# HG changeset patch # User chaieb # Date 1234198443 0 # Node ID 4ac60c7d9b786ede66cc1167cd9f57d4e6a5e5c7 # Parent 86d94bb792261690c71ff230f6c42c58461d0248 (Real) Vectors in Euclidean space, and elementary linear algebra. diff -r 86d94bb79226 -r 4ac60c7d9b78 src/HOL/Library/Euclidean_Space.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Library/Euclidean_Space.thy Mon Feb 09 16:54:03 2009 +0000 @@ -0,0 +1,5171 @@ +(* Title: Library/Euclidean_Space + ID: $Id: + Author: Amine Chaieb, University of Cambridge +*) + +header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*} + +theory Euclidean_Space + imports "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Complex_Main + Finite_Cartesian_Product Glbs Infinite_Set Numeral_Type + uses ("normarith.ML") +begin + +text{* Some common special cases.*} + +lemma forall_1: "(\(i::'a::{order,one}). 1 <= i \ i <= 1 --> P i) \ P 1" + by (metis order_eq_iff) +lemma forall_dimindex_1: "(\i \ {1..dimindex(UNIV:: 1 set)}. P i) \ P 1" + by (simp add: dimindex_def) + +lemma forall_2: "(\(i::nat). 1 <= i \ i <= 2 --> P i) \ P 1 \ P 2" +proof- + have "\i::nat. 1 <= i \ i <= 2 \ i = 1 \ i = 2" by arith + thus ?thesis by metis +qed + +lemma forall_3: "(\(i::nat). 1 <= i \ i <= 3 --> P i) \ P 1 \ P 2 \ P 3" +proof- + have "\i::nat. 1 <= i \ i <= 3 \ i = 1 \ i = 2 \ i = 3" by arith + thus ?thesis by metis +qed + +lemma setsum_singleton[simp]: "setsum f {x} = f x" by simp +lemma setsum_1: "setsum f {(1::'a::{order,one})..1} = f 1" + by (simp add: atLeastAtMost_singleton) + +lemma setsum_2: "setsum f {1::nat..2} = f 1 + f 2" + by (simp add: nat_number atLeastAtMostSuc_conv add_commute) + +lemma setsum_3: "setsum f {1::nat..3} = f 1 + f 2 + f 3" + by (simp add: nat_number atLeastAtMostSuc_conv add_commute) + +section{* Basic componentwise operations on vectors. *} + +instantiation "^" :: (plus,type) plus +begin +definition vector_add_def : "op + \ (\ x y. (\ i. (x$i) + (y$i)))" +instance .. +end + +instantiation "^" :: (times,type) times +begin + definition vector_mult_def : "op * \ (\ x y. (\ i. (x$i) * (y$i)))" + instance .. +end + +instantiation "^" :: (minus,type) minus begin + definition vector_minus_def : "op - \ (\ x y. (\ i. (x$i) - (y$i)))" +instance .. +end + +instantiation "^" :: (uminus,type) uminus begin + definition vector_uminus_def : "uminus \ (\ x. (\ i. - (x$i)))" +instance .. +end +instantiation "^" :: (zero,type) zero begin + definition vector_zero_def : "0 \ (\ i. 0)" +instance .. +end + +instantiation "^" :: (one,type) one begin + definition vector_one_def : "1 \ (\ i. 1)" +instance .. +end + +instantiation "^" :: (ord,type) ord + begin +definition vector_less_eq_def: + "less_eq (x :: 'a ^'b) y = (ALL i : {1 .. dimindex (UNIV :: 'b set)}. + x$i <= y$i)" +definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i : {1 .. + dimindex (UNIV :: 'b set)}. x$i < y$i)" + +instance by (intro_classes) +end + +text{* Also the scalar-vector multiplication. FIXME: We should unify this with the scalar multiplication in real_vector *} + +definition vector_scalar_mult:: "'a::times \ 'a ^'n \ 'a ^ 'n" (infixr "*s" 75) + where "c *s x = (\ i. c * (x$i))" + +text{* Constant Vectors *} + +definition "vec x = (\ i. x)" + +text{* Dot products. *} + +definition dot :: "'a::{comm_monoid_add, times} ^ 'n \ 'a ^ 'n \ 'a" (infix "\" 70) where + "x \ y = setsum (\i. x$i * y$i) {1 .. dimindex (UNIV:: 'n set)}" +lemma dot_1[simp]: "(x::'a::{comm_monoid_add, times}^1) \ y = (x$1) * (y$1)" + by (simp add: dot_def dimindex_def) + +lemma dot_2[simp]: "(x::'a::{comm_monoid_add, times}^2) \ y = (x$1) * (y$1) + (x$2) * (y$2)" + by (simp add: dot_def dimindex_def nat_number) + +lemma dot_3[simp]: "(x::'a::{comm_monoid_add, times}^3) \ y = (x$1) * (y$1) + (x$2) * (y$2) + (x$3) * (y$3)" + by (simp add: dot_def dimindex_def nat_number) + +section {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *} + +lemmas Cart_lambda_beta' = Cart_lambda_beta[rule_format] +method_setup vector = {* +let + val ss1 = HOL_basic_ss addsimps [@{thm dot_def}, @{thm setsum_addf} RS sym, + @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib}, + @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym] + val ss2 = @{simpset} addsimps + [@{thm vector_add_def}, @{thm vector_mult_def}, + @{thm vector_minus_def}, @{thm vector_uminus_def}, + @{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def}, + @{thm Cart_lambda_beta'}, @{thm vector_scalar_mult_def}] + fun vector_arith_tac ths = + simp_tac ss1 + THEN' (fn i => rtac @{thm setsum_cong2} i + ORELSE rtac @{thm setsum_0'} i + ORELSE simp_tac (HOL_basic_ss addsimps [@{thm "Cart_eq"}]) i) + (* THEN' TRY o clarify_tac HOL_cs THEN' (TRY o rtac @{thm iffI}) *) + THEN' asm_full_simp_tac (ss2 addsimps ths) + in + Method.thms_args (Method.SIMPLE_METHOD' o vector_arith_tac) +end +*} "Lifts trivial vector statements to real arith statements" + +lemma vec_0[simp]: "vec 0 = 0" by (vector vector_zero_def) +lemma vec_1[simp]: "vec 1 = 1" by (vector vector_one_def) + + + +text{* Obvious "component-pushing". *} + +lemma vec_component: " i \ {1 .. dimindex (UNIV :: 'n set)} \ (vec x :: 'a ^ 'n)$i = x" + by (vector vec_def) + +lemma vector_add_component: + fixes x y :: "'a::{plus} ^ 'n" assumes i: "i \ {1 .. dimindex(UNIV:: 'n set)}" + shows "(x + y)$i = x$i + y$i" + using i by vector + +lemma vector_minus_component: + fixes x y :: "'a::{minus} ^ 'n" assumes i: "i \ {1 .. dimindex(UNIV:: 'n set)}" + shows "(x - y)$i = x$i - y$i" + using i by vector + +lemma vector_mult_component: + fixes x y :: "'a::{times} ^ 'n" assumes i: "i \ {1 .. dimindex(UNIV:: 'n set)}" + shows "(x * y)$i = x$i * y$i" + using i by vector + +lemma vector_smult_component: + fixes y :: "'a::{times} ^ 'n" assumes i: "i \ {1 .. dimindex(UNIV:: 'n set)}" + shows "(c *s y)$i = c * (y$i)" + using i by vector + +lemma vector_uminus_component: + fixes x :: "'a::{uminus} ^ 'n" assumes i: "i \ {1 .. dimindex(UNIV:: 'n set)}" + shows "(- x)$i = - (x$i)" + using i by vector + +lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector + +lemmas vector_component = vec_component vector_add_component vector_mult_component vector_smult_component vector_minus_component vector_uminus_component cond_component + +subsection {* Some frequently useful arithmetic lemmas over vectors. *} + +instance "^" :: (semigroup_add,type) semigroup_add + apply (intro_classes) by (vector add_assoc) + + +instance "^" :: (monoid_add,type) monoid_add + apply (intro_classes) by vector+ + +instance "^" :: (group_add,type) group_add + apply (intro_classes) by (vector algebra_simps)+ + +instance "^" :: (ab_semigroup_add,type) ab_semigroup_add + apply (intro_classes) by (vector add_commute) + +instance "^" :: (comm_monoid_add,type) comm_monoid_add + apply (intro_classes) by vector + +instance "^" :: (ab_group_add,type) ab_group_add + apply (intro_classes) by vector+ + +instance "^" :: (cancel_semigroup_add,type) cancel_semigroup_add + apply (intro_classes) + by (vector Cart_eq)+ + +instance "^" :: (cancel_ab_semigroup_add,type) cancel_ab_semigroup_add + apply (intro_classes) + by (vector Cart_eq) + +instance "^" :: (semigroup_mult,type) semigroup_mult + apply (intro_classes) by (vector mult_assoc) + +instance "^" :: (monoid_mult,type) monoid_mult + apply (intro_classes) by vector+ + +instance "^" :: (ab_semigroup_mult,type) ab_semigroup_mult + apply (intro_classes) by (vector mult_commute) + +instance "^" :: (ab_semigroup_idem_mult,type) ab_semigroup_idem_mult + apply (intro_classes) by (vector mult_idem) + +instance "^" :: (comm_monoid_mult,type) comm_monoid_mult + apply (intro_classes) by vector + +fun vector_power :: "('a::{one,times} ^'n) \ nat \ 'a^'n" where + "vector_power x 0 = 1" + | "vector_power x (Suc n) = x * vector_power x n" + +instantiation "^" :: (recpower,type) recpower +begin + definition vec_power_def: "op ^ \ vector_power" + instance + apply (intro_classes) by (simp_all add: vec_power_def) +end + +instance "^" :: (semiring,type) semiring + apply (intro_classes) by (vector ring_simps)+ + +instance "^" :: (semiring_0,type) semiring_0 + apply (intro_classes) by (vector ring_simps)+ +instance "^" :: (semiring_1,type) semiring_1 + apply (intro_classes) apply vector using dimindex_ge_1 by auto +instance "^" :: (comm_semiring,type) comm_semiring + apply (intro_classes) by (vector ring_simps)+ + +instance "^" :: (comm_semiring_0,type) comm_semiring_0 by (intro_classes) +instance "^" :: (semiring_0_cancel,type) semiring_0_cancel by (intro_classes) +instance "^" :: (comm_semiring_0_cancel,type) comm_semiring_0_cancel by (intro_classes) +instance "^" :: (ring,type) ring by (intro_classes) +instance "^" :: (semiring_1_cancel,type) semiring_1_cancel by (intro_classes) +instance "^" :: (comm_semiring_1,type) comm_semiring_1 by (intro_classes) +lemma of_nat_index: + "i\{1 .. dimindex (UNIV :: 'n set)} \ (of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n" + apply (induct n) + apply vector + apply vector + done +lemma zero_index[simp]: + "i\{1 .. dimindex (UNIV :: 'n set)} \ (0 :: 'a::zero ^'n)$i = 0" by vector + +lemma one_index[simp]: + "i\{1 .. dimindex (UNIV :: 'n set)} \ (1 :: 'a::one ^'n)$i = 1" by vector + +lemma one_plus_of_nat_neq_0: "(1::'a::semiring_char_0) + of_nat n \ 0" +proof- + have "(1::'a) + of_nat n = 0 \ of_nat 1 + of_nat n = (of_nat 0 :: 'a)" by simp + also have "\ \ 1 + n = 0" by (simp only: of_nat_add[symmetric] of_nat_eq_iff) + finally show ?thesis by simp +qed + +instance "^" :: (semiring_char_0,type) semiring_char_0 +proof (intro_classes) + fix m n ::nat + show "(of_nat m :: 'a^'b) = of_nat n \ m = n" + proof(induct m arbitrary: n) + case 0 thus ?case apply vector + apply (induct n,auto simp add: ring_simps) + using dimindex_ge_1 apply auto + apply vector + by (auto simp add: of_nat_index one_plus_of_nat_neq_0) + next + case (Suc n m) + thus ?case apply vector + apply (induct m, auto simp add: ring_simps of_nat_index zero_index) + using dimindex_ge_1 apply simp apply blast + apply (simp add: one_plus_of_nat_neq_0) + using dimindex_ge_1 apply simp apply blast + apply (simp add: vector_component one_index of_nat_index) + apply (simp only: of_nat.simps(2)[where ?'a = 'a, symmetric] of_nat_eq_iff) + using dimindex_ge_1 apply simp apply blast + apply (simp add: vector_component one_index of_nat_index) + apply (simp only: of_nat.simps(2)[where ?'a = 'a, symmetric] of_nat_eq_iff) + using dimindex_ge_1 apply simp apply blast + apply (simp add: vector_component one_index of_nat_index) + done + qed +qed + +instance "^" :: (comm_ring_1,type) comm_ring_1 by intro_classes + (* FIXME!!! Why does the axclass package complain here !!*) +(* instance "^" :: (ring_char_0,type) ring_char_0 by intro_classes *) + +lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x" + by (vector mult_assoc) +lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x" + by (vector ring_simps) +lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y" + by (vector ring_simps) +lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector +lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector +lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y" + by (vector ring_simps) +lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector +lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector +lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector +lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector +lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x" + by (vector ring_simps) + +lemma vec_eq[simp]: "(vec m = vec n) \ (m = n)" + apply (auto simp add: vec_def Cart_eq vec_component Cart_lambda_beta ) + using dimindex_ge_1 apply auto done + +subsection{* Properties of the dot product. *} + +lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \ y = y \ x" + by (vector mult_commute) +lemma dot_ladd: "((x::'a::ring ^ 'n) + y) \ z = (x \ z) + (y \ z)" + by (vector ring_simps) +lemma dot_radd: "x \ (y + (z::'a::ring ^ 'n)) = (x \ y) + (x \ z)" + by (vector ring_simps) +lemma dot_lsub: "((x::'a::ring ^ 'n) - y) \ z = (x \ z) - (y \ z)" + by (vector ring_simps) +lemma dot_rsub: "(x::'a::ring ^ 'n) \ (y - z) = (x \ y) - (x \ z)" + by (vector ring_simps) +lemma dot_lmult: "(c *s x) \ y = (c::'a::ring) * (x \ y)" by (vector ring_simps) +lemma dot_rmult: "x \ (c *s y) = (c::'a::comm_ring) * (x \ y)" by (vector ring_simps) +lemma dot_lneg: "(-x) \ (y::'a::ring ^ 'n) = -(x \ y)" by vector +lemma dot_rneg: "(x::'a::ring ^ 'n) \ (-y) = -(x \ y)" by vector +lemma dot_lzero[simp]: "0 \ x = (0::'a::{comm_monoid_add, mult_zero})" by vector +lemma dot_rzero[simp]: "x \ 0 = (0::'a::{comm_monoid_add, mult_zero})" by vector +lemma dot_pos_le[simp]: "(0::'a\ordered_ring_strict) <= x \ x" + by (simp add: dot_def setsum_nonneg) + +lemma setsum_squares_eq_0_iff: assumes fS: "finite F" and fp: "\x \ F. f x \ (0 ::'a::pordered_ab_group_add)" shows "setsum f F = 0 \ (ALL x:F. f x = 0)" +using fS fp setsum_nonneg[OF fp] +proof (induct set: finite) + case empty thus ?case by simp +next + case (insert x F) + from insert.prems have Fx: "f x \ 0" and Fp: "\ a \ F. f a \ 0" by simp_all + from insert.hyps Fp setsum_nonneg[OF Fp] + have h: "setsum f F = 0 \ (\a \F. f a = 0)" by metis + from sum_nonneg_eq_zero_iff[OF Fx setsum_nonneg[OF Fp]] insert.hyps(1,2) + show ?case by (simp add: h) +qed + +lemma dot_eq_0: "x \ x = 0 \ (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) = 0" +proof- + {assume f: "finite (UNIV :: 'n set)" + let ?S = "{Suc 0 .. card (UNIV :: 'n set)}" + have fS: "finite ?S" using f by simp + have fp: "\ i\ ?S. x$i * x$i>= 0" by simp + have ?thesis by (vector dimindex_def f setsum_squares_eq_0_iff[OF fS fp])} + moreover + {assume "\ finite (UNIV :: 'n set)" then have ?thesis by (vector dimindex_def)} + ultimately show ?thesis by metis +qed + +lemma dot_pos_lt: "(0 < x \ x) \ (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) \ 0" using dot_eq_0[of x] dot_pos_le[of x] + by (auto simp add: le_less) + +subsection {* Introduce norms, but defer many properties till we get square roots. *} +text{* FIXME : This is ugly *} +defs (overloaded) + real_of_real_def [code inline, simp]: "real == id" + +instantiation "^" :: ("{times, comm_monoid_add}", type) norm begin +definition real_vector_norm_def: "norm \ (\x. sqrt (real (x \ x)))" +instance .. +end + + +subsection{* The collapse of the general concepts to dimention one. *} + +lemma vector_one: "(x::'a ^1) = (\ i. (x$1))" + by (vector dimindex_def) + +lemma forall_one: "(\(x::'a ^1). P x) \ (\x. P(\ i. x))" + apply auto + apply (erule_tac x= "x$1" in allE) + apply (simp only: vector_one[symmetric]) + done + +lemma norm_real: "norm(x::real ^ 1) = abs(x$1)" + by (simp add: real_vector_norm_def) + +text{* Metric *} + +definition dist:: "real ^ 'n \ real ^ 'n \ real" where + "dist x y = norm (x - y)" + +lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))" + using dimindex_ge_1[of "UNIV :: 1 set"] + by (auto simp add: norm_real dist_def vector_component Cart_lambda_beta[where ?'a = "1"] ) + +subsection {* A connectedness or intermediate value lemma with several applications. *} + +lemma connected_real_lemma: + fixes f :: "real \ real ^ 'n" + assumes ab: "a \ b" and fa: "f a \ e1" and fb: "f b \ e2" + and dst: "\e x. a <= x \ x <= b \ 0 < e ==> \d > 0. \y. abs(y - x) < d \ dist(f y) (f x) < e" + and e1: "\y \ e1. \e > 0. \y'. dist y' y < e \ y' \ e1" + and e2: "\y \ e2. \e > 0. \y'. dist y' y < e \ y' \ e2" + and e12: "~(\x \ a. x <= b \ f x \ e1 \ f x \ e2)" + shows "\x \ a. x <= b \ f x \ e1 \ f x \ e2" (is "\ x. ?P x") +proof- + let ?S = "{c. \x \ a. x <= c \ f x \ e1}" + have Se: " \x. x \ ?S" apply (rule exI[where x=a]) by (auto simp add: fa) + have Sub: "\y. isUb UNIV ?S y" + apply (rule exI[where x= b]) + using ab fb e12 by (auto simp add: isUb_def setle_def) + from reals_complete[OF Se Sub] obtain l where + l: "isLub UNIV ?S l"by blast + have alb: "a \ l" "l \ b" using l ab fa fb e12 + apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def) + by (metis linorder_linear) + have ale1: "\z \ a. z < l \ f z \ e1" using l + apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def) + by (metis linorder_linear not_le) + have th1: "\z x e d :: real. z <= x + e \ e < d ==> z < x \ abs(z - x) < d" by arith + have th2: "\e x:: real. 0 < e ==> ~(x + e <= x)" by arith + have th3: "\d::real. d > 0 \ \e > 0. e < d" by dlo + {assume le2: "f l \ e2" + from le2 fa fb e12 alb have la: "l \ a" by metis + hence lap: "l - a > 0" using alb by arith + from e2[rule_format, OF le2] obtain e where + e: "e > 0" "\y. dist y (f l) < e \ y \ e2" by metis + from dst[OF alb e(1)] obtain d where + d: "d > 0" "\y. \y - l\ < d \ dist (f y) (f l) < e" by metis + have "\d'. d' < d \ d' >0 \ l - d' > a" using lap d(1) + apply ferrack by arith + then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis + from d e have th0: "\y. \y - l\ < d \ f y \ e2" by metis + from th0[rule_format, of "l - d'"] d' have "f (l - d') \ e2" by auto + moreover + have "f (l - d') \ e1" using ale1[rule_format, of "l -d'"] d' by auto + ultimately have False using e12 alb d' by auto} + moreover + {assume le1: "f l \ e1" + from le1 fa fb e12 alb have lb: "l \ b" by metis + hence blp: "b - l > 0" using alb by arith + from e1[rule_format, OF le1] obtain e where + e: "e > 0" "\y. dist y (f l) < e \ y \ e1" by metis + from dst[OF alb e(1)] obtain d where + d: "d > 0" "\y. \y - l\ < d \ dist (f y) (f l) < e" by metis + have "\d'. d' < d \ d' >0" using d(1) by dlo + then obtain d' where d': "d' > 0" "d' < d" by metis + from d e have th0: "\y. \y - l\ < d \ f y \ e1" by auto + hence "\y. l \ y \ y \ l + d' \ f y \ e1" using d' by auto + with ale1 have "\y. a \ y \ y \ l + d' \ f y \ e1" by auto + with l d' have False + by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) } + ultimately show ?thesis using alb by metis +qed + +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 *} + +lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)" +proof- + have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith + thus ?thesis by (simp add: ring_simps power2_eq_square) +qed + +lemma square_continuous: "0 < (e::real) ==> \d. 0 < d \ (\y. abs(y - x) < d \ abs(y * y - x * x) < e)" + 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) + apply (rule_tac x="s" in exI) + apply auto + apply (erule_tac x=y in allE) + apply auto + done + +lemma real_le_lsqrt: "0 <= x \ 0 <= y \ x <= y^2 ==> sqrt x <= y" + using real_sqrt_le_iff[of x "y^2"] by simp + +lemma real_le_rsqrt: "x^2 \ y \ x \ sqrt y" + using real_sqrt_le_mono[of "x^2" y] by simp + +lemma real_less_rsqrt: "x^2 < y \ x < sqrt y" + using real_sqrt_less_mono[of "x^2" y] by simp + +lemma sqrt_even_pow2: assumes n: "even n" + shows "sqrt(2 ^ n) = 2 ^ (n div 2)" +proof- + from n obtain m where m: "n = 2*m" unfolding even_nat_equiv_def2 + by (auto simp add: nat_number) + from m have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)" + by (simp only: power_mult[symmetric] mult_commute) + then show ?thesis using m by simp +qed + +lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)" + apply (cases "x = 0", simp_all) + using sqrt_divide_self_eq[of x] + apply (simp add: inverse_eq_divide real_sqrt_ge_0_iff field_simps) + done + +text{* Hence derive more interesting properties of the norm. *} + +lemma norm_0: "norm (0::real ^ 'n) = 0" + by (simp add: real_vector_norm_def dot_eq_0) + +lemma norm_pos_le: "0 <= norm (x::real^'n)" + by (simp add: real_vector_norm_def dot_pos_le) +lemma norm_neg: " norm(-x) = norm (x:: real ^ 'n)" + by (simp add: real_vector_norm_def dot_lneg dot_rneg) +lemma norm_sub: "norm(x - y) = norm(y - (x::real ^ 'n))" + by (metis norm_neg minus_diff_eq) +lemma norm_mul: "norm(a *s x) = abs(a) * norm x" + by (simp add: real_vector_norm_def dot_lmult dot_rmult mult_assoc[symmetric] real_sqrt_mult) +lemma norm_eq_0_dot: "(norm x = 0) \ (x \ x = (0::real))" + by (simp add: real_vector_norm_def) +lemma norm_eq_0: "norm x = 0 \ x = (0::real ^ 'n)" + by (simp add: real_vector_norm_def dot_eq_0) +lemma norm_pos_lt: "0 < norm x \ x \ (0::real ^ 'n)" + by (metis less_le real_vector_norm_def norm_pos_le norm_eq_0) +lemma norm_pow_2: "norm x ^ 2 = x \ x" + by (simp add: real_vector_norm_def dot_pos_le) +lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_0) +lemma norm_le_0: "norm x <= 0 \ x = (0::real ^'n)" + by (metis norm_eq_0 norm_pos_le order_antisym) +lemma vector_mul_eq_0: "(a *s x = 0) \ a = (0::'a::idom) \ x = 0" + by vector +lemma vector_mul_lcancel: "a *s x = a *s y \ a = (0::real) \ x = y" + by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib) +lemma vector_mul_rcancel: "a *s x = b *s x \ (a::real) = b \ x = 0" + by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib) +lemma vector_mul_lcancel_imp: "a \ (0::real) ==> a *s x = a *s y ==> (x = y)" + by (metis vector_mul_lcancel) +lemma vector_mul_rcancel_imp: "x \ 0 \ (a::real) *s x = b *s x ==> a = b" + by (metis vector_mul_rcancel) +lemma norm_cauchy_schwarz: "x \ y <= norm x * norm y" +proof- + {assume "norm x = 0" + hence ?thesis by (simp add: norm_eq_0 dot_lzero dot_rzero norm_0)} + moreover + {assume "norm y = 0" + hence ?thesis by (simp add: norm_eq_0 dot_lzero dot_rzero norm_0)} + moreover + {assume h: "norm x \ 0" "norm y \ 0" + let ?z = "norm y *s x - norm x *s y" + from h have p: "norm x * norm y > 0" by (metis norm_pos_le le_less zero_compare_simps) + from dot_pos_le[of ?z] + have "(norm x * norm y) * (x \ y) \ norm x ^2 * norm y ^2" + apply (simp add: dot_rsub dot_lsub dot_lmult dot_rmult ring_simps) + by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym) + hence "x\y \ (norm x ^2 * norm y ^2) / (norm x * norm y)" using p + by (simp add: field_simps) + hence ?thesis using h by (simp add: power2_eq_square)} + ultimately show ?thesis by metis +qed + +lemma norm_abs[simp]: "abs (norm x) = norm (x::real ^'n)" + using norm_pos_le[of x] by (simp add: real_abs_def linorder_linear) + +lemma norm_cauchy_schwarz_abs: "\x \ y\ \ norm x * norm y" + using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"] + by (simp add: real_abs_def dot_rneg norm_neg) +lemma norm_triangle: "norm(x + y) <= norm x + norm (y::real ^'n)" + unfolding real_vector_norm_def + apply (rule real_le_lsqrt) + apply (auto simp add: dot_pos_le real_vector_norm_def[symmetric] norm_pos_le norm_pow_2[symmetric] intro: add_nonneg_nonneg)[1] + apply (auto simp add: dot_pos_le real_vector_norm_def[symmetric] norm_pos_le norm_pow_2[symmetric] intro: add_nonneg_nonneg)[1] + apply (simp add: dot_ladd dot_radd dot_sym ) + by (simp add: norm_pow_2[symmetric] power2_eq_square ring_simps norm_cauchy_schwarz) + +lemma norm_triangle_sub: "norm (x::real ^'n) <= norm(y) + norm(x - y)" + using norm_triangle[of "y" "x - y"] by (simp add: ring_simps) +lemma norm_triangle_le: "norm(x::real ^'n) + norm y <= e ==> norm(x + y) <= e" + by (metis order_trans norm_triangle) +lemma norm_triangle_lt: "norm(x::real ^'n) + norm(y) < e ==> norm(x + y) < e" + by (metis basic_trans_rules(21) norm_triangle) + +lemma setsum_delta: + assumes fS: "finite S" + shows "setsum (\k. if k=a then b k else 0) S = (if a \ S then b a else 0)" +proof- + let ?f = "(\k. if k=a then b k else 0)" + {assume a: "a \ S" + hence "\ k\ S. ?f k = 0" by simp + hence ?thesis using a by simp} + moreover + {assume a: "a \ S" + let ?A = "S - {a}" + let ?B = "{a}" + have eq: "S = ?A \ ?B" using a by blast + have dj: "?A \ ?B = {}" by simp + from fS have fAB: "finite ?A" "finite ?B" by auto + have "setsum ?f S = setsum ?f ?A + setsum ?f ?B" + using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]] + by simp + then have ?thesis using a by simp} + ultimately show ?thesis by blast +qed + +lemma component_le_norm: "i \ {1 .. dimindex(UNIV :: 'n set)} ==> \x$i\ <= norm (x::real ^ 'n)" +proof(simp add: real_vector_norm_def, rule real_le_rsqrt, clarsimp) + assume i: "Suc 0 \ i" "i \ dimindex (UNIV :: 'n set)" + let ?S = "{1 .. dimindex(UNIV :: 'n set)}" + let ?f = "(\k. if k = i then x$i ^2 else 0)" + have fS: "finite ?S" by simp + from i setsum_delta[OF fS, of i "\k. x$i ^ 2"] + have th: "x$i^2 = setsum ?f ?S" by simp + let ?g = "\k. x$k * x$k" + {fix x assume x: "x \ ?S" have "?f x \ ?g x" by (simp add: power2_eq_square)} + with setsum_mono[of ?S ?f ?g] + have "setsum ?f ?S \ setsum ?g ?S" by blast + then show "x$i ^2 \ x \ (x:: real ^ 'n)" unfolding dot_def th[symmetric] . +qed +lemma norm_bound_component_le: "norm(x::real ^ 'n) <= e + ==> \i \ {1 .. dimindex(UNIV:: 'n set)}. \x$i\ <= e" + by (metis component_le_norm order_trans) + +lemma norm_bound_component_lt: "norm(x::real ^ 'n) < e + ==> \i \ {1 .. dimindex(UNIV:: 'n set)}. \x$i\ < e" + by (metis component_le_norm basic_trans_rules(21)) + +lemma norm_le_l1: "norm (x:: real ^'n) <= setsum(\i. \x$i\) {1..dimindex(UNIV::'n set)}" +proof (simp add: real_vector_norm_def, rule real_le_lsqrt,simp add: dot_pos_le, simp add: setsum_mono, simp add: dot_def, induct "dimindex(UNIV::'n set)") + case 0 thus ?case by simp +next + case (Suc n) + have th: "2 * (\x$(Suc n)\ * (\i = Suc 0..n. \x$i\)) \ 0" + apply simp + apply (rule mult_nonneg_nonneg) + by (simp_all add: setsum_abs_ge_zero) + + from Suc + show ?case using th by (simp add: power2_eq_square ring_simps) +qed + +lemma real_abs_norm: "\ norm x\ = norm (x :: real ^'n)" + by (simp add: norm_pos_le) +lemma real_abs_sub_norm: "\norm(x::real ^'n) - norm y\ <= norm(x - y)" + apply (simp add: abs_le_iff ring_simps) + by (metis norm_triangle_sub norm_sub) +lemma norm_le: "norm(x::real ^ 'n) <= norm(y) \ x \ x <= y \ y" + by (simp add: real_vector_norm_def) +lemma norm_lt: "norm(x::real ^'n) < norm(y) \ x \ x < y \ y" + by (simp add: real_vector_norm_def) +lemma norm_eq: "norm (x::real ^'n) = norm y \ x \ x = y \ y" + by (simp add: order_eq_iff norm_le) +lemma norm_eq_1: "norm(x::real ^ 'n) = 1 \ x \ x = 1" + by (simp add: real_vector_norm_def) + +text{* Squaring equations and inequalities involving norms. *} + +lemma dot_square_norm: "x \ x = norm(x)^2" + by (simp add: real_vector_norm_def dot_pos_le ) + +lemma norm_eq_square: "norm(x) = a \ 0 <= a \ x \ x = a^2" +proof- + have th: "\x y::real. x^2 = y^2 \ x = y \ x = -y" by algebra + show ?thesis using norm_pos_le[of x] + apply (simp add: dot_square_norm th) + apply arith + done +qed + +lemma real_abs_le_square_iff: "\x\ \ \y\ \ (x::real)^2 \ y^2" +proof- + have "x^2 \ y^2 \ (x -y) * (y + x) \ 0" by (simp add: ring_simps power2_eq_square) + also have "\ \ \x\ \ \y\" apply (simp add: zero_compare_simps real_abs_def not_less) by arith +finally show ?thesis .. +qed + +lemma norm_le_square: "norm(x) <= a \ 0 <= a \ x \ x <= a^2" + using norm_pos_le[of x] + apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric]) + apply arith + done + +lemma norm_ge_square: "norm(x) >= a \ a <= 0 \ x \ x >= a ^ 2" + using norm_pos_le[of x] + apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric]) + apply arith + done + +lemma norm_lt_square: "norm(x) < a \ 0 < a \ x \ x < a^2" + by (metis not_le norm_ge_square) +lemma norm_gt_square: "norm(x) > a \ a < 0 \ x \ x > a^2" + by (metis norm_le_square not_less) + +text{* Dot product in terms of the norm rather than conversely. *} + +lemma dot_norm: "x \ y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2" + by (simp add: norm_pow_2 dot_ladd dot_radd dot_sym) + +lemma dot_norm_neg: "x \ y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2" + by (simp add: norm_pow_2 dot_ladd dot_radd dot_lsub dot_rsub dot_sym) + + +text{* Equality of vectors in terms of @{term "op \"} products. *} + +lemma vector_eq: "(x:: real ^ 'n) = y \ x \ x = x \ y\ y \ y = x \ x" (is "?lhs \ ?rhs") +proof + assume "?lhs" then show ?rhs by simp +next + assume ?rhs + then have "x \ x - x \ y = 0 \ x \ y - y\ y = 0" by simp + hence "x \ (x - y) = 0 \ y \ (x - y) = 0" + by (simp add: dot_rsub dot_lsub dot_sym) + then have "(x - y) \ (x - y) = 0" by (simp add: ring_simps dot_lsub dot_rsub) + then show "x = y" by (simp add: dot_eq_0) +qed + + +subsection{* General linear decision procedure for normed spaces. *} + +lemma norm_cmul_rule_thm: "b >= norm(x) ==> \c\ * b >= norm(c *s x)" + apply (clarsimp simp add: norm_mul) + apply (rule mult_mono1) + apply simp_all + done + +lemma norm_add_rule_thm: "b1 >= norm(x1 :: real ^'n) \ b2 >= norm(x2) ==> b1 + b2 >= norm(x1 + x2)" + apply (rule norm_triangle_le) by simp + +lemma ge_iff_diff_ge_0: "(a::'a::ordered_ring) \ b == a - b \ 0" + by (simp add: ring_simps) + +lemma pth_1: "(x::real^'n) == 1 *s x" by (simp only: vector_smult_lid) +lemma pth_2: "x - (y::real^'n) == x + -y" by (atomize (full)) simp +lemma pth_3: "(-x::real^'n) == -1 *s x" by vector +lemma pth_4: "0 *s (x::real^'n) == 0" "c *s 0 = (0::real ^ 'n)" by vector+ +lemma pth_5: "c *s (d *s x) == (c * d) *s (x::real ^ 'n)" by (atomize (full)) vector +lemma pth_6: "(c::real) *s (x + y) == c *s x + c *s y" by (atomize (full)) (vector ring_simps) +lemma pth_7: "0 + x == (x::real^'n)" "x + 0 == x" by simp_all +lemma pth_8: "(c::real) *s x + d *s x == (c + d) *s x" by (atomize (full)) (vector ring_simps) +lemma pth_9: "((c::real) *s x + z) + d *s x == (c + d) *s x + z" + "c *s x + (d *s x + z) == (c + d) *s x + z" + "(c *s x + w) + (d *s x + z) == (c + d) *s x + (w + z)" by ((atomize (full)), vector ring_simps)+ +lemma pth_a: "(0::real) *s x + y == y" by (atomize (full)) vector +lemma pth_b: "(c::real) *s x + d *s y == c *s x + d *s y" + "(c *s x + z) + d *s y == c *s x + (z + d *s y)" + "c *s x + (d *s y + z) == c *s x + (d *s y + z)" + "(c *s x + w) + (d *s y + z) == c *s x + (w + (d *s y + z))" + by ((atomize (full)), vector)+ +lemma pth_c: "(c::real) *s x + d *s y == d *s y + c *s x" + "(c *s x + z) + d *s y == d *s y + (c *s x + z)" + "c *s x + (d *s y + z) == d *s y + (c *s x + z)" + "(c *s x + w) + (d *s y + z) == d *s y + ((c *s x + w) + z)" by ((atomize (full)), vector)+ +lemma pth_d: "x + (0::real ^'n) == x" by (atomize (full)) vector + +lemma norm_imp_pos_and_ge: "norm (x::real ^ 'n) == n \ norm x \ 0 \ n \ norm x" + by (atomize) (auto simp add: norm_pos_le) + +lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \ 0 \ -x \ 0" by arith + +lemma norm_pths: + "(x::real ^'n) = y \ norm (x - y) \ 0" + "x \ y \ \ (norm (x - y) \ 0)" + using norm_pos_le[of "x - y"] by (auto simp add: norm_0 norm_eq_0) + +use "normarith.ML" + +method_setup norm = {* Method.ctxt_args (Method.SIMPLE_METHOD' o NormArith.norm_arith_tac) +*} "Proves simple linear statements about vector norms" + + + +text{* Hence more metric properties. *} + +lemma dist_refl: "dist x x = 0" by norm + +lemma dist_sym: "dist x y = dist y x"by norm + +lemma dist_pos_le: "0 <= dist x y" by norm + +lemma dist_triangle: "dist x z <= dist x y + dist y z" by norm + +lemma dist_triangle_alt: "dist y z <= dist x y + dist x z" by norm + +lemma dist_eq_0: "dist x y = 0 \ x = y" by norm + +lemma dist_pos_lt: "x \ y ==> 0 < dist x y" by norm +lemma dist_nz: "x \ y \ 0 < dist x y" by norm + +lemma dist_triangle_le: "dist x z + dist y z <= e \ dist x y <= e" by norm + +lemma dist_triangle_lt: "dist x z + dist y z < e ==> dist x y < e" by norm + +lemma dist_triangle_half_l: "dist x1 y < e / 2 \ dist x2 y < e / 2 ==> dist x1 x2 < e" by norm + +lemma dist_triangle_half_r: "dist y x1 < e / 2 \ dist y x2 < e / 2 ==> dist x1 x2 < e" by norm + +lemma dist_triangle_add: "dist (x + y) (x' + y') <= dist x x' + dist y y'" + by norm + +lemma dist_mul: "dist (c *s x) (c *s y) = \c\ * dist x y" + unfolding dist_def vector_ssub_ldistrib[symmetric] norm_mul .. + +lemma dist_triangle_add_half: " dist x x' < e / 2 \ dist y y' < e / 2 ==> dist(x + y) (x' + y') < e" by norm + +lemma dist_le_0: "dist x y <= 0 \ x = y" by norm + +instantiation "^" :: (monoid_add,type) monoid_add +begin + instance by (intro_classes) +end + +lemma setsum_eq: "setsum f S = (\ i. setsum (\x. (f x)$i ) S)" + apply vector + apply auto + apply (cases "finite S") + apply (rule finite_induct[of S]) + apply (auto simp add: vector_component zero_index) + done + +lemma setsum_clauses: + shows "setsum f {} = 0" + and "finite S \ setsum f (insert x S) = + (if x \ S then setsum f S else f x + setsum f S)" + by (auto simp add: insert_absorb) + +lemma setsum_cmul: + fixes f:: "'c \ ('a::semiring_1)^'n" + shows "setsum (\x. c *s f x) S = c *s setsum f S" + by (simp add: setsum_eq Cart_eq Cart_lambda_beta vector_component setsum_right_distrib) + +lemma setsum_component: + fixes f:: " 'a \ ('b::semiring_1) ^'n" + assumes i: "i \ {1 .. dimindex(UNIV:: 'n set)}" + shows "(setsum f S)$i = setsum (\x. (f x)$i) S" + using i by (simp add: setsum_eq Cart_lambda_beta) + + (* This needs finiteness assumption due to the definition of fold!!! *) + +lemma setsum_superset: + assumes fb: "finite B" and ab: "A \ B" + and f0: "\x \ B - A. f x = 0" + shows "setsum f B = setsum f A" +proof- + from ab fb have fa: "finite A" by (metis finite_subset) + from fb have fba: "finite (B - A)" by (metis finite_Diff) + have d: "A \ (B - A) = {}" by blast + from ab have b: "B = A \ (B - A)" by blast + from setsum_Un_disjoint[OF fa fba d, of f] b + setsum_0'[OF f0] + show "setsum f B = setsum f A" by simp +qed + +lemma setsum_restrict_set: + assumes fA: "finite A" + shows "setsum f (A \ B) = setsum (\x. if x \ B then f x else 0) A" +proof- + from fA have fab: "finite (A \ B)" by auto + have aba: "A \ B \ A" by blast + let ?g = "\x. if x \ A\B then f x else 0" + from setsum_superset[OF fA aba, of ?g] + show ?thesis by simp +qed + +lemma setsum_cases: + assumes fA: "finite A" + shows "setsum (\x. if x \ B then f x else g x) A = + setsum f (A \ B) + setsum g (A \ - B)" +proof- + have a: "A = A \ B \ A \ -B" "(A \ B) \ (A \ -B) = {}" + by blast+ + from fA + have f: "finite (A \ B)" "finite (A \ -B)" by auto + let ?g = "\x. if x \ B then f x else g x" + from setsum_Un_disjoint[OF f a(2), of ?g] a(1) + show ?thesis by simp +qed + +lemma setsum_norm: + fixes f :: "'a \ 'b::real_normed_vector" + assumes fS: "finite S" + shows "norm (setsum f S) <= setsum (\x. norm(f x)) S" +proof(induct rule: finite_induct[OF fS]) + case 1 thus ?case by (simp add: norm_zero) +next + case (2 x S) + from "2.hyps" have "norm (setsum f (insert x S)) \ norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq) + also have "\ \ norm (f x) + setsum (\x. norm(f x)) S" + using "2.hyps" by simp + finally show ?case using "2.hyps" by simp +qed + +lemma real_setsum_norm: + fixes f :: "'a \ real ^'n" + assumes fS: "finite S" + shows "norm (setsum f S) <= setsum (\x. norm(f x)) S" +proof(induct rule: finite_induct[OF fS]) + case 1 thus ?case by simp norm +next + case (2 x S) + from "2.hyps" have "norm (setsum f (insert x S)) \ norm (f x) + norm (setsum f S)" apply (simp add: norm_triangle_ineq) by norm + also have "\ \ norm (f x) + setsum (\x. norm(f x)) S" + using "2.hyps" by simp + finally show ?case using "2.hyps" by simp +qed + +lemma setsum_norm_le: + fixes f :: "'a \ 'b::real_normed_vector" + assumes fS: "finite S" + and fg: "\x \ S. norm (f x) \ g x" + shows "norm (setsum f S) \ setsum g S" +proof- + from fg have "setsum (\x. norm(f x)) S <= setsum g S" + by - (rule setsum_mono, simp) + then show ?thesis using setsum_norm[OF fS, of f] fg + by arith +qed + +lemma real_setsum_norm_le: + fixes f :: "'a \ real ^ 'n" + assumes fS: "finite S" + and fg: "\x \ S. norm (f x) \ g x" + shows "norm (setsum f S) \ setsum g S" +proof- + from fg have "setsum (\x. norm(f x)) S <= setsum g S" + by - (rule setsum_mono, simp) + then show ?thesis using real_setsum_norm[OF fS, of f] fg + by arith +qed + +lemma setsum_norm_bound: + fixes f :: "'a \ 'b::real_normed_vector" + assumes fS: "finite S" + and K: "\x \ S. norm (f x) \ K" + shows "norm (setsum f S) \ of_nat (card S) * K" + using setsum_norm_le[OF fS K] setsum_constant[symmetric] + by simp + +lemma real_setsum_norm_bound: + fixes f :: "'a \ real ^ 'n" + assumes fS: "finite S" + and K: "\x \ S. norm (f x) \ K" + shows "norm (setsum f S) \ of_nat (card S) * K" + using real_setsum_norm_le[OF fS K] setsum_constant[symmetric] + by simp + +instantiation "^" :: ("{scaleR, one, times}",type) scaleR +begin + +definition vector_scaleR_def: "(scaleR :: real \ 'a ^'b \ 'a ^'b) \ (\ c x . (scaleR c 1) *s x)" +instance .. +end + +instantiation "^" :: ("ring_1",type) ring_1 +begin +instance by intro_classes +end + +instantiation "^" :: (real_algebra_1,type) real_vector +begin + +instance + apply intro_classes + apply (simp_all add: vector_scaleR_def) + apply (simp_all add: vector_sadd_rdistrib vector_add_ldistrib vector_smult_lid vector_smult_assoc scaleR_left_distrib mult_commute) + done +end + +instantiation "^" :: (real_algebra_1,type) real_algebra +begin + +instance + apply intro_classes + apply (simp_all add: vector_scaleR_def ring_simps) + apply vector + apply vector + done +end + +instantiation "^" :: (real_algebra_1,type) real_algebra_1 +begin + +instance .. +end + +lemma setsum_vmul: + fixes f :: "'a \ 'b::{real_normed_vector,semiring, mult_zero}" + assumes fS: "finite S" + shows "setsum f S *s v = setsum (\x. f x *s v) S" +proof(induct rule: finite_induct[OF fS]) + case 1 then show ?case by (simp add: vector_smult_lzero) +next + case (2 x F) + from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v" + by simp + also have "\ = f x *s v + setsum f F *s v" + by (simp add: vector_sadd_rdistrib) + also have "\ = setsum (\x. f x *s v) (insert x F)" using "2.hyps" by simp + finally show ?case . +qed + +(* FIXME : Problem thm setsum_vmul[of _ "f:: 'a \ real ^'n"] --- + Get rid of *s and use real_vector instead! Also prove that ^ creates a real_vector !! *) + +lemma setsum_add_split: assumes mn: "(m::nat) \ n + 1" + shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}" +proof- + let ?A = "{m .. n}" + let ?B = "{n + 1 .. n + p}" + have eq: "{m .. n+p} = ?A \ ?B" using mn by auto + have d: "?A \ ?B = {}" by auto + from setsum_Un_disjoint[of "?A" "?B" f] eq d show ?thesis by auto +qed + +lemma setsum_reindex_nonzero: + assumes fS: "finite S" + and nz: "\ x y. x \ S \ y \ S \ x \ y \ f x = f y \ h (f x) = 0" + shows "setsum h (f ` S) = setsum (h o f) S" +using nz +proof(induct rule: finite_induct[OF fS]) + case 1 thus ?case by simp +next + case (2 x F) + {assume fxF: "f x \ f ` F" hence "\y \ F . f y = f x" by auto + then obtain y where y: "y \ F" "f x = f y" by auto + from "2.hyps" y have xy: "x \ y" by auto + + from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp + have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto + also have "\ = setsum (h o f) (insert x F)" + using "2.hyps" "2.prems" h0 by auto + finally have ?case .} + moreover + {assume fxF: "f x \ f ` F" + have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" + using fxF "2.hyps" by simp + also have "\ = setsum (h o f) (insert x F)" + using "2.hyps" "2.prems" fxF + apply auto apply metis done + finally have ?case .} + ultimately show ?case by blast +qed + +lemma setsum_Un_nonzero: + assumes fS: "finite S" and fF: "finite F" + and f: "\ x\ S \ F . f x = (0::'a::ab_group_add)" + shows "setsum f (S \ F) = setsum f S + setsum f F" + using setsum_Un[OF fS fF, of f] setsum_0'[OF f] by simp + +lemma setsum_natinterval_left: + assumes mn: "(m::nat) <= n" + shows "setsum f {m..n} = f m + setsum f {m + 1..n}" +proof- + from mn have "{m .. n} = insert m {m+1 .. n}" by auto + then show ?thesis by auto +qed + +lemma setsum_natinterval_difff: + fixes f:: "nat \ ('a::ab_group_add)" + shows "setsum (\k. f k - f(k + 1)) {(m::nat) .. n} = + (if m <= n then f m - f(n + 1) else 0)" +by (induct n, auto simp add: ring_simps not_le le_Suc_eq) + +lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def] + +lemma setsum_setsum_restrict: + "finite S \ finite T \ setsum (\x. setsum (\y. f x y) {y. y\ T \ R x y}) S = setsum (\y. setsum (\x. f x y) {x. x \ S \ R x y}) T" + apply (simp add: setsum_restrict_set'[unfolded mem_def] mem_def) + by (rule setsum_commute) + +lemma setsum_image_gen: assumes fS: "finite S" + shows "setsum g S = setsum (\y. setsum g {x. x \ S \ f x = y}) (f ` S)" +proof- + {fix x assume "x \ S" then have "{y. y\ f`S \ f x = y} = {f x}" by auto} + note th0 = this + have "setsum g S = setsum (\x. setsum (\y. g x) {y. y\ f`S \ f x = y}) S" + apply (rule setsum_cong2) + by (simp add: th0) + also have "\ = setsum (\y. setsum g {x. x \ S \ f x = y}) (f ` S)" + apply (rule setsum_setsum_restrict[OF fS]) + by (rule finite_imageI[OF fS]) + finally show ?thesis . +qed + + (* FIXME: Here too need stupid finiteness assumption on T!!! *) +lemma setsum_group: + assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \ T" + shows "setsum (\y. setsum g {x. x\ S \ f x = y}) T = setsum g S" + +apply (subst setsum_image_gen[OF fS, of g f]) +apply (rule setsum_superset[OF fT fST]) +by (auto intro: setsum_0') + +(* FIXME: Change the name to fold_image\ *) +lemma (in comm_monoid_mult) fold_1': "finite S \ (\x\S. f x = 1) \ fold_image op * f 1 S = 1" + apply (induct set: finite) + apply simp by (auto simp add: fold_image_insert) + +lemma (in comm_monoid_mult) fold_union_nonzero: + assumes fS: "finite S" and fT: "finite T" + and I0: "\x \ S\T. f x = 1" + shows "fold_image (op *) f 1 (S \ T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T" +proof- + have "fold_image op * f 1 (S \ T) = 1" + apply (rule fold_1') + using fS fT I0 by auto + with fold_image_Un_Int[OF fS fT] show ?thesis by simp +qed + +lemma setsum_union_nonzero: + assumes fS: "finite S" and fT: "finite T" + and I0: "\x \ S\T. f x = 0" + shows "setsum f (S \ T) = setsum f S + setsum f T" + using fS fT + apply (simp add: setsum_def) + apply (rule comm_monoid_add.fold_union_nonzero) + using I0 by auto + +lemma setprod_union_nonzero: + assumes fS: "finite S" and fT: "finite T" + and I0: "\x \ S\T. f x = 1" + shows "setprod f (S \ T) = setprod f S * setprod f T" + using fS fT + apply (simp add: setprod_def) + apply (rule fold_union_nonzero) + using I0 by auto + +lemma setsum_unions_nonzero: + assumes fS: "finite S" and fSS: "\T \ S. finite T" + and f0: "\T1 T2 x. T1\S \ T2\S \ T1 \ T2 \ x \ T1 \ x \ T2 \ f x = 0" + shows "setsum f (\S) = setsum (\T. setsum f T) S" + using fSS f0 +proof(induct rule: finite_induct[OF fS]) + case 1 thus ?case by simp +next + case (2 T F) + then have fTF: "finite T" "\T\F. finite T" "finite F" and TF: "T \ F" + and H: "setsum f (\ F) = setsum (setsum f) F" by (auto simp add: finite_insert) + from fTF have fUF: "finite (\F)" by (auto intro: finite_Union) + from "2.prems" TF fTF + show ?case + by (auto simp add: H[symmetric] intro: setsum_union_nonzero[OF fTF(1) fUF, of f]) +qed + + (* FIXME : Copied from Pocklington --- should be moved to Finite_Set!!!!!!!! *) + + +lemma (in comm_monoid_mult) fold_related: + assumes Re: "R e e" + and Rop: "\x1 y1 x2 y2. R x1 x2 \ R y1 y2 \ R (x1 * y1) (x2 * y2)" + and fS: "finite S" and Rfg: "\x\S. R (h x) (g x)" + shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)" + using fS by (rule finite_subset_induct) (insert assms, auto) + + (* FIXME: I think we can get rid of the finite assumption!! *) +lemma (in comm_monoid_mult) + fold_eq_general: + assumes fS: "finite S" + and h: "\y\S'. \!x. x\ S \ h(x) = y" + and f12: "\x\S. h x \ S' \ f2(h x) = f1 x" + shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'" +proof- + from h f12 have hS: "h ` S = S'" by auto + {fix x y assume H: "x \ S" "y \ S" "h x = h y" + from f12 h H have "x = y" by auto } + hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast + from f12 have th: "\x. x \ S \ (f2 \ h) x = f1 x" by auto + from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp + also have "\ = fold_image (op *) (f2 o h) e S" + using fold_image_reindex[OF fS hinj, of f2 e] . + also have "\ = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e] + by blast + finally show ?thesis .. +qed + +lemma (in comm_monoid_mult) fold_eq_general_inverses: + assumes fS: "finite S" + and kh: "\y. y \ T \ k y \ S \ h (k y) = y" + and hk: "\x. x \ S \ h x \ T \ k (h x) = x \ g (h x) = f x" + shows "fold_image (op *) f e S = fold_image (op *) g e T" + using fold_eq_general[OF fS, of T h g f e] kh hk by metis + +lemma setsum_eq_general_reverses: + assumes fS: "finite S" and fT: "finite T" + and kh: "\y. y \ T \ k y \ S \ h (k y) = y" + and hk: "\x. x \ S \ h x \ T \ k (h x) = x \ g (h x) = f x" + shows "setsum f S = setsum g T" + apply (simp add: setsum_def fS fT) + apply (rule comm_monoid_add.fold_eq_general_inverses[OF fS]) + apply (erule kh) + apply (erule hk) + done + +lemma vsum_norm_allsubsets_bound: + fixes f:: "'a \ real ^'n" + assumes fP: "finite P" and fPs: "\Q. Q \ P \ norm (setsum f Q) \ e" + shows "setsum (\x. norm (f x)) P \ 2 * real (dimindex(UNIV :: 'n set)) * e" +proof- + let ?d = "real (dimindex (UNIV ::'n set))" + let ?nf = "\x. norm (f x)" + let ?U = "{1 .. dimindex (UNIV :: 'n set)}" + have th0: "setsum (\x. setsum (\i. \f x $ i\) ?U) P = setsum (\i. setsum (\x. \f x $ i\) P) ?U" + by (rule setsum_commute) + have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def) + have "setsum ?nf P \ setsum (\x. setsum (\i. \f x $ i\) ?U) P" + apply (rule setsum_mono) + by (rule norm_le_l1) + also have "\ \ 2 * ?d * e" + unfolding th0 th1 + proof(rule setsum_bounded) + fix i assume i: "i \ ?U" + let ?Pp = "{x. x\ P \ f x $ i \ 0}" + let ?Pn = "{x. x \ P \ f x $ i < 0}" + have thp: "P = ?Pp \ ?Pn" by auto + have thp0: "?Pp \ ?Pn ={}" by auto + have PpP: "?Pp \ P" and PnP: "?Pn \ P" by blast+ + have Ppe:"setsum (\x. \f x $ i\) ?Pp \ e" + using i component_le_norm[OF i, of "setsum (\x. f x) ?Pp"] fPs[OF PpP] + by (auto simp add: setsum_component intro: abs_le_D1) + have Pne: "setsum (\x. \f x $ i\) ?Pn \ e" + using i component_le_norm[OF i, of "setsum (\x. - f x) ?Pn"] fPs[OF PnP] + by (auto simp add: setsum_negf norm_neg setsum_component vector_component intro: abs_le_D1) + have "setsum (\x. \f x $ i\) P = setsum (\x. \f x $ i\) ?Pp + setsum (\x. \f x $ i\) ?Pn" + apply (subst thp) + apply (rule setsum_Un_nonzero) + using fP thp0 by auto + also have "\ \ 2*e" using Pne Ppe by arith + finally show "setsum (\x. \f x $ i\) P \ 2*e" . + qed + finally show ?thesis . +qed + +lemma dot_lsum: "finite S \ setsum f S \ (y::'a::{comm_ring}^'n) = setsum (\x. f x \ y) S " + by (induct rule: finite_induct, auto simp add: dot_lzero dot_ladd) + +lemma dot_rsum: "finite S \ (y::'a::{comm_ring}^'n) \ setsum f S = setsum (\x. y \ f x) S " + by (induct rule: finite_induct, auto simp add: dot_rzero dot_radd) + +subsection{* Basis vectors in coordinate directions. *} + + +definition "basis k = (\ i. if i = k then 1 else 0)" + +lemma delta_mult_idempotent: + "(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) + +lemma norm_basis: + assumes k: "k \ {1 .. dimindex (UNIV :: 'n set)}" + shows "norm (basis k :: real ^'n) = 1" + using k + apply (simp add: basis_def real_vector_norm_def dot_def) + apply (vector delta_mult_idempotent) + using setsum_delta[of "{1 .. dimindex (UNIV :: 'n set)}" "k" "\k. 1::real"] + apply auto + done + +lemma norm_basis_1: "norm(basis 1 :: real ^'n) = 1" + apply (simp add: basis_def real_vector_norm_def dot_def) + apply (vector delta_mult_idempotent) + using setsum_delta[of "{1 .. dimindex (UNIV :: 'n set)}" "1" "\k. 1::real"] dimindex_nonzero[of "UNIV :: 'n set"] + apply auto + done + +lemma vector_choose_size: "0 <= c ==> \(x::real^'n). norm x = c" + apply (rule exI[where x="c *s basis 1"]) + by (simp only: norm_mul norm_basis_1) + +lemma vector_choose_dist: assumes e: "0 <= e" + shows "\(y::real^'n). dist x y = e" +proof- + from vector_choose_size[OF e] obtain c:: "real ^'n" where "norm c = e" + by blast + then have "dist x (x - c) = e" by (simp add: dist_def) + then show ?thesis by blast +qed + +lemma basis_inj: "inj_on (basis :: nat \ real ^'n) {1 .. dimindex (UNIV :: 'n set)}" + by (auto simp add: inj_on_def basis_def Cart_eq Cart_lambda_beta) + +lemma basis_component: "i \ {1 .. dimindex(UNIV:: 'n set)} ==> (basis k ::('a::semiring_1)^'n)$i = (if k=i then 1 else 0)" + by (simp add: basis_def Cart_lambda_beta) + +lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)" + by auto + +lemma basis_expansion: + "setsum (\i. (x$i) *s basis i) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _") + 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) + +lemma basis_expansion_unique: + "setsum (\i. f i *s basis i) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::comm_ring_1) ^'n) \ (\i\{1 .. dimindex(UNIV:: 'n set)}. f i = x$i)" + by (simp add: Cart_eq setsum_component vector_component basis_component setsum_delta cond_value_iff cong del: if_weak_cong) + +lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)" + by auto + +lemma dot_basis: + assumes i: "i \ {1 .. dimindex (UNIV :: 'n set)}" + shows "basis i \ x = x$i" "x \ (basis i :: 'a^'n) = (x$i :: 'a::semiring_1)" + using i + by (auto simp add: dot_def basis_def Cart_lambda_beta cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong) + +lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) \ i \ {1..dimindex(UNIV ::'n set)}" + by (auto simp add: Cart_eq basis_component zero_index) + +lemma basis_nonzero: + assumes k: "k \ {1 .. dimindex(UNIV ::'n set)}" + shows "basis k \ (0:: 'a::semiring_1 ^'n)" + using k by (simp add: basis_eq_0) + +lemma vector_eq_ldot: "(\x. x \ y = x \ z) \ y = (z::'a::semiring_1^'n)" + apply (auto simp add: Cart_eq dot_basis) + apply (erule_tac x="basis i" in allE) + apply (simp add: dot_basis) + apply (subgoal_tac "y = z") + apply simp + apply vector + done + +lemma vector_eq_rdot: "(\z. x \ z = y \ z) \ x = (y::'a::semiring_1^'n)" + apply (auto simp add: Cart_eq dot_basis) + apply (erule_tac x="basis i" in allE) + apply (simp add: dot_basis) + apply (subgoal_tac "x = y") + apply simp + apply vector + done + +subsection{* Orthogonality. *} + +definition "orthogonal x y \ (x \ y = 0)" + +lemma orthogonal_basis: + assumes i:"i \ {1 .. dimindex(UNIV ::'n set)}" + shows "orthogonal (basis i :: 'a^'n) x \ x$i = (0::'a::ring_1)" + using i + 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) + +lemma orthogonal_basis_basis: + assumes i:"i \ {1 .. dimindex(UNIV ::'n set)}" + and j: "j \ {1 .. dimindex(UNIV ::'n set)}" + shows "orthogonal (basis i :: 'a::ring_1^'n) (basis j) \ i \ j" + unfolding orthogonal_basis[OF i] basis_component[OF i] by simp + + (* FIXME : Maybe some of these require less than comm_ring, but not all*) +lemma orthogonal_clauses: + "orthogonal a (0::'a::comm_ring ^'n)" + "orthogonal a x ==> orthogonal a (c *s x)" + "orthogonal a x ==> orthogonal a (-x)" + "orthogonal a x \ orthogonal a y ==> orthogonal a (x + y)" + "orthogonal a x \ orthogonal a y ==> orthogonal a (x - y)" + "orthogonal 0 a" + "orthogonal x a ==> orthogonal (c *s x) a" + "orthogonal x a ==> orthogonal (-x) a" + "orthogonal x a \ orthogonal y a ==> orthogonal (x + y) a" + "orthogonal x a \ orthogonal y a ==> orthogonal (x - y) a" + unfolding orthogonal_def dot_rneg dot_rmult dot_radd dot_rsub + dot_lzero dot_rzero dot_lneg dot_lmult dot_ladd dot_lsub + by simp_all + +lemma orthogonal_commute: "orthogonal (x::'a::{ab_semigroup_mult,comm_monoid_add} ^'n)y \ orthogonal y x" + by (simp add: orthogonal_def dot_sym) + +subsection{* Explicit vector construction from lists. *} + +lemma Cart_lambda_beta_1[simp]: "(Cart_lambda g)$1 = g 1" + apply (rule Cart_lambda_beta[rule_format]) + using dimindex_ge_1 apply auto done + +lemma Cart_lambda_beta_1'[simp]: "(Cart_lambda g)$(Suc 0) = g 1" + by (simp only: One_nat_def[symmetric] Cart_lambda_beta_1) + +definition "vector l = (\ i. if i <= length l then l ! (i - 1) else 0)" + +lemma vector_1: "(vector[x]) $1 = x" + using dimindex_ge_1 + by (auto simp add: vector_def Cart_lambda_beta[rule_format]) +lemma dimindex_2[simp]: "2 \ {1 .. dimindex (UNIV :: 2 set)}" + by (auto simp add: dimindex_def) +lemma dimindex_2'[simp]: "2 \ {Suc 0 .. dimindex (UNIV :: 2 set)}" + by (auto simp add: dimindex_def) +lemma dimindex_3[simp]: "2 \ {1 .. dimindex (UNIV :: 3 set)}" "3 \ {1 .. dimindex (UNIV :: 3 set)}" + by (auto simp add: dimindex_def) + +lemma dimindex_3'[simp]: "2 \ {Suc 0 .. dimindex (UNIV :: 3 set)}" "3 \ {Suc 0 .. dimindex (UNIV :: 3 set)}" + by (auto simp add: dimindex_def) + +lemma vector_2: + "(vector[x,y]) $1 = x" + "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)" + apply (simp add: vector_def) + using Cart_lambda_beta[rule_format, OF dimindex_2, of "\i. if i \ length [x,y] then [x,y] ! (i - 1) else (0::'a)"] + apply (simp only: vector_def ) + apply auto + done + +lemma vector_3: + "(vector [x,y,z] ::('a::zero)^3)$1 = x" + "(vector [x,y,z] ::('a::zero)^3)$2 = y" + "(vector [x,y,z] ::('a::zero)^3)$3 = z" +apply (simp_all add: vector_def Cart_lambda_beta dimindex_3) + using Cart_lambda_beta[rule_format, OF dimindex_3(1), of "\i. if i \ length [x,y,z] then [x,y,z] ! (i - 1) else (0::'a)"] using Cart_lambda_beta[rule_format, OF dimindex_3(2), of "\i. if i \ length [x,y,z] then [x,y,z] ! (i - 1) else (0::'a)"] + by simp_all + +lemma forall_vector_1: "(\v::'a::zero^1. P v) \ (\x. P(vector[x]))" + apply auto + apply (erule_tac x="v$1" in allE) + apply (subgoal_tac "vector [v$1] = v") + apply simp + by (vector vector_def dimindex_def) + +lemma forall_vector_2: "(\v::'a::zero^2. P v) \ (\x y. P(vector[x, y]))" + apply auto + apply (erule_tac x="v$1" in allE) + apply (erule_tac x="v$2" in allE) + apply (subgoal_tac "vector [v$1, v$2] = v") + apply simp + apply (vector vector_def dimindex_def) + apply auto + apply (subgoal_tac "i = 1 \ i =2", auto) + done + +lemma forall_vector_3: "(\v::'a::zero^3. P v) \ (\x y z. P(vector[x, y, z]))" + apply auto + apply (erule_tac x="v$1" in allE) + apply (erule_tac x="v$2" in allE) + apply (erule_tac x="v$3" in allE) + apply (subgoal_tac "vector [v$1, v$2, v$3] = v") + apply simp + apply (vector vector_def dimindex_def) + apply auto + apply (subgoal_tac "i = 1 \ i =2 \ i = 3", auto) + done + +subsection{* Linear functions. *} + +definition "linear f \ (\x y. f(x + y) = f x + f y) \ (\c x. f(c *s x) = c *s f x)" + +lemma linear_compose_cmul: "linear f ==> linear (\x. (c::'a::comm_semiring) *s f x)" + by (vector linear_def Cart_eq Cart_lambda_beta[rule_format] ring_simps) + +lemma linear_compose_neg: "linear (f :: 'a ^'n \ 'a::comm_ring ^'m) ==> linear (\x. -(f(x)))" by (vector linear_def Cart_eq) + +lemma linear_compose_add: "linear (f :: 'a ^'n \ 'a::semiring_1 ^'m) \ linear g ==> linear (\x. f(x) + g(x))" + by (vector linear_def Cart_eq ring_simps) + +lemma linear_compose_sub: "linear (f :: 'a ^'n \ 'a::ring_1 ^'m) \ linear g ==> linear (\x. f x - g x)" + by (vector linear_def Cart_eq ring_simps) + +lemma linear_compose: "linear f \ linear g ==> linear (g o f)" + by (simp add: linear_def) + +lemma linear_id: "linear id" by (simp add: linear_def id_def) + +lemma linear_zero: "linear (\x. 0::'a::semiring_1 ^ 'n)" by (simp add: linear_def) + +lemma linear_compose_setsum: + assumes fS: "finite S" and lS: "\a \ S. linear (f a :: 'a::semiring_1 ^ 'n \ 'a ^ 'm)" + shows "linear(\x. setsum (\a. f a x :: 'a::semiring_1 ^'m) S)" + using lS + apply (induct rule: finite_induct[OF fS]) + by (auto simp add: linear_zero intro: linear_compose_add) + +lemma linear_vmul_component: + fixes f:: "'a::semiring_1^'m \ 'a^'n" + assumes lf: "linear f" and k: "k \ {1 .. dimindex (UNIV :: 'n set)}" + shows "linear (\x. f x $ k *s v)" + using lf k + apply (auto simp add: linear_def ) + by (vector ring_simps)+ + +lemma linear_0: "linear f ==> f 0 = (0::'a::semiring_1 ^'n)" + unfolding linear_def + apply clarsimp + apply (erule allE[where x="0::'a"]) + apply simp + done + +lemma linear_cmul: "linear f ==> f(c*s x) = c *s f x" by (simp add: linear_def) + +lemma linear_neg: "linear (f :: 'a::ring_1 ^'n \ _) ==> f (-x) = - f x" + unfolding vector_sneg_minus1 + using linear_cmul[of f] by auto + +lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def) + +lemma linear_sub: "linear (f::'a::ring_1 ^'n \ _) ==> f(x - y) = f x - f y" + by (simp add: diff_def linear_add linear_neg) + +lemma linear_setsum: + fixes f:: "'a::semiring_1^'n \ _" + assumes lf: "linear f" and fS: "finite S" + shows "f (setsum g S) = setsum (f o g) S" +proof (induct rule: finite_induct[OF fS]) + case 1 thus ?case by (simp add: linear_0[OF lf]) +next + case (2 x F) + have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps" + by simp + also have "\ = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp + also have "\ = setsum (f o g) (insert x F)" using "2.hyps" by simp + finally show ?case . +qed + +lemma linear_setsum_mul: + fixes f:: "'a ^'n \ 'a::semiring_1^'m" + assumes lf: "linear f" and fS: "finite S" + shows "f (setsum (\i. c i *s v i) S) = setsum (\i. c i *s f (v i)) S" + using linear_setsum[OF lf fS, of "\i. c i *s v i" , unfolded o_def] + linear_cmul[OF lf] by simp + +lemma linear_injective_0: + assumes lf: "linear (f:: 'a::ring_1 ^ 'n \ _)" + shows "inj f \ (\x. f x = 0 \ x = 0)" +proof- + have "inj f \ (\ x y. f x = f y \ x = y)" by (simp add: inj_on_def) + also have "\ \ (\ x y. f x - f y = 0 \ x - y = 0)" by simp + also have "\ \ (\ x y. f (x - y) = 0 \ x - y = 0)" + by (simp add: linear_sub[OF lf]) + also have "\ \ (\ x. f x = 0 \ x = 0)" by auto + finally show ?thesis . +qed + +lemma linear_bounded: + fixes f:: "real ^'m \ real ^'n" + assumes lf: "linear f" + shows "\B. \x. norm (f x) \ B * norm x" +proof- + let ?S = "{1..dimindex(UNIV:: 'm set)}" + let ?B = "setsum (\i. norm(f(basis i))) ?S" + have fS: "finite ?S" by simp + {fix x:: "real ^ 'm" + let ?g = "(\i::nat. (x$i) *s (basis i) :: real ^ 'm)" + have "norm (f x) = norm (f (setsum (\i. (x$i) *s (basis i)) ?S))" + by (simp only: basis_expansion) + also have "\ = norm (setsum (\i. (x$i) *s f (basis i))?S)" + using linear_setsum[OF lf fS, of ?g, unfolded o_def] linear_cmul[OF lf] + by auto + finally have th0: "norm (f x) = norm (setsum (\i. (x$i) *s f (basis i))?S)" . + {fix i assume i: "i \ ?S" + from component_le_norm[OF i, of x] + have "norm ((x$i) *s f (basis i :: real ^'m)) \ norm (f (basis i)) * norm x" + unfolding norm_mul + apply (simp only: mult_commute) + apply (rule mult_mono) + by (auto simp add: ring_simps norm_pos_le) } + then have th: "\i\ ?S. norm ((x$i) *s f (basis i :: real ^'m)) \ norm (f (basis i)) * norm x" by metis + from real_setsum_norm_le[OF fS, of "\i. (x$i) *s (f (basis i))", OF th] + have "norm (f x) \ ?B * norm x" unfolding th0 setsum_left_distrib by metis} + then show ?thesis by blast +qed + +lemma linear_bounded_pos: + fixes f:: "real ^'n \ real ^ 'm" + assumes lf: "linear f" + shows "\B > 0. \x. norm (f x) \ B * norm x" +proof- + from linear_bounded[OF lf] obtain B where + B: "\x. norm (f x) \ B * norm x" by blast + let ?K = "\B\ + 1" + have Kp: "?K > 0" by arith + {assume C: "B < 0" + have "norm (1::real ^ 'n) > 0" by (simp add: norm_pos_lt) + with C have "B * norm (1:: real ^ 'n) < 0" + by (simp add: zero_compare_simps) + with B[rule_format, of 1] norm_pos_le[of "f 1"] have False by simp + } + then have Bp: "B \ 0" by ferrack + {fix x::"real ^ 'n" + have "norm (f x) \ ?K * norm x" + using B[rule_format, of x] norm_pos_le[of x] norm_pos_le[of "f x"] Bp + by (auto simp add: ring_simps split add: abs_split) + } + then show ?thesis using Kp by blast +qed + +subsection{* Bilinear functions. *} + +definition "bilinear f \ (\x. linear(\y. f x y)) \ (\y. linear(\x. f x y))" + +lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)" + by (simp add: bilinear_def linear_def) +lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)" + by (simp add: bilinear_def linear_def) + +lemma bilinear_lmul: "bilinear h ==> h (c *s x) y = c *s (h x y)" + by (simp add: bilinear_def linear_def) + +lemma bilinear_rmul: "bilinear h ==> h x (c *s y) = c *s (h x y)" + by (simp add: bilinear_def linear_def) + +lemma bilinear_lneg: "bilinear h ==> h (- (x:: 'a::ring_1 ^ 'n)) y = -(h x y)" + by (simp only: vector_sneg_minus1 bilinear_lmul) + +lemma bilinear_rneg: "bilinear h ==> h x (- (y:: 'a::ring_1 ^ 'n)) = - h x y" + by (simp only: vector_sneg_minus1 bilinear_rmul) + +lemma (in ab_group_add) eq_add_iff: "x = x + y \ y = 0" + using add_imp_eq[of x y 0] by auto + +lemma bilinear_lzero: + fixes h :: "'a::ring^'n \ _" assumes bh: "bilinear h" shows "h 0 x = 0" + using bilinear_ladd[OF bh, of 0 0 x] + by (simp add: eq_add_iff ring_simps) + +lemma bilinear_rzero: + fixes h :: "'a::ring^'n \ _" assumes bh: "bilinear h" shows "h x 0 = 0" + using bilinear_radd[OF bh, of x 0 0 ] + by (simp add: eq_add_iff ring_simps) + +lemma bilinear_lsub: "bilinear h ==> h (x - (y:: 'a::ring_1 ^ 'n)) z = h x z - h y z" + by (simp add: diff_def bilinear_ladd bilinear_lneg) + +lemma bilinear_rsub: "bilinear h ==> h z (x - (y:: 'a::ring_1 ^ 'n)) = h z x - h z y" + by (simp add: diff_def bilinear_radd bilinear_rneg) + +lemma bilinear_setsum: + fixes h:: "'a ^'n \ 'a::semiring_1^'m \ 'a ^ 'k" + assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T" + shows "h (setsum f S) (setsum g T) = setsum (\(i,j). h (f i) (g j)) (S \ T) " +proof- + have "h (setsum f S) (setsum g T) = setsum (\x. h (f x) (setsum g T)) S" + apply (rule linear_setsum[unfolded o_def]) + using bh fS by (auto simp add: bilinear_def) + also have "\ = setsum (\x. setsum (\y. h (f x) (g y)) T) S" + apply (rule setsum_cong, simp) + apply (rule linear_setsum[unfolded o_def]) + using bh fT by (auto simp add: bilinear_def) + finally show ?thesis unfolding setsum_cartesian_product . +qed + +lemma bilinear_bounded: + fixes h:: "real ^'m \ real^'n \ real ^ 'k" + assumes bh: "bilinear h" + shows "\B. \x y. norm (h x y) \ B * norm x * norm y" +proof- + let ?M = "{1 .. dimindex (UNIV :: 'm set)}" + let ?N = "{1 .. dimindex (UNIV :: 'n set)}" + let ?B = "setsum (\(i,j). norm (h (basis i) (basis j))) (?M \ ?N)" + have fM: "finite ?M" and fN: "finite ?N" by simp_all + {fix x:: "real ^ 'm" and y :: "real^'n" + have "norm (h x y) = norm (h (setsum (\i. (x$i) *s basis i) ?M) (setsum (\i. (y$i) *s basis i) ?N))" unfolding basis_expansion .. + also have "\ = norm (setsum (\ (i,j). h ((x$i) *s basis i) ((y$j) *s basis j)) (?M \ ?N))" unfolding bilinear_setsum[OF bh fM fN] .. + finally have th: "norm (h x y) = \" . + have "norm (h x y) \ ?B * norm x * norm y" + apply (simp add: setsum_left_distrib th) + apply (rule real_setsum_norm_le) + using fN fM + apply simp + apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] norm_mul ring_simps) + apply (rule mult_mono) + apply (auto simp add: norm_pos_le zero_le_mult_iff component_le_norm) + apply (rule mult_mono) + apply (auto simp add: norm_pos_le zero_le_mult_iff component_le_norm) + done} + then show ?thesis by metis +qed + +lemma bilinear_bounded_pos: + fixes h:: "real ^'m \ real^'n \ real ^ 'k" + assumes bh: "bilinear h" + shows "\B > 0. \x y. norm (h x y) \ B * norm x * norm y" +proof- + from bilinear_bounded[OF bh] obtain B where + B: "\x y. norm (h x y) \ B * norm x * norm y" by blast + let ?K = "\B\ + 1" + have Kp: "?K > 0" by arith + have KB: "B < ?K" by arith + {fix x::"real ^'m" and y :: "real ^'n" + from KB Kp + have "B * norm x * norm y \ ?K * norm x * norm y" + apply - + apply (rule mult_right_mono, rule mult_right_mono) + by (auto simp add: norm_pos_le) + then have "norm (h x y) \ ?K * norm x * norm y" + using B[rule_format, of x y] by simp} + with Kp show ?thesis by blast +qed + +subsection{* Adjoints. *} + +definition "adjoint f = (SOME f'. \x y. f x \ y = x \ f' y)" + +lemma choice_iff: "(\x. \y. P x y) \ (\f. \x. P x (f x))" by metis + +lemma adjoint_works_lemma: + fixes f:: "'a::ring_1 ^'n \ 'a ^ 'm" + assumes lf: "linear f" + shows "\x y. f x \ y = x \ adjoint f y" +proof- + let ?N = "{1 .. dimindex (UNIV :: 'n set)}" + let ?M = "{1 .. dimindex (UNIV :: 'm set)}" + have fN: "finite ?N" by simp + have fM: "finite ?M" by simp + {fix y:: "'a ^ 'm" + let ?w = "(\ i. (f (basis i) \ y)) :: 'a ^ 'n" + {fix x + have "f x \ y = f (setsum (\i. (x$i) *s basis i) ?N) \ y" + by (simp only: basis_expansion) + also have "\ = (setsum (\i. (x$i) *s f (basis i)) ?N) \ y" + unfolding linear_setsum[OF lf fN] + by (simp add: linear_cmul[OF lf]) + finally have "f x \ y = x \ ?w" + apply (simp only: ) + 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) + done} + } + then show ?thesis unfolding adjoint_def + some_eq_ex[of "\f'. \x y. f x \ y = x \ f' y"] + using choice_iff[of "\a b. \x. f x \ a = x \ b "] + by metis +qed + +lemma adjoint_works: + fixes f:: "'a::ring_1 ^'n \ 'a ^ 'm" + assumes lf: "linear f" + shows "x \ adjoint f y = f x \ y" + using adjoint_works_lemma[OF lf] by metis + + +lemma adjoint_linear: + fixes f :: "'a::comm_ring_1 ^'n \ 'a ^ 'm" + assumes lf: "linear f" + shows "linear (adjoint f)" + by (simp add: linear_def vector_eq_ldot[symmetric] dot_radd dot_rmult adjoint_works[OF lf]) + +lemma adjoint_clauses: + fixes f:: "'a::comm_ring_1 ^'n \ 'a ^ 'm" + assumes lf: "linear f" + shows "x \ adjoint f y = f x \ y" + and "adjoint f y \ x = y \ f x" + by (simp_all add: adjoint_works[OF lf] dot_sym ) + +lemma adjoint_adjoint: + fixes f:: "'a::comm_ring_1 ^ 'n \ _" + assumes lf: "linear f" + shows "adjoint (adjoint f) = f" + apply (rule ext) + by (simp add: vector_eq_ldot[symmetric] adjoint_clauses[OF adjoint_linear[OF lf]] adjoint_clauses[OF lf]) + +lemma adjoint_unique: + fixes f:: "'a::comm_ring_1 ^ 'n \ 'a ^ 'm" + assumes lf: "linear f" and u: "\x y. f' x \ y = x \ f y" + shows "f' = adjoint f" + apply (rule ext) + using u + by (simp add: vector_eq_rdot[symmetric] adjoint_clauses[OF lf]) + +text{* Matrix notation. NB: an MxN matrix is of type 'a^'n^'m, not 'a^'m^'n *} + +consts generic_mult :: "'a \ 'b \ 'c" (infixr "\" 75) + +defs (overloaded) +matrix_matrix_mult_def: "(m:: ('a::semiring_1) ^'n^'m) \ (m' :: 'a ^'p^'n) \ (\ i j. setsum (\k. ((m$i)$k) * ((m'$k)$j)) {1 .. dimindex (UNIV :: 'n set)}) ::'a ^ 'p ^'m" + +abbreviation + matrix_matrix_mult' :: "('a::semiring_1) ^'n^'m \ 'a ^'p^'n \ 'a ^ 'p ^'m" (infixl "**" 70) + where "m ** m' == m\ m'" + +defs (overloaded) + matrix_vector_mult_def: "(m::('a::semiring_1) ^'n^'m) \ (x::'a ^'n) \ (\ i. setsum (\j. ((m$i)$j) * (x$j)) {1..dimindex(UNIV ::'n set)}) :: 'a^'m" + +abbreviation + matrix_vector_mult' :: "('a::semiring_1) ^'n^'m \ 'a ^'n \ 'a ^ 'm" (infixl "*v" 70) + where + "m *v v == m \ v" + +defs (overloaded) + vector_matrix_mult_def: "(x::'a^'m) \ (m::('a::semiring_1) ^'n^'m) \ (\ j. setsum (\i. ((m$i)$j) * (x$i)) {1..dimindex(UNIV :: 'm set)}) :: 'a^'n" + +abbreviation + vactor_matrix_mult' :: "'a ^ 'm \ ('a::semiring_1) ^'n^'m \ 'a ^'n " (infixl "v*" 70) + where + "v v* m == v \ m" + +definition "(mat::'a::zero => 'a ^'n^'m) k = (\ i j. if i = j then k else 0)" +definition "(transp::'a^'n^'m \ 'a^'m^'n) A = (\ i j. ((A$j)$i))" +definition "(row::nat => 'a ^'n^'m \ 'a ^'n) i A = (\ j. ((A$i)$j))" +definition "(column::nat =>'a^'n^'m =>'a^'m) j A = (\ i. ((A$i)$j))" +definition "rows(A::'a^'n^'m) = { row i A | i. i \ {1 .. dimindex(UNIV :: 'm set)}}" +definition "columns(A::'a^'n^'m) = { column i A | i. i \ {1 .. dimindex(UNIV :: 'n set)}}" + +lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def) +lemma matrix_add_ldistrib: "(A ** (B + C)) = (A \ B) + (A \ C)" + by (vector matrix_matrix_mult_def setsum_addf[symmetric] ring_simps) + +lemma setsum_delta': + assumes fS: "finite S" shows + "setsum (\k. if a = k then b k else 0) S = + (if a\ S then b a else 0)" + using setsum_delta[OF fS, of a b, symmetric] + by (auto intro: setsum_cong) + +lemma matrix_mul_lid: "mat 1 ** A = A" + apply (simp add: matrix_matrix_mult_def mat_def) + apply vector + by (auto simp only: cond_value_iff cond_application_beta setsum_delta'[OF finite_atLeastAtMost] mult_1_left mult_zero_left if_True) + + +lemma matrix_mul_rid: "A ** mat 1 = A" + apply (simp add: matrix_matrix_mult_def mat_def) + apply vector + 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) + +lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C" + apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc) + apply (subst setsum_commute) + apply simp + done + +lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x" + apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc) + apply (subst setsum_commute) + apply simp + done + +lemma matrix_vector_mul_lid: "mat 1 *v x = x" + apply (vector matrix_vector_mult_def mat_def) + by (simp add: cond_value_iff cond_application_beta + setsum_delta' cong del: if_weak_cong) + +lemma matrix_transp_mul: "transp(A ** B) = transp B ** transp (A::'a::comm_semiring_1^'m^'n)" + by (simp add: matrix_matrix_mult_def transp_def Cart_eq Cart_lambda_beta mult_commute) + +lemma matrix_eq: "A = B \ (\x. A *v x = B *v x)" (is "?lhs \ ?rhs") + apply auto + apply (subst Cart_eq) + apply clarify + 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) + apply (erule_tac x="basis ia" in allE) + apply (erule_tac x="i" in ballE) + 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) + +lemma matrix_vector_mul_component: + assumes k: "k \ {1.. dimindex (UNIV :: 'm set)}" + shows "((A::'a::semiring_1^'n'^'m) *v x)$k = (A$k) \ x" + using k + by (simp add: matrix_vector_mult_def Cart_lambda_beta dot_def) + +lemma dot_lmul_matrix: "((x::'a::comm_semiring_1 ^'n) v* A) \ y = x \ (A *v y)" + apply (simp add: dot_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib Cart_lambda_beta mult_ac) + apply (subst setsum_commute) + by simp + +lemma transp_mat: "transp (mat n) = mat n" + by (vector transp_def mat_def) + +lemma transp_transp: "transp(transp A) = A" + by (vector transp_def) + +lemma row_transp: + fixes A:: "'a::semiring_1^'n^'m" + assumes i: "i \ {1.. dimindex (UNIV :: 'n set)}" + shows "row i (transp A) = column i A" + using i + by (simp add: row_def column_def transp_def Cart_eq Cart_lambda_beta) + +lemma column_transp: + fixes A:: "'a::semiring_1^'n^'m" + assumes i: "i \ {1.. dimindex (UNIV :: 'm set)}" + shows "column i (transp A) = row i A" + using i + by (simp add: row_def column_def transp_def Cart_eq Cart_lambda_beta) + +lemma rows_transp: "rows(transp (A::'a::semiring_1^'n^'m)) = columns A" +apply (auto simp add: rows_def columns_def row_transp intro: set_ext) +apply (rule_tac x=i in exI) +apply (auto simp add: row_transp) +done + +lemma columns_transp: "columns(transp (A::'a::semiring_1^'n^'m)) = rows A" by (metis transp_transp rows_transp) + +text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *} + +lemma matrix_mult_dot: "A *v x = (\ i. A$i \ x)" + by (simp add: matrix_vector_mult_def dot_def) + +lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\i. (x$i) *s column i A) {1 .. dimindex(UNIV:: 'n set)}" + by (simp add: matrix_vector_mult_def Cart_eq setsum_component Cart_lambda_beta vector_component column_def mult_commute) + +lemma vector_componentwise: + "(x::'a::ring_1^'n) = (\ j. setsum (\i. (x$i) * (basis i :: 'a^'n)$j) {1..dimindex(UNIV :: 'n set)})" + apply (subst basis_expansion[symmetric]) + by (vector Cart_eq Cart_lambda_beta setsum_component) + +lemma linear_componentwise: + fixes f:: "'a::ring_1 ^ 'm \ 'a ^ 'n" + assumes lf: "linear f" and j: "j \ {1 .. dimindex (UNIV :: 'n set)}" + shows "(f x)$j = setsum (\i. (x$i) * (f (basis i)$j)) {1 .. dimindex (UNIV :: 'm set)}" (is "?lhs = ?rhs") +proof- + let ?M = "{1 .. dimindex (UNIV :: 'm set)}" + let ?N = "{1 .. dimindex (UNIV :: 'n set)}" + have fM: "finite ?M" by simp + have "?rhs = (setsum (\i.(x$i) *s f (basis i) ) ?M)$j" + unfolding vector_smult_component[OF j, symmetric] + unfolding setsum_component[OF j, of "(\i.