NSDERIVD1: "[| NSDERIV f (g x) :> Da;
+ ( *f* g) (hypreal_of_real(x) + xa) \<noteq> hypreal_of_real (g x);
+ ( *f* g) (hypreal_of_real(x) + xa) \<approx> 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))
+ \<approx> hypreal_of_real(Da)"
+by (auto simp add: NSDERIV_NSLIM_iff2 NSLIM_def diff_minus [symmetric])
+
+(*--------------------------------------------------------------
+ from other version of differentiability
+
+ f(x + h) - f(x)
+ ----------------- \<approx> Db
+ h
+ --------------------------------------------------------------*)
+lemma NSDERIVD2: "[| NSDERIV g x :> Db; xa \<in> Infinitesimal; xa \<noteq> 0 |]
+ ==> (( *f* g) (hypreal_of_real(x) + xa) + - hypreal_of_real(g x)) / xa
+ \<approx> hypreal_of_real(Db)"
+by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def mem_infmal_iff starfun_lambda_cancel)
+
+lemma lemma_chain: "(z::hypreal) \<noteq> 0 ==> x*y = (x*inverse(z))*(z*y)"
+by auto
+
+(*------------------------------------------------------
+ This proof uses both definitions of differentiability.
+ ------------------------------------------------------*)
+lemma NSDERIV_chain: "[| NSDERIV f (g x) :> Da; NSDERIV g x :> Db |]
+ ==> NSDERIV (f o g) x :> Da * Db"
+apply (simp (no_asm_simp) add: NSDERIV_NSLIM_iff NSLIM_def
+ mem_infmal_iff [symmetric])
+apply clarify
+apply (frule_tac f = g in NSDERIV_approx)
+apply (auto simp add: starfun_lambda_cancel2 starfun_o [symmetric])
+apply (case_tac "( *f* g) (hypreal_of_real (x) + xa) = hypreal_of_real (g x) ")
+apply (drule_tac g = g in NSDERIV_zero)
+apply (auto simp add: divide_inverse)
+apply (rule_tac z1 = "( *f* g) (hypreal_of_real (x) + xa) + -hypreal_of_real (g x) " and y1 = "inverse xa" in lemma_chain [THEN ssubst])
+apply (erule hypreal_not_eq_minus_iff [THEN iffD1])
+apply (rule approx_mult_hypreal_of_real)
+apply (simp_all add: divide_inverse [symmetric])
+apply (blast intro: NSDERIVD1 approx_minus_iff [THEN iffD2])
+apply (blast intro: NSDERIVD2)
+done
+
+(* standard version *)
+lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
+by (simp add: NSDERIV_DERIV_iff [symmetric] NSDERIV_chain)
+
+lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
+by (auto dest: DERIV_chain simp add: o_def)
+
+text{*Differentiation of natural number powers*}
+lemma NSDERIV_Id: "NSDERIV (%x. x) x :> 1"
+by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def starfun_Id)
+declare NSDERIV_Id [simp]
+
+(*derivative of the identity function*)
+lemma DERIV_Id: "DERIV (%x. x) x :> 1"
+by (simp add: NSDERIV_DERIV_iff [symmetric])
+declare DERIV_Id [simp]
+
+lemmas isCont_Id = DERIV_Id [THEN DERIV_isCont, standard]
+
+(*derivative of linear multiplication*)
+lemma DERIV_cmult_Id: "DERIV (op * c) x :> c"
+by (cut_tac c = c and x = x in DERIV_Id [THEN DERIV_cmult], simp)
+declare DERIV_cmult_Id [simp]
+
+lemma NSDERIV_cmult_Id: "NSDERIV (op * c) x :> c"
+by (simp add: NSDERIV_DERIV_iff)
+declare NSDERIV_cmult_Id [simp]
+
+lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
+apply (induct_tac "n")
+apply (drule_tac [2] DERIV_Id [THEN DERIV_mult])
+apply (auto simp add: real_of_nat_Suc left_distrib)
+apply (case_tac "0 < n")
+apply (drule_tac x = x in realpow_minus_mult)
+apply (auto simp add: real_mult_assoc real_add_commute)
+done
+
+(* NS version *)
+lemma NSDERIV_pow: "NSDERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
+by (simp add: NSDERIV_DERIV_iff DERIV_pow)
+
+(*---------------------------------------------------------------
+ Power of -1
+ ---------------------------------------------------------------*)
+
+(*Can't get rid of x \<noteq> 0 because it isn't continuous at zero*)
+lemma NSDERIV_inverse:
+ "x \<noteq> 0 ==> NSDERIV (%x. inverse(x)) x :> (- (inverse x ^ Suc (Suc 0)))"
+apply (simp add: nsderiv_def)
+apply (rule ballI, simp, clarify)
+apply (frule Infinitesimal_add_not_zero)
+prefer 2 apply (simp add: add_commute)
+apply (auto simp add: starfun_inverse_inverse realpow_two
+ simp del: minus_mult_left [symmetric] minus_mult_right [symmetric])
+apply (simp add: inverse_add inverse_mult_distrib [symmetric]
+ inverse_minus_eq [symmetric] add_ac mult_ac
+ del: inverse_mult_distrib inverse_minus_eq
+ minus_mult_left [symmetric] minus_mult_right [symmetric])
+apply (simp (no_asm_simp) add: mult_assoc [symmetric] right_distrib
+ del: minus_mult_left [symmetric] minus_mult_right [symmetric])
+apply (rule_tac y = " inverse (- hypreal_of_real x * hypreal_of_real x) " in approx_trans)
+apply (rule inverse_add_Infinitesimal_approx2)
+apply (auto dest!: hypreal_of_real_HFinite_diff_Infinitesimal
+ simp add: inverse_minus_eq [symmetric] HFinite_minus_iff)
+apply (rule Infinitesimal_HFinite_mult2, auto)
+done
+
+
+
+
+lemma DERIV_inverse: "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
+by (simp add: NSDERIV_inverse NSDERIV_DERIV_iff [symmetric] del: realpow_Suc)
+
+text{*Derivative of inverse*}
+lemma DERIV_inverse_fun: "[| DERIV f x :> d; f(x) \<noteq> 0 |]
+ ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
+apply (simp only: mult_commute [of d] minus_mult_left power_inverse)
+apply (fold o_def)
+apply (blast intro!: DERIV_chain DERIV_inverse)
+done
+
+lemma NSDERIV_inverse_fun: "[| NSDERIV f x :> d; f(x) \<noteq> 0 |]
+ ==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
+by (simp add: NSDERIV_DERIV_iff DERIV_inverse_fun del: realpow_Suc)
+
+text{*Derivative of quotient*}
+lemma DERIV_quotient: "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
+ ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) + -(e*f(x))) / (g(x) ^ Suc (Suc 0))"
+apply (drule_tac f = g in DERIV_inverse_fun)
+apply (drule_tac [2] DERIV_mult)
+apply (assumption+)
+apply (simp add: divide_inverse right_distrib power_inverse minus_mult_left
+ mult_ac
+ del: realpow_Suc minus_mult_right [symmetric] minus_mult_left [symmetric])
+done
+
+lemma NSDERIV_quotient: "[| NSDERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
+ ==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x)
+ + -(e*f(x))) / (g(x) ^ Suc (Suc 0))"
