ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
1 header {*Instanciates the finite cartesian product of euclidean spaces as a euclidean space.*}
3 theory Cartesian_Euclidean_Space
4 imports Finite_Cartesian_Product Integration
7 lemma delta_mult_idempotent:
8 "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)"
12 "\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow>
13 (\<Sum>x\<in>A <+> B. g x) = (\<Sum>x\<in>A. g (Inl x)) + (\<Sum>x\<in>B. g (Inr x))"
15 by (subst setsum_Un_disjoint, auto simp add: setsum_reindex)
17 lemma setsum_UNIV_sum:
18 fixes g :: "'a::finite + 'b::finite \<Rightarrow> _"
19 shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))"
20 apply (subst UNIV_Plus_UNIV [symmetric])
21 apply (rule setsum_Plus [OF finite finite])
24 lemma setsum_mult_product:
25 "setsum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
26 unfolding sumr_group[of h B A, unfolded atLeast0LessThan, symmetric]
27 proof (rule setsum_cong, simp, rule setsum_reindex_cong)
29 show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)
30 show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}"
32 fix j assume "j \<in> {i * B..<i * B + B}"
33 then show "j \<in> (\<lambda>j. j + i * B) ` {..<B}"
34 by (auto intro!: image_eqI[of _ _ "j - i * B"])
39 subsection{* Basic componentwise operations on vectors. *}
41 instantiation vec :: (times, finite) times
44 definition "op * \<equiv> (\<lambda> x y. (\<chi> i. (x$i) * (y$i)))"
49 instantiation vec :: (one, finite) one
52 definition "1 \<equiv> (\<chi> i. 1)"
57 instantiation vec :: (ord, finite) ord
60 definition "x \<le> y \<longleftrightarrow> (\<forall>i. x$i \<le> y$i)"
61 definition "x < (y::'a^'b) \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
66 text{* The ordering on one-dimensional vectors is linear. *}
69 assumes UNIV_one: "card (UNIV \<Colon> 'a set) = Suc 0"
74 from UNIV_one show "finite (UNIV :: 'a set)"
75 by (auto intro!: card_ge_0_finite)
80 instance vec:: (order, finite) order
81 by default (auto simp: less_eq_vec_def less_vec_def vec_eq_iff
82 intro: order.trans order.antisym order.strict_implies_order)
84 instance vec :: (linorder, cart_one) linorder
86 obtain a :: 'b where all: "\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a"
88 have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one)
89 then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq)
90 then have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P b" by auto
91 then show thesis by (auto intro: that)
93 fix x y :: "'a^'b::cart_one"
94 note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps
95 show "x \<le> y \<or> y \<le> x" by auto
98 text{* Constant Vectors *}
100 definition "vec x = (\<chi> i. x)"
102 text{* Also the scalar-vector multiplication. *}
104 definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
105 where "c *s x = (\<chi> i. c * (x$i))"
108 subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *}
110 method_setup vector = {*
113 simpset_of (put_simpset HOL_basic_ss @{context}
114 addsimps [@{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])
118 simpset_of (@{context} addsimps
119 [@{thm plus_vec_def}, @{thm times_vec_def},
120 @{thm minus_vec_def}, @{thm uminus_vec_def},
121 @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def},
122 @{thm scaleR_vec_def},
123 @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}])
124 fun vector_arith_tac ctxt ths =
125 simp_tac (put_simpset ss1 ctxt)
126 THEN' (fn i => rtac @{thm setsum_cong2} i
127 ORELSE rtac @{thm setsum_0'} i
128 ORELSE simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm vec_eq_iff}]) i)
129 (* THEN' TRY o clarify_tac HOL_cs THEN' (TRY o rtac @{thm iffI}) *)
130 THEN' asm_full_simp_tac (put_simpset ss2 ctxt addsimps ths)
132 Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (vector_arith_tac ctxt ths))
134 *} "lift trivial vector statements to real arith statements"
136 lemma vec_0[simp]: "vec 0 = 0" by (vector zero_vec_def)
137 lemma vec_1[simp]: "vec 1 = 1" by (vector one_vec_def)
139 lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
141 lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
143 lemma vec_add: "vec(x + y) = vec x + vec y" by (vector vec_def)
144 lemma vec_sub: "vec(x - y) = vec x - vec y" by (vector vec_def)
145 lemma vec_cmul: "vec(c * x) = c *s vec x " by (vector vec_def)
146 lemma vec_neg: "vec(- x) = - vec x " by (vector vec_def)
150 shows "vec(setsum f S) = setsum (vec o f) S"
154 then show ?case by simp
157 then show ?case by (auto simp add: vec_add)
160 text{* Obvious "component-pushing". *}
162 lemma vec_component [simp]: "vec x $ i = x"
165 lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
168 lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
171 lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
173 lemmas vector_component =
174 vec_component vector_add_component vector_mult_component
175 vector_smult_component vector_minus_component vector_uminus_component
176 vector_scaleR_component cond_component
179 subsection {* Some frequently useful arithmetic lemmas over vectors. *}
181 instance vec :: (semigroup_mult, finite) semigroup_mult
182 by default (vector mult_assoc)
184 instance vec :: (monoid_mult, finite) monoid_mult
187 instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
188 by default (vector mult_commute)
190 instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
193 instance vec :: (semiring, finite) semiring
194 by default (vector field_simps)+
196 instance vec :: (semiring_0, finite) semiring_0
197 by default (vector field_simps)+
198 instance vec :: (semiring_1, finite) semiring_1
200 instance vec :: (comm_semiring, finite) comm_semiring
201 by default (vector field_simps)+
203 instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..