(x$i) *s f (basis i :: 'a^'m))" ?M] + .. + then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion .. +qed + +text{* Inverse matrices (not necessarily square) *} + +definition "invertible(A::'a::semiring_1^'n^'m) \ (\A'::'a^'m^'n. A ** A' = mat 1 \ A' ** A = mat 1)" + +definition "matrix_inv(A:: 'a::semiring_1^'n^'m) = + (SOME A'::'a^'m^'n. A ** A' = mat 1 \ A' ** A = mat 1)" + +text{* Correspondence between matrices and linear operators. *} + +definition matrix:: "('a::{plus,times, one, zero}^'m \ 'a ^ 'n) \ 'a^'m^'n" +where "matrix f = (\ i j. (f(basis j))$i)" + +lemma matrix_vector_mul_linear: "linear(\x. A *v (x::'a::comm_semiring_1 ^ 'n))" + by (simp add: linear_def matrix_vector_mult_def Cart_eq Cart_lambda_beta vector_component ring_simps setsum_right_distrib setsum_addf) + +lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::'a::comm_ring_1 ^ 'n)" +apply (simp add: matrix_def matrix_vector_mult_def Cart_eq Cart_lambda_beta mult_commute del: One_nat_def) +apply clarify +apply (rule linear_componentwise[OF lf, symmetric]) +apply simp +done + +lemma matrix_vector_mul: "linear f ==> f = (\x. matrix f *v (x::'a::comm_ring_1 ^ 'n))" by (simp add: ext matrix_works) + +lemma matrix_of_matrix_vector_mul: "matrix(\x. A *v (x :: 'a:: comm_ring_1 ^ 'n)) = A" + by (simp add: matrix_eq matrix_vector_mul_linear matrix_works) + +lemma matrix_compose: + assumes lf: "linear (f::'a::comm_ring_1^'n \ _)" and lg: "linear g" + shows "matrix (g o f) = matrix g ** matrix f" + using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]] + by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def) + +lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\i. (x$i) *s ((transp A)$i)) {1..dimindex(UNIV:: 'n set)}" + by (simp add: matrix_vector_mult_def transp_def Cart_eq Cart_lambda_beta setsum_component vector_component mult_commute) + +lemma adjoint_matrix: "adjoint(\x. (A::'a::comm_ring_1^'n^'m) *v x) = (\x. transp A *v x)" + apply (rule adjoint_unique[symmetric]) + apply (rule matrix_vector_mul_linear) + apply (simp add: transp_def dot_def Cart_lambda_beta matrix_vector_mult_def setsum_left_distrib setsum_right_distrib) + apply (subst setsum_commute) + apply (auto simp add: mult_ac) + done + +lemma matrix_adjoint: assumes lf: "linear (f :: 'a::comm_ring_1^'n \ 'a ^ 'm)" + shows "matrix(adjoint f) = transp(matrix f)" + apply (subst matrix_vector_mul[OF lf]) + unfolding adjoint_matrix matrix_of_matrix_vector_mul .. + +subsection{* Interlude: Some properties of real sets *} + +lemma seq_mono_lemma: assumes "\(n::nat) \ m. (d n :: real) < e n" and "\n \ m. e n <= e m" + shows "\n \ m. d n < e m" + using prems apply auto + apply (erule_tac x="n" in allE) + apply (erule_tac x="n" in allE) + apply auto + done + + +lemma real_convex_bound_lt: + assumes xa: "(x::real) < a" and ya: "y < a" and u: "0 <= u" and v: "0 <= v" + and uv: "u + v = 1" + shows "u * x + v * y < a" +proof- + have uv': "u = 0 \ v \ 0" using u v uv by arith + have "a = a * (u + v)" unfolding uv by simp + hence th: "u * a + v * a = a" by (simp add: ring_simps) + from xa u have "u \ 0 \ u*x < u*a" by (simp add: mult_compare_simps) + from ya v have "v \ 0 \ v * y < v * a" by (simp add: mult_compare_simps) + from xa ya u v have "u * x + v * y < u * a + v * a" + apply (cases "u = 0", simp_all add: uv') + apply(rule mult_strict_left_mono) + using uv' apply simp_all + + apply (rule add_less_le_mono) + apply(rule mult_strict_left_mono) + apply simp_all + apply (rule mult_left_mono) + apply simp_all + done + thus ?thesis unfolding th . +qed + +lemma real_convex_bound_le: + assumes xa: "(x::real) \ a" and ya: "y \ a" and u: "0 <= u" and v: "0 <= v" + and uv: "u + v = 1" + shows "u * x + v * y \ a" +proof- + from xa ya u v have "u * x + v * y \ u * a + v * a" by (simp add: add_mono mult_left_mono) + also have "\ \ (u + v) * a" by (simp add: ring_simps) + finally show ?thesis unfolding uv by simp +qed + +lemma infinite_enumerate: assumes fS: "infinite S" + shows "\r. subseq r \ (\n. r n \ S)" +unfolding subseq_def +using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto + +lemma approachable_lt_le: "(\(d::real)>0. \x. f x < d \ P x) \ (\d>0. \x. f x \ d \ P x)" +apply auto +apply (rule_tac x="d/2" in exI) +apply auto +done + + +lemma triangle_lemma: + assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2" + shows "x <= y + z" +proof- + have "y^2 + z^2 \ y^2 + 2*y*z + z^2" using z y by (simp add: zero_compare_simps) + with xy have th: "x ^2 \ (y+z)^2" by (simp add: power2_eq_square ring_simps) + from y z have yz: "y + z \ 0" by arith + from power2_le_imp_le[OF th yz] show ?thesis . +qed + + +lemma lambda_skolem: "(\i \ {1 .. dimindex(UNIV :: 'n set)}. \x. P i x) \ + (\x::'a ^ 'n. \i \ {1 .. dimindex(UNIV:: 'n set)}. P i (x$i))" (is "?lhs \ ?rhs") +proof- + let ?S = "{1 .. dimindex(UNIV :: 'n set)}" + {assume H: "?rhs" + then have ?lhs by auto} + moreover + {assume H: "?lhs" + then obtain f where f:"\i\ ?S. P i (f i)" unfolding Ball_def choice_iff by metis + let ?x = "(\ i. (f i)) :: 'a ^ 'n" + {fix i assume i: "i \ ?S" + with f i have "P i (f i)" by metis + then have "P i (?x$i)" using Cart_lambda_beta[of f, rule_format, OF i] by auto + } + hence "\i \ ?S. P i (?x$i)" by metis + hence ?rhs by metis } + ultimately show ?thesis by metis +qed + +(* Supremum and infimum of real sets *) + + +definition rsup:: "real set \ real" where + "rsup S = (SOME a. isLub UNIV S a)" + +lemma rsup_alt: "rsup S = (SOME a. (\x \ S. x \ a) \ (\b. (\x \ S. x \ b) \ a \ b))" by (auto simp add: isLub_def rsup_def leastP_def isUb_def setle_def setge_def) + +lemma rsup: assumes Se: "S \ {}" and b: "\b. S *<= b" + shows "isLub UNIV S (rsup S)" +using Se b +unfolding rsup_def +apply clarify +apply (rule someI_ex) +apply (rule reals_complete) +by (auto simp add: isUb_def setle_def) + +lemma rsup_le: assumes Se: "S \ {}" and Sb: "S *<= b" shows "rsup S \ b" +proof- + from Sb have bu: "isUb UNIV S b" by (simp add: isUb_def setle_def) + from rsup[OF Se] Sb have "isLub UNIV S (rsup S)" by blast + then show ?thesis using bu by (auto simp add: isLub_def leastP_def setle_def setge_def) +qed + +lemma rsup_finite_Max: assumes fS: "finite S" and Se: "S \ {}" + shows "rsup S = Max S" +using fS Se +proof- + let ?m = "Max S" + from Max_ge[OF fS] have Sm: "\ x\ S. x \ ?m" by metis + with rsup[OF Se] have lub: "isLub UNIV S (rsup S)" by (metis setle_def) + from Max_in[OF fS Se] lub have mrS: "?m \ rsup S" + by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def) + moreover + have "rsup S \ ?m" using Sm lub + by (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def) + ultimately show ?thesis by arith +qed + +lemma rsup_finite_in: assumes fS: "finite S" and Se: "S \ {}" + shows "rsup S \ S" + using rsup_finite_Max[OF fS Se] Max_in[OF fS Se] by metis + +lemma rsup_finite_Ub: assumes fS: "finite S" and Se: "S \ {}" + shows "isUb S S (rsup S)" + using rsup_finite_Max[OF fS Se] rsup_finite_in[OF fS Se] Max_ge[OF fS] + unfolding isUb_def setle_def by metis + +lemma rsup_finite_ge_iff: assumes fS: "finite S" and Se: "S \ {}" + shows "a \ rsup S \ (\ x \ S. a \ x)" +using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def) + +lemma rsup_finite_le_iff: assumes fS: "finite S" and Se: "S \ {}" + shows "a \ rsup S \ (\ x \ S. a \ x)" +using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def) + +lemma rsup_finite_gt_iff: assumes fS: "finite S" and Se: "S \ {}" + shows "a < rsup S \ (\ x \ S. a < x)" +using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def) + +lemma rsup_finite_lt_iff: assumes fS: "finite S" and Se: "S \ {}" + shows "a > rsup S \ (\ x \ S. a > x)" +using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def) + +lemma rsup_unique: assumes b: "S *<= b" and S: "\b' < b. \x \ S. b' < x" + shows "rsup S = b" +using b S +unfolding setle_def rsup_alt +apply - +apply (rule some_equality) +apply (metis linorder_not_le order_eq_iff[symmetric])+ +done + +lemma rsup_le_subset: "S\{} \ S \ T \ (\b. T *<= b) \ rsup S \ rsup T" + apply (rule rsup_le) + apply simp + using rsup[of T] by (auto simp add: isLub_def leastP_def setge_def setle_def isUb_def) + +lemma isUb_def': "isUb R S = (\x. S *<= x \ x \ R)" + apply (rule ext) + by (metis isUb_def) + +lemma UNIV_trivial: "UNIV x" using UNIV_I[of x] by (metis mem_def) +lemma rsup_bounds: assumes Se: "S \ {}" and l: "a <=* S" and u: "S *<= b" + shows "a \ rsup S \ rsup S \ b" +proof- + from rsup[OF Se] u have lub: "isLub UNIV S (rsup S)" by blast + hence b: "rsup S \ b" using u by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def') + from Se obtain y where y: "y \ S" by blast + from lub l have "a \ rsup S" apply (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def') + apply (erule ballE[where x=y]) + apply (erule ballE[where x=y]) + apply arith + using y apply auto + done + with b show ?thesis by blast +qed + +lemma rsup_abs_le: "S \ {} \ (\x\S. \x\ \ a) \ \rsup S\ \ a" + unfolding abs_le_interval_iff using rsup_bounds[of S "-a" a] + by (auto simp add: setge_def setle_def) + +lemma rsup_asclose: assumes S:"S \ {}" and b: "\x\S. \x - l\ \ e" shows "\rsup S - l\ \ e" +proof- + have th: "\(x::real) l e. \x - l\ \ e \ l - e \ x \ x \ l + e" by arith + show ?thesis using S b rsup_bounds[of S "l - e" "l+e"] unfolding th + by (auto simp add: setge_def setle_def) +qed + +definition rinf:: "real set \ real" where + "rinf S = (SOME a. isGlb UNIV S a)" + +lemma rinf_alt: "rinf S = (SOME a. (\x \ S. x \ a) \ (\b. (\x \ S. x \ b) \ a \ b))" by (auto simp add: isGlb_def rinf_def greatestP_def isLb_def setle_def setge_def) + +lemma reals_complete_Glb: assumes Se: "\x. x \ S" and lb: "\ y. isLb UNIV S y" + shows "\(t::real). isGlb UNIV S t" +proof- + let ?M = "uminus ` S" + from lb have th: "\y. isUb UNIV ?M y" apply (auto simp add: isUb_def isLb_def setle_def setge_def) + by (rule_tac x="-y" in exI, auto) + from Se have Me: "\x. x \ ?M" by blast + from reals_complete[OF Me th] obtain t where t: "isLub UNIV ?M t" by blast + have "isGlb UNIV S (- t)" using t + apply (auto simp add: isLub_def isGlb_def leastP_def greatestP_def setle_def setge_def isUb_def isLb_def) + apply (erule_tac x="-y" in allE) + apply auto + done + then show ?thesis by metis +qed + +lemma rinf: assumes Se: "S \ {}" and b: "\b. b <=* S" + shows "isGlb UNIV S (rinf S)" +using Se b +unfolding rinf_def +apply clarify +apply (rule someI_ex) +apply (rule reals_complete_Glb) +apply (auto simp add: isLb_def setle_def setge_def) +done + +lemma rinf_ge: assumes Se: "S \ {}" and Sb: "b <=* S" shows "rinf S \ b" +proof- + from Sb have bu: "isLb UNIV S b" by (simp add: isLb_def setge_def) + from rinf[OF Se] Sb have "isGlb UNIV S (rinf S)" by blast + then show ?thesis using bu by (auto simp add: isGlb_def greatestP_def setle_def setge_def) +qed + +lemma rinf_finite_Min: assumes fS: "finite S" and Se: "S \ {}" + shows "rinf S = Min S" +using fS Se +proof- + let ?m = "Min S" + from Min_le[OF fS] have Sm: "\ x\ S. x \ ?m" by metis + with rinf[OF Se] have glb: "isGlb UNIV S (rinf S)" by (metis setge_def) + from Min_in[OF fS Se] glb have mrS: "?m \ rinf S" + by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def) + moreover + have "rinf S \ ?m" using Sm glb + by (auto simp add: isGlb_def greatestP_def isLb_def setle_def setge_def) + ultimately show ?thesis by arith +qed + +lemma rinf_finite_in: assumes fS: "finite S" and Se: "S \ {}" + shows "rinf S \ S" + using rinf_finite_Min[OF fS Se] Min_in[OF fS Se] by metis + +lemma rinf_finite_Lb: assumes fS: "finite S" and Se: "S \ {}" + shows "isLb S S (rinf S)" + using rinf_finite_Min[OF fS Se] rinf_finite_in[OF fS Se] Min_le[OF fS] + unfolding isLb_def setge_def by metis + +lemma rinf_finite_ge_iff: assumes fS: "finite S" and Se: "S \ {}" + shows "a \ rinf S \ (\ x \ S. a \ x)" +using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def) + +lemma rinf_finite_le_iff: assumes fS: "finite S" and Se: "S \ {}" + shows "a \ rinf S \ (\ x \ S. a \ x)" +using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def) + +lemma rinf_finite_gt_iff: assumes fS: "finite S" and Se: "S \ {}" + shows "a < rinf S \ (\ x \ S. a < x)" +using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def) + +lemma rinf_finite_lt_iff: assumes fS: "finite S" and Se: "S \ {}" + shows "a > rinf S \ (\ x \ S. a > x)" +using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def) + +lemma rinf_unique: assumes b: "b <=* S" and S: "\b' > b. \x \ S. b' > x" + shows "rinf S = b" +using b S +unfolding setge_def rinf_alt +apply - +apply (rule some_equality) +apply (metis linorder_not_le order_eq_iff[symmetric])+ +done + +lemma rinf_ge_subset: "S\{} \ S \ T \ (\b. b <=* T) \ rinf S >= rinf T" + apply (rule rinf_ge) + apply simp + using rinf[of T] by (auto simp add: isGlb_def greatestP_def setge_def setle_def isLb_def) + +lemma isLb_def': "isLb R S = (\x. x <=* S \ x \ R)" + apply (rule ext) + by (metis isLb_def) + +lemma rinf_bounds: assumes Se: "S \ {}" and l: "a <=* S" and u: "S *<= b" + shows "a \ rinf S \ rinf S \ b" +proof- + from rinf[OF Se] l have lub: "isGlb UNIV S (rinf S)" by blast + hence b: "a \ rinf S" using l by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def') + from Se obtain y where y: "y \ S" by blast + from lub u have "b \ rinf S" apply (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def') + apply (erule ballE[where x=y]) + apply (erule ballE[where x=y]) + apply arith + using y apply auto + done + with b show ?thesis by blast +qed + +lemma rinf_abs_ge: "S \ {} \ (\x\S. \x\ \ a) \ \rinf S\ \ a" + unfolding abs_le_interval_iff using rinf_bounds[of S "-a" a] + by (auto simp add: setge_def setle_def) + +lemma rinf_asclose: assumes S:"S \ {}" and b: "\x\S. \x - l\ \ e" shows "\rinf S - l\ \ e" +proof- + have th: "\(x::real) l e. \x - l\ \ e \ l - e \ x \ x \ l + e" by arith + show ?thesis using S b rinf_bounds[of S "l - e" "l+e"] unfolding th + by (auto simp add: setge_def setle_def) +qed + + + +subsection{* Operator norm. *} + +definition "onorm f = rsup {norm (f x)| x. norm x = 1}" + +lemma norm_bound_generalize: + fixes f:: "real ^'n \ real^'m" + assumes lf: "linear f" + shows "(\x. norm x = 1 \ norm (f x) \ b) \ (\x. norm (f x) \ b * norm x)" (is "?lhs \ ?rhs") +proof- + {assume H: ?rhs + {fix x :: "real^'n" assume x: "norm x = 1" + from H[rule_format, of x] x have "norm (f x) \ b" by simp} + then have ?lhs by blast } + + moreover + {assume H: ?lhs + from H[rule_format, of "basis 1"] + have bp: "b \ 0" using norm_pos_le[of "f (basis 1)"] dimindex_ge_1[of "UNIV:: 'n set"] + by (auto simp add: norm_basis) + {fix x :: "real ^'n" + {assume "x = 0" + then have "norm (f x) \ b * norm x" by (simp add: linear_0[OF lf] norm_0 bp)} + moreover + {assume x0: "x \ 0" + hence n0: "norm x \ 0" by (metis norm_eq_0) + let ?c = "1/ norm x" + have "norm (?c*s x) = 1" by (simp add: n0 norm_mul) + with H have "norm (f(?c*s x)) \ b" by blast + hence "?c * norm (f x) \ b" + by (simp add: linear_cmul[OF lf] norm_mul) + hence "norm (f x) \ b * norm x" + using n0 norm_pos_le[of x] by (auto simp add: field_simps)} + ultimately have "norm (f x) \ b * norm x" by blast} + then have ?rhs by blast} + ultimately show ?thesis by blast +qed + +lemma onorm: + fixes f:: "real ^'n \ real ^'m" + assumes lf: "linear f" + shows "norm (f x) <= onorm f * norm x" + and "\x. norm (f x) <= b * norm x \ onorm f <= b" +proof- + { + let ?S = "{norm (f x) |x. norm x = 1}" + have Se: "?S \ {}" using norm_basis_1 by auto + from linear_bounded[OF lf] have b: "\ b. ?S *<= b" + unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def) + {from rsup[OF Se b, unfolded onorm_def[symmetric]] + show "norm (f x) <= onorm f * norm x" + apply - + apply (rule spec[where x = x]) + unfolding norm_bound_generalize[OF lf, symmetric] + by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)} + { + show "\x. norm (f x) <= b * norm x \ onorm f <= b" + using rsup[OF Se b, unfolded onorm_def[symmetric]] + unfolding norm_bound_generalize[OF lf, symmetric] + by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)} + } +qed + +lemma onorm_pos_le: assumes lf: "linear (f::real ^'n \ real ^'m)" shows "0 <= onorm f" + using order_trans[OF norm_pos_le onorm(1)[OF lf, of "basis 1"], unfolded norm_basis_1] by simp + +lemma onorm_eq_0: assumes lf: "linear (f::real ^'n \ real ^'m)" + shows "onorm f = 0 \ (\x. f x = 0)" + using onorm[OF lf] + apply (auto simp add: norm_0 onorm_pos_le norm_le_0) + apply atomize + apply (erule allE[where x="0::real"]) + using onorm_pos_le[OF lf] + apply arith + done + +lemma onorm_const: "onorm(\x::real^'n. (y::real ^ 'm)) = norm y" +proof- + let ?f = "\x::real^'n. (y::real ^ 'm)" + have th: "{norm (?f x)| x. norm x = 1} = {norm y}" + by(auto intro: vector_choose_size set_ext) + show ?thesis + unfolding onorm_def th + apply (rule rsup_unique) by (simp_all add: setle_def) +qed + +lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n \ real ^'m)" + shows "0 < onorm f \ ~(\x. f x = 0)" + unfolding onorm_eq_0[OF lf, symmetric] + using onorm_pos_le[OF lf] by arith + +lemma onorm_compose: + assumes lf: "linear (f::real ^'n \ real ^'m)" and lg: "linear g" + shows "onorm (f o g) <= onorm f * onorm g" + apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format]) + unfolding o_def + apply (subst mult_assoc) + apply (rule order_trans) + apply (rule onorm(1)[OF lf]) + apply (rule mult_mono1) + apply (rule onorm(1)[OF lg]) + apply (rule onorm_pos_le[OF lf]) + done + +lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n \ real^'m)" + shows "onorm (\x. - f x) \ onorm f" + using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf] + unfolding norm_neg by metis + +lemma onorm_neg: assumes lf: "linear (f::real ^'n \ real^'m)" + shows "onorm (\x. - f x) = onorm f" + using onorm_neg_lemma[OF lf] onorm_neg_lemma[OF linear_compose_neg[OF lf]] + by simp + +lemma onorm_triangle: + assumes lf: "linear (f::real ^'n \ real ^'m)" and lg: "linear g" + shows "onorm (\x. f x + g x) <= onorm f + onorm g" + apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format]) + apply (rule order_trans) + apply (rule norm_triangle) + apply (simp add: distrib) + apply (rule add_mono) + apply (rule onorm(1)[OF lf]) + apply (rule onorm(1)[OF lg]) + done + +lemma onorm_triangle_le: "linear (f::real ^'n \ real ^'m) \ linear g \ onorm(f) + onorm(g) <= e + \ onorm(\x. f x + g x) <= e" + apply (rule order_trans) + apply (rule onorm_triangle) + apply assumption+ + done + +lemma onorm_triangle_lt: "linear (f::real ^'n \ real ^'m) \ linear g \ onorm(f) + onorm(g) < e + ==> onorm(\x. f x + g x) < e" + apply (rule order_le_less_trans) + apply (rule onorm_triangle) + by assumption+ + +(* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *) + +definition vec1:: "'a \ 'a ^ 1" where "vec1 x = (\ i. x)" + +definition dest_vec1:: "'a ^1 \ 'a" + where "dest_vec1 x = (x$1)" + +lemma vec1_component[simp]: "(vec1 x)$1 = x" + by (simp add: vec1_def) + +lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y" + by (simp_all add: vec1_def dest_vec1_def Cart_eq Cart_lambda_beta dimindex_def del: One_nat_def) + +lemma forall_vec1: "(\x. P x) \ (\x. P (vec1 x))" by (metis vec1_dest_vec1) + +lemma exists_vec1: "(\x. P x) \ (\x. P(vec1 x))" by (metis vec1_dest_vec1) + +lemma forall_dest_vec1: "(\x. P x) \ (\x. P(dest_vec1 x))" by (metis vec1_dest_vec1) + +lemma exists_dest_vec1: "(\x. P x) \ (\x. P(dest_vec1 x))"by (metis vec1_dest_vec1) + +lemma vec1_eq[simp]: "vec1 x = vec1 y \ x = y" by (metis vec1_dest_vec1) + +lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \ x = y" by (metis vec1_dest_vec1) + +lemma vec1_in_image_vec1: "vec1 x \ (vec1 ` S) \ x \ S" by auto + +lemma vec1_vec: "vec1 x = vec x" by (vector vec1_def) + +lemma vec1_add: "vec1(x + y) = vec1 x + vec1 y" by (vector vec1_def) +lemma vec1_sub: "vec1(x - y) = vec1 x - vec1 y" by (vector vec1_def) +lemma vec1_cmul: "vec1(c* x) = c *s vec1 x " by (vector vec1_def) +lemma vec1_neg: "vec1(- x) = - vec1 x " by (vector vec1_def) + +lemma vec1_setsum: assumes fS: "finite S" + shows "vec1(setsum f S) = setsum (vec1 o f) S" + apply (induct rule: finite_induct[OF fS]) + apply (simp add: vec1_vec) + apply (auto simp add: vec1_add) + done + +lemma dest_vec1_lambda: "dest_vec1(\ i. x i) = x 1" + by (simp add: dest_vec1_def) + +lemma dest_vec1_vec: "dest_vec1(vec x) = x" + by (simp add: vec1_vec[symmetric]) + +lemma dest_vec1_add: "dest_vec1(x + y) = dest_vec1 x + dest_vec1 y" + by (metis vec1_dest_vec1 vec1_add) + +lemma dest_vec1_sub: "dest_vec1(x - y) = dest_vec1 x - dest_vec1 y" + by (metis vec1_dest_vec1 vec1_sub) + +lemma dest_vec1_cmul: "dest_vec1(c*sx) = c * dest_vec1 x" + by (metis vec1_dest_vec1 vec1_cmul) + +lemma dest_vec1_neg: "dest_vec1(- x) = - dest_vec1 x" + by (metis vec1_dest_vec1 vec1_neg) + +lemma dest_vec1_0[simp]: "dest_vec1 0 = 0" by (metis vec_0 dest_vec1_vec) + +lemma dest_vec1_sum: assumes fS: "finite S" + shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S" + apply (induct rule: finite_induct[OF fS]) + apply (simp add: dest_vec1_vec) + apply (auto simp add: dest_vec1_add) + done + +lemma norm_vec1: "norm(vec1 x) = abs(x)" + by (simp add: vec1_def norm_real) + +lemma dist_vec1: "dist(vec1 x) (vec1 y) = abs(x - y)" + by (simp only: dist_real vec1_component) +lemma abs_dest_vec1: "norm x = \dest_vec1 x\" + by (metis vec1_dest_vec1 norm_vec1) + +lemma linear_vmul_dest_vec1: + fixes f:: "'a::semiring_1^'n \ 'a^1" + shows "linear f \ linear (\x. dest_vec1(f x) *s v)" + unfolding dest_vec1_def + apply (rule linear_vmul_component) + by (auto simp add: dimindex_def) + +lemma linear_from_scalars: + assumes lf: "linear (f::'a::comm_ring_1 ^1 \ 'a^'n)" + shows "f = (\x. dest_vec1 x *s column 1 (matrix f))" + apply (rule ext) + apply (subst matrix_works[OF lf, symmetric]) + 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 ) + done + +lemma linear_to_scalars: assumes lf: "linear (f::'a::comm_ring_1 ^'n \ 'a^1)" + shows "f = (\x. vec1(row 1 (matrix f) \ x))" + apply (rule ext) + apply (subst matrix_works[OF lf, symmetric]) + 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) + done + +lemma dest_vec1_eq_0: "dest_vec1 x = 0 \ x = 0" + by (simp add: dest_vec1_eq[symmetric]) + +lemma setsum_scalars: assumes fS: "finite S" + shows "setsum f S = vec1 (setsum (dest_vec1 o f) S)" + unfolding vec1_setsum[OF fS] by simp + +lemma dest_vec1_wlog_le: "(\(x::'a::linorder ^ 1) y. P x y \ P y x) \ (\x y. dest_vec1 x <= dest_vec1 y ==> P x y) \ P x y" + apply (cases "dest_vec1 x \ dest_vec1 y") + apply simp + apply (subgoal_tac "dest_vec1 y \ dest_vec1 x") + apply (auto) + done + +text{* Pasting vectors. *} + +lemma linear_fstcart: "linear fstcart" + by (auto simp add: linear_def fstcart_def Cart_eq Cart_lambda_beta vector_component dimindex_finite_sum) + +lemma linear_sndcart: "linear sndcart" + by (auto simp add: linear_def sndcart_def Cart_eq Cart_lambda_beta vector_component dimindex_finite_sum) + +lemma fstcart_vec[simp]: "fstcart(vec x) = vec x" + by (vector fstcart_def vec_def dimindex_finite_sum) + +lemma fstcart_add[simp]:"fstcart(x + y) = fstcart (x::'a::{plus,times}^('b,'c) finite_sum) + fstcart y" + using linear_fstcart[unfolded linear_def] by blast + +lemma fstcart_cmul[simp]:"fstcart(c*s x) = c*s fstcart (x::'a::{plus,times}^('b,'c) finite_sum)" + using linear_fstcart[unfolded linear_def] by blast + +lemma fstcart_neg[simp]:"fstcart(- x) = - fstcart (x::'a::ring_1^('b,'c) finite_sum)" +unfolding vector_sneg_minus1 fstcart_cmul .. + +lemma fstcart_sub[simp]:"fstcart(x - y) = fstcart (x::'a::ring_1^('b,'c) finite_sum) - fstcart y" + unfolding diff_def fstcart_add fstcart_neg .. + +lemma fstcart_setsum: + fixes f:: "'d \ 'a::semiring_1^_" + assumes fS: "finite S" + shows "fstcart (setsum f S) = setsum (\i. fstcart (f i)) S" + by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0) + +lemma sndcart_vec[simp]: "sndcart(vec x) = vec x" + by (vector sndcart_def vec_def dimindex_finite_sum) + +lemma sndcart_add[simp]:"sndcart(x + y) = sndcart (x::'a::{plus,times}^('b,'c) finite_sum) + sndcart y" + using linear_sndcart[unfolded linear_def] by blast + +lemma sndcart_cmul[simp]:"sndcart(c*s x) = c*s sndcart (x::'a::{plus,times}^('b,'c) finite_sum)" + using linear_sndcart[unfolded linear_def] by blast + +lemma sndcart_neg[simp]:"sndcart(- x) = - sndcart (x::'a::ring_1^('b,'c) finite_sum)" +unfolding vector_sneg_minus1 sndcart_cmul .. + +lemma sndcart_sub[simp]:"sndcart(x - y) = sndcart (x::'a::ring_1^('b,'c) finite_sum) - sndcart y" + unfolding diff_def sndcart_add sndcart_neg .. + +lemma sndcart_setsum: + fixes f:: "'d \ 'a::semiring_1^_" + assumes fS: "finite S" + shows "sndcart (setsum f S) = setsum (\i. sndcart (f i)) S" + by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0) + +lemma pastecart_vec[simp]: "pastecart (vec x) (vec x) = vec x" + by (simp add: pastecart_eq fstcart_vec sndcart_vec fstcart_pastecart sndcart_pastecart) + +lemma pastecart_add[simp]:"pastecart (x1::'a::{plus,times}^_) y1 + pastecart x2 y2 = pastecart (x1 + x2) (y1 + y2)" + by (simp add: pastecart_eq fstcart_add sndcart_add fstcart_pastecart sndcart_pastecart) + +lemma pastecart_cmul[simp]: "pastecart (c *s (x1::'a::{plus,times}^_)) (c *s y1) = c *s pastecart x1 y1" + by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart) + +lemma pastecart_neg[simp]: "pastecart (- (x::'a::ring_1^_)) (- y) = - pastecart x y" + unfolding vector_sneg_minus1 pastecart_cmul .. + +lemma pastecart_sub: "pastecart (x1::'a::ring_1^_) y1 - pastecart x2 y2 = pastecart (x1 - x2) (y1 - y2)" + by (simp add: diff_def pastecart_neg[symmetric] del: pastecart_neg) + +lemma pastecart_setsum: + fixes f:: "'d \ 'a::semiring_1^_" + assumes fS: "finite S" + shows "pastecart (setsum f S) (setsum g S) = setsum (\i. pastecart (f i) (g i)) S" + by (simp add: pastecart_eq fstcart_setsum[OF fS] sndcart_setsum[OF fS] fstcart_pastecart sndcart_pastecart) + +lemma norm_fstcart: "norm(fstcart x) <= norm (x::real ^('n,'m) finite_sum)" +proof- + let ?n = "dimindex (UNIV :: 'n set)" + let ?m = "dimindex (UNIV :: 'm set)" + let ?N = "{1 .. ?n}" + let ?M = "{1 .. ?m}" + let ?NM = "{1 .. dimindex (UNIV :: ('n,'m) finite_sum set)}" + have th_0: "1 \ ?n +1" by simp + have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))" + by (simp add: pastecart_fst_snd) + have th1: "fstcart x \ fstcart x \ pastecart (fstcart x) (sndcart x) \ pastecart (fstcart x) (sndcart x)" + 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) + then show ?thesis + unfolding th0 + unfolding real_vector_norm_def real_sqrt_le_iff real_of_real_def id_def + by (simp add: dot_def dimindex_finite_sum Cart_lambda_beta) +qed + +lemma dist_fstcart: "dist(fstcart (x::real^_)) (fstcart y) <= dist x y" + by (metis dist_def fstcart_sub[symmetric] norm_fstcart) + +lemma norm_sndcart: "norm(sndcart x) <= norm (x::real ^('n,'m) finite_sum)" +proof- + let ?n = "dimindex (UNIV :: 'n set)" + let ?m = "dimindex (UNIV :: 'm set)" + let ?N = "{1 .. ?n}" + let ?M = "{1 .. ?m}" + let ?nm = "dimindex (UNIV :: ('n,'m) finite_sum set)" + let ?NM = "{1 .. ?nm}" + have thnm[simp]: "?nm = ?n + ?m" by (simp add: dimindex_finite_sum) + have th_0: "1 \ ?n +1" by simp + have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))" + by (simp add: pastecart_fst_snd) + let ?f = "\n. n - ?n" + let ?S = "{?n+1 .. ?nm}" + have finj:"inj_on ?f ?S" + using dimindex_nonzero[of "UNIV :: 'n set"] dimindex_nonzero[of "UNIV :: 'm set"] + apply (simp add: Ball_def atLeastAtMost_iff inj_on_def dimindex_finite_sum del: One_nat_def) + by arith + have fS: "?f ` ?S = ?M" + apply (rule set_ext) + apply (simp add: image_iff Bex_def) using dimindex_nonzero[of "UNIV :: 'n set"] dimindex_nonzero[of "UNIV :: 'm set"] by arith + have th1: "sndcart x \ sndcart x \ pastecart (fstcart x) (sndcart x) \ pastecart (fstcart x) (sndcart x)" + 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) + then show ?thesis + unfolding th0 + unfolding real_vector_norm_def real_sqrt_le_iff real_of_real_def id_def + by (simp add: dot_def dimindex_finite_sum Cart_lambda_beta) +qed + +lemma dist_sndcart: "dist(sndcart (x::real^_)) (sndcart y) <= dist x y" + by (metis dist_def sndcart_sub[symmetric] norm_sndcart) + +lemma dot_pastecart: "(pastecart (x1::'a::{times,comm_monoid_add}^'n) (x2::'a::{times,comm_monoid_add}^'m)) \ (pastecart y1 y2) = x1 \ y1 + x2 \ y2" +proof- + let ?n = "dimindex (UNIV :: 'n set)" + let ?m = "dimindex (UNIV :: 'm set)" + let ?N = "{1 .. ?n}" + let ?M = "{1 .. ?m}" + let ?nm = "dimindex (UNIV :: ('n,'m) finite_sum set)" + let ?NM = "{1 .. ?nm}" + have thnm: "?nm = ?n + ?m" by (simp add: dimindex_finite_sum) + have th_0: "1 \ ?n +1" by simp + have th_1: "\i. i \ {?m + 1 .. ?nm} \ i - ?m \ ?N" apply (simp add: thnm) by arith + let ?f = "\a b i. (a$i) * (b$i)" + let ?g = "?f (pastecart x1 x2) (pastecart y1 y2)" + let ?S = "{?n +1 .. ?nm}" + {fix i + assume i: "i \ ?N" + have "?g i = ?f x1 y1 i" + using i + apply (simp add: pastecart_def Cart_lambda_beta thnm) done + } + hence th2: "setsum ?g ?N = setsum (?f x1 y1) ?N" + apply - + apply (rule setsum_cong) + apply auto + done + {fix i + assume i: "i \ ?S" + have "?g i = ?f x2 y2 (i - ?n)" + using i + apply (simp add: pastecart_def Cart_lambda_beta thnm) done + } + hence th3: "setsum ?g ?S = setsum (\i. ?f x2 y2 (i -?n)) ?S" + apply - + apply (rule setsum_cong) + apply auto + done + let ?r = "\n. n - ?n" + have rinj: "inj_on ?r ?S" apply (simp add: inj_on_def Ball_def thnm) by arith + have rS: "?r ` ?S = ?M" apply (rule set_ext) + apply (simp add: thnm image_iff Bex_def) by arith + have "pastecart x1 x2 \ (pastecart y1 y2) = setsum ?g ?NM" by (simp add: dot_def) + also have "\ = setsum ?g ?N + setsum ?g ?S" + by (simp add: dot_def thnm setsum_add_split[OF th_0, of _ ?m] del: One_nat_def) + also have "\ = setsum (?f x1 y1) ?N + setsum (?f x2 y2) ?M" + unfolding setsum_reindex[OF rinj, unfolded rS o_def] th2 th3 .. + finally + show ?thesis by (simp add: dot_def) +qed + +lemma norm_pastecart: "norm(pastecart x y) <= norm(x :: real ^ _) + norm(y)" + unfolding real_vector_norm_def dot_pastecart real_sqrt_le_iff real_of_real_def id_def + apply (rule power2_le_imp_le) + apply (simp add: real_sqrt_pow2[OF add_nonneg_nonneg[OF dot_pos_le[of x] dot_pos_le[of y]]]) + apply (auto simp add: power2_eq_square ring_simps) + apply (simp add: power2_eq_square[symmetric]) + apply (rule mult_nonneg_nonneg) + apply (simp_all add: real_sqrt_pow2[OF dot_pos_le]) + apply (rule add_nonneg_nonneg) + apply (simp_all add: real_sqrt_pow2[OF dot_pos_le]) + done + +subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *} + +definition hull :: "'a set set \ 'a set \ 'a set" (infixl "hull" 75) where + "S hull s = Inter {t. t \ S \ s \ t}" + +lemma hull_same: "s \ S \ S hull s = s" + unfolding hull_def by auto + +lemma hull_in: "(\T. T \ S ==> Inter T \ S) ==> (S hull s) \ S" +unfolding hull_def subset_iff by auto + +lemma hull_eq: "(\T. T \ S ==> Inter T \ S) ==> (S hull s) = s \ s \ S" +using hull_same[of s S] hull_in[of S s] by metis + + +lemma hull_hull: "S hull (S hull s) = S hull s" + unfolding hull_def by blast + +lemma hull_subset: "s \ (S hull s)" + unfolding hull_def by blast + +lemma hull_mono: " s \ t ==> (S hull s) \ (S hull t)" + unfolding hull_def by blast + +lemma hull_antimono: "S \ T ==> (T hull s) \ (S hull s)" + unfolding hull_def by blast + +lemma hull_minimal: "s \ t \ t \ S ==> (S hull s) \ t" + unfolding hull_def by blast + +lemma subset_hull: "t \ S ==> S hull s \ t \ s \ t" + unfolding hull_def by blast + +lemma hull_unique: "s \ t \ t \ S \ (\t'. s \ t' \ t' \ S ==> t \ t') + ==> (S hull s = t)" +unfolding hull_def by auto + +lemma hull_induct: "(\x. x\ S \ P x) \ Q {x. P x} \ \x\ Q hull S. P x" + using hull_minimal[of S "{x. P x}" Q] + by (auto simp add: subset_eq Collect_def mem_def) + +lemma hull_inc: "x \ S \ x \ P hull S" by (metis hull_subset subset_eq) + +lemma hull_union_subset: "(S hull s) \ (S hull t) \ (S hull (s \ t))" +unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2) + +lemma hull_union: assumes T: "\T. T \ S ==> Inter T \ S" + shows "S hull (s \ t) = S hull (S hull s \ S hull t)" +apply rule +apply (rule hull_mono) +unfolding Un_subset_iff +apply (metis hull_subset Un_upper1 Un_upper2 subset_trans) +apply (rule hull_minimal) +apply (metis hull_union_subset) +apply (metis hull_in T) +done + +lemma hull_redundant_eq: "a \ (S hull s) \ (S hull (insert a s) = S hull s)" + unfolding hull_def by blast + +lemma hull_redundant: "a \ (S hull s) ==> (S hull (insert a s) = S hull s)" +by (metis hull_redundant_eq) + +text{* Archimedian properties and useful consequences. *} + +lemma real_arch_simple: "\n. x <= real (n::nat)" + using reals_Archimedean2[of x] apply auto by (rule_tac x="Suc n" in exI, auto) +lemmas real_arch_lt = reals_Archimedean2 + +lemmas real_arch = reals_Archimedean3 + +lemma real_arch_inv: "0 < e \ (\n::nat. n \ 0 \ 0 < inverse (real n) \ inverse (real n) < e)" + using reals_Archimedean + apply (auto simp add: field_simps inverse_positive_iff_positive) + apply (subgoal_tac "inverse (real n) > 0") + apply arith + apply simp + done + +lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n" +proof(induct n) + case 0 thus ?case by simp +next + case (Suc n) + hence h: "1 + real n * x \ (1 + x) ^ n" by simp + from h have p: "1 \ (1 + x) ^ n" using Suc.prems by simp + from h have "1 + real n * x + x \ (1 + x) ^ n + x" by simp + also have "\ \ (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric]) + apply (simp add: ring_simps) + using mult_left_mono[OF p Suc.prems] by simp + finally show ?case by (simp add: real_of_nat_Suc ring_simps) +qed + +lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\n. y < x^n" +proof- + from x have x0: "x - 1 > 0" by arith + from real_arch[OF x0, rule_format, of y] + obtain n::nat where n:"y < real n * (x - 1)" by metis + from x0 have x00: "x- 1 \ 0" by arith + from real_pow_lbound[OF x00, of n] n + have "y < x^n" by auto + then show ?thesis by metis +qed + +lemma real_arch_pow2: "\n. (x::real) < 2^ n" + using real_arch_pow[of 2 x] by simp + +lemma real_arch_pow_inv: assumes y: "(y::real) > 0" and x1: "x < 1" + shows "\n. x^n < y" +proof- + {assume x0: "x > 0" + from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps) + from real_arch_pow[OF ix, of "1/y"] + obtain n where n: "1/y < (1/x)^n" by blast + then + have ?thesis using y x0 by (auto simp add: field_simps power_divide) } + moreover + {assume "\ x > 0" with y x1 have ?thesis apply auto by (rule exI[where x=1], auto)} + ultimately show ?thesis by metis +qed + +lemma forall_pos_mono: "(\d e::real. d < e \ P d ==> P e) \ (\n::nat. n \ 0 ==> P(inverse(real n))) \ (\e. 0 < e ==> P e)" + by (metis real_arch_inv) + +lemma forall_pos_mono_1: "(\d e::real. d < e \ P d ==> P e) \ (\n. P(inverse(real (Suc n)))) ==> 0 < e ==> P e" + apply (rule forall_pos_mono) + apply auto + apply (atomize) + apply (erule_tac x="n - 1" in allE) + apply auto + done + +lemma real_archimedian_rdiv_eq_0: assumes x0: "x \ 0" and c: "c \ 0" and xc: "\(m::nat)>0. real m * x \ c" + shows "x = 0" +proof- + {assume "x \ 0" with x0 have xp: "x > 0" by arith + from real_arch[OF xp, rule_format, of c] obtain n::nat where n: "c < real n * x" by blast + with xc[rule_format, of n] have "n = 0" by arith + with n c have False by simp} + then show ?thesis by blast +qed + +(* ------------------------------------------------------------------------- *) +(* Relate max and min to sup and inf. *) +(* ------------------------------------------------------------------------- *) + +lemma real_max_rsup: "max x y = rsup {x,y}" +proof- + have f: "finite {x, y}" "{x,y} \ {}" by simp_all + from rsup_finite_le_iff[OF f, of "max x y"] have "rsup {x,y} \ max x y" by simp + moreover + have "max x y \ rsup {x,y}" using rsup_finite_ge_iff[OF f, of "max x y"] + by (simp add: linorder_linear) + ultimately show ?thesis by arith +qed + +lemma real_min_rinf: "min x y = rinf {x,y}" +proof- + have f: "finite {x, y}" "{x,y} \ {}" by simp_all + from rinf_finite_le_iff[OF f, of "min x y"] have "rinf {x,y} \ min x y" + by (simp add: linorder_linear) + moreover + have "min x y \ rinf {x,y}" using rinf_finite_ge_iff[OF f, of "min x y"] + by simp + ultimately show ?thesis by arith +qed + +(* ------------------------------------------------------------------------- *) +(* Geometric progression. *) +(* ------------------------------------------------------------------------- *) + +lemma sum_gp_basic: "((1::'a::{field, recpower}) - x) * setsum (\i. x^i) {0 .. n} = (1 - x^(Suc n))" + (is "?lhs = ?rhs") +proof- + {assume x1: "x = 1" hence ?thesis by simp} + moreover + {assume x1: "x\1" + hence x1': "x - 1 \ 0" "1 - x \ 0" "x - 1 = - (1 - x)" "- (1 - x) \ 0" by auto + from geometric_sum[OF x1, of "Suc n", unfolded x1'] + have "(- (1 - x)) * setsum (\i. x^i) {0 .. n} = - (1 - x^(Suc n))" + unfolding atLeastLessThanSuc_atLeastAtMost + using x1' apply (auto simp only: field_simps) + apply (simp add: ring_simps) + done + then have ?thesis by (simp add: ring_simps) } + ultimately show ?thesis by metis +qed + +lemma sum_gp_multiplied: assumes mn: "m <= n" + shows "((1::'a::{field, recpower}) - x) * setsum (op ^ x) {m..n} = x^m - x^ Suc n" + (is "?lhs = ?rhs") +proof- + let ?S = "{0..(n - m)}" + from mn have mn': "n - m \ 0" by arith + let ?f = "op + m" + have i: "inj_on ?f ?S" unfolding inj_on_def by auto + have f: "?f ` ?S = {m..n}" + using mn apply (auto simp add: image_iff Bex_def) by arith + have th: "op ^ x o op + m = (\i. x^m * x^i)" + by (rule ext, simp add: power_add power_mult) + from setsum_reindex[OF i, of "op ^ x", unfolded f th setsum_right_distrib[symmetric]] + have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})" by simp + then show ?thesis unfolding sum_gp_basic using mn + by (simp add: ring_simps power_add[symmetric]) +qed + +lemma sum_gp: "setsum (op ^ (x::'a::{field, recpower})) {m .. n} = + (if n < m then 0 else if x = 1 then of_nat ((n + 1) - m) + else (x^ m - x^ (Suc n)) / (1 - x))" +proof- + {assume nm: "n < m" hence ?thesis by simp} + moreover + {assume "\ n < m" hence nm: "m \ n" by arith + {assume x: "x = 1" hence ?thesis by simp} + moreover + {assume x: "x \ 1" hence nz: "1 - x \ 0" by simp + from sum_gp_multiplied[OF nm, of x] nz have ?thesis by (simp add: field_simps)} + ultimately have ?thesis by metis + } + ultimately show ?thesis by metis +qed + +lemma sum_gp_offset: "setsum (op ^ (x::'a::{field,recpower})) {m .. m+n} = + (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))" + unfolding sum_gp[of x m "m + n"] power_Suc + by (simp add: ring_simps power_add) + + +subsection{* A bit of linear algebra. *} + +definition "subspace S \ 0 \ S \ (\x\ S. \y \S. x + y \ S) \ (\c. \x \S. c *s x \S )" +definition "span S = (subspace hull S)" +definition "dependent S \ (\a \ S. a \ span(S - {a}))" +abbreviation "independent s == ~(dependent s)" + +(* Closure properties of subspaces. *) + +lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def) + +lemma subspace_0: "subspace S ==> 0 \ S" by (metis subspace_def) + +lemma subspace_add: "subspace S \ x \ S \ y \ S ==> x + y \ S" + by (metis subspace_def) + +lemma subspace_mul: "subspace S \ x \ S \ c *s x \ S" + by (metis subspace_def) + +lemma subspace_neg: "subspace S \ (x::'a::ring_1^'n) \ S \ - x \ S" + by (metis vector_sneg_minus1 subspace_mul) + +lemma subspace_sub: "subspace S \ (x::'a::ring_1^'n) \ S \ y \ S \ x - y \ S" + by (metis diff_def subspace_add subspace_neg) + +lemma subspace_setsum: + assumes sA: "subspace A" and fB: "finite B" + and f: "\x\ B. f x \ A" + shows "setsum f B \ A" + using fB f sA + apply(induct rule: finite_induct[OF fB]) + by (simp add: subspace_def sA, auto simp add: sA subspace_add) + +lemma subspace_linear_image: + assumes lf: "linear (f::'a::semiring_1^'n \ _)" and sS: "subspace S" + shows "subspace(f ` S)" + using lf sS linear_0[OF lf] + unfolding linear_def subspace_def + apply (auto simp add: image_iff) + apply (rule_tac x="x + y" in bexI, auto) + apply (rule_tac x="c*s x" in bexI, auto) + done + +lemma subspace_linear_preimage: "linear (f::'a::semiring_1^'n \ _) ==> subspace S ==> subspace {x. f x \ S}" + by (auto simp add: subspace_def linear_def linear_0[of f]) + +lemma subspace_trivial: "subspace {0::'a::semiring_1 ^_}" + by (simp add: subspace_def) + +lemma subspace_inter: "subspace A \ subspace B ==> subspace (A \ B)" + by (simp add: subspace_def) + + +lemma span_mono: "A \ B ==> span A \ span B" + by (metis span_def hull_mono) + +lemma subspace_span: "subspace(span S)" + unfolding span_def + apply (rule hull_in[unfolded mem_def]) + apply (simp only: subspace_def Inter_iff Int_iff subset_eq) + apply auto + apply (erule_tac x="X" in ballE) + apply (simp add: mem_def) + apply blast + apply (erule_tac x="X" in ballE) + apply (erule_tac x="X" in ballE) + apply (erule_tac x="X" in ballE) + apply (clarsimp simp add: mem_def) + apply simp + apply simp + apply simp + apply (erule_tac x="X" in ballE) + apply (erule_tac x="X" in ballE) + apply (simp add: mem_def) + apply simp + apply simp + done + +lemma span_clauses: + "a \ S ==> a \ span S" + "0 \ span S" + "x\ span S \ y \ span S ==> x + y \ span S" + "x \ span S \ c *s x \ span S" + by (metis span_def hull_subset subset_eq subspace_span subspace_def)+ + +lemma span_induct: assumes SP: "\x. x \ S ==> P x" + and P: "subspace P" and x: "x \ span S" shows "P x" +proof- + from SP have SP': "S \ P" by (simp add: mem_def subset_eq) + from P have P': "P \ subspace" by (simp add: mem_def) + from x hull_minimal[OF SP' P', unfolded span_def[symmetric]] + show "P x" by (metis mem_def subset_eq) +qed + +lemma span_empty: "span {} = {(0::'a::semiring_0 ^ 'n)}" + apply (simp add: span_def) + apply (rule hull_unique) + apply (auto simp add: mem_def subspace_def) + unfolding mem_def[of "0::'a^'n", symmetric] + apply simp + done + +lemma independent_empty: "independent {}" + by (simp add: dependent_def) + +lemma independent_mono: "independent A \ B \ A ==> independent B" + apply (clarsimp simp add: dependent_def span_mono) + apply (subgoal_tac "span (B - {a}) \ span (A - {a})") + apply force + apply (rule span_mono) + apply auto + done + +lemma span_subspace: "A \ B \ B \ span A \ subspace B \ span A = B" + by (metis order_antisym span_def hull_minimal mem_def) + +lemma span_induct': assumes SP: "\x \ S. P x" + and P: "subspace P" shows "\x \ span S. P x" + using span_induct SP P by blast + +inductive span_induct_alt_help for S:: "'a::semiring_1^'n \ bool" + where + span_induct_alt_help_0: "span_induct_alt_help S 0" + | span_induct_alt_help_S: "x \ S \ span_induct_alt_help S z \ span_induct_alt_help S (c *s x + z)" + +lemma span_induct_alt': + assumes h0: "h (0::'a::semiring_1^'n)" and hS: "\c x y. x \ S \ h y \ h (c*s x + y)" shows "\x \ span S. h x" +proof- + {fix x:: "'a^'n" assume x: "span_induct_alt_help S x" + have "h x" + apply (rule span_induct_alt_help.induct[OF x]) + apply (rule h0) + apply (rule hS, assumption, assumption) + done} + note th0 = this + {fix x assume x: "x \ span S" + + have "span_induct_alt_help S x" + proof(rule span_induct[where x=x and S=S]) + show "x \ span S" using x . + next + fix x assume xS : "x \ S" + from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1] + show "span_induct_alt_help S x" by simp + next + have "span_induct_alt_help S 0" by (rule span_induct_alt_help_0) + moreover + {fix x y assume h: "span_induct_alt_help S x" "span_induct_alt_help S y" + from h + have "span_induct_alt_help S (x + y)" + apply (induct rule: span_induct_alt_help.induct) + apply simp + unfolding add_assoc + apply (rule span_induct_alt_help_S) + apply assumption + apply simp + done} + moreover + {fix c x assume xt: "span_induct_alt_help S x" + then have "span_induct_alt_help S (c*s x)" + apply (induct rule: span_induct_alt_help.induct) + apply (simp add: span_induct_alt_help_0) + apply (simp add: vector_smult_assoc vector_add_ldistrib) + apply (rule span_induct_alt_help_S) + apply assumption + apply simp + done + } + ultimately show "subspace (span_induct_alt_help S)" + unfolding subspace_def mem_def Ball_def by blast + qed} + with th0 show ?thesis by blast +qed + +lemma span_induct_alt: + assumes h0: "h (0::'a::semiring_1^'n)" and hS: "\c x y. x \ S \ h y \ h (c*s x + y)" and x: "x \ span S" + shows "h x" +using span_induct_alt'[of h S] h0 hS x by blast + +(* Individual closure properties. *) + +lemma span_superset: "x \ S ==> x \ span S" by (metis span_clauses) + +lemma span_0: "0 \ span S" by (metis subspace_span subspace_0) + +lemma span_add: "x \ span S \ y \ span S ==> x + y \ span S" + by (metis subspace_add subspace_span) + +lemma span_mul: "x \ span S ==> (c *s x) \ span S" + by (metis subspace_span subspace_mul) + +lemma span_neg: "x \ span S ==> - (x::'a::ring_1^'n) \ span S" + by (metis subspace_neg subspace_span) + +lemma span_sub: "(x::'a::ring_1^'n) \ span S \ y \ span S ==> x - y \ span S" + by (metis subspace_span subspace_sub) + +lemma span_setsum: "finite A \ \x \ A. f x \ span S ==> setsum f A \ span S" + apply (rule subspace_setsum) + by (metis subspace_span subspace_setsum)+ + +lemma span_add_eq: "(x::'a::ring_1^'n) \ span S \ x + y \ span S \ y \ span S" + apply (auto simp only: span_add span_sub) + apply (subgoal_tac "(x + y) - x \ span S", simp) + by (simp only: span_add span_sub) + +(* Mapping under linear image. *) + +lemma span_linear_image: assumes lf: "linear (f::'a::semiring_1 ^ 'n => _)" + shows "span (f ` S) = f ` (span S)" +proof- + {fix x + assume x: "x \ span (f ` S)" + have "x \ f ` span S" + apply (rule span_induct[where x=x and S = "f ` S"]) + apply (clarsimp simp add: image_iff) + apply (frule span_superset) + apply blast + apply (simp only: mem_def) + apply (rule subspace_linear_image[OF lf]) + apply (rule subspace_span) + apply (rule x) + done} + moreover + {fix x assume x: "x \ span S" + have th0:"(\a. f a \ span (f ` S)) = {x. f x \ span (f ` S)}" apply (rule set_ext) + unfolding mem_def Collect_def .. + have "f x \ span (f ` S)" + apply (rule span_induct[where S=S]) + apply (rule span_superset) + apply simp + apply (subst th0) + apply (rule subspace_linear_preimage[OF lf subspace_span, of "f ` S"]) + apply (rule x) + done} + ultimately show ?thesis by blast +qed + +(* The key breakdown property. *) + +lemma span_breakdown: + assumes bS: "(b::'a::ring_1 ^ 'n) \ S" and aS: "a \ span S" + shows "\k. a - k*s b \ span (S - {b})" (is "?P a") +proof- + {fix x assume xS: "x \ S" + {assume ab: "x = b" + then have "?P x" + apply simp + apply (rule exI[where x="1"], simp) + by (rule span_0)} + moreover + {assume ab: "x \ b" + then have "?P x" using xS + apply - + apply (rule exI[where x=0]) + apply (rule span_superset) + by simp} + ultimately have "?P x" by blast} + moreover have "subspace ?P" + unfolding subspace_def + apply auto + apply (simp add: mem_def) + apply (rule exI[where x=0]) + using span_0[of "S - {b}"] + apply (simp add: mem_def) + apply (clarsimp simp add: mem_def) + apply (rule_tac x="k + ka" in exI) + apply (subgoal_tac "x + y - (k + ka) *s b = (x - k*s b) + (y - ka *s b)") + apply (simp only: ) + apply (rule span_add[unfolded mem_def]) + apply assumption+ + apply (vector ring_simps) + apply (clarsimp simp add: mem_def) + apply (rule_tac x= "c*k" in exI) + apply (subgoal_tac "c *s x - (c * k) *s b = c*s (x - k*s b)") + apply (simp only: ) + apply (rule span_mul[unfolded mem_def]) + apply assumption + by (vector ring_simps) + ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis +qed + +lemma span_breakdown_eq: + "(x::'a::ring_1^'n) \ span (insert a S) \ (\k. (x - k *s a) \ span S)" (is "?lhs \ ?rhs") +proof- + {assume x: "x \ span (insert a S)" + from x span_breakdown[of "a" "insert a S" "x"] + have ?rhs apply clarsimp + apply (rule_tac x= "k" in exI) + apply (rule set_rev_mp[of _ "span (S - {a})" _]) + apply assumption + apply (rule span_mono) + apply blast + done} + moreover + { fix k assume k: "x - k *s a \ span S" + have eq: "x = (x - k *s a) + k *s a" by vector + have "(x - k *s a) + k *s a \ span (insert a S)" + apply (rule span_add) + apply (rule set_rev_mp[of _ "span S" _]) + apply (rule k) + apply (rule span_mono) + apply blast + apply (rule span_mul) + apply (rule span_superset) + apply blast + done + then have ?lhs using eq by metis} + ultimately show ?thesis by blast +qed + +(* Hence some "reversal" results.