src/HOL/Library/Euclidean_Space.thy
 author chaieb Mon Feb 09 17:21:46 2009 +0000 (2009-02-09) changeset 29847 af32126ee729 parent 29844 4ac95212efcc child 29881 58f3c48dbbb7 permissions -rw-r--r--
1 (* Title:      Library/Euclidean_Space
2    ID:         \$Id:
3    Author:     Amine Chaieb, University of Cambridge
4 *)
6 header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*}
8 theory Euclidean_Space
9   imports "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Complex_Main
10   Finite_Cartesian_Product Glbs Infinite_Set Numeral_Type
11   uses ("normarith.ML")
12 begin
14 text{* Some common special cases.*}
16 lemma forall_1: "(\<forall>(i::'a::{order,one}). 1 <= i \<and> i <= 1 --> P i) \<longleftrightarrow> P 1"
17   by (metis order_eq_iff)
18 lemma forall_dimindex_1: "(\<forall>i \<in> {1..dimindex(UNIV:: 1 set)}. P i) \<longleftrightarrow> P 1"
21 lemma forall_2: "(\<forall>(i::nat). 1 <= i \<and> i <= 2 --> P i) \<longleftrightarrow> P 1 \<and> P 2"
22 proof-
23   have "\<And>i::nat. 1 <= i \<and> i <= 2 \<longleftrightarrow> i = 1 \<or> i = 2" by arith
24   thus ?thesis by metis
25 qed
27 lemma forall_3: "(\<forall>(i::nat). 1 <= i \<and> i <= 3 --> P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
28 proof-
29   have "\<And>i::nat. 1 <= i \<and> i <= 3 \<longleftrightarrow> i = 1 \<or> i = 2 \<or> i = 3" by arith
30   thus ?thesis by metis
31 qed
33 lemma setsum_singleton[simp]: "setsum f {x} = f x" by simp
34 lemma setsum_1: "setsum f {(1::'a::{order,one})..1} = f 1"
37 lemma setsum_2: "setsum f {1::nat..2} = f 1 + f 2"
40 lemma setsum_3: "setsum f {1::nat..3} = f 1 + f 2 + f 3"
43 section{* Basic componentwise operations on vectors. *}
45 instantiation "^" :: (plus,type) plus
46 begin
47 definition  vector_add_def : "op + \<equiv> (\<lambda> x y.  (\<chi> i. (x\$i) + (y\$i)))"
48 instance ..
49 end
51 instantiation "^" :: (times,type) times
52 begin
53   definition vector_mult_def : "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x\$i) * (y\$i)))"
54   instance ..
55 end
57 instantiation "^" :: (minus,type) minus begin
58   definition vector_minus_def : "op - \<equiv> (\<lambda> x y.  (\<chi> i. (x\$i) - (y\$i)))"
59 instance ..
60 end
62 instantiation "^" :: (uminus,type) uminus begin
63   definition vector_uminus_def : "uminus \<equiv> (\<lambda> x.  (\<chi> i. - (x\$i)))"
64 instance ..
65 end
66 instantiation "^" :: (zero,type) zero begin
67   definition vector_zero_def : "0 \<equiv> (\<chi> i. 0)"
68 instance ..
69 end
71 instantiation "^" :: (one,type) one begin
72   definition vector_one_def : "1 \<equiv> (\<chi> i. 1)"
73 instance ..
74 end
76 instantiation "^" :: (ord,type) ord
77  begin
78 definition vector_less_eq_def:
79   "less_eq (x :: 'a ^'b) y = (ALL i : {1 .. dimindex (UNIV :: 'b set)}.
80   x\$i <= y\$i)"
81 definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i : {1 ..
82   dimindex (UNIV :: 'b set)}. x\$i < y\$i)"
84 instance by (intro_classes)
85 end
87 text{* Also the scalar-vector multiplication. FIXME: We should unify this with the scalar multiplication in real_vector *}
89 definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'n" (infixr "*s" 75)
90   where "c *s x = (\<chi> i. c * (x\$i))"
92 text{* Constant Vectors *}
94 definition "vec x = (\<chi> i. x)"
96 text{* Dot products. *}
98 definition dot :: "'a::{comm_monoid_add, times} ^ 'n \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a" (infix "\<bullet>" 70) where
99   "x \<bullet> y = setsum (\<lambda>i. x\$i * y\$i) {1 .. dimindex (UNIV:: 'n set)}"
100 lemma dot_1[simp]: "(x::'a::{comm_monoid_add, times}^1) \<bullet> y = (x\$1) * (y\$1)"
101   by (simp add: dot_def dimindex_def)
103 lemma dot_2[simp]: "(x::'a::{comm_monoid_add, times}^2) \<bullet> y = (x\$1) * (y\$1) + (x\$2) * (y\$2)"
104   by (simp add: dot_def dimindex_def nat_number)
106 lemma dot_3[simp]: "(x::'a::{comm_monoid_add, times}^3) \<bullet> y = (x\$1) * (y\$1) + (x\$2) * (y\$2) + (x\$3) * (y\$3)"
107   by (simp add: dot_def dimindex_def nat_number)
109 section {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *}
111 lemmas Cart_lambda_beta' = Cart_lambda_beta[rule_format]
112 method_setup vector = {*
113 let
114   val ss1 = HOL_basic_ss addsimps [@{thm dot_def}, @{thm setsum_addf} RS sym,
115   @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
116   @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym]
117   val ss2 = @{simpset} addsimps
119               @{thm vector_minus_def}, @{thm vector_uminus_def},
120               @{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def},
121               @{thm Cart_lambda_beta'}, @{thm vector_scalar_mult_def}]
122  fun vector_arith_tac ths =
123    simp_tac ss1
124    THEN' (fn i => rtac @{thm setsum_cong2} i
125          ORELSE rtac @{thm setsum_0'} i
126          ORELSE simp_tac (HOL_basic_ss addsimps [@{thm "Cart_eq"}]) i)
127    (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
128    THEN' asm_full_simp_tac (ss2 addsimps ths)
129  in
130   Method.thms_args (Method.SIMPLE_METHOD' o vector_arith_tac)
131 end
132 *} "Lifts trivial vector statements to real arith statements"
134 lemma vec_0[simp]: "vec 0 = 0" by (vector vector_zero_def)
135 lemma vec_1[simp]: "vec 1 = 1" by (vector vector_one_def)
139 text{* Obvious "component-pushing". *}
141 lemma vec_component: " i \<in> {1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (vec x :: 'a ^ 'n)\$i = x"
142   by (vector vec_def)
145   fixes x y :: "'a::{plus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
146   shows "(x + y)\$i = x\$i + y\$i"
147   using i by vector
149 lemma vector_minus_component:
150   fixes x y :: "'a::{minus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
151   shows "(x - y)\$i = x\$i - y\$i"
152   using i  by vector
154 lemma vector_mult_component:
155   fixes x y :: "'a::{times} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
156   shows "(x * y)\$i = x\$i * y\$i"
157   using i by vector
159 lemma vector_smult_component:
160   fixes y :: "'a::{times} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
161   shows "(c *s y)\$i = c * (y\$i)"
162   using i by vector
164 lemma vector_uminus_component:
165   fixes x :: "'a::{uminus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
166   shows "(- x)\$i = - (x\$i)"
167   using i by vector
169 lemma cond_component: "(if b then x else y)\$i = (if b then x\$i else y\$i)" by vector
171 lemmas vector_component = vec_component vector_add_component vector_mult_component vector_smult_component vector_minus_component vector_uminus_component cond_component
173 subsection {* Some frequently useful arithmetic lemmas over vectors. *}
176   apply (intro_classes) by (vector add_assoc)
180   apply (intro_classes) by vector+
183   apply (intro_classes) by (vector algebra_simps)+
186   apply (intro_classes) by (vector add_commute)
189   apply (intro_classes) by vector
192   apply (intro_classes) by vector+
195   apply (intro_classes)
196   by (vector Cart_eq)+
199   apply (intro_classes)
200   by (vector Cart_eq)
202 instance "^" :: (semigroup_mult,type) semigroup_mult
203   apply (intro_classes) by (vector mult_assoc)
205 instance "^" :: (monoid_mult,type) monoid_mult
206   apply (intro_classes) by vector+
208 instance "^" :: (ab_semigroup_mult,type) ab_semigroup_mult
209   apply (intro_classes) by (vector mult_commute)
211 instance "^" :: (ab_semigroup_idem_mult,type) ab_semigroup_idem_mult
212   apply (intro_classes) by (vector mult_idem)
214 instance "^" :: (comm_monoid_mult,type) comm_monoid_mult
215   apply (intro_classes) by vector
217 fun vector_power :: "('a::{one,times} ^'n) \<Rightarrow> nat \<Rightarrow> 'a^'n" where
218   "vector_power x 0 = 1"
219   | "vector_power x (Suc n) = x * vector_power x n"
221 instantiation "^" :: (recpower,type) recpower
222 begin
223   definition vec_power_def: "op ^ \<equiv> vector_power"
224   instance
225   apply (intro_classes) by (simp_all add: vec_power_def)
226 end
228 instance "^" :: (semiring,type) semiring
229   apply (intro_classes) by (vector ring_simps)+
231 instance "^" :: (semiring_0,type) semiring_0
232   apply (intro_classes) by (vector ring_simps)+
233 instance "^" :: (semiring_1,type) semiring_1
234   apply (intro_classes) apply vector using dimindex_ge_1 by auto
235 instance "^" :: (comm_semiring,type) comm_semiring
236   apply (intro_classes) by (vector ring_simps)+
238 instance "^" :: (comm_semiring_0,type) comm_semiring_0 by (intro_classes)
239 instance "^" :: (semiring_0_cancel,type) semiring_0_cancel by (intro_classes)
240 instance "^" :: (comm_semiring_0_cancel,type) comm_semiring_0_cancel by (intro_classes)
241 instance "^" :: (ring,type) ring by (intro_classes)
242 instance "^" :: (semiring_1_cancel,type) semiring_1_cancel by (intro_classes)
243 instance "^" :: (comm_semiring_1,type) comm_semiring_1 by (intro_classes)
244 lemma of_nat_index:
245   "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (of_nat n :: 'a::semiring_1 ^'n)\$i = of_nat n"
246   apply (induct n)
247   apply vector
248   apply vector
249   done
250 lemma zero_index[simp]:
251   "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (0 :: 'a::zero ^'n)\$i = 0" by vector
253 lemma one_index[simp]:
254   "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (1 :: 'a::one ^'n)\$i = 1" by vector
256 lemma one_plus_of_nat_neq_0: "(1::'a::semiring_char_0) + of_nat n \<noteq> 0"
257 proof-
258   have "(1::'a) + of_nat n = 0 \<longleftrightarrow> of_nat 1 + of_nat n = (of_nat 0 :: 'a)" by simp
259   also have "\<dots> \<longleftrightarrow> 1 + n = 0" by (simp only: of_nat_add[symmetric] of_nat_eq_iff)
260   finally show ?thesis by simp
261 qed
263 instance "^" :: (semiring_char_0,type) semiring_char_0
264 proof (intro_classes)
265   fix m n ::nat
266   show "(of_nat m :: 'a^'b) = of_nat n \<longleftrightarrow> m = n"
267   proof(induct m arbitrary: n)
268     case 0 thus ?case apply vector
269       apply (induct n,auto simp add: ring_simps)
270       using dimindex_ge_1 apply auto
271       apply vector
272       by (auto simp add: of_nat_index one_plus_of_nat_neq_0)
273   next
274     case (Suc n m)
275     thus ?case  apply vector
276       apply (induct m, auto simp add: ring_simps of_nat_index zero_index)
277       using dimindex_ge_1 apply simp apply blast
279       using dimindex_ge_1 apply simp apply blast
280       apply (simp add: vector_component one_index of_nat_index)
281       apply (simp only: of_nat.simps(2)[where ?'a = 'a, symmetric] of_nat_eq_iff)
282       using  dimindex_ge_1 apply simp apply blast
283       apply (simp add: vector_component one_index of_nat_index)
284       apply (simp only: of_nat.simps(2)[where ?'a = 'a, symmetric] of_nat_eq_iff)
285       using dimindex_ge_1 apply simp apply blast
286       apply (simp add: vector_component one_index of_nat_index)
287       done
288   qed
289 qed
291 instance "^" :: (comm_ring_1,type) comm_ring_1 by intro_classes
292   (* FIXME!!! Why does the axclass package complain here !!*)
293 (* instance "^" :: (ring_char_0,type) ring_char_0 by intro_classes *)
295 lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
296   by (vector mult_assoc)
297 lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
298   by (vector ring_simps)
299 lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
300   by (vector ring_simps)
301 lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
302 lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
303 lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
304   by (vector ring_simps)
305 lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
306 lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
307 lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector
308 lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
309 lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
310   by (vector ring_simps)
312 lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
313   apply (auto simp add: vec_def Cart_eq vec_component Cart_lambda_beta )
314   using dimindex_ge_1 apply auto done
316 subsection{* Properties of the dot product.  *}
318 lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \<bullet> y = y \<bullet> x"
319   by (vector mult_commute)
320 lemma dot_ladd: "((x::'a::ring ^ 'n) + y) \<bullet> z = (x \<bullet> z) + (y \<bullet> z)"
321   by (vector ring_simps)
322 lemma dot_radd: "x \<bullet> (y + (z::'a::ring ^ 'n)) = (x \<bullet> y) + (x \<bullet> z)"
323   by (vector ring_simps)
324 lemma dot_lsub: "((x::'a::ring ^ 'n) - y) \<bullet> z = (x \<bullet> z) - (y \<bullet> z)"
325   by (vector ring_simps)
326 lemma dot_rsub: "(x::'a::ring ^ 'n) \<bullet> (y - z) = (x \<bullet> y) - (x \<bullet> z)"
327   by (vector ring_simps)
328 lemma dot_lmult: "(c *s x) \<bullet> y = (c::'a::ring) * (x \<bullet> y)" by (vector ring_simps)
329 lemma dot_rmult: "x \<bullet> (c *s y) = (c::'a::comm_ring) * (x \<bullet> y)" by (vector ring_simps)
330 lemma dot_lneg: "(-x) \<bullet> (y::'a::ring ^ 'n) = -(x \<bullet> y)" by vector
331 lemma dot_rneg: "(x::'a::ring ^ 'n) \<bullet> (-y) = -(x \<bullet> y)" by vector
332 lemma dot_lzero[simp]: "0 \<bullet> x = (0::'a::{comm_monoid_add, mult_zero})" by vector
333 lemma dot_rzero[simp]: "x \<bullet> 0 = (0::'a::{comm_monoid_add, mult_zero})" by vector
334 lemma dot_pos_le[simp]: "(0::'a\<Colon>ordered_ring_strict) <= x \<bullet> x"
335   by (simp add: dot_def setsum_nonneg)
337 lemma setsum_squares_eq_0_iff: assumes fS: "finite F" and fp: "\<forall>x \<in> F. f x \<ge> (0 ::'a::pordered_ab_group_add)" shows "setsum f F = 0 \<longleftrightarrow> (ALL x:F. f x = 0)"
338 using fS fp setsum_nonneg[OF fp]
339 proof (induct set: finite)
340   case empty thus ?case by simp
341 next
342   case (insert x F)
343   from insert.prems have Fx: "f x \<ge> 0" and Fp: "\<forall> a \<in> F. f a \<ge> 0" by simp_all
344   from insert.hyps Fp setsum_nonneg[OF Fp]
345   have h: "setsum f F = 0 \<longleftrightarrow> (\<forall>a \<in>F. f a = 0)" by metis
346   from sum_nonneg_eq_zero_iff[OF Fx  setsum_nonneg[OF Fp]] insert.hyps(1,2)
347   show ?case by (simp add: h)
348 qed
350 lemma dot_eq_0: "x \<bullet> x = 0 \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) = 0"
351 proof-
352   {assume f: "finite (UNIV :: 'n set)"
353     let ?S = "{Suc 0 .. card (UNIV :: 'n set)}"
354     have fS: "finite ?S" using f by simp
355     have fp: "\<forall> i\<in> ?S. x\$i * x\$i>= 0" by simp
356     have ?thesis by (vector dimindex_def f setsum_squares_eq_0_iff[OF fS fp])}
357   moreover
358   {assume "\<not> finite (UNIV :: 'n set)" then have ?thesis by (vector dimindex_def)}
359   ultimately show ?thesis by metis
360 qed
362 lemma dot_pos_lt: "(0 < x \<bullet> x) \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) \<noteq> 0" using dot_eq_0[of x] dot_pos_le[of x]
363   by (auto simp add: le_less)
365 subsection {* Introduce norms, but defer many properties till we get square roots. *}
366 text{* FIXME : This is ugly *}
368   real_of_real_def [code inline, simp]: "real == id"
370 instantiation "^" :: ("{times, comm_monoid_add}", type) norm begin
371 definition  real_vector_norm_def: "norm \<equiv> (\<lambda>x. sqrt (real (x \<bullet> x)))"
372 instance ..
373 end
376 subsection{* The collapse of the general concepts to dimention one. *}
378 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x\$1))"
379   by (vector dimindex_def)
381 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
382   apply auto
383   apply (erule_tac x= "x\$1" in allE)
384   apply (simp only: vector_one[symmetric])
385   done
387 lemma norm_real: "norm(x::real ^ 1) = abs(x\$1)"
390 text{* Metric *}
392 definition dist:: "real ^ 'n \<Rightarrow> real ^ 'n \<Rightarrow> real" where
393   "dist x y = norm (x - y)"
395 lemma dist_real: "dist(x::real ^ 1) y = abs((x\$1) - (y\$1))"
396   using dimindex_ge_1[of "UNIV :: 1 set"]
397   by (auto simp add: norm_real dist_def vector_component Cart_lambda_beta[where ?'a = "1"] )
399 subsection {* A connectedness or intermediate value lemma with several applications. *}
401 lemma connected_real_lemma:
402   fixes f :: "real \<Rightarrow> real ^ 'n"
403   assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
404   and dst: "\<And>e x. a <= x \<Longrightarrow> x <= b \<Longrightarrow> 0 < e ==> \<exists>d > 0. \<forall>y. abs(y - x) < d \<longrightarrow> dist(f y) (f x) < e"
405   and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
406   and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
407   and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
408   shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
409 proof-
410   let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
411   have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa)
412   have Sub: "\<exists>y. isUb UNIV ?S y"
413     apply (rule exI[where x= b])
414     using ab fb e12 by (auto simp add: isUb_def setle_def)
415   from reals_complete[OF Se Sub] obtain l where
416     l: "isLub UNIV ?S l"by blast
417   have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
418     apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
419     by (metis linorder_linear)
420   have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
421     apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
422     by (metis linorder_linear not_le)
423     have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
424     have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
425     have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by dlo
426     {assume le2: "f l \<in> e2"
427       from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
428       hence lap: "l - a > 0" using alb by arith
429       from e2[rule_format, OF le2] obtain e where
430 	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
431       from dst[OF alb e(1)] obtain d where
432 	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
433       have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" using lap d(1)
434 	apply ferrack by arith
435       then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
436       from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
437       from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
438       moreover
439       have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
440       ultimately have False using e12 alb d' by auto}
441     moreover
442     {assume le1: "f l \<in> e1"
443     from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
444       hence blp: "b - l > 0" using alb by arith
445       from e1[rule_format, OF le1] obtain e where
446 	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
447       from dst[OF alb e(1)] obtain d where
448 	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
449       have "\<exists>d'. d' < d \<and> d' >0" using d(1) by dlo
450       then obtain d' where d': "d' > 0" "d' < d" by metis
451       from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
452       hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
453       with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
454       with l d' have False
455 	by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
456     ultimately show ?thesis using alb by metis
457 qed
459 text{* One immediately useful corollary is the existence of square roots! --- Should help to get rid of all the development of square-root for reals as a special case real ^1 *}
461 lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
462 proof-
463   have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
464   thus ?thesis by (simp add: ring_simps power2_eq_square)
465 qed
467 lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
468   using isCont_power[OF isCont_ident, of 2, unfolded isCont_def LIM_def, rule_format, of e x] apply (auto simp add: power2_eq_square)
469   apply (rule_tac x="s" in exI)
470   apply auto
471   apply (erule_tac x=y in allE)
472   apply auto
473   done
475 lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y^2 ==> sqrt x <= y"
476   using real_sqrt_le_iff[of x "y^2"] by simp
478 lemma real_le_rsqrt: "x^2 \<le> y \<Longrightarrow> x \<le> sqrt y"
479   using real_sqrt_le_mono[of "x^2" y] by simp
481 lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
482   using real_sqrt_less_mono[of "x^2" y] by simp
484 lemma sqrt_even_pow2: assumes n: "even n"
485   shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
486 proof-
487   from n obtain m where m: "n = 2*m" unfolding even_nat_equiv_def2
488     by (auto simp add: nat_number)
489   from m  have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
490     by (simp only: power_mult[symmetric] mult_commute)
491   then show ?thesis  using m by simp
492 qed
494 lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
495   apply (cases "x = 0", simp_all)
496   using sqrt_divide_self_eq[of x]
497   apply (simp add: inverse_eq_divide real_sqrt_ge_0_iff field_simps)
498   done
500 text{* Hence derive more interesting properties of the norm. *}
502 lemma norm_0: "norm (0::real ^ 'n) = 0"
503   by (simp add: real_vector_norm_def dot_eq_0)
505 lemma norm_pos_le: "0 <= norm (x::real^'n)"
506   by (simp add: real_vector_norm_def dot_pos_le)
507 lemma norm_neg: " norm(-x) = norm (x:: real ^ 'n)"
508   by (simp add: real_vector_norm_def dot_lneg dot_rneg)
509 lemma norm_sub: "norm(x - y) = norm(y - (x::real ^ 'n))"
510   by (metis norm_neg minus_diff_eq)
511 lemma norm_mul: "norm(a *s x) = abs(a) * norm x"
512   by (simp add: real_vector_norm_def dot_lmult dot_rmult mult_assoc[symmetric] real_sqrt_mult)
513 lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (x \<bullet> x = (0::real))"
515 lemma norm_eq_0: "norm x = 0 \<longleftrightarrow> x = (0::real ^ 'n)"
516   by (simp add: real_vector_norm_def dot_eq_0)
517 lemma norm_pos_lt: "0 < norm x \<longleftrightarrow> x \<noteq> (0::real ^ 'n)"
518   by (metis less_le real_vector_norm_def norm_pos_le norm_eq_0)
519 lemma norm_pow_2: "norm x ^ 2 = x \<bullet> x"
520   by (simp add: real_vector_norm_def dot_pos_le)
521 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_0)
522 lemma norm_le_0: "norm x <= 0 \<longleftrightarrow> x = (0::real ^'n)"
523   by (metis norm_eq_0 norm_pos_le order_antisym)
524 lemma vector_mul_eq_0: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
525   by vector
526 lemma vector_mul_lcancel: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
527   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
528 lemma vector_mul_rcancel: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
529   by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
530 lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
531   by (metis vector_mul_lcancel)
532 lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
533   by (metis vector_mul_rcancel)
534 lemma norm_cauchy_schwarz: "x \<bullet> y <= norm x * norm y"
535 proof-
536   {assume "norm x = 0"
537     hence ?thesis by (simp add: norm_eq_0 dot_lzero dot_rzero norm_0)}
538   moreover
539   {assume "norm y = 0"
540     hence ?thesis by (simp add: norm_eq_0 dot_lzero dot_rzero norm_0)}
541   moreover
542   {assume h: "norm x \<noteq> 0" "norm y \<noteq> 0"
543     let ?z = "norm y *s x - norm x *s y"
544     from h have p: "norm x * norm y > 0" by (metis norm_pos_le le_less zero_compare_simps)
545     from dot_pos_le[of ?z]
546     have "(norm x * norm y) * (x \<bullet> y) \<le> norm x ^2 * norm y ^2"
547       apply (simp add: dot_rsub dot_lsub dot_lmult dot_rmult ring_simps)
548       by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym)
549     hence "x\<bullet>y \<le> (norm x ^2 * norm y ^2) / (norm x * norm y)" using p
551     hence ?thesis using h by (simp add: power2_eq_square)}
552   ultimately show ?thesis by metis
553 qed
555 lemma norm_abs[simp]: "abs (norm x) = norm (x::real ^'n)"
556   using norm_pos_le[of x] by (simp add: real_abs_def linorder_linear)
558 lemma norm_cauchy_schwarz_abs: "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"
559   using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"]
560   by (simp add: real_abs_def dot_rneg norm_neg)
561 lemma norm_triangle: "norm(x + y) <= norm x + norm (y::real ^'n)"
562   unfolding real_vector_norm_def
563   apply (rule real_le_lsqrt)
567     by (simp add: norm_pow_2[symmetric] power2_eq_square ring_simps norm_cauchy_schwarz)
569 lemma norm_triangle_sub: "norm (x::real ^'n) <= norm(y) + norm(x - y)"
570   using norm_triangle[of "y" "x - y"] by (simp add: ring_simps)
571 lemma norm_triangle_le: "norm(x::real ^'n) + norm y <= e ==> norm(x + y) <= e"
572   by (metis order_trans norm_triangle)
573 lemma norm_triangle_lt: "norm(x::real ^'n) + norm(y) < e ==> norm(x + y) < e"
574   by (metis basic_trans_rules(21) norm_triangle)
576 lemma setsum_delta:
577   assumes fS: "finite S"
578   shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
579 proof-
580   let ?f = "(\<lambda>k. if k=a then b k else 0)"
581   {assume a: "a \<notin> S"
582     hence "\<forall> k\<in> S. ?f k = 0" by simp
583     hence ?thesis  using a by simp}
584   moreover
585   {assume a: "a \<in> S"
586     let ?A = "S - {a}"
587     let ?B = "{a}"
588     have eq: "S = ?A \<union> ?B" using a by blast
589     have dj: "?A \<inter> ?B = {}" by simp
590     from fS have fAB: "finite ?A" "finite ?B" by auto
591     have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
592       using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
593       by simp
594     then have ?thesis  using a by simp}
595   ultimately show ?thesis by blast
596 qed
598 lemma component_le_norm: "i \<in> {1 .. dimindex(UNIV :: 'n set)} ==> \<bar>x\$i\<bar> <= norm (x::real ^ 'n)"
599 proof(simp add: real_vector_norm_def, rule real_le_rsqrt, clarsimp)
600   assume i: "Suc 0 \<le> i" "i \<le> dimindex (UNIV :: 'n set)"
601   let ?S = "{1 .. dimindex(UNIV :: 'n set)}"
602   let ?f = "(\<lambda>k. if k = i then x\$i ^2 else 0)"
603   have fS: "finite ?S" by simp
604   from i setsum_delta[OF fS, of i "\<lambda>k. x\$i ^ 2"]
605   have th: "x\$i^2 = setsum ?f ?S" by simp
606   let ?g = "\<lambda>k. x\$k * x\$k"
607   {fix x assume x: "x \<in> ?S" have "?f x \<le> ?g x" by (simp add: power2_eq_square)}
608   with setsum_mono[of ?S ?f ?g]
609   have "setsum ?f ?S \<le> setsum ?g ?S" by blast
610   then show "x\$i ^2 \<le> x \<bullet> (x:: real ^ 'n)" unfolding dot_def th[symmetric] .
611 qed
612 lemma norm_bound_component_le: "norm(x::real ^ 'n) <= e
613                 ==> \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. \<bar>x\$i\<bar> <= e"
614   by (metis component_le_norm order_trans)
616 lemma norm_bound_component_lt: "norm(x::real ^ 'n) < e
617                 ==> \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. \<bar>x\$i\<bar> < e"
618   by (metis component_le_norm basic_trans_rules(21))
620 lemma norm_le_l1: "norm (x:: real ^'n) <= setsum(\<lambda>i. \<bar>x\$i\<bar>) {1..dimindex(UNIV::'n set)}"
622   case 0 thus ?case by simp
623 next
624   case (Suc n)
625   have th: "2 * (\<bar>x\$(Suc n)\<bar> * (\<Sum>i = Suc 0..n. \<bar>x\$i\<bar>)) \<ge> 0"
626     apply simp
627     apply (rule mult_nonneg_nonneg)
630   from Suc
631   show ?case using th by (simp add: power2_eq_square ring_simps)
632 qed
634 lemma real_abs_norm: "\<bar> norm x\<bar> = norm (x :: real ^'n)"
636 lemma real_abs_sub_norm: "\<bar>norm(x::real ^'n) - norm y\<bar> <= norm(x - y)"
637   apply (simp add: abs_le_iff ring_simps)
638   by (metis norm_triangle_sub norm_sub)
639 lemma norm_le: "norm(x::real ^ 'n) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
641 lemma norm_lt: "norm(x::real ^'n) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
643 lemma norm_eq: "norm (x::real ^'n) = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
644   by (simp add: order_eq_iff norm_le)
645 lemma norm_eq_1: "norm(x::real ^ 'n) = 1 \<longleftrightarrow> x \<bullet> x = 1"
648 text{* Squaring equations and inequalities involving norms.  *}
650 lemma dot_square_norm: "x \<bullet> x = norm(x)^2"
651   by (simp add: real_vector_norm_def  dot_pos_le )
653 lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
654 proof-
655   have th: "\<And>x y::real. x^2 = y^2 \<longleftrightarrow> x = y \<or> x = -y" by algebra
656   show ?thesis using norm_pos_le[of x]
657   apply (simp add: dot_square_norm th)
658   apply arith
659   done
660 qed
662 lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
663 proof-
664   have "x^2 \<le> y^2 \<longleftrightarrow> (x -y) * (y + x) \<le> 0" by (simp add: ring_simps power2_eq_square)
665   also have "\<dots> \<longleftrightarrow> \<bar>x\<bar> \<le> \<bar>y\<bar>" apply (simp add: zero_compare_simps real_abs_def not_less) by arith
666 finally show ?thesis ..