204 instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
205 instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..
206 instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..
207 instance vec :: (ring, finite) ring ..
208 instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..
209 instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..
211 instance vec :: (ring_1, finite) ring_1 ..
213 instance vec :: (real_algebra, finite) real_algebra
214 by default (simp_all add: vec_eq_iff)
216 instance vec :: (real_algebra_1, finite) real_algebra_1 ..
218 lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
221 then show ?case by vector
224 then show ?case by vector
227 lemma one_index [simp]: "(1 :: 'a :: one ^ 'n) $ i = 1"
230 lemma neg_one_index [simp]: "(- 1 :: 'a :: {one, uminus} ^ 'n) $ i = - 1"
233 instance vec :: (semiring_char_0, finite) semiring_char_0
236 show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"
237 by (auto intro!: injI simp add: vec_eq_iff of_nat_index)
240 instance vec :: (numeral, finite) numeral ..
241 instance vec :: (semiring_numeral, finite) semiring_numeral ..
243 lemma numeral_index [simp]: "numeral w $ i = numeral w"
244 by (induct w) (simp_all only: numeral.simps vector_add_component one_index)
246 lemma neg_numeral_index [simp]: "- numeral w $ i = - numeral w"
247 by (simp only: vector_uminus_component numeral_index)
249 instance vec :: (comm_ring_1, finite) comm_ring_1 ..
250 instance vec :: (ring_char_0, finite) ring_char_0 ..
252 lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
253 by (vector mult_assoc)
254 lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
255 by (vector field_simps)
256 lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
257 by (vector field_simps)
258 lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
259 lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
260 lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
261 by (vector field_simps)
262 lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
263 lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
264 lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector
265 lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
266 lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
267 by (vector field_simps)
269 lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
270 by (simp add: vec_eq_iff)
272 lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
273 lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
275 lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
276 by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
277 lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
278 by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
279 lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==> a *s x = a *s y ==> (x = y)"
280 by (metis vector_mul_lcancel)
281 lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
282 by (metis vector_mul_rcancel)
284 lemma component_le_norm_cart: "\<bar>x$i\<bar> <= norm x"
285 apply (simp add: norm_vec_def)
286 apply (rule member_le_setL2, simp_all)
289 lemma norm_bound_component_le_cart: "norm x <= e ==> \<bar>x$i\<bar> <= e"
290 by (metis component_le_norm_cart order_trans)
292 lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
293 by (metis component_le_norm_cart le_less_trans)
295 lemma norm_le_l1_cart: "norm x <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
296 by (simp add: norm_vec_def setL2_le_setsum)
298 lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
299 unfolding scaleR_vec_def vector_scalar_mult_def by simp
301 lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
302 unfolding dist_norm scalar_mult_eq_scaleR
303 unfolding scaleR_right_diff_distrib[symmetric] by simp
305 lemma setsum_component [simp]:
306 fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
307 shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
308 proof (cases "finite S")
310 then show ?thesis by induct simp_all
313 then show ?thesis by simp
316 lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
317 by (simp add: vec_eq_iff)
320 fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
321 shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
322 by (simp add: vec_eq_iff setsum_right_distrib)
324 lemma setsum_norm_allsubsets_bound_cart:
325 fixes f:: "'a \<Rightarrow> real ^'n"
326 assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
327 shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) * e"
328 using setsum_norm_allsubsets_bound[OF assms]
329 by (simp add: DIM_cart Basis_real_def)
331 instantiation vec :: (ordered_euclidean_space, finite) ordered_euclidean_space
334 definition "inf x y = (\<chi> i. inf (x $ i) (y $ i))"
335 definition "sup x y = (\<chi> i. sup (x $ i) (y $ i))"
336 definition "Inf X = (\<chi> i. (INF x:X. x $ i))"
337 definition "Sup X = (\<chi> i. (SUP x:X. x $ i))"
338 definition "abs x = (\<chi> i. abs (x $ i))"
342 unfolding euclidean_representation_setsum'
343 apply (auto simp: less_eq_vec_def inf_vec_def sup_vec_def Inf_vec_def Sup_vec_def inner_axis
344 Basis_vec_def inner_Basis_inf_left inner_Basis_sup_left inner_Basis_INF_left
345 inner_Basis_SUP_left eucl_le[where 'a='a] less_le_not_le abs_vec_def abs_inner)
350 subsection {* Matrix operations *}
352 text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
354 definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
356 where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
358 definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
360 where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
362 definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
364 where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
366 definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
367 definition transpose where
368 "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
369 definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
370 definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
371 definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
372 definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
