# HG changeset patch # User huffman # Date 1236875243 25200 # Node ID 5d7d0add17410719bd8a27bb0659ed116550f3f2 # Parent 5c4c3a9e91022a764855491b716af5769d4380ca remove trailing spaces diff -r 5c4c3a9e9102 -r 5d7d0add1741 src/HOL/Library/Determinants.thy --- a/src/HOL/Library/Determinants.thy Thu Mar 12 08:57:03 2009 -0700 +++ b/src/HOL/Library/Determinants.thy Thu Mar 12 09:27:23 2009 -0700 @@ -40,7 +40,7 @@ lemma setprod_singleton_nat_seg: "setprod f {n..n} = f (n::'a::order)" by simp lemma setprod_numseg: "setprod f {m..0} = (if m=0 then f 0 else 1)" - "setprod f {m .. Suc n} = (if m \ Suc n then f (Suc n) * setprod f {m..n} + "setprod f {m .. Suc n} = (if m \ Suc n then f (Suc n) * setprod f {m..n} else setprod f {m..n})" by (auto simp add: atLeastAtMostSuc_conv) @@ -98,20 +98,20 @@ (* A few general lemmas we need below. *) (* ------------------------------------------------------------------------- *) -lemma Cart_lambda_beta_perm: assumes p: "p permutes {1..dimindex(UNIV::'n set)}" - and i: "i \ {1..dimindex(UNIV::'n set)}" +lemma Cart_lambda_beta_perm: assumes p: "p permutes {1..dimindex(UNIV::'n set)}" + and i: "i \ {1..dimindex(UNIV::'n set)}" shows "Cart_nth (Cart_lambda g ::'a^'n) (p i) = g(p i)" using permutes_in_image[OF p] i by (simp add: Cart_lambda_beta permutes_in_image[OF p]) lemma setprod_permute: - assumes p: "p permutes S" + assumes p: "p permutes S" shows "setprod f S = setprod (f o p) S" proof- {assume "\ finite S" hence ?thesis by simp} moreover {assume fS: "finite S" - then have ?thesis + then have ?thesis apply (simp add: setprod_def) apply (rule ab_semigroup_mult.fold_image_permute) apply (auto simp add: p) @@ -134,9 +134,9 @@ have fU: "finite ?U" by blast {fix p assume p: "p \ {p. p permutes ?U}" from p have pU: "p permutes ?U" by blast - have sth: "sign (inv p) = sign p" + have sth: "sign (inv p) = sign p" by (metis sign_inverse fU p mem_def Collect_def permutation_permutes) - from permutes_inj[OF pU] + from permutes_inj[OF pU] have pi: "inj_on p ?U" by (blast intro: subset_inj_on) from permutes_image[OF pU] have "setprod (\i. ?di (transp A) i (inv p i)) ?U = setprod (\i. ?di (transp A) i (inv p i)) (p ` ?U)" by simp @@ -148,7 +148,7 @@ from i permutes_inv_o[OF pU] permutes_in_image[OF pU] have "((\i. ?di (transp A) i (inv p i)) o p) i = ?di A i (p i)" unfolding transp_def by (simp add: Cart_lambda_beta expand_fun_eq)} - then show "setprod ((\i. ?di (transp A) i (inv p i)) o p) ?U = setprod (\i. ?di A i (p i)) ?U" by (auto intro: setprod_cong) + then show "setprod ((\i. ?di (transp A) i (inv p i)) o p) ?U = setprod (\i. ?di A i (p i)) ?U" by (auto intro: setprod_cong) qed finally have "of_int (sign (inv p)) * (setprod (\i. ?di (transp A) i (inv p i)) ?U) = of_int (sign p) * (setprod (\i. ?di A i (p i)) ?U)" using sth by simp} @@ -156,7 +156,7 @@ apply (rule setsum_cong2) by blast qed -lemma det_lowerdiagonal: +lemma det_lowerdiagonal: fixes A :: "'a::comm_ring_1^'n^'n" assumes ld: "\i j. i \ {1 .. dimindex (UNIV:: 'n set)} \ j \ {1 .. dimindex(UNIV:: 'n set)} \ i < j \ A$i$j = 0" shows "det A = setprod (\i. A$i$i) {1..dimindex(UNIV:: 'n set)}" @@ -179,7 +179,7 @@ unfolding det_def by (simp add: sign_id) qed -lemma det_upperdiagonal: +lemma det_upperdiagonal: fixes A :: "'a::comm_ring_1^'n^'n" assumes ld: "\i j. i \ {1 .. dimindex (UNIV:: 'n set)} \ j \ {1 .. dimindex(UNIV:: 'n set)} \ i > j \ A$i$j = 0" shows "det A = setprod (\i. A$i$i) {1..dimindex(UNIV:: 'n set)}" @@ -216,7 +216,7 @@ then have "det ?A = setprod (\i. ?f i i) ?U" using det_lowerdiagonal by blast also have "\ = 1" unfolding th setprod_1 .. - finally show ?thesis . + finally show ?thesis . qed lemma det_0: "det (mat 0 :: 'a::comm_ring_1^'n^'n) = 0" @@ -235,7 +235,7 @@ then have "det ?A = setprod (\i. ?f i i) ?U" using det_lowerdiagonal by blast also have "\ = 0" unfolding th .. - finally show ?thesis . + finally show ?thesis . qed lemma det_permute_rows: @@ -243,7 +243,7 @@ assumes p: "p permutes {1 .. dimindex (UNIV :: 'n set)}" shows "det(\ i. A$p i :: 'a^'n^'n) = of_int (sign p) * det A" apply (simp add: det_def setsum_right_distrib mult_assoc[symmetric] del: One_nat_def) - apply (subst sum_permutations_compose_right[OF p]) + apply (subst sum_permutations_compose_right[OF p]) proof(rule setsum_cong2) let ?U = "{1 .. dimindex (UNIV :: 'n set)}" let ?PU = "{p. p permutes ?U}" @@ -259,14 +259,14 @@ have "?Ap$i$ (q o p) i = A $ p i $ (q o p) i " by simp} hence "setprod (\i. ?Ap$i$ (q o p) i) ?U = setprod (\i. A$p i$(q o p) i) ?U" by (auto intro: setprod_cong) - also have "\ = setprod ((\i. A$p i$(q o p) i) o inv p) ?U" + also have "\ = setprod ((\i. A$p i$(q o p) i) o inv p) ?U" by (simp only: setprod_permute[OF ip, symmetric]) also have "\ = setprod (\i. A $ (p o inv p) i $ (q o (p o inv p)) i) ?U" by (simp only: o_def) also have "\ = setprod (\i. A$i$q i) ?U" by (simp only: o_def permutes_inverses[OF p]) - finally have thp: "setprod (\i. ?Ap$i$ (q o p) i) ?U = setprod (\i. A$i$q i) ?U" + finally have thp: "setprod (\i. ?Ap$i$ (q o p) i) ?U = setprod (\i. A$i$q i) ?U" by blast - show "of_int (sign (q o p)) * setprod (\i. ?Ap$i$ (q o p) i) ?U = of_int (sign p) * of_int (sign q) * setprod (\i. A$i$q i) ?U" + show "of_int (sign (q o p)) * setprod (\i. ?Ap$i$ (q o p) i) ?U = of_int (sign p) * of_int (sign q) * setprod (\i. A$i$q i) ?U" by (simp only: thp sign_compose[OF qp pp] mult_commute of_int_mult) qed @@ -282,18 +282,18 @@ moreover have "?Ap = transp (\ i. transp A $ p i)" by (simp add: transp_def Cart_eq Cart_lambda_beta Cart_lambda_beta_perm[OF p]) - ultimately show ?thesis by simp + ultimately show ?thesis by simp qed lemma det_identical_rows: fixes A :: "'a::ordered_idom^'n^'n" - assumes i: "i\{1 .. dimindex (UNIV :: 'n set)}" + assumes i: "i\{1 .. dimindex (UNIV :: 'n set)}" and j: "j\{1 .. dimindex (UNIV :: 'n set)}" and ij: "i \ j" and r: "row i A = row j A" shows "det A = 0" proof- - have tha: "\(a::'a) b. a = b ==> b = - a ==> a = 0" + have tha: "\(a::'a) b. a = b ==> b = - a ==> a = 0" by simp have th1: "of_int (-1) = - 1" by (metis of_int_1 of_int_minus number_of_Min) let ?p = "Fun.swap i j id" @@ -302,12 +302,12 @@ hence "det A = det ?A" by simp moreover have "det A = - det ?A" by (simp add: det_permute_rows[OF permutes_swap_id[OF i j]] sign_swap_id ij th1) - ultimately show "det A = 0" by (metis tha) + ultimately show "det A = 0" by (metis tha) qed lemma det_identical_columns: fixes A :: "'a::ordered_idom^'n^'n" - assumes i: "i\{1 .. dimindex (UNIV :: 'n set)}" + assumes i: "i\{1 .. dimindex (UNIV :: 'n set)}" and j: "j\{1 .. dimindex (UNIV :: 'n set)}" and ij: "i \ j" and r: "column i A = column j A" @@ -316,9 +316,9 @@ apply (rule det_identical_rows[OF i j ij]) by (metis row_transp i j r) -lemma det_zero_row: +lemma det_zero_row: fixes A :: "'a::{idom, ring_char_0}^'n^'n" - assumes i: "i\{1 .. dimindex (UNIV :: 'n set)}" + assumes i: "i\{1 .. dimindex (UNIV :: 'n set)}" and r: "row i A = 0" shows "det A = 0" using i r @@ -332,16 +332,16 @@ apply (subgoal_tac "(0\'a ^ 'n) $ a i = 0") apply simp apply (rule zero_index) -apply (drule permutes_in_image[of _ _ i]) +apply (drule permutes_in_image[of _ _ i]) apply simp -apply (drule permutes_in_image[of _ _ i]) +apply (drule permutes_in_image[of _ _ i]) apply simp apply simp done lemma det_zero_column: fixes A :: "'a::{idom,ring_char_0}^'n^'n" - assumes i: "i\{1 .. dimindex (UNIV :: 'n set)}" + assumes i: "i\{1 .. dimindex (UNIV :: 'n set)}" and r: "column i A = 0" shows "det A = 0" apply (subst det_transp[symmetric]) @@ -361,7 +361,7 @@ proof(rule setprod_cong[OF refl]) let ?U = "{1 .. dimindex (UNIV :: 'n set)}" fix i assume i: "i \ ?U" - from Cart_lambda_beta'[OF i, of g] have + from Cart_lambda_beta'[OF i, of g] have "((\ i. g i) :: 'a^'n^'n) $ i = g i" . hence "((\ i. g i) :: 'a^'n^'n) $ i $ f i = g i $ f i" by simp then @@ -369,7 +369,7 @@ qed lemma det_row_add: - assumes k: "k \ {1 .. dimindex (UNIV :: 'n set)}" + assumes k: "k \ {1 .. dimindex (UNIV :: 'n set)}" shows "det((\ i. if i = k then a i + b i else c i)::'a::comm_ring_1^'n^'n) = det((\ i. if i = k then a i else c i)::'a::comm_ring_1^'n^'n) + det((\ i. if i = k then b i else c i)::'a::comm_ring_1^'n^'n)" @@ -387,7 +387,7 @@ note pin[simp] = permutes_in_image[OF pU] have kU: "?U = insert k ?Uk" using k by blast {fix j assume j: "j \ ?Uk" - from j have "?f j $ p j = ?g j $ p j" and "?f j $ p j= ?h j $ p j" + from j have "?f j $ p j = ?g j $ p j" and "?f j $ p j= ?h j $ p j" by simp_all} then have th1: "setprod (\i. ?f i $ p i) ?Uk = setprod (\i. ?g i $ p i) ?Uk" and th2: "setprod (\i. ?f i $ p i) ?Uk = setprod (\i. ?h i $ p i) ?Uk" @@ -411,7 +411,7 @@ qed lemma det_row_mul: - assumes k: "k \ {1 .. dimindex (UNIV :: 'n set)}" + assumes k: "k \ {1 .. dimindex (UNIV :: 'n set)}" shows "det((\ i. if i = k then c *s a i else b i)::'a::comm_ring_1^'n^'n) = c* det((\ i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)" @@ -451,7 +451,7 @@ qed lemma det_row_0: - assumes k: "k \ {1 .. dimindex (UNIV :: 'n set)}" + assumes k: "k \ {1 .. dimindex (UNIV :: 'n set)}" shows "det((\ i. if i = k then 0 else b i)::'a::comm_ring_1^'n^'n) = 0" using det_row_mul[OF k, of 0 "\i. 1" b] apply (simp) @@ -483,8 +483,8 @@ let ?S = "{row j A |j. j\ ?U \ j\ i}" let ?d = "\x. det (\ k. if k = i then x else row k A)" let ?P = "\x. ?d (row i A + x) = det A" - {fix k - + {fix k + have "(if k = i then row i A + 0 else row k A) = row k A" by simp} then have P0: "?P 0" apply - @@ -499,10 +499,10 @@ apply (rule det_identical_rows[OF i j(2,3)]) using i j by (vector row_def) have "?d (row i A + (c*s z + y)) = ?d (?w + c*s z)" unfolding th0 .. - then have "?P (c*s z + y)" unfolding thz Py det_row_mul[OF i] det_row_add[OF i] + then have "?P (c*s z + y)" unfolding thz Py det_row_mul[OF i] det_row_add[OF i] by simp } - ultimately show ?thesis + ultimately show ?thesis apply - apply (rule span_induct_alt[of ?P ?S, OF P0]) apply blast @@ -524,7 +524,7 @@ from d obtain i where i: "i \ ?U" "row i A \ span (rows A - {row i A})" unfolding dependent_def rows_def by blast {fix j k assume j: "j \?U" and k: "k \ ?U" and jk: "j \ k" - and c: "row j A = row k A" + and c: "row j A = row k A" from det_identical_rows[OF j k jk c] have ?thesis .} moreover {assume H: "\ i j. i\ ?U \ j \ ?U \ i \ j \ row i A \ row j A" @@ -537,7 +537,7 @@ from det_row_span[OF i(1) th0] have "det A = det (\ k. if k = i then 0 *s 1 else row k A)" unfolding right_minus vector_smult_lzero .. - with det_row_mul[OF i(1), of "0::'a" "\i. 1"] + with det_row_mul[OF i(1), of "0::'a" "\i. 1"] have "det A = 0" by simp} ultimately show ?thesis by blast qed @@ -552,7 +552,7 @@ lemma Cart_lambda_cong: "(\x. x \ {1 .. dimindex (UNIV :: 'n set)} \ f x = g x) \ (Cart_lambda f::'a^'n) = (Cart_lambda g :: 'a^'n)" apply (rule iffD1[OF Cart_lambda_unique]) by vector -lemma det_linear_row_setsum: +lemma det_linear_row_setsum: assumes fS: "finite S" and k: "k \ {1 .. dimindex (UNIV :: 'n set)}" shows "det ((\ i. if i = k then setsum (a i) S else c i)::'a::comm_ring_1^'n^'n) = setsum (\j. det ((\ i. if i = k then a i j else c i)::'a^'n^'n)) S" using k @@ -567,7 +567,7 @@ assumes fS: "finite S" shows "finite {f. (\i \ {1.. (k::nat)}. f i \ S) \ (\i. i \ {1 .. k} \ f i = i)}" proof(induct k) - case 0 + case 0 have th: "{f. \i. f i = i} = {id}" by (auto intro: ext) show ?case by (auto simp add: th) next @@ -581,7 +581,7 @@ apply (auto intro: ext) done with finite_imageI[OF finite_cartesian_product[OF fS Suc.hyps(1)], of ?f] - show ?case by metis + show ?case by metis qed @@ -608,9 +608,9 @@ from Suc.prems have k': "k \ dimindex (UNIV :: 'n set)" by arith have thif: "\a b c d. (if b \ a then c else d) = (if a then c else if b then c else d)" by simp have thif2: "\a b c d e. (if a then b else if c then d else e) = - (if c then (if a then b else d) else (if a then b else e))" by simp - have "det (\ i. if i \ Suc k then setsum (a i) S else c i) = - det (\ i. if i = Suc k then setsum (a i) S + (if c then (if a then b else d) else (if a then b else e))" by simp + have "det (\ i. if i \ Suc k then setsum (a i) S else c i) = + det (\ i. if i = Suc k then setsum (a i) S else if i \ k then setsum (a i) S else c i)" unfolding le_Suc_eq thif .. also have "\ = (\j\S. det (\ i. if i \ k then setsum (a i) S @@ -618,14 +618,14 @@ unfolding det_linear_row_setsum[OF fS Sk] apply (subst thif2) by (simp cong del: if_weak_cong cong add: if_cong) - finally have tha: - "det (\ i. if i \ Suc k then setsum (a i) S else c i) = + finally have tha: + "det (\ i. if i \ Suc k then setsum (a i) S else c i) = (\(j, f)\S \ ?F k. det (\ i. if i \ k then a i (f i) else if i = Suc k then a i j - else c i))" + else c i))" unfolding Suc.hyps[OF k'] unfolding setsum_cartesian_product by blast show ?case unfolding tha - apply(rule setsum_eq_general_reverses[where h= "?h" and k= "?k"], + apply(rule setsum_eq_general_reverses[where h= "?h" and k= "?k"], blast intro: finite_cartesian_product fS finite_bounded_functions[OF fS], blast intro: finite_cartesian_product fS finite_bounded_functions[OF fS], auto intro: ext) apply (rule cong[OF refl[of det]]) @@ -637,7 +637,7 @@ shows "det (\ i. setsum (a i) S) = setsum (\f. det (\ i. a i (f i) :: 'a::comm_ring_1 ^ 'n^'n)) {f. (\i \ {1 .. dimindex (UNIV :: 'n set)}. f i \ S) \ (\i. i \ {1.. dimindex (UNIV :: 'n set)} \ f i = i)}" proof- have th0: "\x y. ((\ i. if i <= dimindex(UNIV:: 'n set) then x i else y i) :: 'a^'n^'n) = (\ i. x i)" by vector - + from det_linear_rows_setsum_lemma[OF fS, of "dimindex (UNIV :: 'n set)" a, unfolded th0, OF order_refl] show ?thesis by blast qed @@ -674,25 +674,25 @@ have fU: "finite ?U" by simp have fF: "finite ?F" using finite_bounded_functions[OF fU] . {fix p assume p: "p permutes ?U" - + have "p \ ?F" unfolding mem_Collect_eq permutes_in_image[OF p] using p[unfolded permutes_def] by simp} - then have PUF: "?PU \ ?F" by blast + then have PUF: "?PU \ ?F" by blast {fix f assume fPU: "f \ ?F - ?PU" have fUU: "f ` ?U \ ?U" using fPU by auto from fPU have f: "\i \ ?U. f i \ ?U" - "\i. i \ ?U \ f i = i" "\(\y. \!x. f x = y)" unfolding permutes_def + "\i. i \ ?U \ f i = i" "\(\y. \!x. f x = y)" unfolding permutes_def by auto - + let ?A = "(\ i. A$i$f i *s B$f i) :: 'a^'n^'n" let ?B = "(\ i. B$f i) :: 'a^'n^'n" {assume fni: "\ inj_on f ?U" then obtain i j where ij: "i \ ?U" "j \ ?U" "f i = f j" "i \ j" unfolding inj_on_def by blast - from ij + from ij have rth: "row i ?B = row j ?B" by (vector row_def) - from det_identical_rows[OF ij(1,2,4) rth] - have "det (\ i. A$i$f i *s B$f i) = 0" + from det_identical_rows[OF ij(1,2,4) rth] + have "det (\ i. A$i$f i *s B$f i) = 0" unfolding det_rows_mul by simp} moreover {assume fi: "inj_on f ?U" @@ -701,7 +701,7 @@ apply (case_tac "i \ ?U") apply (case_tac "j \ ?U") by metis+ note fs = fi[unfolded surjective_iff_injective_gen[OF fU fU refl fUU, symmetric]] - + {fix y from fs f have "\x. f x = y" by (cases "y \ ?U") blast+ then obtain x where x: "f x = y" by blast @@ -724,17 +724,17 @@ fix q assume qU: "q \ ?PU" hence q: "q permutes ?U" by blast from p q have pp: "permutation p" and pq: "permutation q" - unfolding permutation_permutes by auto - have th00: "of_int (sign p) * of_int (sign p) = (1::'a)" - "\a. of_int (sign p) * (of_int (sign p) * a) = a" - unfolding mult_assoc[symmetric] unfolding of_int_mult[symmetric] + unfolding permutation_permutes by auto + have th00: "of_int (sign p) * of_int (sign p) = (1::'a)" + "\a. of_int (sign p) * (of_int (sign p) * a) = a" + unfolding mult_assoc[symmetric] unfolding of_int_mult[symmetric] by (simp_all add: sign_idempotent) have ths: "?s q = ?s p * ?s (q o inv p)" using pp pq permutation_inverse[OF pp] sign_inverse[OF pp] by (simp add: th00 mult_ac sign_idempotent sign_compose) have th001: "setprod (\i. B$i$ q (inv p i)) ?U = setprod ((\i. B$i$ q (inv p i)) o p) ?U" by (rule setprod_permute[OF p]) - have thp: "setprod (\i. (\ i. A$i$p i *s B$p i :: 'a^'n^'n) $i $ q i) ?U = setprod (\i. A$i$p i) ?U * setprod (\i. B$i$ q (inv p i)) ?U" + have thp: "setprod (\i. (\ i. A$i$p i *s B$p i :: 'a^'n^'n) $i $ q i) ?U = setprod (\i. A$i$p i) ?U * setprod (\i. B$i$ q (inv p i)) ?U" unfolding th001 setprod_timesf[symmetric] o_def permutes_inverses[OF p] apply (rule setprod_cong[OF refl]) using permutes_in_image[OF q] by vector @@ -743,16 +743,16 @@ by (simp add: sign_nz th00 ring_simps sign_idempotent sign_compose) qed } - then have th2: "setsum (\f. det (\ i. A$i$f i *s B$f i)) ?PU = det A * det B" + then have th2: "setsum (\f. det (\ i. A$i$f i *s B$f i)) ?PU = det A * det B" unfolding det_def setsum_product - by (rule setsum_cong2) + by (rule setsum_cong2) have "det (A**B) = setsum (\f. det (\ i. A $ i $ f i *s B $ f i)) ?F" - unfolding matrix_mul_setsum_alt det_linear_rows_setsum[OF fU] .. + unfolding matrix_mul_setsum_alt det_linear_rows_setsum[OF fU] .. also have "\ = setsum (\f. det (\ i. A$i$f i *s B$f i)) ?PU" - using setsum_mono_zero_cong_left[OF fF PUF zth, symmetric] + using setsum_mono_zero_cong_left[OF fF PUF zth, symmetric] unfolding det_rows_mul by auto finally show ?thesis unfolding th2 . -qed +qed (* ------------------------------------------------------------------------- *) (* Relation to invertibility. *) @@ -768,7 +768,7 @@ shows "invertible A \ (\(B::real^'n^'n). A** B = mat 1)" by (metis invertible_def matrix_left_right_inverse) -lemma invertible_det_nz: +lemma invertible_det_nz: fixes A::"real ^'n^'n" shows "invertible A \ det A \ 0" proof- @@ -782,7 +782,7 @@ {assume H: "\ invertible A" let ?U = "{1 .. dimindex(UNIV :: 'n set)}" have fU: "finite ?U" by simp - from H obtain c i where c: "setsum (\i. c i *s row i A) ?U = 0" + from H obtain c i where c: "setsum (\i. c i *s row i A) ?U = 0" and iU: "i \ ?U" and ci: "c i \ 0" unfolding invertible_righ_inverse unfolding matrix_right_invertible_independent_rows by blast @@ -791,14 +791,14 @@ apply (simp only: ab_left_minus add_assoc[symmetric]) apply simp done - from c ci + from c ci have thr0: "- row i A = setsum (\j. (1/ c i) *s c j *s row j A) (?U - {i})" - unfolding setsum_diff1'[OF fU iU] setsum_cmul + unfolding setsum_diff1'[OF fU iU] setsum_cmul apply (simp add: field_simps) apply (rule vector_mul_lcancel_imp[OF ci]) apply (auto simp add: vector_smult_assoc vector_smult_rneg field_simps) unfolding stupid .. - have thr: "- row i A \ span {row j A| j. j\ ?U \ j \ i}" + have thr: "- row i A \ span {row j A| j. j\ ?U \ j \ i}" unfolding thr0 apply (rule span_setsum) apply simp @@ -808,8 +808,8 @@ apply auto done let ?B = "(\ k. if k = i then 0 else row k A) :: real ^'n^'n" - have thrb: "row i ?B = 0" using iU by (vector row_def) - have "det A = 0" + have thrb: "row i ?B = 0" using iU by (vector row_def) + have "det A = 0" unfolding det_row_span[OF iU thr, symmetric] right_minus unfolding det_zero_row[OF iU thrb] ..} ultimately show ?thesis by blast @@ -823,8 +823,8 @@ fixes A:: "'a::ordered_idom^'n^'n" and x :: "'a ^'n" assumes k: "k \ {1 .. dimindex(UNIV ::'n set)}" shows "det ((\ i. if i = k then setsum (\i. x$i *s row i A) {1 .. dimindex(UNIV::'n set)} - else row i A)::'a^'n^'n) = x$k * det A" - (is "?lhs = ?rhs") + else row i A)::'a^'n^'n) = x$k * det A" + (is "?lhs = ?rhs") proof- let ?U = "{1 .. dimindex (UNIV :: 'n set)}" let ?Uk = "?U - {k}" @@ -835,7 +835,7 @@ by (vector ring_simps) have th001: "\f k . (\x. if x = k then f k else f x) = f" by (auto intro: ext) have "(\ i. row i A) = A" by (vector row_def) - then have thd1: "det (\ i. row i A) = det A" by simp + then have thd1: "det (\ i. row i A) = det A" by simp have thd0: "det (\ i. if i = k then row k A + (\i \ ?Uk. x $ i *s row i A) else row i A) = det A" apply (rule det_row_span[OF k]) apply (rule span_setsum[OF fUk]) @@ -846,7 +846,7 @@ done show "?lhs = x$k * det A" apply (subst U) - unfolding setsum_insert[OF fUk kUk] + unfolding setsum_insert[OF fUk kUk] apply (subst th00) unfolding add_assoc apply (subst det_row_add[OF k]) @@ -863,8 +863,8 @@ proof- have stupid: "\c. setsum (\i. c i *s row i (transp A)) ?U = setsum (\i. c i *s column i A) ?U" by (auto simp add: row_transp intro: setsum_cong2) - show ?thesis - unfolding matrix_mult_vsum + show ?thesis + unfolding matrix_mult_vsum unfolding cramer_lemma_transp[OF k, of x "transp A", unfolded det_transp, symmetric] unfolding stupid[of "\i. x$i"] apply (subst det_transp[symmetric]) @@ -873,10 +873,10 @@ lemma cramer: fixes A ::"real^'n^'n" - assumes d0: "det A \ 0" + assumes d0: "det A \ 0" shows "A *v x = b \ x = (\ k. det(\ i j. if j=k then b$i else A$i$j :: real^'n^'n) / det A)" proof- - from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1" + from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1" unfolding invertible_det_nz[symmetric] invertible_def by blast have "(A ** B) *v b = b" by (simp add: B matrix_vector_mul_lid) hence "A *v (B *v b) = b" by (simp add: matrix_vector_mul_assoc) @@ -896,10 +896,10 @@ lemma orthogonal_transformation: "orthogonal_transformation f \ linear f \ (\(v::real ^'n). norm (f v) = norm v)" unfolding orthogonal_transformation_def - apply auto + apply auto apply (erule_tac x=v in allE)+ apply (simp add: real_vector_norm_def) - by (simp add: dot_norm linear_add[symmetric]) + by (simp add: dot_norm linear_add[symmetric]) definition "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \ transp Q ** Q = mat 1 \ Q ** transp Q = mat 1" @@ -909,12 +909,12 @@ lemma orthogonal_matrix_id: "orthogonal_matrix (mat 1)" by (simp add: orthogonal_matrix_def transp_mat matrix_mul_lid) -lemma orthogonal_matrix_mul: +lemma orthogonal_matrix_mul: fixes A :: "real ^'n^'n" assumes oA : "orthogonal_matrix A" - and oB: "orthogonal_matrix B" + and oB: "orthogonal_matrix B" shows "orthogonal_matrix(A ** B)" - using oA oB + using oA oB unfolding orthogonal_matrix matrix_transp_mul apply (subst matrix_mul_assoc) apply (subst matrix_mul_assoc[symmetric]) @@ -939,7 +939,7 @@ "\b (x::'a::comm_ring_1). x*(if b then 1 else 0) = (if b then x else 0)" by simp_all from fd[rule_format, of "basis i" "basis j", unfolded matrix_works[OF lf, symmetric] dot_matrix_vector_mul] i j - have "?A$i$j = ?m1 $ i $ j" + have "?A$i$j = ?m1 $ i $ j" by (simp add: Cart_lambda_beta' dot_def matrix_matrix_mult_def columnvector_def rowvector_def basis_def th0 setsum_delta[OF fU] mat_def del: One_nat_def)} hence "orthogonal_matrix ?mf" unfolding orthogonal_matrix by vector with lf have ?rhs by blast} @@ -953,17 +953,17 @@ ultimately show ?thesis by blast qed -lemma det_orthogonal_matrix: +lemma det_orthogonal_matrix: fixes Q:: "'a::ordered_idom^'n^'n" assumes oQ: "orthogonal_matrix Q" shows "det Q = 1 \ det Q = - 1" proof- - - have th: "\x::'a. x = 1 \ x = - 1 \ x*x = 1" (is "\x::'a. ?ths x") - proof- + + have th: "\x::'a. x = 1 \ x = - 1 \ x*x = 1" (is "\x::'a. ?ths x") + proof- fix x:: 'a have th0: "x*x - 1 = (x - 1)*(x + 1)" by (simp add: ring_simps) - have th1: "\(x::'a) y. x = - y \ x + y = 0" + have th1: "\(x::'a) y. x = - y \ x + y = 0" apply (subst eq_iff_diff_eq_0) by simp have "x*x = 1 \ x*x - 1 = 0" by simp also have "\ \ x = 1 \ x = - 1" unfolding th0 th1 by simp @@ -972,27 +972,27 @@ from oQ have "Q ** transp Q = mat 1" by (metis orthogonal_matrix_def) hence "det (Q ** transp Q) = det (mat 1:: 'a^'n^'n)" by simp hence "det Q * det Q = 1" by (simp add: det_mul det_I det_transp) - then show ?thesis unfolding th . + then show ?thesis unfolding th . qed (* ------------------------------------------------------------------------- *) (* Linearity of scaling, and hence isometry, that preserves origin. *) (* ------------------------------------------------------------------------- *) -lemma scaling_linear: +lemma scaling_linear: fixes f :: "real ^'n \ real ^'n" assumes f0: "f 0 = 0" and fd: "\x y. dist (f x) (f y) = c * dist x y" shows "linear f" proof- - {fix v w + {fix v w {fix x note fd[rule_format, of x 0, unfolded dist_def f0 diff_0_right] } note th0 = this - have "f v \ f w = c^2 * (v \ w)" + have "f v \ f w = c^2 * (v \ w)" unfolding dot_norm_neg dist_def[symmetric] unfolding th0 fd[rule_format] by (simp add: power2_eq_square field_simps)} note fc = this show ?thesis unfolding linear_def vector_eq by (simp add: dot_lmult dot_ladd dot_rmult dot_radd fc ring_simps) -qed +qed lemma isometry_linear: "f (0:: real^'n) = (0:: real^'n) \ \x y. dist(f x) (f y) = dist x y @@ -1005,7 +1005,7 @@ lemma orthogonal_transformation_isometry: "orthogonal_transformation f \ f(0::real^'n) = (0::real^'n) \ (\x y. dist(f x) (f y) = dist x y)" - unfolding orthogonal_transformation + unfolding orthogonal_transformation apply (rule iffI) apply clarify apply (clarsimp simp add: linear_0 linear_sub[symmetric] dist_def) @@ -1028,12 +1028,12 @@ and fd1: "\ x y. norm x = 1 \ norm y = 1 \ dist (f x) (f y) = dist x y" shows "\g. orthogonal_transformation g \ (\x. norm x = 1 \ g x = f x)" proof- - {fix x y x' y' x0 y0 x0' y0' :: "real ^'n" + {fix x y x' y' x0 y0 x0' y0' :: "real ^'n" assume H: "x = norm x *s x0" "y = norm y *s y0" - "x' = norm x *s x0'" "y' = norm y *s y0'" + "x' = norm x *s x0'" "y' = norm y *s y0'" "norm x0 = 1" "norm x0' = 1" "norm y0 = 1" "norm y0' = 1" "norm(x0' - y0') = norm(x0 - y0)" - + have "norm(x' - y') = norm(x - y)" apply (subst H(1)) apply (subst H(2)) @@ -1055,13 +1055,13 @@ then have "dist (?g x) (?g y) = dist x y" by simp } moreover {assume "x = 0" "y \ 0" - then have "dist (?g x) (?g y) = dist x y" + then have "dist (?g x) (?g y) = dist x y" apply (simp add: dist_def norm_mul) apply (rule f1[rule_format]) by(simp add: norm_mul field_simps)} moreover {assume "x \ 0" "y = 0" - then have "dist (?g x) (?g y) = dist x y" + then have "dist (?g x) (?g y) = dist x y" apply (simp add: dist_def norm_mul) apply (rule f1[rule_format]) by(simp add: norm_mul field_simps)} @@ -1077,14 +1077,14 @@ norm (inverse (norm x) *s x - inverse (norm y) *s y)" using z by (auto simp add: vector_smult_assoc field_simps norm_mul intro: f1[rule_format] fd1[rule_format, unfolded dist_def]) - from z th0[OF th00] have "dist (?g x) (?g y) = dist x y" + from z th0[OF th00] have "dist (?g x) (?g y) = dist x y" by (simp add: dist_def)} ultimately have "dist (?g x) (?g y) = dist x y" by blast} note thd = this - show ?thesis + show ?thesis apply (rule exI[where x= ?g]) unfolding orthogonal_transformation_isometry - using g0 thfg thd by metis + using g0 thfg thd by metis qed (* ------------------------------------------------------------------------- *) @@ -1094,7 +1094,7 @@ definition "rotation_matrix Q \ orthogonal_matrix Q \ det Q = 1" definition "rotoinversion_matrix Q \ orthogonal_matrix Q \ det Q = - 1" -lemma orthogonal_rotation_or_rotoinversion: +lemma orthogonal_rotation_or_rotoinversion: fixes Q :: "'a::ordered_idom^'n^'n" shows " orthogonal_matrix Q \ rotation_matrix Q \ rotoinversion_matrix Q" by (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix) @@ -1104,9 +1104,9 @@ lemma setprod_1: "setprod f {(1::nat)..1} = f 1" by simp -lemma setprod_2: "setprod f {(1::nat)..2} = f 1 * f 2" +lemma setprod_2: "setprod f {(1::nat)..2} = f 1 * f 2" by (simp add: nat_number setprod_numseg mult_commute) -lemma setprod_3: "setprod f {(1::nat)..3} = f 1 * f 2 * f 3" +lemma setprod_3: "setprod f {(1::nat)..3} = f 1 * f 2 * f 3" by (simp add: nat_number setprod_numseg mult_commute) lemma det_1: "det (A::'a::comm_ring_1^1^1) = A$1$1" @@ -1116,7 +1116,7 @@ proof- have f12: "finite {2::nat}" "1 \ {2::nat}" by auto have th12: "{1 .. 2} = insert (1::nat) {2}" by auto - show ?thesis + show ?thesis apply (simp add: det_def dimindex_def th12 del: One_nat_def) unfolding setsum_over_permutations_insert[OF f12] unfolding permutes_sing @@ -1124,7 +1124,7 @@ by (simp add: arith_simps(31)[symmetric] of_int_minus of_int_1 del: arith_simps(31)) qed -lemma det_3: "det (A::'a::comm_ring_1^3^3) = +lemma det_3: "det (A::'a::comm_ring_1^3^3) = A$1$1 * A$2$2 * A$3$3 + A$1$2 * A$2$3 * A$3$1 + A$1$3 * A$2$1 * A$3$2 - @@ -1136,7 +1136,7 @@ have f23: "finite {(3::nat)}" "2 \ {(3::nat)}" by auto have th12: "{1 .. 3} = insert (1::nat) (insert 2 {3})" by auto - show ?thesis + show ?thesis apply (simp add: det_def dimindex_def th12 del: One_nat_def) unfolding setsum_over_permutations_insert[OF f123] unfolding setsum_over_permutations_insert[OF f23] diff -r 5c4c3a9e9102 -r 5d7d0add1741 src/HOL/Library/Euclidean_Space.thy --- a/src/HOL/Library/Euclidean_Space.thy Thu Mar 12 08:57:03 2009 -0700 +++ b/src/HOL/Library/Euclidean_Space.thy Thu Mar 12 09:27:23 2009 -0700 @@ -5,7 +5,7 @@ header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*} theory Euclidean_Space - imports "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Complex_Main + imports "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Complex_Main Finite_Cartesian_Product Glbs Infinite_Set Numeral_Type Inner_Product uses ("normarith.ML") @@ -31,26 +31,26 @@ qed lemma setsum_singleton[simp]: "setsum f {x} = f x" by simp -lemma setsum_1: "setsum f {(1::'a::{order,one})..1} = f 1" +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" +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" +lemma setsum_3: "setsum f {1::nat..3} = f 1 + f 2 + f 3" by (simp add: nat_number atLeastAtMostSuc_conv add_commute) subsection{* Basic componentwise operations on vectors. *} instantiation "^" :: (plus,type) plus begin -definition vector_add_def : "op + \ (\ x y. (\ i. (x$i) + (y$i)))" +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)))" + definition vector_mult_def : "op * \ (\ x y. (\ i. (x$i) * (y$i)))" instance .. end @@ -64,12 +64,12 @@ instance .. end instantiation "^" :: (zero,type) zero begin - definition vector_zero_def : "0 \ (\ i. 0)" + definition vector_zero_def : "0 \ (\ i. 0)" instance .. end instantiation "^" :: (one,type) one begin - definition vector_one_def : "1 \ (\ i. 1)" + definition vector_one_def : "1 \ (\ i. 1)" instance .. end @@ -80,13 +80,13 @@ 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 instantiation "^" :: (scaleR, type) scaleR begin -definition vector_scaleR_def: "scaleR = (\ r x. (\ i. scaleR r (x$i)))" +definition vector_scaleR_def: "scaleR = (\ r x. (\ i. scaleR r (x$i)))" instance .. end @@ -117,19 +117,19 @@ 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}, + 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}, + 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 vector_scaleR_def}, @{thm Cart_lambda_beta'}, @{thm vector_scalar_mult_def}] - fun vector_arith_tac ths = + fun vector_arith_tac ths = simp_tac ss1 THEN' (fn i => rtac @{thm setsum_cong2} i - ORELSE rtac @{thm setsum_0'} 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) @@ -145,30 +145,30 @@ 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: +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: +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: +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: +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: +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 @@ -188,26 +188,26 @@ subsection {* Some frequently useful arithmetic lemmas over vectors. *} -instance "^" :: (semigroup_add,type) semigroup_add +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 +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 +instance "^" :: (ab_group_add,type) ab_group_add apply (intro_classes) by vector+ -instance "^" :: (cancel_semigroup_add,type) cancel_semigroup_add +instance "^" :: (cancel_semigroup_add,type) cancel_semigroup_add apply (intro_classes) by (vector Cart_eq)+ @@ -218,30 +218,30 @@ instance "^" :: (real_vector, type) real_vector by default (vector scaleR_left_distrib scaleR_right_distrib)+ -instance "^" :: (semigroup_mult,type) semigroup_mult +instance "^" :: (semigroup_mult,type) semigroup_mult apply (intro_classes) by (vector mult_assoc) -instance "^" :: (monoid_mult,type) monoid_mult +instance "^" :: (monoid_mult,type) monoid_mult apply (intro_classes) by vector+ -instance "^" :: (ab_semigroup_mult,type) ab_semigroup_mult +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 +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 +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 +instantiation "^" :: (recpower,type) recpower begin definition vec_power_def: "op ^ \ vector_power" - instance - apply (intro_classes) by (simp_all add: vec_power_def) + instance + apply (intro_classes) by (simp_all add: vec_power_def) end instance "^" :: (semiring,type) semiring @@ -250,16 +250,16 @@ 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 + 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 "^" :: (comm_semiring_0,type) comm_semiring_0 by (intro_classes) instance "^" :: (cancel_comm_monoid_add, type) cancel_comm_monoid_add .. -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 "^" :: (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) instance "^" :: (ring_1,type) ring_1 .. @@ -273,31 +273,31 @@ instance "^" :: (real_algebra_1,type) real_algebra_1 .. -lemma of_nat_index: +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]: +lemma zero_index[simp]: "i\{1 .. dimindex (UNIV :: 'n set)} \ (0 :: 'a::zero ^'n)$i = 0" by vector -lemma one_index[simp]: +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 + 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) +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 + case 0 thus ?case apply vector apply (induct n,auto simp add: ring_simps) using dimindex_ge_1 apply auto apply vector @@ -323,24 +323,24 @@ instance "^" :: (comm_ring_1,type) comm_ring_1 by intro_classes 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" +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" +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" +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" +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" +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)" +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 @@ -581,15 +581,15 @@ subsection{* Properties of the dot product. *} -lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \ y = y \ x" +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)" +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)" +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)" +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) @@ -625,8 +625,8 @@ ultimately show ?thesis by metis qed -lemma dot_pos_lt[simp]: "(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) +lemma dot_pos_lt[simp]: "(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{* The collapse of the general concepts to dimension one. *} @@ -642,13 +642,13 @@ lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)" by (simp add: vector_norm_def dimindex_def) -lemma norm_real: "norm(x::real ^ 1) = abs(x$1)" +lemma norm_real: "norm(x::real ^ 1) = abs(x$1)" by (simp add: norm_vector_1) text{* Metric *} text {* FIXME: generalize to arbitrary @{text real_normed_vector} types *} -definition dist:: "real ^ 'n \ real ^ 'n \ real" where +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))" @@ -667,14 +667,14 @@ 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" + 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 + 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) + 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) @@ -685,11 +685,11 @@ {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 + 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 + 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) + 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 @@ -701,16 +701,16 @@ {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 + 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 + 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 + 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 + 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 @@ -719,7 +719,7 @@ 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 + 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 @@ -740,14 +740,14 @@ 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" +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 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 + then show ?thesis using m by simp qed lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)" @@ -786,7 +786,7 @@ {assume "norm x = 0" hence ?thesis by (simp add: dot_lzero dot_rzero)} moreover - {assume "norm y = 0" + {assume "norm y = 0" hence ?thesis by (simp add: dot_lzero dot_rzero)} moreover {assume h: "norm x \ 0" "norm y \ 0" @@ -829,7 +829,7 @@ lemma norm_le_l1: "norm (x:: real ^'n) <= setsum(\i. \x$i\) {1..dimindex(UNIV::'n set)}" by (simp add: vector_norm_def setL2_le_setsum) -lemma real_abs_norm[simp]: "\ norm x\ = norm (x :: real ^'n)" +lemma real_abs_norm[simp]: "\ norm x\ = norm (x :: real ^'n)" by (rule abs_norm_cancel) lemma real_abs_sub_norm: "\norm(x::real ^'n) - norm y\ <= norm(x - y)" by (rule norm_triangle_ineq3) @@ -863,7 +863,7 @@ apply arith done -lemma norm_ge_square: "norm(x) >= a \ a <= 0 \ x \ x >= a ^ 2" +lemma norm_ge_square: "norm(x) >= a \ a <= 0 \ x \ x >= a ^ 2" apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric]) using norm_ge_zero[of x] apply arith @@ -891,7 +891,7 @@ 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" + 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) @@ -919,13 +919,13 @@ 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_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" +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))" @@ -941,7 +941,7 @@ lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \ 0 \ -x \ 0" by arith -lemma norm_pths: +lemma norm_pths: "(x::real ^'n) = y \ norm (x - y) \ 0" "x \ y \ \ (norm (x - y) \ 0)" using norm_ge_zero[of "x - y"] by auto @@ -967,26 +967,26 @@ lemma dist_eq_0[simp]: "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_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[simp]: "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[simp]: "dist x y <= 0 \ x = y" by norm + by norm + +lemma dist_mul[simp]: "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[simp]: "dist x y <= 0 \ x = y" by norm lemma setsum_eq: "setsum f S = (\ i. setsum (\x. (f x)$i ) S)" apply vector @@ -996,24 +996,24 @@ apply (auto simp add: vector_component zero_index) done -lemma setsum_clauses: +lemma setsum_clauses: shows "setsum f {} = 0" and "finite S \ setsum f (insert x S) = (if x \ S then setsum f S else f x + setsum f S)" by (auto simp add: insert_absorb) -lemma setsum_cmul: +lemma setsum_cmul: fixes f:: "'c \ ('a::semiring_1)^'n" shows "setsum (\x. c *s f x) S = c *s setsum f S" by (simp add: setsum_eq Cart_eq Cart_lambda_beta vector_component setsum_right_distrib) -lemma setsum_component: +lemma setsum_component: fixes f:: " 'a \ ('b::semiring_1) ^'n" assumes i: "i \ {1 .. dimindex(UNIV:: 'n set)}" shows "(setsum f S)$i = setsum (\x. (f x)$i) S" using i by (simp add: setsum_eq Cart_lambda_beta) -lemma setsum_norm: +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" @@ -1027,7 +1027,7 @@ finally show ?case using "2.hyps" by simp qed -lemma real_setsum_norm: +lemma real_setsum_norm: fixes f :: "'a \ real ^'n" assumes fS: "finite S" shows "norm (setsum f S) <= setsum (\x. norm(f x)) S" @@ -1041,25 +1041,25 @@ finally show ?case using "2.hyps" by simp qed -lemma setsum_norm_le: +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" + 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: +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" + 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 @@ -1089,9 +1089,9 @@ 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" + 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" + 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 . @@ -1105,20 +1105,20 @@ proof- let ?A = "{m .. n}" let ?B = "{n + 1 .. n + p}" - have eq: "{m .. n+p} = ?A \ ?B" using mn by auto + 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_natinterval_left: - assumes mn: "(m::nat) <= n" + 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: +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)" @@ -1136,8 +1136,8 @@ 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) + 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]) @@ -1149,14 +1149,14 @@ 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_mono_zero_right[OF fT fST]) by (auto intro: setsum_0') lemma vsum_norm_allsubsets_bound: fixes f:: "'a \ real ^'n" - assumes fP: "finite P" and fPs: "\Q. Q \ P \ norm (setsum f Q) \ e" + 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))" @@ -1183,9 +1183,9 @@ 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 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" + 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_zero) + apply (rule setsum_Un_zero) using fP thp0 by auto also have "\ \ 2*e" using Pne Ppe by arith finally show "setsum (\x. \f x $ i\) P \ 2*e" . @@ -1204,13 +1204,13 @@ definition "basis k = (\ i. if i = k then 1 else 0)" -lemma delta_mult_idempotent: +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 + 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"] @@ -1228,7 +1228,7 @@ 