667 qed
669 lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
670   using norm_pos_le[of x]
671   apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
672   apply arith
673   done
675 lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2"
676   using norm_pos_le[of x]
677   apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
678   apply arith
679   done
681 lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a^2"
682   by (metis not_le norm_ge_square)
683 lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a^2"
684   by (metis norm_le_square not_less)
686 text{* Dot product in terms of the norm rather than conversely. *}
688 lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"
691 lemma dot_norm_neg: "x \<bullet> y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2"
695 text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
697 lemma vector_eq: "(x:: real ^ 'n) = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y\<and> y \<bullet> y = x \<bullet> x" (is "?lhs \<longleftrightarrow> ?rhs")
698 proof
699   assume "?lhs" then show ?rhs by simp
700 next
701   assume ?rhs
702   then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y\<bullet> y = 0" by simp
703   hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
704     by (simp add: dot_rsub dot_lsub dot_sym)
705   then have "(x - y) \<bullet> (x - y) = 0" by (simp add: ring_simps dot_lsub dot_rsub)
706   then show "x = y" by (simp add: dot_eq_0)
707 qed
710 subsection{* General linear decision procedure for normed spaces. *}
712 lemma norm_cmul_rule_thm: "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(c *s x)"
713   apply (clarsimp simp add: norm_mul)
714   apply (rule mult_mono1)
715   apply simp_all
716   done
718 lemma norm_add_rule_thm: "b1 >= norm(x1 :: real ^'n) \<Longrightarrow> b2 >= norm(x2) ==> b1 + b2 >= norm(x1 + x2)"
719   apply (rule norm_triangle_le) by simp
721 lemma ge_iff_diff_ge_0: "(a::'a::ordered_ring) \<ge> b == a - b \<ge> 0"
724 lemma pth_1: "(x::real^'n) == 1 *s x" by (simp only: vector_smult_lid)
725 lemma pth_2: "x - (y::real^'n) == x + -y" by (atomize (full)) simp
726 lemma pth_3: "(-x::real^'n) == -1 *s x" by vector
727 lemma pth_4: "0 *s (x::real^'n) == 0" "c *s 0 = (0::real ^ 'n)" by vector+
728 lemma pth_5: "c *s (d *s x) == (c * d) *s (x::real ^ 'n)" by (atomize (full)) vector
729 lemma pth_6: "(c::real) *s (x + y) == c *s x + c *s y" by (atomize (full)) (vector ring_simps)
730 lemma pth_7: "0 + x == (x::real^'n)" "x + 0 == x" by simp_all
731 lemma pth_8: "(c::real) *s x + d *s x == (c + d) *s x" by (atomize (full)) (vector ring_simps)
732 lemma pth_9: "((c::real) *s x + z) + d *s x == (c + d) *s x + z"
733   "c *s x + (d *s x + z) == (c + d) *s x + z"
734   "(c *s x + w) + (d *s x + z) == (c + d) *s x + (w + z)" by ((atomize (full)), vector ring_simps)+
735 lemma pth_a: "(0::real) *s x + y == y" by (atomize (full)) vector
736 lemma pth_b: "(c::real) *s x + d *s y == c *s x + d *s y"
737   "(c *s x + z) + d *s y == c *s x + (z + d *s y)"
738   "c *s x + (d *s y + z) == c *s x + (d *s y + z)"
739   "(c *s x + w) + (d *s y + z) == c *s x + (w + (d *s y + z))"
740   by ((atomize (full)), vector)+
741 lemma pth_c: "(c::real) *s x + d *s y == d *s y + c *s x"
742   "(c *s x + z) + d *s y == d *s y + (c *s x + z)"
743   "c *s x + (d *s y + z) == d *s y + (c *s x + z)"
744   "(c *s x + w) + (d *s y + z) == d *s y + ((c *s x + w) + z)" by ((atomize (full)), vector)+
745 lemma pth_d: "x + (0::real ^'n) == x" by (atomize (full)) vector
747 lemma norm_imp_pos_and_ge: "norm (x::real ^ 'n) == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
748   by (atomize) (auto simp add: norm_pos_le)
750 lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
752 lemma norm_pths:
753   "(x::real ^'n) = y \<longleftrightarrow> norm (x - y) \<le> 0"
754   "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
755   using norm_pos_le[of "x - y"] by (auto simp add: norm_0 norm_eq_0)
757 use "normarith.ML"
759 method_setup norm = {* Method.ctxt_args (Method.SIMPLE_METHOD' o NormArith.norm_arith_tac)
760 *} "Proves simple linear statements about vector norms"
764 text{* Hence more metric properties. *}
766 lemma dist_refl: "dist x x = 0" by norm
768 lemma dist_sym: "dist x y = dist y x"by norm
770 lemma dist_pos_le: "0 <= dist x y" by norm
772 lemma dist_triangle: "dist x z <= dist x y + dist y z" by norm
774 lemma dist_triangle_alt: "dist y z <= dist x y + dist x z" by norm
776 lemma dist_eq_0: "dist x y = 0 \<longleftrightarrow> x = y" by norm
778 lemma dist_pos_lt: "x \<noteq> y ==> 0 < dist x y" by norm
779 lemma dist_nz:  "x \<noteq> y \<longleftrightarrow> 0 < dist x y" by norm
781 lemma dist_triangle_le: "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e" by norm
783 lemma dist_triangle_lt: "dist x z + dist y z < e ==> dist x y < e" by norm
785 lemma dist_triangle_half_l: "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 ==> dist x1 x2 < e" by norm
787 lemma dist_triangle_half_r: "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 ==> dist x1 x2 < e" by norm
789 lemma dist_triangle_add: "dist (x + y) (x' + y') <= dist x x' + dist y y'"
790   by norm
792 lemma dist_mul: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
793   unfolding dist_def vector_ssub_ldistrib[symmetric] norm_mul ..
795 lemma dist_triangle_add_half: " dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 ==> dist(x + y) (x' + y') < e" by norm
797 lemma dist_le_0: "dist x y <= 0 \<longleftrightarrow> x = y" by norm
800 begin
801   instance by (intro_classes)
802 end
804 lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)\$i ) S)"
805   apply vector
806   apply auto
807   apply (cases "finite S")
808   apply (rule finite_induct[of S])
809   apply (auto simp add: vector_component zero_index)
810   done
812 lemma setsum_clauses:
813   shows "setsum f {} = 0"
814   and "finite S \<Longrightarrow> setsum f (insert x S) =
815                  (if x \<in> S then setsum f S else f x + setsum f S)"
816   by (auto simp add: insert_absorb)
818 lemma setsum_cmul:
819   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
820   shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
821   by (simp add: setsum_eq Cart_eq Cart_lambda_beta vector_component setsum_right_distrib)
823 lemma setsum_component:
824   fixes f:: " 'a \<Rightarrow> ('b::semiring_1) ^'n"
825   assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
826   shows "(setsum f S)\$i = setsum (\<lambda>x. (f x)\$i) S"
827   using i by (simp add: setsum_eq Cart_lambda_beta)
829   (* This needs finiteness assumption due to the definition of fold!!! *)
831 lemma setsum_superset:
832   assumes fb: "finite B" and ab: "A \<subseteq> B"
833   and f0: "\<forall>x \<in> B - A. f x = 0"
834   shows "setsum f B = setsum f A"
835 proof-
836   from ab fb have fa: "finite A" by (metis finite_subset)
837   from fb have fba: "finite (B - A)" by (metis finite_Diff)
838   have d: "A \<inter> (B - A) = {}" by blast
839   from ab have b: "B = A \<union> (B - A)" by blast
840   from setsum_Un_disjoint[OF fa fba d, of f] b
841     setsum_0'[OF f0]
842   show "setsum f B = setsum f A" by simp
843 qed
845 lemma setsum_restrict_set:
846   assumes fA: "finite A"
847   shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
848 proof-
849   from fA have fab: "finite (A \<inter> B)" by auto
850   have aba: "A \<inter> B \<subseteq> A" by blast
851   let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
852   from setsum_superset[OF fA aba, of ?g]
853   show ?thesis by simp
854 qed
856 lemma setsum_cases:
857   assumes fA: "finite A"
858   shows "setsum (\<lambda>x. if x \<in> B then f x else g x) A =
859          setsum f (A \<inter> B) + setsum g (A \<inter> - B)"
860 proof-
861   have a: "A = A \<inter> B \<union> A \<inter> -B" "(A \<inter> B) \<inter> (A \<inter> -B) = {}"
862     by blast+
863   from fA
864   have f: "finite (A \<inter> B)" "finite (A \<inter> -B)" by auto
865   let ?g = "\<lambda>x. if x \<in> B then f x else g x"
866   from setsum_Un_disjoint[OF f a(2), of ?g] a(1)
867   show ?thesis by simp
868 qed
870 lemma setsum_norm:
871   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
872   assumes fS: "finite S"
873   shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
874 proof(induct rule: finite_induct[OF fS])
875   case 1 thus ?case by (simp add: norm_zero)
876 next
877   case (2 x S)
878   from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
879   also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
880     using "2.hyps" by simp
881   finally  show ?case  using "2.hyps" by simp
882 qed
884 lemma real_setsum_norm:
885   fixes f :: "'a \<Rightarrow> real ^'n"
886   assumes fS: "finite S"
887   shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
888 proof(induct rule: finite_induct[OF fS])
889   case 1 thus ?case by simp norm
890 next
891   case (2 x S)
892   from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" apply (simp add: norm_triangle_ineq) by norm
893   also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
894     using "2.hyps" by simp
895   finally  show ?case  using "2.hyps" by simp
896 qed
898 lemma setsum_norm_le:
899   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
900   assumes fS: "finite S"
901   and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
902   shows "norm (setsum f S) \<le> setsum g S"
903 proof-
904   from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
905     by - (rule setsum_mono, simp)
906   then show ?thesis using setsum_norm[OF fS, of f] fg
907     by arith
908 qed
910 lemma real_setsum_norm_le:
911   fixes f :: "'a \<Rightarrow> real ^ 'n"
912   assumes fS: "finite S"
913   and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
914   shows "norm (setsum f S) \<le> setsum g S"
915 proof-
916   from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
917     by - (rule setsum_mono, simp)
918   then show ?thesis using real_setsum_norm[OF fS, of f] fg
919     by arith
920 qed
922 lemma setsum_norm_bound:
923   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
924   assumes fS: "finite S"
925   and K: "\<forall>x \<in> S. norm (f x) \<le> K"
926   shows "norm (setsum f S) \<le> of_nat (card S) * K"
927   using setsum_norm_le[OF fS K] setsum_constant[symmetric]
928   by simp
930 lemma real_setsum_norm_bound:
931   fixes f :: "'a \<Rightarrow> real ^ 'n"
932   assumes fS: "finite S"
933   and K: "\<forall>x \<in> S. norm (f x) \<le> K"
934   shows "norm (setsum f S) \<le> of_nat (card S) * K"
935   using real_setsum_norm_le[OF fS K] setsum_constant[symmetric]
936   by simp
938 instantiation "^" :: ("{scaleR, one, times}",type) scaleR
939 begin
941 definition vector_scaleR_def: "(scaleR :: real \<Rightarrow> 'a ^'b \<Rightarrow> 'a ^'b) \<equiv> (\<lambda> c x . (scaleR c 1) *s x)"
942 instance ..
943 end
945 instantiation "^" :: ("ring_1",type) ring_1
946 begin
947 instance by intro_classes
948 end
950 instantiation "^" :: (real_algebra_1,type) real_vector
951 begin
953 instance
954   apply intro_classes
957   done
958 end
960 instantiation "^" :: (real_algebra_1,type) real_algebra
961 begin
963 instance
964   apply intro_classes
965   apply (simp_all add: vector_scaleR_def ring_simps)
966   apply vector
967   apply vector
968   done
969 end
971 instantiation "^" :: (real_algebra_1,type) real_algebra_1
972 begin
974 instance ..
975 end
977 lemma setsum_vmul:
978   fixes f :: "'a \<Rightarrow> 'b::{real_normed_vector,semiring, mult_zero}"
979   assumes fS: "finite S"
980   shows "setsum f S *s v = setsum (\<lambda>x. f x *s v) S"
981 proof(induct rule: finite_induct[OF fS])
982   case 1 then show ?case by (simp add: vector_smult_lzero)
983 next
984   case (2 x F)
985   from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v"
986     by simp
987   also have "\<dots> = f x *s v + setsum f F *s v"
989   also have "\<dots> = setsum (\<lambda>x. f x *s v) (insert x F)" using "2.hyps" by simp
990   finally show ?case .
991 qed
993 (* FIXME : Problem thm setsum_vmul[of _ "f:: 'a \<Rightarrow> real ^'n"]  ---
994  Get rid of *s and use real_vector instead! Also prove that ^ creates a real_vector !! *)
996 lemma setsum_add_split: assumes mn: "(m::nat) \<le> n + 1"
997   shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
998 proof-
999   let ?A = "{m .. n}"
1000   let ?B = "{n + 1 .. n + p}"
1001   have eq: "{m .. n+p} = ?A \<union> ?B" using mn by auto
1002   have d: "?A \<inter> ?B = {}" by auto
1003   from setsum_Un_disjoint[of "?A" "?B" f] eq d show ?thesis by auto
1004 qed
1006 lemma setsum_reindex_nonzero:
1007   assumes fS: "finite S"
1008   and nz: "\<And> x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"
1009   shows "setsum h (f ` S) = setsum (h o f) S"
1010 using nz
1011 proof(induct rule: finite_induct[OF fS])
1012   case 1 thus ?case by simp
1013 next
1014   case (2 x F)
1015   {assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
1016     then obtain y where y: "y \<in> F" "f x = f y" by auto
1017     from "2.hyps" y have xy: "x \<noteq> y" by auto
1019     from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
1020     have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
1021     also have "\<dots> = setsum (h o f) (insert x F)"
1022       using "2.hyps" "2.prems" h0  by auto
1023     finally have ?case .}
1024   moreover
1025   {assume fxF: "f x \<notin> f ` F"
1026     have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)"
1027       using fxF "2.hyps" by simp
1028     also have "\<dots> = setsum (h o f) (insert x F)"
1029       using "2.hyps" "2.prems" fxF
1030       apply auto apply metis done
1031     finally have ?case .}
1032   ultimately show ?case by blast
1033 qed
1035 lemma setsum_Un_nonzero:
1036   assumes fS: "finite S" and fF: "finite F"
1037   and f: "\<forall> x\<in> S \<inter> F . f x = (0::'a::ab_group_add)"
1038   shows "setsum f (S \<union> F) = setsum f S + setsum f F"
1039   using setsum_Un[OF fS fF, of f] setsum_0'[OF f] by simp
1041 lemma setsum_natinterval_left:
1042   assumes mn: "(m::nat) <= n"
1043   shows "setsum f {m..n} = f m + setsum f {m + 1..n}"
1044 proof-
1045   from mn have "{m .. n} = insert m {m+1 .. n}" by auto
1046   then show ?thesis by auto
1047 qed
1049 lemma setsum_natinterval_difff:
1050   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
1051   shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
1052           (if m <= n then f m - f(n + 1) else 0)"
1053 by (induct n, auto simp add: ring_simps not_le le_Suc_eq)
1055 lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]
1057 lemma setsum_setsum_restrict:
1058   "finite S \<Longrightarrow> finite T \<Longrightarrow> setsum (\<lambda>x. setsum (\<lambda>y. f x y) {y. y\<in> T \<and> R x y}) S = setsum (\<lambda>y. setsum (\<lambda>x. f x y) {x. x \<in> S \<and> R x y}) T"
1059   apply (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)
1060   by (rule setsum_commute)
1062 lemma setsum_image_gen: assumes fS: "finite S"
1063   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
1064 proof-
1065   {fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto}
1066   note th0 = this
1067   have "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
1068     apply (rule setsum_cong2)
1070   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
1071     apply (rule setsum_setsum_restrict[OF fS])
1072     by (rule finite_imageI[OF fS])
1073   finally show ?thesis .
1074 qed
1076     (* FIXME: Here too need stupid finiteness assumption on T!!! *)
1077 lemma setsum_group:
1078   assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
1079   shows "setsum (\<lambda>y. setsum g {x. x\<in> S \<and> f x = y}) T = setsum g S"
1081 apply (subst setsum_image_gen[OF fS, of g f])
1082 apply (rule setsum_superset[OF fT fST])
1083 by (auto intro: setsum_0')
1085 (* FIXME: Change the name to fold_image\<dots> *)
1086 lemma (in comm_monoid_mult) fold_1': "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
1087   apply (induct set: finite)
1088   apply simp by (auto simp add: fold_image_insert)
1090 lemma (in comm_monoid_mult) fold_union_nonzero:
1091   assumes fS: "finite S" and fT: "finite T"
1092   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
1093   shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
1094 proof-
1095   have "fold_image op * f 1 (S \<inter> T) = 1"
1096     apply (rule fold_1')
1097     using fS fT I0 by auto
1098   with fold_image_Un_Int[OF fS fT] show ?thesis by simp
1099 qed
1101 lemma setsum_union_nonzero:
1102   assumes fS: "finite S" and fT: "finite T"
1103   and I0: "\<forall>x \<in> S\<inter>T. f x = 0"
1104   shows "setsum f (S \<union> T) = setsum f S  + setsum f T"
1105   using fS fT
1108   using I0 by auto
1110 lemma setprod_union_nonzero:
1111   assumes fS: "finite S" and fT: "finite T"
1112   and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
1113   shows "setprod f (S \<union> T) = setprod f S  * setprod f T"
1114   using fS fT
1116   apply (rule fold_union_nonzero)
1117   using I0 by auto
1119 lemma setsum_unions_nonzero:
1120   assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
1121   and f0: "\<And>T1 T2 x. T1\<in>S \<Longrightarrow> T2\<in>S \<Longrightarrow> T1 \<noteq> T2 \<Longrightarrow> x \<in> T1 \<Longrightarrow> x \<in> T2 \<Longrightarrow> f x = 0"
1122   shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
1123   using fSS f0
1124 proof(induct rule: finite_induct[OF fS])
1125   case 1 thus ?case by simp
1126 next
1127   case (2 T F)
1128   then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F"
1129     and H: "setsum f (\<Union> F) = setsum (setsum f) F" by (auto simp add: finite_insert)
1130   from fTF have fUF: "finite (\<Union>F)" by (auto intro: finite_Union)
1131   from "2.prems" TF fTF
1132   show ?case
1133     by (auto simp add: H[symmetric] intro: setsum_union_nonzero[OF fTF(1) fUF, of f])
1134 qed
1136   (* FIXME : Copied from Pocklington --- should be moved to Finite_Set!!!!!!!! *)
1139 lemma (in comm_monoid_mult) fold_related:
1140   assumes Re: "R e e"
1141   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
1142   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
1143   shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
1144   using fS by (rule finite_subset_induct) (insert assms, auto)
1146   (* FIXME: I think we can get rid of the finite assumption!! *)
1147 lemma (in comm_monoid_mult)
1148   fold_eq_general:
1149   assumes fS: "finite S"
1150   and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y"
1151   and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
1152   shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
1153 proof-
1154   from h f12 have hS: "h ` S = S'" by auto
1155   {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
1156     from f12 h H  have "x = y" by auto }
1157   hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
1158   from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
1159   from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
1160   also have "\<dots> = fold_image (op *) (f2 o h) e S"
1161     using fold_image_reindex[OF fS hinj, of f2 e] .
1162   also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
1163     by blast
1164   finally show ?thesis ..
1165 qed
1167 lemma (in comm_monoid_mult) fold_eq_general_inverses:
1168   assumes fS: "finite S"
1169   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
1170   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
1171   shows "fold_image (op *) f e S = fold_image (op *) g e T"
1172   using fold_eq_general[OF fS, of T h g f e] kh hk by metis
1174 lemma setsum_eq_general_reverses:
1175   assumes fS: "finite S" and fT: "finite T"
1176   and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
1177   and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x  \<and> g (h x) = f x"
1178   shows "setsum f S = setsum g T"
1179   apply (simp add: setsum_def fS fT)
1181   apply (erule kh)
1182   apply (erule hk)
1183   done
1185 lemma vsum_norm_allsubsets_bound:
1186   fixes f:: "'a \<Rightarrow> real ^'n"
1187   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
1188   shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real (dimindex(UNIV :: 'n set)) *  e"
1189 proof-
1190   let ?d = "real (dimindex (UNIV ::'n set))"
1191   let ?nf = "\<lambda>x. norm (f x)"
1192   let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
1193   have th0: "setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x \$ i\<bar>) ?U) P = setsum (\<lambda>i. setsum (\<lambda>x. \<bar>f x \$ i\<bar>) P) ?U"
1194     by (rule setsum_commute)
1195   have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
1196   have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x \$ i\<bar>) ?U) P"
1197     apply (rule setsum_mono)
1198     by (rule norm_le_l1)
1199   also have "\<dots> \<le> 2 * ?d * e"
1200     unfolding th0 th1
1201   proof(rule setsum_bounded)
1202     fix i assume i: "i \<in> ?U"
1203     let ?Pp = "{x. x\<in> P \<and> f x \$ i \<ge> 0}"
1204     let ?Pn = "{x. x \<in> P \<and> f x \$ i < 0}"
1205     have thp: "P = ?Pp \<union> ?Pn" by auto
1206     have thp0: "?Pp \<inter> ?Pn ={}" by auto
1207     have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
1208     have Ppe:"setsum (\<lambda>x. \<bar>f x \$ i\<bar>) ?Pp \<le> e"
1209       using i component_le_norm[OF i, of "setsum (\<lambda>x. f x) ?Pp"]  fPs[OF PpP]
1210       by (auto simp add: setsum_component intro: abs_le_D1)
1211     have Pne: "setsum (\<lambda>x. \<bar>f x \$ i\<bar>) ?Pn \<le> e"
1212       using i component_le_norm[OF i, of "setsum (\<lambda>x. - f x) ?Pn"]  fPs[OF PnP]
1213       by (auto simp add: setsum_negf norm_neg setsum_component vector_component intro: abs_le_D1)
1214     have "setsum (\<lambda>x. \<bar>f x \$ i\<bar>) P = setsum (\<lambda>x. \<bar>f x \$ i\<bar>) ?Pp + setsum (\<lambda>x. \<bar>f x \$ i\<bar>) ?Pn"
1215       apply (subst thp)
1216       apply (rule setsum_Un_nonzero)
1217       using fP thp0 by auto
1218     also have "\<dots> \<le> 2*e" using Pne Ppe by arith
1219     finally show "setsum (\<lambda>x. \<bar>f x \$ i\<bar>) P \<le> 2*e" .
1220   qed
1221   finally show ?thesis .
1222 qed
1224 lemma dot_lsum: "finite S \<Longrightarrow> setsum f S \<bullet> (y::'a::{comm_ring}^'n) = setsum (\<lambda>x. f x \<bullet> y) S "
1227 lemma dot_rsum: "finite S \<Longrightarrow> (y::'a::{comm_ring}^'n) \<bullet> setsum f S = setsum (\<lambda>x. y \<bullet> f x) S "
1230 subsection{* Basis vectors in coordinate directions. *}
1233 definition "basis k = (\<chi> i. if i = k then 1 else 0)"
1235 lemma delta_mult_idempotent:
1236   "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)" by (cases "k=a", auto)
1238 lemma norm_basis:
1239   assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
1240   shows "norm (basis k :: real ^'n) = 1"
1241   using k
1242   apply (simp add: basis_def real_vector_norm_def dot_def)
1243   apply (vector delta_mult_idempotent)
1244   using setsum_delta[of "{1 .. dimindex (UNIV :: 'n set)}" "k" "\<lambda>k. 1::real"]
1245   apply auto
1246   done
1248 lemma norm_basis_1: "norm(basis 1 :: real ^'n) = 1"
1249   apply (simp add: basis_def real_vector_norm_def dot_def)
1250   apply (vector delta_mult_idempotent)
1251   using setsum_delta[of "{1 .. dimindex (UNIV :: 'n set)}" "1" "\<lambda>k. 1::real"] dimindex_nonzero[of "UNIV :: 'n set"]
1252   apply auto
1253   done
1255 lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n). norm x = c"
1256   apply (rule exI[where x="c *s basis 1"])
1257   by (simp only: norm_mul norm_basis_1)
1259 lemma vector_choose_dist: assumes e: "0 <= e"
1260   shows "\<exists>(y::real^'n). dist x y = e"
1261 proof-
1262   from vector_choose_size[OF e] obtain c:: "real ^'n"  where "norm c = e"
1263     by blast
1264   then have "dist x (x - c) = e" by (simp add: dist_def)
1265   then show ?thesis by blast
1266 qed
1268 lemma basis_inj: "inj_on (basis :: nat \<Rightarrow> real ^'n) {1 .. dimindex (UNIV :: 'n set)}"
1269   by (auto simp add: inj_on_def basis_def Cart_eq Cart_lambda_beta)
1271 lemma basis_component: "i \<in> {1 .. dimindex(UNIV:: 'n set)} ==> (basis k ::('a::semiring_1)^'n)\$i = (if k=i then 1 else 0)"
1272   by (simp add: basis_def Cart_lambda_beta)
1274 lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
1275   by auto
1277 lemma basis_expansion:
1278   "setsum (\<lambda>i. (x\$i) *s basis i) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
1279   by (auto simp add: Cart_eq basis_component[where ?'n = "'n"] setsum_component vector_component cond_value_iff setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
1281 lemma basis_expansion_unique:
1282   "setsum (\<lambda>i. f i *s basis i) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::comm_ring_1) ^'n) \<longleftrightarrow> (\<forall>i\<in>{1 .. dimindex(UNIV:: 'n set)}. f i = x\$i)"
1283   by (simp add: Cart_eq setsum_component vector_component basis_component setsum_delta cond_value_iff cong del: if_weak_cong)
1285 lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
1286   by auto
1288 lemma dot_basis:
1289   assumes i: "i \<in> {1 .. dimindex (UNIV :: 'n set)}"
1290   shows "basis i \<bullet> x = x\$i" "x \<bullet> (basis i :: 'a^'n) = (x\$i :: 'a::semiring_1)"
1291   using i
1292   by (auto simp add: dot_def basis_def Cart_lambda_beta cond_application_beta  cond_value_iff setsum_delta cong del: if_weak_cong)
1294 lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> i \<notin> {1..dimindex(UNIV ::'n set)}"
1295   by (auto simp add: Cart_eq basis_component zero_index)
1297 lemma basis_nonzero:
1298   assumes k: "k \<in> {1 .. dimindex(UNIV ::'n set)}"
1299   shows "basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
1300   using k by (simp add: basis_eq_0)
1302 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = (z::'a::semiring_1^'n)"
1303   apply (auto simp add: Cart_eq dot_basis)
1304   apply (erule_tac x="basis i" in allE)
1306   apply (subgoal_tac "y = z")
1307   apply simp
1308   apply vector
1309   done
1311 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = (y::'a::semiring_1^'n)"
1312   apply (auto simp add: Cart_eq dot_basis)
1313   apply (erule_tac x="basis i" in allE)
1315   apply (subgoal_tac "x = y")
1316   apply simp
1317   apply vector
1318   done
1320 subsection{* Orthogonality. *}
1322 definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
1324 lemma orthogonal_basis:
1325   assumes i:"i \<in> {1 .. dimindex(UNIV ::'n set)}"
1326   shows "orthogonal (basis i :: 'a^'n) x \<longleftrightarrow> x\$i = (0::'a::ring_1)"
1327   using i
1328   by (auto simp add: orthogonal_def dot_def basis_def Cart_lambda_beta cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)
1330 lemma orthogonal_basis_basis:
1331   assumes i:"i \<in> {1 .. dimindex(UNIV ::'n set)}"
1332   and j: "j \<in> {1 .. dimindex(UNIV ::'n set)}"
1333   shows "orthogonal (basis i :: 'a::ring_1^'n) (basis j) \<longleftrightarrow> i \<noteq> j"
1334   unfolding orthogonal_basis[OF i] basis_component[OF i] by simp
1336   (* FIXME : Maybe some of these require less than comm_ring, but not all*)
1337 lemma orthogonal_clauses:
1338   "orthogonal a (0::'a::comm_ring ^'n)"
1339   "orthogonal a x ==> orthogonal a (c *s x)"
1340   "orthogonal a x ==> orthogonal a (-x)"
1341   "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x + y)"
1342   "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x - y)"
1343   "orthogonal 0 a"
1344   "orthogonal x a ==> orthogonal (c *s x) a"
1345   "orthogonal x a ==> orthogonal (-x) a"
1346   "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x + y) a"
1347   "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x - y) a"
1348   unfolding orthogonal_def dot_rneg dot_rmult dot_radd dot_rsub
1349   dot_lzero dot_rzero dot_lneg dot_lmult dot_ladd dot_lsub
1350   by simp_all
1352 lemma orthogonal_commute: "orthogonal (x::'a::{ab_semigroup_mult,comm_monoid_add} ^'n)y \<longleftrightarrow> orthogonal y x"
1353   by (simp add: orthogonal_def dot_sym)
1355 subsection{* Explicit vector construction from lists. *}
1357 lemma Cart_lambda_beta_1[simp]: "(Cart_lambda g)\$1 = g 1"
1358   apply (rule Cart_lambda_beta[rule_format])
1359   using dimindex_ge_1 apply auto done
1361 lemma Cart_lambda_beta_1'[simp]: "(Cart_lambda g)\$(Suc 0) = g 1"
1362   by (simp only: One_nat_def[symmetric] Cart_lambda_beta_1)
1364 definition "vector l = (\<chi> i. if i <= length l then l ! (i - 1) else 0)"
1366 lemma vector_1: "(vector[x]) \$1 = x"
1367   using dimindex_ge_1
1368   by (auto simp add: vector_def Cart_lambda_beta[rule_format])
1369 lemma dimindex_2[simp]: "2 \<in> {1 .. dimindex (UNIV :: 2 set)}"
1370   by (auto simp add: dimindex_def)
1371 lemma dimindex_2'[simp]: "2 \<in> {Suc 0 .. dimindex (UNIV :: 2 set)}"
1372   by (auto simp add: dimindex_def)
1373 lemma dimindex_3[simp]: "2 \<in> {1 .. dimindex (UNIV :: 3 set)}" "3 \<in> {1 .. dimindex (UNIV :: 3 set)}"
1374   by (auto simp add: dimindex_def)
1376 lemma dimindex_3'[simp]: "2 \<in> {Suc 0 .. dimindex (UNIV :: 3 set)}" "3 \<in> {Suc 0 .. dimindex (UNIV :: 3 set)}"
1377   by (auto simp add: dimindex_def)
1379 lemma vector_2:
1380  "(vector[x,y]) \$1 = x"
1381  "(vector[x,y] :: 'a^2)\$2 = (y::'a::zero)"
1383   using Cart_lambda_beta[rule_format, OF dimindex_2, of "\<lambda>i. if i \<le> length [x,y] then [x,y] ! (i - 1) else (0::'a)"]
1384   apply (simp only: vector_def )
1385   apply auto
1386   done
1388 lemma vector_3:
1389  "(vector [x,y,z] ::('a::zero)^3)\$1 = x"
1390  "(vector [x,y,z] ::('a::zero)^3)\$2 = y"
1391  "(vector [x,y,z] ::('a::zero)^3)\$3 = z"
1392 apply (simp_all add: vector_def Cart_lambda_beta dimindex_3)
1393   using Cart_lambda_beta[rule_format, OF dimindex_3(1), of "\<lambda>i. if i \<le> length [x,y,z] then [x,y,z] ! (i - 1) else (0::'a)"]   using Cart_lambda_beta[rule_format, OF dimindex_3(2), of "\<lambda>i. if i \<le> length [x,y,z] then [x,y,z] ! (i - 1) else (0::'a)"]
1394   by simp_all
1396 lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
1397   apply auto
1398   apply (erule_tac x="v\$1" in allE)
1399   apply (subgoal_tac "vector [v\$1] = v")
1400   apply simp
1401   by (vector vector_def dimindex_def)
1403 lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
1404   apply auto
1405   apply (erule_tac x="v\$1" in allE)
1406   apply (erule_tac x="v\$2" in allE)
1407   apply (subgoal_tac "vector [v\$1, v\$2] = v")
1408   apply simp
1409   apply (vector vector_def dimindex_def)
1410   apply auto
1411   apply (subgoal_tac "i = 1 \<or> i =2", auto)
1412   done
1414 lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
1415   apply auto
1416   apply (erule_tac x="v\$1" in allE)
1417   apply (erule_tac x="v\$2" in allE)
1418   apply (erule_tac x="v\$3" in allE)
1419   apply (subgoal_tac "vector [v\$1, v\$2, v\$3] = v")
1420   apply simp
1421   apply (vector vector_def dimindex_def)
1422   apply auto
1423   apply (subgoal_tac "i = 1 \<or> i =2 \<or> i = 3", auto)
1424   done
1426 subsection{* Linear functions. *}
1428 definition "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *s x) = c *s f x)"
1430 lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. (c::'a::comm_semiring) *s f x)"
1431   by (vector linear_def Cart_eq Cart_lambda_beta[rule_format] ring_simps)
1433 lemma linear_compose_neg: "linear (f :: 'a ^'n \<Rightarrow> 'a::comm_ring ^'m) ==> linear (\<lambda>x. -(f(x)))" by (vector linear_def Cart_eq)
1435 lemma linear_compose_add: "linear (f :: 'a ^'n \<Rightarrow> 'a::semiring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
1436   by (vector linear_def Cart_eq ring_simps)
1438 lemma linear_compose_sub: "linear (f :: 'a ^'n \<Rightarrow> 'a::ring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
1439   by (vector linear_def Cart_eq ring_simps)
1441 lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
1444 lemma linear_id: "linear id" by (simp add: linear_def id_def)
1446 lemma linear_zero: "linear (\<lambda>x. 0::'a::semiring_1 ^ 'n)" by (simp add: linear_def)
1448 lemma linear_compose_setsum:
1449   assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a :: 'a::semiring_1 ^ 'n \<Rightarrow> 'a ^ 'm)"
1450   shows "linear(\<lambda>x. setsum (\<lambda>a. f a x :: 'a::semiring_1 ^'m) S)"
1451   using lS
1452   apply (induct rule: finite_induct[OF fS])
1455 lemma linear_vmul_component:
1456   fixes f:: "'a::semiring_1^'m \<Rightarrow> 'a^'n"
1457   assumes lf: "linear f" and k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
1458   shows "linear (\<lambda>x. f x \$ k *s v)"
1459   using lf k
1460   apply (auto simp add: linear_def )
1461   by (vector ring_simps)+
1463 lemma linear_0: "linear f ==> f 0 = (0::'a::semiring_1 ^'n)"
1464   unfolding linear_def
1465   apply clarsimp
1466   apply (erule allE[where x="0::'a"])
1467   apply simp
1468   done
1470 lemma linear_cmul: "linear f ==> f(c*s x) = c *s f x" by (simp add: linear_def)
1472 lemma linear_neg: "linear (f :: 'a::ring_1 ^'n \<Rightarrow> _) ==> f (-x) = - f x"
1473   unfolding vector_sneg_minus1
1474   using linear_cmul[of f] by auto
1476 lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)
1478 lemma linear_sub: "linear (f::'a::ring_1 ^'n \<Rightarrow> _) ==> f(x - y) = f x - f y"
1481 lemma linear_setsum:
1482   fixes f:: "'a::semiring_1^'n \<Rightarrow> _"
1483   assumes lf: "linear f" and fS: "finite S"
1484   shows "f (setsum g S) = setsum (f o g) S"
1485 proof (induct rule: finite_induct[OF fS])
1486   case 1 thus ?case by (simp add: linear_0[OF lf])
1487 next
1488   case (2 x F)
1489   have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps"
1490     by simp
1491   also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
1492   also have "\<dots> = setsum (f o g) (insert x F)" using "2.hyps" by simp
1493   finally show ?case .