+by (simp add: NSDERIV_DERIV_iff DERIV_quotient del: realpow_Suc)
+
+(* ------------------------------------------------------------------------ *)
+(* Caratheodory formulation of derivative at a point: standard proof *)
+(* ------------------------------------------------------------------------ *)
+
+lemma CARAT_DERIV:
+ "(DERIV f x :> l) =
+ (\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isCont g x & g x = l)"
+ (is "?lhs = ?rhs")
+proof
+ assume der: "DERIV f x :> l"
+ show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
+ proof (intro exI conjI)
+ let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
+ show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
+ show "isCont ?g x" using der
+ by (simp add: isCont_iff DERIV_iff diff_minus
+ cong: LIM_equal [rule_format])
+ show "?g x = l" by simp
+ qed
+next
+ assume "?rhs"
+ then obtain g where
+ "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
+ thus "(DERIV f x :> l)"
+ by (auto simp add: isCont_iff DERIV_iff diff_minus
+ cong: LIM_equal [rule_format])
+qed
+
+
+lemma CARAT_NSDERIV: "NSDERIV f x :> l ==>
+ \<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isNSCont g x & g x = l"
+by (auto simp add: NSDERIV_DERIV_iff isNSCont_isCont_iff CARAT_DERIV)
+
+lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
+by auto
+
+lemma CARAT_DERIVD:
+ assumes all: "\<forall>z. f z - f x = g z * (z-x)"
+ and nsc: "isNSCont g x"
+ shows "NSDERIV f x :> g x"
+proof -
+ from nsc
+ have "\<forall>w. w \<noteq> hypreal_of_real x \<and> w \<approx> hypreal_of_real x \<longrightarrow>
+ ( *f* g) w * (w - hypreal_of_real x) / (w - hypreal_of_real x) \<approx>
+ hypreal_of_real (g x)"
+ by (simp add: diff_minus isNSCont_def)
+ thus ?thesis using all
+ by (simp add: NSDERIV_iff2 starfun_if_eq cong: if_cong)
+qed
+
+(*--------------------------------------------------------------------------*)
+(* Lemmas about nested intervals and proof by bisection (cf.Harrison) *)
+(* All considerably tidied by lcp *)
+(*--------------------------------------------------------------------------*)
+
+lemma lemma_f_mono_add [rule_format (no_asm)]: "(\<forall>n. (f::nat=>real) n \<le> f (Suc n)) --> f m \<le> f(m + no)"
+apply (induct_tac "no")
+apply (auto intro: order_trans)
+done
+
+lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
+ \<forall>n. g(Suc n) \<le> g(n);
+ \<forall>n. f(n) \<le> g(n) |]
+ ==> Bseq f"
+apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
+apply (induct_tac "n")
+apply (auto intro: order_trans)
+apply (rule_tac y = "g (Suc na) " in order_trans)
+apply (induct_tac [2] "na")
+apply (auto intro: order_trans)
+done
+
+lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
+ \<forall>n. g(Suc n) \<le> g(n);
+ \<forall>n. f(n) \<le> g(n) |]
+ ==> Bseq g"
+apply (subst Bseq_minus_iff [symmetric])
+apply (rule_tac g = "%x. - (f x) " in f_inc_g_dec_Beq_f)
+apply auto
+done
+
+lemma f_inc_imp_le_lim: "[| \<forall>n. f n \<le> f (Suc n); convergent f |] ==> f n \<le> lim f"
+apply (rule linorder_not_less [THEN iffD1])
+apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
+apply (drule real_less_sum_gt_zero)
+apply (drule_tac x = "f n + - lim f" in spec, safe)
+apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
+apply (subgoal_tac "lim f \<le> f (no + n) ")
+apply (induct_tac [2] "no")
+apply (auto intro: order_trans simp add: diff_minus real_abs_def)
+apply (drule_tac no=no and m=n in lemma_f_mono_add)
+apply (auto simp add: add_commute)
+done
+
+lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
+apply (rule LIMSEQ_minus [THEN limI])
+apply (simp add: convergent_LIMSEQ_iff)
+done
+
+lemma g_dec_imp_lim_le: "[| \<forall>n. g(Suc n) \<le> g(n); convergent g |] ==> lim g \<le> g n"
+apply (subgoal_tac "- (g n) \<le> - (lim g) ")
+apply (cut_tac [2] f = "%x. - (g x) " in f_inc_imp_le_lim)
+apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
+done
+
+lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
+ \<forall>n. g(Suc n) \<le> g(n);
+ \<forall>n. f(n) \<le> g(n) |]
+ ==> \<exists>l m. l \<le> m & ((\<forall>n. f(n) \<le> l) & f ----> l) &
+ ((\<forall>n. m \<le> g(n)) & g ----> m)"
+apply (subgoal_tac "monoseq f & monoseq g")
+prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
+apply (subgoal_tac "Bseq f & Bseq g")
+prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
+apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
+apply (rule_tac x = "lim f" in exI)
+apply (rule_tac x = "lim g" in exI)
+apply (auto intro: LIMSEQ_le)
+apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
+done
+
+lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
+ \<forall>n. g(Suc n) \<le> g(n);
+ \<forall>n. f(n) \<le> g(n);
+ (%n. f(n) - g(n)) ----> 0 |]
+ ==> \<exists>l. ((\<forall>n. f(n) \<le> l) & f ----> l) &
+ ((\<forall>n. l \<le> g(n)) & g ----> l)"
+apply (drule lemma_nest, auto)
+apply (subgoal_tac "l = m")
+apply (drule_tac [2] X = f in LIMSEQ_diff)
+apply (auto intro: LIMSEQ_unique)
+done
+
+text{*The universal quantifiers below are required for the declaration
+ of @{text Bolzano_nest_unique} below.*}
+
+lemma Bolzano_bisect_le:
+ "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
+apply (rule allI)
+apply (induct_tac "n")
+apply (auto simp add: Let_def split_def)
+done
+
+lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
+ \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
+apply (rule allI)
+apply (induct_tac "n")
+apply (auto simp add: Bolzano_bisect_le Let_def split_def)
+done
+
+lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
+ \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
+apply (rule allI)
+apply (induct_tac "n")
+apply (auto simp add: Bolzano_bisect_le Let_def split_def)
+done
+
+lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
+apply auto
+apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
+apply auto
+done
+
+lemma Bolzano_bisect_diff:
+ "a \<le> b ==>
+ snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
+ (b-a) / (2 ^ n)"
+apply (induct_tac "n")
+apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
+apply (auto simp add: add_ac Bolzano_bisect_le diff_minus)
+done
+
+lemmas Bolzano_nest_unique =
+ lemma_nest_unique
+ [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
+
+
+lemma not_P_Bolzano_bisect:
+ assumes P: "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
+ and notP: "~ P(a,b)"
+ and le: "a \<le> b"
+ shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
+proof (induct n)
+ case 0 thus ?case by simp
+ next
+ case (Suc n)
+ thus ?case
+ by (auto simp del: surjective_pairing [symmetric]
+ simp add: Let_def split_def Bolzano_bisect_le [OF le]
+ P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
+qed
+
+(*Now we re-package P_prem as a formula*)
+lemma not_P_Bolzano_bisect':
+ "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
+ ~ P(a,b); a \<le> b |] ==>
+ \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
+by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
+
+
+
+lemma lemma_BOLZANO:
+ "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
+ \<forall>x. \<exists>d::real. 0 < d &
+ (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
+ a \<le> b |]
+ ==> P(a,b)"
+apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
+apply (rule LIMSEQ_minus_cancel)
+apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
+apply (rule ccontr)
+apply (drule not_P_Bolzano_bisect', assumption+)
+apply (rename_tac "l")
+apply (drule_tac x = l in spec, clarify)
+apply (simp add: LIMSEQ_def)
+apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
+apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
+apply (drule real_less_half_sum, auto)
+apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
+apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
+apply safe
+apply (simp_all (no_asm_simp))
+apply (rule_tac y = "abs (fst (Bolzano_bisect P a b (no + noa)) - l) + abs (snd (Bolzano_bisect P a b (no + noa)) - l) " in order_le_less_trans)
+apply (simp (no_asm_simp) add: abs_if)
+apply (rule real_sum_of_halves [THEN subst])
+apply (rule add_strict_mono)
+apply (simp_all add: diff_minus [symmetric])
+done
+
+
+lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
+ (\<forall>x. \<exists>d::real. 0 < d &
+ (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
+ --> (\<forall>a b. a \<le> b --> P(a,b))"
+apply clarify
+apply (blast intro: lemma_BOLZANO)
+done
+
+
+subsection{*Intermediate Value Theorem: Prove Contrapositive by Bisection*}
+
+lemma IVT: "[| f(a) \<le> y; y \<le> f(b);
+ a \<le> b;
+ (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
+ ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
+apply (rule contrapos_pp, assumption)
+apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> ~ (f (u) \<le> y & y \<le> f (v))" in lemma_BOLZANO2)
+apply safe
+apply simp_all
+apply (simp add: isCont_iff LIM_def)
+apply (rule ccontr)
+apply (subgoal_tac "a \<le> x & x \<le> b")
+ prefer 2
+ apply simp
+ apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
+apply (drule_tac x = x in spec)+
+apply simp
+apply (drule_tac P = "%r. ?P r --> (\<exists>s. 0<s & ?Q r s) " and x = "\<bar>y - f x\<bar> " in spec)
+apply safe
+apply simp
+apply (drule_tac x = s in spec, clarify)
+apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
+apply (drule_tac x = "ba-x" in spec)
+apply (simp_all add: abs_if)
+apply (drule_tac x = "aa-x" in spec)
+apply (case_tac "x \<le> aa", simp_all)
+apply (drule_tac x = x and y = aa in order_antisym)
+apply (assumption, simp)
+done
+
+lemma IVT2: "[| f(b) \<le> y; y \<le> f(a);
+ a \<le> b;
+ (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
+ |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
+apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
+apply (drule IVT [where f = "%x. - f x"], assumption)
+apply (auto intro: isCont_minus)
+done
+
+(*HOL style here: object-level formulations*)
+lemma IVT_objl: "(f(a) \<le> y & y \<le> f(b) & a \<le> b &
+ (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
+ --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
+apply (blast intro: IVT)
+done
+
+lemma IVT2_objl: "(f(b) \<le> y & y \<le> f(a) & a \<le> b &
+ (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
+ --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
+apply (blast intro: IVT2)
+done
+
+(*---------------------------------------------------------------------------*)
+(* By bisection, function continuous on closed interval is bounded above *)
+(*---------------------------------------------------------------------------*)
+
+
+lemma isCont_bounded:
+ "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
+ ==> \<exists>M. \<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M"
+apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> (\<exists>M. \<forall>x. u \<le> x & x \<le> v --> f x \<le> M) " in lemma_BOLZANO2)
+apply safe
+apply simp_all
+apply (rename_tac x xa ya M Ma)
+apply (cut_tac x = M and y = Ma in linorder_linear, safe)
+apply (rule_tac x = Ma in exI, clarify)
+apply (cut_tac x = xb and y = xa in linorder_linear, force)
+apply (rule_tac x = M in exI, clarify)
+apply (cut_tac x = xb and y = xa in linorder_linear, force)
+apply (case_tac "a \<le> x & x \<le> b")
+apply (rule_tac [2] x = 1 in exI)
+prefer 2 apply force
+apply (simp add: LIM_def isCont_iff)
+apply (drule_tac x = x in spec, auto)
+apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
+apply (drule_tac x = 1 in spec, auto)
+apply (rule_tac x = s in exI, clarify)
+apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
+apply (drule_tac x = "xa-x" in spec)
+apply (auto simp add: abs_ge_self, arith+)
+done
+
+(*----------------------------------------------------------------------------*)
+(* Refine the above to existence of least upper bound *)
+(*----------------------------------------------------------------------------*)
+
+lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
+ (\<exists>t. isLub UNIV S t)"
+apply (blast intro: reals_complete)
+done
+
+lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
