author | paulson |
Fri, 15 Dec 2000 17:41:38 +0100 | |
changeset 10677 | 36625483213f |
parent 10670 | 4b0e346c8ca3 |
child 10690 | cd80241125b0 |
permissions | -rw-r--r-- |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1 |
(* Title : Lim.ML |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
2 |
Author : Jacques D. Fleuriot |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
3 |
Copyright : 1998 University of Cambridge |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
4 |
Description : Theory of limits, continuity and |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
5 |
differentiation of real=>real functions |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
6 |
*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
7 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
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|
8 |
|
10677 | 9 |
(*DELETE?????*) |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
10 |
fun ARITH_PROVE str = prove_goal thy str |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
11 |
(fn prems => [cut_facts_tac prems 1,arith_tac 1]); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
12 |
|
10648 | 13 |
Goal "(x + - a = (#0::real)) = (x=a)"; |
14 |
by (arith_tac 1); |
|
15 |
qed "real_add_minus_iff"; |
|
16 |
Addsimps [real_add_minus_iff]; |
|
17 |
||
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
18 |
(*--------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
19 |
Theory of limits, continuity and differentiation of |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
20 |
real=>real functions |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
21 |
----------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
22 |
Goalw [LIM_def] "(f -- a --> L) = (ALL r. #0 < r --> \ |
10648 | 23 |
\ (EX s. #0 < s & (ALL x. (x ~= a & (abs(x + -a) < s) \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
24 |
\ --> abs(f x + -L) < r))))"; |
10648 | 25 |
by Auto_tac; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
26 |
qed "LIM_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
27 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
28 |
Goalw [LIM_def] |
10677 | 29 |
"[| f -- a --> L; #0 < r |] \ |
10648 | 30 |
\ ==> (EX s. #0 < s & (ALL x. (x ~= a \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
31 |
\ & (abs(x + -a) < s) --> abs(f x + -L) < r)))"; |
10648 | 32 |
by Auto_tac; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
33 |
qed "LIMD"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
34 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
35 |
Goalw [LIM_def] "(%x. k) -- x --> k"; |
10648 | 36 |
by Auto_tac; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
37 |
qed "LIM_const"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
38 |
Addsimps [LIM_const]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
39 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
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|
40 |
(***-----------------------------------------------------------***) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
41 |
(*** Some Purely Standard Proofs - Can be used for comparison ***) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
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|
42 |
(***-----------------------------------------------------------***) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
43 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
44 |
(*--------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
45 |
LIM_add |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
46 |
---------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
47 |
Goalw [LIM_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
48 |
"[| f -- x --> l; g -- x --> m |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
49 |
\ ==> (%x. f(x) + g(x)) -- x --> (l + m)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
50 |
by (Step_tac 1); |
10677 | 51 |
by (REPEAT(dres_inst_tac [("x","r/#2")] spec 1)); |
10607 | 52 |
by (dtac (rename_numerals (real_zero_less_two RS real_inverse_gt_zero |
10648 | 53 |
RSN (2,real_mult_less_mono2))) 1); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
54 |
by (Asm_full_simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
55 |
by (Clarify_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
56 |
by (res_inst_tac [("R1.0","s"),("R2.0","sa")] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
57 |
real_linear_less2 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
58 |
by (res_inst_tac [("x","s")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
59 |
by (res_inst_tac [("x","sa")] exI 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
60 |
by (res_inst_tac [("x","sa")] exI 3); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
61 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
62 |
by (REPEAT(dres_inst_tac [("x","xa")] spec 1) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
63 |
THEN step_tac (claset() addSEs [real_less_trans]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
64 |
by (REPEAT(dres_inst_tac [("x","xa")] spec 2) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
65 |
THEN step_tac (claset() addSEs [real_less_trans]) 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
66 |
by (REPEAT(dres_inst_tac [("x","xa")] spec 3) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
67 |
THEN step_tac (claset() addSEs [real_less_trans]) 3); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
68 |
by (ALLGOALS(rtac (abs_sum_triangle_ineq RS real_le_less_trans))); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
69 |
by (ALLGOALS(rtac (real_sum_of_halves RS subst))); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
70 |
by (auto_tac (claset() addIs [real_add_less_mono],simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
71 |
qed "LIM_add"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
72 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
73 |
Goalw [LIM_def] "f -- a --> L ==> (%x. -f(x)) -- a --> -L"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
74 |
by (full_simp_tac (simpset() addsimps [real_minus_add_distrib RS sym, |
10648 | 75 |
abs_minus_cancel] |
76 |
delsimps [real_minus_add_distrib,real_minus_minus]) 1); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
77 |
qed "LIM_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
78 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
79 |
(*---------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
80 |
LIM_add_minus |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
81 |
----------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
82 |
Goal "[| f -- x --> l; g -- x --> m |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
83 |
\ ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
84 |
by (blast_tac (claset() addDs [LIM_add,LIM_minus]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
85 |
qed "LIM_add_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
86 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
87 |
(*---------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
88 |
LIM_zero |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
89 |
----------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
90 |
Goal "f -- a --> l ==> (%x. f(x) + -l) -- a --> #0"; |
10648 | 91 |
by (res_inst_tac [("z1","l")] (rename_numerals (real_add_minus RS subst)) 1); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
92 |
by (rtac LIM_add_minus 1 THEN Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
93 |
qed "LIM_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
94 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
95 |
(*-------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
96 |
Limit not zero |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
97 |
--------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
98 |
Goalw [LIM_def] "k ~= #0 ==> ~ ((%x. k) -- x --> #0)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
99 |
by (res_inst_tac [("R1.0","k"),("R2.0","#0")] real_linear_less2 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
100 |
by (auto_tac (claset(),simpset() addsimps [abs_minus_eqI2,abs_eqI2])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
101 |
by (dtac (rename_numerals (real_minus_zero_less_iff RS iffD2)) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
102 |
by (res_inst_tac [("x","-k")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
103 |
by (res_inst_tac [("x","k")] exI 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
104 |
by Auto_tac; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
105 |
by (ALLGOALS(dres_inst_tac [("y","s")] real_dense)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
106 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
107 |
by (ALLGOALS(res_inst_tac [("x","r + x")] exI)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
108 |
by (auto_tac (claset(),simpset() addsimps [real_add_assoc,abs_eqI2])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
109 |
qed "LIM_not_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
110 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
111 |
(* [| k ~= #0; (%x. k) -- x --> #0 |] ==> R *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
112 |
bind_thm("LIM_not_zeroE", (LIM_not_zero RS notE)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
113 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
114 |
Goal "(%x. k) -- x --> L ==> k = L"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
115 |
by (rtac ccontr 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
116 |
by (dtac LIM_zero 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
117 |
by (rtac LIM_not_zeroE 1 THEN assume_tac 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
118 |
by (arith_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
119 |
qed "LIM_const_eq"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
120 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
121 |
(*------------------------ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
122 |
Limit is Unique |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
123 |
------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
124 |
Goal "[| f -- x --> L; f -- x --> M |] ==> L = M"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
125 |
by (dtac LIM_minus 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
126 |
by (dtac LIM_add 1 THEN assume_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
127 |
by (auto_tac (claset() addSDs [LIM_const_eq RS sym], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
128 |
simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
129 |
qed "LIM_unique"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
130 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
131 |
(*------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
132 |
LIM_mult_zero |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
133 |
-------------*) |
10677 | 134 |
Goalw [LIM_def] "[| f -- x --> #0; g -- x --> #0 |] \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
135 |
\ ==> (%x. f(x)*g(x)) -- x --> #0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
136 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
137 |
by (dres_inst_tac [("x","#1")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
138 |
by (dres_inst_tac [("x","r")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
139 |
by (cut_facts_tac [real_zero_less_one] 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
140 |
by (asm_full_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
141 |
[abs_mult]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
142 |
by (Clarify_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
143 |
by (res_inst_tac [("R1.0","s"),("R2.0","sa")] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
144 |
real_linear_less2 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
145 |
by (res_inst_tac [("x","s")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
146 |
by (res_inst_tac [("x","sa")] exI 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
147 |
by (res_inst_tac [("x","sa")] exI 3); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
148 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
149 |
by (REPEAT(dres_inst_tac [("x","xa")] spec 1) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
150 |
THEN step_tac (claset() addSEs [real_less_trans]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
151 |
by (REPEAT(dres_inst_tac [("x","xa")] spec 2) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
152 |
THEN step_tac (claset() addSEs [real_less_trans]) 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
153 |
by (REPEAT(dres_inst_tac [("x","xa")] spec 3) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
154 |
THEN step_tac (claset() addSEs [real_less_trans]) 3); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
155 |
by (ALLGOALS(res_inst_tac [("t","r")] (real_mult_1 RS subst))); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
156 |
by (ALLGOALS(rtac abs_mult_less2)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
157 |
by Auto_tac; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
158 |
qed "LIM_mult_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
159 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
160 |
Goalw [LIM_def] "(%x. x) -- a --> a"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
161 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
162 |
qed "LIM_self"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
163 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
164 |
(*-------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
165 |
Limits are equal for functions equal except at limit point |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
166 |
--------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
167 |
Goalw [LIM_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
168 |
"[| ALL x. x ~= a --> (f x = g x) |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
169 |
\ ==> (f -- a --> l) = (g -- a --> l)"; |
10648 | 170 |
by (auto_tac (claset(), simpset() addsimps [real_add_minus_iff])); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
171 |
qed "LIM_equal"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
172 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
173 |
Goal "[| (%x. f(x) + -g(x)) -- a --> #0; \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
174 |
\ g -- a --> l |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
175 |
\ ==> f -- a --> l"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
176 |
by (dtac LIM_add 1 THEN assume_tac 1); |
10648 | 177 |
by (auto_tac (claset(), simpset() addsimps [real_add_assoc])); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
178 |
qed "LIM_trans"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
179 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
180 |
(***-------------------------------------------------------------***) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
181 |
(*** End of Purely Standard Proofs ***) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
182 |
(***-------------------------------------------------------------***) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
183 |
(*-------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
184 |
Standard and NS definitions of Limit |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
185 |
--------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
186 |
Goalw [LIM_def,NSLIM_def,inf_close_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
187 |
"f -- x --> L ==> f -- x --NS> L"; |
10648 | 188 |
by (asm_full_simp_tac |
189 |
(simpset() addsimps [Infinitesimal_FreeUltrafilterNat_iff]) 1); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
190 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
191 |
by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1); |
10648 | 192 |
by (auto_tac (claset(), |
193 |
simpset() addsimps [real_add_minus_iff, |
|
194 |
starfun, hypreal_minus,hypreal_of_real_minus RS sym, |
|
195 |
hypreal_of_real_def, hypreal_add])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
196 |
by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2 THEN Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
197 |
by (dres_inst_tac [("x","u")] spec 1 THEN Clarify_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
198 |
by (dres_inst_tac [("x","s")] spec 1 THEN Clarify_tac 1); |
10648 | 199 |
by (subgoal_tac "ALL n::nat. (xa n) ~= x & \ |
200 |
\ abs ((xa n) + - x) < s --> abs (f (xa n) + - L) < u" 1); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
201 |
by (Blast_tac 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
202 |
by (dtac FreeUltrafilterNat_all 1); |
10648 | 203 |
by (Ultra_tac 1); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
204 |
qed "LIM_NSLIM"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
205 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
206 |
(*--------------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
207 |
Limit: NS definition ==> standard definition |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
208 |
---------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
209 |
|
10648 | 210 |
Goal "ALL s. #0 < s --> (EX xa. xa ~= x & \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
211 |
\ abs (xa + - x) < s & r <= abs (f xa + -L)) \ |
10648 | 212 |
\ ==> ALL n::nat. EX xa. xa ~= x & \ |
10607 | 213 |
\ abs(xa + -x) < inverse(real_of_posnat n) & r <= abs(f xa + -L)"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
214 |
by (Step_tac 1); |
10607 | 215 |
by (cut_inst_tac [("n1","n")] (real_of_posnat_gt_zero RS real_inverse_gt_zero) 1); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
216 |
by Auto_tac; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
217 |
val lemma_LIM = result(); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
218 |
|
10648 | 219 |
Goal "ALL s. #0 < s --> (EX xa. xa ~= x & \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
220 |
\ abs (xa + - x) < s & r <= abs (f xa + -L)) \ |
10648 | 221 |
\ ==> EX X. ALL n::nat. X n ~= x & \ |
10607 | 222 |
\ abs(X n + -x) < inverse(real_of_posnat n) & r <= abs(f (X n) + -L)"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
223 |
by (dtac lemma_LIM 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
224 |
by (dtac choice 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
225 |
by (Blast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
226 |
val lemma_skolemize_LIM2 = result(); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
227 |
|
10677 | 228 |
Goal "ALL n. X n ~= x & \ |
10607 | 229 |
\ abs (X n + - x) < inverse (real_of_posnat n) & \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
230 |
\ r <= abs (f (X n) + - L) ==> \ |
10607 | 231 |
\ ALL n. abs (X n + - x) < inverse (real_of_posnat n)"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
232 |
by (Auto_tac ); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
233 |
val lemma_simp = result(); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
234 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
235 |
(*------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
236 |
NSLIM => LIM |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
237 |
-------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
238 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
239 |
Goalw [LIM_def,NSLIM_def,inf_close_def] |
10677 | 240 |
"f -- x --NS> L ==> f -- x --> L"; |
10648 | 241 |
by (asm_full_simp_tac |
242 |
(simpset() addsimps [Infinitesimal_FreeUltrafilterNat_iff]) 1); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
243 |
by (EVERY1[Step_tac, rtac ccontr, Asm_full_simp_tac]); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
244 |
by (fold_tac [real_le_def]); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
245 |
by (dtac lemma_skolemize_LIM2 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
246 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
247 |
by (dres_inst_tac [("x","Abs_hypreal(hyprel^^{X})")] spec 1); |
10648 | 248 |
by (asm_full_simp_tac |
249 |
(simpset() addsimps [starfun, |
|
250 |
hypreal_minus,hypreal_of_real_minus RS sym, |
|
251 |
hypreal_of_real_def,hypreal_add]) 1); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
252 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
253 |
by (dtac (lemma_simp RS real_seq_to_hypreal_Infinitesimal) 1); |
10648 | 254 |
by (asm_full_simp_tac |
255 |
(simpset() addsimps |
|
256 |
[Infinitesimal_FreeUltrafilterNat_iff,hypreal_of_real_def, |
|
257 |
hypreal_minus,hypreal_of_real_minus,hypreal_add]) 1); |
|
258 |
by (Blast_tac 1); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
259 |
by (rotate_tac 2 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
260 |
by (dres_inst_tac [("x","r")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
261 |
by (Clarify_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
262 |