1494 qed
1496 lemma linear_setsum_mul:
1497   fixes f:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m"
1498   assumes lf: "linear f" and fS: "finite S"
1499   shows "f (setsum (\<lambda>i. c i *s v i) S) = setsum (\<lambda>i. c i *s f (v i)) S"
1500   using linear_setsum[OF lf fS, of "\<lambda>i. c i *s v i" , unfolded o_def]
1501   linear_cmul[OF lf] by simp
1503 lemma linear_injective_0:
1504   assumes lf: "linear (f:: 'a::ring_1 ^ 'n \<Rightarrow> _)"
1505   shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
1506 proof-
1507   have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
1508   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
1509   also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
1510     by (simp add: linear_sub[OF lf])
1511   also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
1512   finally show ?thesis .
1513 qed
1515 lemma linear_bounded:
1516   fixes f:: "real ^'m \<Rightarrow> real ^'n"
1517   assumes lf: "linear f"
1518   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
1519 proof-
1520   let ?S = "{1..dimindex(UNIV:: 'm set)}"
1521   let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
1522   have fS: "finite ?S" by simp
1523   {fix x:: "real ^ 'm"
1524     let ?g = "(\<lambda>i::nat. (x\$i) *s (basis i) :: real ^ 'm)"
1525     have "norm (f x) = norm (f (setsum (\<lambda>i. (x\$i) *s (basis i)) ?S))"
1526       by (simp only:  basis_expansion)
1527     also have "\<dots> = norm (setsum (\<lambda>i. (x\$i) *s f (basis i))?S)"
1528       using linear_setsum[OF lf fS, of ?g, unfolded o_def] linear_cmul[OF lf]
1529       by auto
1530     finally have th0: "norm (f x) = norm (setsum (\<lambda>i. (x\$i) *s f (basis i))?S)" .
1531     {fix i assume i: "i \<in> ?S"
1532       from component_le_norm[OF i, of x]
1533       have "norm ((x\$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x"
1534       unfolding norm_mul
1535       apply (simp only: mult_commute)
1536       apply (rule mult_mono)
1537       by (auto simp add: ring_simps norm_pos_le) }
1538     then have th: "\<forall>i\<in> ?S. norm ((x\$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x" by metis
1539     from real_setsum_norm_le[OF fS, of "\<lambda>i. (x\$i) *s (f (basis i))", OF th]
1540     have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
1541   then show ?thesis by blast
1542 qed
1544 lemma linear_bounded_pos:
1545   fixes f:: "real ^'n \<Rightarrow> real ^ 'm"
1546   assumes lf: "linear f"
1547   shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
1548 proof-
1549   from linear_bounded[OF lf] obtain B where
1550     B: "\<forall>x. norm (f x) \<le> B * norm x" by blast
1551   let ?K = "\<bar>B\<bar> + 1"
1552   have Kp: "?K > 0" by arith
1553     {assume C: "B < 0"
1554       have "norm (1::real ^ 'n) > 0" by (simp add: norm_pos_lt)
1555       with C have "B * norm (1:: real ^ 'n) < 0"
1557       with B[rule_format, of 1] norm_pos_le[of "f 1"] have False by simp
1558     }
1559     then have Bp: "B \<ge> 0" by ferrack
1560     {fix x::"real ^ 'n"
1561       have "norm (f x) \<le> ?K *  norm x"
1562       using B[rule_format, of x] norm_pos_le[of x] norm_pos_le[of "f x"] Bp
1564   }
1565   then show ?thesis using Kp by blast
1566 qed
1568 subsection{* Bilinear functions. *}
1570 definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
1572 lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)"
1573   by (simp add: bilinear_def linear_def)
1574 lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
1575   by (simp add: bilinear_def linear_def)
1577 lemma bilinear_lmul: "bilinear h ==> h (c *s x) y = c *s (h x y)"
1578   by (simp add: bilinear_def linear_def)
1580 lemma bilinear_rmul: "bilinear h ==> h x (c *s y) = c *s (h x y)"
1581   by (simp add: bilinear_def linear_def)
1583 lemma bilinear_lneg: "bilinear h ==> h (- (x:: 'a::ring_1 ^ 'n)) y = -(h x y)"
1584   by (simp only: vector_sneg_minus1 bilinear_lmul)
1586 lemma bilinear_rneg: "bilinear h ==> h x (- (y:: 'a::ring_1 ^ 'n)) = - h x y"
1587   by (simp only: vector_sneg_minus1 bilinear_rmul)
1589 lemma  (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
1590   using add_imp_eq[of x y 0] by auto
1592 lemma bilinear_lzero:
1593   fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h 0 x = 0"
1594   using bilinear_ladd[OF bh, of 0 0 x]
1597 lemma bilinear_rzero:
1598   fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h x 0 = 0"
1599   using bilinear_radd[OF bh, of x 0 0 ]
1602 lemma bilinear_lsub: "bilinear h ==> h (x - (y:: 'a::ring_1 ^ 'n)) z = h x z - h y z"
1605 lemma bilinear_rsub: "bilinear h ==> h z (x - (y:: 'a::ring_1 ^ 'n)) = h z x - h z y"
1608 lemma bilinear_setsum:
1609   fixes h:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m \<Rightarrow> 'a ^ 'k"
1610   assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T"
1611   shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
1612 proof-
1613   have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
1614     apply (rule linear_setsum[unfolded o_def])
1615     using bh fS by (auto simp add: bilinear_def)
1616   also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
1617     apply (rule setsum_cong, simp)
1618     apply (rule linear_setsum[unfolded o_def])
1619     using bh fT by (auto simp add: bilinear_def)
1620   finally show ?thesis unfolding setsum_cartesian_product .
1621 qed
1623 lemma bilinear_bounded:
1624   fixes h:: "real ^'m \<Rightarrow> real^'n \<Rightarrow> real ^ 'k"
1625   assumes bh: "bilinear h"
1626   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
1627 proof-
1628   let ?M = "{1 .. dimindex (UNIV :: 'm set)}"
1629   let ?N = "{1 .. dimindex (UNIV :: 'n set)}"
1630   let ?B = "setsum (\<lambda>(i,j). norm (h (basis i) (basis j))) (?M \<times> ?N)"
1631   have fM: "finite ?M" and fN: "finite ?N" by simp_all
1632   {fix x:: "real ^ 'm" and  y :: "real^'n"
1633     have "norm (h x y) = norm (h (setsum (\<lambda>i. (x\$i) *s basis i) ?M) (setsum (\<lambda>i. (y\$i) *s basis i) ?N))" unfolding basis_expansion ..
1634     also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x\$i) *s basis i) ((y\$j) *s basis j)) (?M \<times> ?N))"  unfolding bilinear_setsum[OF bh fM fN] ..
1635     finally have th: "norm (h x y) = \<dots>" .
1636     have "norm (h x y) \<le> ?B * norm x * norm y"
1637       apply (simp add: setsum_left_distrib th)
1638       apply (rule real_setsum_norm_le)
1639       using fN fM
1640       apply simp
1641       apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] norm_mul ring_simps)
1642       apply (rule mult_mono)
1643       apply (auto simp add: norm_pos_le zero_le_mult_iff component_le_norm)
1644       apply (rule mult_mono)
1645       apply (auto simp add: norm_pos_le zero_le_mult_iff component_le_norm)
1646       done}
1647   then show ?thesis by metis
1648 qed
1650 lemma bilinear_bounded_pos:
1651   fixes h:: "real ^'m \<Rightarrow> real^'n \<Rightarrow> real ^ 'k"
1652   assumes bh: "bilinear h"
1653   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
1654 proof-
1655   from bilinear_bounded[OF bh] obtain B where
1656     B: "\<forall>x y. norm (h x y) \<le> B * norm x * norm y" by blast
1657   let ?K = "\<bar>B\<bar> + 1"
1658   have Kp: "?K > 0" by arith
1659   have KB: "B < ?K" by arith
1660   {fix x::"real ^'m" and y :: "real ^'n"
1661     from KB Kp
1662     have "B * norm x * norm y \<le> ?K * norm x * norm y"
1663       apply -
1664       apply (rule mult_right_mono, rule mult_right_mono)
1665       by (auto simp add: norm_pos_le)
1666     then have "norm (h x y) \<le> ?K * norm x * norm y"
1667       using B[rule_format, of x y] by simp}
1668   with Kp show ?thesis by blast
1669 qed
1673 definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
1675 lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
1678   fixes f:: "'a::ring_1 ^'n \<Rightarrow> 'a ^ 'm"
1679   assumes lf: "linear f"
1680   shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"
1681 proof-
1682   let ?N = "{1 .. dimindex (UNIV :: 'n set)}"
1683   let ?M = "{1 .. dimindex (UNIV :: 'm set)}"
1684   have fN: "finite ?N" by simp
1685   have fM: "finite ?M" by simp
1686   {fix y:: "'a ^ 'm"
1687     let ?w = "(\<chi> i. (f (basis i) \<bullet> y)) :: 'a ^ 'n"
1688     {fix x
1689       have "f x \<bullet> y = f (setsum (\<lambda>i. (x\$i) *s basis i) ?N) \<bullet> y"
1690 	by (simp only: basis_expansion)
1691       also have "\<dots> = (setsum (\<lambda>i. (x\$i) *s f (basis i)) ?N) \<bullet> y"
1692 	unfolding linear_setsum[OF lf fN]
1693 	by (simp add: linear_cmul[OF lf])
1694       finally have "f x \<bullet> y = x \<bullet> ?w"
1695 	apply (simp only: )
1696 	apply (simp add: dot_def setsum_component Cart_lambda_beta setsum_left_distrib setsum_right_distrib vector_component setsum_commute[of _ ?M ?N] ring_simps del: One_nat_def)
1697 	done}
1698   }
1699   then show ?thesis unfolding adjoint_def
1700     some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
1701     using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]
1702     by metis
1703 qed
1706   fixes f:: "'a::ring_1 ^'n \<Rightarrow> 'a ^ 'm"
1707   assumes lf: "linear f"
1708   shows "x \<bullet> adjoint f y = f x \<bullet> y"
1709   using adjoint_works_lemma[OF lf] by metis
1713   fixes f :: "'a::comm_ring_1 ^'n \<Rightarrow> 'a ^ 'm"
1714   assumes lf: "linear f"
1719   fixes f:: "'a::comm_ring_1 ^'n \<Rightarrow> 'a ^ 'm"
1720   assumes lf: "linear f"
1721   shows "x \<bullet> adjoint f y = f x \<bullet> y"
1722   and "adjoint f y \<bullet> x = y \<bullet> f x"
1726   fixes f:: "'a::comm_ring_1 ^ 'n \<Rightarrow> _"
1727   assumes lf: "linear f"
1729   apply (rule ext)
1733   fixes f:: "'a::comm_ring_1 ^ 'n \<Rightarrow> 'a ^ 'm"
1734   assumes lf: "linear f" and u: "\<forall>x y. f' x \<bullet> y = x \<bullet> f y"
1735   shows "f' = adjoint f"
1736   apply (rule ext)
1737   using u
1740 text{* Matrix notation. NB: an MxN matrix is of type 'a^'n^'m, not 'a^'m^'n *}
1742 consts generic_mult :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" (infixr "\<star>" 75)
1745 matrix_matrix_mult_def: "(m:: ('a::semiring_1) ^'n^'m) \<star> (m' :: 'a ^'p^'n) \<equiv> (\<chi> i j. setsum (\<lambda>k. ((m\$i)\$k) * ((m'\$k)\$j)) {1 .. dimindex (UNIV :: 'n set)}) ::'a ^ 'p ^'m"
1747 abbreviation
1748   matrix_matrix_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"  (infixl "**" 70)
1749   where "m ** m' == m\<star> m'"
1752   matrix_vector_mult_def: "(m::('a::semiring_1) ^'n^'m) \<star> (x::'a ^'n) \<equiv> (\<chi> i. setsum (\<lambda>j. ((m\$i)\$j) * (x\$j)) {1..dimindex(UNIV ::'n set)}) :: 'a^'m"
1754 abbreviation
1755   matrix_vector_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"  (infixl "*v" 70)
1756   where
1757   "m *v v == m \<star> v"
1760   vector_matrix_mult_def: "(x::'a^'m) \<star> (m::('a::semiring_1) ^'n^'m) \<equiv> (\<chi> j. setsum (\<lambda>i. ((m\$i)\$j) * (x\$i)) {1..dimindex(UNIV :: 'm set)}) :: 'a^'n"
1762 abbreviation
1763   vactor_matrix_mult' :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "  (infixl "v*" 70)
1764   where
1765   "v v* m == v \<star> m"
1767 definition "(mat::'a::zero => 'a ^'n^'m) k = (\<chi> i j. if i = j then k else 0)"
1768 definition "(transp::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A\$j)\$i))"
1769 definition "(row::nat => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A\$i)\$j))"
1770 definition "(column::nat =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A\$i)\$j))"
1771 definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> {1 .. dimindex(UNIV :: 'm set)}}"
1772 definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> {1 .. dimindex(UNIV :: 'n set)}}"
1774 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
1775 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A \<star> B) + (A \<star> C)"
1776   by (vector matrix_matrix_mult_def setsum_addf[symmetric] ring_simps)
1778 lemma setsum_delta':
1779   assumes fS: "finite S" shows
1780   "setsum (\<lambda>k. if a = k then b k else 0) S =
1781      (if a\<in> S then b a else 0)"
1782   using setsum_delta[OF fS, of a b, symmetric]
1783   by (auto intro: setsum_cong)
1785 lemma matrix_mul_lid: "mat 1 ** A = A"
1786   apply (simp add: matrix_matrix_mult_def mat_def)
1787   apply vector
1788   by (auto simp only: cond_value_iff cond_application_beta setsum_delta'[OF finite_atLeastAtMost]  mult_1_left mult_zero_left if_True)
1791 lemma matrix_mul_rid: "A ** mat 1 = A"
1792   apply (simp add: matrix_matrix_mult_def mat_def)
1793   apply vector
1794   by (auto simp only: cond_value_iff cond_application_beta setsum_delta[OF finite_atLeastAtMost]  mult_1_right mult_zero_right if_True cong: if_cong)
1796 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
1797   apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
1798   apply (subst setsum_commute)
1799   apply simp
1800   done
1802 lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
1803   apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
1804   apply (subst setsum_commute)
1805   apply simp
1806   done
1808 lemma matrix_vector_mul_lid: "mat 1 *v x = x"
1809   apply (vector matrix_vector_mult_def mat_def)
1810   by (simp add: cond_value_iff cond_application_beta
1811     setsum_delta' cong del: if_weak_cong)
1813 lemma matrix_transp_mul: "transp(A ** B) = transp B ** transp (A::'a::comm_semiring_1^'m^'n)"
1814   by (simp add: matrix_matrix_mult_def transp_def Cart_eq Cart_lambda_beta mult_commute)
1816 lemma matrix_eq: "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
1817   apply auto
1818   apply (subst Cart_eq)
1819   apply clarify
1820   apply (clarsimp simp add: matrix_vector_mult_def basis_def cond_value_iff cond_application_beta Cart_eq Cart_lambda_beta cong del: if_weak_cong)
1821   apply (erule_tac x="basis ia" in allE)
1822   apply (erule_tac x="i" in ballE)
1823   by (auto simp add: basis_def cond_value_iff cond_application_beta Cart_lambda_beta setsum_delta[OF finite_atLeastAtMost] cong del: if_weak_cong)
1825 lemma matrix_vector_mul_component:
1826   assumes k: "k \<in> {1.. dimindex (UNIV :: 'm set)}"
1827   shows "((A::'a::semiring_1^'n'^'m) *v x)\$k = (A\$k) \<bullet> x"
1828   using k
1829   by (simp add: matrix_vector_mult_def Cart_lambda_beta dot_def)
1831 lemma dot_lmul_matrix: "((x::'a::comm_semiring_1 ^'n) v* A) \<bullet> y = x \<bullet> (A *v y)"
1832   apply (simp add: dot_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib Cart_lambda_beta mult_ac)
1833   apply (subst setsum_commute)
1834   by simp
1836 lemma transp_mat: "transp (mat n) = mat n"
1837   by (vector transp_def mat_def)
1839 lemma transp_transp: "transp(transp A) = A"
1840   by (vector transp_def)
1842 lemma row_transp:
1843   fixes A:: "'a::semiring_1^'n^'m"
1844   assumes i: "i \<in> {1.. dimindex (UNIV :: 'n set)}"
1845   shows "row i (transp A) = column i A"
1846   using i
1847   by (simp add: row_def column_def transp_def Cart_eq Cart_lambda_beta)
1849 lemma column_transp:
1850   fixes A:: "'a::semiring_1^'n^'m"
1851   assumes i: "i \<in> {1.. dimindex (UNIV :: 'm set)}"
1852   shows "column i (transp A) = row i A"
1853   using i
1854   by (simp add: row_def column_def transp_def Cart_eq Cart_lambda_beta)
1856 lemma rows_transp: "rows(transp (A::'a::semiring_1^'n^'m)) = columns A"
1857 apply (auto simp add: rows_def columns_def row_transp intro: set_ext)
1858 apply (rule_tac x=i in exI)
1859 apply (auto simp add: row_transp)
1860 done
1862 lemma columns_transp: "columns(transp (A::'a::semiring_1^'n^'m)) = rows A" by (metis transp_transp rows_transp)
1864 text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
1866 lemma matrix_mult_dot: "A *v x = (\<chi> i. A\$i \<bullet> x)"
1867   by (simp add: matrix_vector_mult_def dot_def)
1869 lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x\$i) *s column i A) {1 .. dimindex(UNIV:: 'n set)}"
1870   by (simp add: matrix_vector_mult_def Cart_eq setsum_component Cart_lambda_beta vector_component column_def mult_commute)
1872 lemma vector_componentwise:
1873   "(x::'a::ring_1^'n) = (\<chi> j. setsum (\<lambda>i. (x\$i) * (basis i :: 'a^'n)\$j) {1..dimindex(UNIV :: 'n set)})"
1874   apply (subst basis_expansion[symmetric])
1875   by (vector Cart_eq Cart_lambda_beta setsum_component)
1877 lemma linear_componentwise:
1878   fixes f:: "'a::ring_1 ^ 'm \<Rightarrow> 'a ^ 'n"
1879   assumes lf: "linear f" and j: "j \<in> {1 .. dimindex (UNIV :: 'n set)}"
1880   shows "(f x)\$j = setsum (\<lambda>i. (x\$i) * (f (basis i)\$j)) {1 .. dimindex (UNIV :: 'm set)}" (is "?lhs = ?rhs")
1881 proof-
1882   let ?M = "{1 .. dimindex (UNIV :: 'm set)}"
1883   let ?N = "{1 .. dimindex (UNIV :: 'n set)}"
1884   have fM: "finite ?M" by simp
1885   have "?rhs = (setsum (\<lambda>i.(x\$i) *s f (basis i) ) ?M)\$j"
1886     unfolding vector_smult_component[OF j, symmetric]
1887     unfolding setsum_component[OF j, of "(\<lambda>i.(x\$i) *s f (basis i :: 'a^'m))" ?M]
1888     ..
1889   then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion ..
1890 qed
1892 text{* Inverse matrices  (not necessarily square) *}
1894 definition "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
1896 definition "matrix_inv(A:: 'a::semiring_1^'n^'m) =
1897         (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
1899 text{* Correspondence between matrices and linear operators. *}
1901 definition matrix:: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
1902 where "matrix f = (\<chi> i j. (f(basis j))\$i)"
1904 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::'a::comm_semiring_1 ^ 'n))"
1905   by (simp add: linear_def matrix_vector_mult_def Cart_eq Cart_lambda_beta vector_component ring_simps setsum_right_distrib setsum_addf)
1907 lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::'a::comm_ring_1 ^ 'n)"
1908 apply (simp add: matrix_def matrix_vector_mult_def Cart_eq Cart_lambda_beta mult_commute del: One_nat_def)
1909 apply clarify
1910 apply (rule linear_componentwise[OF lf, symmetric])
1911 apply simp
1912 done
1914 lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::'a::comm_ring_1 ^ 'n))" by (simp add: ext matrix_works)
1916 lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: 'a:: comm_ring_1 ^ 'n)) = A"
1917   by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
1919 lemma matrix_compose:
1920   assumes lf: "linear (f::'a::comm_ring_1^'n \<Rightarrow> _)" and lg: "linear g"
1921   shows "matrix (g o f) = matrix g ** matrix f"
1922   using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
1923   by (simp  add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
1925 lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x\$i) *s ((transp A)\$i)) {1..dimindex(UNIV:: 'n set)}"
1926   by (simp add: matrix_vector_mult_def transp_def Cart_eq Cart_lambda_beta setsum_component vector_component mult_commute)
1928 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::'a::comm_ring_1^'n^'m) *v x) = (\<lambda>x. transp A *v x)"
1930   apply (rule matrix_vector_mul_linear)
1931   apply (simp add: transp_def dot_def Cart_lambda_beta matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
1932   apply (subst setsum_commute)
1933   apply (auto simp add: mult_ac)
1934   done
1936 lemma matrix_adjoint: assumes lf: "linear (f :: 'a::comm_ring_1^'n \<Rightarrow> 'a ^ 'm)"
1937   shows "matrix(adjoint f) = transp(matrix f)"
1938   apply (subst matrix_vector_mul[OF lf])
1941 subsection{* Interlude: Some properties of real sets *}
1943 lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
1944   shows "\<forall>n \<ge> m. d n < e m"
1945   using prems apply auto
1946   apply (erule_tac x="n" in allE)
1947   apply (erule_tac x="n" in allE)
1948   apply auto
1949   done
1952 lemma real_convex_bound_lt:
1953   assumes xa: "(x::real) < a" and ya: "y < a" and u: "0 <= u" and v: "0 <= v"
1954   and uv: "u + v = 1"
1955   shows "u * x + v * y < a"
1956 proof-
1957   have uv': "u = 0 \<longrightarrow> v \<noteq> 0" using u v uv by arith
1958   have "a = a * (u + v)" unfolding uv  by simp
1959   hence th: "u * a + v * a = a" by (simp add: ring_simps)
1960   from xa u have "u \<noteq> 0 \<Longrightarrow> u*x < u*a" by (simp add: mult_compare_simps)
1961   from ya v have "v \<noteq> 0 \<Longrightarrow> v * y < v * a" by (simp add: mult_compare_simps)
1962   from xa ya u v have "u * x + v * y < u * a + v * a"
1963     apply (cases "u = 0", simp_all add: uv')
1964     apply(rule mult_strict_left_mono)
1965     using uv' apply simp_all
1968     apply(rule mult_strict_left_mono)
1969     apply simp_all
1970     apply (rule mult_left_mono)
1971     apply simp_all
1972     done
1973   thus ?thesis unfolding th .
1974 qed
1976 lemma real_convex_bound_le:
1977   assumes xa: "(x::real) \<le> a" and ya: "y \<le> a" and u: "0 <= u" and v: "0 <= v"
1978   and uv: "u + v = 1"
1979   shows "u * x + v * y \<le> a"
1980 proof-
1981   from xa ya u v have "u * x + v * y \<le> u * a + v * a" by (simp add: add_mono mult_left_mono)
1982   also have "\<dots> \<le> (u + v) * a" by (simp add: ring_simps)
1983   finally show ?thesis unfolding uv by simp
1984 qed
1986 lemma infinite_enumerate: assumes fS: "infinite S"
1987   shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
1988 unfolding subseq_def
1989 using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
1991 lemma approachable_lt_le: "(\<exists>(d::real)>0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
1992 apply auto
1993 apply (rule_tac x="d/2" in exI)
1994 apply auto
1995 done
1998 lemma triangle_lemma:
1999   assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
2000   shows "x <= y + z"
2001 proof-
2002   have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y  by (simp add: zero_compare_simps)
2003   with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square ring_simps)
2004   from y z have yz: "y + z \<ge> 0" by arith
2005   from power2_le_imp_le[OF th yz] show ?thesis .