+ ==> \<exists>M. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
+ (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
+apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x) " in lemma_reals_complete)
+apply auto
+apply (drule isCont_bounded, assumption)
+apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
+apply (rule exI, auto)
+apply (auto dest!: spec simp add: linorder_not_less)
+done
+
+(*----------------------------------------------------------------------------*)
+(* Now show that it attains its upper bound *)
+(*----------------------------------------------------------------------------*)
+
+lemma isCont_eq_Ub:
+ assumes le: "a \<le> b"
+ and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
+ shows "\<exists>M. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
+ (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
+proof -
+ from isCont_has_Ub [OF le con]
+ obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
+ and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x" by blast
+ show ?thesis
+ proof (intro exI, intro conjI)
+ show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
+ show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
+ proof (rule ccontr)
+ assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
+ with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
+ by (auto simp add: linorder_not_le [symmetric] intro: order_antisym)
+ hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
+ by (auto simp add: isCont_inverse isCont_diff con)
+ from isCont_bounded [OF le this]
+ obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
+ have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
+ by (simp add: M3)
+ have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
+ by (auto intro: order_le_less_trans [of _ k])
+ with Minv
+ have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
+ by (intro strip less_imp_inverse_less, simp_all)
+ hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
+ by simp
+ have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
+ by (simp, arith)
+ from M2 [OF this]
+ obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
+ thus False using invlt [of x] by force
+ qed
+ qed
+qed
+
+
+
+(*----------------------------------------------------------------------------*)
+(* Same theorem for lower bound *)
+(*----------------------------------------------------------------------------*)
+
+lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
+ ==> \<exists>M. (\<forall>x. a \<le> x & x \<le> b --> M \<le> f(x)) &
+ (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
+apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
+prefer 2 apply (blast intro: isCont_minus)
+apply (drule_tac f = " (%x. - (f x))" in isCont_eq_Ub)
+apply safe
+apply auto
+done
+
+
+(* ------------------------------------------------------------------------- *)
+(* Another version. *)
+(* ------------------------------------------------------------------------- *)
+
+lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
+ ==> \<exists>L M. (\<forall>x. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
+ (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
+apply (frule isCont_eq_Lb)
+apply (frule_tac [2] isCont_eq_Ub)
+apply (assumption+, safe)
+apply (rule_tac x = "f x" in exI)
+apply (rule_tac x = "f xa" in exI, simp, safe)
+apply (cut_tac x = x and y = xa in linorder_linear, safe)
+apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
+apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
+apply (rule_tac [2] x = xb in exI)
+apply (rule_tac [4] x = xb in exI, simp_all)
+done
+
+(*----------------------------------------------------------------------------*)
+(* If f'(x) > 0 then x is locally strictly increasing at the right *)
+(*----------------------------------------------------------------------------*)
+
+lemma DERIV_left_inc:
+ "[| DERIV f x :> l; 0 < l |]
+ ==> \<exists>d. 0 < d & (\<forall>h. 0 < h & h < d --> f(x) < f(x + h))"
+apply (simp add: deriv_def LIM_def)
+apply (drule spec, auto)
+apply (rule_tac x = s in exI, auto)
+apply (subgoal_tac "0 < l*h")
+ prefer 2 apply (simp add: zero_less_mult_iff)
+apply (drule_tac x = h in spec)
+apply (simp add: real_abs_def pos_le_divide_eq pos_less_divide_eq
+ split add: split_if_asm)
+done
+
+lemma DERIV_left_dec:
+ assumes der: "DERIV f x :> l"
+ and l: "l < 0"
+ shows "\<exists>d. 0 < d & (\<forall>h. 0 < h & h < d --> f(x) < f(x-h))"
+proof -
+ from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
+ have "\<exists>s. 0 < s \<and>
+ (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l)"
+ by (simp add: diff_minus)
+ then obtain s
+ where s: "0 < s"
+ and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
+ by auto
+ thus ?thesis
+ proof (intro exI conjI strip)
+ show "0<s" .
+ fix h::real
+ assume "0 < h \<and> h < s"
+ with all [of "-h"] show "f x < f (x-h)"
+ proof (simp add: real_abs_def pos_less_divide_eq diff_minus [symmetric]
+ split add: split_if_asm)
+ assume "~ l \<le> - ((f (x-h) - f x) / h)" and h: "0 < h"
+ with l
+ have "0 < (f (x-h) - f x) / h" by arith
+ thus "f x < f (x-h)"
+ by (simp add: pos_less_divide_eq h)
+ qed
+ qed
+qed
+
+lemma DERIV_local_max:
+ assumes der: "DERIV f x :> l"
+ and d: "0 < d"
+ and le: "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
+ shows "l = 0"
+proof (cases rule: linorder_cases [of l 0])
+ case equal show ?thesis .
+next
+ case less
+ from DERIV_left_dec [OF der less]
+ obtain d' where d': "0 < d'"
+ and lt: "\<forall>h. 0 < h \<and> h < d' \<longrightarrow> f x < f (x-h)" by blast
+ from real_lbound_gt_zero [OF d d']
+ obtain e where "0 < e \<and> e < d \<and> e < d'" ..
+ with lt le [THEN spec [where x="x-e"]]
+ show ?thesis by (auto simp add: abs_if)
+next
+ case greater
+ from DERIV_left_inc [OF der greater]
+ obtain d' where d': "0 < d'"
+ and lt: "\<forall>h. 0 < h \<and> h < d' \<longrightarrow> f x < f (x + h)" by blast
+ from real_lbound_gt_zero [OF d d']
+ obtain e where "0 < e \<and> e < d \<and> e < d'" ..