by (dtac FreeUltrafilterNat_all 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
263 |
by (Ultra_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
264 |
qed "NSLIM_LIM"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
265 |
|
10648 | 266 |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
267 |
(**** Key result ****) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
268 |
Goal "(f -- x --> L) = (f -- x --NS> L)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
269 |
by (blast_tac (claset() addIs [LIM_NSLIM,NSLIM_LIM]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
270 |
qed "LIM_NSLIM_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
271 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
272 |
(*-------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
273 |
(* Proving properties of limits using nonstandard definition and *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
274 |
(* hence, the properties hold for standard limits as well *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
275 |
(*-------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
276 |
(*------------------------------------------------ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
277 |
NSLIM_mult and hence (trivially) LIM_mult |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
278 |
------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
279 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
280 |
Goalw [NSLIM_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
281 |
"[| f -- x --NS> l; g -- x --NS> m |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
282 |
\ ==> (%x. f(x) * g(x)) -- x --NS> (l * m)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
283 |
by (auto_tac (claset() addSIs [starfun_mult_HFinite_inf_close], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
284 |
simpset() addsimps [hypreal_of_real_mult])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
285 |
qed "NSLIM_mult"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
286 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
287 |
Goal "[| f -- x --> l; g -- x --> m |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
288 |
\ ==> (%x. f(x) * g(x)) -- x --> (l * m)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
289 |
by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
290 |
NSLIM_mult]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
291 |
qed "LIM_mult2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
292 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
293 |
(*---------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
294 |
NSLIM_add and hence (trivially) LIM_add |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
295 |
Note the much shorter proof |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
296 |
----------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
297 |
Goalw [NSLIM_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
298 |
"[| f -- x --NS> l; g -- x --NS> m |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
299 |
\ ==> (%x. f(x) + g(x)) -- x --NS> (l + m)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
300 |
by (auto_tac (claset() addSIs [starfun_add_inf_close], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
301 |
simpset() addsimps [hypreal_of_real_add])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
302 |
qed "NSLIM_add"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
303 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
304 |
Goal "[| f -- x --> l; g -- x --> m |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
305 |
\ ==> (%x. f(x) + g(x)) -- x --> (l + m)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
306 |
by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
307 |
NSLIM_add]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
308 |
qed "LIM_add2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
309 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
310 |
(*---------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
311 |
NSLIM_const |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
312 |
----------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
313 |
Goalw [NSLIM_def] "(%x. k) -- x --NS> k"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
314 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
315 |
qed "NSLIM_const"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
316 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
317 |
Addsimps [NSLIM_const]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
318 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
319 |
Goal "(%x. k) -- x --> k"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
320 |
by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
321 |
qed "LIM_const2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
322 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
323 |
(*---------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
324 |
NSLIM_minus |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
325 |
----------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
326 |
Goalw [NSLIM_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
327 |
"f -- a --NS> L ==> (%x. -f(x)) -- a --NS> -L"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
328 |
by (auto_tac (claset() addIs [inf_close_minus], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
329 |
simpset() addsimps [starfun_minus RS sym, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
330 |
hypreal_of_real_minus])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
331 |
qed "NSLIM_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
332 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
333 |
Goal "f -- a --> L ==> (%x. -f(x)) -- a --> -L"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
334 |
by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
335 |
NSLIM_minus]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
336 |
qed "LIM_minus2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
337 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
338 |
(*---------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
339 |
NSLIM_add_minus |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
340 |
----------------------------------------------*) |
10677 | 341 |
Goal "[| f -- x --NS> l; g -- x --NS> m |] \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
342 |
\ ==> (%x. f(x) + -g(x)) -- x --NS> (l + -m)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
343 |
by (blast_tac (claset() addDs [NSLIM_add,NSLIM_minus]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
344 |
qed "NSLIM_add_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
345 |
|
10677 | 346 |
Goal "[| f -- x --> l; g -- x --> m |] \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
347 |
\ ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
348 |
by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
349 |
NSLIM_add_minus]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
350 |
qed "LIM_add_minus2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
351 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
352 |
(*----------------------------- |
10607 | 353 |
NSLIM_inverse |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
354 |
-----------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
355 |
Goalw [NSLIM_def] |
10677 | 356 |
"[| f -- a --NS> L; L ~= #0 |] \ |
357 |
\ ==> (%x. inverse(f(x))) -- a --NS> (inverse L)"; |
|
358 |
by (Clarify_tac 1); |
|
359 |
by (dtac spec 1); |
|
360 |
by (auto_tac (claset(), |
|
361 |
simpset() addsimps [hypreal_of_real_not_zero_iff RS sym, |
|
362 |
hypreal_of_real_inverse RS sym])); |
|
363 |
by (forward_tac [inf_close_hypreal_of_real_not_zero] 1 THEN assume_tac 1); |
|
364 |
by (dtac hypreal_of_real_HFinite_diff_Infinitesimal 1); |
|
365 |
by (stac (starfun_inverse RS sym) 1); |
|
366 |
by (auto_tac (claset() addSEs [inf_close_inverse], |
|
367 |
simpset() addsimps [hypreal_of_real_zero_iff])); |
|
10607 | 368 |
qed "NSLIM_inverse"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
369 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
370 |
Goal "[| f -- a --> L; \ |
10607 | 371 |
\ L ~= #0 |] ==> (%x. inverse(f(x))) -- a --> (inverse L)"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
372 |
by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, |
10607 | 373 |
NSLIM_inverse]) 1); |
374 |
qed "LIM_inverse"; |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
375 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
376 |
(*------------------------------ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
377 |
NSLIM_zero |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
378 |
------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
379 |
Goal "f -- a --NS> l ==> (%x. f(x) + -l) -- a --NS> #0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
380 |
by (res_inst_tac [("z1","l")] (rename_numerals |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
381 |
(real_add_minus RS subst)) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
382 |
by (rtac NSLIM_add_minus 1 THEN Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
383 |
qed "NSLIM_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
384 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
385 |
Goal "f -- a --> l ==> (%x. f(x) + -l) -- a --> #0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
386 |
by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
387 |
NSLIM_zero]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
388 |
qed "LIM_zero2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
389 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
390 |
Goal "(%x. f(x) - l) -- x --NS> #0 ==> f -- x --NS> l"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
391 |
by (dres_inst_tac [("g","%x. l"),("m","l")] NSLIM_add 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
392 |
by (auto_tac (claset(),simpset() addsimps [real_diff_def, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
393 |
real_add_assoc])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
394 |
qed "NSLIM_zero_cancel"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
395 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
396 |
Goal "(%x. f(x) - l) -- x --> #0 ==> f -- x --> l"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
397 |
by (dres_inst_tac [("g","%x. l"),("m","l")] LIM_add 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
398 |
by (auto_tac (claset(),simpset() addsimps [real_diff_def, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
399 |
real_add_assoc])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
400 |
qed "LIM_zero_cancel"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
401 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
402 |
(*-------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
403 |
NSLIM_not_zero |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
404 |
--------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
405 |
Goalw [NSLIM_def] "k ~= #0 ==> ~ ((%x. k) -- x --NS> #0)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
406 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
407 |
by (res_inst_tac [("x","hypreal_of_real x + ehr")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
408 |
by (auto_tac (claset() addIs [Infinitesimal_add_inf_close_self |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
409 |
RS inf_close_sym],simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
410 |
by (dres_inst_tac [("x1","-hypreal_of_real x")] (hypreal_add_left_cancel RS iffD2) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
411 |
by (asm_full_simp_tac (simpset() addsimps [hypreal_add_assoc RS sym, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
412 |
hypreal_epsilon_not_zero]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
413 |
qed "NSLIM_not_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
414 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
415 |
(* [| k ~= #0; (%x. k) -- x --NS> #0 |] ==> R *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
416 |
bind_thm("NSLIM_not_zeroE", (NSLIM_not_zero RS notE)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
417 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
418 |
Goal "k ~= #0 ==> ~ ((%x. k) -- x --> #0)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
419 |
by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
420 |
NSLIM_not_zero]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
421 |
qed "LIM_not_zero2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
422 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
423 |
(*------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
424 |
NSLIM of constant function |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
425 |
-------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
426 |
Goal "(%x. k) -- x --NS> L ==> k = L"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
427 |
by (rtac ccontr 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
428 |
by (dtac NSLIM_zero 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
429 |
by (rtac NSLIM_not_zeroE 1 THEN assume_tac 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
430 |
by (arith_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
431 |
qed "NSLIM_const_eq"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
432 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
433 |
Goal "(%x. k) -- x --> L ==> k = L"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
434 |
by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
435 |
NSLIM_const_eq]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
436 |
qed "LIM_const_eq2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
437 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
438 |
(*------------------------ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
439 |
NS Limit is Unique |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
440 |
------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
441 |
(* can actually be proved more easily by unfolding def! *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
442 |
Goal "[| f -- x --NS> L; f -- x --NS> M |] ==> L = M"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
443 |
by (dtac NSLIM_minus 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
444 |
by (dtac NSLIM_add 1 THEN assume_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
445 |
by (auto_tac (claset() addSDs [NSLIM_const_eq RS sym], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
446 |
simpset() addsimps [real_add_minus])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
447 |
qed "NSLIM_unique"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
448 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
449 |
Goal "[| f -- x --> L; f -- x --> M |] ==> L = M"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
450 |
by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
451 |
NSLIM_unique]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
452 |
qed "LIM_unique2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
453 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
454 |
(*-------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
455 |
NSLIM_mult_zero |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
456 |
--------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
457 |
Goal "[| f -- x --NS> #0; g -- x --NS> #0 |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
458 |
\ ==> (%x. f(x)*g(x)) -- x --NS> #0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
459 |
by (dtac NSLIM_mult 1 THEN Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
460 |
qed "NSLIM_mult_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
461 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
462 |
(* we can use the corresponding thm LIM_mult2 *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
463 |
(* for standard definition of limit *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
464 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
465 |
Goal "[| f -- x --> #0; g -- x --> #0 |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
466 |
\ ==> (%x. f(x)*g(x)) -- x --> #0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
467 |
by (dtac LIM_mult2 1 THEN Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
468 |
qed "LIM_mult_zero2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
469 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
470 |
(*---------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
471 |
NSLIM_self |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
472 |
----------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
473 |
Goalw [NSLIM_def] "(%x. x) -- a --NS> a"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
474 |
by (auto_tac (claset() addIs [starfun_Idfun_inf_close],simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
475 |
qed "NSLIM_self"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
476 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
477 |
Goal "(%x. x) -- a --> a"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
478 |
by (simp_tac (simpset() addsimps [LIM_NSLIM_iff,NSLIM_self]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
479 |
qed "LIM_self2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
480 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
481 |
(*----------------------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
482 |
Derivatives and Continuity - NS and Standard properties |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
483 |
-----------------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
484 |
(*--------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
485 |
Continuity |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
486 |
---------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
487 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
488 |
Goalw [isNSCont_def] |
10677 | 489 |
"[| isNSCont f a; y @= hypreal_of_real a |] \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
490 |
\ ==> (*f* f) y @= hypreal_of_real (f a)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
491 |
by (Blast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
492 |
qed "isNSContD"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
493 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
494 |
Goalw [isNSCont_def,NSLIM_def] |
10677 | 495 |
"isNSCont f a ==> f -- a --NS> (f a) "; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
496 |
by (Blast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
497 |
qed "isNSCont_NSLIM"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
498 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
499 |
Goalw [isNSCont_def,NSLIM_def] |
10677 | 500 |
"f -- a --NS> (f a) ==> isNSCont f a"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
501 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
502 |
by (res_inst_tac [("Q","y = hypreal_of_real a")] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
503 |
(excluded_middle RS disjE) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
504 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
505 |
qed "NSLIM_isNSCont"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
506 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
507 |
(*----------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
508 |
NS continuity can be defined using NS Limit in |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
509 |
similar fashion to standard def of continuity |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
510 |
-----------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
511 |
Goal "(isNSCont f a) = (f -- a --NS> (f a))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
512 |
by (blast_tac (claset() addIs [isNSCont_NSLIM,NSLIM_isNSCont]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
513 |
qed "isNSCont_NSLIM_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
514 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
515 |
(*---------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
516 |
Hence, NS continuity can be given |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
517 |
in terms of standard limit |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
518 |
---------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
519 |
Goal "(isNSCont f a) = (f -- a --> (f a))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
520 |
by (asm_full_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
521 |
[LIM_NSLIM_iff,isNSCont_NSLIM_iff]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
522 |
qed "isNSCont_LIM_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
523 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
524 |
(*----------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
525 |
Moreover, it's trivial now that NS continuity |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
526 |
is equivalent to standard continuity |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
527 |
-----------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