2006 qed
2009 lemma lambda_skolem: "(\<forall>i \<in> {1 .. dimindex(UNIV :: 'n set)}. \<exists>x. P i x) \<longleftrightarrow>
2010    (\<exists>x::'a ^ 'n. \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. P i (x\$i))" (is "?lhs \<longleftrightarrow> ?rhs")
2011 proof-
2012   let ?S = "{1 .. dimindex(UNIV :: 'n set)}"
2013   {assume H: "?rhs"
2014     then have ?lhs by auto}
2015   moreover
2016   {assume H: "?lhs"
2017     then obtain f where f:"\<forall>i\<in> ?S. P i (f i)" unfolding Ball_def choice_iff by metis
2018     let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
2019     {fix i assume i: "i \<in> ?S"
2020       with f i have "P i (f i)" by metis
2021       then have "P i (?x\$i)" using Cart_lambda_beta[of f, rule_format, OF i] by auto
2022     }
2023     hence "\<forall>i \<in> ?S. P i (?x\$i)" by metis
2024     hence ?rhs by metis }
2025   ultimately show ?thesis by metis
2026 qed
2028 (* Supremum and infimum of real sets *)
2031 definition rsup:: "real set \<Rightarrow> real" where
2032   "rsup S = (SOME a. isLub UNIV S a)"
2034 lemma rsup_alt: "rsup S = (SOME a. (\<forall>x \<in> S. x \<le> a) \<and> (\<forall>b. (\<forall>x \<in> S. x \<le> b) \<longrightarrow> a \<le> b))"  by (auto simp  add: isLub_def rsup_def leastP_def isUb_def setle_def setge_def)
2036 lemma rsup: assumes Se: "S \<noteq> {}" and b: "\<exists>b. S *<= b"
2037   shows "isLub UNIV S (rsup S)"
2038 using Se b
2039 unfolding rsup_def
2040 apply clarify
2041 apply (rule someI_ex)
2042 apply (rule reals_complete)
2043 by (auto simp add: isUb_def setle_def)
2045 lemma rsup_le: assumes Se: "S \<noteq> {}" and Sb: "S *<= b" shows "rsup S \<le> b"
2046 proof-
2047   from Sb have bu: "isUb UNIV S b" by (simp add: isUb_def setle_def)
2048   from rsup[OF Se] Sb have "isLub UNIV S (rsup S)"  by blast
2049   then show ?thesis using bu by (auto simp add: isLub_def leastP_def setle_def setge_def)
2050 qed
2052 lemma rsup_finite_Max: assumes fS: "finite S" and Se: "S \<noteq> {}"
2053   shows "rsup S = Max S"
2054 using fS Se
2055 proof-
2056   let ?m = "Max S"
2057   from Max_ge[OF fS] have Sm: "\<forall> x\<in> S. x \<le> ?m" by metis
2058   with rsup[OF Se] have lub: "isLub UNIV S (rsup S)" by (metis setle_def)
2059   from Max_in[OF fS Se] lub have mrS: "?m \<le> rsup S"
2060     by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def)
2061   moreover
2062   have "rsup S \<le> ?m" using Sm lub
2063     by (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
2064   ultimately  show ?thesis by arith
2065 qed
2067 lemma rsup_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
2068   shows "rsup S \<in> S"
2069   using rsup_finite_Max[OF fS Se] Max_in[OF fS Se] by metis
2071 lemma rsup_finite_Ub: assumes fS: "finite S" and Se: "S \<noteq> {}"
2072   shows "isUb S S (rsup S)"
2073   using rsup_finite_Max[OF fS Se] rsup_finite_in[OF fS Se] Max_ge[OF fS]
2074   unfolding isUb_def setle_def by metis
2076 lemma rsup_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
2077   shows "a \<le> rsup S \<longleftrightarrow> (\<exists> x \<in> S. a \<le> x)"
2078 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
2080 lemma rsup_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
2081   shows "a \<ge> rsup S \<longleftrightarrow> (\<forall> x \<in> S. a \<ge> x)"
2082 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
2084 lemma rsup_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
2085   shows "a < rsup S \<longleftrightarrow> (\<exists> x \<in> S. a < x)"
2086 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
2088 lemma rsup_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
2089   shows "a > rsup S \<longleftrightarrow> (\<forall> x \<in> S. a > x)"
2090 using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
2092 lemma rsup_unique: assumes b: "S *<= b" and S: "\<forall>b' < b. \<exists>x \<in> S. b' < x"
2093   shows "rsup S = b"
2094 using b S
2095 unfolding setle_def rsup_alt
2096 apply -
2097 apply (rule some_equality)
2098 apply (metis  linorder_not_le order_eq_iff[symmetric])+
2099 done
2101 lemma rsup_le_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. T *<= b) \<Longrightarrow> rsup S \<le> rsup T"
2102   apply (rule rsup_le)
2103   apply simp
2104   using rsup[of T] by (auto simp add: isLub_def leastP_def setge_def setle_def isUb_def)
2106 lemma isUb_def': "isUb R S = (\<lambda>x. S *<= x \<and> x \<in> R)"
2107   apply (rule ext)
2108   by (metis isUb_def)
2110 lemma UNIV_trivial: "UNIV x" using UNIV_I[of x] by (metis mem_def)
2111 lemma rsup_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
2112   shows "a \<le> rsup S \<and> rsup S \<le> b"
2113 proof-
2114   from rsup[OF Se] u have lub: "isLub UNIV S (rsup S)" by blast
2115   hence b: "rsup S \<le> b" using u by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
2116   from Se obtain y where y: "y \<in> S" by blast
2117   from lub l have "a \<le> rsup S" apply (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
2118     apply (erule ballE[where x=y])
2119     apply (erule ballE[where x=y])
2120     apply arith
2121     using y apply auto
2122     done
2123   with b show ?thesis by blast
2124 qed
2126 lemma rsup_abs_le: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rsup S\<bar> \<le> a"
2127   unfolding abs_le_interval_iff  using rsup_bounds[of S "-a" a]
2128   by (auto simp add: setge_def setle_def)
2130 lemma rsup_asclose: assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>rsup S - l\<bar> \<le> e"
2131 proof-
2132   have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith
2133   show ?thesis using S b rsup_bounds[of S "l - e" "l+e"] unfolding th
2134     by  (auto simp add: setge_def setle_def)
2135 qed
2137 definition rinf:: "real set \<Rightarrow> real" where
2138   "rinf S = (SOME a. isGlb UNIV S a)"
2140 lemma rinf_alt: "rinf S = (SOME a. (\<forall>x \<in> S. x \<ge> a) \<and> (\<forall>b. (\<forall>x \<in> S. x \<ge> b) \<longrightarrow> a \<ge> b))"  by (auto simp  add: isGlb_def rinf_def greatestP_def isLb_def setle_def setge_def)
2142 lemma reals_complete_Glb: assumes Se: "\<exists>x. x \<in> S" and lb: "\<exists> y. isLb UNIV S y"
2143   shows "\<exists>(t::real). isGlb UNIV S t"
2144 proof-
2145   let ?M = "uminus ` S"
2146   from lb have th: "\<exists>y. isUb UNIV ?M y" apply (auto simp add: isUb_def isLb_def setle_def setge_def)
2147     by (rule_tac x="-y" in exI, auto)
2148   from Se have Me: "\<exists>x. x \<in> ?M" by blast
2149   from reals_complete[OF Me th] obtain t where t: "isLub UNIV ?M t" by blast
2150   have "isGlb UNIV S (- t)" using t
2151     apply (auto simp add: isLub_def isGlb_def leastP_def greatestP_def setle_def setge_def isUb_def isLb_def)
2152     apply (erule_tac x="-y" in allE)
2153     apply auto
2154     done
2155   then show ?thesis by metis
2156 qed
2158 lemma rinf: assumes Se: "S \<noteq> {}" and b: "\<exists>b. b <=* S"
2159   shows "isGlb UNIV S (rinf S)"
2160 using Se b
2161 unfolding rinf_def
2162 apply clarify
2163 apply (rule someI_ex)
2164 apply (rule reals_complete_Glb)
2165 apply (auto simp add: isLb_def setle_def setge_def)
2166 done
2168 lemma rinf_ge: assumes Se: "S \<noteq> {}" and Sb: "b <=* S" shows "rinf S \<ge> b"
2169 proof-
2170   from Sb have bu: "isLb UNIV S b" by (simp add: isLb_def setge_def)
2171   from rinf[OF Se] Sb have "isGlb UNIV S (rinf S)"  by blast
2172   then show ?thesis using bu by (auto simp add: isGlb_def greatestP_def setle_def setge_def)
2173 qed
2175 lemma rinf_finite_Min: assumes fS: "finite S" and Se: "S \<noteq> {}"
2176   shows "rinf S = Min S"
2177 using fS Se
2178 proof-
2179   let ?m = "Min S"
2180   from Min_le[OF fS] have Sm: "\<forall> x\<in> S. x \<ge> ?m" by metis
2181   with rinf[OF Se] have glb: "isGlb UNIV S (rinf S)" by (metis setge_def)
2182   from Min_in[OF fS Se] glb have mrS: "?m \<ge> rinf S"
2183     by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def)
2184   moreover
2185   have "rinf S \<ge> ?m" using Sm glb
2186     by (auto simp add: isGlb_def greatestP_def isLb_def setle_def setge_def)
2187   ultimately  show ?thesis by arith
2188 qed
2190 lemma rinf_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
2191   shows "rinf S \<in> S"
2192   using rinf_finite_Min[OF fS Se] Min_in[OF fS Se] by metis
2194 lemma rinf_finite_Lb: assumes fS: "finite S" and Se: "S \<noteq> {}"
2195   shows "isLb S S (rinf S)"
2196   using rinf_finite_Min[OF fS Se] rinf_finite_in[OF fS Se] Min_le[OF fS]
2197   unfolding isLb_def setge_def by metis
2199 lemma rinf_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
2200   shows "a \<le> rinf S \<longleftrightarrow> (\<forall> x \<in> S. a \<le> x)"
2201 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
2203 lemma rinf_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
2204   shows "a \<ge> rinf S \<longleftrightarrow> (\<exists> x \<in> S. a \<ge> x)"
2205 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
2207 lemma rinf_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
2208   shows "a < rinf S \<longleftrightarrow> (\<forall> x \<in> S. a < x)"
2209 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
2211 lemma rinf_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
2212   shows "a > rinf S \<longleftrightarrow> (\<exists> x \<in> S. a > x)"
2213 using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
2215 lemma rinf_unique: assumes b: "b <=* S" and S: "\<forall>b' > b. \<exists>x \<in> S. b' > x"
2216   shows "rinf S = b"
2217 using b S
2218 unfolding setge_def rinf_alt
2219 apply -
2220 apply (rule some_equality)
2221 apply (metis  linorder_not_le order_eq_iff[symmetric])+
2222 done
2224 lemma rinf_ge_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. b <=* T) \<Longrightarrow> rinf S >= rinf T"
2225   apply (rule rinf_ge)
2226   apply simp
2227   using rinf[of T] by (auto simp add: isGlb_def greatestP_def setge_def setle_def isLb_def)
2229 lemma isLb_def': "isLb R S = (\<lambda>x. x <=* S \<and> x \<in> R)"
2230   apply (rule ext)
2231   by (metis isLb_def)
2233 lemma rinf_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
2234   shows "a \<le> rinf S \<and> rinf S \<le> b"
2235 proof-
2236   from rinf[OF Se] l have lub: "isGlb UNIV S (rinf S)" by blast
2237   hence b: "a \<le> rinf S" using l by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')
2238   from Se obtain y where y: "y \<in> S" by blast
2239   from lub u have "b \<ge> rinf S" apply (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')
2240     apply (erule ballE[where x=y])
2241     apply (erule ballE[where x=y])
2242     apply arith
2243     using y apply auto
2244     done
2245   with b show ?thesis by blast
2246 qed
2248 lemma rinf_abs_ge: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rinf S\<bar> \<le> a"
2249   unfolding abs_le_interval_iff  using rinf_bounds[of S "-a" a]
2250   by (auto simp add: setge_def setle_def)
2252 lemma rinf_asclose: assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>rinf S - l\<bar> \<le> e"
2253 proof-
2254   have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith
2255   show ?thesis using S b rinf_bounds[of S "l - e" "l+e"] unfolding th
2256     by  (auto simp add: setge_def setle_def)
2257 qed
2261 subsection{* Operator norm. *}
2263 definition "onorm f = rsup {norm (f x)| x. norm x = 1}"
2265 lemma norm_bound_generalize:
2266   fixes f:: "real ^'n \<Rightarrow> real^'m"
2267   assumes lf: "linear f"
2268   shows "(\<forall>x. norm x = 1 \<longrightarrow> norm (f x) \<le> b) \<longleftrightarrow> (\<forall>x. norm (f x) \<le> b * norm x)" (is "?lhs \<longleftrightarrow> ?rhs")
2269 proof-
2270   {assume H: ?rhs
2271     {fix x :: "real^'n" assume x: "norm x = 1"
2272       from H[rule_format, of x] x have "norm (f x) \<le> b" by simp}
2273     then have ?lhs by blast }
2275   moreover
2276   {assume H: ?lhs
2277     from H[rule_format, of "basis 1"]
2278     have bp: "b \<ge> 0" using norm_pos_le[of "f (basis 1)"] dimindex_ge_1[of "UNIV:: 'n set"]
2279       by (auto simp add: norm_basis)
2280     {fix x :: "real ^'n"
2281       {assume "x = 0"
2282 	then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] norm_0 bp)}
2283       moreover
2284       {assume x0: "x \<noteq> 0"
2285 	hence n0: "norm x \<noteq> 0" by (metis norm_eq_0)
2286 	let ?c = "1/ norm x"
2287 	have "norm (?c*s x) = 1" by (simp add: n0 norm_mul)
2288 	with H have "norm (f(?c*s x)) \<le> b" by blast
2289 	hence "?c * norm (f x) \<le> b"
2290 	  by (simp add: linear_cmul[OF lf] norm_mul)
2291 	hence "norm (f x) \<le> b * norm x"
2292 	  using n0 norm_pos_le[of x] by (auto simp add: field_simps)}
2293       ultimately have "norm (f x) \<le> b * norm x" by blast}
2294     then have ?rhs by blast}
2295   ultimately show ?thesis by blast
2296 qed
2298 lemma onorm:
2299   fixes f:: "real ^'n \<Rightarrow> real ^'m"
2300   assumes lf: "linear f"
2301   shows "norm (f x) <= onorm f * norm x"
2302   and "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
2303 proof-
2304   {
2305     let ?S = "{norm (f x) |x. norm x = 1}"
2306     have Se: "?S \<noteq> {}" using  norm_basis_1 by auto
2307     from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b"
2308       unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def)
2309     {from rsup[OF Se b, unfolded onorm_def[symmetric]]
2310       show "norm (f x) <= onorm f * norm x"
2311 	apply -
2312 	apply (rule spec[where x = x])
2313 	unfolding norm_bound_generalize[OF lf, symmetric]
2314 	by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
2315     {
2316       show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
2317 	using rsup[OF Se b, unfolded onorm_def[symmetric]]
2318 	unfolding norm_bound_generalize[OF lf, symmetric]
2319 	by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
2320   }
2321 qed
2323 lemma onorm_pos_le: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" shows "0 <= onorm f"
2324   using order_trans[OF norm_pos_le onorm(1)[OF lf, of "basis 1"], unfolded norm_basis_1] by simp
2326 lemma onorm_eq_0: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)"
2327   shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
2328   using onorm[OF lf]
2329   apply (auto simp add: norm_0 onorm_pos_le norm_le_0)
2330   apply atomize
2331   apply (erule allE[where x="0::real"])
2332   using onorm_pos_le[OF lf]
2333   apply arith
2334   done
2336 lemma onorm_const: "onorm(\<lambda>x::real^'n. (y::real ^ 'm)) = norm y"
2337 proof-
2338   let ?f = "\<lambda>x::real^'n. (y::real ^ 'm)"
2339   have th: "{norm (?f x)| x. norm x = 1} = {norm y}"
2340     by(auto intro: vector_choose_size set_ext)
2341   show ?thesis
2342     unfolding onorm_def th
2343     apply (rule rsup_unique) by (simp_all  add: setle_def)
2344 qed
2346 lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n \<Rightarrow> real ^'m)"
2347   shows "0 < onorm f \<longleftrightarrow> ~(\<forall>x. f x = 0)"
2348   unfolding onorm_eq_0[OF lf, symmetric]
2349   using onorm_pos_le[OF lf] by arith
2351 lemma onorm_compose:
2352   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and lg: "linear g"
2353   shows "onorm (f o g) <= onorm f * onorm g"
2354   apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format])
2355   unfolding o_def
2356   apply (subst mult_assoc)
2357   apply (rule order_trans)
2358   apply (rule onorm(1)[OF lf])
2359   apply (rule mult_mono1)
2360   apply (rule onorm(1)[OF lg])
2361   apply (rule onorm_pos_le[OF lf])
2362   done
2364 lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"
2365   shows "onorm (\<lambda>x. - f x) \<le> onorm f"
2366   using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf]
2367   unfolding norm_neg by metis
2369 lemma onorm_neg: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"
2370   shows "onorm (\<lambda>x. - f x) = onorm f"
2371   using onorm_neg_lemma[OF lf] onorm_neg_lemma[OF linear_compose_neg[OF lf]]
2372   by simp
2374 lemma onorm_triangle:
2375   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and lg: "linear g"
2376   shows "onorm (\<lambda>x. f x + g x) <= onorm f + onorm g"
2377   apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format])
2378   apply (rule order_trans)
2379   apply (rule norm_triangle)
2382   apply (rule onorm(1)[OF lf])
2383   apply (rule onorm(1)[OF lg])
2384   done
2386 lemma onorm_triangle_le: "linear (f::real ^'n \<Rightarrow> real ^'m) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) <= e
2387   \<Longrightarrow> onorm(\<lambda>x. f x + g x) <= e"
2388   apply (rule order_trans)
2389   apply (rule onorm_triangle)
2390   apply assumption+
2391   done
2393 lemma onorm_triangle_lt: "linear (f::real ^'n \<Rightarrow> real ^'m) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) < e
2394   ==> onorm(\<lambda>x. f x + g x) < e"
2395   apply (rule order_le_less_trans)
2396   apply (rule onorm_triangle)
2397   by assumption+
2399 (* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *)
2401 definition vec1:: "'a \<Rightarrow> 'a ^ 1" where "vec1 x = (\<chi> i. x)"
2403 definition dest_vec1:: "'a ^1 \<Rightarrow> 'a"
2404   where "dest_vec1 x = (x\$1)"
2406 lemma vec1_component[simp]: "(vec1 x)\$1 = x"
2409 lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"
2410   by (simp_all add: vec1_def dest_vec1_def Cart_eq Cart_lambda_beta dimindex_def del: One_nat_def)
2412 lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))" by (metis vec1_dest_vec1)
2414 lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))" by (metis vec1_dest_vec1)
2416 lemma forall_dest_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P(dest_vec1 x))"  by (metis vec1_dest_vec1)
2418 lemma exists_dest_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(dest_vec1 x))"by (metis vec1_dest_vec1)
2420 lemma vec1_eq[simp]:  "vec1 x = vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
2422 lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
2424 lemma vec1_in_image_vec1: "vec1 x \<in> (vec1 ` S) \<longleftrightarrow> x \<in> S" by auto
2426 lemma vec1_vec: "vec1 x = vec x" by (vector vec1_def)
2428 lemma vec1_add: "vec1(x + y) = vec1 x + vec1 y" by (vector vec1_def)
2429 lemma vec1_sub: "vec1(x - y) = vec1 x - vec1 y" by (vector vec1_def)
2430 lemma vec1_cmul: "vec1(c* x) = c *s vec1 x " by (vector vec1_def)
2431 lemma vec1_neg: "vec1(- x) = - vec1 x " by (vector vec1_def)
2433 lemma vec1_setsum: assumes fS: "finite S"
2434   shows "vec1(setsum f S) = setsum (vec1 o f) S"
2435   apply (induct rule: finite_induct[OF fS])
2438   done
2440 lemma dest_vec1_lambda: "dest_vec1(\<chi> i. x i) = x 1"
2443 lemma dest_vec1_vec: "dest_vec1(vec x) = x"
2446 lemma dest_vec1_add: "dest_vec1(x + y) = dest_vec1 x + dest_vec1 y"
2449 lemma dest_vec1_sub: "dest_vec1(x - y) = dest_vec1 x - dest_vec1 y"
2450  by (metis vec1_dest_vec1 vec1_sub)
2452 lemma dest_vec1_cmul: "dest_vec1(c*sx) = c * dest_vec1 x"
2453  by (metis vec1_dest_vec1 vec1_cmul)
2455 lemma dest_vec1_neg: "dest_vec1(- x) = - dest_vec1 x"
2456  by (metis vec1_dest_vec1 vec1_neg)
2458 lemma dest_vec1_0[simp]: "dest_vec1 0 = 0" by (metis vec_0 dest_vec1_vec)
2460 lemma dest_vec1_sum: assumes fS: "finite S"
2461   shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S"
2462   apply (induct rule: finite_induct[OF fS])
2465   done
2467 lemma norm_vec1: "norm(vec1 x) = abs(x)"
2468   by (simp add: vec1_def norm_real)
2470 lemma dist_vec1: "dist(vec1 x) (vec1 y) = abs(x - y)"
2471   by (simp only: dist_real vec1_component)
2472 lemma abs_dest_vec1: "norm x = \<bar>dest_vec1 x\<bar>"
2473   by (metis vec1_dest_vec1 norm_vec1)
2475 lemma linear_vmul_dest_vec1:
2476   fixes f:: "'a::semiring_1^'n \<Rightarrow> 'a^1"
2477   shows "linear f \<Longrightarrow> linear (\<lambda>x. dest_vec1(f x) *s v)"
2478   unfolding dest_vec1_def
2479   apply (rule linear_vmul_component)
2480   by (auto simp add: dimindex_def)
2482 lemma linear_from_scalars:
2483   assumes lf: "linear (f::'a::comm_ring_1 ^1 \<Rightarrow> 'a^'n)"
2484   shows "f = (\<lambda>x. dest_vec1 x *s column 1 (matrix f))"
2485   apply (rule ext)
2486   apply (subst matrix_works[OF lf, symmetric])
2487   apply (auto simp add: Cart_eq matrix_vector_mult_def dest_vec1_def column_def Cart_lambda_beta vector_component dimindex_def mult_commute del: One_nat_def )
2488   done
2490 lemma linear_to_scalars: assumes lf: "linear (f::'a::comm_ring_1 ^'n \<Rightarrow> 'a^1)"
2491   shows "f = (\<lambda>x. vec1(row 1 (matrix f) \<bullet> x))"
2492   apply (rule ext)
2493   apply (subst matrix_works[OF lf, symmetric])
2494   apply (auto simp add: Cart_eq matrix_vector_mult_def vec1_def row_def Cart_lambda_beta vector_component dimindex_def dot_def mult_commute)
2495   done
2497 lemma dest_vec1_eq_0: "dest_vec1 x = 0 \<longleftrightarrow> x = 0"
2500 lemma setsum_scalars: assumes fS: "finite S"
2501   shows "setsum f S = vec1 (setsum (dest_vec1 o f) S)"
2502   unfolding vec1_setsum[OF fS] by simp
2504 lemma dest_vec1_wlog_le: "(\<And>(x::'a::linorder ^ 1) y. P x y \<longleftrightarrow> P y x)  \<Longrightarrow> (\<And>x y. dest_vec1 x <= dest_vec1 y ==> P x y) \<Longrightarrow> P x y"
2505   apply (cases "dest_vec1 x \<le> dest_vec1 y")
2506   apply simp
2507   apply (subgoal_tac "dest_vec1 y \<le> dest_vec1 x")
2508   apply (auto)
2509   done
2511 text{* Pasting vectors. *}
2513 lemma linear_fstcart: "linear fstcart"
2514   by (auto simp add: linear_def fstcart_def Cart_eq Cart_lambda_beta vector_component dimindex_finite_sum)
2516 lemma linear_sndcart: "linear sndcart"
2517   by (auto simp add: linear_def sndcart_def Cart_eq Cart_lambda_beta vector_component dimindex_finite_sum)
2519 lemma fstcart_vec[simp]: "fstcart(vec x) = vec x"
2520   by (vector fstcart_def vec_def dimindex_finite_sum)
2522 lemma fstcart_add[simp]:"fstcart(x + y) = fstcart (x::'a::{plus,times}^('b,'c) finite_sum) + fstcart y"
2523   using linear_fstcart[unfolded linear_def] by blast
2525 lemma fstcart_cmul[simp]:"fstcart(c*s x) = c*s fstcart (x::'a::{plus,times}^('b,'c) finite_sum)"
2526   using linear_fstcart[unfolded linear_def] by blast
2528 lemma fstcart_neg[simp]:"fstcart(- x) = - fstcart (x::'a::ring_1^('b,'c) finite_sum)"
2529 unfolding vector_sneg_minus1 fstcart_cmul ..
2531 lemma fstcart_sub[simp]:"fstcart(x - y) = fstcart (x::'a::ring_1^('b,'c) finite_sum) - fstcart y"
2532   unfolding diff_def fstcart_add fstcart_neg  ..
2534 lemma fstcart_setsum:
2535   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
2536   assumes fS: "finite S"
2537   shows "fstcart (setsum f S) = setsum (\<lambda>i. fstcart (f i)) S"
2538   by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
2540 lemma sndcart_vec[simp]: "sndcart(vec x) = vec x"
2541   by (vector sndcart_def vec_def dimindex_finite_sum)
2543 lemma sndcart_add[simp]:"sndcart(x + y) = sndcart (x::'a::{plus,times}^('b,'c) finite_sum) + sndcart y"
2544   using linear_sndcart[unfolded linear_def] by blast
2546 lemma sndcart_cmul[simp]:"sndcart(c*s x) = c*s sndcart (x::'a::{plus,times}^('b,'c) finite_sum)"
2547   using linear_sndcart[unfolded linear_def] by blast
2549 lemma sndcart_neg[simp]:"sndcart(- x) = - sndcart (x::'a::ring_1^('b,'c) finite_sum)"
2550 unfolding vector_sneg_minus1 sndcart_cmul ..
2552 lemma sndcart_sub[simp]:"sndcart(x - y) = sndcart (x::'a::ring_1^('b,'c) finite_sum) - sndcart y"
2553   unfolding diff_def sndcart_add sndcart_neg  ..
2555 lemma sndcart_setsum:
2556   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
2557   assumes fS: "finite S"
2558   shows "sndcart (setsum f S) = setsum (\<lambda>i. sndcart (f i)) S"
2559   by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
2561 lemma pastecart_vec[simp]: "pastecart (vec x) (vec x) = vec x"
2562   by (simp add: pastecart_eq fstcart_vec sndcart_vec fstcart_pastecart sndcart_pastecart)
2564 lemma pastecart_add[simp]:"pastecart (x1::'a::{plus,times}^_) y1 + pastecart x2 y2 = pastecart (x1 + x2) (y1 + y2)"
2567 lemma pastecart_cmul[simp]: "pastecart (c *s (x1::'a::{plus,times}^_)) (c *s y1) = c *s pastecart x1 y1"
2568   by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
2570 lemma pastecart_neg[simp]: "pastecart (- (x::'a::ring_1^_)) (- y) = - pastecart x y"
2571   unfolding vector_sneg_minus1 pastecart_cmul ..