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" +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" @@ -1250,7 +1250,7 @@ "setsum (\i. (x$i) *s basis i) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _") by (auto simp add: Cart_eq basis_component[where ?'n = "'n"] setsum_component vector_component cond_value_iff setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong) -lemma basis_expansion_unique: +lemma basis_expansion_unique: "setsum (\i. f i *s basis i) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::comm_ring_1) ^'n) \ (\i\{1 .. dimindex(UNIV:: 'n set)}. f i = x$i)" by (simp add: Cart_eq setsum_component vector_component basis_component setsum_delta cond_value_iff cong del: if_weak_cong) @@ -1266,7 +1266,7 @@ 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: +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) @@ -1294,15 +1294,15 @@ definition "orthogonal x y \ (x \ y = 0)" lemma orthogonal_basis: - assumes i:"i \ {1 .. dimindex(UNIV ::'n set)}" + 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" + 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*) @@ -1443,14 +1443,14 @@ 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) + 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: +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" @@ -1470,7 +1470,7 @@ 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 + linear_cmul[OF lf] by simp lemma linear_injective_0: assumes lf: "linear (f:: 'a::ring_1 ^ 'n \ _)" @@ -1478,7 +1478,7 @@ 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)" + 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 . @@ -1518,7 +1518,7 @@ assumes lf: "linear f" shows "\B > 0. \x. norm (f x) \ B * norm x" proof- - from linear_bounded[OF lf] obtain B where + 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 @@ -1562,15 +1562,15 @@ 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: + +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] + using bilinear_ladd[OF bh, of 0 0 x] by (simp add: eq_add_iff ring_simps) -lemma bilinear_rzero: +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 ] + 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" @@ -1583,7 +1583,7 @@ 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- +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) @@ -1598,7 +1598,7 @@ 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- +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)" @@ -1626,7 +1626,7 @@ 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 + 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 @@ -1634,11 +1634,11 @@ {fix x::"real ^'m" and y :: "real ^'n" from KB Kp have "B * norm x * norm y \ ?K * norm x * norm y" - apply - + apply - apply (rule mult_right_mono, rule mult_right_mono) by (auto simp add: norm_ge_zero) then have "norm (h x y) \ ?K * norm x * norm y" - using B[rule_format, of x y] by simp} + using B[rule_format, of x y] by simp} with Kp show ?thesis by blast qed @@ -1663,14 +1663,14 @@ 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] + 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 + 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 @@ -1715,27 +1715,27 @@ consts generic_mult :: "'a \ 'b \ 'c" (infixr "\" 75) -defs (overloaded) +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 +abbreviation matrix_matrix_mult' :: "('a::semiring_1) ^'n^'m \ 'a ^'p^'n \ 'a ^ 'p ^'m" (infixl "**" 70) where "m ** m' == m\ m'" -defs (overloaded) +defs (overloaded) matrix_vector_mult_def: "(m::('a::semiring_1) ^'n^'m) \ (x::'a ^'n) \ (\ i. setsum (\j. ((m$i)$j) * (x$j)) {1..dimindex(UNIV ::'n set)}) :: 'a^'m" -abbreviation +abbreviation matrix_vector_mult' :: "('a::semiring_1) ^'n^'m \ 'a ^'n \ 'a ^ 'm" (infixl "*v" 70) - where + where "m *v v == m \ v" -defs (overloaded) +defs (overloaded) vector_matrix_mult_def: "(x::'a^'m) \ (m::('a::semiring_1) ^'n^'m) \ (\ j. setsum (\i. ((m$i)$j) * (x$i)) {1..dimindex(UNIV :: 'm set)}) :: 'a^'n" -abbreviation +abbreviation vactor_matrix_mult' :: "'a ^ 'm \ ('a::semiring_1) ^'n^'m \ 'a ^'n " (infixl "v*" 70) - where + where "v v* m == v \ m" definition "(mat::'a::zero => 'a ^'n^'m) k = (\ i j. if i = j then k else 0)" @@ -1749,11 +1749,11 @@ 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 = +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] + using setsum_delta[OF fS, of a b, symmetric] by (auto intro: setsum_cong) lemma matrix_mul_lid: "mat 1 ** A = A" @@ -1781,7 +1781,7 @@ 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 + 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)" @@ -1796,7 +1796,7 @@ 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: +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 @@ -1813,18 +1813,18 @@ lemma transp_transp: "transp(transp A) = A" by (vector transp_def) -lemma row_transp: +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 + 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 + 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" @@ -1890,8 +1890,8 @@ 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" +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) @@ -1923,9 +1923,9 @@ done -lemma real_convex_bound_lt: +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" + and uv: "u + v = 1" shows "u * x + v * y < a" proof- have uv': "u = 0 \ v \ 0" using u v uv by arith @@ -1937,7 +1937,7 @@ 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 @@ -1947,9 +1947,9 @@ thus ?thesis unfolding th . qed -lemma real_convex_bound_le: +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" + 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) @@ -1969,7 +1969,7 @@ done -lemma triangle_lemma: +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- @@ -1992,12 +1992,12 @@ 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$i)" using Cart_lambda_beta[of f, rule_format, OF i] by auto + then have "P i (?x$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 +qed (* Supremum and infimum of real sets *) @@ -2019,7 +2019,7 @@ 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 + 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 @@ -2030,12 +2030,12 @@ 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" + 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 + 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 + ultimately show ?thesis by arith qed lemma rsup_finite_in: assumes fS: "finite S" and Se: "S \ {}" @@ -2065,7 +2065,7 @@ lemma rsup_unique: assumes b: "S *<= b" and S: "\b' < b. \x \ S. b' < x" shows "rsup S = b" -using b S +using b S unfolding setle_def rsup_alt apply - apply (rule some_equality) @@ -2104,7 +2104,7 @@ 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 + show ?thesis using S b rsup_bounds[of S "l - e" "l+e"] unfolding th by (auto simp add: setge_def setle_def) qed @@ -2142,7 +2142,7 @@ 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 + 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 @@ -2153,12 +2153,12 @@ 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" + 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 + 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 + ultimately show ?thesis by arith qed lemma rinf_finite_in: assumes fS: "finite S" and Se: "S \ {}" @@ -2188,7 +2188,7 @@ lemma rinf_unique: assumes b: "b <=* S" and S: "\b' > b. \x \ S. b' > x" shows "rinf S = b" -using b S +using b S unfolding setge_def rinf_alt apply - apply (rule some_equality) @@ -2226,7 +2226,7 @@ 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 + show ?thesis using S b rinf_bounds[of S "l - e" "l+e"] unfolding th by (auto simp add: setge_def setle_def) qed @@ -2248,7 +2248,7 @@ moreover {assume H: ?lhs - from H[rule_format, of "basis 1"] + from H[rule_format, of "basis 1"] have bp: "b \ 0" using norm_ge_zero[of "f (basis 1)"] dimindex_ge_1[of "UNIV:: 'n set"] by (auto simp add: norm_basis elim: order_trans [OF norm_ge_zero]) {fix x :: "real ^'n" @@ -2260,9 +2260,9 @@ let ?c = "1/ norm x" have "norm (?c*s x) = 1" using x0 by (simp add: n0 norm_mul) with H have "norm (f(?c*s x)) \ b" by blast - hence "?c * norm (f x) \ b" + hence "?c * norm (f x) \ b" by (simp add: linear_cmul[OF lf] norm_mul) - hence "norm (f x) \ b * norm x" + hence "norm (f x) \ b * norm x" using n0 norm_ge_zero[of x] by (auto simp