*) + +lemma in_span_insert: + assumes a: "(a::'a::field^'n) \ span (insert b S)" and na: "a \ span S" + shows "b \ span (insert a S)" +proof- + from span_breakdown[of b "insert b S" a, OF insertI1 a] + obtain k where k: "a - k*s b \ span (S - {b})" by auto + {assume k0: "k = 0" + with k have "a \ span S" + apply (simp) + apply (rule set_rev_mp) + apply assumption + apply (rule span_mono) + apply blast + done + with na have ?thesis by blast} + moreover + {assume k0: "k \ 0" + have eq: "b = (1/k) *s a - ((1/k) *s a - b)" by vector + from k0 have eq': "(1/k) *s (a - k*s b) = (1/k) *s a - b" + by (vector field_simps) + from k have "(1/k) *s (a - k*s b) \ span (S - {b})" + by (rule span_mul) + hence th: "(1/k) *s a - b \ span (S - {b})" + unfolding eq' . + + from k + have ?thesis + apply (subst eq) + apply (rule span_sub) + apply (rule span_mul) + apply (rule span_superset) + apply blast + apply (rule set_rev_mp) + apply (rule th) + apply (rule span_mono) + using na by blast} + ultimately show ?thesis by blast +qed + +lemma in_span_delete: + assumes a: "(a::'a::field^'n) \ span S" + and na: "a \ span (S-{b})" + shows "b \ span (insert a (S - {b}))" + apply (rule in_span_insert) + apply (rule set_rev_mp) + apply (rule a) + apply (rule span_mono) + apply blast + apply (rule na) + done + +(* Transitivity property. *) + +lemma span_trans: + assumes x: "(x::'a::ring_1^'n) \ span S" and y: "y \ span (insert x S)" + shows "y \ span S" +proof- + from span_breakdown[of x "insert x S" y, OF insertI1 y] + obtain k where k: "y -k*s x \ span (S - {x})" by auto + have eq: "y = (y - k *s x) + k *s x" by vector + show ?thesis + apply (subst eq) + apply (rule span_add) + apply (rule set_rev_mp) + apply (rule k) + apply (rule span_mono) + apply blast + apply (rule span_mul) + by (rule x) +qed + +(* ------------------------------------------------------------------------- *) +(* An explicit expansion is sometimes needed. *) +(* ------------------------------------------------------------------------- *) + +lemma span_explicit: + "span P = {y::'a::semiring_1^'n. \S u. finite S \ S \ P \ setsum (\v. u v *s v) S = y}" + (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \S u. ?Q S u y}") +proof- + {fix x assume x: "x \ ?E" + then obtain S u where fS: "finite S" and SP: "S\P" and u: "setsum (\v. u v *s v) S = x" + by blast + have "x \ span P" + unfolding u[symmetric] + apply (rule span_setsum[OF fS]) + using span_mono[OF SP] + by (auto intro: span_superset span_mul)} + moreover + have "\x \ span P. x \ ?E" + unfolding mem_def Collect_def + proof(rule span_induct_alt') + show "?h 0" + apply (rule exI[where x="{}"]) by simp + next + fix c x y + assume x: "x \ P" and hy: "?h y" + from hy obtain S u where fS: "finite S" and SP: "S\P" + and u: "setsum (\v. u v *s v) S = y" by blast + let ?S = "insert x S" + let ?u = "\y. if y = x then (if x \ S then u y + c else c) + else u y" + from fS SP x have th0: "finite (insert x S)" "insert x S \ P" by blast+ + {assume xS: "x \ S" + have S1: "S = (S - {x}) \ {x}" + and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \ {x} = {}" using xS fS by auto + have "setsum (\v. ?u v *s v) ?S =(\v\S - {x}. u v *s v) + (u x + c) *s x" + using xS + by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]] + setsum_clauses(2)[OF fS] cong del: if_weak_cong) + also have "\ = (\v\S. u v *s v) + c *s x" + apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]]) + by (vector ring_simps) + also have "\ = c*s x + y" + by (simp add: add_commute u) + finally have "setsum (\v. ?u v *s v) ?S = c*s x + y" . + then have "?Q ?S ?u (c*s x + y)" using th0 by blast} + moreover + {assume xS: "x \ S" + have th00: "(\v\S. (if v = x then c else u v) *s v) = y" + unfolding u[symmetric] + apply (rule setsum_cong2) + using xS by auto + have "?Q ?S ?u (c*s x + y)" using fS xS th0 + by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)} + ultimately have "?Q ?S ?u (c*s x + y)" + by (cases "x \ S", simp, simp) + then show "?h (c*s x + y)" + apply - + apply (rule exI[where x="?S"]) + apply (rule exI[where x="?u"]) by metis + qed + ultimately show ?thesis by blast +qed + +lemma dependent_explicit: + "dependent P \ (\S u. finite S \ S \ P \ (\(v::'a::{idom,field}^'n) \S. u v \ 0 \ setsum (\v. u v *s v) S = 0))" (is "?lhs = ?rhs") +proof- + {assume dP: "dependent P" + then obtain a S u where aP: "a \ P" and fS: "finite S" + and SP: "S \ P - {a}" and ua: "setsum (\v. u v *s v) S = a" + unfolding dependent_def span_explicit by blast + let ?S = "insert a S" + let ?u = "\y. if y = a then - 1 else u y" + let ?v = a + from aP SP have aS: "a \ S" by blast + from fS SP aP have th0: "finite ?S" "?S \ P" "?v \ ?S" "?u ?v \ 0" by auto + have s0: "setsum (\v. ?u v *s v) ?S = 0" + using fS aS + apply (simp add: vector_smult_lneg vector_smult_lid setsum_clauses ring_simps ) + apply (subst (2) ua[symmetric]) + apply (rule setsum_cong2) + by auto + with th0 have ?rhs + apply - + apply (rule exI[where x= "?S"]) + apply (rule exI[where x= "?u"]) + by clarsimp} + moreover + {fix S u v assume fS: "finite S" + and SP: "S \ P" and vS: "v \ S" and uv: "u v \ 0" + and u: "setsum (\v. u v *s v) S = 0" + let ?a = v + let ?S = "S - {v}" + let ?u = "\i. (- u i) / u v" + have th0: "?a \ P" "finite ?S" "?S \ P" using fS SP vS by auto + have "setsum (\v. ?u v *s v) ?S = setsum (\v. (- (inverse (u ?a))) *s (u v *s v)) S - ?u v *s v" + using fS vS uv + by (simp add: setsum_diff1 vector_smult_lneg divide_inverse + vector_smult_assoc field_simps) + also have "\ = ?a" + unfolding setsum_cmul u + using uv by (simp add: vector_smult_lneg) + finally have "setsum (\v. ?u v *s v) ?S = ?a" . + with th0 have ?lhs + unfolding dependent_def span_explicit + apply - + apply (rule bexI[where x= "?a"]) + apply simp_all + apply (rule exI[where x= "?S"]) + by auto} + ultimately show ?thesis by blast +qed + + +lemma span_finite: + assumes fS: "finite S" + shows "span S = {(y::'a::semiring_1^'n). \u. setsum (\v. u v *s v) S = y}" + (is "_ = ?rhs") +proof- + {fix y assume y: "y \ span S" + from y obtain S' u where fS': "finite S'" and SS': "S' \ S" and + u: "setsum (\v. u v *s v) S' = y" unfolding span_explicit by blast + let ?u = "\x. if x \ S' then u x else 0" + from setsum_restrict_set[OF fS, of "\v. u v *s v" S', symmetric] SS' + have "setsum (\v. ?u v *s v) S = setsum (\v. u v *s v) S'" + unfolding cond_value_iff cond_application_beta + apply (simp add: cond_value_iff cong del: if_weak_cong) + apply (rule setsum_cong) + apply auto + done + hence "setsum (\v. ?u v *s v) S = y" by (metis u) + hence "y \ ?rhs" by auto} + moreover + {fix y u assume u: "setsum (\v. u v *s v) S = y" + then have "y \ span S" using fS unfolding span_explicit by auto} + ultimately show ?thesis by blast +qed + + +(* Standard bases are a spanning set, and obviously finite. *) + +lemma span_stdbasis:"span {basis i :: 'a::ring_1^'n | i. i \ {1 .. dimindex(UNIV :: 'n set)}} = UNIV" +apply (rule set_ext) +apply auto +apply (subst basis_expansion[symmetric]) +apply (rule span_setsum) +apply simp +apply auto +apply (rule span_mul) +apply (rule span_superset) +apply (auto simp add: Collect_def mem_def) +done + + +lemma has_size_stdbasis: "{basis i ::real ^'n | i. i \ {1 .. dimindex (UNIV :: 'n set)}} hassize (dimindex(UNIV :: 'n set))" (is "?S hassize ?n") +proof- + have eq: "?S = basis ` {1 .. ?n}" by blast + show ?thesis unfolding eq + apply (rule hassize_image_inj[OF basis_inj]) + by (simp add: hassize_def) +qed + +lemma finite_stdbasis: "finite {basis i ::real^'n |i. i\ {1 .. dimindex(UNIV:: 'n set)}}" + using has_size_stdbasis[unfolded hassize_def] + .. + +lemma card_stdbasis: "card {basis i ::real^'n |i. i\ {1 .. dimindex(UNIV :: 'n set)}} = dimindex(UNIV :: 'n set)" + using has_size_stdbasis[unfolded hassize_def] + .. + +lemma independent_stdbasis_lemma: + assumes x: "(x::'a::semiring_1 ^ 'n) \ span (basis ` S)" + and i: "i \ {1 .. dimindex (UNIV :: 'n set)}" + and iS: "i \ S" + shows "(x$i) = 0" +proof- + let ?n = "dimindex (UNIV :: 'n set)" + let ?U = "{1 .. ?n}" + let ?B = "basis ` S" + let ?P = "\(x::'a^'n). \i\ ?U. i \ S \ x$i =0" + {fix x::"'a^'n" assume xS: "x\ ?B" + from xS have "?P x" by (auto simp add: basis_component)} + moreover + have "subspace ?P" + by (auto simp add: subspace_def Collect_def mem_def zero_index vector_component) + ultimately show ?thesis + using x span_induct[of ?B ?P x] i iS by blast +qed + +lemma independent_stdbasis: "independent {basis i ::real^'n |i. i\ {1 .. dimindex(UNIV :: 'n set)}}" +proof- + let ?n = "dimindex (UNIV :: 'n set)" + let ?I = "{1 .. ?n}" + let ?b = "basis :: nat \ real ^'n" + let ?B = "?b ` ?I" + have eq: "{?b i|i. i \ ?I} = ?B" + by auto + {assume d: "dependent ?B" + then obtain k where k: "k \ ?I" "?b k \ span (?B - {?b k})" + unfolding dependent_def by auto + have eq1: "?B - {?b k} = ?B - ?b ` {k}" by simp + have eq2: "?B - {?b k} = ?b ` (?I - {k})" + unfolding eq1 + apply (rule inj_on_image_set_diff[symmetric]) + apply (rule basis_inj) using k(1) by auto + from k(2) have th0: "?b k \ span (?b ` (?I - {k}))" unfolding eq2 . + from independent_stdbasis_lemma[OF th0 k(1), simplified] + have False by (simp add: basis_component[OF k(1), of k])} + then show ?thesis unfolding eq dependent_def .. +qed + +(* This is useful for building a basis step-by-step. *) + +lemma independent_insert: + "independent(insert (a::'a::field ^'n) S) \ + (if a \ S then independent S + else independent S \ a \ span S)" (is "?lhs \ ?rhs") +proof- + {assume aS: "a \ S" + hence ?thesis using insert_absorb[OF aS] by simp} + moreover + {assume aS: "a \ S" + {assume i: ?lhs + then have ?rhs using aS + apply simp + apply (rule conjI) + apply (rule independent_mono) + apply assumption + apply blast + by (simp add: dependent_def)} + moreover + {assume i: ?rhs + have ?lhs using i aS + apply simp + apply (auto simp add: dependent_def) + apply (case_tac "aa = a", auto) + apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})") + apply simp + apply (subgoal_tac "a \ span (insert aa (S - {aa}))") + apply (subgoal_tac "insert aa (S - {aa}) = S") + apply simp + apply blast + apply (rule in_span_insert) + apply assumption + apply blast + apply blast + done} + ultimately have ?thesis by blast} + ultimately show ?thesis by blast +qed + +(* The degenerate case of the Exchange Lemma. *) + +lemma mem_delete: "x \ (A - {a}) \ x \ a \ x \ A" + by blast + +lemma span_span: "span (span A) = span A" + unfolding span_def hull_hull .. + +lemma span_inc: "S \ span S" + by (metis subset_eq span_superset) + +lemma spanning_subset_independent: + assumes BA: "B \ A" and iA: "independent (A::('a::field ^'n) set)" + and AsB: "A \ span B" + shows "A = B" +proof + from BA show "B \ A" . +next + from span_mono[OF BA] span_mono[OF AsB] + have sAB: "span A = span B" unfolding span_span by blast + + {fix x assume x: "x \ A" + from iA have th0: "x \ span (A - {x})" + unfolding dependent_def using x by blast + from x have xsA: "x \ span A" by (blast intro: span_superset) + have "A - {x} \ A" by blast + hence th1:"span (A - {x}) \ span A" by (metis span_mono) + {assume xB: "x \ B" + from xB BA have "B \ A -{x}" by blast + hence "span B \ span (A - {x})" by (metis span_mono) + with th1 th0 sAB have "x \ span A" by blast + with x have False by (metis span_superset)} + then have "x \ B" by blast} + then show "A \ B" by blast +qed + +(* The general case of the Exchange Lemma, the key to what follows. *) + +lemma exchange_lemma: + assumes f:"finite (t:: ('a::field^'n) set)" and i: "independent s" + and sp:"s \ span t" + shows "\t'. (t' hassize card t) \ s \ t' \ t' \ s \ t \ s \ span t'" +using f i sp +proof(induct c\"card(t - s)" arbitrary: s t rule: nat_less_induct) + fix n:: nat and s t :: "('a ^'n) set" + assume H: " \m(x:: ('a ^'n) set) xa. + finite xa \ + independent x \ + x \ span xa \ + m = card (xa - x) \ + (\t'. (t' hassize card xa) \ + x \ t' \ t' \ x \ xa \ x \ span t')" + and ft: "finite t" and s: "independent s" and sp: "s \ span t" + and n: "n = card (t - s)" + let ?P = "\t'. (t' hassize card t) \ s \ t' \ t' \ s \ t \ s \ span t'" + let ?ths = "\t'. ?P t'" + {assume st: "s \ t" + from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t]) + by (auto simp add: hassize_def intro: span_superset)} + moreover + {assume st: "t \ s" + + from spanning_subset_independent[OF st s sp] + st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t]) + by (auto simp add: hassize_def intro: span_superset)} + moreover + {assume st: "\ s \ t" "\ t \ s" + from st(2) obtain b where b: "b \ t" "b \ s" by blast + from b have "t - {b} - s \ t - s" by blast + then have cardlt: "card (t - {b} - s) < n" using n ft + by (auto intro: psubset_card_mono) + from b ft have ct0: "card t \ 0" by auto + {assume stb: "s \ span(t -{b})" + from ft have ftb: "finite (t -{b})" by auto + from H[rule_format, OF cardlt ftb s stb] + obtain u where u: "u hassize card (t-{b})" "s \ u" "u \ s \ (t - {b})" "s \ span u" by blast + let ?w = "insert b u" + have th0: "s \ insert b u" using u by blast + from u(3) b have "u \ s \ t" by blast + then have th1: "insert b u \ s \ t" using u b by blast + have bu: "b \ u" using b u by blast + from u(1) have fu: "finite u" by (simp add: hassize_def) + from u(1) ft b have "u hassize (card t - 1)" by auto + then + have th2: "insert b u hassize card t" + using card_insert_disjoint[OF fu bu] ct0 by (auto simp add: hassize_def) + from u(4) have "s \ span u" . + also have "\ \ span (insert b u)" apply (rule span_mono) by blast + finally have th3: "s \ span (insert b u)" . from th0 th1 th2 th3 have th: "?P ?w" by blast + from th have ?ths by blast} + moreover + {assume stb: "\ s \ span(t -{b})" + from stb obtain a where a: "a \ s" "a \ span (t - {b})" by blast + have ab: "a \ b" using a b by blast + have at: "a \ t" using a ab span_superset[of a "t- {b}"] by auto + have mlt: "card ((insert a (t - {b})) - s) < n" + using cardlt ft n a b by auto + have ft': "finite (insert a (t - {b}))" using ft by auto + {fix x assume xs: "x \ s" + have t: "t \ (insert b (insert a (t -{b})))" using b by auto + from b(1) have "b \ span t" by (simp add: span_superset) + have bs: "b \ span (insert a (t - {b}))" + by (metis in_span_delete a sp mem_def subset_eq) + from xs sp have "x \ span t" by blast + with span_mono[OF t] + have x: "x \ span (insert b (insert a (t - {b})))" .. + from span_trans[OF bs x] have "x \ span (insert a (t - {b}))" .} + then have sp': "s \ span (insert a (t - {b}))" by blast + + from H[rule_format, OF mlt ft' s sp' refl] obtain u where + u: "u hassize card (insert a (t -{b}))" "s \ u" "u \ s \ insert a (t -{b})" + "s \ span u" by blast + from u a b ft at ct0 have "?P u" by (auto simp add: hassize_def) + then have ?ths by blast } + ultimately have ?ths by blast + } + ultimately + show ?ths by blast +qed + +(* This implies corresponding size bounds. *) + +lemma independent_span_bound: + assumes f: "finite t" and i: "independent (s::('a::field^'n) set)" and sp:"s \ span t" + shows "finite s \ card s \ card t" + by (metis exchange_lemma[OF f i sp] hassize_def finite_subset card_mono) + +lemma finite_Atleast_Atmost[simp]: "finite {f x |x. x\ {(i::'a::finite_intvl_succ) .. j}}" +proof- + have eq: "{f x |x. x\ {i .. j}} = f ` {i .. j}" by auto + show ?thesis unfolding eq + apply (rule finite_imageI) + apply (rule finite_intvl) + done +qed + +lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\ {(i::nat) .. j}}" +proof- + have eq: "{f x |x. x\ {i .. j}} = f ` {i .. j}" by auto + show ?thesis unfolding eq + apply (rule finite_imageI) + apply (rule finite_atLeastAtMost) + done +qed + + +lemma independent_bound: + fixes S:: "(real^'n) set" + shows "independent S \ finite S \ card S <= dimindex(UNIV :: 'n set)" + apply (subst card_stdbasis[symmetric]) + apply (rule independent_span_bound) + apply (rule finite_Atleast_Atmost_nat) + apply assumption + unfolding span_stdbasis + apply (rule subset_UNIV) + done + +lemma dependent_biggerset: "(finite (S::(real ^'n) set) ==> card S > dimindex(UNIV:: 'n set)) ==> dependent S" + by (metis independent_bound not_less) + +(* Hence we can create a maximal independent subset. *) + +lemma maximal_independent_subset_extend: + assumes sv: "(S::(real^'n) set) \ V" and iS: "independent S" + shows "\B. S \ B \ B \ V \ independent B \ V \ span B" + using sv iS +proof(induct d\ "dimindex (UNIV :: 'n set) - card S" arbitrary: S rule: nat_less_induct) + fix n and S:: "(real^'n) set" + assume H: "\mS \ V. independent S \ m = dimindex (UNIV::'n set) - card S \ + (\B. S \ B \ B \ V \ independent B \ V \ span B)" + and sv: "S \ V" and i: "independent S" and n: "n = dimindex (UNIV :: 'n set) - card S" + let ?P = "\B. S \ B \ B \ V \ independent B \ V \ span B" + let ?ths = "\x. ?P x" + let ?d = "dimindex (UNIV :: 'n set)" + {assume "V \ span S" + then have ?ths using sv i by blast } + moreover + {assume VS: "\ V \ span S" + from VS obtain a where a: "a \ V" "a \ span S" by blast + from a have aS: "a \ S" by (auto simp add: span_superset) + have th0: "insert a S \ V" using a sv by blast + from independent_insert[of a S] i a + have th1: "independent (insert a S)" by auto + have mlt: "?d - card (insert a S) < n" + using aS a n independent_bound[OF th1] dimindex_ge_1[of "UNIV :: 'n set"] + by auto + + from H[rule_format, OF mlt th0 th1 refl] + obtain B where B: "insert a S \ B" "B \ V" "independent B" " V \ span B" + by blast + from B have "?P B" by auto + then have ?ths by blast} + ultimately show ?ths by blast +qed + +lemma maximal_independent_subset: + "\(B:: (real ^'n) set). B\ V \ independent B \ V \ span B" + by (metis maximal_independent_subset_extend[of "{}:: (real ^'n) set"] empty_subsetI independent_empty) + +(* Notion of dimension. *) + +definition "dim V = (SOME n. \B. B \ V \ independent B \ V \ span B \ (B hassize n))" + +lemma basis_exists: "\B. (B :: (real ^'n) set) \ V \ independent B \ V \ span B \ (B hassize dim V)" +unfolding dim_def some_eq_ex[of "\n. \B. B \ V \ independent B \ V \ span B \ (B hassize n)"] +unfolding hassize_def +using maximal_independent_subset[of V] independent_bound +by auto + +(* Consequences of independence or spanning for cardinality. *) + +lemma independent_card_le_dim: "(B::(real ^'n) set) \ V \ independent B \ finite B \ card B \ dim V" +by (metis basis_exists[of V] independent_span_bound[where ?'a=real] hassize_def subset_trans) + +lemma span_card_ge_dim: "(B::(real ^'n) set) \ V \ V \ span B \ finite B \ dim V \ card B" + by (metis basis_exists[of V] independent_span_bound hassize_def subset_trans) + +lemma basis_card_eq_dim: + "B \ (V:: (real ^'n) set) \ V \ span B \ independent B \ finite B \ card B = dim V" + by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_mono) + +lemma dim_unique: "(B::(real ^'n) set) \ V \ V \ span B \ independent B \ B hassize n \ dim V = n" + by (metis basis_card_eq_dim hassize_def) + +(* More lemmas about dimension. *) + +lemma dim_univ: "dim (UNIV :: (real^'n) set) = dimindex (UNIV :: 'n set)" + apply (rule dim_unique[of "{basis i |i. i\ {1 .. dimindex (UNIV :: 'n set)}}"]) + by (auto simp only: span_stdbasis has_size_stdbasis independent_stdbasis) + +lemma dim_subset: + "(S:: (real ^'n) set) \ T \ dim S \ dim T" + using basis_exists[of T] basis_exists[of S] + by (metis independent_span_bound[where ?'a = real and ?'n = 'n] subset_eq hassize_def) + +lemma dim_subset_univ: "dim (S:: (real^'n) set) \ dimindex (UNIV :: 'n set)" + by (metis dim_subset subset_UNIV dim_univ) + +(* Converses to those. *) + +lemma card_ge_dim_independent: + assumes BV:"(B::(real ^'n) set) \ V" and iB:"independent B" and dVB:"dim V \ card B" + shows "V \ span B" +proof- + {fix a assume aV: "a \ V" + {assume aB: "a \ span B" + then have iaB: "independent (insert a B)" using iB aV BV by (simp add: independent_insert) + from aV BV have th0: "insert a B \ V" by blast + from aB have "a \B" by (auto simp add: span_superset) + with independent_card_le_dim[OF th0 iaB] dVB have False by auto} + then have "a \ span B" by blast} + then show ?thesis by blast +qed + +lemma card_le_dim_spanning: + assumes BV: "(B:: (real ^'n) set) \ V" and VB: "V \ span B" + and fB: "finite B" and dVB: "dim V \ card B" + shows "independent B" +proof- + {fix a assume a: "a \ B" "a \ span (B -{a})" + from a fB have c0: "card B \ 0" by auto + from a fB have cb: "card (B -{a}) = card B - 1" by auto + from BV a have th0: "B -{a} \ V" by blast + {fix x assume x: "x \ V" + from a have eq: "insert a (B -{a}) = B" by blast + from x VB have x': "x \ span B" by blast + from span_trans[OF a(2), unfolded eq, OF x'] + have "x \ span (B -{a})" . } + then have th1: "V \ span (B -{a})" by blast + have th2: "finite (B -{a})" using fB by auto + from span_card_ge_dim[OF th0 th1 th2] + have c: "dim V \ card (B -{a})" . + from c c0 dVB cb have False by simp} + then show ?thesis unfolding dependent_def by blast +qed + +lemma card_eq_dim: "(B:: (real ^'n) set) \ V \ B hassize dim V \ independent B \ V \ span B" + by (metis hassize_def order_eq_iff card_le_dim_spanning + card_ge_dim_independent) + +(* ------------------------------------------------------------------------- *) +(* More general size bound lemmas. *) +(* ------------------------------------------------------------------------- *) + +lemma independent_bound_general: + "independent (S:: (real^'n) set) \ finite S \ card S \ dim S" + by (metis independent_card_le_dim independent_bound subset_refl) + +lemma dependent_biggerset_general: "(finite (S:: (real^'n) set) \ card S > dim S) \ dependent S" + using independent_bound_general[of S] by (metis linorder_not_le) + +lemma dim_span: "dim (span (S:: (real ^'n) set)) = dim S" +proof- + have th0: "dim S \ dim (span S)" + by (auto simp add: subset_eq intro: dim_subset span_superset) + from basis_exists[of S] + obtain B where B: "B \ S" "independent B" "S \ span B" "B hassize dim S" by blast + from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+ + have bSS: "B \ span S" using B(1) by (metis subset_eq span_inc) + have sssB: "span S \ span B" using span_mono[OF B(3)] by (simp add: span_span) + from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis + using fB(2) by arith +qed + +lemma subset_le_dim: "(S:: (real ^'n) set) \ span T \ dim S \ dim T" + by (metis dim_span dim_subset) + +lemma span_eq_dim: "span (S:: (real ^'n) set) = span T ==> dim S = dim T" + by (metis dim_span) + +lemma spans_image: + assumes lf: "linear (f::'a::semiring_1^'n \ _)" and VB: "V \ span B" + shows "f ` V \ span (f ` B)" + unfolding span_linear_image[OF lf] + by (metis VB image_mono) + +lemma dim_image_le: assumes lf: "linear f" shows "dim (f ` S) \ dim (S:: (real ^'n) set)" +proof- + from basis_exists[of S] obtain B where + B: "B \ S" "independent B" "S \ span B" "B hassize dim S" by blast + from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+ + have "dim (f ` S) \ card (f ` B)" + apply (rule span_card_ge_dim) + using lf B fB by (auto simp add: span_linear_image spans_image subset_image_iff) + also have "\ \ dim S" using card_image_le[OF fB(1)] fB by simp + finally show ?thesis . +qed + +(* Relation between bases and injectivity/surjectivity of map. *) + +lemma spanning_surjective_image: + assumes us: "UNIV \ span (S:: ('a::semiring_1 ^'n) set)" + and lf: "linear f" and sf: "surj f" + shows "UNIV \ span (f ` S)" +proof- + have "UNIV \ f ` UNIV" using sf by (auto simp add: surj_def) + also have " \ \ span (f ` S)" using spans_image[OF lf us] . +finally show ?thesis . +qed + +lemma independent_injective_image: + assumes iS: "independent (S::('a::semiring_1^'n) set)" and lf: "linear f" and fi: "inj f" + shows "independent (f ` S)" +proof- + {fix a assume a: "a \ S" "f a \ span (f ` S - {f a})" + have eq: "f ` S - {f a} = f ` (S - {a})" using fi + by (auto simp add: inj_on_def) + from a have "f a \ f ` span (S -{a})" + unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast + hence "a \ span (S -{a})" using fi by (auto simp add: inj_on_def) + with a(1) iS have False by (simp add: dependent_def) } + then show ?thesis unfolding dependent_def by blast +qed + +(* ------------------------------------------------------------------------- *) +(* Picking an orthogonal replacement for a spanning set. *) +(* ------------------------------------------------------------------------- *) + (* FIXME : Move to some general theory ?