2573 lemma pastecart_sub: "pastecart (x1::'a::ring_1^_) y1 - pastecart x2 y2 = pastecart (x1 - x2) (y1 - y2)"
2574   by (simp add: diff_def pastecart_neg[symmetric] del: pastecart_neg)
2576 lemma pastecart_setsum:
2577   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
2578   assumes fS: "finite S"
2579   shows "pastecart (setsum f S) (setsum g S) = setsum (\<lambda>i. pastecart (f i) (g i)) S"
2580   by (simp  add: pastecart_eq fstcart_setsum[OF fS] sndcart_setsum[OF fS] fstcart_pastecart sndcart_pastecart)
2582 lemma norm_fstcart: "norm(fstcart x) <= norm (x::real ^('n,'m) finite_sum)"
2583 proof-
2584   let ?n = "dimindex (UNIV :: 'n set)"
2585   let ?m = "dimindex (UNIV :: 'm set)"
2586   let ?N = "{1 .. ?n}"
2587   let ?M = "{1 .. ?m}"
2588   let ?NM = "{1 .. dimindex (UNIV :: ('n,'m) finite_sum set)}"
2589   have th_0: "1 \<le> ?n +1" by simp
2590   have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
2592   have th1: "fstcart x \<bullet> fstcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
2593     by (simp add: dot_def setsum_add_split[OF th_0, of _ ?m] pastecart_def dimindex_finite_sum Cart_lambda_beta setsum_nonneg zero_le_square del: One_nat_def)
2594   then show ?thesis
2595     unfolding th0
2596     unfolding real_vector_norm_def real_sqrt_le_iff real_of_real_def id_def
2597     by (simp add: dot_def dimindex_finite_sum Cart_lambda_beta)
2598 qed
2600 lemma dist_fstcart: "dist(fstcart (x::real^_)) (fstcart y) <= dist x y"
2601   by (metis dist_def fstcart_sub[symmetric] norm_fstcart)
2603 lemma norm_sndcart: "norm(sndcart x) <= norm (x::real ^('n,'m) finite_sum)"
2604 proof-
2605   let ?n = "dimindex (UNIV :: 'n set)"
2606   let ?m = "dimindex (UNIV :: 'm set)"
2607   let ?N = "{1 .. ?n}"
2608   let ?M = "{1 .. ?m}"
2609   let ?nm = "dimindex (UNIV :: ('n,'m) finite_sum set)"
2610   let ?NM = "{1 .. ?nm}"
2611   have thnm[simp]: "?nm = ?n + ?m" by (simp add: dimindex_finite_sum)
2612   have th_0: "1 \<le> ?n +1" by simp
2613   have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
2615   let ?f = "\<lambda>n. n - ?n"
2616   let ?S = "{?n+1 .. ?nm}"
2617   have finj:"inj_on ?f ?S"
2618     using dimindex_nonzero[of "UNIV :: 'n set"] dimindex_nonzero[of "UNIV :: 'm set"]
2619     apply (simp add: Ball_def atLeastAtMost_iff inj_on_def dimindex_finite_sum del: One_nat_def)
2620     by arith
2621   have fS: "?f ` ?S = ?M"
2622     apply (rule set_ext)
2623     apply (simp add: image_iff Bex_def) using dimindex_nonzero[of "UNIV :: 'n set"] dimindex_nonzero[of "UNIV :: 'm set"] by arith
2624   have th1: "sndcart x \<bullet> sndcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
2625     by (simp add: dot_def setsum_add_split[OF th_0, of _ ?m] pastecart_def dimindex_finite_sum Cart_lambda_beta setsum_nonneg zero_le_square setsum_reindex[OF finj, unfolded fS] del: One_nat_def)
2626   then show ?thesis
2627     unfolding th0
2628     unfolding real_vector_norm_def real_sqrt_le_iff real_of_real_def id_def
2629     by (simp add: dot_def dimindex_finite_sum Cart_lambda_beta)
2630 qed
2632 lemma dist_sndcart: "dist(sndcart (x::real^_)) (sndcart y) <= dist x y"
2633   by (metis dist_def sndcart_sub[symmetric] norm_sndcart)
2635 lemma dot_pastecart: "(pastecart (x1::'a::{times,comm_monoid_add}^'n) (x2::'a::{times,comm_monoid_add}^'m)) \<bullet> (pastecart y1 y2) =  x1 \<bullet> y1 + x2 \<bullet> y2"
2636 proof-
2637   let ?n = "dimindex (UNIV :: 'n set)"
2638   let ?m = "dimindex (UNIV :: 'm set)"
2639   let ?N = "{1 .. ?n}"
2640   let ?M = "{1 .. ?m}"
2641   let ?nm = "dimindex (UNIV :: ('n,'m) finite_sum set)"
2642   let ?NM = "{1 .. ?nm}"
2643   have thnm: "?nm = ?n + ?m" by (simp add: dimindex_finite_sum)
2644   have th_0: "1 \<le> ?n +1" by simp
2645   have th_1: "\<And>i. i \<in> {?m + 1 .. ?nm} \<Longrightarrow> i - ?m \<in> ?N" apply (simp add: thnm) by arith
2646   let ?f = "\<lambda>a b i. (a\$i) * (b\$i)"
2647   let ?g = "?f (pastecart x1 x2) (pastecart y1 y2)"
2648   let ?S = "{?n +1 .. ?nm}"
2649   {fix i
2650     assume i: "i \<in> ?N"
2651     have "?g i = ?f x1 y1 i"
2652       using i
2653       apply (simp add: pastecart_def Cart_lambda_beta thnm) done
2654   }
2655   hence th2: "setsum ?g ?N = setsum (?f x1 y1) ?N"
2656     apply -
2657     apply (rule setsum_cong)
2658     apply auto
2659     done
2660   {fix i
2661     assume i: "i \<in> ?S"
2662     have "?g i = ?f x2 y2 (i - ?n)"
2663       using i
2664       apply (simp add: pastecart_def Cart_lambda_beta thnm) done
2665   }
2666   hence th3: "setsum ?g ?S = setsum (\<lambda>i. ?f x2 y2 (i -?n)) ?S"
2667     apply -
2668     apply (rule setsum_cong)
2669     apply auto
2670     done
2671   let ?r = "\<lambda>n. n - ?n"
2672   have rinj: "inj_on ?r ?S" apply (simp add: inj_on_def Ball_def thnm) by arith
2673   have rS: "?r ` ?S = ?M" apply (rule set_ext)
2674     apply (simp add: thnm image_iff Bex_def) by arith
2675   have "pastecart x1 x2 \<bullet> (pastecart y1 y2) = setsum ?g ?NM" by (simp add: dot_def)
2676   also have "\<dots> = setsum ?g ?N + setsum ?g ?S"
2677     by (simp add: dot_def thnm setsum_add_split[OF th_0, of _ ?m] del: One_nat_def)
2678   also have "\<dots> = setsum (?f x1 y1) ?N + setsum (?f x2 y2) ?M"
2679     unfolding setsum_reindex[OF rinj, unfolded rS o_def] th2 th3 ..
2680   finally
2681   show ?thesis by (simp add: dot_def)
2682 qed
2684 lemma norm_pastecart: "norm(pastecart x y) <= norm(x :: real ^ _) + norm(y)"
2685   unfolding real_vector_norm_def dot_pastecart real_sqrt_le_iff real_of_real_def id_def
2686   apply (rule power2_le_imp_le)
2688   apply (auto simp add: power2_eq_square ring_simps)
2690   apply (rule mult_nonneg_nonneg)
2691   apply (simp_all add: real_sqrt_pow2[OF dot_pos_le])
2693   apply (simp_all add: real_sqrt_pow2[OF dot_pos_le])
2694   done
2696 subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
2698 definition hull :: "'a set set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75) where
2699   "S hull s = Inter {t. t \<in> S \<and> s \<subseteq> t}"
2701 lemma hull_same: "s \<in> S \<Longrightarrow> S hull s = s"
2702   unfolding hull_def by auto
2704 lemma hull_in: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) \<in> S"
2705 unfolding hull_def subset_iff by auto
2707 lemma hull_eq: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) = s \<longleftrightarrow> s \<in> S"
2708 using hull_same[of s S] hull_in[of S s] by metis
2711 lemma hull_hull: "S hull (S hull s) = S hull s"
2712   unfolding hull_def by blast
2714 lemma hull_subset: "s \<subseteq> (S hull s)"
2715   unfolding hull_def by blast
2717 lemma hull_mono: " s \<subseteq> t ==> (S hull s) \<subseteq> (S hull t)"
2718   unfolding hull_def by blast
2720 lemma hull_antimono: "S \<subseteq> T ==> (T hull s) \<subseteq> (S hull s)"
2721   unfolding hull_def by blast
2723 lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> t \<in> S ==> (S hull s) \<subseteq> t"
2724   unfolding hull_def by blast
2726 lemma subset_hull: "t \<in> S ==> S hull s \<subseteq> t \<longleftrightarrow>  s \<subseteq> t"
2727   unfolding hull_def by blast
2729 lemma hull_unique: "s \<subseteq> t \<Longrightarrow> t \<in> S \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> t' \<in> S ==> t \<subseteq> t')
2730            ==> (S hull s = t)"
2731 unfolding hull_def by auto
2733 lemma hull_induct: "(\<And>x. x\<in> S \<Longrightarrow> P x) \<Longrightarrow> Q {x. P x} \<Longrightarrow> \<forall>x\<in> Q hull S. P x"
2734   using hull_minimal[of S "{x. P x}" Q]
2735   by (auto simp add: subset_eq Collect_def mem_def)
2737 lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S" by (metis hull_subset subset_eq)
2739 lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
2740 unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
2742 lemma hull_union: assumes T: "\<And>T. T \<subseteq> S ==> Inter T \<in> S"
2743   shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
2744 apply rule
2745 apply (rule hull_mono)
2746 unfolding Un_subset_iff
2747 apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
2748 apply (rule hull_minimal)
2749 apply (metis hull_union_subset)
2750 apply (metis hull_in T)
2751 done
2753 lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)"
2754   unfolding hull_def by blast
2756 lemma hull_redundant: "a \<in> (S hull s) ==> (S hull (insert a s) = S hull s)"
2757 by (metis hull_redundant_eq)
2759 text{* Archimedian properties and useful consequences. *}
2761 lemma real_arch_simple: "\<exists>n. x <= real (n::nat)"
2762   using reals_Archimedean2[of x] apply auto by (rule_tac x="Suc n" in exI, auto)
2763 lemmas real_arch_lt = reals_Archimedean2
2765 lemmas real_arch = reals_Archimedean3
2767 lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
2768   using reals_Archimedean
2769   apply (auto simp add: field_simps inverse_positive_iff_positive)
2770   apply (subgoal_tac "inverse (real n) > 0")
2771   apply arith
2772   apply simp
2773   done
2775 lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
2776 proof(induct n)
2777   case 0 thus ?case by simp
2778 next
2779   case (Suc n)
2780   hence h: "1 + real n * x \<le> (1 + x) ^ n" by simp
2781   from h have p: "1 \<le> (1 + x) ^ n" using Suc.prems by simp
2782   from h have "1 + real n * x + x \<le> (1 + x) ^ n + x" by simp
2783   also have "\<dots> \<le> (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric])
2785     using mult_left_mono[OF p Suc.prems] by simp
2786   finally show ?case  by (simp add: real_of_nat_Suc ring_simps)
2787 qed
2789 lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\<exists>n. y < x^n"
2790 proof-
2791   from x have x0: "x - 1 > 0" by arith
2792   from real_arch[OF x0, rule_format, of y]
2793   obtain n::nat where n:"y < real n * (x - 1)" by metis
2794   from x0 have x00: "x- 1 \<ge> 0" by arith
2795   from real_pow_lbound[OF x00, of n] n
2796   have "y < x^n" by auto
2797   then show ?thesis by metis
2798 qed
2800 lemma real_arch_pow2: "\<exists>n. (x::real) < 2^ n"
2801   using real_arch_pow[of 2 x] by simp
2803 lemma real_arch_pow_inv: assumes y: "(y::real) > 0" and x1: "x < 1"
2804   shows "\<exists>n. x^n < y"
2805 proof-
2806   {assume x0: "x > 0"
2807     from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps)
2808     from real_arch_pow[OF ix, of "1/y"]
2809     obtain n where n: "1/y < (1/x)^n" by blast
2810     then
2811     have ?thesis using y x0 by (auto simp add: field_simps power_divide) }
2812   moreover
2813   {assume "\<not> x > 0" with y x1 have ?thesis apply auto by (rule exI[where x=1], auto)}
2814   ultimately show ?thesis by metis
2815 qed
2817 lemma forall_pos_mono: "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow> (\<And>n::nat. n \<noteq> 0 ==> P(inverse(real n))) \<Longrightarrow> (\<And>e. 0 < e ==> P e)"
2818   by (metis real_arch_inv)
2820 lemma forall_pos_mono_1: "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow> (\<And>n. P(inverse(real (Suc n)))) ==> 0 < e ==> P e"
2821   apply (rule forall_pos_mono)
2822   apply auto
2823   apply (atomize)
2824   apply (erule_tac x="n - 1" in allE)
2825   apply auto
2826   done
2828 lemma real_archimedian_rdiv_eq_0: assumes x0: "x \<ge> 0" and c: "c \<ge> 0" and xc: "\<forall>(m::nat)>0. real m * x \<le> c"
2829   shows "x = 0"
2830 proof-
2831   {assume "x \<noteq> 0" with x0 have xp: "x > 0" by arith
2832     from real_arch[OF xp, rule_format, of c] obtain n::nat where n: "c < real n * x"  by blast
2833     with xc[rule_format, of n] have "n = 0" by arith
2834     with n c have False by simp}
2835   then show ?thesis by blast
2836 qed
2838 (* ------------------------------------------------------------------------- *)
2839 (* Relate max and min to sup and inf.                                        *)
2840 (* ------------------------------------------------------------------------- *)
2842 lemma real_max_rsup: "max x y = rsup {x,y}"
2843 proof-
2844   have f: "finite {x, y}" "{x,y} \<noteq> {}"  by simp_all
2845   from rsup_finite_le_iff[OF f, of "max x y"] have "rsup {x,y} \<le> max x y" by simp
2846   moreover
2847   have "max x y \<le> rsup {x,y}" using rsup_finite_ge_iff[OF f, of "max x y"]
2849   ultimately show ?thesis by arith
2850 qed
2852 lemma real_min_rinf: "min x y = rinf {x,y}"
2853 proof-
2854   have f: "finite {x, y}" "{x,y} \<noteq> {}"  by simp_all
2855   from rinf_finite_le_iff[OF f, of "min x y"] have "rinf {x,y} \<le> min x y"
2857   moreover
2858   have "min x y \<le> rinf {x,y}" using rinf_finite_ge_iff[OF f, of "min x y"]
2859     by simp
2860   ultimately show ?thesis by arith
2861 qed
2863 (* ------------------------------------------------------------------------- *)
2864 (* Geometric progression.                                                    *)
2865 (* ------------------------------------------------------------------------- *)
2867 lemma sum_gp_basic: "((1::'a::{field, recpower}) - x) * setsum (\<lambda>i. x^i) {0 .. n} = (1 - x^(Suc n))"
2868   (is "?lhs = ?rhs")
2869 proof-
2870   {assume x1: "x = 1" hence ?thesis by simp}
2871   moreover
2872   {assume x1: "x\<noteq>1"
2873     hence x1': "x - 1 \<noteq> 0" "1 - x \<noteq> 0" "x - 1 = - (1 - x)" "- (1 - x) \<noteq> 0" by auto
2874     from geometric_sum[OF x1, of "Suc n", unfolded x1']
2875     have "(- (1 - x)) * setsum (\<lambda>i. x^i) {0 .. n} = - (1 - x^(Suc n))"
2876       unfolding atLeastLessThanSuc_atLeastAtMost
2877       using x1' apply (auto simp only: field_simps)
2879       done
2880     then have ?thesis by (simp add: ring_simps) }
2881   ultimately show ?thesis by metis
2882 qed
2884 lemma sum_gp_multiplied: assumes mn: "m <= n"
2885   shows "((1::'a::{field, recpower}) - x) * setsum (op ^ x) {m..n} = x^m - x^ Suc n"
2886   (is "?lhs = ?rhs")
2887 proof-
2888   let ?S = "{0..(n - m)}"
2889   from mn have mn': "n - m \<ge> 0" by arith
2890   let ?f = "op + m"
2891   have i: "inj_on ?f ?S" unfolding inj_on_def by auto
2892   have f: "?f ` ?S = {m..n}"
2893     using mn apply (auto simp add: image_iff Bex_def) by arith
2894   have th: "op ^ x o op + m = (\<lambda>i. x^m * x^i)"
2896   from setsum_reindex[OF i, of "op ^ x", unfolded f th setsum_right_distrib[symmetric]]
2897   have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})" by simp
2898   then show ?thesis unfolding sum_gp_basic using mn
2900 qed
2902 lemma sum_gp: "setsum (op ^ (x::'a::{field, recpower})) {m .. n} =
2903    (if n < m then 0 else if x = 1 then of_nat ((n + 1) - m)
2904                     else (x^ m - x^ (Suc n)) / (1 - x))"
2905 proof-
2906   {assume nm: "n < m" hence ?thesis by simp}
2907   moreover
2908   {assume "\<not> n < m" hence nm: "m \<le> n" by arith
2909     {assume x: "x = 1"  hence ?thesis by simp}
2910     moreover
2911     {assume x: "x \<noteq> 1" hence nz: "1 - x \<noteq> 0" by simp
2912       from sum_gp_multiplied[OF nm, of x] nz have ?thesis by (simp add: field_simps)}
2913     ultimately have ?thesis by metis
2914   }
2915   ultimately show ?thesis by metis
2916 qed
2918 lemma sum_gp_offset: "setsum (op ^ (x::'a::{field,recpower})) {m .. m+n} =
2919   (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
2920   unfolding sum_gp[of x m "m + n"] power_Suc
2924 subsection{* A bit of linear algebra. *}
2926 definition "subspace S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>x\<in> S. \<forall>y \<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x \<in>S. c *s x \<in>S )"
2927 definition "span S = (subspace hull S)"
2928 definition "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span(S - {a}))"
2929 abbreviation "independent s == ~(dependent s)"
2931 (* Closure properties of subspaces.                                          *)
2933 lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def)
2935 lemma subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
2937 lemma subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S"
2938   by (metis subspace_def)
2940 lemma subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *s x \<in> S"
2941   by (metis subspace_def)
2943 lemma subspace_neg: "subspace S \<Longrightarrow> (x::'a::ring_1^'n) \<in> S \<Longrightarrow> - x \<in> S"
2944   by (metis vector_sneg_minus1 subspace_mul)
2946 lemma subspace_sub: "subspace S \<Longrightarrow> (x::'a::ring_1^'n) \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
2947   by (metis diff_def subspace_add subspace_neg)
2949 lemma subspace_setsum:
2950   assumes sA: "subspace A" and fB: "finite B"
2951   and f: "\<forall>x\<in> B. f x \<in> A"
2952   shows "setsum f B \<in> A"
2953   using  fB f sA
2954   apply(induct rule: finite_induct[OF fB])
2957 lemma subspace_linear_image:
2958   assumes lf: "linear (f::'a::semiring_1^'n \<Rightarrow> _)" and sS: "subspace S"
2959   shows "subspace(f ` S)"
2960   using lf sS linear_0[OF lf]
2961   unfolding linear_def subspace_def
2962   apply (auto simp add: image_iff)
2963   apply (rule_tac x="x + y" in bexI, auto)
2964   apply (rule_tac x="c*s x" in bexI, auto)
2965   done
2967 lemma subspace_linear_preimage: "linear (f::'a::semiring_1^'n \<Rightarrow> _) ==> subspace S ==> subspace {x. f x \<in> S}"
2968   by (auto simp add: subspace_def linear_def linear_0[of f])
2970 lemma subspace_trivial: "subspace {0::'a::semiring_1 ^_}"
2973 lemma subspace_inter: "subspace A \<Longrightarrow> subspace B ==> subspace (A \<inter> B)"
2977 lemma span_mono: "A \<subseteq> B ==> span A \<subseteq> span B"
2978   by (metis span_def hull_mono)
2980 lemma subspace_span: "subspace(span S)"
2981   unfolding span_def
2982   apply (rule hull_in[unfolded mem_def])
2983   apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
2984   apply auto
2985   apply (erule_tac x="X" in ballE)
2987   apply blast
2988   apply (erule_tac x="X" in ballE)
2989   apply (erule_tac x="X" in ballE)
2990   apply (erule_tac x="X" in ballE)
2991   apply (clarsimp simp add: mem_def)
2992   apply simp
2993   apply simp
2994   apply simp
2995   apply (erule_tac x="X" in ballE)
2996   apply (erule_tac x="X" in ballE)
2998   apply simp
2999   apply simp
3000   done
3002 lemma span_clauses:
3003   "a \<in> S ==> a \<in> span S"
3004   "0 \<in> span S"
3005   "x\<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
3006   "x \<in> span S \<Longrightarrow> c *s x \<in> span S"
3007   by (metis span_def hull_subset subset_eq subspace_span subspace_def)+
3009 lemma span_induct: assumes SP: "\<And>x. x \<in> S ==> P x"
3010   and P: "subspace P" and x: "x \<in> span S" shows "P x"
3011 proof-
3012   from SP have SP': "S \<subseteq> P" by (simp add: mem_def subset_eq)
3013   from P have P': "P \<in> subspace" by (simp add: mem_def)
3014   from x hull_minimal[OF SP' P', unfolded span_def[symmetric]]
3015   show "P x" by (metis mem_def subset_eq)
3016 qed
3018 lemma span_empty: "span {} = {(0::'a::semiring_0 ^ 'n)}"
3020   apply (rule hull_unique)
3021   apply (auto simp add: mem_def subspace_def)
3022   unfolding mem_def[of "0::'a^'n", symmetric]
3023   apply simp
3024   done
3026 lemma independent_empty: "independent {}"
3029 lemma independent_mono: "independent A \<Longrightarrow> B \<subseteq> A ==> independent B"
3030   apply (clarsimp simp add: dependent_def span_mono)
3031   apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
3032   apply force
3033   apply (rule span_mono)
3034   apply auto
3035   done
3037 lemma span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
3038   by (metis order_antisym span_def hull_minimal mem_def)
3040 lemma span_induct': assumes SP: "\<forall>x \<in> S. P x"
3041   and P: "subspace P" shows "\<forall>x \<in> span S. P x"
3042   using span_induct SP P by blast
3044 inductive span_induct_alt_help for S:: "'a::semiring_1^'n \<Rightarrow> bool"
3045   where
3046   span_induct_alt_help_0: "span_induct_alt_help S 0"
3047   | span_induct_alt_help_S: "x \<in> S \<Longrightarrow> span_induct_alt_help S z \<Longrightarrow> span_induct_alt_help S (c *s x + z)"
3049 lemma span_induct_alt':
3050   assumes h0: "h (0::'a::semiring_1^'n)" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c*s x + y)" shows "\<forall>x \<in> span S. h x"
3051 proof-
3052   {fix x:: "'a^'n" assume x: "span_induct_alt_help S x"
3053     have "h x"
3054       apply (rule span_induct_alt_help.induct[OF x])
3055       apply (rule h0)
3056       apply (rule hS, assumption, assumption)
3057       done}
3058   note th0 = this
3059   {fix x assume x: "x \<in> span S"
3061     have "span_induct_alt_help S x"
3062       proof(rule span_induct[where x=x and S=S])
3063 	show "x \<in> span S" using x .
3064       next
3065 	fix x assume xS : "x \<in> S"
3066 	  from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
3067 	  show "span_induct_alt_help S x" by simp
3068 	next
3069 	have "span_induct_alt_help S 0" by (rule span_induct_alt_help_0)
3070 	moreover
3071 	{fix x y assume h: "span_induct_alt_help S x" "span_induct_alt_help S y"
3072 	  from h
3073 	  have "span_induct_alt_help S (x + y)"
3074 	    apply (induct rule: span_induct_alt_help.induct)
3075 	    apply simp
3077 	    apply (rule span_induct_alt_help_S)
3078 	    apply assumption
3079 	    apply simp
3080 	    done}
3081 	moreover
3082 	{fix c x assume xt: "span_induct_alt_help S x"
3083 	  then have "span_induct_alt_help S (c*s x)"
3084 	    apply (induct rule: span_induct_alt_help.induct)
3087 	    apply (rule span_induct_alt_help_S)
3088 	    apply assumption
3089 	    apply simp
3090 	    done
3091 	}
3092 	ultimately show "subspace (span_induct_alt_help S)"
3093 	  unfolding subspace_def mem_def Ball_def by blast
3094       qed}
3095   with th0 show ?thesis by blast
3096 qed
3098 lemma span_induct_alt:
3099   assumes h0: "h (0::'a::semiring_1^'n)" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c*s x + y)" and x: "x \<in> span S"
3100   shows "h x"
3101 using span_induct_alt'[of h S] h0 hS x by blast
3103 (* Individual closure properties. *)
3105 lemma span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses)
3107 lemma span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
3109 lemma span_add: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
3112 lemma span_mul: "x \<in> span S ==> (c *s x) \<in> span S"
3113   by (metis subspace_span subspace_mul)
3115 lemma span_neg: "x \<in> span S ==> - (x::'a::ring_1^'n) \<in> span S"
3116   by (metis subspace_neg subspace_span)
3118 lemma span_sub: "(x::'a::ring_1^'n) \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
3119   by (metis subspace_span subspace_sub)
3121 lemma span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
3122   apply (rule subspace_setsum)
3123   by (metis subspace_span subspace_setsum)+
3125 lemma span_add_eq: "(x::'a::ring_1^'n) \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
3126   apply (auto simp only: span_add span_sub)
3127   apply (subgoal_tac "(x + y) - x \<in> span S", simp)
3128   by (simp only: span_add span_sub)
3130 (* Mapping under linear image. *)
3132 lemma span_linear_image: assumes lf: "linear (f::'a::semiring_1 ^ 'n => _)"
3133   shows "span (f ` S) = f ` (span S)"
3134 proof-
3135   {fix x
3136     assume x: "x \<in> span (f ` S)"
3137     have "x \<in> f ` span S"
3138       apply (rule span_induct[where x=x and S = "f ` S"])
3139       apply (clarsimp simp add: image_iff)
3140       apply (frule span_superset)
3141       apply blast
3142       apply (simp only: mem_def)
3143       apply (rule subspace_linear_image[OF lf])
3144       apply (rule subspace_span)
3145       apply (rule x)
3146       done}
3147   moreover
3148   {fix x assume x: "x \<in> span S"
3149     have th0:"(\<lambda>a. f a \<in> span (f ` S)) = {x. f x \<in> span (f ` S)}" apply (rule set_ext)
3150       unfolding mem_def Collect_def ..
3151     have "f x \<in> span (f ` S)"
3152       apply (rule span_induct[where S=S])
3153       apply (rule span_superset)
3154       apply simp
3155       apply (subst th0)
3156       apply (rule subspace_linear_preimage[OF lf subspace_span, of "f ` S"])
3157       apply (rule x)
3158       done}
3159   ultimately show ?thesis by blast
3160 qed
3162 (* The key breakdown property. *)
3164 lemma span_breakdown:
3165   assumes bS: "(b::'a::ring_1 ^ 'n) \<in> S" and aS: "a \<in> span S"
3166   shows "\<exists>k. a - k*s b \<in> span (S - {b})" (is "?P a")
3167 proof-
3168   {fix x assume xS: "x \<in> S"
3169     {assume ab: "x = b"
3170       then have "?P x"
3171 	apply simp
3172 	apply (rule exI[where x="1"], simp)
3173 	by (rule span_0)}
3174     moreover
3175     {assume ab: "x \<noteq> b"
3176       then have "?P x"  using xS
3177 	apply -
3178 	apply (rule exI[where x=0])
3179 	apply (rule span_superset)
3180 	by simp}
3181     ultimately have "?P x" by blast}
3182   moreover have "subspace ?P"
3183     unfolding subspace_def
3184     apply auto
3186     apply (rule exI[where x=0])
3187     using span_0[of "S - {b}"]
3189     apply (clarsimp simp add: mem_def)
3190     apply (rule_tac x="k + ka" in exI)
3191     apply (subgoal_tac "x + y - (k + ka) *s b = (x - k*s b) + (y - ka *s b)")
3192     apply (simp only: )
3194     apply assumption+
3195     apply (vector ring_simps)
3196     apply (clarsimp simp add: mem_def)
3197     apply (rule_tac x= "c*k" in exI)
3198     apply (subgoal_tac "c *s x - (c * k) *s b = c*s (x - k*s b)")
3199     apply (simp only: )
3200     apply (rule span_mul[unfolded mem_def])
3201     apply assumption
3202     by (vector ring_simps)
3203   ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis
3204 qed
3206 lemma span_breakdown_eq:
3207   "(x::'a::ring_1^'n) \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. (x - k *s a) \<in> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
3208 proof-
3209   {assume x: "x \<in> span (insert a S)"
3210     from x span_breakdown[of "a" "insert a S" "x"]
3211     have ?rhs apply clarsimp
3212       apply (rule_tac x= "k" in exI)
3213       apply (rule set_rev_mp[of _ "span (S - {a})" _])
3214       apply assumption
3215       apply (rule span_mono)
3216       apply blast
3217       done}
3218   moreover
3219   { fix k assume k: "x - k *s a \<in> span S"
3220     have eq: "x = (x - k *s a) + k *s a" by vector
3221     have "(x - k *s a) + k *s a \<in> span (insert a S)"
3223       apply (rule set_rev_mp[of _ "span S" _])
3224       apply (rule k)
3225       apply (rule span_mono)
3226       apply blast
3227       apply (rule span_mul)
3228       apply (rule span_superset)
3229       apply blast
3230       done
3231     then have ?lhs using eq by metis}
3232   ultimately show ?thesis by blast
3233 qed
3235 (* Hence some "reversal" results.*)
3237 lemma in_span_insert:
3238   assumes a: "(a::'a::field^'n) \<in> span (insert b S)" and na: "a \<notin> span S"
3239   shows "b \<in> span (insert a S)"
3240 proof-
3241   from span_breakdown[of b "insert b S" a, OF insertI1 a]
3242   obtain k where k: "a - k*s b \<in> span (S - {b})" by auto
3243   {assume k0: "k = 0"
3244     with k have "a \<in> span S"
3245       apply (simp)
3246       apply (rule set_rev_mp)
3247       apply assumption
3248       apply (rule span_mono)
3249       apply blast
3250       done
3251     with na  have ?thesis by blast}
3252   moreover
3253   {assume k0: "k \<noteq> 0"
3254     have eq: "b = (1/k) *s a - ((1/k) *s a - b)" by vector
3255     from k0 have eq': "(1/k) *s (a - k*s b) = (1/k) *s a - b"
3256       by (vector field_simps)
3257     from k have "(1/k) *s (a - k*s b) \<in> span (S - {b})"
3258       by (rule span_mul)
3259     hence th: "(1/k) *s a - b \<in> span (S - {b})"
3260       unfolding eq' .