+ with lt le [THEN spec [where x="x+e"]]
+ show ?thesis by (auto simp add: abs_if)
+qed
+
+
+text{*Similar theorem for a local minimum*}
+lemma DERIV_local_min:
+ "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
+by (drule DERIV_minus [THEN DERIV_local_max], auto)
+
+
+text{*In particular, if a function is locally flat*}
+lemma DERIV_local_const:
+ "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
+by (auto dest!: DERIV_local_max)
+
+text{*Lemma about introducing open ball in open interval*}
+lemma lemma_interval_lt:
+ "[| a < x; x < b |]
+ ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
+apply (simp add: abs_interval_iff)
+apply (insert linorder_linear [of "x-a" "b-x"], safe)
+apply (rule_tac x = "x-a" in exI)
+apply (rule_tac [2] x = "b-x" in exI, auto)
+done
+
+lemma lemma_interval: "[| a < x; x < b |] ==>
+ \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
+apply (drule lemma_interval_lt, auto)
+apply (auto intro!: exI)
+done
+
+text{*Rolle's Theorem.
+ If @{term f} is defined and continuous on the closed interval
+ @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
+ and @{term "f(a) = f(b)"},
+ then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
+theorem Rolle:
+ assumes lt: "a < b"
+ and eq: "f(a) = f(b)"
+ and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
+ and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
+ shows "\<exists>z. a < z & z < b & DERIV f z :> 0"
+proof -
+ have le: "a \<le> b" using lt by simp
+ from isCont_eq_Ub [OF le con]
+ obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
+ and alex: "a \<le> x" and xleb: "x \<le> b"
+ by blast
+ from isCont_eq_Lb [OF le con]
+ obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
+ and alex': "a \<le> x'" and x'leb: "x' \<le> b"
+ by blast
+ show ?thesis
+ proof cases
+ assume axb: "a < x & x < b"
+ --{*@{term f} attains its maximum within the interval*}
+ hence ax: "a<x" and xb: "x<b" by auto
+ from lemma_interval [OF ax xb]
+ obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
+ by blast
+ hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
+ by blast
+ from differentiableD [OF dif [OF axb]]
+ obtain l where der: "DERIV f x :> l" ..
+ have "l=0" by (rule DERIV_local_max [OF der d bound'])
+ --{*the derivative at a local maximum is zero*}
+ thus ?thesis using ax xb der by auto
+ next
+ assume notaxb: "~ (a < x & x < b)"
+ hence xeqab: "x=a | x=b" using alex xleb by arith
+ hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
+ show ?thesis
+ proof cases
+ assume ax'b: "a < x' & x' < b"
+ --{*@{term f} attains its minimum within the interval*}
+ hence ax': "a<x'" and x'b: "x'<b" by auto
+ from lemma_interval [OF ax' x'b]
+ obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
+ by blast
+ hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
+ by blast
+ from differentiableD [OF dif [OF ax'b]]
+ obtain l where der: "DERIV f x' :> l" ..
+ have "l=0" by (rule DERIV_local_min [OF der d bound'])
+ --{*the derivative at a local minimum is zero*}
+ thus ?thesis using ax' x'b der by auto
+ next
+ assume notax'b: "~ (a < x' & x' < b)"
+ --{*@{term f} is constant througout the interval*}
+ hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
+ hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
+ from dense [OF lt]
+ obtain r where ar: "a < r" and rb: "r < b" by blast
+ from lemma_interval [OF ar rb]
+ obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
+ by blast
+ have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
+ proof (clarify)
+ fix z::real
+ assume az: "a \<le> z" and zb: "z \<le> b"
+ show "f z = f b"
+ proof (rule order_antisym)
+ show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
+ show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
+ qed
+ qed
+ have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
+ proof (intro strip)
+ fix y::real
+ assume lt: "\<bar>r-y\<bar> < d"
+ hence "f y = f b" by (simp add: eq_fb bound)
+ thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
+ qed
+ from differentiableD [OF dif [OF conjI [OF ar rb]]]
+ obtain l where der: "DERIV f r :> l" ..
+ have "l=0" by (rule DERIV_local_const [OF der d bound'])
+ --{*the derivative of a constant function is zero*}
+ thus ?thesis using ar rb der by auto
+ qed
+ qed
+qed
+
+
+subsection{*Mean Value Theorem*}
+
+lemma lemma_MVT:
+ "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
+proof cases
+ assume "a=b" thus ?thesis by simp
+next
+ assume "a\<noteq>b"
+ hence ba: "b-a \<noteq> 0" by arith
+ show ?thesis
+ by (rule real_mult_left_cancel [OF ba, THEN iffD1],
+ simp add: right_diff_distrib, simp add: left_diff_distrib)
+qed
+
+theorem MVT:
+ assumes lt: "a < b"
+ and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
+ and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
+ shows "\<exists>l z. a < z & z < b & DERIV f z :> l &
+ (f(b) - f(a) = (b-a) * l)"
+proof -
+ let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
+ have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
+ by (fast intro: isCont_diff isCont_const isCont_mult isCont_Id)
+ have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
+ proof (clarify)
+ fix x::real
+ assume ax: "a < x" and xb: "x < b"
+ from differentiableD [OF dif [OF conjI [OF ax xb]]]
+ obtain l where der: "DERIV f x :> l" ..
+ show "?F differentiable x"
+ by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
+ blast intro: DERIV_diff DERIV_cmult_Id der)
+ qed
+ from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
+ obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
+ by blast
+ have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
+ by (rule DERIV_cmult_Id)
+ hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
+ :> 0 + (f b - f a) / (b - a)"
+ by (rule DERIV_add [OF der])
+ show ?thesis
+ proof (intro exI conjI)
+ show "a < z" .
+ show "z < b" .