374 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
375 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
376 by (vector matrix_matrix_mult_def setsum_addf[symmetric] field_simps)
378 lemma matrix_mul_lid:
379 fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
380 shows "mat 1 ** A = A"
381 apply (simp add: matrix_matrix_mult_def mat_def)
383 apply (auto simp only: if_distrib cond_application_beta setsum_delta'[OF finite]
384 mult_1_left mult_zero_left if_True UNIV_I)
388 lemma matrix_mul_rid:
389 fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
390 shows "A ** mat 1 = A"
391 apply (simp add: matrix_matrix_mult_def mat_def)
393 apply (auto simp only: if_distrib cond_application_beta setsum_delta[OF finite]
394 mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
397 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
398 apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
399 apply (subst setsum_commute)
403 lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
404 apply (vector matrix_matrix_mult_def matrix_vector_mult_def
405 setsum_right_distrib setsum_left_distrib mult_assoc)
406 apply (subst setsum_commute)
410 lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
411 apply (vector matrix_vector_mult_def mat_def)
412 apply (simp add: if_distrib cond_application_beta setsum_delta' cong del: if_weak_cong)
415 lemma matrix_transpose_mul:
416 "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
417 by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult_commute)
420 fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
421 shows "A = B \<longleftrightarrow> (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
423 apply (subst vec_eq_iff)
425 apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
426 apply (erule_tac x="axis ia 1" in allE)
427 apply (erule_tac x="i" in allE)
428 apply (auto simp add: if_distrib cond_application_beta axis_def
429 setsum_delta[OF finite] cong del: if_weak_cong)
432 lemma matrix_vector_mul_component: "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
433 by (simp add: matrix_vector_mult_def inner_vec_def)
435 lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
436 apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
437 apply (subst setsum_commute)
441 lemma transpose_mat: "transpose (mat n) = mat n"
442 by (vector transpose_def mat_def)
444 lemma transpose_transpose: "transpose(transpose A) = A"
445 by (vector transpose_def)
448 fixes A:: "'a::semiring_1^_^_"
449 shows "row i (transpose A) = column i A"
450 by (simp add: row_def column_def transpose_def vec_eq_iff)
452 lemma column_transpose:
453 fixes A:: "'a::semiring_1^_^_"
454 shows "column i (transpose A) = row i A"
455 by (simp add: row_def column_def transpose_def vec_eq_iff)
457 lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
458 by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
460 lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
461 by (metis transpose_transpose rows_transpose)
463 text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
465 lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
466 by (simp add: matrix_vector_mult_def inner_vec_def)
468 lemma matrix_mult_vsum:
469 "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
470 by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult_commute)
472 lemma vector_componentwise:
473 "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"
474 by (simp add: axis_def if_distrib setsum_cases vec_eq_iff)
476 lemma basis_expansion: "setsum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
477 by (auto simp add: axis_def vec_eq_iff if_distrib setsum_cases cong del: if_weak_cong)
479 lemma linear_componentwise:
480 fixes f:: "real ^'m \<Rightarrow> real ^ _"
481 assumes lf: "linear f"
482 shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
484 let ?M = "(UNIV :: 'm set)"
485 let ?N = "(UNIV :: 'n set)"
486 have fM: "finite ?M" by simp
487 have "?rhs = (setsum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j"
488 unfolding setsum_component by simp
490 unfolding linear_setsum_mul[OF lf fM, symmetric]
491 unfolding scalar_mult_eq_scaleR[symmetric]
492 unfolding basis_expansion
496 text{* Inverse matrices (not necessarily square) *}
499 "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
502 "matrix_inv(A:: 'a::semiring_1^'n^'m) =
503 (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
505 text{* Correspondence between matrices and linear operators. *}
507 definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
508 where "matrix f = (\<chi> i j. (f(axis j 1))$i)"
510 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
511 by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff
512 field_simps setsum_right_distrib setsum_addf)
515 assumes lf: "linear f"
516 shows "matrix f *v x = f (x::real ^ 'n)"
517 apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult_commute)
519 apply (rule linear_componentwise[OF lf, symmetric])
522 lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
523 by (simp add: ext matrix_works)
525 lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
526 by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
528 lemma matrix_compose:
529 assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
530 and lg: "linear (g::real^'m \<Rightarrow> real^_)"
531 shows "matrix (g o f) = matrix g ** matrix f"
532 using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
533 by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
535 lemma matrix_vector_column:
536 "(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
537 by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult_commute)
539 lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
540 apply (rule adjoint_unique)
541 apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
542 setsum_left_distrib setsum_right_distrib)
543 apply (subst setsum_commute)
544 apply (auto simp add: mult_ac)
547 lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
548 shows "matrix(adjoint f) = transpose(matrix f)"
549 apply (subst matrix_vector_mul[OF lf])
550 unfolding adjoint_matrix matrix_of_matrix_vector_mul
555 subsection {* lambda skolemization on cartesian products *}