3262     from k
3263     have ?thesis
3264       apply (subst eq)
3265       apply (rule span_sub)
3266       apply (rule span_mul)
3267       apply (rule span_superset)
3268       apply blast
3269       apply (rule set_rev_mp)
3270       apply (rule th)
3271       apply (rule span_mono)
3272       using na by blast}
3273   ultimately show ?thesis by blast
3274 qed
3276 lemma in_span_delete:
3277   assumes a: "(a::'a::field^'n) \<in> span S"
3278   and na: "a \<notin> span (S-{b})"
3279   shows "b \<in> span (insert a (S - {b}))"
3280   apply (rule in_span_insert)
3281   apply (rule set_rev_mp)
3282   apply (rule a)
3283   apply (rule span_mono)
3284   apply blast
3285   apply (rule na)
3286   done
3288 (* Transitivity property. *)
3290 lemma span_trans:
3291   assumes x: "(x::'a::ring_1^'n) \<in> span S" and y: "y \<in> span (insert x S)"
3292   shows "y \<in> span S"
3293 proof-
3294   from span_breakdown[of x "insert x S" y, OF insertI1 y]
3295   obtain k where k: "y -k*s x \<in> span (S - {x})" by auto
3296   have eq: "y = (y - k *s x) + k *s x" by vector
3297   show ?thesis
3298     apply (subst eq)
3300     apply (rule set_rev_mp)
3301     apply (rule k)
3302     apply (rule span_mono)
3303     apply blast
3304     apply (rule span_mul)
3305     by (rule x)
3306 qed
3308 (* ------------------------------------------------------------------------- *)
3309 (* An explicit expansion is sometimes needed.                                *)
3310 (* ------------------------------------------------------------------------- *)
3312 lemma span_explicit:
3313   "span P = {y::'a::semiring_1^'n. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *s v) S = y}"
3314   (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
3315 proof-
3316   {fix x assume x: "x \<in> ?E"
3317     then obtain S u where fS: "finite S" and SP: "S\<subseteq>P" and u: "setsum (\<lambda>v. u v *s v) S = x"
3318       by blast
3319     have "x \<in> span P"
3320       unfolding u[symmetric]
3321       apply (rule span_setsum[OF fS])
3322       using span_mono[OF SP]
3323       by (auto intro: span_superset span_mul)}
3324   moreover
3325   have "\<forall>x \<in> span P. x \<in> ?E"
3326     unfolding mem_def Collect_def
3327   proof(rule span_induct_alt')
3328     show "?h 0"
3329       apply (rule exI[where x="{}"]) by simp
3330   next
3331     fix c x y
3332     assume x: "x \<in> P" and hy: "?h y"
3333     from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
3334       and u: "setsum (\<lambda>v. u v *s v) S = y" by blast
3335     let ?S = "insert x S"
3336     let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c)
3337                   else u y"
3338     from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P" by blast+
3339     {assume xS: "x \<in> S"
3340       have S1: "S = (S - {x}) \<union> {x}"
3341 	and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}" using xS fS by auto
3342       have "setsum (\<lambda>v. ?u v *s v) ?S =(\<Sum>v\<in>S - {x}. u v *s v) + (u x + c) *s x"
3343 	using xS
3344 	by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]]
3345 	  setsum_clauses(2)[OF fS] cong del: if_weak_cong)
3346       also have "\<dots> = (\<Sum>v\<in>S. u v *s v) + c *s x"
3347 	apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]])
3348 	by (vector ring_simps)
3349       also have "\<dots> = c*s x + y"
3351       finally have "setsum (\<lambda>v. ?u v *s v) ?S = c*s x + y" .
3352     then have "?Q ?S ?u (c*s x + y)" using th0 by blast}
3353   moreover
3354   {assume xS: "x \<notin> S"
3355     have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *s v) = y"
3356       unfolding u[symmetric]
3357       apply (rule setsum_cong2)
3358       using xS by auto
3359     have "?Q ?S ?u (c*s x + y)" using fS xS th0
3361   ultimately have "?Q ?S ?u (c*s x + y)"
3362     by (cases "x \<in> S", simp, simp)
3363     then show "?h (c*s x + y)"
3364       apply -
3365       apply (rule exI[where x="?S"])
3366       apply (rule exI[where x="?u"]) by metis
3367   qed
3368   ultimately show ?thesis by blast
3369 qed
3371 lemma dependent_explicit:
3372   "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>(v::'a::{idom,field}^'n) \<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *s v) S = 0))" (is "?lhs = ?rhs")
3373 proof-
3374   {assume dP: "dependent P"
3375     then obtain a S u where aP: "a \<in> P" and fS: "finite S"
3376       and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *s v) S = a"
3377       unfolding dependent_def span_explicit by blast
3378     let ?S = "insert a S"
3379     let ?u = "\<lambda>y. if y = a then - 1 else u y"
3380     let ?v = a
3381     from aP SP have aS: "a \<notin> S" by blast
3382     from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" by auto
3383     have s0: "setsum (\<lambda>v. ?u v *s v) ?S = 0"
3384       using fS aS
3385       apply (simp add: vector_smult_lneg vector_smult_lid setsum_clauses ring_simps )
3386       apply (subst (2) ua[symmetric])
3387       apply (rule setsum_cong2)
3388       by auto
3389     with th0 have ?rhs
3390       apply -
3391       apply (rule exI[where x= "?S"])
3392       apply (rule exI[where x= "?u"])
3393       by clarsimp}
3394   moreover
3395   {fix S u v assume fS: "finite S"
3396       and SP: "S \<subseteq> P" and vS: "v \<in> S" and uv: "u v \<noteq> 0"
3397     and u: "setsum (\<lambda>v. u v *s v) S = 0"
3398     let ?a = v
3399     let ?S = "S - {v}"
3400     let ?u = "\<lambda>i. (- u i) / u v"
3401     have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"       using fS SP vS by auto
3402     have "setsum (\<lambda>v. ?u v *s v) ?S = setsum (\<lambda>v. (- (inverse (u ?a))) *s (u v *s v)) S - ?u v *s v"
3403       using fS vS uv
3404       by (simp add: setsum_diff1 vector_smult_lneg divide_inverse
3405 	vector_smult_assoc field_simps)
3406     also have "\<dots> = ?a"
3407       unfolding setsum_cmul u
3408       using uv by (simp add: vector_smult_lneg)
3409     finally  have "setsum (\<lambda>v. ?u v *s v) ?S = ?a" .
3410     with th0 have ?lhs
3411       unfolding dependent_def span_explicit
3412       apply -
3413       apply (rule bexI[where x= "?a"])
3414       apply simp_all
3415       apply (rule exI[where x= "?S"])
3416       by auto}
3417   ultimately show ?thesis by blast
3418 qed
3421 lemma span_finite:
3422   assumes fS: "finite S"
3423   shows "span S = {(y::'a::semiring_1^'n). \<exists>u. setsum (\<lambda>v. u v *s v) S = y}"
3424   (is "_ = ?rhs")
3425 proof-
3426   {fix y assume y: "y \<in> span S"
3427     from y obtain S' u where fS': "finite S'" and SS': "S' \<subseteq> S" and
3428       u: "setsum (\<lambda>v. u v *s v) S' = y" unfolding span_explicit by blast
3429     let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
3430     from setsum_restrict_set[OF fS, of "\<lambda>v. u v *s v" S', symmetric] SS'
3431     have "setsum (\<lambda>v. ?u v *s v) S = setsum (\<lambda>v. u v *s v) S'"
3432       unfolding cond_value_iff cond_application_beta
3433       apply (simp add: cond_value_iff cong del: if_weak_cong)
3434       apply (rule setsum_cong)
3435       apply auto
3436       done
3437     hence "setsum (\<lambda>v. ?u v *s v) S = y" by (metis u)
3438     hence "y \<in> ?rhs" by auto}
3439   moreover
3440   {fix y u assume u: "setsum (\<lambda>v. u v *s v) S = y"
3441     then have "y \<in> span S" using fS unfolding span_explicit by auto}
3442   ultimately show ?thesis by blast
3443 qed
3446 (* Standard bases are a spanning set, and obviously finite.                  *)
3448 lemma span_stdbasis:"span {basis i :: 'a::ring_1^'n | i. i \<in> {1 .. dimindex(UNIV :: 'n set)}} = UNIV"
3449 apply (rule set_ext)
3450 apply auto
3451 apply (subst basis_expansion[symmetric])
3452 apply (rule span_setsum)
3453 apply simp
3454 apply auto
3455 apply (rule span_mul)
3456 apply (rule span_superset)
3457 apply (auto simp add: Collect_def mem_def)
3458 done
3461 lemma has_size_stdbasis: "{basis i ::real ^'n | i. i \<in> {1 .. dimindex (UNIV :: 'n set)}} hassize (dimindex(UNIV :: 'n set))" (is "?S hassize ?n")
3462 proof-
3463   have eq: "?S = basis ` {1 .. ?n}" by blast
3464   show ?thesis unfolding eq
3465     apply (rule hassize_image_inj[OF basis_inj])
3467 qed
3469 lemma finite_stdbasis: "finite {basis i ::real^'n |i. i\<in> {1 .. dimindex(UNIV:: 'n set)}}"
3470   using has_size_stdbasis[unfolded hassize_def]
3471   ..
3473 lemma card_stdbasis: "card {basis i ::real^'n |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}} = dimindex(UNIV :: 'n set)"
3474   using has_size_stdbasis[unfolded hassize_def]
3475   ..
3477 lemma independent_stdbasis_lemma:
3478   assumes x: "(x::'a::semiring_1 ^ 'n) \<in> span (basis ` S)"
3479   and i: "i \<in> {1 .. dimindex (UNIV :: 'n set)}"
3480   and iS: "i \<notin> S"
3481   shows "(x\$i) = 0"
3482 proof-
3483   let ?n = "dimindex (UNIV :: 'n set)"
3484   let ?U = "{1 .. ?n}"
3485   let ?B = "basis ` S"
3486   let ?P = "\<lambda>(x::'a^'n). \<forall>i\<in> ?U. i \<notin> S \<longrightarrow> x\$i =0"
3487  {fix x::"'a^'n" assume xS: "x\<in> ?B"
3488    from xS have "?P x" by (auto simp add: basis_component)}
3489  moreover
3490  have "subspace ?P"
3491    by (auto simp add: subspace_def Collect_def mem_def zero_index vector_component)
3492  ultimately show ?thesis
3493    using x span_induct[of ?B ?P x] i iS by blast
3494 qed
3496 lemma independent_stdbasis: "independent {basis i ::real^'n |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}}"
3497 proof-
3498   let ?n = "dimindex (UNIV :: 'n set)"
3499   let ?I = "{1 .. ?n}"
3500   let ?b = "basis :: nat \<Rightarrow> real ^'n"
3501   let ?B = "?b ` ?I"
3502   have eq: "{?b i|i. i \<in> ?I} = ?B"
3503     by auto
3504   {assume d: "dependent ?B"
3505     then obtain k where k: "k \<in> ?I" "?b k \<in> span (?B - {?b k})"
3506       unfolding dependent_def by auto
3507     have eq1: "?B - {?b k} = ?B - ?b ` {k}"  by simp
3508     have eq2: "?B - {?b k} = ?b ` (?I - {k})"
3509       unfolding eq1
3510       apply (rule inj_on_image_set_diff[symmetric])
3511       apply (rule basis_inj) using k(1) by auto
3512     from k(2) have th0: "?b k \<in> span (?b ` (?I - {k}))" unfolding eq2 .
3513     from independent_stdbasis_lemma[OF th0 k(1), simplified]
3514     have False by (simp add: basis_component[OF k(1), of k])}
3515   then show ?thesis unfolding eq dependent_def ..
3516 qed
3518 (* This is useful for building a basis step-by-step.                         *)
3520 lemma independent_insert:
3521   "independent(insert (a::'a::field ^'n) S) \<longleftrightarrow>
3522       (if a \<in> S then independent S
3523                 else independent S \<and> a \<notin> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
3524 proof-
3525   {assume aS: "a \<in> S"
3526     hence ?thesis using insert_absorb[OF aS] by simp}
3527   moreover
3528   {assume aS: "a \<notin> S"
3529     {assume i: ?lhs
3530       then have ?rhs using aS
3531 	apply simp
3532 	apply (rule conjI)
3533 	apply (rule independent_mono)
3534 	apply assumption
3535 	apply blast
3537     moreover
3538     {assume i: ?rhs
3539       have ?lhs using i aS
3540 	apply simp
3541 	apply (auto simp add: dependent_def)
3542 	apply (case_tac "aa = a", auto)
3543 	apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})")
3544 	apply simp
3545 	apply (subgoal_tac "a \<in> span (insert aa (S - {aa}))")
3546 	apply (subgoal_tac "insert aa (S - {aa}) = S")
3547 	apply simp
3548 	apply blast
3549 	apply (rule in_span_insert)
3550 	apply assumption
3551 	apply blast
3552 	apply blast
3553 	done}
3554     ultimately have ?thesis by blast}
3555   ultimately show ?thesis by blast
3556 qed
3558 (* The degenerate case of the Exchange Lemma.  *)
3560 lemma mem_delete: "x \<in> (A - {a}) \<longleftrightarrow> x \<noteq> a \<and> x \<in> A"
3561   by blast
3563 lemma span_span: "span (span A) = span A"
3564   unfolding span_def hull_hull ..
3566 lemma span_inc: "S \<subseteq> span S"
3567   by (metis subset_eq span_superset)
3569 lemma spanning_subset_independent:
3570   assumes BA: "B \<subseteq> A" and iA: "independent (A::('a::field ^'n) set)"
3571   and AsB: "A \<subseteq> span B"
3572   shows "A = B"
3573 proof
3574   from BA show "B \<subseteq> A" .
3575 next
3576   from span_mono[OF BA] span_mono[OF AsB]
3577   have sAB: "span A = span B" unfolding span_span by blast
3579   {fix x assume x: "x \<in> A"
3580     from iA have th0: "x \<notin> span (A - {x})"
3581       unfolding dependent_def using x by blast
3582     from x have xsA: "x \<in> span A" by (blast intro: span_superset)
3583     have "A - {x} \<subseteq> A" by blast
3584     hence th1:"span (A - {x}) \<subseteq> span A" by (metis span_mono)
3585     {assume xB: "x \<notin> B"
3586       from xB BA have "B \<subseteq> A -{x}" by blast
3587       hence "span B \<subseteq> span (A - {x})" by (metis span_mono)
3588       with th1 th0 sAB have "x \<notin> span A" by blast
3589       with x have False by (metis span_superset)}
3590     then have "x \<in> B" by blast}
3591   then show "A \<subseteq> B" by blast
3592 qed
3594 (* The general case of the Exchange Lemma, the key to what follows.  *)
3596 lemma exchange_lemma:
3597   assumes f:"finite (t:: ('a::field^'n) set)" and i: "independent s"
3598   and sp:"s \<subseteq> span t"
3599   shows "\<exists>t'. (t' hassize card t) \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
3600 using f i sp
3601 proof(induct c\<equiv>"card(t - s)" arbitrary: s t rule: nat_less_induct)
3602   fix n:: nat and s t :: "('a ^'n) set"
3603   assume H: " \<forall>m<n. \<forall>(x:: ('a ^'n) set) xa.
3604                 finite xa \<longrightarrow>
3605                 independent x \<longrightarrow>
3606                 x \<subseteq> span xa \<longrightarrow>
3607                 m = card (xa - x) \<longrightarrow>
3608                 (\<exists>t'. (t' hassize card xa) \<and>
3609                       x \<subseteq> t' \<and> t' \<subseteq> x \<union> xa \<and> x \<subseteq> span t')"
3610     and ft: "finite t" and s: "independent s" and sp: "s \<subseteq> span t"
3611     and n: "n = card (t - s)"
3612   let ?P = "\<lambda>t'. (t' hassize card t) \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
3613   let ?ths = "\<exists>t'. ?P t'"
3614   {assume st: "s \<subseteq> t"
3615     from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
3616       by (auto simp add: hassize_def intro: span_superset)}
3617   moreover
3618   {assume st: "t \<subseteq> s"
3620     from spanning_subset_independent[OF st s sp]
3621       st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
3622       by (auto simp add: hassize_def intro: span_superset)}
3623   moreover
3624   {assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
3625     from st(2) obtain b where b: "b \<in> t" "b \<notin> s" by blast
3626       from b have "t - {b} - s \<subset> t - s" by blast
3627       then have cardlt: "card (t - {b} - s) < n" using n ft
3628  	by (auto intro: psubset_card_mono)
3629       from b ft have ct0: "card t \<noteq> 0" by auto
3630     {assume stb: "s \<subseteq> span(t -{b})"
3631       from ft have ftb: "finite (t -{b})" by auto
3632       from H[rule_format, OF cardlt ftb s stb]
3633       obtain u where u: "u hassize card (t-{b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u" by blast
3634       let ?w = "insert b u"
3635       have th0: "s \<subseteq> insert b u" using u by blast
3636       from u(3) b have "u \<subseteq> s \<union> t" by blast
3637       then have th1: "insert b u \<subseteq> s \<union> t" using u b by blast
3638       have bu: "b \<notin> u" using b u by blast
3639       from u(1) have fu: "finite u" by (simp add: hassize_def)
3640       from u(1) ft b have "u hassize (card t - 1)" by auto
3641       then
3642       have th2: "insert b u hassize card t"
3643 	using  card_insert_disjoint[OF fu bu] ct0 by (auto simp add: hassize_def)
3644       from u(4) have "s \<subseteq> span u" .
3645       also have "\<dots> \<subseteq> span (insert b u)" apply (rule span_mono) by blast
3646       finally have th3: "s \<subseteq> span (insert b u)" .      from th0 th1 th2 th3 have th: "?P ?w"  by blast
3647       from th have ?ths by blast}
3648     moreover
3649     {assume stb: "\<not> s \<subseteq> span(t -{b})"
3650       from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})" by blast
3651       have ab: "a \<noteq> b" using a b by blast
3652       have at: "a \<notin> t" using a ab span_superset[of a "t- {b}"] by auto
3653       have mlt: "card ((insert a (t - {b})) - s) < n"
3654 	using cardlt ft n  a b by auto
3655       have ft': "finite (insert a (t - {b}))" using ft by auto
3656       {fix x assume xs: "x \<in> s"
3657 	have t: "t \<subseteq> (insert b (insert a (t -{b})))" using b by auto
3658 	from b(1) have "b \<in> span t" by (simp add: span_superset)
3659 	have bs: "b \<in> span (insert a (t - {b}))"
3660 	  by (metis in_span_delete a sp mem_def subset_eq)
3661 	from xs sp have "x \<in> span t" by blast
3662 	with span_mono[OF t]
3663 	have x: "x \<in> span (insert b (insert a (t - {b})))" ..
3664 	from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))"  .}
3665       then have sp': "s \<subseteq> span (insert a (t - {b}))" by blast
3667       from H[rule_format, OF mlt ft' s sp' refl] obtain u where
3668 	u: "u hassize card (insert a (t -{b}))" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
3669 	"s \<subseteq> span u" by blast
3670       from u a b ft at ct0 have "?P u" by (auto simp add: hassize_def)
3671       then have ?ths by blast }
3672     ultimately have ?ths by blast
3673   }
3674   ultimately
3675   show ?ths  by blast
3676 qed
3678 (* This implies corresponding size bounds.                                   *)
3680 lemma independent_span_bound:
3681   assumes f: "finite t" and i: "independent (s::('a::field^'n) set)" and sp:"s \<subseteq> span t"
3682   shows "finite s \<and> card s \<le> card t"
3683   by (metis exchange_lemma[OF f i sp] hassize_def finite_subset card_mono)
3685 lemma finite_Atleast_Atmost[simp]: "finite {f x |x. x\<in> {(i::'a::finite_intvl_succ) .. j}}"
3686 proof-
3687   have eq: "{f x |x. x\<in> {i .. j}} = f ` {i .. j}" by auto
3688   show ?thesis unfolding eq
3689     apply (rule finite_imageI)
3690     apply (rule finite_intvl)
3691     done
3692 qed
3694 lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> {(i::nat) .. j}}"
3695 proof-
3696   have eq: "{f x |x. x\<in> {i .. j}} = f ` {i .. j}" by auto
3697   show ?thesis unfolding eq
3698     apply (rule finite_imageI)
3699     apply (rule finite_atLeastAtMost)
3700     done
3701 qed
3704 lemma independent_bound:
3705   fixes S:: "(real^'n) set"
3706   shows "independent S \<Longrightarrow> finite S \<and> card S <= dimindex(UNIV :: 'n set)"
3707   apply (subst card_stdbasis[symmetric])
3708   apply (rule independent_span_bound)
3709   apply (rule finite_Atleast_Atmost_nat)
3710   apply assumption
3711   unfolding span_stdbasis
3712   apply (rule subset_UNIV)
3713   done
3715 lemma dependent_biggerset: "(finite (S::(real ^'n) set) ==> card S > dimindex(UNIV:: 'n set)) ==> dependent S"
3716   by (metis independent_bound not_less)
3718 (* Hence we can create a maximal independent subset.                         *)
3720 lemma maximal_independent_subset_extend:
3721   assumes sv: "(S::(real^'n) set) \<subseteq> V" and iS: "independent S"
3722   shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
3723   using sv iS
3724 proof(induct d\<equiv> "dimindex (UNIV :: 'n set) - card S" arbitrary: S rule: nat_less_induct)
3725   fix n and S:: "(real^'n) set"
3726   assume H: "\<forall>m<n. \<forall>S \<subseteq> V. independent S \<longrightarrow> m = dimindex (UNIV::'n set) - card S \<longrightarrow>
3727               (\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B)"
3728     and sv: "S \<subseteq> V" and i: "independent S" and n: "n = dimindex (UNIV :: 'n set) - card S"
3729   let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
3730   let ?ths = "\<exists>x. ?P x"
3731   let ?d = "dimindex (UNIV :: 'n set)"
3732   {assume "V \<subseteq> span S"
3733     then have ?ths  using sv i by blast }
3734   moreover
3735   {assume VS: "\<not> V \<subseteq> span S"
3736     from VS obtain a where a: "a \<in> V" "a \<notin> span S" by blast
3737     from a have aS: "a \<notin> S" by (auto simp add: span_superset)
3738     have th0: "insert a S \<subseteq> V" using a sv by blast
3739     from independent_insert[of a S]  i a
3740     have th1: "independent (insert a S)" by auto
3741     have mlt: "?d - card (insert a S) < n"
3742       using aS a n independent_bound[OF th1] dimindex_ge_1[of "UNIV :: 'n set"]
3743       by auto
3745     from H[rule_format, OF mlt th0 th1 refl]
3746     obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
3747       by blast
3748     from B have "?P B" by auto
3749     then have ?ths by blast}
3750   ultimately show ?ths by blast
3751 qed
3753 lemma maximal_independent_subset:
3754   "\<exists>(B:: (real ^'n) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
3755   by (metis maximal_independent_subset_extend[of "{}:: (real ^'n) set"] empty_subsetI independent_empty)
3757 (* Notion of dimension.                                                      *)
3759 definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize n))"
3761 lemma basis_exists:  "\<exists>B. (B :: (real ^'n) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize dim V)"
3762 unfolding dim_def some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize n)"]
3763 unfolding hassize_def
3764 using maximal_independent_subset[of V] independent_bound
3765 by auto
3767 (* Consequences of independence or spanning for cardinality.                 *)
3769 lemma independent_card_le_dim: "(B::(real ^'n) set) \<subseteq> V \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B \<le> dim V"
3770 by (metis basis_exists[of V] independent_span_bound[where ?'a=real] hassize_def subset_trans)
3772 lemma span_card_ge_dim:  "(B::(real ^'n) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
3773   by (metis basis_exists[of V] independent_span_bound hassize_def subset_trans)
3775 lemma basis_card_eq_dim:
3776   "B \<subseteq> (V:: (real ^'n) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
3777   by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_mono)
3779 lemma dim_unique: "(B::(real ^'n) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> B hassize n \<Longrightarrow> dim V = n"
3780   by (metis basis_card_eq_dim hassize_def)
3782 (* More lemmas about dimension.                                              *)
3784 lemma dim_univ: "dim (UNIV :: (real^'n) set) = dimindex (UNIV :: 'n set)"
3785   apply (rule dim_unique[of "{basis i |i. i\<in> {1 .. dimindex (UNIV :: 'n set)}}"])
3786   by (auto simp only: span_stdbasis has_size_stdbasis independent_stdbasis)
3788 lemma dim_subset:
3789   "(S:: (real ^'n) set) \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
3790   using basis_exists[of T] basis_exists[of S]
3791   by (metis independent_span_bound[where ?'a = real and ?'n = 'n] subset_eq hassize_def)
3793 lemma dim_subset_univ: "dim (S:: (real^'n) set) \<le> dimindex (UNIV :: 'n set)"
3794   by (metis dim_subset subset_UNIV dim_univ)
3796 (* Converses to those.                                                       *)
3798 lemma card_ge_dim_independent:
3799   assumes BV:"(B::(real ^'n) set) \<subseteq> V" and iB:"independent B" and dVB:"dim V \<le> card B"
3800   shows "V \<subseteq> span B"
3801 proof-
3802   {fix a assume aV: "a \<in> V"
3803     {assume aB: "a \<notin> span B"
3804       then have iaB: "independent (insert a B)" using iB aV  BV by (simp add: independent_insert)
3805       from aV BV have th0: "insert a B \<subseteq> V" by blast
3806       from aB have "a \<notin>B" by (auto simp add: span_superset)
3807       with independent_card_le_dim[OF th0 iaB] dVB  have False by auto}
3808     then have "a \<in> span B"  by blast}
3809   then show ?thesis by blast
3810 qed
3812 lemma card_le_dim_spanning:
3813   assumes BV: "(B:: (real ^'n) set) \<subseteq> V" and VB: "V \<subseteq> span B"
3814   and fB: "finite B" and dVB: "dim V \<ge> card B"
3815   shows "independent B"
3816 proof-
3817   {fix a assume a: "a \<in> B" "a \<in> span (B -{a})"
3818     from a fB have c0: "card B \<noteq> 0" by auto
3819     from a fB have cb: "card (B -{a}) = card B - 1" by auto
3820     from BV a have th0: "B -{a} \<subseteq> V" by blast
3821     {fix x assume x: "x \<in> V"
3822       from a have eq: "insert a (B -{a}) = B" by blast
3823       from x VB have x': "x \<in> span B" by blast
3824       from span_trans[OF a(2), unfolded eq, OF x']
3825       have "x \<in> span (B -{a})" . }
3826     then have th1: "V \<subseteq> span (B -{a})" by blast
3827     have th2: "finite (B -{a})" using fB by auto
3828     from span_card_ge_dim[OF th0 th1 th2]
3829     have c: "dim V \<le> card (B -{a})" .
3830     from c c0 dVB cb have False by simp}
3831   then show ?thesis unfolding dependent_def by blast
3832 qed
3834 lemma card_eq_dim: "(B:: (real ^'n) set) \<subseteq> V \<Longrightarrow> B hassize dim V \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
3835   by (metis hassize_def order_eq_iff card_le_dim_spanning
3836     card_ge_dim_independent)
3838 (* ------------------------------------------------------------------------- *)
3839 (* More general size bound lemmas.                                           *)
3840 (* ------------------------------------------------------------------------- *)
3842 lemma independent_bound_general:
3843   "independent (S:: (real^'n) set) \<Longrightarrow> finite S \<and> card S \<le> dim S"
3844   by (metis independent_card_le_dim independent_bound subset_refl)
3846 lemma dependent_biggerset_general: "(finite (S:: (real^'n) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
3847   using independent_bound_general[of S] by (metis linorder_not_le)
3849 lemma dim_span: "dim (span (S:: (real ^'n) set)) = dim S"
3850 proof-
3851   have th0: "dim S \<le> dim (span S)"
3852     by (auto simp add: subset_eq intro: dim_subset span_superset)
3853   from basis_exists[of S]
3854   obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
3855   from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+
3856   have bSS: "B \<subseteq> span S" using B(1) by (metis subset_eq span_inc)
3857   have sssB: "span S \<subseteq> span B" using span_mono[OF B(3)] by (simp add: span_span)
3858   from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
3859     using fB(2)  by arith
3860 qed
3862 lemma subset_le_dim: "(S:: (real ^'n) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
3863   by (metis dim_span dim_subset)
3865 lemma span_eq_dim: "span (S:: (real ^'n) set) = span T ==> dim S = dim T"
3866   by (metis dim_span)
3868 lemma spans_image:
3869   assumes lf: "linear (f::'a::semiring_1^'n \<Rightarrow> _)" and VB: "V \<subseteq> span B"
3870   shows "f ` V \<subseteq> span (f ` B)"
3871   unfolding span_linear_image[OF lf]
3872   by (metis VB image_mono)
3874 lemma dim_image_le: assumes lf: "linear f" shows "dim (f ` S) \<le> dim (S:: (real ^'n) set)"
3875 proof-
3876   from basis_exists[of S] obtain B where
3877     B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
3878   from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+
3879   have "dim (f ` S) \<le> card (f ` B)"
3880     apply (rule span_card_ge_dim)
3881     using lf B fB by (auto simp add: span_linear_image spans_image subset_image_iff)
3882   also have "\<dots> \<le> dim S" using card_image_le[OF fB(1)] fB by simp
3883   finally show ?thesis .
3884 qed
3886 (* Relation between bases and injectivity/surjectivity of map.               *)
3888 lemma spanning_surjective_image:
3889   assumes us: "UNIV \<subseteq> span (S:: ('a::semiring_1 ^'n) set)"
3890   and lf: "linear f" and sf: "surj f"
3891   shows "UNIV \<subseteq> span (f ` S)"
3892 proof-
3893   have "UNIV \<subseteq> f ` UNIV" using sf by (auto simp add: surj_def)
3894   also have " \<dots> \<subseteq> span (f ` S)" using spans_image[OF lf us] .
3895 finally show ?thesis .