add: field_simps)} ultimately have "norm (f x) \ b * norm x" by blast} then have ?rhs by blast} @@ -2278,16 +2278,16 @@ { 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" + 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 - + 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" + 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)} @@ -2297,7 +2297,7 @@ lemma onorm_pos_le: assumes lf: "linear (f::real ^'n \ real ^'m)" shows "0 <= onorm f" using order_trans[OF norm_ge_zero 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)" +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: onorm_pos_le) @@ -2317,7 +2317,7 @@ apply (rule rsup_unique) by (simp_all add: setle_def) qed -lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n \ real ^'m)" +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 @@ -2374,7 +2374,7 @@ definition vec1:: "'a \ 'a ^ 1" where "vec1 x = (\ i. x)" -definition dest_vec1:: "'a ^1 \ 'a" +definition dest_vec1:: "'a ^1 \ 'a" where "dest_vec1 x = (x$1)" lemma vec1_component[simp]: "(vec1 x)$1 = x" @@ -2385,7 +2385,7 @@ 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 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) @@ -2446,7 +2446,7 @@ lemma abs_dest_vec1: "norm x = \dest_vec1 x\" by (metis vec1_dest_vec1 norm_vec1) -lemma linear_vmul_dest_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 @@ -2563,10 +2563,10 @@ 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)" + 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 th0 unfolding real_vector_norm_def real_sqrt_le_iff id_def by (simp add: dot_def dimindex_finite_sum Cart_lambda_beta) qed @@ -2592,13 +2592,13 @@ 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" + 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) + 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 th0 unfolding real_vector_norm_def real_sqrt_le_iff id_def by (simp add: dot_def dimindex_finite_sum Cart_lambda_beta) qed @@ -2644,14 +2644,14 @@ 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) + 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 + finally show ?thesis by (simp add: dot_def) qed @@ -2679,7 +2679,7 @@ 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 +using hull_same[of s S] hull_in[of S s] by metis lemma hull_hull: "S hull (S hull s) = S hull s" @@ -2749,12 +2749,12 @@ lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n" proof(induct n) case 0 thus ?case by simp -next +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]) + 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) @@ -2763,13 +2763,13 @@ 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] + 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 + from real_pow_lbound[OF x00, of n] n have "y < x^n" by auto then show ?thesis by metis -qed +qed lemma real_arch_pow2: "\n. (x::real) < 2^ n" using real_arch_pow[of 2 x] by simp @@ -2777,13 +2777,13 @@ lemma real_arch_pow_inv: assumes y: "(y::real) > 0" and x1: "x < 1" shows "\n. x^n < y" proof- - {assume x0: "x > 0" + {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 + then have ?thesis using y x0 by (auto simp add: field_simps power_divide) } - moreover + moreover {assume "\ x > 0" with y x1 have ?thesis apply auto by (rule exI[where x=1], auto)} ultimately show ?thesis by metis qed @@ -2821,18 +2821,18 @@ 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 +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" + 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 +qed (* ------------------------------------------------------------------------- *) (* Geometric progression. *) @@ -2863,9 +2863,9 @@ 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}" + 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)" + 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 @@ -2873,8 +2873,8 @@ 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) +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} @@ -2889,7 +2889,7 @@ ultimately show ?thesis by metis qed -lemma sum_gp_offset: "setsum (op ^ (x::'a::{field,recpower})) {m .. m+n} = +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) @@ -2908,7 +2908,7 @@ lemma subspace_0: "subspace S ==> 0 \ S" by (metis subspace_def) -lemma subspace_add: "subspace S \ x \ S \ y \ S ==> x + y \ S" +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" @@ -2926,10 +2926,10 @@ 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" + 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 @@ -2986,7 +2986,7 @@ 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) + show "P x" by (metis mem_def subset_eq) qed lemma span_empty: "span {} = {(0::'a::semiring_0 ^ 'n)}" @@ -3016,11 +3016,11 @@ using span_induct SP P by blast inductive span_induct_alt_help for S:: "'a::semiring_1^'n \ bool" - where + 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': +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" @@ -3031,7 +3031,7 @@ 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 . @@ -3043,7 +3043,7 @@ 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 + from h have "span_induct_alt_help S (x + y)" apply (induct rule: span_induct_alt_help.induct) apply simp @@ -3054,7 +3054,7 @@ done} moreover {fix c x assume xt: "span_induct_alt_help S x" - then have "span_induct_alt_help S (c*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) @@ -3063,13 +3063,13 @@ apply simp done } - ultimately show "subspace (span_induct_alt_help S)" + 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: +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 @@ -3118,9 +3118,9 @@ apply (rule subspace_span) apply (rule x) done} - moreover + 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) + 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]) @@ -3146,15 +3146,15 @@ apply (rule exI[where x="1"], simp) by (rule span_0)} moreover - {assume ab: "x \ b" + {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 + moreover have "subspace ?P" + unfolding subspace_def apply auto apply (simp add: mem_def) apply (rule exI[where x=0]) @@ -3174,7 +3174,7 @@ 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 + ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis qed lemma span_breakdown_eq: @@ -3186,7 +3186,7 @@ apply (rule_tac x= "k" in exI) apply (rule set_rev_mp[of _ "span (S - {a})" _]) apply assumption - apply (rule span_mono) + apply (rule span_mono) apply blast done} moreover @@ -3196,7 +3196,7 @@ apply (rule span_add) apply (rule set_rev_mp[of _ "span S" _]) apply (rule k) - apply (rule span_mono) + apply (rule span_mono) apply blast apply (rule span_mul) apply (rule span_superset) @@ -3224,7 +3224,7 @@ done with na have ?thesis by blast} moreover - {assume k0: "k \ 0" + {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) @@ -3247,8 +3247,8 @@ ultimately show ?thesis by blast qed -lemma in_span_delete: - assumes a: "(a::'a::field^'n) \ span S" +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) @@ -3268,7 +3268,7 @@ 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 + show ?thesis apply (subst eq) apply (rule span_add) apply (rule set_rev_mp) @@ -3304,18 +3304,18 @@ 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" + 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}" + 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]] + 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]]) @@ -3324,7 +3324,7 @@ 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 + moreover {assume xS: "x \ S" have th00: "(\v\S. (if v = x then c else u v) *s v) = y" unfolding u[symmetric] @@ -3334,7 +3334,7 @@ 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)" + then show "?h (c*s x + y)" apply - apply (rule exI[where x="?S"]) apply (rule exI[where x="?u"]) by metis @@ -3346,11 +3346,11 @@ "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" + 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 ?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 @@ -3366,16 +3366,16 @@ 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" + {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 ?