528 |
Goalw [isCont_def] "(isNSCont f a) = (isCont f a)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
529 |
by (rtac isNSCont_LIM_iff 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
530 |
qed "isNSCont_isCont_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
531 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
532 |
(*---------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
533 |
Standard continuity ==> NS continuity |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
534 |
----------------------------------------*) |
10677 | 535 |
Goal "isCont f a ==> isNSCont f a"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
536 |
by (etac (isNSCont_isCont_iff RS iffD2) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
537 |
qed "isCont_isNSCont"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
538 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
539 |
(*---------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
540 |
NS continuity ==> Standard continuity |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
541 |
----------------------------------------*) |
10677 | 542 |
Goal "isNSCont f a ==> isCont f a"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
543 |
by (etac (isNSCont_isCont_iff RS iffD1) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
544 |
qed "isNSCont_isCont"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
545 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
546 |
(*-------------------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
547 |
Alternative definition of continuity |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
548 |
--------------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
549 |
(* Prove equivalence between NS limits - *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
550 |
(* seems easier than using standard def *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
551 |
Goalw [NSLIM_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
552 |
"(f -- a --NS> L) = ((%h. f(a + h)) -- #0 --NS> L)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
553 |
by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_zero])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
554 |
by (dres_inst_tac [("x","hypreal_of_real a + x")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
555 |
by (dres_inst_tac [("x","-hypreal_of_real a + x")] spec 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
556 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
557 |
by (dtac (sym RS (hypreal_eq_minus_iff4 RS iffD1)) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
558 |
by (rtac ((mem_infmal_iff RS iffD2) RS |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
559 |
(Infinitesimal_add_inf_close_self RS inf_close_sym)) 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
560 |
by (rtac (inf_close_minus_iff2 RS iffD1) 5); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
561 |
by (asm_full_simp_tac (simpset() addsimps [hypreal_add_commute]) 4); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
562 |
by (dtac (sym RS (hypreal_eq_minus_iff RS iffD2)) 4); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
563 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 3); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
564 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 6); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
565 |
by (auto_tac (claset(),simpset() addsimps [starfun, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
566 |
hypreal_of_real_def,hypreal_minus,hypreal_add, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
567 |
real_add_assoc,inf_close_refl,hypreal_zero_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
568 |
qed "NSLIM_h_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
569 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
570 |
Goal "(f -- a --NS> f a) = ((%h. f(a + h)) -- #0 --NS> f a)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
571 |
by (rtac NSLIM_h_iff 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
572 |
qed "NSLIM_isCont_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
573 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
574 |
Goal "(f -- a --> f a) = ((%h. f(a + h)) -- #0 --> f(a))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
575 |
by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
576 |
NSLIM_isCont_iff]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
577 |
qed "LIM_isCont_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
578 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
579 |
Goalw [isCont_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
580 |
"(isCont f x) = ((%h. f(x + h)) -- #0 --> f(x))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
581 |
by (simp_tac (simpset() addsimps [LIM_isCont_iff]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
582 |
qed "isCont_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
583 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
584 |
(*-------------------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
585 |
Immediate application of nonstandard criterion for continuity can offer |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
586 |
very simple proofs of some standard property of continuous functions |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
587 |
--------------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
588 |
(*------------------------ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
589 |
sum continuous |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
590 |
------------------------*) |
10677 | 591 |
Goal "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) + g(x)) a"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
592 |
by (auto_tac (claset() addIs [starfun_add_inf_close],simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
593 |
[isNSCont_isCont_iff RS sym,isNSCont_def,hypreal_of_real_add])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
594 |
qed "isCont_add"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
595 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
596 |
(*------------------------ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
597 |
mult continuous |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
598 |
------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
599 |
Goal "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) * g(x)) a"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
600 |
by (auto_tac (claset() addSIs [starfun_mult_HFinite_inf_close],simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
601 |
[isNSCont_isCont_iff RS sym,isNSCont_def,hypreal_of_real_mult])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
602 |
qed "isCont_mult"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
603 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
604 |
(*------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
605 |
composition of continuous functions |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
606 |
Note very short straightforard proof! |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
607 |
------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
608 |
Goal "[| isCont f a; isCont g (f a) |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
609 |
\ ==> isCont (g o f) a"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
610 |
by (auto_tac (claset(),simpset() addsimps [isNSCont_isCont_iff RS sym, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
611 |
isNSCont_def,starfun_o RS sym])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
612 |
qed "isCont_o"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
613 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
614 |
Goal "[| isCont f a; isCont g (f a) |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
615 |
\ ==> isCont (%x. g (f x)) a"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
616 |
by (auto_tac (claset() addDs [isCont_o],simpset() addsimps [o_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
617 |
qed "isCont_o2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
618 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
619 |
Goalw [isNSCont_def] "isNSCont f a ==> isNSCont (%x. - f x) a"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
620 |
by (auto_tac (claset(),simpset() addsimps [starfun_minus RS sym, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
621 |
hypreal_of_real_minus])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
622 |
qed "isNSCont_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
623 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
624 |
Goal "isCont f a ==> isCont (%x. - f x) a"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
625 |
by (auto_tac (claset(),simpset() addsimps [isNSCont_isCont_iff RS sym, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
626 |
isNSCont_minus])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
627 |
qed "isCont_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
628 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
629 |
Goalw [isCont_def] |
10607 | 630 |
"[| isCont f x; f x ~= #0 |] ==> isCont (%x. inverse (f x)) x"; |
631 |
by (blast_tac (claset() addIs [LIM_inverse]) 1); |
|
632 |
qed "isCont_inverse"; |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
633 |
|
10607 | 634 |
Goal "[| isNSCont f x; f x ~= #0 |] ==> isNSCont (%x. inverse (f x)) x"; |
635 |
by (auto_tac (claset() addIs [isCont_inverse],simpset() addsimps |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
636 |
[isNSCont_isCont_iff])); |
10607 | 637 |
qed "isNSCont_inverse"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
638 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
639 |
Goalw [real_diff_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
640 |
"[| isCont f a; isCont g a |] ==> isCont (%x. f(x) - g(x)) a"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
641 |
by (auto_tac (claset() addIs [isCont_add,isCont_minus],simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
642 |
qed "isCont_diff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
643 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
644 |
Goalw [isCont_def] "isCont (%x. k) a"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
645 |
by (Simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
646 |
qed "isCont_const"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
647 |
Addsimps [isCont_const]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
648 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
649 |
Goalw [isNSCont_def] "isNSCont (%x. k) a"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
650 |
by (Simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
651 |
qed "isNSCont_const"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
652 |
Addsimps [isNSCont_const]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
653 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
654 |
Goalw [isNSCont_def] "isNSCont abs a"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
655 |
by (auto_tac (claset() addIs [inf_close_hrabs],simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
656 |
[hypreal_of_real_hrabs RS sym,starfun_rabs_hrabs])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
657 |
qed "isNSCont_rabs"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
658 |
Addsimps [isNSCont_rabs]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
659 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
660 |
Goal "isCont abs a"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
661 |
by (auto_tac (claset(),simpset() addsimps [isNSCont_isCont_iff RS sym])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
662 |
qed "isCont_rabs"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
663 |
Addsimps [isCont_rabs]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
664 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
665 |
(**************************************************************** |
10677 | 666 |
(%* Leave as commented until I add topology theory or remove? *%) |
667 |
(%*------------------------------------------------------------ |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
668 |
Elementary topology proof for a characterisation of |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
669 |
continuity now: a function f is continuous if and only |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
670 |
if the inverse image, {x. f(x) : A}, of any open set A |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
671 |
is always an open set |
10677 | 672 |
------------------------------------------------------------*%) |
673 |
Goal "[| isNSopen A; ALL x. isNSCont f x |] \ |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
674 |
\ ==> isNSopen {x. f x : A}"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
675 |
by (auto_tac (claset(),simpset() addsimps [isNSopen_iff1])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
676 |
by (dtac (mem_monad_inf_close RS inf_close_sym) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
677 |
by (dres_inst_tac [("x","a")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
678 |
by (dtac isNSContD 1 THEN assume_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
679 |
by (dtac bspec 1 THEN assume_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
680 |
by (dres_inst_tac [("x","( *f* f) x")] inf_close_mem_monad2 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
681 |
by (blast_tac (claset() addIs [starfun_mem_starset]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
682 |
qed "isNSCont_isNSopen"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
683 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
684 |
Goalw [isNSCont_def] |
10677 | 685 |
"ALL A. isNSopen A --> isNSopen {x. f x : A} \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
686 |
\ ==> isNSCont f x"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
687 |
by (auto_tac (claset() addSIs [(mem_infmal_iff RS iffD1) RS |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
688 |
(inf_close_minus_iff RS iffD2)],simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
689 |
[Infinitesimal_def,SReal_iff])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
690 |
by (dres_inst_tac [("x","{z. abs(z + -f(x)) < ya}")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
691 |
by (etac (isNSopen_open_interval RSN (2,impE)) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
692 |
by (auto_tac (claset(),simpset() addsimps [isNSopen_def,isNSnbhd_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
693 |
by (dres_inst_tac [("x","x")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
694 |
by (auto_tac (claset() addDs [inf_close_sym RS inf_close_mem_monad], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
695 |
simpset() addsimps [hypreal_of_real_zero RS sym,STAR_starfun_rabs_add_minus])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
696 |
qed "isNSopen_isNSCont"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
697 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
698 |
Goal "(ALL x. isNSCont f x) = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
699 |
\ (ALL A. isNSopen A --> isNSopen {x. f(x) : A})"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
700 |
by (blast_tac (claset() addIs [isNSCont_isNSopen, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
701 |
isNSopen_isNSCont]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
702 |
qed "isNSCont_isNSopen_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
703 |
|
10677 | 704 |
(%*------- Standard version of same theorem --------*%) |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
705 |
Goal "(ALL x. isCont f x) = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
706 |
\ (ALL A. isopen A --> isopen {x. f(x) : A})"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
707 |
by (auto_tac (claset() addSIs [isNSCont_isNSopen_iff], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
708 |
simpset() addsimps [isNSopen_isopen_iff RS sym, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
709 |
isNSCont_isCont_iff RS sym])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
710 |
qed "isCont_isopen_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
711 |
*******************************************************************) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
712 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
713 |
(*----------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
714 |
Uniform continuity |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
715 |
------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
716 |
Goalw [isNSUCont_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
717 |
"[| isNSUCont f; x @= y|] ==> (*f* f) x @= (*f* f) y"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
718 |
by (Blast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
719 |
qed "isNSUContD"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
720 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
721 |
Goalw [isUCont_def,isCont_def,LIM_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
722 |
"isUCont f ==> EX x. isCont f x"; |
10648 | 723 |
by (Force_tac 1); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
724 |
qed "isUCont_isCont"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
725 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
726 |
Goalw [isNSUCont_def,isUCont_def,inf_close_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
727 |
"isUCont f ==> isNSUCont f"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
728 |
by (asm_full_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
729 |
[Infinitesimal_FreeUltrafilterNat_iff]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
730 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
731 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
732 |
by (res_inst_tac [("z","y")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
733 |
by (auto_tac (claset(),simpset() addsimps [starfun, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
734 |
hypreal_minus, hypreal_add])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
735 |
by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2 THEN Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
736 |
by (dres_inst_tac [("x","u")] spec 1 THEN Clarify_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
737 |
by (dres_inst_tac [("x","s")] spec 1 THEN Clarify_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
738 |
by (subgoal_tac "ALL n::nat. abs ((xa n) + - (xb n)) < s --> abs (f (xa n) + - f (xb n)) < u" 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
739 |
by (Blast_tac 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
740 |
by (thin_tac "ALL x y. abs (x + - y) < s --> abs (f x + - f y) < u" 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
741 |
by (dtac FreeUltrafilterNat_all 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
742 |
by (Ultra_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
743 |
qed "isUCont_isNSUCont"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
744 |
|
10677 | 745 |
Goal "ALL s. #0 < s --> \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
746 |
\ (EX xa y. abs (xa + - y) < s & \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
747 |
\ r <= abs (f xa + -f y)) ==> \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
748 |
\ ALL n::nat. EX xa y. \ |
10607 | 749 |
\ abs(xa + -y) < inverse(real_of_posnat n) & \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
750 |
\ r <= abs(f xa + -f y)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
751 |
by (Step_tac 1); |
10607 | 752 |
by (cut_inst_tac [("n1","n")] (real_of_posnat_gt_zero RS real_inverse_gt_zero) 1); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
753 |
by Auto_tac; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
754 |
val lemma_LIMu = result(); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
755 |
|
10677 | 756 |
Goal "ALL s. #0 < s --> \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
757 |
\ (EX xa y. abs (xa + - y) < s & \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
758 |
\ r <= abs (f xa + -f y)) ==> \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
759 |
\ EX X Y. ALL n::nat. \ |
10607 | 760 |
\ abs(X n + -(Y n)) < inverse(real_of_posnat n) & \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
761 |
\ r <= abs(f (X n) + -f (Y n))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
762 |
by (dtac lemma_LIMu 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
763 |
by (dtac choice 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
764 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
765 |
by (dtac choice 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
766 |
by (Blast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
767 |
val lemma_skolemize_LIM2u = result(); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
768 |
|
10607 | 769 |