3896 qed
3898 lemma independent_injective_image:
3899   assumes iS: "independent (S::('a::semiring_1^'n) set)" and lf: "linear f" and fi: "inj f"
3900   shows "independent (f ` S)"
3901 proof-
3902   {fix a assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
3903     have eq: "f ` S - {f a} = f ` (S - {a})" using fi
3904       by (auto simp add: inj_on_def)
3905     from a have "f a \<in> f ` span (S -{a})"
3906       unfolding eq span_linear_image[OF lf, of "S - {a}"]  by blast
3907     hence "a \<in> span (S -{a})" using fi by (auto simp add: inj_on_def)
3908     with a(1) iS  have False by (simp add: dependent_def) }
3909   then show ?thesis unfolding dependent_def by blast
3910 qed
3912 (* ------------------------------------------------------------------------- *)
3913 (* Picking an orthogonal replacement for a spanning set.                     *)
3914 (* ------------------------------------------------------------------------- *)
3915     (* FIXME : Move to some general theory ?*)
3916 definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
3918 lemma vector_sub_project_orthogonal: "(b::'a::ordered_field^'n) \<bullet> (x - ((b \<bullet> x) / (b\<bullet>b)) *s b) = 0"
3919   apply (cases "b = 0", simp)
3920   apply (simp add: dot_rsub dot_rmult)
3921   unfolding times_divide_eq_right[symmetric]
3922   by (simp add: field_simps dot_eq_0)
3924 lemma basis_orthogonal:
3925   fixes B :: "(real ^'n) set"
3926   assumes fB: "finite B"
3927   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
3928   (is " \<exists>C. ?P B C")
3929 proof(induct rule: finite_induct[OF fB])
3930   case 1 thus ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def)
3931 next
3932   case (2 a B)
3933   note fB = `finite B` and aB = `a \<notin> B`
3934   from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C`
3935   obtain C where C: "finite C" "card C \<le> card B"
3936     "span C = span B" "pairwise orthogonal C" by blast
3937   let ?a = "a - setsum (\<lambda>x. (x\<bullet>a / (x\<bullet>x)) *s x) C"
3938   let ?C = "insert ?a C"
3939   from C(1) have fC: "finite ?C" by simp
3940   from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)" by (simp add: card_insert_if)
3941   {fix x k
3942     have th0: "\<And>(a::'b::comm_ring) b c. a - (b - c) = c + (a - b)" by (simp add: ring_simps)
3943     have "x - k *s (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *s x)) \<in> span C \<longleftrightarrow> x - k *s a \<in> span C"
3944       apply (simp only: vector_ssub_ldistrib th0)
3946       apply (rule span_mul)
3947       apply (rule span_setsum[OF C(1)])
3948       apply clarify
3949       apply (rule span_mul)
3950       by (rule span_superset)}
3951   then have SC: "span ?C = span (insert a B)"
3952     unfolding expand_set_eq span_breakdown_eq C(3)[symmetric] by auto
3953   thm pairwise_def
3954   {fix x y assume xC: "x \<in> ?C" and yC: "y \<in> ?C" and xy: "x \<noteq> y"
3955     {assume xa: "x = ?a" and ya: "y = ?a"
3956       have "orthogonal x y" using xa ya xy by blast}
3957     moreover
3958     {assume xa: "x = ?a" and ya: "y \<noteq> ?a" "y \<in> C"
3959       from ya have Cy: "C = insert y (C - {y})" by blast
3960       have fth: "finite (C - {y})" using C by simp
3961       have "orthogonal x y"
3962 	using xa ya
3963 	unfolding orthogonal_def xa dot_lsub dot_rsub diff_eq_0_iff_eq
3964 	apply simp
3965 	apply (subst Cy)
3966 	using C(1) fth
3967 	apply (simp only: setsum_clauses)
3968 	apply (auto simp add: dot_ladd dot_lmult dot_eq_0 dot_sym[of y a] dot_lsum[OF fth])
3969 	apply (rule setsum_0')
3970 	apply clarsimp
3971 	apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
3972 	by auto}
3973     moreover
3974     {assume xa: "x \<noteq> ?a" "x \<in> C" and ya: "y = ?a"
3975       from xa have Cx: "C = insert x (C - {x})" by blast
3976       have fth: "finite (C - {x})" using C by simp
3977       have "orthogonal x y"
3978 	using xa ya
3979 	unfolding orthogonal_def ya dot_rsub dot_lsub diff_eq_0_iff_eq
3980 	apply simp
3981 	apply (subst Cx)
3982 	using C(1) fth
3983 	apply (simp only: setsum_clauses)
3984 	apply (subst dot_sym[of x])
3985 	apply (auto simp add: dot_radd dot_rmult dot_eq_0 dot_sym[of x a] dot_rsum[OF fth])
3986 	apply (rule setsum_0')
3987 	apply clarsimp
3988 	apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
3989 	by auto}
3990     moreover
3991     {assume xa: "x \<in> C" and ya: "y \<in> C"
3992       have "orthogonal x y" using xa ya xy C(4) unfolding pairwise_def by blast}
3993     ultimately have "orthogonal x y" using xC yC by blast}
3994   then have CPO: "pairwise orthogonal ?C" unfolding pairwise_def by blast
3995   from fC cC SC CPO have "?P (insert a B) ?C" by blast
3996   then show ?case by blast
3997 qed
3999 lemma orthogonal_basis_exists:
4000   fixes V :: "(real ^'n) set"
4001   shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (B hassize dim V) \<and> pairwise orthogonal B"
4002 proof-
4003   from basis_exists[of V] obtain B where B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "B hassize dim V" by blast
4004   from B have fB: "finite B" "card B = dim V" by (simp_all add: hassize_def)
4005   from basis_orthogonal[OF fB(1)] obtain C where
4006     C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C" by blast
4007   from C B
4008   have CSV: "C \<subseteq> span V" by (metis span_inc span_mono subset_trans)
4009   from span_mono[OF B(3)]  C have SVC: "span V \<subseteq> span C" by (simp add: span_span)
4010   from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
4011   have iC: "independent C" by (simp add: dim_span)
4012   from C fB have "card C \<le> dim V" by simp
4013   moreover have "dim V \<le> card C" using span_card_ge_dim[OF CSV SVC C(1)]
4015   ultimately have CdV: "C hassize dim V" unfolding hassize_def using C(1) by simp
4016   from C B CSV CdV iC show ?thesis by auto
4017 qed
4019 lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
4020   by (metis set_eq_subset span_mono span_span span_inc)
4022 (* ------------------------------------------------------------------------- *)
4023 (* Low-dimensional subset is in a hyperplane (weak orthogonal complement).   *)
4024 (* ------------------------------------------------------------------------- *)
4026 lemma span_not_univ_orthogonal:
4027   assumes sU: "span S \<noteq> UNIV"
4028   shows "\<exists>(a:: real ^'n). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
4029 proof-
4030   from sU obtain a where a: "a \<notin> span S" by blast
4031   from orthogonal_basis_exists obtain B where
4032     B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "B hassize dim S" "pairwise orthogonal B"
4033     by blast
4034   from B have fB: "finite B" "card B = dim S" by (simp_all add: hassize_def)
4035   from span_mono[OF B(2)] span_mono[OF B(3)]
4036   have sSB: "span S = span B" by (simp add: span_span)
4037   let ?a = "a - setsum (\<lambda>b. (a\<bullet>b / (b\<bullet>b)) *s b) B"
4038   have "setsum (\<lambda>b. (a\<bullet>b / (b\<bullet>b)) *s b) B \<in> span S"
4039     unfolding sSB
4040     apply (rule span_setsum[OF fB(1)])
4041     apply clarsimp
4042     apply (rule span_mul)
4043     by (rule span_superset)
4044   with a have a0:"?a  \<noteq> 0" by auto
4045   have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
4046   proof(rule span_induct')
4047     show "subspace (\<lambda>x. ?a \<bullet> x = 0)"
4049   next
4050     {fix x assume x: "x \<in> B"
4051       from x have B': "B = insert x (B - {x})" by blast
4052       have fth: "finite (B - {x})" using fB by simp
4053       have "?a \<bullet> x = 0"
4054 	apply (subst B') using fB fth
4055 	unfolding setsum_clauses(2)[OF fth]
4056 	apply simp
4058 	apply (rule setsum_0', rule ballI)
4059 	unfolding dot_sym
4060 	by (auto simp add: x field_simps dot_eq_0 intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])}
4061     then show "\<forall>x \<in> B. ?a \<bullet> x = 0" by blast
4062   qed
4063   with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"])
4064 qed
4066 lemma span_not_univ_subset_hyperplane:
4067   assumes SU: "span S \<noteq> (UNIV ::(real^'n) set)"
4068   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
4069   using span_not_univ_orthogonal[OF SU] by auto
4071 lemma lowdim_subset_hyperplane:
4072   assumes d: "dim S < dimindex (UNIV :: 'n set)"
4073   shows "\<exists>(a::real ^'n). a  \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
4074 proof-
4075   {assume "span S = UNIV"
4076     hence "dim (span S) = dim (UNIV :: (real ^'n) set)" by simp
4077     hence "dim S = dimindex (UNIV :: 'n set)" by (simp add: dim_span dim_univ)
4078     with d have False by arith}
4079   hence th: "span S \<noteq> UNIV" by blast
4080   from span_not_univ_subset_hyperplane[OF th] show ?thesis .
4081 qed
4083 (* We can extend a linear basis-basis injection to the whole set.            *)
4085 lemma linear_indep_image_lemma:
4086   assumes lf: "linear f" and fB: "finite B"
4087   and ifB: "independent (f ` B)"
4088   and fi: "inj_on f B" and xsB: "x \<in> span B"
4089   and fx: "f (x::'a::field^'n) = 0"
4090   shows "x = 0"
4091   using fB ifB fi xsB fx
4092 proof(induct arbitrary: x rule: finite_induct[OF fB])
4093   case 1 thus ?case by (auto simp add:  span_empty)
4094 next
4095   case (2 a b x)
4096   have fb: "finite b" using "2.prems" by simp
4097   have th0: "f ` b \<subseteq> f ` (insert a b)"
4098     apply (rule image_mono) by blast
4099   from independent_mono[ OF "2.prems"(2) th0]
4100   have ifb: "independent (f ` b)"  .
4101   have fib: "inj_on f b"
4102     apply (rule subset_inj_on [OF "2.prems"(3)])
4103     by blast
4104   from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
4105   obtain k where k: "x - k*s a \<in> span (b -{a})" by blast
4106   have "f (x - k*s a) \<in> span (f ` b)"
4107     unfolding span_linear_image[OF lf]
4108     apply (rule imageI)
4109     using k span_mono[of "b-{a}" b] by blast
4110   hence "f x - k*s f a \<in> span (f ` b)"
4111     by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
4112   hence th: "-k *s f a \<in> span (f ` b)"
4113     using "2.prems"(5) by (simp add: vector_smult_lneg)
4114   {assume k0: "k = 0"
4115     from k0 k have "x \<in> span (b -{a})" by simp
4116     then have "x \<in> span b" using span_mono[of "b-{a}" b]
4117       by blast}
4118   moreover
4119   {assume k0: "k \<noteq> 0"
4120     from span_mul[OF th, of "- 1/ k"] k0
4121     have th1: "f a \<in> span (f ` b)"
4122       by (auto simp add: vector_smult_assoc)
4123     from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
4124     have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
4125     from "2.prems"(2)[unfolded dependent_def bex_simps(10), rule_format, of "f a"]
4126     have "f a \<notin> span (f ` b)" using tha
4127       using "2.hyps"(2)
4128       "2.prems"(3) by auto
4129     with th1 have False by blast
4130     then have "x \<in> span b" by blast}
4131   ultimately have xsb: "x \<in> span b" by blast
4132   from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)]
4133   show "x = 0" .
4134 qed
4136 (* We can extend a linear mapping from basis.                                *)
4138 lemma linear_independent_extend_lemma:
4139   assumes fi: "finite B" and ib: "independent B"
4140   shows "\<exists>g. (\<forall>x\<in> span B. \<forall>y\<in> span B. g ((x::'a::field^'n) + y) = g x + g y)
4141            \<and> (\<forall>x\<in> span B. \<forall>c. g (c*s x) = c *s g x)
4142            \<and> (\<forall>x\<in> B. g x = f x)"
4143 using ib fi
4144 proof(induct rule: finite_induct[OF fi])
4145   case 1 thus ?case by (auto simp add: span_empty)
4146 next
4147   case (2 a b)
4148   from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
4150   from "2.hyps"(3)[OF ibf] obtain g where
4151     g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
4152     "\<forall>x\<in>span b. \<forall>c. g (c *s x) = c *s g x" "\<forall>x\<in>b. g x = f x" by blast
4153   let ?h = "\<lambda>z. SOME k. (z - k *s a) \<in> span b"
4154   {fix z assume z: "z \<in> span (insert a b)"
4155     have th0: "z - ?h z *s a \<in> span b"
4156       apply (rule someI_ex)
4157       unfolding span_breakdown_eq[symmetric]
4158       using z .
4159     {fix k assume k: "z - k *s a \<in> span b"
4160       have eq: "z - ?h z *s a - (z - k*s a) = (k - ?h z) *s a"
4162       from span_sub[OF th0 k]
4163       have khz: "(k - ?h z) *s a \<in> span b" by (simp add: eq)
4164       {assume "k \<noteq> ?h z" hence k0: "k - ?h z \<noteq> 0" by simp
4165 	from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
4166 	have "a \<in> span b" by (simp add: vector_smult_assoc)
4167 	with "2.prems"(1) "2.hyps"(2) have False
4168 	  by (auto simp add: dependent_def)}
4169       then have "k = ?h z" by blast}
4170     with th0 have "z - ?h z *s a \<in> span b \<and> (\<forall>k. z - k *s a \<in> span b \<longrightarrow> k = ?h z)" by blast}
4171   note h = this
4172   let ?g = "\<lambda>z. ?h z *s f a + g (z - ?h z *s a)"
4173   {fix x y assume x: "x \<in> span (insert a b)" and y: "y \<in> span (insert a b)"
4174     have tha: "\<And>(x::'a^'n) y a k l. (x + y) - (k + l) *s a = (x - k *s a) + (y - l *s a)"
4175       by (vector ring_simps)
4176     have addh: "?h (x + y) = ?h x + ?h y"
4177       apply (rule conjunct2[OF h, rule_format, symmetric])
4178       apply (rule span_add[OF x y])
4179       unfolding tha
4180       by (metis span_add x y conjunct1[OF h, rule_format])
4181     have "?g (x + y) = ?g x + ?g y"
4183       g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
4185   moreover
4186   {fix x:: "'a^'n" and c:: 'a  assume x: "x \<in> span (insert a b)"
4187     have tha: "\<And>(x::'a^'n) c k a. c *s x - (c * k) *s a = c *s (x - k *s a)"
4188       by (vector ring_simps)
4189     have hc: "?h (c *s x) = c * ?h x"
4190       apply (rule conjunct2[OF h, rule_format, symmetric])
4191       apply (metis span_mul x)
4192       by (metis tha span_mul x conjunct1[OF h])
4193     have "?g (c *s x) = c*s ?g x"
4194       unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
4195       by (vector ring_simps)}
4196   moreover
4197   {fix x assume x: "x \<in> (insert a b)"
4198     {assume xa: "x = a"
4199       have ha1: "1 = ?h a"
4200 	apply (rule conjunct2[OF h, rule_format])
4201 	apply (metis span_superset insertI1)
4202 	using conjunct1[OF h, OF span_superset, OF insertI1]
4203 	by (auto simp add: span_0)
4205       from xa ha1[symmetric] have "?g x = f x"
4206 	apply simp
4207 	using g(2)[rule_format, OF span_0, of 0]
4208 	by simp}
4209     moreover
4210     {assume xb: "x \<in> b"
4211       have h0: "0 = ?h x"
4212 	apply (rule conjunct2[OF h, rule_format])
4213 	apply (metis  span_superset insertI1 xb x)
4214 	apply simp
4215 	apply (metis span_superset xb)
4216 	done
4217       have "?g x = f x"
4218 	by (simp add: h0[symmetric] g(3)[rule_format, OF xb])}
4219     ultimately have "?g x = f x" using x by blast }
4220   ultimately show ?case apply - apply (rule exI[where x="?g"]) by blast
4221 qed
4223 lemma linear_independent_extend:
4224   assumes iB: "independent (B:: (real ^'n) set)"
4225   shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
4226 proof-
4227   from maximal_independent_subset_extend[of B "UNIV"] iB
4228   obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C" by auto
4230   from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
4231   obtain g where g: "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y)
4232            \<and> (\<forall>x\<in> span C. \<forall>c. g (c*s x) = c *s g x)
4233            \<and> (\<forall>x\<in> C. g x = f x)" by blast
4234   from g show ?thesis unfolding linear_def using C
4235     apply clarsimp by blast
4236 qed
4238 (* Can construct an isomorphism between spaces of same dimension.            *)
4240 lemma card_le_inj: assumes fA: "finite A" and fB: "finite B"
4241   and c: "card A \<le> card B" shows "(\<exists>f. f ` A \<subseteq> B \<and> inj_on f A)"
4242 using fB c
4243 proof(induct arbitrary: B rule: finite_induct[OF fA])
4244   case 1 thus ?case by simp
4245 next
4246   case (2 x s t)
4247   thus ?case
4248   proof(induct rule: finite_induct[OF "2.prems"(1)])
4249     case 1    then show ?case by simp
4250   next
4251     case (2 y t)
4252     from "2.prems"(1,2,5) "2.hyps"(1,2) have cst:"card s \<le> card t" by simp
4253     from "2.prems"(3) [OF "2.hyps"(1) cst] obtain f where
4254       f: "f ` s \<subseteq> t \<and> inj_on f s" by blast
4255     from f "2.prems"(2) "2.hyps"(2) show ?case
4256       apply -
4257       apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
4258       by (auto simp add: inj_on_def)
4259   qed
4260 qed
4262 lemma card_subset_eq: assumes fB: "finite B" and AB: "A \<subseteq> B" and
4263   c: "card A = card B"
4264   shows "A = B"
4265 proof-
4266   from fB AB have fA: "finite A" by (auto intro: finite_subset)
4267   from fA fB have fBA: "finite (B - A)" by auto
4268   have e: "A \<inter> (B - A) = {}" by blast
4269   have eq: "A \<union> (B - A) = B" using AB by blast
4270   from card_Un_disjoint[OF fA fBA e, unfolded eq c]
4271   have "card (B - A) = 0" by arith
4272   hence "B - A = {}" unfolding card_eq_0_iff using fA fB by simp
4273   with AB show "A = B" by blast
4274 qed
4276 lemma subspace_isomorphism:
4277   assumes s: "subspace (S:: (real ^'n) set)" and t: "subspace T"
4278   and d: "dim S = dim T"
4279   shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
4280 proof-
4281   from basis_exists[of S] obtain B where
4282     B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
4283   from basis_exists[of T] obtain C where
4284     C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "C hassize dim T" by blast
4285   from B(4) C(4) card_le_inj[of B C] d obtain f where
4286     f: "f ` B \<subseteq> C" "inj_on f B" unfolding hassize_def by auto
4287   from linear_independent_extend[OF B(2)] obtain g where
4288     g: "linear g" "\<forall>x\<in> B. g x = f x" by blast
4289   from B(4) have fB: "finite B" by (simp add: hassize_def)
4290   from C(4) have fC: "finite C" by (simp add: hassize_def)
4291   from inj_on_iff_eq_card[OF fB, of f] f(2)
4292   have "card (f ` B) = card B" by simp
4293   with B(4) C(4) have ceq: "card (f ` B) = card C" using d
4295   have "g ` B = f ` B" using g(2)
4296     by (auto simp add: image_iff)
4297   also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
4298   finally have gBC: "g ` B = C" .
4299   have gi: "inj_on g B" using f(2) g(2)
4300     by (auto simp add: inj_on_def)
4301   note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
4302   {fix x y assume x: "x \<in> S" and y: "y \<in> S" and gxy:"g x = g y"
4303     from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" by blast+
4304     from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])
4305     have th1: "x - y \<in> span B" using x' y' by (metis span_sub)
4306     have "x=y" using g0[OF th1 th0] by simp }
4307   then have giS: "inj_on g S"
4308     unfolding inj_on_def by blast
4309   from span_subspace[OF B(1,3) s]
4310   have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])
4311   also have "\<dots> = span C" unfolding gBC ..
4312   also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
4313   finally have gS: "g ` S = T" .
4314   from g(1) gS giS show ?thesis by blast
4315 qed
4317 (* linear functions are equal on a subspace if they are on a spanning set.   *)
4319 lemma subspace_kernel:
4320   assumes lf: "linear (f::'a::semiring_1 ^'n \<Rightarrow> _)"
4321   shows "subspace {x. f x = 0}"
4325 lemma linear_eq_0_span:
4326   assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
4327   shows "\<forall>x \<in> span B. f x = (0::'a::semiring_1 ^'n)"
4328 proof
4329   fix x assume x: "x \<in> span B"
4330   let ?P = "\<lambda>x. f x = 0"
4331   from subspace_kernel[OF lf] have "subspace ?P" unfolding Collect_def .
4332   with x f0 span_induct[of B "?P" x] show "f x = 0" by blast
4333 qed
4335 lemma linear_eq_0:
4336   assumes lf: "linear f" and SB: "S \<subseteq> span B" and f0: "\<forall>x\<in>B. f x = 0"
4337   shows "\<forall>x \<in> S. f x = (0::'a::semiring_1^'n)"
4338   by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
4340 lemma linear_eq:
4341   assumes lf: "linear (f::'a::ring_1^'n \<Rightarrow> _)" and lg: "linear g" and S: "S \<subseteq> span B"
4342   and fg: "\<forall> x\<in> B. f x = g x"
4343   shows "\<forall>x\<in> S. f x = g x"
4344 proof-
4345   let ?h = "\<lambda>x. f x - g x"
4346   from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
4347   from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
4348   show ?thesis by simp
4349 qed
4351 lemma linear_eq_stdbasis:
4352   assumes lf: "linear (f::'a::ring_1^'m \<Rightarrow> 'a^'n)" and lg: "linear g"
4353   and fg: "\<forall>i \<in> {1 .. dimindex(UNIV :: 'm set)}. f (basis i) = g(basis i)"
4354   shows "f = g"
4355 proof-
4356   let ?U = "UNIV :: 'm set"
4357   let ?I = "{basis i:: 'a^'m|i. i \<in> {1 .. dimindex ?U}}"
4358   {fix x assume x: "x \<in> (UNIV :: ('a^'m) set)"
4359     from equalityD2[OF span_stdbasis]
4360     have IU: " (UNIV :: ('a^'m) set) \<subseteq> span ?I" by blast
4361     from linear_eq[OF lf lg IU] fg x
4362     have "f x = g x" unfolding Collect_def  Ball_def mem_def by metis}
4363   then show ?thesis by (auto intro: ext)
4364 qed
4366 (* Similar results for bilinear functions.                                   *)
4368 lemma bilinear_eq:
4369   assumes bf: "bilinear (f:: 'a::ring^'m \<Rightarrow> 'a^'n \<Rightarrow> 'a^'p)"
4370   and bg: "bilinear g"
4371   and SB: "S \<subseteq> span B" and TC: "T \<subseteq> span C"
4372   and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
4373   shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
4374 proof-
4375   let ?P = "\<lambda>x. \<forall>y\<in> span C. f x y = g x y"
4376   from bf bg have sp: "subspace ?P"
4377     unfolding bilinear_def linear_def subspace_def bf bg
4380   have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
4381     apply -
4382     apply (rule ballI)
4383     apply (rule span_induct[of B ?P])
4384     defer
4385     apply (rule sp)
4386     apply assumption
4387     apply (clarsimp simp add: Ball_def)
4388     apply (rule_tac P="\<lambda>y. f xa y = g xa y" and S=C in span_induct)
4389     using fg
4390     apply (auto simp add: subspace_def)
4391     using bf bg unfolding bilinear_def linear_def
4393   then show ?thesis using SB TC by (auto intro: ext)
4394 qed
4396 lemma bilinear_eq_stdbasis:
4397   assumes bf: "bilinear (f:: 'a::ring_1^'m \<Rightarrow> 'a^'n \<Rightarrow> 'a^'p)"
4398   and bg: "bilinear g"
4399   and fg: "\<forall>i\<in> {1 .. dimindex (UNIV :: 'm set)}. \<forall>j\<in>  {1 .. dimindex (UNIV :: 'n set)}. f (basis i) (basis j) = g (basis i) (basis j)"
4400   shows "f = g"
4401 proof-
4402   from fg have th: "\<forall>x \<in> {basis i| i. i\<in> {1 .. dimindex (UNIV :: 'm set)}}. \<forall>y\<in>  {basis j |j. j \<in> {1 .. dimindex (UNIV :: 'n set)}}. f x y = g x y" by blast
4403   from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] show ?thesis by (blast intro: ext)
4404 qed
4406 (* Detailed theorems about left and right invertibility in general case.     *)
4408 lemma left_invertible_transp:
4409   "(\<exists>(B::'a^'n^'m). B ** transp (A::'a^'n^'m) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B::'a^'m^'n). A ** B = mat 1)"
4410   by (metis matrix_transp_mul transp_mat transp_transp)
4412 lemma right_invertible_transp:
4413   "(\<exists>(B::'a^'n^'m). transp (A::'a^'n^'m) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B::'a^'m^'n). B ** A = mat 1)"
4414   by (metis matrix_transp_mul transp_mat transp_transp)
4416 lemma linear_injective_left_inverse:
4417   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and fi: "inj f"
4418   shows "\<exists>g. linear g \<and> g o f = id"
4419 proof-
4420   from linear_independent_extend[OF independent_injective_image, OF independent_stdbasis, OF lf fi]
4421   obtain h:: "real ^'m \<Rightarrow> real ^'n" where h: "linear h" " \<forall>x \<in> f ` {basis i|i. i \<in> {1 .. dimindex (UNIV::'n set)}}. h x = inv f x" by blast
4422   from h(2)
4423   have th: "\<forall>i\<in>{1..dimindex (UNIV::'n set)}. (h \<circ> f) (basis i) = id (basis i)"
4424     using inv_o_cancel[OF fi, unfolded stupid_ext[symmetric] id_def o_def]
4425     apply auto
4426     apply (erule_tac x="basis i" in allE)
4427     by auto
4429   from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
4430   have "h o f = id" .
4431   then show ?thesis using h(1) by blast
4432 qed
4434 lemma linear_surjective_right_inverse:
4435   assumes lf: "linear (f:: real ^'m \<Rightarrow> real ^'n)" and sf: "surj f"
4436   shows "\<exists>g. linear g \<and> f o g = id"
4437 proof-
4438   from linear_independent_extend[OF independent_stdbasis]
4439   obtain h:: "real ^'n \<Rightarrow> real ^'m" where
4440     h: "linear h" "\<forall> x\<in> {basis i| i. i\<in> {1 .. dimindex (UNIV :: 'n set)}}. h x = inv f x" by blast
4441   from h(2)
4442   have th: "\<forall>i\<in>{1..dimindex (UNIV::'n set)}. (f o h) (basis i) = id (basis i)"
4443     using sf
4444     apply (auto simp add: surj_iff o_def stupid_ext[symmetric])
4445     apply (erule_tac x="basis i" in allE)
4446     by auto
4448   from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
4449   have "f o h = id" .
4450   then show ?thesis using h(1) by blast
4451 qed
4453 lemma matrix_left_invertible_injective:
4454 "(\<exists>B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"
4455 proof-
4456   {fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
4457     from xy have "B*v (A *v x) = B *v (A*v y)" by simp
4458     hence "x = y"
4459       unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid .}
4460   moreover
4461   {assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
4462     hence i: "inj (op *v A)" unfolding inj_on_def by auto
4463     from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
4464     obtain g where g: "linear g" "g o op *v A = id" by blast
4465     have "matrix g ** A = mat 1"
4466       unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
4467       using g(2) by (simp add: o_def id_def stupid_ext)
4468     then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast}
4469   ultimately show ?thesis by blast
4470 qed
4472 lemma matrix_left_invertible_ker:
4473   "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
4474   unfolding matrix_left_invertible_injective
4475   using linear_injective_0[OF matrix_vector_mul_linear, of A]
4478 lemma matrix_right_invertible_surjective:
4479 "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
4480 proof-
4481   {fix B :: "real ^'m^'n"  assume AB: "A ** B = mat 1"
4482     {fix x :: "real ^ 'm"
4483       have "A *v (B *v x) = x"
4484 	by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB)}
4485     hence "surj (op *v A)" unfolding surj_def by metis }
4486   moreover
4487   {assume sf: "surj (op *v A)"
4488     from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
4489     obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
4490       by blast
4492     have "A ** (matrix g) = mat 1"
4493       unfolding matrix_eq  matrix_vector_mul_lid
4494 	matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
4495       using g(2) unfolding o_def stupid_ext[symmetric] id_def
4496       .