+ show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by simp
+ show "DERIV f z :> ((f b - f a)/(b-a))" using derF by simp
+ qed
+qed
+
+
+text{*A function is constant if its derivative is 0 over an interval.*}
+
+lemma DERIV_isconst_end: "[| a < b;
+ \<forall>x. a \<le> x & x \<le> b --> isCont f x;
+ \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
+ ==> (f b = f a)"
+apply (drule MVT, assumption)
+apply (blast intro: differentiableI)
+apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
+done
+
+lemma DERIV_isconst1: "[| a < b;
+ \<forall>x. a \<le> x & x \<le> b --> isCont f x;
+ \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
+ ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
+apply safe
+apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
+apply (drule_tac b = x in DERIV_isconst_end, auto)
+done
+
+lemma DERIV_isconst2: "[| a < b;
+ \<forall>x. a \<le> x & x \<le> b --> isCont f x;
+ \<forall>x. a < x & x < b --> DERIV f x :> 0;
+ a \<le> x; x \<le> b |]
+ ==> f x = f a"
+apply (blast dest: DERIV_isconst1)
+done
+
+lemma DERIV_isconst_all: "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
+apply (rule linorder_cases [of x y])
+apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
+done
+
+lemma DERIV_const_ratio_const:
+ "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
+apply (rule linorder_cases [of a b], auto)
+apply (drule_tac [!] f = f in MVT)
+apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
+apply (auto dest: DERIV_unique simp add: left_distrib diff_minus)
+done
+
+lemma DERIV_const_ratio_const2:
+ "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
+apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
+apply (auto dest!: DERIV_const_ratio_const simp add: real_mult_assoc)
+done
+
+lemma real_average_minus_first: "((a + b) /2 - a) = (b-a)/(2::real)"
+by auto
+declare real_average_minus_first [simp]
+
+lemma real_average_minus_second: "((b + a)/2 - a) = (b-a)/(2::real)"
+by auto
+declare real_average_minus_second [simp]
+
+text{*Gallileo's "trick": average velocity = av. of end velocities*}
+
+lemma DERIV_const_average:
+ assumes neq: "a \<noteq> (b::real)"
+ and der: "\<forall>x. DERIV v x :> k"
+ shows "v ((a + b)/2) = (v a + v b)/2"
+proof (cases rule: linorder_cases [of a b])
+ case equal with neq show ?thesis by simp
+next
+ case less
+ have "(v b - v a) / (b - a) = k"
+ by (rule DERIV_const_ratio_const2 [OF neq der])
+ hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
+ moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
+ by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
+ ultimately show ?thesis using neq by force
+next
+ case greater
+ have "(v b - v a) / (b - a) = k"
+ by (rule DERIV_const_ratio_const2 [OF neq der])
+ hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
+ moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
+ by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
+ ultimately show ?thesis using neq by (force simp add: add_commute)
+qed
+
+
+text{*Dull lemma: an continuous injection on an interval must have a
+strict maximum at an end point, not in the middle.*}
+
+lemma lemma_isCont_inj:
+ assumes d: "0 < d"
+ and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
+ and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
+ shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
+proof (rule ccontr)
+ assume "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
+ hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
+ show False
+ proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
+ case le
+ from d cont all [of "x+d"]
+ have flef: "f(x+d) \<le> f x"
+ and xlex: "x - d \<le> x"
+ and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
+ by (auto simp add: abs_if)
+ from IVT [OF le flef xlex cont']
+ obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
+ moreover
+ hence "g(f x') = g (f(x+d))" by simp
+ ultimately show False using d inj [of x'] inj [of "x+d"]
+ by (simp add: abs_le_interval_iff)
+ next
+ case ge
+ from d cont all [of "x-d"]
+ have flef: "f(x-d) \<le> f x"
+ and xlex: "x \<le> x+d"
+ and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
+ by (auto simp add: abs_if)
+ from IVT2 [OF ge flef xlex cont']
+ obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
+ moreover
+ hence "g(f x') = g (f(x-d))" by simp
+ ultimately show False using d inj [of x'] inj [of "x-d"]
+ by (simp add: abs_le_interval_iff)
+ qed
+qed
+
+
+text{*Similar version for lower bound.*}
+
+lemma lemma_isCont_inj2:
+ "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
+ \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
+ ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
+apply (insert lemma_isCont_inj
+ [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
+apply (simp add: isCont_minus linorder_not_le)
+done
+
+text{*Show there's an interval surrounding @{term "f(x)"} in
+@{text "f[[x - d, x + d]]"} .*}
+
+lemma isCont_inj_range:
+ assumes d: "0 < d"
+ and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
+ and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
+ shows "\<exists>e. 0<e & (\<forall>y. \<bar>y - f x\<bar> \<le> e --> (\<exists>z. \<bar>z-x\<bar> \<le> d & f z = y))"
+proof -
+ have "x-d \<le> x+d" "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d
+ by (auto simp add: abs_le_interval_iff)
+ from isCont_Lb_Ub [OF this]
+ obtain L M
+ where all1 [rule_format]: "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> L \<le> f z \<and> f z \<le> M"
+ and all2 [rule_format]:
+ "\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>z. x-d \<le> z \<and> z \<le> x+d \<and> f z = y)"
+ by auto
+ with d have "L \<le> f x & f x \<le> M" by simp
+ moreover have "L \<noteq> f x"
+ proof -
+ from lemma_isCont_inj2 [OF d inj cont]
+ obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x" by auto
+ thus ?thesis using all1 [of u] by arith
+ qed
+ moreover have "f x \<noteq> M"
+ proof -
+ from lemma_isCont_inj [OF d inj cont]
+ obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u" by auto
+ thus ?thesis using all1 [of u] by arith
+ qed
+ ultimately have "L < f x & f x < M" by arith
+ hence "0 < f x - L" "0 < M - f x" by arith+
+ from real_lbound_gt_zero [OF this]
+ obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
+ thus ?thesis
+ proof (intro exI conjI)
+ show "0<e" .
+ show "\<forall>y. \<bar>y - f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y)"
+ proof (intro strip)
+ fix y::real
+ assume "\<bar>y - f x\<bar> \<le> e"
+ with e have "L \<le> y \<and> y \<le> M" by arith
+ from all2 [OF this]
+ obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
+ thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y"
+ by (force simp add: abs_le_interval_iff)
+ qed
+ qed
+qed
+
+
+text{*Continuity of inverse function*}
+
+lemma isCont_inverse_function:
+ assumes d: "0 < d"
+ and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
+ and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
+ shows "isCont g (f x)"
+proof (simp add: isCont_iff LIM_eq)
+ show "\<forall>r. 0 < r \<longrightarrow>
+ (\<exists>s. 0<s \<and> (\<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r))"
+ proof (intro strip)
+ fix r::real
+ assume r: "0<r"
+ from real_lbound_gt_zero [OF r d]
+ obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
+ with inj cont
+ have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
+ "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z" by auto
+ from isCont_inj_range [OF e this]
+ obtain e' where e': "0 < e'"
+ and all: "\<forall>y. \<bar>y - f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)"
+ by blast
+ show "\<exists>s. 0<s \<and> (\<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r)"
+ proof (intro exI conjI)
+ show "0<e'" .