Goal "ALL n. abs (X n + -Y n) < inverse (real_of_posnat n) & \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
770 |
\ r <= abs (f (X n) + - f(Y n)) ==> \ |
10607 | 771 |
\ ALL n. abs (X n + - Y n) < inverse (real_of_posnat n)"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
772 |
by (Auto_tac ); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
773 |
val lemma_simpu = result(); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
774 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
775 |
Goal "{n. f (X n) + - f(Y n) = Ya n} Int \ |
10607 | 776 |
\ {n. abs (X n + - Y n) < inverse (real_of_posnat n) & \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
777 |
\ r <= abs (f (X n) + - f(Y n))} <= \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
778 |
\ {n. r <= abs (Ya n)}"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
779 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
780 |
val lemma_Intu = result (); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
781 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
782 |
Goalw [isNSUCont_def,isUCont_def,inf_close_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
783 |
"isNSUCont f ==> isUCont f"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
784 |
by (asm_full_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
785 |
[Infinitesimal_FreeUltrafilterNat_iff]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
786 |
by (EVERY1[Step_tac, rtac ccontr, Asm_full_simp_tac]); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
787 |
by (fold_tac [real_le_def]); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
788 |
by (dtac lemma_skolemize_LIM2u 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
789 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
790 |
by (dres_inst_tac [("x","Abs_hypreal(hyprel^^{X})")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
791 |
by (dres_inst_tac [("x","Abs_hypreal(hyprel^^{Y})")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
792 |
by (asm_full_simp_tac (simpset() addsimps [starfun, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
793 |
hypreal_minus,hypreal_add]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
794 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
795 |
by (dtac (lemma_simpu RS real_seq_to_hypreal_Infinitesimal2) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
796 |
by (asm_full_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
797 |
[Infinitesimal_FreeUltrafilterNat_iff, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
798 |
hypreal_minus,hypreal_add]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
799 |
by (Blast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
800 |
by (rotate_tac 2 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
801 |
by (dres_inst_tac [("x","r")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
802 |
by (Clarify_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
803 |
by (dtac FreeUltrafilterNat_all 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
804 |
by (Ultra_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
805 |
qed "isNSUCont_isUCont"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
806 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
807 |
(*------------------------------------------------------------------ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
808 |
Derivatives |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
809 |
------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
810 |
Goalw [deriv_def] |
10677 | 811 |
"(DERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- #0 --> D)"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
812 |
by (Blast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
813 |
qed "DERIV_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
814 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
815 |
Goalw [deriv_def] |
10677 | 816 |
"(DERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- #0 --NS> D)"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
817 |
by (simp_tac (simpset() addsimps [LIM_NSLIM_iff]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
818 |
qed "DERIV_NS_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
819 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
820 |
Goalw [deriv_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
821 |
"DERIV f x :> D \ |
10677 | 822 |
\ ==> (%h. (f(x + h) + - f(x))/h) -- #0 --> D"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
823 |
by (Blast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
824 |
qed "DERIVD"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
825 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
826 |
Goalw [deriv_def] "DERIV f x :> D ==> \ |
10677 | 827 |
\ (%h. (f(x + h) + - f(x))/h) -- #0 --NS> D"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
828 |
by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
829 |
qed "NS_DERIVD"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
830 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
831 |
(* Uniqueness *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
832 |
Goalw [deriv_def] |
10677 | 833 |
"[| DERIV f x :> D; DERIV f x :> E |] ==> D = E"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
834 |
by (blast_tac (claset() addIs [LIM_unique]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
835 |
qed "DERIV_unique"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
836 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
837 |
Goalw [nsderiv_def] |
10677 | 838 |
"[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
839 |
by (cut_facts_tac [Infinitesimal_epsilon, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
840 |
hypreal_epsilon_not_zero] 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
841 |
by (auto_tac (claset() addSDs [bspec] addSIs |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
842 |
[inj_hypreal_of_real RS injD] addDs [inf_close_trans3],simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
843 |
qed "NSDeriv_unique"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
844 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
845 |
(*------------------------------------------------------------------------ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
846 |
Differentiable |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
847 |
------------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
848 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
849 |
Goalw [differentiable_def] |
10677 | 850 |
"f differentiable x ==> EX D. DERIV f x :> D"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
851 |
by (assume_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
852 |
qed "differentiableD"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
853 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
854 |
Goalw [differentiable_def] |
10677 | 855 |
"DERIV f x :> D ==> f differentiable x"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
856 |
by (Blast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
857 |
qed "differentiableI"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
858 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
859 |
Goalw [NSdifferentiable_def] |
10677 | 860 |
"f NSdifferentiable x ==> EX D. NSDERIV f x :> D"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
861 |
by (assume_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
862 |
qed "NSdifferentiableD"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
863 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
864 |
Goalw [NSdifferentiable_def] |
10677 | 865 |
"NSDERIV f x :> D ==> f NSdifferentiable x"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
866 |
by (Blast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
867 |
qed "NSdifferentiableI"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
868 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
869 |
(*-------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
870 |
Alternative definition for differentiability |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
871 |
-------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
872 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
873 |
Goalw [LIM_def] |
10677 | 874 |
"((%h. (f(a + h) + - f(a))/h) -- #0 --> D) = \ |
875 |
\ ((%x. (f(x) + -f(a)) / (x + -a)) -- a --> D)"; |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
876 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
877 |
by (ALLGOALS(dtac spec)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
878 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
879 |
by (Blast_tac 1 THEN Blast_tac 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
880 |
by (ALLGOALS(res_inst_tac [("x","s")] exI)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
881 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
882 |
by (dres_inst_tac [("x","x + -a")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
883 |
by (dres_inst_tac [("x","x + a")] spec 2); |
10677 | 884 |
by (auto_tac (claset(), simpset() addsimps real_add_ac)); |
885 |
by (arith_tac 1); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
886 |
qed "DERIV_LIM_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
887 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
888 |
Goalw [deriv_def] "(DERIV f x :> D) = \ |
10677 | 889 |
\ ((%z. (f(z) + -f(x)) / (z + -x)) -- x --> D)"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
890 |
by (simp_tac (simpset() addsimps [DERIV_LIM_iff]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
891 |
qed "DERIV_iff2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
892 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
893 |
(*-------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
894 |
Equivalence of NS and standard defs of differentiation |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
895 |
-------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
896 |
(*------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
897 |
First NSDERIV in terms of NSLIM |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
898 |
-------------------------------------------*) |
10677 | 899 |
|
900 |
Addsimps [starfun_Id]; (*???????Star.ML?????*) |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
901 |
(*--- first equivalence ---*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
902 |
Goalw [nsderiv_def,NSLIM_def] |
10677 | 903 |
"(NSDERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- #0 --NS> D)"; |
904 |
by (auto_tac (claset(), simpset() addsimps [hypreal_of_real_zero])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
905 |
by (dres_inst_tac [("x","xa")] bspec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
906 |
by (rtac ccontr 3); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
907 |
by (dres_inst_tac [("x","h")] spec 3); |
10677 | 908 |
by (auto_tac (claset(), |
909 |
simpset() addsimps [mem_infmal_iff, |
|
910 |
starfun_divide RS sym, starfun_add RS sym, |
|
911 |
hypreal_of_real_minus, starfun_lambda_cancel])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
912 |
qed "NSDERIV_NSLIM_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
913 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
914 |
(*--- second equivalence ---*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
915 |
Goal "(NSDERIV f x :> D) = \ |
10677 | 916 |
\ ((%z. (f(z) + -f(x)) / (z + -x)) -- x --NS> D)"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
917 |
by (full_simp_tac (simpset() addsimps |
10677 | 918 |
[NSDERIV_NSLIM_iff, DERIV_LIM_iff, LIM_NSLIM_iff RS sym]) 1); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
919 |
qed "NSDERIV_NSLIM_iff2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
920 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
921 |
(* while we're at it! *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
922 |
Goalw [real_diff_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
923 |
"(NSDERIV f x :> D) = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
924 |
\ (ALL xa. \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
925 |
\ xa ~= hypreal_of_real x & xa @= hypreal_of_real x --> \ |
10677 | 926 |
\ (*f* (%z. (f z - f x) / (z - x))) xa @= hypreal_of_real D)"; |
927 |
by (auto_tac (claset(), simpset() addsimps [NSDERIV_NSLIM_iff2, NSLIM_def])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
928 |
qed "NSDERIV_iff2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
929 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
930 |
Addsimps [inf_close_refl]; |
10670 | 931 |
|
10677 | 932 |
|
933 |
(*FIXME: replace both of these by simprocs for cancellation of common factors*) |
|
934 |
Goal "h ~= (0::hypreal) ==> (x/h)*h = x"; |
|
935 |
by (asm_simp_tac |
|
936 |
(simpset() addsimps [hypreal_divide_def, hypreal_mult_inverse_left, |
|
937 |
hypreal_mult_assoc]) 1); |
|
938 |
qed "hypreal_divide_mult_self_eq"; |
|
939 |
Addsimps [hypreal_divide_mult_self_eq]; |
|
940 |
||
941 |
Goal "h ~= (0::real) ==> (x/h)*h = x"; |
|
942 |
by (asm_full_simp_tac |
|
943 |
(simpset() addsimps [real_divide_def, real_mult_inv_left, |
|
944 |
real_mult_assoc]) 1); |
|
945 |
qed "real_divide_mult_self_eq"; |
|
946 |
Addsimps [real_divide_mult_self_eq]; |
|
947 |
||
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
948 |
Goal "(NSDERIV f x :> D) ==> \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
949 |
\ (ALL xa. \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
950 |
\ xa @= hypreal_of_real x --> \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
951 |
\ (*f* (%z. f z - f x)) xa @= hypreal_of_real D * (xa - hypreal_of_real x))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
952 |
by (auto_tac (claset(),simpset() addsimps [NSDERIV_iff2])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
953 |
by (case_tac "xa = hypreal_of_real x" 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
954 |
by (auto_tac (claset(),simpset() addsimps [hypreal_diff_def, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
955 |
hypreal_of_real_zero])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
956 |
by (dres_inst_tac [("x","xa")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
957 |
by Auto_tac; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
958 |
by (dres_inst_tac [("c","xa - hypreal_of_real x"),("b","hypreal_of_real D")] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
959 |
inf_close_mult1 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
960 |
by (ALLGOALS(dtac (hypreal_not_eq_minus_iff RS iffD1))); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
961 |
by (subgoal_tac "(*f* (%z. z - x)) xa ~= (0::hypreal)" 2); |
10677 | 962 |
by (rotate_tac ~1 2); |
963 |
by (auto_tac (claset(), |
|
964 |
simpset() addsimps [starfun_divide RS sym, |
|
965 |
starfun_add RS sym, real_diff_def, |
|
966 |
hypreal_of_real_minus, hypreal_diff_def, |
|
967 |
(inf_close_minus_iff RS iffD1) RS (mem_infmal_iff RS iffD2), |
|
968 |
Infinitesimal_subset_HFinite RS subsetD])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
969 |
qed "NSDERIVD5"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
970 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
971 |
Goal "(NSDERIV f x :> D) ==> \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
972 |
\ (ALL h: Infinitesimal. \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
973 |
\ ((*f* f)(hypreal_of_real x + h) - \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
974 |
\ hypreal_of_real (f x))@= (hypreal_of_real D) * h)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
975 |
by (auto_tac (claset(),simpset() addsimps [nsderiv_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
976 |
by (case_tac "h = (0::hypreal)" 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
977 |
by (auto_tac (claset(),simpset() addsimps [hypreal_diff_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
978 |
by (dres_inst_tac [("x","h")] bspec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
979 |
by (dres_inst_tac [("c","h")] inf_close_mult1 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
980 |
by (auto_tac (claset() addIs [Infinitesimal_subset_HFinite RS subsetD], |
10677 | 981 |
simpset() addsimps [hypreal_diff_def])); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
982 |
qed "NSDERIVD4"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
983 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
984 |
Goal "(NSDERIV f x :> D) ==> \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
985 |
\ (ALL h: Infinitesimal - {0}. \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
986 |
\ ((*f* f)(hypreal_of_real x + h) - \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
987 |
\ hypreal_of_real (f x))@= (hypreal_of_real D) * h)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
988 |
by (auto_tac (claset(),simpset() addsimps [nsderiv_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
989 |
by (rtac ccontr 1 THEN dres_inst_tac [("x","h")] bspec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
990 |
by (dres_inst_tac [("c","h")] inf_close_mult1 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
991 |
by (auto_tac (claset() addIs [Infinitesimal_subset_HFinite RS subsetD], |
10677 | 992 |
simpset() addsimps [hypreal_mult_assoc, hypreal_diff_def])); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
993 |
qed "NSDERIVD3"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
994 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
995 |
(*-------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
996 |
Now equivalence between NSDERIV and DERIV |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
997 |
-------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
998 |
Goalw [deriv_def] "(NSDERIV f x :> D) = (DERIV f x :> D)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
999 |
by (simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff,LIM_NSLIM_iff]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1000 |
qed "NSDERIV_DERIV_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1001 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1002 |
(*--------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1003 |
Differentiability implies continuity |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1004 |
nice and simple "algebraic" proof |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1005 |
--------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1006 |
Goalw [nsderiv_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1007 |
"NSDERIV f x :> D ==> isNSCont f x"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1008 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1009 |
[isNSCont_NSLIM_iff,NSLIM_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1010 |
by (dtac (inf_close_minus_iff RS iffD1) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1011 |
by (dtac (hypreal_not_eq_minus_iff RS iffD1) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1012 |
by (dres_inst_tac [("x","-hypreal_of_real x + xa")] bspec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1013 |
by (asm_full_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1014 |
[hypreal_add_assoc RS sym]) 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1015 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1016 |
[mem_infmal_iff RS sym,hypreal_add_commute])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1017 |
by (dres_inst_tac [("c","xa + -hypreal_of_real x")] inf_close_mult1 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1018 |
by (auto_tac (claset() addIs [Infinitesimal_subset_HFinite |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1019 |
RS subsetD],simpset() addsimps [hypreal_mult_assoc])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1020 |
by (dres_inst_tac [("x3","D")] (HFinite_hypreal_of_real RSN |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1021 |
(2,Infinitesimal_HFinite_mult) RS (mem_infmal_iff RS iffD1)) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1022 |
by (blast_tac (claset() addIs [inf_close_trans, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1023 |
hypreal_mult_commute RS subst, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1024 |
(inf_close_minus_iff RS iffD2)]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1025 |
qed "NSDERIV_isNSCont"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1026 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1027 |
(* Now Sandard proof *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1028 |
Goal "DERIV f x :> D ==> isCont f x"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1029 |
by (asm_full_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1030 |
[NSDERIV_DERIV_iff RS sym, isNSCont_isCont_iff RS sym, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1031 |
NSDERIV_isNSCont]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1032 |
qed "DERIV_isCont"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1033 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1034 |
(*---------------------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1035 |