*) +definition "pairwise R S \ (\x \ S. \y\ S. x\y \ R x y)" + +lemma vector_sub_project_orthogonal: "(b::'a::ordered_field^'n) \ (x - ((b \ x) / (b\b)) *s b) = 0" + apply (cases "b = 0", simp) + apply (simp add: dot_rsub dot_rmult) + unfolding times_divide_eq_right[symmetric] + by (simp add: field_simps dot_eq_0) + +lemma basis_orthogonal: + fixes B :: "(real ^'n) set" + assumes fB: "finite B" + shows "\C. finite C \ card C \ card B \ span C = span B \ pairwise orthogonal C" + (is " \C. ?P B C") +proof(induct rule: finite_induct[OF fB]) + case 1 thus ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def) +next + case (2 a B) + note fB = `finite B` and aB = `a \ B` + from `\C. finite C \ card C \ card B \ span C = span B \ pairwise orthogonal C` + obtain C where C: "finite C" "card C \ card B" + "span C = span B" "pairwise orthogonal C" by blast + let ?a = "a - setsum (\x. (x\a / (x\x)) *s x) C" + let ?C = "insert ?a C" + from C(1) have fC: "finite ?C" by simp + from fB aB C(1,2) have cC: "card ?C \ card (insert a B)" by (simp add: card_insert_if) + {fix x k + have th0: "\(a::'b::comm_ring) b c. a - (b - c) = c + (a - b)" by (simp add: ring_simps) + have "x - k *s (a - (\x\C. (x \ a / (x \ x)) *s x)) \ span C \ x - k *s a \ span C" + apply (simp only: vector_ssub_ldistrib th0) + apply (rule span_add_eq) + apply (rule span_mul) + apply (rule span_setsum[OF C(1)]) + apply clarify + apply (rule span_mul) + by (rule span_superset)} + then have SC: "span ?C = span (insert a B)" + unfolding expand_set_eq span_breakdown_eq C(3)[symmetric] by auto + thm pairwise_def + {fix x y assume xC: "x \ ?C" and yC: "y \ ?C" and xy: "x \ y" + {assume xa: "x = ?a" and ya: "y = ?a" + have "orthogonal x y" using xa ya xy by blast} + moreover + {assume xa: "x = ?a" and ya: "y \ ?a" "y \ C" + from ya have Cy: "C = insert y (C - {y})" by blast + have fth: "finite (C - {y})" using C by simp + have "orthogonal x y" + using xa ya + unfolding orthogonal_def xa dot_lsub dot_rsub diff_eq_0_iff_eq + apply simp + apply (subst Cy) + using C(1) fth + apply (simp only: setsum_clauses) + apply (auto simp add: dot_ladd dot_lmult dot_eq_0 dot_sym[of y a] dot_lsum[OF fth]) + apply (rule setsum_0') + apply clarsimp + apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format]) + by auto} + moreover + {assume xa: "x \ ?a" "x \ C" and ya: "y = ?a" + from xa have Cx: "C = insert x (C - {x})" by blast + have fth: "finite (C - {x})" using C by simp + have "orthogonal x y" + using xa ya + unfolding orthogonal_def ya dot_rsub dot_lsub diff_eq_0_iff_eq + apply simp + apply (subst Cx) + using C(1) fth + apply (simp only: setsum_clauses) + apply (subst dot_sym[of x]) + apply (auto simp add: dot_radd dot_rmult dot_eq_0 dot_sym[of x a] dot_rsum[OF fth]) + apply (rule setsum_0') + apply clarsimp + apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format]) + by auto + have "orthogonal x y" using xa ya sorry} + moreover + {assume xa: "x \ C" and ya: "y \ C" + have "orthogonal x y" using xa ya xy C(4) unfolding pairwise_def by blast} + ultimately have "orthogonal x y" using xC yC by blast} + then have CPO: "pairwise orthogonal ?C" unfolding pairwise_def by blast + from fC cC SC CPO have "?P (insert a B) ?C" by blast + then show ?case by blast +qed + +lemma orthogonal_basis_exists: + fixes V :: "(real ^'n) set" + shows "\B. independent B \ B \ span V \ V \ span B \ (B hassize dim V) \ pairwise orthogonal B" +proof- + from basis_exists[of V] obtain B where B: "B \ V" "independent B" "V \ span B" "B hassize dim V" by blast + from B have fB: "finite B" "card B = dim V" by (simp_all add: hassize_def) + from basis_orthogonal[OF fB(1)] obtain C where + C: "finite C" "card C \ card B" "span C = span B" "pairwise orthogonal C" by blast + from C B + have CSV: "C \ span V" by (metis span_inc span_mono subset_trans) + from span_mono[OF B(3)] C have SVC: "span V \ span C" by (simp add: span_span) + from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB + have iC: "independent C" by (simp add: dim_span) + from C fB have "card C \ dim V" by simp + moreover have "dim V \ card C" using span_card_ge_dim[OF CSV SVC C(1)] + by (simp add: dim_span) + ultimately have CdV: "C hassize dim V" unfolding hassize_def using C(1) by simp + from C B CSV CdV iC show ?thesis by auto +qed + +lemma span_eq: "span S = span T \ S \ span T \ T \ span S" + by (metis set_eq_subset span_mono span_span span_inc) + +(* ------------------------------------------------------------------------- *) +(* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *) +(* ------------------------------------------------------------------------- *) + +lemma span_not_univ_orthogonal: + assumes sU: "span S \ UNIV" + shows "\(a:: real ^'n). a \0 \ (\x \ span S. a \ x = 0)" +proof- + from sU obtain a where a: "a \ span S" by blast + from orthogonal_basis_exists obtain B where + B: "independent B" "B \ span S" "S \ span B" "B hassize dim S" "pairwise orthogonal B" + by blast + from B have fB: "finite B" "card B = dim S" by (simp_all add: hassize_def) + from span_mono[OF B(2)] span_mono[OF B(3)] + have sSB: "span S = span B" by (simp add: span_span) + let ?a = "a - setsum (\b. (a\b / (b\b)) *s b) B" + have "setsum (\b. (a\b / (b\b)) *s b) B \ span S" + unfolding sSB + apply (rule span_setsum[OF fB(1)]) + apply clarsimp + apply (rule span_mul) + by (rule span_superset) + with a have a0:"?a \ 0" by auto + have "\x\span B. ?a \ x = 0" + proof(rule span_induct') + show "subspace (\x. ?a \ x = 0)" + by (auto simp add: subspace_def mem_def dot_radd dot_rmult) + next + {fix x assume x: "x \ B" + from x have B': "B = insert x (B - {x})" by blast + have fth: "finite (B - {x})" using fB by simp + have "?a \ x = 0" + apply (subst B') using fB fth + unfolding setsum_clauses(2)[OF fth] + apply simp + apply (clarsimp simp add: dot_lsub dot_ladd dot_lmult dot_lsum dot_eq_0) + apply (rule setsum_0', rule ballI) + unfolding dot_sym + by (auto simp add: x field_simps dot_eq_0 intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])} + then show "\x \ B. ?a \ x = 0" by blast + qed + with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"]) +qed + +lemma span_not_univ_subset_hyperplane: + assumes SU: "span S \ (UNIV ::(real^'n) set)" + shows "\ a. a \0 \ span S \ {x. a \ x = 0}" + using span_not_univ_orthogonal[OF SU] by auto + +lemma lowdim_subset_hyperplane: + assumes d: "dim S < dimindex (UNIV :: 'n set)" + shows "\(a::real ^'n). a \ 0 \ span S \ {x. a \ x = 0}" +proof- + {assume "span S = UNIV" + hence "dim (span S) = dim (UNIV :: (real ^'n) set)" by simp + hence "dim S = dimindex (UNIV :: 'n set)" by (simp add: dim_span dim_univ) + with d have False by arith} + hence th: "span S \ UNIV" by blast + from span_not_univ_subset_hyperplane[OF th] show ?thesis . +qed + +(* We can extend a linear basis-basis injection to the whole set. *) + +lemma linear_indep_image_lemma: + assumes lf: "linear f" and fB: "finite B" + and ifB: "independent (f ` B)" + and fi: "inj_on f B" and xsB: "x \ span B" + and fx: "f (x::'a::field^'n) = 0" + shows "x = 0" + using fB ifB fi xsB fx +proof(induct arbitrary: x rule: finite_induct[OF fB]) + case 1 thus ?case by (auto simp add: span_empty) +next + case (2 a b x) + have fb: "finite b" using "2.prems" by simp + have th0: "f ` b \ f ` (insert a b)" + apply (rule image_mono) by blast + from independent_mono[ OF "2.prems"(2) th0] + have ifb: "independent (f ` b)" . + have fib: "inj_on f b" + apply (rule subset_inj_on [OF "2.prems"(3)]) + by blast + from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)] + obtain k where k: "x - k*s a \ span (b -{a})" by blast + have "f (x - k*s a) \ span (f ` b)" + unfolding span_linear_image[OF lf] + apply (rule imageI) + using k span_mono[of "b-{a}" b] by blast + hence "f x - k*s f a \ span (f ` b)" + by (simp add: linear_sub[OF lf] linear_cmul[OF lf]) + hence th: "-k *s f a \ span (f ` b)" + using "2.prems"(5) by (simp add: vector_smult_lneg) + {assume k0: "k = 0" + from k0 k have "x \ span (b -{a})" by simp + then have "x \ span b" using span_mono[of "b-{a}" b] + by blast} + moreover + {assume k0: "k \ 0" + from span_mul[OF th, of "- 1/ k"] k0 + have th1: "f a \ span (f ` b)" + by (auto simp add: vector_smult_assoc) + from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric] + have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast + from "2.prems"(2)[unfolded dependent_def bex_simps(10), rule_format, of "f a"] + have "f a \ span (f ` b)" using tha + using "2.hyps"(2) + "2.prems"(3) by auto + with th1 have False by blast + then have "x \ span b" by blast} + ultimately have xsb: "x \ span b" by blast + from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] + show "x = 0" . +qed + +(* We can extend a linear mapping from basis. *) + +lemma linear_independent_extend_lemma: + assumes fi: "finite B" and ib: "independent B" + shows "\g. (\x\ span B. \y\ span B. g ((x::'a::field^'n) + y) = g x + g y) + \ (\x\ span B. \c. g (c*s x) = c *s g x) + \ (\x\ B. g x = f x)" +using ib fi +proof(induct rule: finite_induct[OF fi]) + case 1 thus ?case by (auto simp add: span_empty) +next + case (2 a b) + from "2.prems" "2.hyps" have ibf: "independent b" "finite b" + by (simp_all add: independent_insert) + from "2.hyps"(3)[OF ibf] obtain g where + g: "\x\span b. \y\span b. g (x + y) = g x + g y" + "\x\span b. \c. g (c *s x) = c *s g x" "\x\b. g x = f x" by blast + let ?h = "\z. SOME k. (z - k *s a) \ span b" + {fix z assume z: "z \ span (insert a b)" + have th0: "z - ?h z *s a \ span b" + apply (rule someI_ex) + unfolding span_breakdown_eq[symmetric] + using z . + {fix k assume k: "z - k *s a \ span b" + have eq: "z - ?h z *s a - (z - k*s a) = (k - ?h z) *s a" + by (simp add: ring_simps vector_sadd_rdistrib[symmetric]) + from span_sub[OF th0 k] + have khz: "(k - ?h z) *s a \ span b" by (simp add: eq) + {assume "k \ ?h z" hence k0: "k - ?h z \ 0" by simp + from k0 span_mul[OF khz, of "1 /(k - ?h z)"] + have "a \ span b" by (simp add: vector_smult_assoc) + with "2.prems"(1) "2.hyps"(2) have False + by (auto simp add: dependent_def)} + then have "k = ?h z" by blast} + with th0 have "z - ?h z *s a \ span b \ (\k. z - k *s a \ span b \ k = ?h z)" by blast} + note h = this + let ?g = "\z. ?h z *s f a + g (z - ?h z *s a)" + {fix x y assume x: "x \ span (insert a b)" and y: "y \ span (insert a b)" + have tha: "\(x::'a^'n) y a k l. (x + y) - (k + l) *s a = (x - k *s a) + (y - l *s a)" + by (vector ring_simps) + have addh: "?h (x + y) = ?h x + ?h y" + apply (rule conjunct2[OF h, rule_format, symmetric]) + apply (rule span_add[OF x y]) + unfolding tha + by (metis span_add x y conjunct1[OF h, rule_format]) + have "?g (x + y) = ?g x + ?g y" + unfolding addh tha + g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]] + by (simp add: vector_sadd_rdistrib)} + moreover + {fix x:: "'a^'n" and c:: 'a assume x: "x \ span (insert a b)" + have tha: "\(x::'a^'n) c k a. c *s x - (c * k) *s a = c *s (x - k *s a)" + by (vector ring_simps) + have hc: "?h (c *s x) = c * ?h x" + apply (rule conjunct2[OF h, rule_format, symmetric]) + apply (metis span_mul x) + by (metis tha span_mul x conjunct1[OF h]) + have "?g (c *s x) = c*s ?g x" + unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]] + by (vector ring_simps)} + moreover + {fix x assume x: "x \ (insert a b)" + {assume xa: "x = a" + have ha1: "1 = ?h a" + apply (rule conjunct2[OF h, rule_format]) + apply (metis span_superset insertI1) + using conjunct1[OF h, OF span_superset, OF insertI1] + by (auto simp add: span_0) + + from xa ha1[symmetric] have "?g x = f x" + apply simp + using g(2)[rule_format, OF span_0, of 0] + by simp} + moreover + {assume xb: "x \ b" + have h0: "0 = ?h x" + apply (rule conjunct2[OF h, rule_format]) + apply (metis span_superset insertI1 xb x) + apply simp + apply (metis span_superset xb) + done + have "?g x = f x" + by (simp add: h0[symmetric] g(3)[rule_format, OF xb])} + ultimately have "?g x = f x" using x by blast } + ultimately show ?case apply - apply (rule exI[where x="?g"]) by blast +qed + +lemma linear_independent_extend: + assumes iB: "independent (B:: (real ^'n) set)" + shows "\g. linear g \ (\x\B. g x = f x)" +proof- + from maximal_independent_subset_extend[of B "UNIV"] iB + obtain C where C: "B \ C" "independent C" "\x. x \ span C" by auto + + from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f] + obtain g where g: "(\x\ span C. \y\ span C. g (x + y) = g x + g y) + \ (\x\ span C. \c. g (c*s x) = c *s g x) + \ (\x\ C. g x = f x)" by blast + from g show ?thesis unfolding linear_def using C + apply clarsimp by blast +qed + +(* Can construct an isomorphism between spaces of same dimension. *) + +lemma card_le_inj: assumes fA: "finite A" and fB: "finite B" + and c: "card A \ card B" shows "(\f. f ` A \ B \ inj_on f A)" +using fB c +proof(induct arbitrary: B rule: finite_induct[OF fA]) + case 1 thus ?case by simp +next + case (2 x s t) + thus ?case + proof(induct rule: finite_induct[OF "2.prems"(1)]) + case 1 then show ?case by simp + next + case (2 y t) + from "2.prems"(1,2,5) "2.hyps"(1,2) have cst:"card s \ card t" by simp + from "2.prems"(3) [OF "2.hyps"(1) cst] obtain f where + f: "f ` s \ t \ inj_on f s" by blast + from f "2.prems"(2) "2.hyps"(2) show ?case + apply - + apply (rule exI[where x = "\z. if z = x then y else f z"]) + by (auto simp add: inj_on_def) + qed +qed + +lemma card_subset_eq: assumes fB: "finite B" and AB: "A \ B" and + c: "card A = card B" + shows "A = B" +proof- + from fB AB have fA: "finite A" by (auto intro: finite_subset) + from fA fB have fBA: "finite (B - A)" by auto + have e: "A \ (B - A) = {}" by blast + have eq: "A \ (B - A) = B" using AB by blast + from card_Un_disjoint[OF fA fBA e, unfolded eq c] + have "card (B - A) = 0" by arith + hence "B - A = {}" unfolding card_eq_0_iff using fA fB by simp + with AB show "A = B" by blast +qed + +lemma subspace_isomorphism: + assumes s: "subspace (S:: (real ^'n) set)" and t: "subspace T" + and d: "dim S = dim T" + shows "\f. linear f \ f ` S = T \ inj_on f S" +proof- + from basis_exists[of S] obtain B where + B: "B \ S" "independent B" "S \ span B" "B hassize dim S" by blast + from basis_exists[of T] obtain C where + C: "C \ T" "independent C" "T \ span C" "C hassize dim T" by blast + from B(4) C(4) card_le_inj[of B C] d obtain f where + f: "f ` B \ C" "inj_on f B" unfolding hassize_def by auto + from linear_independent_extend[OF B(2)] obtain g where + g: "linear g" "\x\ B. g x = f x" by blast + from B(4) have fB: "finite B" by (simp add: hassize_def) + from C(4) have fC: "finite C" by (simp add: hassize_def) + from inj_on_iff_eq_card[OF fB, of f] f(2) + have "card (f ` B) = card B" by simp + with B(4) C(4) have ceq: "card (f ` B) = card C" using d + by (simp add: hassize_def) + have "g ` B = f ` B" using g(2) + by (auto simp add: image_iff) + also have "\ = C" using card_subset_eq[OF fC f(1) ceq] . + finally have gBC: "g ` B = C" . + have gi: "inj_on g B" using f(2) g(2) + by (auto simp add: inj_on_def) + note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi] + {fix x y assume x: "x \ S" and y: "y \ S" and gxy:"g x = g y" + from B(3) x y have x': "x \ span B" and y': "y \ span B" by blast+ + from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)]) + have th1: "x - y \ span B" using x' y' by (metis span_sub) + have "x=y" using g0[OF th1 th0] by simp } + then have giS: "inj_on g S" + unfolding inj_on_def by blast + from span_subspace[OF B(1,3) s] + have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)]) + also have "\ = span C" unfolding gBC .. + also have "\ = T" using span_subspace[OF C(1,3) t] . + finally have gS: "g ` S = T" . + from g(1) gS giS show ?thesis by blast +qed + +(* linear functions are equal on a subspace if they are on a spanning set. *) + +lemma subspace_kernel: + assumes lf: "linear (f::'a::semiring_1 ^'n \ _)" + shows "subspace {x. f x = 0}" +apply (simp add: subspace_def) +by (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf]) + +lemma linear_eq_0_span: + assumes lf: "linear f" and f0: "\x\B. f x = 0" + shows "\x \ span B. f x = (0::'a::semiring_1 ^'n)" +proof + fix x assume x: "x \ span B" + let ?P = "\x. f x = 0" + from subspace_kernel[OF lf] have "subspace ?P" unfolding Collect_def . + with x f0 span_induct[of B "?P" x] show "f x = 0" by blast +qed + +lemma linear_eq_0: + assumes lf: "linear f" and SB: "S \ span B" and f0: "\x\B. f x = 0" + shows "\x \ S. f x = (0::'a::semiring_1^'n)" + by (metis linear_eq_0_span[OF lf] subset_eq SB f0) + +lemma linear_eq: + assumes lf: "linear (f::'a::ring_1^'n \ _)" and lg: "linear g" and S: "S \ span B" + and fg: "\ x\ B. f x = g x" + shows "\x\ S. f x = g x" +proof- + let ?h = "\x. f x - g x" + from fg have fg': "\x\ B. ?h x = 0" by simp + from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg'] + show ?thesis by simp +qed + +lemma linear_eq_stdbasis: + assumes lf: "linear (f::'a::ring_1^'m \ 'a^'n)" and lg: "linear g" + and fg: "\i \ {1 .. dimindex(UNIV :: 'm set)}. f (basis i) = g(basis i)" + shows "f = g" +proof- + let ?U = "UNIV :: 'm set" + let ?I = "{basis i:: 'a^'m|i. i \ {1 .. dimindex ?U}}" + {fix x assume x: "x \ (UNIV :: ('a^'m) set)" + from equalityD2[OF span_stdbasis] + have IU: " (UNIV :: ('a^'m) set) \ span ?I" by blast + from linear_eq[OF lf lg IU] fg x + have "f x = g x" unfolding Collect_def Ball_def mem_def by metis} + then show ?thesis by (auto intro: ext) +qed + +(* Similar results for bilinear functions. *) + +lemma bilinear_eq: + assumes bf: "bilinear (f:: 'a::ring^'m \ 'a^'n \ 'a^'p)" + and bg: "bilinear g" + and SB: "S \ span B" and TC: "T \ span C" + and fg: "\x\ B. \y\ C. f x y = g x y" + shows "\x\S. \y\T. f x y = g x y " +proof- + let ?P = "\x. \y\ span C. f x y = g x y" + from bf bg have sp: "subspace ?P" + unfolding bilinear_def linear_def subspace_def bf bg + by(auto simp add: span_0 mem_def bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def intro: bilinear_ladd[OF bf]) + + have "\x \ span B. \y\ span C. f x y = g x y" + apply - + apply (rule ballI) + apply (rule span_induct[of B ?P]) + defer + apply (rule sp) + apply assumption + apply (clarsimp simp add: Ball_def) + apply (rule_tac P="\y. f xa y = g xa y" and S=C in span_induct) + using fg + apply (auto simp add: subspace_def) + using bf bg unfolding bilinear_def linear_def + by(auto simp add: span_0 mem_def bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def intro: bilinear_ladd[OF bf]) + then show ?thesis using SB TC by (auto intro: ext) +qed + +lemma bilinear_eq_stdbasis: + assumes bf: "bilinear (f:: 'a::ring_1^'m \ 'a^'n \ 'a^'p)" + and bg: "bilinear g" + and fg: "\i\ {1 .. dimindex (UNIV :: 'm set)}. \j\ {1 .. dimindex (UNIV :: 'n set)}. f (basis i) (basis j) = g (basis i) (basis j)" + shows "f = g" +proof- + from fg have th: "\x \ {basis i| i. i\ {1 .. dimindex (UNIV :: 'm set)}}. \y\ {basis j |j. j \ {1 .. dimindex (UNIV :: 'n set)}}. f x y = g x y" by blast + from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] show ?thesis by (blast intro: ext) +qed + +(* Detailed theorems about left and right invertibility in general case. *) + +lemma left_invertible_transp: + "(\(B::'a^'n^'m). B ** transp (A::'a^'n^'m) = mat (1::'a::comm_semiring_1)) \ (\(B::'a^'m^'n). A ** B = mat 1)" + by (metis matrix_transp_mul transp_mat transp_transp) + +lemma right_invertible_transp: + "(\(B::'a^'n^'m). transp (A::'a^'n^'m) ** B = mat (1::'a::comm_semiring_1)) \ (\(B::'a^'m^'n). B ** A = mat 1)" + by (metis matrix_transp_mul transp_mat transp_transp) + +lemma linear_injective_left_inverse: + assumes lf: "linear (f::real ^'n \ real ^'m)" and fi: "inj f" + shows "\g. linear g \ g o f = id" +proof- + from linear_independent_extend[OF independent_injective_image, OF independent_stdbasis, OF lf fi] + obtain h:: "real ^'m \ real ^'n" where h: "linear h" " \x \ f ` {basis i|i. i \ {1 .. dimindex (UNIV::'n set)}}. h x = inv f x" by blast + from h(2) + have th: "\i\{1..dimindex (UNIV::'n set)}. (h \ f) (basis i) = id (basis i)" + using inv_o_cancel[OF fi, unfolded stupid_ext[symmetric] id_def o_def] + apply auto + apply (erule_tac x="basis i" in allE) + by auto + + from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th] + have "h o f = id" . + then show ?thesis using h(1) by blast +qed + +lemma linear_surjective_right_inverse: + assumes lf: "linear (f:: real ^'m \ real ^'n)" and sf: "surj f" + shows "\g. linear g \ f o g = id" +proof- + from linear_independent_extend[OF independent_stdbasis] + obtain h:: "real ^'n \ real ^'m" where + h: "linear h" "\ x\ {basis i| i. i\ {1 .. dimindex (UNIV :: 'n set)}}. h x = inv f x" by blast + from h(2) + have th: "\i\{1..dimindex (UNIV::'n set)}. (f o h) (basis i) = id (basis i)" + using sf + apply (auto simp add: surj_iff o_def stupid_ext[symmetric]) + apply (erule_tac x="basis i" in allE) + by auto + + from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th] + have "f o h = id" . + then show ?thesis using h(1) by blast +qed + +lemma matrix_left_invertible_injective: +"(\B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \ (\x y. A *v x = A *v y \ x = y)" +proof- + {fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y" + from xy have "B*v (A *v x) = B *v (A*v y)" by simp + hence "x = y" + unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid .} + moreover + {assume A: "\x y. A *v x = A *v y \ x = y" + hence i: "inj (op *v A)" unfolding inj_on_def by auto + from linear_injective_left_inverse[OF matrix_vector_mul_linear i] + obtain g where g: "linear g" "g o op *v A = id" by blast + have "matrix g ** A = mat 1" + unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] + using g(2) by (simp add: o_def id_def stupid_ext) + then have "\B. (B::real ^'m^'n) ** A = mat 1" by blast} + ultimately show ?thesis by blast +qed + +lemma matrix_left_invertible_ker: + "(\B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \ (\x. A *v x = 0 \ x = 0)" + unfolding matrix_left_invertible_injective + using linear_injective_0[OF matrix_vector_mul_linear, of A] + by (simp add: inj_on_def) + +lemma matrix_right_invertible_surjective: +"(\B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \ surj (\x. A *v x)" +proof- + {fix B :: "real ^'m^'n" assume AB: "A ** B = mat 1" + {fix x :: "real ^ 'm" + have "A *v (B *v x) = x" + by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB)} + hence "surj (op *v A)" unfolding surj_def by metis } + moreover + {assume sf: "surj (op *v A)" + from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf] + obtain g:: "real ^'m \ real ^'n" where g: "linear g" "op *v A o g = id" + by blast + + have "A ** (matrix g) = mat 1" + unfolding matrix_eq matrix_vector_mul_lid + matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] + using g(2) unfolding o_def stupid_ext[symmetric] id_def + . + hence "\B. A ** (B::real^'m^'n) = mat 1" by blast + } + ultimately show ?thesis unfolding surj_def by blast +qed + +lemma matrix_left_invertible_independent_columns: + fixes A :: "real^'n^'m" + shows "(\(B::real ^'m^'n). B ** A = mat 1) \ (\c. setsum (\i. c i *s column i A) {1 .. dimindex(UNIV :: 'n set)} = 0 \ (\i\ {1 .. dimindex (UNIV :: 'n set)}. c i = 0))" + (is "?lhs \ ?rhs") +proof- + let ?U = "{1 .. dimindex(UNIV :: 'n set)}" + {assume k: "\x. A *v x = 0 \ x = 0" + {fix c i assume c: "setsum (\i. c i *s column i A) ?U = 0" + and i: "i \ ?U" + let ?x = "\ i. c i" + have th0:"A *v ?x = 0" + using c + unfolding matrix_mult_vsum Cart_eq + by (auto simp add: vector_component zero_index setsum_component Cart_lambda_beta) + from k[rule_format, OF th0] i + have "c i = 0" by (vector Cart_eq)} + hence ?rhs by blast} + moreover + {assume H: ?rhs + {fix x assume x: "A *v x = 0" + let ?c = "\i. ((x$i ):: real)" + from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x] + have "x = 0" by vector}} + ultimately show ?thesis unfolding matrix_left_invertible_ker by blast +qed + +lemma matrix_right_invertible_independent_rows: + fixes A :: "real^'n^'m" + shows "(\(B::real^'m^'n). A ** B = mat 1) \ (\c. setsum (\i. c i *s row i A) {1 .. dimindex(UNIV :: 'm set)} = 0 \ (\i\ {1 .. dimindex (UNIV :: 'm set)}. c i = 0))" + unfolding left_invertible_transp[symmetric] + matrix_left_invertible_independent_columns + by (simp add: column_transp) + +lemma matrix_right_invertible_span_columns: + "(\(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \ span (columns A) = UNIV" (is "?lhs = ?rhs") +proof- + let ?U = "{1 .. dimindex (UNIV :: 'm set)}" + have fU: "finite ?U" by simp + have lhseq: "?lhs \ (\y. \(x::real^'m). setsum (\i. (x$i) *s column i A) ?U = y)" + unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def + apply (subst eq_commute) .. + have rhseq: "?rhs \ (\x. x \ span (columns A))" by blast + {assume h: ?lhs + {fix x:: "real ^'n" + from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m" + where y: "setsum (\i. (y$i) *s column i A) ?U = x" by blast + have "x \ span (columns A)" + unfolding y[symmetric] + apply (rule span_setsum[OF fU]) + apply clarify + apply (rule span_mul) + apply (rule span_superset) + unfolding columns_def + by blast} + then have ?rhs unfolding rhseq by blast} + moreover + {assume h:?rhs + let ?P = "\(y::real ^'n). \(x::real^'m). setsum (\i. (x$i) *s column i A) ?U = y" + {fix y have "?P y" + proof(rule span_induct_alt[of ?P "columns A"]) + show "\x\real ^ 'm. setsum (\i. (x$i) *s column i A) ?U = 0" + apply (rule exI[where x=0]) + by (simp add: zero_index vector_smult_lzero) + next + fix c y1 y2 assume y1: "y1 \ columns A" and y2: "?P y2" + from y1 obtain i where i: "i \ ?U" "y1 = column i A" + unfolding columns_def by blast + from y2 obtain x:: "real ^'m" where + x: "setsum (\i. (x$i) *s column i A) ?U = y2" by blast + let ?x = "(\ j. if j = i then c + (x$i) else (x$j))::real^'m" + show "?P (c*s y1 + y2)" + proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric]Cart_lambda_beta setsum_component cond_value_iff right_distrib cond_application_beta vector_component cong del: if_weak_cong, simp only: One_nat_def[symmetric]) + fix j + have th: "\xa \ ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j) + else (x$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) + by (simp add: ring_simps) + have "setsum (\xa. if xa = i then (c + (x$i)) * ((column xa A)$j) + else (x$xa) * ((column xa A$j))) ?U = setsum (\xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U" + apply (rule setsum_cong[OF refl]) + using th by blast + also have "\ = setsum (\xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\xa. ((x$xa) * ((column xa A)$j))) ?U" + by (simp add: setsum_addf) + also have "\ = c * ((column i A)$j) + setsum (\xa. ((x$xa) * ((column xa A)$j))) ?U" + unfolding setsum_delta[OF fU] + using i(1) by simp + finally show "setsum (\xa. if xa = i then (c + (x$i)) * ((column xa A)$j) + else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\xa. ((x$xa) * ((column xa A)$j))) ?U" . + qed + next + show "y \ span (columns A)" unfolding h by blast + qed} + then have ?lhs unfolding lhseq ..} + ultimately show ?thesis by blast +qed + +lemma matrix_left_invertible_span_rows: + "(\(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \ span (rows A) = UNIV" + unfolding right_invertible_transp[symmetric] + unfolding columns_transp[symmetric] + unfolding matrix_right_invertible_span_columns + .. + +(* An injective map real^'n->real^'n is also surjective. *) + +lemma linear_injective_imp_surjective: + assumes lf: "linear (f:: real ^'n \ real ^'n)" and fi: "inj f" + shows "surj f" +proof- + let ?U = "UNIV :: (real ^'n) set" + from basis_exists[of ?U] obtain B + where B: "B \ ?U" "independent B" "?U \ span B" "B hassize dim ?U" + by blast + from B(4) have d: "dim ?U = card B" by (simp add: hassize_def) + have th: "?U \ span (f ` B)" + apply (rule card_ge_dim_independent) + apply blast + apply (rule independent_injective_image[OF B(2) lf fi]) + apply (rule order_eq_refl) + apply (rule sym) + unfolding d + apply (rule card_image) + apply (rule subset_inj_on[OF fi]) + by blast + from th show ?thesis + unfolding span_linear_image[OF lf] surj_def + using B(3) by blast +qed + +(* And vice versa. *) + +lemma surjective_iff_injective_gen: + assumes fS: "finite S" and fT: "finite T" and c: "card S = card T" + and ST: "f ` S \ T" + shows "(\y \ T. \x \ S. f x = y) \ inj_on f S" (is "?lhs \ ?rhs") +proof- + {assume h: "?lhs" + {fix x y assume x: "x \ S" and y: "y \ S" and f: "f x = f y" + from x fS have S0: "card S \ 0" by auto + {assume xy: "x \ y" + have th: "card S \ card (f ` (S - {y}))" + unfolding c + apply (rule card_mono) + apply (rule finite_imageI) + using fS apply simp + using h xy x y f unfolding subset_eq image_iff + apply auto + apply (case_tac "xa = f x") + apply (rule bexI[where x=x]) + apply auto + done + also have " \ \ card (S -{y})" + apply (rule card_image_le) + using fS by simp + also have "\ \ card S - 1" using y fS by simp + finally have False using S0 by arith } + then have "x = y" by blast} + then have ?rhs unfolding inj_on_def by blast} + moreover + {assume h: ?rhs + have "f ` S = T" + apply (rule card_subset_eq[OF fT ST]) + unfolding card_image[OF h] using c . + then have ?lhs by blast} + ultimately show ?thesis by blast +qed + +lemma linear_surjective_imp_injective: + assumes lf: "linear (f::real ^'n => real ^'n)" and sf: "surj f" + shows "inj f" +proof- + let ?U = "UNIV :: (real ^'n) set" + from basis_exists[of ?U] obtain B + where B: "B \ ?U" "independent B" "?U \ span B" "B hassize dim ?U" + by blast + {fix x assume x: "x \ span B" and fx: "f x = 0" + from B(4) have fB: "finite B" by (simp add: hassize_def) + from B(4) have d: "dim ?U = card B" by (simp add: hassize_def) + have fBi: "independent (f ` B)" + apply (rule card_le_dim_spanning[of "f ` B" ?U]) + apply blast + using sf B(3) + unfolding span_linear_image[OF lf] surj_def subset_eq image_iff + apply blast + using fB apply (blast intro: finite_imageI) + unfolding d + apply (rule card_image_le) + apply (rule fB) + done + have th0: "dim ?U \ card (f ` B)" + apply (rule span_card_ge_dim) + apply blast + unfolding span_linear_image[OF lf] + apply (rule subset_trans[where B = "f ` UNIV"]) + using sf unfolding surj_def apply blast + apply (rule image_mono) + apply (rule B(3)) + apply (metis finite_imageI fB) + done + + moreover have "card (f ` B) \ card B" + by (rule card_image_le, rule fB) + ultimately have th1: "card B = card (f ` B)" unfolding d by arith + have fiB: "inj_on f B" + unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric] by blast + from linear_indep_image_lemma[OF lf fB fBi fiB x] fx + have "x = 0" by blast} + note th = this + from th show ?thesis unfolding linear_injective_0[OF lf] + using B(3) by blast +qed + +(* Hence either is enough for isomorphism. *) + +lemma left_right_inverse_eq: + assumes fg: "f o g = id" and gh: "g o h = id" + shows "f = h" +proof- + have "f = f o (g o h)" unfolding gh by simp + also have "\ = (f o g) o h" by (simp add: o_assoc) + finally show "f = h" unfolding fg by simp +qed + +lemma isomorphism_expand: + "f o g = id \ g o f = id \ (\x. f(g x) = x) \ (\x. g(f x) = x)" + by (simp add: expand_fun_eq o_def id_def) + +lemma linear_injective_isomorphism: + assumes lf: "linear (f :: real^'n \ real ^'n)" and fi: "inj f" + shows "\f'. linear f' \ (\x. f' (f x) = x) \ (\x. f (f' x) = x)" +unfolding isomorphism_expand[symmetric] +using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]] linear_injective_left_inverse[OF lf fi] +by (metis left_right_inverse_eq) + +lemma linear_surjective_isomorphism: + assumes lf: "linear (f::real ^'n \ real ^'n)" and sf: "surj f" + shows "\f'. linear f' \ (\x. f' (f x) = x) \ (\x. f (f' x) = x)" +unfolding isomorphism_expand[symmetric] +using linear_surjective_right_inverse[OF lf sf] linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]] +by (metis left_right_inverse_eq) + +(* Left and right inverses are the same for R^N->R^N. *) + +lemma linear_inverse_left: + assumes lf: "linear (f::real ^'n \ real ^'n)" and lf': "linear f'" + shows "f o f' = id \ f' o f = id" +proof- + {fix f f':: "real ^'n \ real ^'n" + assume lf: "linear f" "linear f'" and f: "f o f' = id" + from f have sf: "surj f" + + apply (auto simp add: o_def stupid_ext[symmetric] id_def surj_def) + by metis + from linear_surjective_isomorphism[OF lf(1) sf] lf f + have "f' o f = id" unfolding stupid_ext[symmetric] o_def id_def + by metis} + then show ?thesis using lf lf' by metis +qed + +(* Moreover, a one-sided inverse is automatically linear. *) + +lemma left_inverse_linear: + assumes lf: "linear (f::real ^'n \ real ^'n)" and gf: "g o f = id" + shows "linear g" +proof- + from gf have fi: "inj f" apply (auto simp add: inj_on_def o_def id_def stupid_ext[symmetric]) + by metis + from linear_injective_isomorphism[OF lf fi] + obtain h:: "real ^'n \ real ^'n" where + h: "linear h" "\x. h (f x) = x" "\x. f (h x) = x" by blast + have "h = g" apply (rule ext) using gf h(2,3) + apply (simp add: o_def id_def stupid_ext[symmetric]) + by metis + with h(1) show ?thesis by blast +qed + +lemma right_inverse_linear: + assumes lf: "linear (f:: real ^'n \ real ^'n)" and gf: "f o g = id" + shows "linear g" +proof- + from gf have fi: "surj f" apply (auto simp add: surj_def o_def id_def stupid_ext[symmetric]) + by metis + from linear_surjective_isomorphism[OF lf fi] + obtain h:: "real ^'n \ real ^'n" where + h: "linear h" "\x. h (f x) = x" "\x. f (h x) = x" by blast + have "h = g" apply (rule ext) using gf h(2,3) + apply (simp add: o_def id_def stupid_ext[symmetric]) + by metis + with h(1) show ?thesis by blast +qed + +(* The same result in terms of square matrices. *) + +lemma matrix_left_right_inverse: + fixes A A' :: "real ^'n^'n" + shows "A ** A' = mat 1 \ A' ** A = mat 1" +proof- + {fix A A' :: "real ^'n^'n" assume AA': "A ** A' = mat 1" + have sA: "surj (op *v A)" + unfolding surj_def + apply clarify + apply (rule_tac x="(A' *v y)" in exI) + by (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid) + from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA] + obtain f' :: "real ^'n \ real ^'n" + where f': "linear f'" "\x. f' (A *v x) = x" "\x. A *v f' x = x" by blast + have th: "matrix f' ** A = mat 1" + by (simp add: matrix_eq matrix_works[OF f'(1)] matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format]) + hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp + hence "matrix f' = A'" by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid) + hence "matrix f' ** A = A' ** A" by simp + hence "A' ** A = mat 1" by (simp add: th)} + then show ?thesis by blast +qed + +(* Considering an n-element vector as an n-by-1 or 1-by-n matrix. *) + +definition "rowvector v = (\ i j. (v$j))" + +definition "columnvector v = (\ i j. (v$i))" + +lemma transp_columnvector: + "transp(columnvector v) = rowvector v" + by (simp add: transp_def rowvector_def columnvector_def Cart_eq Cart_lambda_beta) + +lemma transp_rowvector: "transp(rowvector v) = columnvector v" + by (simp add: transp_def columnvector_def rowvector_def Cart_eq Cart_lambda_beta) + +lemma dot_rowvector_columnvector: + "columnvector (A *v v) = A ** columnvector v" + by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def) + +lemma dot_matrix_product: "(x::'a::semiring_1^'n) \ y = (((rowvector x ::'a^'n^1) ** (columnvector y :: 'a^1^'n))$1)$1" + apply (vector matrix_matrix_mult_def rowvector_def columnvector_def dot_def) + by (simp add: Cart_lambda_beta) + +lemma dot_matrix_vector_mul: + fixes A B :: "real ^'n ^'n" and x y :: "real ^'n" + shows "(A *v x) \ (B *v y) = + (((rowvector x :: real^'n^1) ** ((transp A ** B) ** (columnvector y :: real ^1^'n)))$1)$1" +unfolding dot_matrix_product transp_columnvector[symmetric] + dot_rowvector_columnvector matrix_transp_mul matrix_mul_assoc .. + +(* Infinity norm. *) + +definition "infnorm (x::real^'n) = rsup {abs(x$i) |i. i\ {1 .. dimindex(UNIV :: 'n set)}}" + +lemma numseg_dimindex_nonempty: "\i. i \ {1 .. dimindex (UNIV :: 'n set)}" + using dimindex_ge_1 by auto + +lemma infnorm_set_image: + "{abs(x$i) |i. i\ {1 .. dimindex(UNIV :: 'n set)}} = + (\i. abs(x$i)) ` {1 .. dimindex(UNIV :: 'n set)}" by blast + +lemma infnorm_set_lemma: + shows "finite {abs((x::'a::abs ^'n)$i) |i. i\ {1 .. dimindex(UNIV :: 'n set)}}" + and "{abs(x$i) |i. i\ {1 .. dimindex(UNIV :: 'n set)}} \ {}" + unfolding infnorm_set_image + using dimindex_ge_1[of "UNIV :: 'n set"] + by (auto intro: finite_imageI) + +lemma infnorm_pos_le: "0 \ infnorm x" + unfolding infnorm_def + unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma] + unfolding infnorm_set_image + using dimindex_ge_1 + by auto + +lemma infnorm_triangle: "infnorm ((x::real^'n) + y) \ infnorm x + infnorm y" +proof- + have th: "\x y (z::real). x - y <= z \ x - z <= y" by arith + have th1: "\S f. f ` S = { f i| i. i \ S}" by blast + have th2: "\x (y::real). abs(x + y) - abs(x) <= abs(y)" by arith + show ?thesis + unfolding infnorm_def + unfolding rsup_finite_le_iff[ OF infnorm_set_lemma] + apply (subst diff_le_eq[symmetric]) + unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma] + unfolding infnorm_set_image bex_simps + apply (subst th) + unfolding th1 + unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma] + + unfolding infnorm_set_image ball_simps bex_simps + apply (simp add: vector_add_component) + apply (metis numseg_dimindex_nonempty th2) + done +qed + +lemma infnorm_eq_0: "infnorm x = 0 \ (x::real ^'n) = 0" +proof- + have "infnorm x <= 0 \ x = 0" + unfolding infnorm_def + unfolding rsup_finite_le_iff[OF infnorm_set_lemma] + unfolding infnorm_set_image ball_simps + by vector + then show ?thesis using infnorm_pos_le[of x] by simp +qed + +lemma infnorm_0: "infnorm 0 = 0" + by (simp add: infnorm_eq_0) + +lemma infnorm_neg: "infnorm (- x) = infnorm x" + unfolding infnorm_def + apply (rule cong[of "rsup" "rsup"]) + apply blast + apply (rule set_ext) + apply (auto simp add: vector_component abs_minus_cancel) + apply (rule_tac x="i" in exI) + apply (simp add: vector_component) + done + +lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)" +proof- + have "y - x = - (x - y)" by simp + then show ?thesis by (metis infnorm_neg) +qed + +lemma real_abs_sub_infnorm: "\ infnorm x - infnorm y\ \ infnorm (x - y)" +proof- + have th: "\(nx::real) n ny. nx <= n + ny \ ny <= n + nx ==> \nx - ny\ <= n" + by arith + from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"] + have ths: "infnorm x \ infnorm (x - y) + infnorm y" + "infnorm y \ infnorm (x - y) + infnorm x" + by (simp_all add: ring_simps infnorm_neg diff_def[symmetric]) + from th[OF ths] show ?thesis . +qed + +lemma real_abs_infnorm: " \infnorm x\ = infnorm x" + using infnorm_pos_le[of x] by arith + +lemma component_le_infnorm: assumes i: "i \ {1 .. dimindex (UNIV :: 'n set)}" + shows "\x$i\ \ infnorm (x::real^'n)" +proof- + let ?U = "{1 .. dimindex (UNIV :: 'n set)}" + let ?S = "{\x$i\ |i. i\ ?U}" + have fS: "finite ?S" unfolding image_Collect[symmetric] + apply (rule finite_imageI) unfolding Collect_def mem_def by simp + have S0: "?S \ {}" using numseg_dimindex_nonempty by blast + have th1: "\S f. f ` S = { f i| i. i \ S}" by blast + from rsup_finite_in[OF fS S0] rsup_finite_Ub[OF fS S0] i + show ?thesis unfolding infnorm_def isUb_def setle_def + unfolding infnorm_set_image ball_simps by auto +qed + +lemma infnorm_mul_lemma: "infnorm(a *s x) <= \a\ * infnorm x" + apply (subst infnorm_def) + unfolding rsup_finite_le_iff[OF infnorm_set_lemma] + unfolding infnorm_set_image ball_simps + apply (simp add: abs_mult vector_component del: One_nat_def) + apply (rule ballI) + apply (drule component_le_infnorm[of _ x]) + apply (rule mult_mono) + apply auto + done + +lemma infnorm_mul: "infnorm(a *s x) = abs a * infnorm x" +proof- + {assume a0: "a = 0" hence ?thesis by (simp add: infnorm_0) } + moreover + {assume a0: "a \ 0" + from a0 have th: "(1/a) *s (a *s x) = x" + by (simp add: vector_smult_assoc) + from a0 have ap: "\a\ > 0" by arith + from infnorm_mul_lemma[of "1/a" "a *s x"] + have "infnorm x \ 1/\a\ * infnorm (a*s x)" + unfolding th by simp + with ap have "\a\ * infnorm x \ \a\ * (1/\a\ * infnorm (a *s x))" by (simp add: field_simps) + then have "\a\ * infnorm x \ infnorm (a*s x)" + using ap by (simp add: field_simps) + with infnorm_mul_lemma[of a x] have ?thesis by arith } + ultimately show ?thesis by blast +qed + +lemma infnorm_pos_lt: "infnorm x > 0 \ x \ 0" + using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith + +(* Prove that it differs only up to a bound from Euclidean norm. *) + +lemma infnorm_le_norm: "infnorm x \ norm x" + unfolding infnorm_def rsup_finite_le_iff[OF infnorm_set_lemma] + unfolding infnorm_set_image ball_simps + by (metis component_le_norm) +lemma card_enum: "card {1 .. n} = n" by auto +lemma norm_le_infnorm: "norm(x) <= sqrt(real (dimindex(UNIV ::'n set))) * infnorm(x::real ^'n)" +proof- + let ?d = "dimindex(UNIV ::'n set)" + have d: "?d = card {1 .. ?d}" by auto + have "real ?d \ 0" by simp + hence d2: "(sqrt (real ?d))^2 = real ?d" + by (auto intro: real_sqrt_pow2) + have th: "sqrt (real ?d) * infnorm x \ 0" + by (simp add: dimindex_ge_1 zero_le_mult_iff real_sqrt_ge_0_iff infnorm_pos_le) + have th1: "x\x \ (sqrt (real ?d) * infnorm x)^2" + unfolding power_mult_distrib d2 + apply (subst d) + apply (subst power2_abs[symmetric]) + unfolding real_of_nat_def dot_def power2_eq_square[symmetric] + apply (subst power2_abs[symmetric]) + apply (rule setsum_bounded) + apply (rule power_mono) + unfolding abs_of_nonneg[OF infnorm_pos_le] + unfolding infnorm_def rsup_finite_ge_iff[OF infnorm_set_lemma] + unfolding infnorm_set_image bex_simps + apply blast + by (rule abs_ge_zero) + from real_le_lsqrt[OF dot_pos_le th th1] + show ?thesis unfolding real_vector_norm_def real_of_real_def id_def . +qed + +(* Equality in Cauchy-Schwarz and triangle inequalities. *) + +lemma norm_cauchy_schwarz_eq: "(x::real ^'n) \ y = norm x * norm y \ norm x *s y = norm y *s x" (is "?lhs \ ?rhs") +proof- + {assume h: "x = 0" + hence ?thesis by (simp add: norm_0)} + moreover + {assume h: "y = 0" + hence ?thesis by (simp add: norm_0)} + moreover + {assume x: "x \ 0" and y: "y \ 0" + from dot_eq_0[of "norm y *s x - norm x *s y"] + have "?rhs \ (norm y * (norm y * norm x * norm x - norm x * (x \ y)) - norm x * (norm y * (y \ x) - norm x * norm y * norm y) = 0)" + using x y + unfolding dot_rsub dot_lsub dot_lmult dot_rmult + unfolding norm_pow_2[symmetric] power2_eq_square diff_eq_0_iff_eq apply (simp add: dot_sym) + apply (simp add: ring_simps) + apply metis + done + also have "\ \ (2 * norm x * norm y * (norm x * norm y - x \ y) = 0)" using x y + by (simp add: ring_simps dot_sym) + also have "\ \ ?lhs" using x y + apply (simp add: norm_eq_0) + by metis + finally have ?thesis by blast} + ultimately show ?thesis by blast +qed + +lemma norm_cauchy_schwarz_abs_eq: "abs(x \ y) = norm x * norm y \ + norm x *s y = norm y *s x \ norm(x) *s y = - norm y *s x" (is "?lhs \ ?rhs") +proof- + have th: "\(x::real) a. a \ 0 \ abs x = a \ x = a \ x = - a" by arith + have "?rhs \ norm x *s y = norm y *s x \ norm (- x) *s y = norm y *s (- x)" + apply (simp add: norm_neg) by vector + also have "\ \(x \ y = norm x * norm y \ + (-x) \ y = norm x * norm y)" + unfolding norm_cauchy_schwarz_eq[symmetric] + unfolding norm_neg + norm_mul by blast + also have "\ \ ?lhs" + unfolding th[OF mult_nonneg_nonneg, OF norm_pos_le[of x] norm_pos_le[of y]] dot_lneg + by arith + finally show ?thesis .. +qed + +lemma norm_triangle_eq: "norm(x + y) = norm x + norm y \ norm x *s y = norm y *s x" +proof- + {assume x: "x =0 \ y =0" + hence ?thesis by (cases "x=0", simp_all add: norm_0)} + moreover + {assume x: "x \ 0" and y: "y \ 0" + hence "norm x \ 0" "norm y \ 0" + by (simp_all add: norm_eq_0) + hence n: "norm x > 0" "norm y > 0" + using norm_pos_le[of x] norm_pos_le[of y] + by arith+ + have th: "\(a::real) b c. a + b + c \ 0 ==> (a = b + c \ a^2 = (b + c)^2)" by algebra + have "norm(x + y) = norm x + norm y \ norm(x + y)^ 2 = (norm x + norm y) ^2" + apply (rule th) using n norm_pos_le[of "x + y"] + by arith + also have "\ \ norm x *s y = norm y *s x" + unfolding norm_cauchy_schwarz_eq[symmetric] + unfolding norm_pow_2 dot_ladd dot_radd + by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym ring_simps) + finally have ?thesis .} + ultimately show ?thesis by blast +qed + +(* Collinearity.*) + +definition "collinear S \ (\u. \x \ S. \ y \ S. \c. x - y = c *s u)" + +lemma collinear_empty: "collinear {}" by (simp add: collinear_def) + +lemma collinear_sing: "collinear {(x::'a::ring_1^'n)}" + apply (simp add: collinear_def) + apply (rule exI[where x=0]) + by simp + +lemma collinear_2: "collinear {(x::'a::ring_1^'n),y}" + apply (simp add: collinear_def) + apply (rule exI[where x="x - y"]) + apply auto + apply (rule exI[where x=0], simp) + apply (rule exI[where x=1], simp) + apply (rule exI[where x="- 1"], simp add: vector_sneg_minus1[symmetric]) + apply (rule exI[where x=0], simp) + done + +lemma collinear_lemma: "collinear {(0::real^'n),x,y} \ x = 0 \ y = 0 \ (\c. y = c *s x)" (is "?lhs \ ?rhs") +proof- + {assume "x=0 \ y = 0" hence ?thesis + by (cases "x = 0", simp_all add: collinear_2 insert_commute)} + moreover + {assume x: "x \ 0" and y: "y \ 0" + {assume h: "?lhs" + then obtain u where u: "\ x\ {0,x,y}. \y\ {0,x,y}. \c. x - y = c *s u" unfolding collinear_def by blast + from u[rule_format, of x 0] u[rule_format, of y 0] + obtain cx and cy where + cx: "x = cx*s u" and cy: "y = cy*s u" + by auto + from cx x have cx0: "cx \ 0" by auto + from cy y have cy0: "cy \ 0" by auto + let ?d = "cy / cx" + from cx cy cx0 have "y = ?d *s x" + by (simp add: vector_smult_assoc) + hence ?rhs using x y by blast} + moreover + {assume h: "?rhs" + then obtain c where c: "y = c*s x" using x y by blast + have ?lhs unfolding collinear_def c + apply (rule exI[where x=x]) + apply auto + apply (rule exI[where x=0], simp) + apply (rule exI[where x="- 1"], simp only: vector_smult_lneg vector_smult_lid) + apply (rule exI[where x= "-c"], simp only: vector_smult_lneg) + apply (rule exI[where x=1], simp) + apply (rule exI[where x=0], simp) + apply (rule exI[where x="1 - c"], simp add: vector_smult_lneg vector_sub_rdistrib) + apply (rule exI[where x="c - 1"], simp add: vector_smult_lneg vector_sub_rdistrib) + apply (rule exI[where x=0], simp) + done} + ultimately have ?thesis by blast} + ultimately show ?thesis by blast +qed + +lemma norm_cauchy_schwarz_equal: "abs(x \ y) = norm x * norm y \ collinear {(0::real^'n),x,y}" +unfolding norm_cauchy_schwarz_abs_eq +apply (cases "x=0", simp_all add: collinear_2 norm_0) +apply (cases "y=0", simp_all add: collinear_2 norm_0 insert_commute) +unfolding collinear_lemma +apply simp +apply (subgoal_tac "norm x \ 0") +apply (subgoal_tac "norm y \ 0") +apply (rule iffI) +apply (cases "norm x *s y = norm y *s x") +apply (rule exI[where x="(1/norm x) * norm y"]) +apply (drule sym) +unfolding vector_smult_assoc[symmetric] +apply (simp add: vector_smult_assoc field_simps) +apply (rule exI[where x="(1/norm x) * - norm y"]) +apply clarify +apply (drule sym) +unfolding vector_smult_assoc[symmetric] +apply (simp add: vector_smult_assoc field_simps) +apply (erule exE) +apply (erule ssubst) +unfolding vector_smult_assoc +unfolding norm_mul +apply (subgoal_tac "norm x * c = \c\ * norm x \ norm x * c = - \c\ * norm x") +apply (case_tac "c <= 0", simp add: ring_simps) +apply (simp add: ring_simps) +apply (case_tac "c <= 0", simp add: ring_simps) +apply (simp add: ring_simps) +apply (simp add: norm_eq_0) +apply (simp add: norm_eq_0) +done + +end \ No newline at end of file