+ show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
+ proof (intro strip)
+ fix z::real
+ assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
+ with e e_lt e_simps all [rule_format, of "f x + z"]
+ show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
+ qed
+ qed
+ qed
+qed
+
+ML
+{*
+val LIM_def = thm"LIM_def";
+val NSLIM_def = thm"NSLIM_def";
+val isCont_def = thm"isCont_def";
+val isNSCont_def = thm"isNSCont_def";
+val deriv_def = thm"deriv_def";
+val nsderiv_def = thm"nsderiv_def";
+val differentiable_def = thm"differentiable_def";
+val NSdifferentiable_def = thm"NSdifferentiable_def";
+val increment_def = thm"increment_def";
+val isUCont_def = thm"isUCont_def";
+val isNSUCont_def = thm"isNSUCont_def";
+
+val half_gt_zero_iff = thm "half_gt_zero_iff";
+val half_gt_zero = thms "half_gt_zero";
+val abs_diff_triangle_ineq = thm "abs_diff_triangle_ineq";
+val LIM_eq = thm "LIM_eq";
+val LIM_D = thm "LIM_D";
+val LIM_const = thm "LIM_const";
+val LIM_add = thm "LIM_add";
+val LIM_minus = thm "LIM_minus";
+val LIM_add_minus = thm "LIM_add_minus";
+val LIM_diff = thm "LIM_diff";
+val LIM_const_not_eq = thm "LIM_const_not_eq";
+val LIM_const_eq = thm "LIM_const_eq";
+val LIM_unique = thm "LIM_unique";
+val LIM_mult_zero = thm "LIM_mult_zero";
+val LIM_self = thm "LIM_self";
+val LIM_equal = thm "LIM_equal";
+val LIM_trans = thm "LIM_trans";
+val LIM_NSLIM = thm "LIM_NSLIM";
+val NSLIM_LIM = thm "NSLIM_LIM";
+val LIM_NSLIM_iff = thm "LIM_NSLIM_iff";
+val NSLIM_mult = thm "NSLIM_mult";
+val LIM_mult2 = thm "LIM_mult2";
+val NSLIM_add = thm "NSLIM_add";
+val LIM_add2 = thm "LIM_add2";
+val NSLIM_const = thm "NSLIM_const";
+val LIM_const2 = thm "LIM_const2";
+val NSLIM_minus = thm "NSLIM_minus";
+val LIM_minus2 = thm "LIM_minus2";
+val NSLIM_add_minus = thm "NSLIM_add_minus";
+val LIM_add_minus2 = thm "LIM_add_minus2";
+val NSLIM_inverse = thm "NSLIM_inverse";
+val LIM_inverse = thm "LIM_inverse";
+val NSLIM_zero = thm "NSLIM_zero";
+val LIM_zero2 = thm "LIM_zero2";
+val NSLIM_zero_cancel = thm "NSLIM_zero_cancel";
+val LIM_zero_cancel = thm "LIM_zero_cancel";
+val NSLIM_not_zero = thm "NSLIM_not_zero";
+val NSLIM_const_not_eq = thm "NSLIM_const_not_eq";
+val NSLIM_const_eq = thm "NSLIM_const_eq";
+val NSLIM_unique = thm "NSLIM_unique";
+val LIM_unique2 = thm "LIM_unique2";
+val NSLIM_mult_zero = thm "NSLIM_mult_zero";
+val LIM_mult_zero2 = thm "LIM_mult_zero2";
+val NSLIM_self = thm "NSLIM_self";
+val isNSContD = thm "isNSContD";
+val isNSCont_NSLIM = thm "isNSCont_NSLIM";
+val NSLIM_isNSCont = thm "NSLIM_isNSCont";
+val isNSCont_NSLIM_iff = thm "isNSCont_NSLIM_iff";
+val isNSCont_LIM_iff = thm "isNSCont_LIM_iff";
+val isNSCont_isCont_iff = thm "isNSCont_isCont_iff";
+val isCont_isNSCont = thm "isCont_isNSCont";
+val isNSCont_isCont = thm "isNSCont_isCont";
+val NSLIM_h_iff = thm "NSLIM_h_iff";
+val NSLIM_isCont_iff = thm "NSLIM_isCont_iff";
+val LIM_isCont_iff = thm "LIM_isCont_iff";
+val isCont_iff = thm "isCont_iff";
+val isCont_add = thm "isCont_add";
+val isCont_mult = thm "isCont_mult";
+val isCont_o = thm "isCont_o";
+val isCont_o2 = thm "isCont_o2";
+val isNSCont_minus = thm "isNSCont_minus";
+val isCont_minus = thm "isCont_minus";
+val isCont_inverse = thm "isCont_inverse";
+val isNSCont_inverse = thm "isNSCont_inverse";
+val isCont_diff = thm "isCont_diff";
+val isCont_const = thm "isCont_const";
+val isNSCont_const = thm "isNSCont_const";
+val isNSCont_rabs = thm "isNSCont_rabs";
+val isCont_rabs = thm "isCont_rabs";
+val isNSUContD = thm "isNSUContD";
+val isUCont_isCont = thm "isUCont_isCont";
+val isUCont_isNSUCont = thm "isUCont_isNSUCont";
+val isNSUCont_isUCont = thm "isNSUCont_isUCont";
+val DERIV_iff = thm "DERIV_iff";
+val DERIV_NS_iff = thm "DERIV_NS_iff";
+val DERIV_D = thm "DERIV_D";
+val NS_DERIV_D = thm "NS_DERIV_D";
+val DERIV_unique = thm "DERIV_unique";
+val NSDeriv_unique = thm "NSDeriv_unique";
+val differentiableD = thm "differentiableD";
+val differentiableI = thm "differentiableI";
+val NSdifferentiableD = thm "NSdifferentiableD";
+val NSdifferentiableI = thm "NSdifferentiableI";
+val LIM_I = thm "LIM_I";
+val DERIV_LIM_iff = thm "DERIV_LIM_iff";
+val DERIV_iff2 = thm "DERIV_iff2";