Differentiation rules for combinations of functions |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1036 |
follow from clear, straightforard, algebraic |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1037 |
manipulations |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1038 |
----------------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1039 |
(*------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1040 |
Constant function |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1041 |
------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1042 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1043 |
(* use simple constant nslimit theorem *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1044 |
Goal "(NSDERIV (%x. k) x :> #0)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1045 |
by (simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1046 |
[NSDERIV_NSLIM_iff,real_add_minus]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1047 |
qed "NSDERIV_const"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1048 |
Addsimps [NSDERIV_const]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1049 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1050 |
Goal "(DERIV (%x. k) x :> #0)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1051 |
by (simp_tac (simpset() addsimps [NSDERIV_DERIV_iff RS sym]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1052 |
qed "DERIV_const"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1053 |
Addsimps [DERIV_const]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1054 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1055 |
(*----------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1056 |
Sum of functions- proved easily |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1057 |
----------------------------------------------------*) |
10677 | 1058 |
|
1059 |
||
1060 |
Goal "hypreal_of_real (z1 / z2) = hypreal_of_real z1 / hypreal_of_real z2"; |
|
1061 |
by (simp_tac (simpset() addsimps [hypreal_divide_def, real_divide_def, |
|
1062 |
hypreal_of_real_mult, hypreal_of_real_inverse]) 1); |
|
1063 |
qed "hypreal_of_real_divide"; (**??????HYPERDEF.ML??????*) |
|
1064 |
||
1065 |
||
1066 |
(**??????HYPERDEF.ML???after inverse_distrib???*) |
|
1067 |
Goal "(x::hypreal) * (y/z) = (x*y)/z"; |
|
1068 |
by (simp_tac (simpset() addsimps [hypreal_divide_def, hypreal_mult_assoc]) 1); |
|
1069 |
qed "hypreal_times_divide1_eq"; |
|
1070 |
||
1071 |
Goal "(y/z) * (x::hypreal) = (y*x)/z"; |
|
1072 |
by (simp_tac (simpset() addsimps [hypreal_divide_def]@hypreal_mult_ac) 1); |
|
1073 |
qed "hypreal_times_divide2_eq"; |
|
1074 |
||
1075 |
Addsimps [hypreal_times_divide1_eq, hypreal_times_divide2_eq]; |
|
1076 |
||
1077 |
Goal "(x::hypreal) / (y/z) = (x*z)/y"; |
|
1078 |
by (simp_tac (simpset() addsimps [hypreal_divide_def, hypreal_inverse_distrib]@ |
|
1079 |
hypreal_mult_ac) 1); |
|
1080 |
qed "hypreal_divide_divide1_eq"; |
|
1081 |
||
1082 |
Goal "((x::hypreal) / y) / z = x/(y*z)"; |
|
1083 |
by (simp_tac (simpset() addsimps [hypreal_divide_def, hypreal_inverse_distrib, |
|
1084 |
hypreal_mult_assoc]) 1); |
|
1085 |
qed "hypreal_divide_divide2_eq"; |
|
1086 |
||
1087 |
Addsimps [hypreal_divide_divide1_eq, hypreal_divide_divide2_eq]; |
|
1088 |
||
1089 |
(** As with multiplication, pull minus signs OUT of the / operator **) |
|
1090 |
||
1091 |
Goal "(-x) / (y::hypreal) = - (x/y)"; |
|
1092 |
by (simp_tac (simpset() addsimps [hypreal_divide_def]) 1); |
|
1093 |
qed "hypreal_minus_divide_eq"; |
|
1094 |
Addsimps [hypreal_minus_divide_eq]; |
|
1095 |
||
1096 |
Goal "(x / -(y::hypreal)) = - (x/y)"; |
|
1097 |
by (simp_tac (simpset() addsimps [hypreal_divide_def, hypreal_minus_inverse]) 1); |
|
1098 |
qed "hypreal_divide_minus_eq"; |
|
1099 |
Addsimps [hypreal_divide_minus_eq]; |
|
1100 |
||
1101 |
Goal "(x+y)/(z::hypreal) = x/z + y/z"; |
|
1102 |
by (simp_tac (simpset() addsimps [hypreal_divide_def, hypreal_add_mult_distrib]) 1); |
|
1103 |
qed "hypreal_add_divide_distrib"; |
|
1104 |
||
1105 |
||
1106 |
Goal "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] \ |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1107 |
\ ==> NSDERIV (%x. f x + g x) x :> Da + Db"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1108 |
by (asm_full_simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff, |
10677 | 1109 |
NSLIM_def]) 1 THEN REPEAT(Step_tac 1)); |
1110 |
by (auto_tac (claset(), |
|
1111 |
simpset() addsimps [starfun_divide RS sym, starfun_add RS sym, |
|
1112 |
hypreal_of_real_minus, hypreal_of_real_add, hypreal_of_real_divide, |
|
1113 |
hypreal_add_divide_distrib])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1114 |
by (thin_tac "xa @= hypreal_of_real #0" 1 THEN dtac inf_close_add 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1115 |
by (auto_tac (claset(),simpset() addsimps hypreal_add_ac)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1116 |
qed "NSDERIV_add"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1117 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1118 |
(* Standard theorem *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1119 |
Goal "[| DERIV f x :> Da; DERIV g x :> Db |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1120 |
\ ==> DERIV (%x. f x + g x) x :> Da + Db"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1121 |
by (asm_full_simp_tac (simpset() addsimps [NSDERIV_add, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1122 |
NSDERIV_DERIV_iff RS sym]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1123 |
qed "DERIV_add"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1124 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1125 |
(*----------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1126 |
Product of functions - Proof is trivial but tedious |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1127 |
and long due to rearrangement of terms |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1128 |
----------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1129 |
(* lemma - terms manipulation tedious as usual*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1130 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1131 |
Goal "((a::real)*b) + -(c*d) = (b*(a + -c)) + \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1132 |
\ (c*(b + -d))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1133 |
by (full_simp_tac (simpset() addsimps [real_add_mult_distrib2, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1134 |
real_minus_mult_eq2 RS sym,real_mult_commute]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1135 |
val lemma_nsderiv1 = result(); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1136 |
|
10677 | 1137 |
(*FIXME: replace both of these by simprocs for cancellation of common factors*) |
1138 |
Goal "h ~= (0::hypreal) ==> (x*h)/h = x"; |
|
1139 |
by (asm_simp_tac |
|
1140 |
(simpset() addsimps [hypreal_divide_def, hypreal_mult_inverse_left, |
|
1141 |
hypreal_mult_assoc]) 1); |
|
1142 |
qed "hypreal_divide_mult2_self_eq"; |
|
1143 |
Addsimps [hypreal_divide_mult2_self_eq]; |
|
1144 |
||
1145 |
Goal "h ~= (0::real) ==> (x*h)/h = x"; |
|
1146 |
by (asm_full_simp_tac |
|
1147 |
(simpset() addsimps [real_divide_def, real_mult_inv_left, |
|
1148 |
real_mult_assoc]) 1); |
|
1149 |
qed "real_divide_mult2_self_eq"; |
|
1150 |
Addsimps [real_divide_mult2_self_eq]; |
|
1151 |
||
1152 |
Goal "[| (x + y) / z = hypreal_of_real D + yb; z ~= 0; \ |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1153 |
\ z : Infinitesimal; yb : Infinitesimal |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1154 |
\ ==> x + y @= 0"; |
10677 | 1155 |
by (forw_inst_tac [("c1","z")] (hypreal_mult_right_cancel RS iffD2) 1 |
1156 |
THEN assume_tac 1); |
|
1157 |
by (thin_tac "(x + y) / z = hypreal_of_real D + yb" 1); |
|
1158 |
by (auto_tac (claset() addSIs [Infinitesimal_HFinite_mult2, HFinite_add], |
|
1159 |
simpset() addsimps [hypreal_mult_assoc, mem_infmal_iff RS sym])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1160 |
by (etac (Infinitesimal_subset_HFinite RS subsetD) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1161 |
val lemma_nsderiv2 = result(); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1162 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1163 |
Goal "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1164 |
\ ==> NSDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"; |
10677 | 1165 |
by (asm_full_simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff, NSLIM_def]) 1 |
1166 |
THEN REPEAT(Step_tac 1)); |
|
1167 |
by (auto_tac (claset(), |
|
1168 |
simpset() addsimps [starfun_divide RS sym, starfun_mult RS sym, |
|
1169 |
starfun_add RS sym,starfun_lambda_cancel,hypreal_of_real_zero, |
|
1170 |
lemma_nsderiv1])); |
|
1171 |
by (simp_tac (simpset() addsimps [hypreal_add_divide_distrib]) 1); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1172 |
by (REPEAT(dtac (bex_Infinitesimal_iff2 RS iffD2) 1)); |
10677 | 1173 |
by (auto_tac (claset(), |
1174 |
simpset() delsimps [hypreal_times_divide1_eq] |
|
1175 |
addsimps [hypreal_times_divide1_eq RS sym, |
|
1176 |
hypreal_of_real_minus])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1177 |
by (dres_inst_tac [("D","Db")] lemma_nsderiv2 1); |
10677 | 1178 |
by (dtac (inf_close_minus_iff RS iffD2 RS (bex_Infinitesimal_iff2 RS iffD2)) 4); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1179 |
by (auto_tac (claset() addSIs [inf_close_add_mono1], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1180 |
simpset() addsimps [hypreal_of_real_add,hypreal_add_mult_distrib, |
10677 | 1181 |
hypreal_add_mult_distrib2,hypreal_of_real_mult,hypreal_mult_commute, |
1182 |
hypreal_add_assoc])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1183 |
by (res_inst_tac [("w1","hypreal_of_real Db * hypreal_of_real (f x)")] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1184 |
(hypreal_add_commute RS subst) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1185 |
by (auto_tac (claset() addSIs [Infinitesimal_add_inf_close_self2 RS |
10677 | 1186 |
inf_close_sym,Infinitesimal_add,Infinitesimal_mult, |
1187 |
Infinitesimal_hypreal_of_real_mult, |
|
1188 |
Infinitesimal_hypreal_of_real_mult2 ], |
|
1189 |
simpset() addsimps [hypreal_add_assoc RS sym])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1190 |
qed "NSDERIV_mult"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1191 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1192 |
Goal "[| DERIV f x :> Da; DERIV g x :> Db |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1193 |
\ ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1194 |
by (asm_full_simp_tac (simpset() addsimps [NSDERIV_mult, |
10677 | 1195 |
NSDERIV_DERIV_iff RS sym]) 1); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1196 |
qed "DERIV_mult"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1197 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1198 |
(*---------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1199 |
Multiplying by a constant |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1200 |
---------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1201 |
Goal "NSDERIV f x :> D \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1202 |
\ ==> NSDERIV (%x. c * f x) x :> c*D"; |
10677 | 1203 |
by (asm_full_simp_tac |
1204 |
(simpset() addsimps [real_times_divide1_eq RS sym, NSDERIV_NSLIM_iff, |
|
1205 |
real_minus_mult_eq2, real_add_mult_distrib2 RS sym] |
|
1206 |
delsimps [real_times_divide1_eq, real_minus_mult_eq2 RS sym]) 1); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1207 |
by (etac (NSLIM_const RS NSLIM_mult) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1208 |
qed "NSDERIV_cmult"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1209 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1210 |
(* let's do the standard proof though theorem *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1211 |
(* LIM_mult2 follows from a NS proof *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1212 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1213 |
Goalw [deriv_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1214 |
"DERIV f x :> D \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1215 |
\ ==> DERIV (%x. c * f x) x :> c*D"; |
10677 | 1216 |
by (asm_full_simp_tac |
1217 |
(simpset() addsimps [real_times_divide1_eq RS sym, NSDERIV_NSLIM_iff, |
|
1218 |
real_minus_mult_eq2, real_add_mult_distrib2 RS sym] |
|
1219 |
delsimps [real_times_divide1_eq, real_minus_mult_eq2 RS sym]) 1); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1220 |
by (etac (LIM_const RS LIM_mult2) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1221 |
qed "DERIV_cmult"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1222 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1223 |
(*-------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1224 |
Negation of function |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1225 |
-------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1226 |
Goal "NSDERIV f x :> D \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1227 |
\ ==> NSDERIV (%x. -(f x)) x :> -D"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1228 |
by (asm_full_simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1229 |
by (res_inst_tac [("t","f x")] (real_minus_minus RS subst) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1230 |
by (asm_simp_tac (simpset() addsimps [real_minus_add_distrib RS sym, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1231 |
real_minus_mult_eq1 RS sym] delsimps [real_minus_add_distrib, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1232 |
real_minus_minus]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1233 |
by (etac NSLIM_minus 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1234 |
qed "NSDERIV_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1235 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1236 |
Goal "DERIV f x :> D \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1237 |
\ ==> DERIV (%x. -(f x)) x :> -D"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1238 |
by (asm_full_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1239 |
[NSDERIV_minus,NSDERIV_DERIV_iff RS sym]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1240 |
qed "DERIV_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1241 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1242 |
(*------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1243 |
Subtraction |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1244 |
------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1245 |
Goal "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1246 |
\ ==> NSDERIV (%x. f x + -g x) x :> Da + -Db"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1247 |
by (blast_tac (claset() addDs [NSDERIV_add,NSDERIV_minus]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1248 |
qed "NSDERIV_add_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1249 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1250 |
Goal "[| DERIV f x :> Da; DERIV g x :> Db |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1251 |
\ ==> DERIV (%x. f x + -g x) x :> Da + -Db"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1252 |
by (blast_tac (claset() addDs [DERIV_add,DERIV_minus]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1253 |
qed "DERIV_add_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1254 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1255 |
Goalw [real_diff_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1256 |
"[| NSDERIV f x :> Da; NSDERIV g x :> Db |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1257 |
\ ==> NSDERIV (%x. f x - g x) x :> Da - Db"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1258 |
by (blast_tac (claset() addIs [NSDERIV_add_minus]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1259 |
qed "NSDERIV_diff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1260 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1261 |
Goalw [real_diff_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1262 |
"[| DERIV f x :> Da; DERIV g x :> Db |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1263 |
\ ==> DERIV (%x. f x - g x) x :> Da - Db"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1264 |
by (blast_tac (claset() addIs [DERIV_add_minus]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1265 |
qed "DERIV_diff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1266 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1267 |
(*--------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1268 |
(NS) Increment |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1269 |
---------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1270 |
Goalw [increment_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1271 |
"f NSdifferentiable x ==> \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1272 |
\ increment f x h = (*f* f) (hypreal_of_real(x) + h) + \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1273 |
\ -hypreal_of_real (f x)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1274 |
by (Blast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1275 |
qed "incrementI"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1276 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1277 |
Goal "NSDERIV f x :> D ==> \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1278 |
\ increment f x h = (*f* f) (hypreal_of_real(x) + h) + \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1279 |
\ -hypreal_of_real (f x)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1280 |
by (etac (NSdifferentiableI RS incrementI) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1281 |
qed "incrementI2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1282 |
|
10677 | 1283 |
(*FIXME: replace by simprocs for cancellation of common factors*) |
1284 |
Goal "h ~= (0::hypreal) ==> h/h = 1hr"; |
|
1285 |
by (asm_simp_tac |
|
1286 |
(simpset() addsimps [hypreal_divide_def, hypreal_mult_inverse_left]) 1); |
|
1287 |
qed "hypreal_divide_self_eq"; |
|
1288 |
Addsimps [hypreal_divide_self_eq]; |
|
1289 |
||
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1290 |
(* The Increment theorem -- Keisler p. 65 *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1291 |
Goal "[| NSDERIV f x :> D; h: Infinitesimal; h ~= 0 |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1292 |
\ ==> EX e: Infinitesimal. increment f x h = hypreal_of_real(D)*h + e*h"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1293 |
by (forw_inst_tac [("h","h")] incrementI2 1 THEN rewtac nsderiv_def); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1294 |
by (dtac bspec 1 THEN Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1295 |
by (dtac (bex_Infinitesimal_iff2 RS iffD2) 1 THEN Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1296 |
by (forw_inst_tac [("b1","hypreal_of_real(D) + y")] |
10677 | 1297 |
(hypreal_mult_right_cancel RS iffD2) 1); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1298 |
by (thin_tac "((*f* f) (hypreal_of_real(x) + h) + \ |
10677 | 1299 |
\ - hypreal_of_real (f x)) / h = hypreal_of_real(D) + y" 2); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1300 |
by (assume_tac 1); |
10677 | 1301 |
by (asm_full_simp_tac (simpset() addsimps [hypreal_times_divide1_eq RS sym] |
1302 |
delsimps [hypreal_times_divide1_eq]) 1); |
|
1303 |
by (auto_tac (claset(), |
|
1304 |
simpset() addsimps [hypreal_add_mult_distrib])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1305 |
qed "increment_thm"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1306 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1307 |
Goal "[| NSDERIV f x :> D; h @= 0; h ~= 0 |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1308 |
\ ==> EX e: Infinitesimal. increment f x h = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1309 |
\ hypreal_of_real(D)*h + e*h"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1310 |
by (blast_tac (claset() addSDs [mem_infmal_iff RS iffD2] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1311 |
addSIs [increment_thm]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1312 |
qed "increment_thm2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1313 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1314 |
Goal "[| NSDERIV f x :> D; h @= 0; h ~= 0 |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1315 |
\ ==> increment f x h @= 0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1316 |
by (dtac increment_thm2 1 THEN auto_tac (claset() addSIs |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1317 |
[Infinitesimal_HFinite_mult2,HFinite_add],simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1318 |
[hypreal_add_mult_distrib RS sym,mem_infmal_iff RS sym])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1319 |
by (etac (Infinitesimal_subset_HFinite RS subsetD) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1320 |