557 (* FIXME: rename do choice_cart *)
559 lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
560 (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
562 let ?S = "(UNIV :: 'n set)"
564 then have ?lhs by auto }
567 then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
568 let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
570 from f have "P i (f i)" by metis
571 then have "P i (?x $ i)" by auto
573 hence "\<forall>i. P i (?x$i)" by metis
574 hence ?rhs by metis }
575 ultimately show ?thesis by metis
578 lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
579 unfolding inner_simps scalar_mult_eq_scaleR by auto
581 lemma left_invertible_transpose:
582 "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
583 by (metis matrix_transpose_mul transpose_mat transpose_transpose)
585 lemma right_invertible_transpose:
586 "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
587 by (metis matrix_transpose_mul transpose_mat transpose_transpose)
589 lemma matrix_left_invertible_injective:
590 "(\<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)"
592 { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
593 from xy have "B*v (A *v x) = B *v (A*v y)" by simp
595 unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }
597 { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
598 hence i: "inj (op *v A)" unfolding inj_on_def by auto
599 from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
600 obtain g where g: "linear g" "g o op *v A = id" by blast
601 have "matrix g ** A = mat 1"
602 unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
603 using g(2) by (simp add: fun_eq_iff)
604 then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast }
605 ultimately show ?thesis by blast
608 lemma matrix_left_invertible_ker:
609 "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
610 unfolding matrix_left_invertible_injective
611 using linear_injective_0[OF matrix_vector_mul_linear, of A]
612 by (simp add: inj_on_def)
614 lemma matrix_right_invertible_surjective:
615 "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
617 { fix B :: "real ^'m^'n"
618 assume AB: "A ** B = mat 1"
619 { fix x :: "real ^ 'm"
620 have "A *v (B *v x) = x"
621 by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
622 hence "surj (op *v A)" unfolding surj_def by metis }
624 { assume sf: "surj (op *v A)"
625 from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
626 obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
629 have "A ** (matrix g) = mat 1"
630 unfolding matrix_eq matrix_vector_mul_lid
631 matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
632 using g(2) unfolding o_def fun_eq_iff id_def
634 hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
636 ultimately show ?thesis unfolding surj_def by blast
639 lemma matrix_left_invertible_independent_columns:
640 fixes A :: "real^'n^'m"
641 shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
642 (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
643 (is "?lhs \<longleftrightarrow> ?rhs")
645 let ?U = "UNIV :: 'n set"
646 { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
648 assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
649 let ?x = "\<chi> i. c i"
650 have th0:"A *v ?x = 0"
652 unfolding matrix_mult_vsum vec_eq_iff
654 from k[rule_format, OF th0] i
655 have "c i = 0" by (vector vec_eq_iff)}
656 hence ?rhs by blast }
659 { fix x assume x: "A *v x = 0"
660 let ?c = "\<lambda>i. ((x$i ):: real)"
661 from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
662 have "x = 0" by vector }
664 ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
667 lemma matrix_right_invertible_independent_rows:
668 fixes A :: "real^'n^'m"
669 shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
670 (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
671 unfolding left_invertible_transpose[symmetric]
672 matrix_left_invertible_independent_columns
673 by (simp add: column_transpose)
675 lemma matrix_right_invertible_span_columns:
676 "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
677 span (columns A) = UNIV" (is "?lhs = ?rhs")
679 let ?U = "UNIV :: 'm set"
680 have fU: "finite ?U" by simp
681 have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
682 unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
683 apply (subst eq_commute)
686 have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
689 from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
690 where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
691 have "x \<in> span (columns A)"
692 unfolding y[symmetric]
693 apply (rule span_setsum[OF fU])
695 unfolding scalar_mult_eq_scaleR
696 apply (rule span_mul)
697 apply (rule span_superset)
698 unfolding columns_def
702 then have ?rhs unfolding rhseq by blast }
705 let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
708 proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
709 show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
710 by (rule exI[where x=0], simp)
713 assume y1: "y1 \<in> columns A" and y2: "?P y2"
714 from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
715 unfolding columns_def by blast
716 from y2 obtain x:: "real ^'m" where
717 x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
718 let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
719 show "?P (c*s y1 + y2)"
720 proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib distrib_left cond_application_beta cong del: if_weak_cong)
722 have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
723 else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))"
724 using i(1) by (simp add: field_simps)
725 have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
726 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)$j))) ?U"
727 apply (rule setsum_cong[OF refl])
730 also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
731 by (simp add: setsum_addf)
732 also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
733 unfolding setsum_delta[OF fU]
735 finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
736 else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
739 show "y \<in> span (columns A)"
743 then have ?lhs unfolding lhseq ..