4497     hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
4498   }
4499   ultimately show ?thesis unfolding surj_def by blast
4500 qed
4502 lemma matrix_left_invertible_independent_columns:
4503   fixes A :: "real^'n^'m"
4504   shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s column i A) {1 .. dimindex(UNIV :: 'n set)} = 0 \<longrightarrow> (\<forall>i\<in> {1 .. dimindex (UNIV :: 'n set)}. c i = 0))"
4505    (is "?lhs \<longleftrightarrow> ?rhs")
4506 proof-
4507   let ?U = "{1 .. dimindex(UNIV :: 'n set)}"
4508   {assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
4509     {fix c i assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0"
4510       and i: "i \<in> ?U"
4511       let ?x = "\<chi> i. c i"
4512       have th0:"A *v ?x = 0"
4513 	using c
4514 	unfolding matrix_mult_vsum Cart_eq
4515 	by (auto simp add: vector_component zero_index setsum_component Cart_lambda_beta)
4516       from k[rule_format, OF th0] i
4517       have "c i = 0" by (vector Cart_eq)}
4518     hence ?rhs by blast}
4519   moreover
4520   {assume H: ?rhs
4521     {fix x assume x: "A *v x = 0"
4522       let ?c = "\<lambda>i. ((x\$i ):: real)"
4523       from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
4524       have "x = 0" by vector}}
4525   ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
4526 qed
4528 lemma matrix_right_invertible_independent_rows:
4529   fixes A :: "real^'n^'m"
4530   shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s row i A) {1 .. dimindex(UNIV :: 'm set)} = 0 \<longrightarrow> (\<forall>i\<in> {1 .. dimindex (UNIV :: 'm set)}. c i = 0))"
4531   unfolding left_invertible_transp[symmetric]
4532     matrix_left_invertible_independent_columns
4535 lemma matrix_right_invertible_span_columns:
4536   "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> span (columns A) = UNIV" (is "?lhs = ?rhs")
4537 proof-
4538   let ?U = "{1 .. dimindex (UNIV :: 'm set)}"
4539   have fU: "finite ?U" by simp
4540   have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x\$i) *s column i A) ?U = y)"
4541     unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
4542     apply (subst eq_commute) ..
4543   have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
4544   {assume h: ?lhs
4545     {fix x:: "real ^'n"
4546 	from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m"
4547 	  where y: "setsum (\<lambda>i. (y\$i) *s column i A) ?U = x" by blast
4548 	have "x \<in> span (columns A)"
4549 	  unfolding y[symmetric]
4550 	  apply (rule span_setsum[OF fU])
4551 	  apply clarify
4552 	  apply (rule span_mul)
4553 	  apply (rule span_superset)
4554 	  unfolding columns_def
4555 	  by blast}
4556     then have ?rhs unfolding rhseq by blast}
4557   moreover
4558   {assume h:?rhs
4559     let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x\$i) *s column i A) ?U = y"
4560     {fix y have "?P y"
4561       proof(rule span_induct_alt[of ?P "columns A"])
4562 	show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x\$i) *s column i A) ?U = 0"
4563 	  apply (rule exI[where x=0])
4564 	  by (simp add: zero_index vector_smult_lzero)
4565       next
4566 	fix c y1 y2 assume y1: "y1 \<in> columns A" and y2: "?P y2"
4567 	from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
4568 	  unfolding columns_def by blast
4569 	from y2 obtain x:: "real ^'m" where
4570 	  x: "setsum (\<lambda>i. (x\$i) *s column i A) ?U = y2" by blast
4571 	let ?x = "(\<chi> j. if j = i then c + (x\$i) else (x\$j))::real^'m"
4572 	show "?P (c*s y1 + y2)"
4573 	  proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric]Cart_lambda_beta setsum_component cond_value_iff right_distrib cond_application_beta vector_component cong del: if_weak_cong, simp only: One_nat_def[symmetric])
4574 	    fix j
4575 	    have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x\$i)) * ((column xa A)\$j)
4576            else (x\$xa) * ((column xa A\$j))) = (if xa = i then c * ((column i A)\$j) else 0) + ((x\$xa) * ((column xa A)\$j))" using i(1)
4578 	    have "setsum (\<lambda>xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j)
4579            else (x\$xa) * ((column xa A\$j))) ?U = setsum (\<lambda>xa. (if xa = i then c * ((column i A)\$j) else 0) + ((x\$xa) * ((column xa A)\$j))) ?U"
4580 	      apply (rule setsum_cong[OF refl])
4581 	      using th by blast
4582 	    also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)\$j) else 0) ?U + setsum (\<lambda>xa. ((x\$xa) * ((column xa A)\$j))) ?U"
4584 	    also have "\<dots> = c * ((column i A)\$j) + setsum (\<lambda>xa. ((x\$xa) * ((column xa A)\$j))) ?U"
4585 	      unfolding setsum_delta[OF fU]
4586 	      using i(1) by simp
4587 	    finally show "setsum (\<lambda>xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j)
4588            else (x\$xa) * ((column xa A\$j))) ?U = c * ((column i A)\$j) + setsum (\<lambda>xa. ((x\$xa) * ((column xa A)\$j))) ?U" .
4589 	  qed
4590 	next
4591 	  show "y \<in> span (columns A)" unfolding h by blast
4592 	qed}
4593     then have ?lhs unfolding lhseq ..}
4594   ultimately show ?thesis by blast
4595 qed
4597 lemma matrix_left_invertible_span_rows:
4598   "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
4599   unfolding right_invertible_transp[symmetric]
4600   unfolding columns_transp[symmetric]
4601   unfolding matrix_right_invertible_span_columns
4602  ..
4604 (* An injective map real^'n->real^'n is also surjective.                       *)
4606 lemma linear_injective_imp_surjective:
4607   assumes lf: "linear (f:: real ^'n \<Rightarrow> real ^'n)" and fi: "inj f"
4608   shows "surj f"
4609 proof-
4610   let ?U = "UNIV :: (real ^'n) set"
4611   from basis_exists[of ?U] obtain B
4612     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "B hassize dim ?U"
4613     by blast
4614   from B(4) have d: "dim ?U = card B" by (simp add: hassize_def)
4615   have th: "?U \<subseteq> span (f ` B)"
4616     apply (rule card_ge_dim_independent)
4617     apply blast
4618     apply (rule independent_injective_image[OF B(2) lf fi])
4619     apply (rule order_eq_refl)
4620     apply (rule sym)
4621     unfolding d
4622     apply (rule card_image)
4623     apply (rule subset_inj_on[OF fi])
4624     by blast
4625   from th show ?thesis
4626     unfolding span_linear_image[OF lf] surj_def
4627     using B(3) by blast
4628 qed
4630 (* And vice versa.                                                           *)
4632 lemma surjective_iff_injective_gen:
4633   assumes fS: "finite S" and fT: "finite T" and c: "card S = card T"
4634   and ST: "f ` S \<subseteq> T"
4635   shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S" (is "?lhs \<longleftrightarrow> ?rhs")
4636 proof-
4637   {assume h: "?lhs"
4638     {fix x y assume x: "x \<in> S" and y: "y \<in> S" and f: "f x = f y"
4639       from x fS have S0: "card S \<noteq> 0" by auto
4640       {assume xy: "x \<noteq> y"
4641 	have th: "card S \<le> card (f ` (S - {y}))"
4642 	  unfolding c
4643 	  apply (rule card_mono)
4644 	  apply (rule finite_imageI)
4645 	  using fS apply simp
4646 	  using h xy x y f unfolding subset_eq image_iff
4647 	  apply auto
4648 	  apply (case_tac "xa = f x")
4649 	  apply (rule bexI[where x=x])
4650 	  apply auto
4651 	  done
4652 	also have " \<dots> \<le> card (S -{y})"
4653 	  apply (rule card_image_le)
4654 	  using fS by simp
4655 	also have "\<dots> \<le> card S - 1" using y fS by simp
4656 	finally have False  using S0 by arith }
4657       then have "x = y" by blast}
4658     then have ?rhs unfolding inj_on_def by blast}
4659   moreover
4660   {assume h: ?rhs
4661     have "f ` S = T"
4662       apply (rule card_subset_eq[OF fT ST])
4663       unfolding card_image[OF h] using c .
4664     then have ?lhs by blast}
4665   ultimately show ?thesis by blast
4666 qed
4668 lemma linear_surjective_imp_injective:
4669   assumes lf: "linear (f::real ^'n => real ^'n)" and sf: "surj f"
4670   shows "inj f"
4671 proof-
4672   let ?U = "UNIV :: (real ^'n) set"
4673   from basis_exists[of ?U] obtain B
4674     where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "B hassize dim ?U"
4675     by blast
4676   {fix x assume x: "x \<in> span B" and fx: "f x = 0"
4677     from B(4) have fB: "finite B" by (simp add: hassize_def)
4678     from B(4) have d: "dim ?U = card B" by (simp add: hassize_def)
4679     have fBi: "independent (f ` B)"
4680       apply (rule card_le_dim_spanning[of "f ` B" ?U])
4681       apply blast
4682       using sf B(3)
4683       unfolding span_linear_image[OF lf] surj_def subset_eq image_iff
4684       apply blast
4685       using fB apply (blast intro: finite_imageI)
4686       unfolding d
4687       apply (rule card_image_le)
4688       apply (rule fB)
4689       done
4690     have th0: "dim ?U \<le> card (f ` B)"
4691       apply (rule span_card_ge_dim)
4692       apply blast
4693       unfolding span_linear_image[OF lf]
4694       apply (rule subset_trans[where B = "f ` UNIV"])
4695       using sf unfolding surj_def apply blast
4696       apply (rule image_mono)
4697       apply (rule B(3))
4698       apply (metis finite_imageI fB)
4699       done
4701     moreover have "card (f ` B) \<le> card B"
4702       by (rule card_image_le, rule fB)
4703     ultimately have th1: "card B = card (f ` B)" unfolding d by arith
4704     have fiB: "inj_on f B"
4705       unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric] by blast
4706     from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
4707     have "x = 0" by blast}
4708   note th = this
4709   from th show ?thesis unfolding linear_injective_0[OF lf]
4710     using B(3) by blast
4711 qed
4713 (* Hence either is enough for isomorphism.                                   *)
4715 lemma left_right_inverse_eq:
4716   assumes fg: "f o g = id" and gh: "g o h = id"
4717   shows "f = h"
4718 proof-
4719   have "f = f o (g o h)" unfolding gh by simp
4720   also have "\<dots> = (f o g) o h" by (simp add: o_assoc)
4721   finally show "f = h" unfolding fg by simp
4722 qed
4724 lemma isomorphism_expand:
4725   "f o g = id \<and> g o f = id \<longleftrightarrow> (\<forall>x. f(g x) = x) \<and> (\<forall>x. g(f x) = x)"
4726   by (simp add: expand_fun_eq o_def id_def)
4728 lemma linear_injective_isomorphism:
4729   assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'n)" and fi: "inj f"
4730   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
4731 unfolding isomorphism_expand[symmetric]
4732 using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]] linear_injective_left_inverse[OF lf fi]
4733 by (metis left_right_inverse_eq)
4735 lemma linear_surjective_isomorphism:
4736   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and sf: "surj f"
4737   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
4738 unfolding isomorphism_expand[symmetric]
4739 using linear_surjective_right_inverse[OF lf sf] linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
4740 by (metis left_right_inverse_eq)
4742 (* Left and right inverses are the same for R^N->R^N.                        *)
4744 lemma linear_inverse_left:
4745   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and lf': "linear f'"
4746   shows "f o f' = id \<longleftrightarrow> f' o f = id"
4747 proof-
4748   {fix f f':: "real ^'n \<Rightarrow> real ^'n"
4749     assume lf: "linear f" "linear f'" and f: "f o f' = id"
4750     from f have sf: "surj f"
4752       apply (auto simp add: o_def stupid_ext[symmetric] id_def surj_def)
4753       by metis
4754     from linear_surjective_isomorphism[OF lf(1) sf] lf f
4755     have "f' o f = id" unfolding stupid_ext[symmetric] o_def id_def
4756       by metis}
4757   then show ?thesis using lf lf' by metis
4758 qed
4760 (* Moreover, a one-sided inverse is automatically linear.                    *)
4762 lemma left_inverse_linear:
4763   assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and gf: "g o f = id"
4764   shows "linear g"
4765 proof-
4766   from gf have fi: "inj f" apply (auto simp add: inj_on_def o_def id_def stupid_ext[symmetric])
4767     by metis
4768   from linear_injective_isomorphism[OF lf fi]
4769   obtain h:: "real ^'n \<Rightarrow> real ^'n" where
4770     h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
4771   have "h = g" apply (rule ext) using gf h(2,3)
4772     apply (simp add: o_def id_def stupid_ext[symmetric])
4773     by metis
4774   with h(1) show ?thesis by blast
4775 qed
4777 lemma right_inverse_linear:
4778   assumes lf: "linear (f:: real ^'n \<Rightarrow> real ^'n)" and gf: "f o g = id"
4779   shows "linear g"
4780 proof-
4781   from gf have fi: "surj f" apply (auto simp add: surj_def o_def id_def stupid_ext[symmetric])
4782     by metis
4783   from linear_surjective_isomorphism[OF lf fi]
4784   obtain h:: "real ^'n \<Rightarrow> real ^'n" where
4785     h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
4786   have "h = g" apply (rule ext) using gf h(2,3)
4787     apply (simp add: o_def id_def stupid_ext[symmetric])
4788     by metis
4789   with h(1) show ?thesis by blast
4790 qed
4792 (* The same result in terms of square matrices.                              *)
4794 lemma matrix_left_right_inverse:
4795   fixes A A' :: "real ^'n^'n"
4796   shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
4797 proof-
4798   {fix A A' :: "real ^'n^'n" assume AA': "A ** A' = mat 1"
4799     have sA: "surj (op *v A)"
4800       unfolding surj_def
4801       apply clarify
4802       apply (rule_tac x="(A' *v y)" in exI)
4803       by (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
4804     from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
4805     obtain f' :: "real ^'n \<Rightarrow> real ^'n"
4806       where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
4807     have th: "matrix f' ** A = mat 1"
4808       by (simp add: matrix_eq matrix_works[OF f'(1)] matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
4809     hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
4810     hence "matrix f' = A'" by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
4811     hence "matrix f' ** A = A' ** A" by simp
4812     hence "A' ** A = mat 1" by (simp add: th)}
4813   then show ?thesis by blast
4814 qed
4816 (* Considering an n-element vector as an n-by-1 or 1-by-n matrix.            *)
4818 definition "rowvector v = (\<chi> i j. (v\$j))"
4820 definition "columnvector v = (\<chi> i j. (v\$i))"
4822 lemma transp_columnvector:
4823  "transp(columnvector v) = rowvector v"
4824   by (simp add: transp_def rowvector_def columnvector_def Cart_eq Cart_lambda_beta)
4826 lemma transp_rowvector: "transp(rowvector v) = columnvector v"
4827   by (simp add: transp_def columnvector_def rowvector_def Cart_eq Cart_lambda_beta)
4829 lemma dot_rowvector_columnvector:
4830   "columnvector (A *v v) = A ** columnvector v"
4831   by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
4833 lemma dot_matrix_product: "(x::'a::semiring_1^'n) \<bullet> y = (((rowvector x ::'a^'n^1) ** (columnvector y :: 'a^1^'n))\$1)\$1"
4834   apply (vector matrix_matrix_mult_def rowvector_def columnvector_def dot_def)
4837 lemma dot_matrix_vector_mul:
4838   fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
4839   shows "(A *v x) \<bullet> (B *v y) =
4840       (((rowvector x :: real^'n^1) ** ((transp A ** B) ** (columnvector y :: real ^1^'n)))\$1)\$1"
4841 unfolding dot_matrix_product transp_columnvector[symmetric]
4842   dot_rowvector_columnvector matrix_transp_mul matrix_mul_assoc ..
4844 (* Infinity norm.                                                            *)
4846 definition "infnorm (x::real^'n) = rsup {abs(x\$i) |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}}"
4848 lemma numseg_dimindex_nonempty: "\<exists>i. i \<in> {1 .. dimindex (UNIV :: 'n set)}"
4849   using dimindex_ge_1 by auto
4851 lemma infnorm_set_image:
4852   "{abs(x\$i) |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}} =
4853   (\<lambda>i. abs(x\$i)) ` {1 .. dimindex(UNIV :: 'n set)}" by blast
4855 lemma infnorm_set_lemma:
4856   shows "finite {abs((x::'a::abs ^'n)\$i) |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}}"
4857   and "{abs(x\$i) |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}} \<noteq> {}"
4858   unfolding infnorm_set_image
4859   using dimindex_ge_1[of "UNIV :: 'n set"]
4860   by (auto intro: finite_imageI)
4862 lemma infnorm_pos_le: "0 \<le> infnorm x"
4863   unfolding infnorm_def
4864   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
4865   unfolding infnorm_set_image
4866   using dimindex_ge_1
4867   by auto
4869 lemma infnorm_triangle: "infnorm ((x::real^'n) + y) \<le> infnorm x + infnorm y"
4870 proof-
4871   have th: "\<And>x y (z::real). x - y <= z \<longleftrightarrow> x - z <= y" by arith
4872   have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
4873   have th2: "\<And>x (y::real). abs(x + y) - abs(x) <= abs(y)" by arith
4874   show ?thesis
4875   unfolding infnorm_def
4876   unfolding rsup_finite_le_iff[ OF infnorm_set_lemma]
4877   apply (subst diff_le_eq[symmetric])
4878   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
4879   unfolding infnorm_set_image bex_simps
4880   apply (subst th)
4881   unfolding th1
4882   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
4884   unfolding infnorm_set_image ball_simps bex_simps
4886   apply (metis numseg_dimindex_nonempty th2)
4887   done
4888 qed
4890 lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::real ^'n) = 0"
4891 proof-
4892   have "infnorm x <= 0 \<longleftrightarrow> x = 0"
4893     unfolding infnorm_def
4894     unfolding rsup_finite_le_iff[OF infnorm_set_lemma]
4895     unfolding infnorm_set_image ball_simps
4896     by vector
4897   then show ?thesis using infnorm_pos_le[of x] by simp
4898 qed
4900 lemma infnorm_0: "infnorm 0 = 0"
4903 lemma infnorm_neg: "infnorm (- x) = infnorm x"
4904   unfolding infnorm_def
4905   apply (rule cong[of "rsup" "rsup"])
4906   apply blast
4907   apply (rule set_ext)
4908   apply (auto simp add: vector_component abs_minus_cancel)
4909   apply (rule_tac x="i" in exI)
4911   done
4913 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
4914 proof-
4915   have "y - x = - (x - y)" by simp
4916   then show ?thesis  by (metis infnorm_neg)
4917 qed
4919 lemma real_abs_sub_infnorm: "\<bar> infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
4920 proof-
4921   have th: "\<And>(nx::real) n ny. nx <= n + ny \<Longrightarrow> ny <= n + nx ==> \<bar>nx - ny\<bar> <= n"
4922     by arith
4923   from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
4924   have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
4925     "infnorm y \<le> infnorm (x - y) + infnorm x"
4926     by (simp_all add: ring_simps infnorm_neg diff_def[symmetric])
4927   from th[OF ths]  show ?thesis .
4928 qed
4930 lemma real_abs_infnorm: " \<bar>infnorm x\<bar> = infnorm x"
4931   using infnorm_pos_le[of x] by arith
4933 lemma component_le_infnorm: assumes i: "i \<in> {1 .. dimindex (UNIV :: 'n set)}"
4934   shows "\<bar>x\$i\<bar> \<le> infnorm (x::real^'n)"
4935 proof-
4936   let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
4937   let ?S = "{\<bar>x\$i\<bar> |i. i\<in> ?U}"
4938   have fS: "finite ?S" unfolding image_Collect[symmetric]
4939     apply (rule finite_imageI) unfolding Collect_def mem_def by simp
4940   have S0: "?S \<noteq> {}" using numseg_dimindex_nonempty by blast
4941   have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
4942   from rsup_finite_in[OF fS S0] rsup_finite_Ub[OF fS S0] i
4943   show ?thesis unfolding infnorm_def isUb_def setle_def
4944     unfolding infnorm_set_image ball_simps by auto
4945 qed
4947 lemma infnorm_mul_lemma: "infnorm(a *s x) <= \<bar>a\<bar> * infnorm x"
4948   apply (subst infnorm_def)
4949   unfolding rsup_finite_le_iff[OF infnorm_set_lemma]
4950   unfolding infnorm_set_image ball_simps
4951   apply (simp add: abs_mult vector_component del: One_nat_def)
4952   apply (rule ballI)
4953   apply (drule component_le_infnorm[of _ x])
4954   apply (rule mult_mono)
4955   apply auto
4956   done
4958 lemma infnorm_mul: "infnorm(a *s x) = abs a * infnorm x"
4959 proof-
4960   {assume a0: "a = 0" hence ?thesis by (simp add: infnorm_0) }
4961   moreover
4962   {assume a0: "a \<noteq> 0"
4963     from a0 have th: "(1/a) *s (a *s x) = x"
4965     from a0 have ap: "\<bar>a\<bar> > 0" by arith
4966     from infnorm_mul_lemma[of "1/a" "a *s x"]
4967     have "infnorm x \<le> 1/\<bar>a\<bar> * infnorm (a*s x)"
4968       unfolding th by simp
4969     with ap have "\<bar>a\<bar> * infnorm x \<le> \<bar>a\<bar> * (1/\<bar>a\<bar> * infnorm (a *s x))" by (simp add: field_simps)
4970     then have "\<bar>a\<bar> * infnorm x \<le> infnorm (a*s x)"
4971       using ap by (simp add: field_simps)
4972     with infnorm_mul_lemma[of a x] have ?thesis by arith }
4973   ultimately show ?thesis by blast
4974 qed
4976 lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
4977   using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
4979 (* Prove that it differs only up to a bound from Euclidean norm.             *)
4981 lemma infnorm_le_norm: "infnorm x \<le> norm x"
4982   unfolding infnorm_def rsup_finite_le_iff[OF infnorm_set_lemma]
4983   unfolding infnorm_set_image  ball_simps
4984   by (metis component_le_norm)
4985 lemma card_enum: "card {1 .. n} = n" by auto
4986 lemma norm_le_infnorm: "norm(x) <= sqrt(real (dimindex(UNIV ::'n set))) * infnorm(x::real ^'n)"
4987 proof-
4988   let ?d = "dimindex(UNIV ::'n set)"
4989   have d: "?d = card {1 .. ?d}" by auto
4990   have "real ?d \<ge> 0" by simp
4991   hence d2: "(sqrt (real ?d))^2 = real ?d"
4992     by (auto intro: real_sqrt_pow2)
4993   have th: "sqrt (real ?d) * infnorm x \<ge> 0"
4994     by (simp add: dimindex_ge_1 zero_le_mult_iff real_sqrt_ge_0_iff infnorm_pos_le)
4995   have th1: "x\<bullet>x \<le> (sqrt (real ?d) * infnorm x)^2"
4996     unfolding power_mult_distrib d2
4997     apply (subst d)
4998     apply (subst power2_abs[symmetric])
4999     unfolding real_of_nat_def dot_def power2_eq_square[symmetric]
5000     apply (subst power2_abs[symmetric])
5001     apply (rule setsum_bounded)
5002     apply (rule power_mono)
5003     unfolding abs_of_nonneg[OF infnorm_pos_le]
5004     unfolding infnorm_def  rsup_finite_ge_iff[OF infnorm_set_lemma]
5005     unfolding infnorm_set_image bex_simps
5006     apply blast
5007     by (rule abs_ge_zero)
5008   from real_le_lsqrt[OF dot_pos_le th th1]
5009   show ?thesis unfolding real_vector_norm_def  real_of_real_def id_def .
5010 qed
5012 (* Equality in Cauchy-Schwarz and triangle inequalities.                     *)
5014 lemma norm_cauchy_schwarz_eq: "(x::real ^'n) \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *s y = norm y *s x" (is "?lhs \<longleftrightarrow> ?rhs")
5015 proof-
5016   {assume h: "x = 0"
5017     hence ?thesis by (simp add: norm_0)}
5018   moreover
5019   {assume h: "y = 0"
5020     hence ?thesis by (simp add: norm_0)}
5021   moreover
5022   {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
5023     from dot_eq_0[of "norm y *s x - norm x *s y"]
5024     have "?rhs \<longleftrightarrow> (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) - norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
5025       using x y
5026       unfolding dot_rsub dot_lsub dot_lmult dot_rmult
5027       unfolding norm_pow_2[symmetric] power2_eq_square diff_eq_0_iff_eq apply (simp add: dot_sym)
5029       apply metis
5030       done
5031     also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
5032       by (simp add: ring_simps dot_sym)
5033     also have "\<dots> \<longleftrightarrow> ?lhs" using x y
5035       by metis
5036     finally have ?thesis by blast}
5037   ultimately show ?thesis by blast
5038 qed
5040 lemma norm_cauchy_schwarz_abs_eq: "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow>
5041                 norm x *s y = norm y *s x \<or> norm(x) *s y = - norm y *s x" (is "?lhs \<longleftrightarrow> ?rhs")
5042 proof-
5043   have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a" by arith
5044   have "?rhs \<longleftrightarrow> norm x *s y = norm y *s x \<or> norm (- x) *s y = norm y *s (- x)"
5045     apply (simp add: norm_neg) by vector
5046   also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or>
5047      (-x) \<bullet> y = norm x * norm y)"
5048     unfolding norm_cauchy_schwarz_eq[symmetric]
5049     unfolding norm_neg
5050       norm_mul by blast
5051   also have "\<dots> \<longleftrightarrow> ?lhs"
5052     unfolding th[OF mult_nonneg_nonneg, OF norm_pos_le[of x] norm_pos_le[of y]] dot_lneg
5053     by arith
5054   finally show ?thesis ..
5055 qed
5057 lemma norm_triangle_eq: "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *s y = norm y *s x"
5058 proof-
5059   {assume x: "x =0 \<or> y =0"
5060     hence ?thesis by (cases "x=0", simp_all add: norm_0)}
5061   moreover
5062   {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
5063     hence "norm x \<noteq> 0" "norm y \<noteq> 0"
5065     hence n: "norm x > 0" "norm y > 0"
5066       using norm_pos_le[of x] norm_pos_le[of y]
5067       by arith+
5068     have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a^2 = (b + c)^2)" by algebra
5069     have "norm(x + y) = norm x + norm y \<longleftrightarrow> norm(x + y)^ 2 = (norm x + norm y) ^2"
5070       apply (rule th) using n norm_pos_le[of "x + y"]
5071       by arith
5072     also have "\<dots> \<longleftrightarrow> norm x *s y = norm y *s x"
5073       unfolding norm_cauchy_schwarz_eq[symmetric]
5075       by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym ring_simps)
5076     finally have ?thesis .}
5077   ultimately show ?thesis by blast
5078 qed
5080 (* Collinearity.*)
5082 definition "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *s u)"
5084 lemma collinear_empty:  "collinear {}" by (simp add: collinear_def)
5086 lemma collinear_sing: "collinear {(x::'a::ring_1^'n)}"
5088   apply (rule exI[where x=0])
5089   by simp
5091 lemma collinear_2: "collinear {(x::'a::ring_1^'n),y}"
5093   apply (rule exI[where x="x - y"])
5094   apply auto
5095   apply (rule exI[where x=0], simp)
5096   apply (rule exI[where x=1], simp)
5097   apply (rule exI[where x="- 1"], simp add: vector_sneg_minus1[symmetric])
5098   apply (rule exI[where x=0], simp)
5099   done
5101 lemma collinear_lemma: "collinear {(0::real^'n),x,y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *s x)" (is "?lhs \<longleftrightarrow> ?rhs")
5102 proof-
5103   {assume "x=0 \<or> y = 0" hence ?thesis
5104       by (cases "x = 0", simp_all add: collinear_2 insert_commute)}
5105   moreover
5106   {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
5107     {assume h: "?lhs"
5108       then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *s u" unfolding collinear_def by blast
5109       from u[rule_format, of x 0] u[rule_format, of y 0]
5110       obtain cx and cy where
5111 	cx: "x = cx*s u" and cy: "y = cy*s u"
5112 	by auto
5113       from cx x have cx0: "cx \<noteq> 0" by auto
5114       from cy y have cy0: "cy \<noteq> 0" by auto
5115       let ?d = "cy / cx"
5116       from cx cy cx0 have "y = ?d *s x"
5118       hence ?rhs using x y by blast}
5119     moreover
5120     {assume h: "?rhs"
5121       then obtain c where c: "y = c*s x" using x y by blast
5122       have ?lhs unfolding collinear_def c
5123 	apply (rule exI[where x=x])
5124 	apply auto
5125 	apply (rule exI[where x=0], simp)
5126 	apply (rule exI[where x="- 1"], simp only: vector_smult_lneg vector_smult_lid)
5127 	apply (rule exI[where x= "-c"], simp only: vector_smult_lneg)
5128 	apply (rule exI[where x=1], simp)
5129 	apply (rule exI[where x=0], simp)
5130 	apply (rule exI[where x="1 - c"], simp add: vector_smult_lneg vector_sub_rdistrib)
5131 	apply (rule exI[where x="c - 1"], simp add: vector_smult_lneg vector_sub_rdistrib)
5132 	apply (rule exI[where x=0], simp)
5133 	done}
5134     ultimately have ?thesis by blast}
5135   ultimately show ?thesis by blast
5136 qed
5138 lemma norm_cauchy_schwarz_equal: "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {(0::real^'n),x,y}"
5139 unfolding norm_cauchy_schwarz_abs_eq
5140 apply (cases "x=0", simp_all add: collinear_2 norm_0)
5141 apply (cases "y=0", simp_all add: collinear_2 norm_0 insert_commute)
5142 unfolding collinear_lemma
5143 apply simp
5144 apply (subgoal_tac "norm x \<noteq> 0")
5145 apply (subgoal_tac "norm y \<noteq> 0")
5146 apply (rule iffI)
5147 apply (cases "norm x *s y = norm y *s x")
5148 apply (rule exI[where x="(1/norm x) * norm y"])
5149 apply (drule sym)
5150 unfolding vector_smult_assoc[symmetric]
5151 apply (simp add: vector_smult_assoc field_simps)
5152 apply (rule exI[where x="(1/norm x) * - norm y"])
5153 apply clarify
5154 apply (drule sym)
5155 unfolding vector_smult_assoc[symmetric]
5156 apply (simp add: vector_smult_assoc field_simps)
5157 apply (erule exE)
5158 apply (erule ssubst)
5159 unfolding vector_smult_assoc
5160 unfolding norm_mul
5161 apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
5162 apply (case_tac "c <= 0", simp add: ring_simps)