qed "increment_inf_close_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1321 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1322 |
(*--------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1323 |
Similarly to the above, the chain rule admits an entirely |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1324 |
straightforward derivation. Compare this with Harrison's |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1325 |
HOL proof of the chain rule, which proved to be trickier and |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1326 |
required an alternative characterisation of differentiability- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1327 |
the so-called Carathedory derivative. Our main problem is |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1328 |
manipulation of terms. |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1329 |
--------------------------------------------------------------*) |
10677 | 1330 |
|
1331 |
(*???HYPERDEF.ML???*) |
|
1332 |
Goal "(0::hypreal)/x = 0"; |
|
1333 |
by (simp_tac (simpset() addsimps [hypreal_divide_def]) 1); |
|
1334 |
qed "hypreal_zero_divide"; |
|
1335 |
||
1336 |
Goal "x/1hr = x"; |
|
1337 |
by (simp_tac (simpset() addsimps [hypreal_divide_def]) 1); |
|
1338 |
qed "hypreal_divide_one"; |
|
1339 |
Addsimps [hypreal_zero_divide, hypreal_divide_one]; |
|
1340 |
||
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1341 |
(* lemmas *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1342 |
Goalw [nsderiv_def] |
10677 | 1343 |
"[| NSDERIV g x :> D; \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1344 |
\ (*f* g) (hypreal_of_real(x) + xa) = hypreal_of_real(g x);\ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1345 |
\ xa : Infinitesimal;\ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1346 |
\ xa ~= 0 \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1347 |
\ |] ==> D = #0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1348 |
by (dtac bspec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1349 |
by (Auto_tac); |
10677 | 1350 |
by (asm_full_simp_tac (simpset() addsimps [hypreal_of_real_zero RS sym]) 1); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1351 |
qed "NSDERIV_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1352 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1353 |
(* can be proved differently using NSLIM_isCont_iff *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1354 |
Goalw [nsderiv_def] |
10677 | 1355 |
"[| NSDERIV f x :> D; \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1356 |
\ h: Infinitesimal; h ~= 0 \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1357 |
\ |] ==> (*f* f) (hypreal_of_real(x) + h) + -hypreal_of_real(f x) @= 0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1358 |
by (asm_full_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1359 |
[mem_infmal_iff RS sym]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1360 |
by (rtac Infinitesimal_ratio 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1361 |
by (rtac inf_close_hypreal_of_real_HFinite 3); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1362 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1363 |
qed "NSDERIV_inf_close"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1364 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1365 |
(*--------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1366 |
from one version of differentiability |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1367 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1368 |
f(x) - f(a) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1369 |
--------------- @= Db |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1370 |
x - a |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1371 |
---------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1372 |
Goal "[| NSDERIV f (g x) :> Da; \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1373 |
\ (*f* g) (hypreal_of_real(x) + xa) ~= hypreal_of_real (g x); \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1374 |
\ (*f* g) (hypreal_of_real(x) + xa) @= hypreal_of_real (g x) \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1375 |
\ |] ==> ((*f* f) ((*f* g) (hypreal_of_real(x) + xa)) \ |
10677 | 1376 |
\ + - hypreal_of_real (f (g x))) \ |
1377 |
\ / ((*f* g) (hypreal_of_real(x) + xa) + - hypreal_of_real (g x)) \ |
|
1378 |
\ @= hypreal_of_real(Da)"; |
|
1379 |
by (auto_tac (claset(), |
|
1380 |
simpset() addsimps [NSDERIV_NSLIM_iff2, NSLIM_def, starfun_divide RS sym, |
|
1381 |
hypreal_of_real_minus, |
|
1382 |
starfun_add RS sym])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1383 |
qed "NSDERIVD1"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1384 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1385 |
(*-------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1386 |
from other version of differentiability |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1387 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1388 |
f(x + h) - f(x) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1389 |
----------------- @= Db |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1390 |
h |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1391 |
--------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1392 |
Goal "[| NSDERIV g x :> Db; xa: Infinitesimal; xa ~= 0 |] \ |
10677 | 1393 |
\ ==> ((*f* g) (hypreal_of_real(x) + xa) + - hypreal_of_real(g x)) / xa \ |
1394 |
\ @= hypreal_of_real(Db)"; |
|
1395 |
by (auto_tac (claset(), |
|
1396 |
simpset() addsimps [NSDERIV_NSLIM_iff, NSLIM_def, starfun_divide RS sym, |
|
1397 |
starfun_add RS sym, hypreal_of_real_zero, mem_infmal_iff, |
|
1398 |
starfun_lambda_cancel,hypreal_of_real_minus])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1399 |
qed "NSDERIVD2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1400 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1401 |
(*--------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1402 |
This proof uses both possible definitions |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1403 |
for differentiability. |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1404 |
--------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1405 |
Goal "[| NSDERIV f (g x) :> Da; NSDERIV g x :> Db |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1406 |
\ ==> NSDERIV (f o g) x :> Da * Db"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1407 |
by (asm_simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1408 |
NSLIM_def,hypreal_of_real_zero,mem_infmal_iff RS sym]) 1 THEN Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1409 |
by (forw_inst_tac [("f","g")] NSDERIV_inf_close 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1410 |
by (auto_tac (claset(),simpset() addsimps [starfun_add RS sym, |
10677 | 1411 |
hypreal_of_real_minus, hypreal_of_real_mult, |
1412 |
starfun_lambda_cancel2,starfun_o RS sym, starfun_divide RS sym])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1413 |
by (case_tac "(*f* g) (hypreal_of_real(x) + xa) = hypreal_of_real (g x)" 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1414 |
by (dres_inst_tac [("g","g")] NSDERIV_zero 1); |
10677 | 1415 |
by (auto_tac (claset(), |
1416 |
simpset() addsimps [hypreal_divide_def, hypreal_of_real_zero])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1417 |
by (res_inst_tac [("z1","(*f* g) (hypreal_of_real(x) + xa) + -hypreal_of_real (g x)"), |
10607 | 1418 |
("y1","inverse xa")] (lemma_chain RS ssubst) 1); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1419 |
by (etac (hypreal_not_eq_minus_iff RS iffD1) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1420 |
by (rtac inf_close_mult_hypreal_of_real 1); |
10677 | 1421 |
by (fold_tac [hypreal_divide_def]); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1422 |
by (blast_tac (claset() addIs [NSDERIVD1, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1423 |
inf_close_minus_iff RS iffD2]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1424 |
by (blast_tac (claset() addIs [NSDERIVD2]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1425 |
qed "NSDERIV_chain"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1426 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1427 |
(* standard version *) |
10677 | 1428 |
Goal "[| DERIV f (g x) :> Da; \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1429 |
\ DERIV g x :> Db \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1430 |
\ |] ==> DERIV (f o g) x :> Da * Db"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1431 |
by (asm_full_simp_tac (simpset() addsimps [NSDERIV_DERIV_iff RS sym, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1432 |
NSDERIV_chain]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1433 |
qed "DERIV_chain"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1434 |
|
10648 | 1435 |
Goal "[| DERIV f (g x) :> Da; DERIV g x :> Db |] \ |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1436 |
\ ==> DERIV (%x. f (g x)) x :> Da * Db"; |
10648 | 1437 |
by (auto_tac (claset() addDs [DERIV_chain], simpset() addsimps [o_def])); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1438 |
qed "DERIV_chain2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1439 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1440 |
(*------------------------------------------------------------------ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1441 |
Differentiation of natural number powers |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1442 |
------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1443 |
Goal "NSDERIV (%x. x) x :> #1"; |
10677 | 1444 |
by (auto_tac (claset(), |
1445 |
simpset() addsimps [NSDERIV_NSLIM_iff, |
|
1446 |
NSLIM_def, starfun_divide RS sym,starfun_Id, hypreal_of_real_zero, |
|
1447 |
hypreal_of_real_one])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1448 |
qed "NSDERIV_Id"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1449 |
Addsimps [NSDERIV_Id]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1450 |
|
10670 | 1451 |
(*derivative of the identity function*) |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1452 |
Goal "DERIV (%x. x) x :> #1"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1453 |
by (simp_tac (simpset() addsimps [NSDERIV_DERIV_iff RS sym]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1454 |
qed "DERIV_Id"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1455 |
Addsimps [DERIV_Id]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1456 |
|
10677 | 1457 |
bind_thm ("isCont_Id", DERIV_Id RS DERIV_isCont); |
1458 |
||
10670 | 1459 |
(*derivative of linear multiplication*) |
10648 | 1460 |
Goal "DERIV (op * c) x :> c"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1461 |
by (cut_inst_tac [("c","c"),("x","x")] (DERIV_Id RS DERIV_cmult) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1462 |
by (Asm_full_simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1463 |
qed "DERIV_cmult_Id"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1464 |
Addsimps [DERIV_cmult_Id]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1465 |
|
10648 | 1466 |
Goal "NSDERIV (op * c) x :> c"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1467 |
by (simp_tac (simpset() addsimps [NSDERIV_DERIV_iff]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1468 |
qed "NSDERIV_cmult_Id"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1469 |
Addsimps [NSDERIV_cmult_Id]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1470 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1471 |
Goal "DERIV (%x. x ^ n) x :> real_of_nat n * (x ^ (n - 1))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1472 |
by (induct_tac "n" 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1473 |
by (dtac (DERIV_Id RS DERIV_mult) 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1474 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1475 |
[real_add_mult_distrib])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1476 |
by (case_tac "0 < n" 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1477 |
by (dres_inst_tac [("x","x")] realpow_minus_mult 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1478 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1479 |
[real_mult_assoc,real_add_commute])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1480 |
qed "DERIV_pow"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1481 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1482 |
(* NS version *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1483 |
Goal "NSDERIV (%x. x ^ n) x :> real_of_nat n * (x ^ (n - 1))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1484 |
by (simp_tac (simpset() addsimps [NSDERIV_DERIV_iff,DERIV_pow]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1485 |
qed "NSDERIV_pow"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1486 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1487 |
(*--------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1488 |
Power of -1 |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1489 |
---------------------------------------------------------------*) |
10677 | 1490 |
|
1491 |
||
1492 |
(*FIXME: replace by simprocs for cancellation of common factors*) |
|
1493 |
Goal "h ~= (0::hypreal) ==> (h*x)/h = x"; |
|
1494 |
by (asm_simp_tac |
|
1495 |
(simpset() addsimps [inst "z" "h" hypreal_mult_commute]) 1); |
|
1496 |
qed "hypreal_times_divide_self_eq"; |
|
1497 |
Addsimps [hypreal_times_divide_self_eq]; |
|
1498 |
||
1499 |
(*Can't get rid of x ~= #0 because it isn't continuous at zero*) |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1500 |
Goalw [nsderiv_def] |
10607 | 1501 |
"x ~= #0 ==> NSDERIV (%x. inverse(x)) x :> (- (inverse x ^ 2))"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1502 |
by (rtac ballI 1 THEN Asm_full_simp_tac 1 THEN Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1503 |
by (forward_tac [Infinitesimal_add_not_zero] 1); |
10677 | 1504 |
by (auto_tac (claset(), |
1505 |
simpset() addsimps [starfun_inverse_inverse, |
|
10607 | 1506 |
hypreal_of_real_inverse RS sym,hypreal_of_real_minus,realpow_two, |
10677 | 1507 |
hypreal_of_real_mult, |
1508 |
hypreal_of_real_divide] |
|
1509 |
delsimps [hypreal_minus_mult_eq1 RS sym, |
|
1510 |
hypreal_minus_mult_eq2 RS sym])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1511 |
by (dtac (hypreal_of_real_not_zero_iff RS iffD2) 1); |
10607 | 1512 |
by (asm_full_simp_tac (simpset() addsimps [hypreal_inverse_add, |
1513 |
inverse_mult_eq RS sym, hypreal_minus_inverse RS sym] |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1514 |
@ hypreal_add_ac @ hypreal_mult_ac delsimps [hypreal_minus_mult_eq1 RS |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1515 |
sym,hypreal_minus_mult_eq2 RS sym] ) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1516 |
by (asm_simp_tac (simpset() addsimps [hypreal_mult_assoc RS sym, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1517 |
hypreal_add_mult_distrib2] delsimps [hypreal_minus_mult_eq1 RS |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1518 |
sym,hypreal_minus_mult_eq2 RS sym]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1519 |
by (dres_inst_tac [("x3","x") ] ((HFinite_hypreal_of_real RSN |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1520 |
(2,Infinitesimal_HFinite_mult2)) RS |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1521 |
(Infinitesimal_minus_iff RS iffD1)) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1522 |
by (forw_inst_tac [("x","hypreal_of_real x"),("y","hypreal_of_real x")] hypreal_mult_not_0 1); |
10607 | 1523 |
by (res_inst_tac [("y"," inverse(- hypreal_of_real x * hypreal_of_real x)")] inf_close_trans 2); |
1524 |
by (rtac inverse_add_Infinitesimal_inf_close2 2); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1525 |
by (auto_tac (claset() addIs [HFinite_minus_iff RS iffD1] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1526 |
addSDs [Infinitesimal_minus_iff RS iffD2, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1527 |
hypreal_of_real_HFinite_diff_Infinitesimal], simpset() addsimps |
10607 | 1528 |
[hypreal_minus_inverse RS sym,hypreal_of_real_mult RS sym])); |
1529 |
qed "NSDERIV_inverse"; |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1530 |
|
10607 | 1531 |
Goal "x ~= #0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ 2))"; |
1532 |
by (asm_simp_tac (simpset() addsimps [NSDERIV_inverse, |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1533 |
NSDERIV_DERIV_iff RS sym] delsimps [thm "realpow_Suc"]) 1); |
10607 | 1534 |
qed "DERIV_inverse"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1535 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1536 |
(*-------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1537 |
Derivative of inverse |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1538 |
-------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1539 |
Goal "[| DERIV f x :> d; f(x) ~= #0 |] \ |
10607 | 1540 |
\ ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ 2)))"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1541 |
by (rtac (real_mult_commute RS subst) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1542 |
by (asm_simp_tac (simpset() addsimps [real_minus_mult_eq1, |
10607 | 1543 |
realpow_inverse] delsimps [thm "realpow_Suc", |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1544 |
real_minus_mult_eq1 RS sym]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1545 |
by (fold_goals_tac [o_def]); |
10607 | 1546 |
by (blast_tac (claset() addSIs [DERIV_chain,DERIV_inverse]) 1); |
1547 |
qed "DERIV_inverse_fun"; |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1548 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1549 |
Goal "[| NSDERIV f x :> d; f(x) ~= #0 |] \ |
10607 | 1550 |
\ ==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ 2)))"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1551 |
by (asm_full_simp_tac (simpset() addsimps [NSDERIV_DERIV_iff, |
10607 | 1552 |
DERIV_inverse_fun] delsimps [thm "realpow_Suc"]) 1); |
1553 |
qed "NSDERIV_inverse_fun"; |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1554 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1555 |
(*-------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1556 |
Derivative of quotient |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1557 |
-------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1558 |
Goal "[| DERIV f x :> d; DERIV g x :> e; g(x) ~= #0 |] \ |
10677 | 1559 |
\ ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) + -(e*f(x))) / (g(x) ^ 2)"; |
10607 | 1560 |
by (dres_inst_tac [("f","g")] DERIV_inverse_fun 1); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1561 |
by (dtac DERIV_mult 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1562 |
by (REPEAT(assume_tac 1)); |
10677 | 1563 |
by (asm_full_simp_tac |
1564 |
(simpset() addsimps [real_divide_def, real_add_mult_distrib2, |
|
1565 |
realpow_inverse,real_minus_mult_eq1] @ real_mult_ac |
|
1566 |
delsimps [thm "realpow_Suc", real_minus_mult_eq1 RS sym, |
|
1567 |
real_minus_mult_eq2 RS sym]) 1); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1568 |
qed "DERIV_quotient"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1569 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1570 |
Goal "[| NSDERIV f x :> d; DERIV g x :> e; g(x) ~= #0 |] \ |
10677 | 1571 |
\ ==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x) \ |
1572 |
\ + -(e*f(x))) / (g(x) ^ 2)"; |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1573 |
by (asm_full_simp_tac (simpset() addsimps [NSDERIV_DERIV_iff, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1574 |
DERIV_quotient] delsimps [thm "realpow_Suc"]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1575 |
qed "NSDERIV_quotient"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1576 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1577 |
(* ------------------------------------------------------------------------ *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1578 |
(* Caratheodory formulation of derivative at a point: standard proof *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1579 |
(* ------------------------------------------------------------------------ *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1580 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1581 |
Goal "(DERIV f x :> l) = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1582 |
\ (EX g. (ALL z. f z - f x = g z * (z - x)) & isCont g x & g x = l)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1583 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1584 |
by (res_inst_tac |
10677 | 1585 |
[("x","%z. if z = x then l else (f(z) - f(x)) / (z - x)")] exI 1); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1586 |
by (auto_tac (claset(),simpset() addsimps [real_mult_assoc, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1587 |
ARITH_PROVE "z ~= x ==> z - x ~= (#0::real)"])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1588 |