745 ultimately show ?thesis by blast
748 lemma matrix_left_invertible_span_rows:
749 "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
750 unfolding right_invertible_transpose[symmetric]
751 unfolding columns_transpose[symmetric]
752 unfolding matrix_right_invertible_span_columns
755 text {* The same result in terms of square matrices. *}
757 lemma matrix_left_right_inverse:
758 fixes A A' :: "real ^'n^'n"
759 shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
761 { fix A A' :: "real ^'n^'n"
762 assume AA': "A ** A' = mat 1"
763 have sA: "surj (op *v A)"
766 apply (rule_tac x="(A' *v y)" in exI)
767 apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
769 from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
770 obtain f' :: "real ^'n \<Rightarrow> real ^'n"
771 where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
772 have th: "matrix f' ** A = mat 1"
773 by (simp add: matrix_eq matrix_works[OF f'(1)]
774 matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
775 hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
776 hence "matrix f' = A'"
777 by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
778 hence "matrix f' ** A = A' ** A" by simp
779 hence "A' ** A = mat 1" by (simp add: th)
781 then show ?thesis by blast
784 text {* Considering an n-element vector as an n-by-1 or 1-by-n matrix. *}
786 definition "rowvector v = (\<chi> i j. (v$j))"
788 definition "columnvector v = (\<chi> i j. (v$i))"
790 lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
791 by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
793 lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
794 by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
796 lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
797 by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
799 lemma dot_matrix_product:
800 "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
801 by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
803 lemma dot_matrix_vector_mul:
804 fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
805 shows "(A *v x) \<bullet> (B *v y) =
806 (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
807 unfolding dot_matrix_product transpose_columnvector[symmetric]
808 dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
811 lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x$i) |i. i\<in>UNIV}"
812 by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
814 lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
815 using Basis_le_infnorm[of "axis i 1" x]
816 by (simp add: Basis_vec_def axis_eq_axis inner_axis)
818 lemma continuous_component: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
819 unfolding continuous_def by (rule tendsto_vec_nth)
821 lemma continuous_on_component: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
822 unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
824 lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
825 by (simp add: Collect_all_eq closed_INT closed_Collect_le)
827 lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
828 unfolding bounded_def
830 apply (rule_tac x="x $ i" in exI)
831 apply (rule_tac x="e" in exI)
833 apply (rule order_trans [OF dist_vec_nth_le], simp)
836 lemma compact_lemma_cart:
837 fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
838 assumes f: "bounded (range f)"
840 \<exists>l r. subseq r \<and>
841 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
844 have "finite d" by simp
845 thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
846 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
849 thus ?case unfolding subseq_def by auto
852 obtain l1::"'a^'n" and r1 where r1:"subseq r1"
853 and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
854 using insert(3) by auto
855 have s': "bounded ((\<lambda>x. x $ k) ` range f)" using `bounded (range f)`
856 by (auto intro!: bounded_component_cart)
857 have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` range f" by simp
858 have "bounded (range (\<lambda>i. f (r1 i) $ k))"
859 by (metis (lifting) bounded_subset image_subsetI f' s')
860 then obtain l2 r2 where r2: "subseq r2"
861 and lr2: "((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
862 using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) $ k"] by (auto simp: o_def)
863 def r \<equiv> "r1 \<circ> r2"
865 using r1 and r2 unfolding r_def o_def subseq_def by auto
867 def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n"
868 { fix e :: real assume "e > 0"
869 from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
871 from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially"
873 from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $ i) (l1 $ i) < e) sequentially"
874 by (rule eventually_subseq)
875 have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $ i) (l $ i) < e) sequentially"
876 using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
878 ultimately show ?case by auto
882 instance vec :: (heine_borel, finite) heine_borel
884 fix f :: "nat \<Rightarrow> 'a ^ 'b"
885 assume f: "bounded (range f)"
886 then obtain l r where r: "subseq r"
887 and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
888 using compact_lemma_cart [OF f] by blast