+val NSDERIV_NSLIM_iff = thm "NSDERIV_NSLIM_iff";
+val NSDERIV_NSLIM_iff2 = thm "NSDERIV_NSLIM_iff2";
+val NSDERIV_iff2 = thm "NSDERIV_iff2";
+val hypreal_not_eq_minus_iff = thm "hypreal_not_eq_minus_iff";
+val NSDERIVD5 = thm "NSDERIVD5";
+val NSDERIVD4 = thm "NSDERIVD4";
+val NSDERIVD3 = thm "NSDERIVD3";
+val NSDERIV_DERIV_iff = thm "NSDERIV_DERIV_iff";
+val NSDERIV_isNSCont = thm "NSDERIV_isNSCont";
+val DERIV_isCont = thm "DERIV_isCont";
+val NSDERIV_const = thm "NSDERIV_const";
+val DERIV_const = thm "DERIV_const";
+val NSDERIV_add = thm "NSDERIV_add";
+val DERIV_add = thm "DERIV_add";
+val NSDERIV_mult = thm "NSDERIV_mult";
+val DERIV_mult = thm "DERIV_mult";
+val NSDERIV_cmult = thm "NSDERIV_cmult";
+val DERIV_cmult = thm "DERIV_cmult";
+val NSDERIV_minus = thm "NSDERIV_minus";
+val DERIV_minus = thm "DERIV_minus";
+val NSDERIV_add_minus = thm "NSDERIV_add_minus";
+val DERIV_add_minus = thm "DERIV_add_minus";
+val NSDERIV_diff = thm "NSDERIV_diff";
+val DERIV_diff = thm "DERIV_diff";
+val incrementI = thm "incrementI";
+val incrementI2 = thm "incrementI2";
+val increment_thm = thm "increment_thm";
+val increment_thm2 = thm "increment_thm2";
+val increment_approx_zero = thm "increment_approx_zero";
+val NSDERIV_zero = thm "NSDERIV_zero";
+val NSDERIV_approx = thm "NSDERIV_approx";
+val NSDERIVD1 = thm "NSDERIVD1";
+val NSDERIVD2 = thm "NSDERIVD2";
+val NSDERIV_chain = thm "NSDERIV_chain";
+val DERIV_chain = thm "DERIV_chain";
+val DERIV_chain2 = thm "DERIV_chain2";
+val NSDERIV_Id = thm "NSDERIV_Id";
+val DERIV_Id = thm "DERIV_Id";
+val isCont_Id = thms "isCont_Id";
+val DERIV_cmult_Id = thm "DERIV_cmult_Id";
+val NSDERIV_cmult_Id = thm "NSDERIV_cmult_Id";
+val DERIV_pow = thm "DERIV_pow";
+val NSDERIV_pow = thm "NSDERIV_pow";
+val NSDERIV_inverse = thm "NSDERIV_inverse";
+val DERIV_inverse = thm "DERIV_inverse";
+val DERIV_inverse_fun = thm "DERIV_inverse_fun";
+val NSDERIV_inverse_fun = thm "NSDERIV_inverse_fun";
+val DERIV_quotient = thm "DERIV_quotient";
+val NSDERIV_quotient = thm "NSDERIV_quotient";
+val CARAT_DERIV = thm "CARAT_DERIV";
+val CARAT_NSDERIV = thm "CARAT_NSDERIV";
+val hypreal_eq_minus_iff3 = thm "hypreal_eq_minus_iff3";
+val starfun_if_eq = thm "starfun_if_eq";
+val CARAT_DERIVD = thm "CARAT_DERIVD";
+val f_inc_g_dec_Beq_f = thm "f_inc_g_dec_Beq_f";
+val f_inc_g_dec_Beq_g = thm "f_inc_g_dec_Beq_g";
+val f_inc_imp_le_lim = thm "f_inc_imp_le_lim";
+val lim_uminus = thm "lim_uminus";
+val g_dec_imp_lim_le = thm "g_dec_imp_lim_le";
+val Bolzano_bisect_le = thm "Bolzano_bisect_le";
+val Bolzano_bisect_fst_le_Suc = thm "Bolzano_bisect_fst_le_Suc";
+val Bolzano_bisect_Suc_le_snd = thm "Bolzano_bisect_Suc_le_snd";
+val eq_divide_2_times_iff = thm "eq_divide_2_times_iff";
+val Bolzano_bisect_diff = thm "Bolzano_bisect_diff";
+val Bolzano_nest_unique = thms "Bolzano_nest_unique";
+val not_P_Bolzano_bisect = thm "not_P_Bolzano_bisect";
+val not_P_Bolzano_bisect = thm "not_P_Bolzano_bisect";
+val lemma_BOLZANO2 = thm "lemma_BOLZANO2";
+val IVT = thm "IVT";
+val IVT2 = thm "IVT2";
+val IVT_objl = thm "IVT_objl";
+val IVT2_objl = thm "IVT2_objl";
+val isCont_bounded = thm "isCont_bounded";
+val isCont_has_Ub = thm "isCont_has_Ub";
+val isCont_eq_Ub = thm "isCont_eq_Ub";
+val isCont_eq_Lb = thm "isCont_eq_Lb";
+val isCont_Lb_Ub = thm "isCont_Lb_Ub";
+val DERIV_left_inc = thm "DERIV_left_inc";
+val DERIV_left_dec = thm "DERIV_left_dec";
+val DERIV_local_max = thm "DERIV_local_max";
+val DERIV_local_min = thm "DERIV_local_min";
+val DERIV_local_const = thm "DERIV_local_const";
+val Rolle = thm "Rolle";
+val MVT = thm "MVT";
+val DERIV_isconst_end = thm "DERIV_isconst_end";
+val DERIV_isconst1 = thm "DERIV_isconst1";
+val DERIV_isconst2 = thm "DERIV_isconst2";
+val DERIV_isconst_all = thm "DERIV_isconst_all";
+val DERIV_const_ratio_const = thm "DERIV_const_ratio_const";
+val DERIV_const_ratio_const2 = thm "DERIV_const_ratio_const2";
+val real_average_minus_first = thm "real_average_minus_first";
+val real_average_minus_second = thm "real_average_minus_second";
+val DERIV_const_average = thm "DERIV_const_average";
+val isCont_inj_range = thm "isCont_inj_range";
+val isCont_inverse_function = thm "isCont_inverse_function";
+*}
+
end