by (auto_tac (claset(),simpset() addsimps [isCont_iff,DERIV_iff])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1589 |
by (ALLGOALS(rtac (LIM_equal RS iffD1))); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1590 |
by (auto_tac (claset(),simpset() addsimps [real_diff_def,real_mult_assoc])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1591 |
qed "CARAT_DERIV"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1592 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1593 |
Goal "NSDERIV f x :> l ==> \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1594 |
\ EX g. (ALL z. f z - f x = g z * (z - x)) & isNSCont g x & g x = l"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1595 |
by (auto_tac (claset(),simpset() addsimps [NSDERIV_DERIV_iff, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1596 |
isNSCont_isCont_iff,CARAT_DERIV])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1597 |
qed "CARAT_NSDERIV"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1598 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1599 |
(* How about a NS proof? *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1600 |
Goal "(ALL z. f z - f x = g z * (z - x)) & isNSCont g x & g x = l \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1601 |
\ ==> NSDERIV f x :> l"; |
10677 | 1602 |
by (auto_tac (claset(), |
1603 |
simpset() addsimps [NSDERIV_iff2, starfun_mult RS sym, |
|
1604 |
starfun_divide RS sym])); |
|
1605 |
by (auto_tac (claset(), |
|
1606 |
simpset() addsimps [hypreal_mult_assoc, |
|
1607 |
starfun_diff RS sym])); |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1608 |
by (asm_full_simp_tac (simpset() addsimps [hypreal_eq_minus_iff3 RS sym, |
10677 | 1609 |
hypreal_diff_def]) 1); |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1610 |
by (dtac (CLAIM_SIMP "x ~= y ==> x - y ~= (0::hypreal)" [hypreal_eq_minus_iff3 RS sym, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1611 |
hypreal_diff_def]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1612 |
by (asm_full_simp_tac (simpset() addsimps [isNSCont_def]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1613 |
qed "CARAT_DERIVD"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1614 |
|
10648 | 1615 |
|
1616 |
||
1617 |
(*----------------------------------------------------------------------------*) |
|
1618 |
(* Useful lemmas about nested intervals and proof by bisection (cf.Harrison) *) |
|
1619 |
(*----------------------------------------------------------------------------*) |
|
1620 |
||
1621 |
Goal "(ALL n. (f::nat=>real) n <= f (Suc n)) --> f m <= f(m + no)"; |
|
1622 |
by (induct_tac "no" 1); |
|
1623 |
by (auto_tac (claset() addIs [real_le_trans],simpset())); |
|
1624 |
qed_spec_mp "lemma_f_mono_add"; |
|
1625 |
||
1626 |
||
1627 |
(* IMPROVE?: long proof! *) |
|
1628 |
Goal "[| ALL n. f(n) <= f(Suc n); \ |
|
1629 |
\ ALL n. g(Suc n) <= g(n); \ |
|
1630 |
\ ALL n. f(n) <= g(n) |] \ |
|
1631 |
\ ==> EX l m. l <= m & ((ALL n. f(n) <= l) & f ----> l) & \ |
|
1632 |
\ ((ALL n. m <= g(n)) & g ----> m)"; |
|
1633 |
by (subgoal_tac "Bseq f" 1); |
|
1634 |
by (res_inst_tac [("k","f 0"),("K","g 0")] BseqI2 2 THEN rtac allI 2); |
|
1635 |
by (induct_tac "n" 2); |
|
1636 |
by (auto_tac (claset() addIs [real_le_trans],simpset())); |
|
1637 |
by (res_inst_tac [("j","g(Suc na)")] real_le_trans 2); |
|
1638 |
by (induct_tac "na" 3); |
|
1639 |
by (auto_tac (claset() addIs [real_le_trans],simpset())); |
|
1640 |
by (subgoal_tac "monoseq f" 1); |
|
1641 |
by (subgoal_tac "monoseq g" 1); |
|
1642 |
by (subgoal_tac "Bseq g" 1); |
|
1643 |
by (res_inst_tac [("k","f 0"),("K","g 0")] BseqI2 2 THEN rtac allI 2); |
|
1644 |
by (induct_tac "n" 2); |
|
1645 |
by (auto_tac (claset() addIs [real_le_trans],simpset())); |
|
1646 |
by (res_inst_tac [("j","f(Suc na)")] real_le_trans 2); |
|
1647 |
by (induct_tac "na" 2); |
|
1648 |
by (auto_tac (claset() addIs [real_le_trans],simpset())); |
|
1649 |
by (auto_tac (claset() addSDs [Bseq_monoseq_convergent], |
|
1650 |
simpset() addsimps [convergent_LIMSEQ_iff])); |
|
1651 |
by (res_inst_tac [("x","lim f")] exI 1); |
|
1652 |
by (res_inst_tac [("x","lim g")] exI 1); |
|
1653 |
by (auto_tac (claset() addIs [LIMSEQ_le],simpset())); |
|
1654 |
by (rtac real_leI 1 THEN rtac real_leI 2); |
|
1655 |
by (auto_tac (claset(),simpset() addsimps [LIMSEQ_iff,monoseq_Suc])); |
|
1656 |
by (ALLGOALS (dtac real_less_sum_gt_zero)); |
|
1657 |
by (dres_inst_tac [("x","f n + - lim f")] spec 1); |
|
1658 |
by (rotate_tac 4 2); |
|
1659 |
by (dres_inst_tac [("x","lim g + - g n")] spec 2); |
|
1660 |
by (Step_tac 1); |
|
1661 |
by (ALLGOALS(rotate_tac 5)); |
|
1662 |
by (ALLGOALS(dres_inst_tac [("x","no + n")] spec)); |
|
1663 |
by (Auto_tac); |
|
1664 |
by (subgoal_tac "0 <= f(no + n) - lim f" 1); |
|
1665 |
by (induct_tac "no" 2); |
|
1666 |
by (auto_tac (claset() addIs [real_le_trans], |
|
1667 |
simpset() addsimps [real_diff_def])); |
|
10677 | 1668 |
by (asm_full_simp_tac (simpset() addsimps [inst "r" "f ?x + ?y" abs_real_def]) 1); |
1669 |
by (dres_inst_tac [("i"," f (no + n)"),("no1","no")] |
|
1670 |
(lemma_f_mono_add RSN (2,real_less_le_trans)) 1); |
|
1671 |
by (subgoal_tac "g(no + n) <= lim g" 3); |
|
10648 | 1672 |
by (induct_tac "no" 4); |
1673 |
by (auto_tac (claset() addIs [real_le_trans], |
|
1674 |
simpset() addsimps [real_diff_def,abs_minus_add_cancel])); |
|
10677 | 1675 |
by (asm_full_simp_tac (simpset() addsimps [inst "r" "lim g + ?y" abs_real_def]) 2); |
1676 |
by (cut_inst_tac [("f", "%x. -(g x)"), ("m","n"), ("no","no")] |
|
1677 |
lemma_f_mono_add 2); |
|
1678 |
by (auto_tac (claset(), simpset() addsimps [add_commute])); |
|
10648 | 1679 |
qed "lemma_nest"; |
1680 |
||
1681 |
Goal "[| ALL n. f(n) <= f(Suc n); \ |
|
1682 |
\ ALL n. g(Suc n) <= g(n); \ |
|
1683 |
\ ALL n. f(n) <= g(n); \ |
|
1684 |
\ (%n. f(n) - g(n)) ----> 0 |] \ |
|
1685 |
\ ==> EX l. ((ALL n. f(n) <= l) & f ----> l) & \ |
|
1686 |
\ ((ALL n. l <= g(n)) & g ----> l)"; |
|
1687 |
by (dtac lemma_nest 1 THEN Auto_tac); |
|
1688 |
by (subgoal_tac "l = m" 1); |
|
1689 |
by (dres_inst_tac [("X","f")] LIMSEQ_diff 2); |
|
1690 |
by (auto_tac (claset() addIs [LIMSEQ_unique],simpset())); |
|
1691 |
qed "lemma_nest_unique"; |
|
1692 |
||
1693 |
Goal "EX! fn. (fn 0 = e) & (ALL n. fn (Suc n) = f n (fn n))"; |
|
1694 |
by (rtac ex1I 1); |
|
1695 |
by (rtac conjI 1 THEN rtac allI 2); |
|
1696 |
by (rtac def_nat_rec_0 1 THEN rtac def_nat_rec_Suc 2); |
|
1697 |
by (Auto_tac); |
|
1698 |
by (rtac ext 1 THEN induct_tac "n" 1); |
|
1699 |
by (Auto_tac); |
|
1700 |
qed "nat_Axiom"; |
|
1701 |
||
1702 |
||
10677 | 1703 |
(*?????????OBSOLETE????????***) |
10648 | 1704 |
Goal "((x::real) = y / (#2 * z)) = (#2 * x = y/z)"; |
1705 |
by Auto_tac; |
|
1706 |
by (dres_inst_tac [("f","%u. (#1/#2)*u")] arg_cong 1); |
|
1707 |
by Auto_tac; |
|
1708 |
qed "eq_divide_2_times_iff"; |
|
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1709 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1710 |
|
10648 | 1711 |
val [prem1,prem2] = |
1712 |
Goal "[| !!a b c. [| a <= b; b <= c; P(a,b); P(b,c)|] ==> P(a,c); \ |
|
1713 |
\ ALL x. EX d::real. 0 < d & \ |
|
1714 |
\ (ALL a b. a <= x & x <= b & (b - a) < d --> P(a,b)) \ |
|
1715 |
\ |] ==> ALL a b. a <= b --> P(a,b)"; |
|
1716 |
by (Step_tac 1); |
|
1717 |
by (cut_inst_tac [("e","(a,b)"), |
|
1718 |
("f","%n fn. if P(fst fn, (fst fn + snd fn)/#2) \ |
|
1719 |
\ then ((fst fn + snd fn) /#2,snd fn) \ |
|
1720 |
\ else (fst fn,(fst fn + snd fn)/#2)")] |
|
1721 |
nat_Axiom 1); |
|
1722 |
by (etac ex1E 1 THEN etac conjE 1 THEN rtac ccontr 1); |
|
1723 |
(* set up 1st premise of lemma_nest_unique *) |
|
1724 |
by (subgoal_tac "ALL n. fst((fn::nat =>(real*real)) n) <= snd (fn n)" 1); |
|
1725 |
by (rtac allI 2); |
|
1726 |
by (induct_tac "n" 2); |
|
1727 |
by (Asm_simp_tac 2); |
|
1728 |
by (dres_inst_tac [("x","na")] spec 2); |
|
1729 |
by (case_tac "P (fst (fn na), (fst (fn na) + snd (fn na)) / #2)" 2); |
|
1730 |
by (Asm_simp_tac 3); |
|
1731 |
by (Asm_simp_tac 2); |
|
1732 |
(* 2nd premise *) |
|
1733 |
by (subgoal_tac "ALL n. fst(fn n) <= fst (fn (Suc n))" 1); |
|
1734 |
by (rtac allI 2); |
|
1735 |
by (induct_tac "n" 2); |
|
1736 |
by (dres_inst_tac [("x","0")] spec 2); |
|
1737 |
by (Asm_simp_tac 3); (*super slow, but proved!*) |
|
1738 |
by (Asm_simp_tac 2); |
|
1739 |
(* 3rd premise? [lcp] *) |
|
1740 |
by (subgoal_tac "ALL n. ~P(fst(fn n),snd(fn n))" 1); |
|
1741 |
by (rtac allI 2); |
|
1742 |
by (induct_tac "n" 2); |
|
1743 |
by (Asm_simp_tac 2); |
|
1744 |
by (Asm_simp_tac 2 THEN rtac impI 2); |
|
1745 |
by (rtac ccontr 2); |
|
1746 |
by (res_inst_tac [("Pa","P (fst (fn na), snd (fn na))")] swap 2); |
|
1747 |
by (assume_tac 2); |
|
1748 |
by (res_inst_tac [("b","(fst(fn na) + snd(fn na))/#2")] prem1 2); |
|
1749 |
by (Asm_full_simp_tac 4); |
|
1750 |
by (Asm_full_simp_tac 4); |
|
1751 |
by (Asm_full_simp_tac 2); |
|
1752 |
by (Asm_simp_tac 2); |
|
1753 |
(* 3rd premise [looks like the 4th to lcp!] *) |
|
1754 |
by (subgoal_tac "ALL n. snd(fn (Suc n)) <= snd (fn n)" 1); |
|
1755 |
by (rtac allI 2); |
|
1756 |
by (induct_tac "n" 2); |
|
1757 |
by (Asm_simp_tac 2); |
|
1758 |
by (Asm_simp_tac 2); |
|
1759 |
(* FIXME: simplification takes a very long time! *) |
|
1760 |
by (ALLGOALS(thin_tac "ALL y. \ |
|
1761 |
\ y 0 = (a, b) & \ |
|
1762 |
\ (ALL n. \ |
|
1763 |
\ y (Suc n) = \ |
|
1764 |
\ (if P (fst (y n), (fst (y n) + snd (y n)) /#2) \ |
|
1765 |
\ then ((fst (y n) + snd (y n)) /#2, snd (y n)) \ |
|
1766 |
\ else (fst (y n),\ |
|
1767 |
\ (fst (y n) + snd (y n)) /#2))) --> \ |
|
1768 |
\ y = fn")); |
|
1769 |
(*another premise? the 5th? lcp*) |
|
1770 |
by (subgoal_tac "ALL n. snd(fn n) - fst(fn n) = \ |
|
1771 |
\ (b - a) * inverse(#2 ^ n)" 1); |
|
1772 |
by (rtac allI 2); |
|
1773 |
by (induct_tac "n" 2); |
|
1774 |
by (Asm_simp_tac 2); |
|
1775 |
by (asm_full_simp_tac |
|
1776 |
(simpset() addsimps [eq_divide_2_times_iff, real_inverse_eq_divide, |
|
1777 |
real_diff_mult_distrib2]) 2); |
|
1778 |
(*end of the premises [lcp]*) |
|
1779 |
by (dtac lemma_nest_unique 1); |
|
1780 |
by (REPEAT(assume_tac 1)); |
|
1781 |
by (Step_tac 2); |
|
1782 |
by (rtac LIMSEQ_minus_cancel 1); |
|
1783 |
by (asm_simp_tac (simpset() addsimps [CLAIM "-((a::real) - b) = (b - a)", |
|
1784 |
realpow_inverse]) 1); |
|
1785 |
by (subgoal_tac "#0 = (b-a) * #0" 1); |
|
1786 |
by (eres_inst_tac [("t","#0")] ssubst 1); |
|
1787 |
by (rtac (LIMSEQ_const RS LIMSEQ_mult) 1); |
|
1788 |
by (rtac LIMSEQ_realpow_zero 1); |
|
1789 |
by (Asm_full_simp_tac 3); |
|
1790 |
by (EVERY1[Simp_tac, Simp_tac]); |
|
1791 |
by (cut_facts_tac [prem2] 1); |
|
1792 |
by (dres_inst_tac [("x","l")] spec 1 THEN Step_tac 1); |
|
1793 |
by (rewtac LIMSEQ_def); |
|
1794 |
by (dres_inst_tac [("x","d/#2")] spec 1); |
|
1795 |
by (dres_inst_tac [("x","d/#2")] spec 1); |
|
1796 |
by (dtac real_less_half_sum 1); |
|
1797 |
by (thin_tac "ALL n. \ |
|
1798 |
\ fn (Suc n) = \ |
|
1799 |
\ (if P (fst (fn n), (fst (fn n) + snd (fn n))/#2) \ |
|
1800 |
\ then ((fst (fn n) + snd (fn n))/#2, snd (fn n)) \ |
|
1801 |
\ else (fst (fn n), \ |
|
1802 |
\ (fst (fn n) + snd (fn n))/#2))" 1); |
|
1803 |
by (Step_tac 1); |
|
1804 |
by (ALLGOALS(Asm_full_simp_tac)); |
|
1805 |
by (dres_inst_tac [("x","fst(fn (no + noa))")] spec 1); |
|
1806 |
by (dres_inst_tac [("x","snd(fn (no + noa))")] spec 1); |
|
1807 |
by (Step_tac 1); |
|
1808 |
by (ALLGOALS(Asm_full_simp_tac)); |
|
1809 |
by (res_inst_tac [("j","abs(fst(fn(no + noa)) - l) + \ |
|
1810 |
\ abs(snd(fn(no + noa)) - l)")] real_le_less_trans 1); |
|
1811 |
by (rtac (real_sum_of_halves RS subst) 2); |
|
1812 |
by (rewtac real_diff_def); |
|
1813 |
by (rtac real_add_less_mono 2); |
|
1814 |
||
1815 |
by (Asm_full_simp_tac 2); |
|
1816 |
by (Asm_full_simp_tac 2); |
|
1817 |
by (res_inst_tac [("y1","fst(fn (no + noa)) ")] |
|
1818 |
(abs_minus_add_cancel RS subst) 1); |
|
1819 |
by (subgoal_tac "#0 <= (l + - fst (fn (no + noa)))" 1); |
|
1820 |
by (subgoal_tac "#0 <= (snd (fn (no + noa)) + - l)" 1); |
|
1821 |
by (asm_simp_tac (simpset() addsimps [abs_eqI1] @ real_add_ac) 1); |
|
10677 | 1822 |
by (ALLGOALS(asm_simp_tac (simpset() addsimps real_add_divide_distrib::real_add_ac))); |
10648 | 1823 |
qed "lemma_BOLZANO"; |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1824 |
|
10648 | 1825 |
Goal "((ALL a b c. (a <= b & b <= c & P(a,b) & P(b,c)) --> P(a,c)) & \ |
1826 |
\ (ALL x. EX d::real. 0 < d & \ |
|
1827 |
\ (ALL a b. a <= x & x <= b & (b - a) < d --> P(a,b)))) \ |
|
1828 |
\ --> (ALL a b. a <= b --> P(a,b))"; |
|
1829 |
by (Step_tac 1); |
|
1830 |
by (rtac (lemma_BOLZANO RS allE) 1); |
|
1831 |
by (assume_tac 2); |
|
1832 |
by (ALLGOALS(Blast_tac)); |
|
1833 |
qed "lemma_BOLZANO2"; |
|
1834 |
||
1835 |
(*----------------------------------------------------------------------------*) |
|
1836 |
(* Intermediate Value Theorem (prove contrapositive by bisection) *) |
|
1837 |
(*----------------------------------------------------------------------------*) |
|
1838 |
||
1839 |
Goal "[| f(a) <= y & y <= f(b); \ |
|
1840 |
\ a <= b; \ |
|
1841 |
\ (ALL x. a <= x & x <= b --> isCont f x) \ |
|
1842 |
\ |] ==> EX x. a <= x & x <= b & f(x) = y"; |
|
10659 | 1843 |
by (rtac contrapos_pp 1); |
1844 |
by (assume_tac 1); |
|
10648 | 1845 |
by (cut_inst_tac [("P","%(u,v). a <= u & u <= v & v <= b --> \ |
1846 |
\ ~(f(u) <= y & y <= f(v))")] lemma_BOLZANO2 1); |
|
1847 |
by (Step_tac 1); |
|
1848 |
by (ALLGOALS(Asm_full_simp_tac)); |
|
1849 |
by (Blast_tac 2); |
|
1850 |
by (asm_full_simp_tac (simpset() addsimps [isCont_iff,LIM_def]) 1); |
|
1851 |
by (rtac ccontr 1); |
|
1852 |
by (subgoal_tac "a <= x & x <= b" 1); |
|
1853 |
by (Asm_full_simp_tac 2); |
|
1854 |
by (rotate_tac 3 2); |
|
1855 |
by (dres_inst_tac [("x","1r")] spec 2); |
|
1856 |
by (Step_tac 2); |
|
10659 | 1857 |
by (Asm_full_simp_tac 2); |
1858 |
by (Asm_full_simp_tac 2); |
|
10648 | 1859 |
by (REPEAT(blast_tac (claset() addIs [real_le_trans]) 2)); |
1860 |
by (REPEAT(dres_inst_tac [("x","x")] spec 1)); |
|
1861 |
by (Asm_full_simp_tac 1); |
|
1862 |
(**) |
|
1863 |
by (rotate_tac 4 1); |
|
1864 |
by (dres_inst_tac [("x","abs(y - f x)")] spec 1); |
|
1865 |
by (Step_tac 1); |
|
1866 |
by (asm_full_simp_tac (simpset() addsimps [real_less_le, |
|
1867 |
abs_ge_zero,real_diff_def]) 1); |
|
1868 |
by (dtac (sym RS (abs_zero_iff RS iffD2)) 1); |
|
1869 |
by (arith_tac 1); |
|
1870 |
by (dres_inst_tac [("x","s")] spec 1); |
|
1871 |
by (Step_tac 1); |
|
1872 |
by (cut_inst_tac [("R1.0","f x"),("R2.0","y")] real_linear 1); |
|
1873 |
by (Auto_tac); |
|
1874 |
by (dres_inst_tac [("x","ba - x")] spec 1); |
|
10659 | 1875 |
by (auto_tac (claset(), |
1876 |
simpset() |
|
1877 |
addsimps [abs_iff, real_diff_le_eq, |
|
1878 |
(real_diff_def RS meta_eq_to_obj_eq) RS sym, |
|
1879 |
real_less_le_iff, real_le_diff_eq, |
|
1880 |
CLAIM "(~(x::real) <= y) = (y < x)", |
|
10648 | 1881 |
CLAIM "(-x < -y) = ((y::real) < x)", |
10659 | 1882 |
CLAIM "(-(y - x)) = (x - (y::real))", |
1883 |
CLAIM "(x-y=#0) = (x = (y::real))"])); |
|
10648 | 1884 |
by (dres_inst_tac [("x","aa - x")] spec 1); |
1885 |
by (case_tac "x <= aa" 1); |
|
1886 |
by (auto_tac (claset(),simpset() addsimps [real_minus_zero_le_iff2, |
|
1887 |
real_le_diff_eq,real_diff_le_eq,(real_le_def RS meta_eq_to_obj_eq) |
|
1888 |
RS sym])); |
|
1889 |
by (dres_inst_tac [("z","x"),("w","aa")] real_le_anti_sym 1); |
|
1890 |
by (assume_tac 1 THEN Asm_full_simp_tac 1); |
|
1891 |
qed "IVT"; |
|
1892 |
||
1893 |
Goal "[| f(b) <= y & y <= f(a); \ |
|
1894 |
\ a <= b; \ |
|
1895 |
\ (ALL x. a <= x & x <= b --> isCont f x) \ |
|
1896 |
\ |] ==> EX x. a <= x & x <= b & f(x) = y"; |
|
1897 |
by (subgoal_tac "- f a <= -y & -y <= - f b" 1); |
|
1898 |
by (thin_tac "f b <= y & y <= f a" 1); |
|
1899 |
by (dres_inst_tac [("f","%x. - f x")] IVT 1); |
|
1900 |
by (auto_tac (claset() addIs [isCont_minus],simpset())); |
|
1901 |
qed "IVT2"; |
|
1902 |
||
1903 |
Goal "(f(a) <= y & y <= f(b) & a <= b & \ |
|
1904 |
\ (ALL x. a <= x & x <= b --> isCont f x)) \ |
|
1905 |
\ --> (EX x. a <= x & x <= b & f(x) = y)"; |
|
1906 |
by (blast_tac (claset() addIs [IVT]) 1); |
|
1907 |
qed "IVT_objl"; |
|
1908 |
||
1909 |
Goal "(f(b) <= y & y <= f(a) & a <= b & \ |
|
1910 |
\ (ALL x. a <= x & x <= b --> isCont f x)) \ |
|
1911 |
\ --> (EX x. a <= x & x <= b & f(x) = y)"; |
|
1912 |
by (blast_tac (claset() addIs [IVT2]) 1); |
|
1913 |
qed "IVT2_objl"; |
|
1914 |
||
1915 |
(*----------------------------------------------------------------------------*) |
|
1916 |
(* By bisection, function continuous on closed interval is bounded above *) |
|
1917 |
(*----------------------------------------------------------------------------*) |
|
1918 |
||
1919 |
Goal "abs (real_of_nat x) = real_of_nat x"; |
|
1920 |
by (auto_tac (claset() addIs [abs_eqI1],simpset())); |
|
1921 |
qed "abs_real_of_nat_cancel"; |
|
1922 |
Addsimps [abs_real_of_nat_cancel]; |
|
1923 |
||
1924 |
Goal "~ abs(x) + 1r < x"; |
|
1925 |
by (rtac real_leD 1); |
|
1926 |
by (auto_tac (claset() addIs [abs_ge_self RS real_le_trans],simpset())); |
|
1927 |
qed "abs_add_one_not_less_self"; |
|
1928 |
Addsimps [abs_add_one_not_less_self]; |
|
1929 |
||
1930 |
||
1931 |
Goal "[| a <= b; ALL x. a <= x & x <= b --> isCont f x |]\ |
|
1932 |
\ ==> EX M. ALL x. a <= x & x <= b --> f(x) <= M"; |
|
1933 |
by (cut_inst_tac [("P","%(u,v). a <= u & u <= v & v <= b --> \ |
|
1934 |
\ (EX M. ALL x. u <= x & x <= v --> f x <= M)")] |
|
1935 |
lemma_BOLZANO2 1); |
|
1936 |
by (Step_tac 1); |
|
1937 |
by (ALLGOALS(Asm_full_simp_tac)); |
|
1938 |
by (cut_inst_tac [("x","M"),("y","Ma")] |
|
1939 |
(CLAIM "x <= y | y <= (x::real)") 1); |
|
1940 |
by (Step_tac 1); |
|
1941 |
by (res_inst_tac [("x","Ma")] exI 1); |
|
1942 |
by (Step_tac 1); |
|
1943 |
by (cut_inst_tac [("x","xb"),("y","xa")] |
|
1944 |
(CLAIM "x <= y | y <= (x::real)") 1); |
|
1945 |
by (Step_tac 1); |
|
1946 |
by (rtac real_le_trans 1 THEN assume_tac 2); |
|
1947 |
by (Asm_full_simp_tac 1); |
|
1948 |
by (Asm_full_simp_tac 1); |
|
1949 |
by (res_inst_tac [("x","M")] exI 1); |
|
1950 |
by (Step_tac 1); |
|
1951 |
by (cut_inst_tac [("x","xb"),("y","xa")] |
|
1952 |
(CLAIM "x <= y | y <= (x::real)") 1); |
|
1953 |
by (Step_tac 1); |
|
1954 |
by (Asm_full_simp_tac 1); |
|
1955 |
by (rtac real_le_trans 1 THEN assume_tac 2); |
|
1956 |
by (Asm_full_simp_tac 1); |
|
1957 |
by (case_tac "a <= x & x <= b" 1); |
|
1958 |
by (res_inst_tac [("x","#1")] exI 2); |
|
1959 |
by (auto_tac (claset(), |
|
1960 |
simpset() addsimps [LIM_def,isCont_iff])); |
|
1961 |
by (dres_inst_tac [("x","x")] spec 1 THEN Auto_tac); |
|
1962 |
by (thin_tac "ALL M. EX x. a <= x & x <= b & ~ f x <= M" 1); |
|
1963 |
by (dres_inst_tac [("x","#1")] spec 1); |
|
1964 |
by Auto_tac; |
|
1965 |
by (res_inst_tac [("x","s")] exI 1 THEN Auto_tac); |
|
1966 |
by (res_inst_tac [("x","abs(f x) + #1")] exI 1 THEN Step_tac 1); |
|
1967 |
by (dres_inst_tac [("x","xa - x")] spec 1 THEN Auto_tac); |
|
1968 |
by (res_inst_tac [("j","abs(f xa)")] real_le_trans 3); |
|
1969 |
by (res_inst_tac [("j","abs(f x) + abs(f(xa) - f(x))")] real_le_trans 4); |
|
1970 |
by (auto_tac (claset() addIs [abs_triangle_ineq RSN (2, real_le_trans)], |
|
1971 |
simpset() addsimps [real_diff_def,abs_ge_self])); |
|
1972 |
(*arith_tac problem: this step should not be needed*) |
|
1973 |
by (asm_full_simp_tac (simpset() addsimps [rename_numerals (real_eq_minus_iff2 RS sym)]) 1); |
|
1974 |
by (auto_tac (claset(), |
|
1975 |
simpset() addsimps [abs_real_def] addsplits [split_if_asm])); |
|
1976 |
qed "isCont_bounded"; |
|
1977 |
||
1978 |
(*----------------------------------------------------------------------------*) |
|
1979 |
(* Refine the above to existence of least upper bound *) |
|
1980 |
(*----------------------------------------------------------------------------*) |
|
1981 |
||
1982 |
Goal "((EX x. x : S) & (EX y. isUb UNIV S (y::real))) --> \ |
|
1983 |
\ (EX t. isLub UNIV S t)"; |
|
1984 |
by (blast_tac (claset() addIs [reals_complete]) 1); |
|
1985 |
qed "lemma_reals_complete"; |
|
1986 |
||
1987 |
Goal "[| a <= b; ALL x. a <= x & x <= b --> isCont f x |] \ |
|
1988 |
\ ==> EX M. (ALL x. a <= x & x <= b --> f(x) <= M) & \ |
|
1989 |
\ (ALL N. N < M --> (EX x. a <= x & x <= b & N < f(x)))"; |
|
1990 |
by (cut_inst_tac [("S","Collect (%y. EX x. a <= x & x <= b & y = f x)")] |
|
1991 |
lemma_reals_complete 1); |
|
1992 |
by (Auto_tac); |
|
1993 |
by (dtac isCont_bounded 1 THEN assume_tac 1); |
|
1994 |
by (auto_tac (claset(),simpset() addsimps [isUb_def,leastP_def, |
|
1995 |
isLub_def,setge_def,setle_def])); |
|
1996 |
by (rtac exI 1 THEN Auto_tac); |
|
1997 |
by (REPEAT(dtac spec 1) THEN Auto_tac); |
|
1998 |
by (dres_inst_tac [("x","x")] spec 1); |
|
1999 |
by (auto_tac (claset() addSIs [real_leI],simpset())); |
|
2000 |
qed "isCont_has_Ub"; |
|
2001 |
||
2002 |
(*----------------------------------------------------------------------------*) |
|
2003 |
(* Now show that it attains its upper bound *) |
|
2004 |