889 let ?d = "UNIV::'b set"
890 { fix e::real assume "e>0"
891 hence "0 < e / (real_of_nat (card ?d))"
892 using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
893 with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
897 assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
898 have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
899 unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum)
900 also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
901 by (rule setsum_strict_mono) (simp_all add: n)
902 finally have "dist (f (r n)) l < e" by simp
904 ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
905 by (rule eventually_elim1)
907 hence "((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
908 with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
913 shows "box a b = {x::real^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}"
914 and "{a .. b} = {x::real^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
915 by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_interval Basis_vec_def inner_axis)
917 lemma mem_interval_cart:
919 shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
920 and "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
921 using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
923 lemma interval_eq_empty_cart:
925 shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1)
926 and "({a .. b} = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
928 { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>box a b"
929 hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval_cart by auto
930 hence "a$i < b$i" by auto
931 hence False using as by auto }
933 { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
934 let ?x = "(1/2) *\<^sub>R (a + b)"
936 have "a$i < b$i" using as[THEN spec[where x=i]] by auto
937 hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
938 unfolding vector_smult_component and vector_add_component
940 hence "box a b \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto }
941 ultimately show ?th1 by blast
943 { fix i x assume as:"b$i < a$i" and x:"x\<in>{a .. b}"
944 hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval_cart by auto
945 hence "a$i \<le> b$i" by auto
946 hence False using as by auto }
948 { assume as:"\<forall>i. \<not> (b$i < a$i)"
949 let ?x = "(1/2) *\<^sub>R (a + b)"
951 have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
952 hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
953 unfolding vector_smult_component and vector_add_component
955 hence "{a .. b} \<noteq> {}" using mem_interval_cart(2)[of "?x" a b] by auto }
956 ultimately show ?th2 by blast
959 lemma interval_ne_empty_cart:
961 shows "{a .. b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"
962 and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
963 unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
964 (* BH: Why doesn't just "auto" work here? *)
966 lemma subset_interval_imp_cart:
968 shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}"
969 and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> {c .. d} \<subseteq> box a b"
970 and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> {a .. b}"
971 and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> box a b"
972 unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart
973 by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
976 fixes a :: "'a::linorder^'n"
977 shows "{a .. a} = {a} \<and> {a<..<a} = {}"
978 apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
981 lemma interval_open_subset_closed_cart:
983 shows "box a b \<subseteq> {a .. b}"
984 proof (simp add: subset_eq, rule)
986 assume x: "x \<in>box a b"
988 have "a $ i \<le> x $ i"
989 using x order_less_imp_le[of "a$i" "x$i"]
990 by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff mem_interval Basis_vec_def inner_axis)
994 have "x $ i \<le> b $ i"
995 using x order_less_imp_le[of "x$i" "b$i"]
996 by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff mem_interval Basis_vec_def inner_axis)
999 show "a \<le> x \<and> x \<le> b"
1000 by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
1003 lemma subset_interval_cart:
1004 fixes a :: "real^'n"
1005 shows "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1)
1006 and "{c .. d} \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
1007 and "box c d \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3)
1008 and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
1009 using subset_interval[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
1011 lemma disjoint_interval_cart:
1013 shows "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1)
1014 and "{a .. b} \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)
1015 and "box a b \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3)
1016 and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
1017 using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
1019 lemma inter_interval_cart:
1020 fixes a :: "real^'n"
1021 shows "{a .. b} \<inter> {c .. d} = {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
1022 unfolding set_eq_iff and Int_iff and mem_interval_cart
1025 lemma closed_interval_left_cart:
1026 fixes b :: "real^'n"
1027 shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
1028 by (simp add: Collect_all_eq closed_INT closed_Collect_le)
1030 lemma closed_interval_right_cart:
1032 shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
1033 by (simp add: Collect_all_eq closed_INT closed_Collect_le)
1035 lemma is_interval_cart:
1036 "is_interval (s::(real^'n) set) \<longleftrightarrow>
1037 (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
1038 by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
1040 lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
1041 by (simp add: closed_Collect_le)
1043 lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
1044 by (simp add: closed_Collect_le)
1046 lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
1047 by (simp add: open_Collect_less)
1049 lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i > a}"
1050 by (simp add: open_Collect_less)
1052 lemma Lim_component_le_cart:
1053 fixes f :: "'a \<Rightarrow> real^'n"
1054 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. f x $i \<le> b) net"
1056 by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
1058 lemma Lim_component_ge_cart:
1059 fixes f :: "'a \<Rightarrow> real^'n"
1060 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$i) net"
1062 by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
1064 lemma Lim_component_eq_cart:
1065 fixes f :: "'a \<Rightarrow> real^'n"
1066 assumes net: "(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
1068 using ev[unfolded order_eq_iff eventually_conj_iff] and
1069 Lim_component_ge_cart[OF net, of b i] and
1070 Lim_component_le_cart[OF net, of i b] by auto
1072 lemma connected_ivt_component_cart:
1073 fixes x :: "real^'n"
1074 shows "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s. z$k = a)"
1075 using connected_ivt_hyperplane[of s x y "axis k 1" a]
1076 by (auto simp add: inner_axis inner_commute)
1078 lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
1079 unfolding subspace_def by auto
1081 lemma closed_substandard_cart:
1082 "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}"
1085 have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"
1086 by (cases "P i") (simp_all add: closed_Collect_eq) }
1088 unfolding Collect_all_eq by (simp add: closed_INT)
1091 lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
1094 let ?a = "\<lambda>x. axis x 1 :: real^'n"
1095 have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"
1096 by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
1097 have "?a ` d \<subseteq> Basis"
1098 by (auto simp: Basis_vec_def)
1100 using dim_substandard[of "?a ` d"] card_image[of ?a d]
1101 by (auto simp: axis_eq_axis inj_on_def *)
1104 lemma affinity_inverses:
1105 assumes m0: "m \<noteq> (0::'a::field)"
1106 shows "(\<lambda>x. m *s x + c) o (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
1107 "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) o (\<lambda>x. m *s x + c) = id"
1109 apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
1110 apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric])
1113 lemma vector_affinity_eq:
1114 assumes m0: "(m::'a::field) \<noteq> 0"
1115 shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
1117 assume h: "m *s x + c = y"
1118 hence "m *s x = y - c" by (simp add: field_simps)
1119 hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
1120 then show "x = inverse m *s y + - (inverse m *s c)"
1121 using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
1123 assume h: "x = inverse m *s y + - (inverse m *s c)"
1124 show "m *s x + c = y" unfolding h
1125 using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
1128 lemma vector_eq_affinity:
1129 "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
1130 using vector_affinity_eq[where m=m and x=x and y=y and c=c]
1134 fixes f :: "real^'n \<Rightarrow> real"
1135 shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
1136 unfolding euclidean_eq_iff[where 'a="real^'n"]
1137 by simp (simp add: Basis_vec_def inner_axis)
1139 lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
1140 by (rule vector_cart)
1142 subsection "Convex Euclidean Space"
1144 lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
1145 using const_vector_cart[of 1] by (simp add: one_vec_def)
1147 declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
1148 declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
1150 lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
1152 lemma convex_box_cart:
1153 assumes "\<And>i. convex {x. P i x}"
1154 shows "convex {x. \<forall>i. P i (x$i)}"
1155 using assms unfolding convex_def by auto
1157 lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
1158 by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval)
1160 lemma unit_interval_convex_hull_cart:
1161 "{0::real^'n .. 1} = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}" (is "?int = convex hull ?points")
1162 unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"]
1163 by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
1165 lemma cube_convex_hull_cart:
1167 obtains s::"(real^'n) set"
1168 where "finite s" "{x - (\<chi> i. d) .. x + (\<chi> i. d)} = convex hull s"
1170 from cube_convex_hull [OF assms, of x] guess s .