(*----------------------------------------------------------------------------*) |
|
10662
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2005 |
|
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2006 |
Goal "[| a <= b; ALL x. a <= x & x <= b --> isCont f x |] \ |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2007 |
\ ==> EX M. (ALL x. a <= x & x <= b --> f(x) <= M) & \ |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2008 |
\ (EX x. a <= x & x <= b & f(x) = M)"; |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2009 |
by (ftac isCont_has_Ub 1 THEN assume_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2010 |
by (Clarify_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2011 |
by (res_inst_tac [("x","M")] exI 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2012 |
by (Asm_full_simp_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2013 |
by (rtac ccontr 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2014 |
by (subgoal_tac "ALL x. a <= x & x <= b --> f x < M" 1 THEN Step_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2015 |
by (rtac ccontr 2 THEN dtac real_leI 2); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2016 |
by (dres_inst_tac [("z","M")] real_le_anti_sym 2); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2017 |
by (REPEAT(Blast_tac 2)); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2018 |
by (subgoal_tac "ALL x. a <= x & x <= b --> isCont (%x. inverse(M - f x)) x" 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2019 |
by (Step_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2020 |
by (EVERY[rtac isCont_inverse 2, rtac isCont_diff 2, rtac notI 4]); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2021 |
by (ALLGOALS(asm_full_simp_tac (simpset() addsimps [real_diff_eq_eq]))); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2022 |
by (Blast_tac 2); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2023 |
by (subgoal_tac |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2024 |
"EX k. ALL x. a <= x & x <= b --> (%x. inverse(M - (f x))) x <= k" 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2025 |
by (rtac isCont_bounded 2); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2026 |
by (Step_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2027 |
by (subgoal_tac "ALL x. a <= x & x <= b --> #0 < inverse(M - f(x))" 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2028 |
by (Asm_full_simp_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2029 |
by (Step_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2030 |
by (asm_full_simp_tac (simpset() addsimps [real_less_diff_eq]) 2); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2031 |
by (subgoal_tac |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2032 |
"ALL x. a <= x & x <= b --> (%x. inverse(M - (f x))) x < (k + #1)" 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2033 |
by (Step_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2034 |
by (res_inst_tac [("j","k")] real_le_less_trans 2); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2035 |
by (asm_full_simp_tac (simpset() addsimps [real_zero_less_one]) 3); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2036 |
by (Asm_full_simp_tac 2); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2037 |
by (subgoal_tac "ALL x. a <= x & x <= b --> \ |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2038 |
\ inverse(k + #1) < inverse((%x. inverse(M - (f x))) x)" 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2039 |
by (Step_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2040 |
by (rtac real_inverse_less_swap 2); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2041 |
by (ALLGOALS Asm_full_simp_tac); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2042 |
by (dres_inst_tac [("P", "%N. N<M --> ?Q N"), |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2043 |
("x","M - inverse(k + #1)")] spec 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2044 |
by (Step_tac 1 THEN dtac real_leI 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2045 |
by (dtac (real_le_diff_eq RS iffD1) 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2046 |
by (REPEAT(dres_inst_tac [("x","a")] spec 1)); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2047 |
by (Asm_full_simp_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2048 |
by (asm_full_simp_tac |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2049 |
(simpset() addsimps [real_inverse_eq_divide, pos_real_divide_le_eq]) 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2050 |
by (cut_inst_tac [("x","k"),("y","M-f a")] real_0_less_mult_iff 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2051 |
by (Asm_full_simp_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2052 |
(*last one*) |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2053 |
by (REPEAT(dres_inst_tac [("x","x")] spec 1)); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2054 |
by (Asm_full_simp_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2055 |
qed "isCont_eq_Ub"; |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2056 |
|
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2057 |
|
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2058 |
(*----------------------------------------------------------------------------*) |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2059 |
(* Same theorem for lower bound *) |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2060 |
(*----------------------------------------------------------------------------*) |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2061 |
|
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2062 |
Goal "[| a <= b; ALL x. a <= x & x <= b --> isCont f x |] \ |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2063 |
\ ==> EX M. (ALL x. a <= x & x <= b --> M <= f(x)) & \ |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2064 |
\ (EX x. a <= x & x <= b & f(x) = M)"; |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2065 |
by (subgoal_tac "ALL x. a <= x & x <= b --> isCont (%x. -(f x)) x" 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2066 |
by (blast_tac (claset() addIs [isCont_minus]) 2); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2067 |
by (dres_inst_tac [("f","(%x. -(f x))")] isCont_eq_Ub 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2068 |
by (Step_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2069 |
by (Auto_tac); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2070 |
qed "isCont_eq_Lb"; |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2071 |
|
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2072 |
|
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2073 |
(* ------------------------------------------------------------------------- *) |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2074 |
(* Another version. *) |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2075 |
(* ------------------------------------------------------------------------- *) |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2076 |
|
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2077 |
Goal "[|a <= b; ALL x. a <= x & x <= b --> isCont f x |] \ |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2078 |
\ ==> EX L M. (ALL x. a <= x & x <= b --> L <= f(x) & f(x) <= M) & \ |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2079 |
\ (ALL y. L <= y & y <= M --> (EX x. a <= x & x <= b & (f(x) = y)))"; |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2080 |
by (ftac isCont_eq_Lb 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2081 |
by (ftac isCont_eq_Ub 2); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2082 |
by (REPEAT(assume_tac 1)); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2083 |
by (Step_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2084 |
by (res_inst_tac [("x","f x")] exI 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2085 |
by (res_inst_tac [("x","f xa")] exI 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2086 |
by (Asm_full_simp_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2087 |
by (Step_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2088 |
by (cut_inst_tac [("x","x"),("y","xa")] (CLAIM "x <= y | y <= (x::real)") 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2089 |
by (Step_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2090 |
by (cut_inst_tac [("f","f"),("a","x"),("b","xa"),("y","y")] IVT_objl 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2091 |
by (cut_inst_tac [("f","f"),("a","xa"),("b","x"),("y","y")] IVT2_objl 2); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2092 |
by (Step_tac 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2093 |
by (res_inst_tac [("x","xb")] exI 2); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2094 |
by (res_inst_tac [("x","xb")] exI 4); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2095 |
by (ALLGOALS(Asm_full_simp_tac)); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2096 |
qed "isCont_Lb_Ub"; |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2097 |
|
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2098 |
(*----------------------------------------------------------------------------*) |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2099 |
(* If f'(x) > 0 then x is locally strictly increasing at the right *) |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2100 |
(*----------------------------------------------------------------------------*) |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2101 |
|
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2102 |
Goalw [deriv_def,LIM_def] |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2103 |
"[| DERIV f x :> l; #0 < l |] ==> \ |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2104 |
\ EX d. #0 < d & (ALL h. #0 < h & h < d --> f(x) < f(x + h))"; |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2105 |
by (dtac spec 1 THEN Auto_tac); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2106 |
by (res_inst_tac [("x","s")] exI 1 THEN Auto_tac); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2107 |
by (subgoal_tac "#0 < l*h" 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2108 |
by (asm_full_simp_tac (simpset() addsimps [real_0_less_mult_iff]) 2); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2109 |
by (dres_inst_tac [("x","h")] spec 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2110 |
by (asm_full_simp_tac |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2111 |
(simpset() addsimps [abs_real_def, real_inverse_eq_divide, |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2112 |
pos_real_le_divide_eq, pos_real_less_divide_eq] |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2113 |
addsplits [split_if_asm]) 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2114 |
qed "DERIV_left_inc"; |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2115 |
|
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2116 |
Goalw [deriv_def,LIM_def] |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2117 |
"[| DERIV f x :> l; l < #0 |] ==> \ |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2118 |
\ EX d. #0 < d & (ALL h. #0 < h & h < d --> f(x) < f(x - h))"; |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2119 |
by (dres_inst_tac [("x","-l")] spec 1 THEN Auto_tac); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2120 |
by (res_inst_tac [("x","s")] exI 1 THEN Auto_tac); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2121 |
by (subgoal_tac "l*h < #0" 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2122 |
by (asm_full_simp_tac (simpset() addsimps [real_mult_less_0_iff]) 2); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2123 |
by (dres_inst_tac [("x","-h")] spec 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2124 |
by (asm_full_simp_tac |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2125 |
(simpset() addsimps [abs_real_def, real_inverse_eq_divide, |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2126 |
pos_real_less_divide_eq, |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2127 |
symmetric real_diff_def] |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2128 |
addsplits [split_if_asm]) 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2129 |
by (subgoal_tac "#0 < (f (x - h) - f x)/h" 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2130 |
by (arith_tac 2); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2131 |
by (asm_full_simp_tac |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2132 |
(simpset() addsimps [pos_real_less_divide_eq]) 1); |
cf6be1804912
new material, including default simprules that should be introduced earlier
paulson
parents:
10659
diff
changeset
|
2133 |
qed "DERIV_left_dec"; |
10670 | 2134 |
|
2135 |
||
2136 |
Goal "[| DERIV f x :> l; \ |
|
2137 |
\ EX d. #0 < d & (ALL y. abs(x - y) < d --> f(y) <= f(x)) |] \ |
|
2138 |
\ ==> l = #0"; |
|
2139 |
by (res_inst_tac [("R1.0","l"),("R2.0","#0")] real_linear_less2 1); |
|
2140 |
by (Step_tac 1); |
|
2141 |
by (dtac DERIV_left_dec 1); |
|
2142 |
by (dtac DERIV_left_inc 3); |
|
2143 |
by (Step_tac 1); |
|
2144 |
by (dres_inst_tac [("d1.0","d"),("d2.0","da")] real_lbound_gt_zero 1); |
|
2145 |
by (dres_inst_tac [("d1.0","d"),("d2.0","da")] real_lbound_gt_zero 3); |
|
2146 |
by (Step_tac 1); |
|
2147 |
by (dres_inst_tac [("x","x - e")] spec 1); |
|
2148 |
by (dres_inst_tac [("x","x + e")] spec 2); |
|
2149 |
by (auto_tac (claset(),simpset() addsimps [abs_eqI2,abs_minus_cancel])); |
|
2150 |
qed "DERIV_local_max"; |
|
2151 |
||
2152 |
(*----------------------------------------------------------------------------*) |
|
2153 |
(* Similar theorem for a local minimum *) |
|
2154 |
(*----------------------------------------------------------------------------*) |
|
2155 |
||
2156 |
Goal "[| DERIV f x :> l; \ |
|
2157 |
\ EX d::real. #0 < d & (ALL y. abs(x - y) < d --> f(x) <= f(y)) |] \ |
|
2158 |
\ ==> l = #0"; |
|
2159 |
by (dtac (DERIV_minus RS DERIV_local_max) 1); |
|
2160 |
by Auto_tac; |
|
2161 |
qed "DERIV_local_min"; |
|
2162 |
||
2163 |
(*----------------------------------------------------------------------------*) |
|
2164 |
(* In particular if a function is locally flat *) |
|
2165 |
(*----------------------------------------------------------------------------*) |
|
2166 |
||
2167 |
Goal "[| DERIV f x :> l; \ |
|
2168 |
\ EX d. #0 < d & (ALL y. abs(x - y) < d --> f(x) = f(y)) |] \ |
|
2169 |
\ ==> l = #0"; |
|
2170 |
by (auto_tac (claset() addSDs [DERIV_local_max],simpset())); |
|
2171 |
qed "DERIV_local_const"; |
|
2172 |
||
2173 |
(*----------------------------------------------------------------------------*) |
|
2174 |
(* Lemma about introducing open ball in open interval *) |
|
2175 |
(*----------------------------------------------------------------------------*) |
|
2176 |
||
2177 |
Goal "[| a < x; x < b |] ==> \ |
|
2178 |
\ EX d::real. #0 < d & (ALL y. abs(x - y) < d --> a < y & y < b)"; |
|
2179 |
by (simp_tac (simpset() addsimps [abs_interval_iff]) 1); |
|
2180 |
by (cut_inst_tac [("x","x - a"),("y","b - x")] |
|
2181 |
(CLAIM "x <= y | y <= (x::real)") 1); |
|
2182 |
by (Step_tac 1); |
|
2183 |
by (res_inst_tac [("x","x - a")] exI 1); |
|
2184 |
by (res_inst_tac [("x","b - x")] exI 2); |
|
2185 |
by (Auto_tac); |
|
2186 |
by (auto_tac (claset(),simpset() addsimps [real_less_diff_eq])); |
|
2187 |
qed "lemma_interval_lt"; |
|
2188 |
||
2189 |
Goal "[| a < x; x < b |] ==> \ |
|
2190 |
\ EX d::real. #0 < d & (ALL y. abs(x - y) < d --> a <= y & y <= b)"; |
|
2191 |
by (dtac lemma_interval_lt 1); |
|
2192 |
by (Auto_tac); |
|
2193 |
by (auto_tac (claset() addSIs [exI] ,simpset())); |
|
2194 |
qed "lemma_interval"; |
|
2195 |
||
2196 |
(*----------------------------------------------------------------------- |
|
2197 |
Rolle's Theorem |
|
2198 |
If f is defined and continuous on the finite closed interval [a,b] |
|
2199 |
and differentiable a least on the open interval (a,b), and f(a) = f(b), |
|
2200 |
then x0 : (a,b) such that f'(x0) = #0 |
|
2201 |
----------------------------------------------------------------------*) |
|
2202 |
||
2203 |
Goal "[| a < b; f(a) = f(b); \ |
|
2204 |
\ ALL x. a <= x & x <= b --> isCont f x; \ |
|
2205 |
\ ALL x. a < x & x < b --> f differentiable x \ |
|
2206 |
\ |] ==> EX z. a < z & z < b & DERIV f z :> #0"; |
|
2207 |
by (ftac (real_less_imp_le RS isCont_eq_Ub) 1); |
|
2208 |
by (EVERY1[assume_tac,Step_tac]); |
|
2209 |
by (ftac (real_less_imp_le RS isCont_eq_Lb) 1); |
|
2210 |
by (EVERY1[assume_tac,Step_tac]); |
|
2211 |
by (case_tac "a < x & x < b" 1 THEN etac conjE 1); |
|
2212 |
by (Asm_full_simp_tac 2); |
|
2213 |
by (forw_inst_tac [("a","a"),("x","x")] lemma_interval 1); |
|
2214 |
by (EVERY1[assume_tac,etac exE]); |
|
2215 |
by (res_inst_tac [("x","x")] exI 1 THEN Asm_full_simp_tac 1); |
|
2216 |
by (subgoal_tac "(EX l. DERIV f x :> l) & \ |
|
2217 |
\ (EX d. #0 < d & (ALL y. abs(x - y) < d --> f(y) <= f(x)))" 1); |
|
2218 |
by (Clarify_tac 1 THEN rtac conjI 2); |
|
2219 |
by (blast_tac (claset() addIs [differentiableD]) 2); |
|
2220 |
by (Blast_tac 2); |
|
2221 |
by (ftac DERIV_local_max 1); |
|
2222 |
by (EVERY1[Blast_tac,Blast_tac]); |
|
2223 |
by (case_tac "a < xa & xa < b" 1 THEN etac conjE 1); |
|
2224 |
by (Asm_full_simp_tac 2); |
|
2225 |
by (forw_inst_tac [("a","a"),("x","xa")] lemma_interval 1); |
|
2226 |
by (EVERY1[assume_tac,etac exE]); |
|
2227 |
by (res_inst_tac [("x","xa")] exI 1 THEN Asm_full_simp_tac 1); |
|
2228 |
by (subgoal_tac "(EX l. DERIV f xa :> l) & \ |
|
2229 |
\ (EX d. #0 < d & (ALL y. abs(xa - y) < d --> f(xa) <= f(y)))" 1); |
|
2230 |
by (Clarify_tac 1 THEN rtac conjI 2); |
|
2231 |
by (blast_tac (claset() addIs [differentiableD]) 2); |
|
2232 |
by (Blast_tac 2); |
|
2233 |
by (ftac DERIV_local_min 1); |
|
2234 |
by (EVERY1[Blast_tac,Blast_tac]); |
|
2235 |
by (subgoal_tac "ALL x. a <= x & x <= b --> f(x) = f(b)" 1); |
|
2236 |
by (Clarify_tac 2); |
|
2237 |
by (rtac real_le_anti_sym 2); |
|
2238 |
by (subgoal_tac "f b = f x" 2); |
|
2239 |
by (Asm_full_simp_tac 2); |
|
2240 |
by (res_inst_tac [("x1","a"),("y1","x")] (real_le_imp_less_or_eq RS disjE) 2); |
|
2241 |
by (assume_tac 2); |
|
2242 |
by (dres_inst_tac [("z","x"),("w","b")] real_le_anti_sym 2); |
|
2243 |
by (subgoal_tac "f b = f xa" 5); |
|
2244 |
by (Asm_full_simp_tac 5); |
|
2245 |
by (res_inst_tac [("x1","a"),("y1","xa")] (real_le_imp_less_or_eq RS disjE) 5); |
|
2246 |
by (assume_tac 5); |
|
2247 |
by (dres_inst_tac [("z","xa"),("w","b")] real_le_anti_sym 5); |
|
2248 |
by (REPEAT(Asm_full_simp_tac 2)); |
|
2249 |
by (dtac real_dense 1 THEN etac exE 1); |
|
2250 |
by (res_inst_tac [("x","r")] exI 1 THEN Asm_full_simp_tac 1); |
|
2251 |
by (etac conjE 1); |
|
2252 |
by (forw_inst_tac [("a","a"),("x","r")] lemma_interval 1); |
|
2253 |
by (EVERY1[assume_tac, etac exE]); |
|
2254 |
by (subgoal_tac "(EX l. DERIV f r :> l) & \ |
|
2255 |
\ (EX d. #0 < d & (ALL y. abs(r - y) < d --> f(r) = f(y)))" 1); |
|
2256 |
by (Clarify_tac 1 THEN rtac conjI 2); |
|
2257 |
by (blast_tac (claset() addIs [differentiableD]) 2); |
|
2258 |
by (EVERY1[ftac DERIV_local_const, Blast_tac, Blast_tac]); |
|
2259 |
by (res_inst_tac [("x","d")] exI 1); |
|
2260 |
by (EVERY1[rtac conjI, Blast_tac, rtac allI, rtac impI]); |
|
2261 |
by (res_inst_tac [("s","f b")] trans 1); |
|
2262 |
by (blast_tac (claset() addSDs [real_less_imp_le]) 1); |
|
2263 |
by (rtac sym 1 THEN Blast_tac 1); |
|
2264 |
qed "Rolle"; |
|
2265 |
||
2266 |
(*----------------------------------------------------------------------------*) |
|
2267 |
(* Mean value theorem *) |
|
2268 |
(*----------------------------------------------------------------------------*) |
|
2269 |
||
2270 |
||
10677 | 2271 |
(*????????TO BE OBSOLETE?????????*) |
10670 | 2272 |
Goal "k~=#0 ==> (k*m) / k = (m::real)"; |
2273 |
by (dres_inst_tac [("n","#1")] real_mult_div_cancel1 1); |
|
2274 |
by (Asm_full_simp_tac 1); |
|
2275 |
qed "real_mult_div_self1"; |
|
2276 |
Addsimps [real_mult_div_self1]; |
|
2277 |
||
2278 |
||
10677 | 2279 |
(**???FIXME: replace by simproc*??*) |
2280 |
Goal "h ~= (0::real) ==> (h*x)/h = x"; |
|
2281 |
by (asm_full_simp_tac |
|
2282 |
(simpset() addsimps [real_divide_mult2_self_eq, |
|
2283 |
real_mult_commute]) 1); |
|
2284 |
qed "real_divide_mult3_self_eq"; |
|
2285 |
Delsimps [real_divide_mult3_self_eq]; |
|
10670 | 2286 |
|
2287 |
||
2288 |
Goal "f a - (f b - f a)/(b - a) * a = \ |
|
2289 |
\ f b - (f b - f a)/(b - a) * (b::real)"; |
|
2290 |
by (case_tac "a = b" 1); |
|
2291 |
by (asm_full_simp_tac (simpset() addsimps [REAL_DIVIDE_ZERO]) 1); |
|
2292 |
by (res_inst_tac [("c1","b - a")] (real_mult_left_cancel RS iffD1) 1); |
|
2293 |
by (arith_tac 1); |
|
2294 |
by (auto_tac (claset(), |
|
2295 |
simpset() addsimps [real_diff_mult_distrib2, real_diff_eq_eq, |
|
10677 | 2296 |
inst "a" "a" eq_commute])); |
10670 | 2297 |
by (auto_tac (claset(), |
10677 | 2298 |
simpset() addsimps [real_diff_mult_distrib2, real_mult_commute])); |
10670 | 2299 |
qed "lemma_MVT"; |