1171 with that[of s] show thesis by (simp add: const_vector_cart)
1175 subsection "Derivative"
1177 lemma differentiable_at_imp_differentiable_on:
1178 "(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
1179 by (metis differentiable_at_withinI differentiable_on_def)
1181 definition "jacobian f net = matrix(frechet_derivative f net)"
1183 lemma jacobian_works:
1184 "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
1185 (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
1187 unfolding jacobian_def
1188 apply (simp only: matrix_works[OF linear_frechet_derivative]) defer
1189 apply (rule differentiableI)
1191 unfolding frechet_derivative_works
1196 subsection {* Component of the differential must be zero if it exists at a local
1197 maximum or minimum for that corresponding component. *}
1199 lemma differential_zero_maxmin_cart:
1200 fixes f::"real^'a \<Rightarrow> real^'b"
1201 assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$k \<le> (f x)$k) \<or> (\<forall>y\<in>ball x e. (f x)$k \<le> (f y)$k))"
1202 "f differentiable (at x)"
1203 shows "jacobian f (at x) $ k = 0"
1204 using differential_zero_maxmin_component[of "axis k 1" e x f] assms
1205 vector_cart[of "\<lambda>j. frechet_derivative f (at x) j $ k"]
1206 by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
1208 subsection {* Lemmas for working on @{typ "real^1"} *}
1210 lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
1211 by (metis (full_types) num1_eq_iff)
1213 lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
1214 by auto (metis (full_types) num1_eq_iff)
1218 shows "x = 1 \<or> x = 2"
1221 then have "0 <= z" and "z < 2" by simp_all
1222 then have "z = 0 | z = 1" by arith
1223 then show ?case by auto
1226 lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
1227 by (metis exhaust_2)
1231 shows "x = 1 \<or> x = 2 \<or> x = 3"
1234 then have "0 <= z" and "z < 3" by simp_all
1235 then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
1236 then show ?case by auto
1239 lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
1240 by (metis exhaust_3)
1242 lemma UNIV_1 [simp]: "UNIV = {1::1}"
1243 by (auto simp add: num1_eq_iff)
1245 lemma UNIV_2: "UNIV = {1::2, 2::2}"
1246 using exhaust_2 by auto
1248 lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
1249 using exhaust_3 by auto
1251 lemma setsum_1: "setsum f (UNIV::1 set) = f 1"
1252 unfolding UNIV_1 by simp
1254 lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"
1255 unfolding UNIV_2 by simp
1257 lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
1258 unfolding UNIV_3 by (simp add: add_ac)
1260 instantiation num1 :: cart_one
1265 show "CARD(1) = Suc 0" by auto
1270 subsection{* The collapse of the general concepts to dimension one. *}
1272 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
1273 by (simp add: vec_eq_iff)
1275 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
1277 apply (erule_tac x= "x$1" in allE)
1278 apply (simp only: vector_one[symmetric])
1281 lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
1282 by (simp add: norm_vec_def)
1284 lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
1285 by (simp add: norm_vector_1)
1287 lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
1288 by (auto simp add: norm_real dist_norm)
1291 subsection{* Explicit vector construction from lists. *}
1293 definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
1295 lemma vector_1: "(vector[x]) $1 = x"
1296 unfolding vector_def by simp
1299 "(vector[x,y]) $1 = x"
1300 "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
1301 unfolding vector_def by simp_all
1304 "(vector [x,y,z] ::('a::zero)^3)$1 = x"
1305 "(vector [x,y,z] ::('a::zero)^3)$2 = y"
1306 "(vector [x,y,z] ::('a::zero)^3)$3 = z"
1307 unfolding vector_def by simp_all
1309 lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
1311 apply (erule_tac x="v$1" in allE)
1312 apply (subgoal_tac "vector [v$1] = v")
1314 apply (vector vector_def)
1318 lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
1320 apply (erule_tac x="v$1" in allE)
1321 apply (erule_tac x="v$2" in allE)
1322 apply (subgoal_tac "vector [v$1, v$2] = v")
1324 apply (vector vector_def)
1325 apply (simp add: forall_2)
1328 lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
1330 apply (erule_tac x="v$1" in allE)
1331 apply (erule_tac x="v$2" in allE)
1332 apply (erule_tac x="v$3" in allE)
1333 apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
1335 apply (vector vector_def)
1336 apply (simp add: forall_3)
1339 lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"
1340 apply (rule bounded_linearI[where K=1])
1341 using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
1343 lemma integral_component_eq_cart[simp]:
1344 fixes f :: "'n::ordered_euclidean_space \<Rightarrow> real^'m"
1345 assumes "f integrable_on s"
1346 shows "integral s (\<lambda>x. f x $ k) = integral s f $ k"
1347 using integral_linear[OF assms(1) bounded_linear_component_cart,unfolded o_def] .
1349 lemma interval_split_cart:
1350 "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
1351 "{a..b} \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
1352 apply (rule_tac[!] set_eqI)
1353 unfolding Int_iff mem_interval_cart mem_Collect_eq
1354 unfolding vec_lambda_beta
1357 lemma interval_bij_bij_cart: fixes x::"real^'n" assumes "\<forall>i. a$i < b$i \<and> u$i < v$i"
1358 shows "interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x"
1359 using assms by (intro interval_bij_bij) (auto simp: Basis_vec_def inner_axis)