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 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 + 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 @@ -3398,7 +3398,7 @@ (is "_ = ?rhs") proof- {fix y assume y: "y \ span S" - from y obtain S' u where fS': "finite S'" and SS': "S' \ S" and + 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' @@ -3410,7 +3410,7 @@ done hence "setsum (\v. ?u v *s v) S = y" by (metis u) hence "y \ ?rhs" by auto} - moreover + 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 @@ -3431,7 +3431,7 @@ 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 @@ -3461,10 +3461,10 @@ {fix x::"'a^'n" assume xS: "x\ ?B" from xS have "?P x" by (auto simp add: basis_component)} moreover - have "subspace ?P" + 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 + 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)}}" @@ -3508,7 +3508,7 @@ apply assumption apply blast by (simp add: dependent_def)} - moreover + moreover {assume i: ?rhs have ?lhs using i aS apply simp @@ -3541,7 +3541,7 @@ by (metis subset_eq span_superset) lemma spanning_subset_independent: - assumes BA: "B \ A" and iA: "independent (A::('a::field ^'n) set)" + assumes BA: "B \ A" and iA: "independent (A::('a::field ^'n) set)" and AsB: "A \ span B" shows "A = B" proof @@ -3569,7 +3569,7 @@ lemma exchange_lemma: assumes f:"finite (t:: ('a::field^'n) set)" and i: "independent s" - and sp:"s \ span t" + 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) @@ -3584,15 +3584,15 @@ 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]) + 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]) + + 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" @@ -3603,28 +3603,28 @@ 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] + 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 + 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" + 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})" + {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" + 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" @@ -3637,15 +3637,15 @@ 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 + + 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 + ultimately show ?ths by blast qed @@ -3659,7 +3659,7 @@ 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 + show ?thesis unfolding eq apply (rule finite_imageI) apply (rule finite_intvl) done @@ -3668,7 +3668,7 @@ 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 + show ?thesis unfolding eq apply (rule finite_imageI) apply (rule finite_atLeastAtMost) done @@ -3682,7 +3682,7 @@ apply (rule independent_span_bound) apply (rule finite_Atleast_Atmost_nat) apply assumption - unfolding span_stdbasis + unfolding span_stdbasis apply (rule subset_UNIV) done @@ -3710,14 +3710,14 @@ 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 + 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" + 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} @@ -3732,7 +3732,7 @@ 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)" +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 @@ -3784,7 +3784,7 @@ qed lemma card_le_dim_spanning: - assumes BV: "(B:: (real ^'n) set) \ V" and VB: "V \ span B" + 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- @@ -3794,10 +3794,10 @@ 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 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 + 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})" . @@ -3806,7 +3806,7 @@ 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 + by (metis hassize_def order_eq_iff card_le_dim_spanning card_ge_dim_independent) (* ------------------------------------------------------------------------- *) @@ -3818,18 +3818,18 @@ 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) + 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)" + have th0: "dim S \ dim (span S)" by (auto simp add: subset_eq intro: dim_subset span_superset) - from basis_exists[of S] + 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 + 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 @@ -3847,7 +3847,7 @@ 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 + 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)" @@ -3860,7 +3860,7 @@ (* Relation between bases and injectivity/surjectivity of map. *) lemma spanning_surjective_image: - assumes us: "UNIV \ span (S:: ('a::semiring_1 ^'n) set)" + assumes us: "UNIV \ span (S:: ('a::semiring_1 ^'n) set)" and lf: "linear f" and sf: "surj f" shows "UNIV \ span (f ` S)" proof- @@ -3881,7 +3881,7 @@ 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 +qed (* ------------------------------------------------------------------------- *) (* Picking an orthogonal replacement for a spanning set. *) @@ -3904,15 +3904,15 @@ 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" + 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 + {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) @@ -3924,18 +3924,18 @@ 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 + 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" + {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" + {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 simp apply (subst Cy) using C(1) fth apply (simp only: setsum_clauses) @@ -3946,13 +3946,13 @@ apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format]) by auto} moreover - {assume xa: "x \ ?a" "x \ C" and ya: "y = ?a" + {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 simp apply (subst Cx) using C(1) fth apply (simp only: setsum_clauses) @@ -3963,12 +3963,12 @@ apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format]) by auto} moreover - {assume xa: "x \ C" and ya: "y \ C" + {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 + then show ?case by blast qed lemma orthogonal_basis_exists: @@ -3977,18 +3977,18 @@ 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 + 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 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) + 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 + from C B CSV CdV iC show ?thesis by auto qed lemma span_eq: "span S = span T \ S \ span T \ T \ span S" @@ -4003,8 +4003,8 @@ 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" + 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)] @@ -4020,12 +4020,12 @@ 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) + 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" + have "?a \ x = 0" apply (subst B') using fB fth unfolding setsum_clauses(2)[OF fth] apply simp @@ -4038,7 +4038,7 @@ with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"]) qed -lemma span_not_univ_subset_hyperplane: +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 @@ -4058,9 +4058,9 @@ (* 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" + 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 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 @@ -4070,11 +4070,11 @@ 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 + 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)]) + 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 @@ -4084,16 +4084,16 @@ 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)" + hence th: "-k *s f a \ span (f ` b)" using "2.prems"(5) by (simp add: vector_smult_lneg) - {assume k0: "k = 0" + {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)" + 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 @@ -4112,17 +4112,17 @@ 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) + 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) + 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 + 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" @@ -4132,12 +4132,12 @@ 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" + 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)"] + 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)} @@ -4146,26 +4146,26 @@ 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)" + 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" + 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)" + 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" + 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" + 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 @@ -4177,7 +4177,7 @@ using conjunct1[OF h, OF span_superset, OF insertI1] by (auto simp add: span_0) - from xa ha1[symmetric] have "?g x = f x" + from xa ha1[symmetric] have "?g x = f x" apply simp using g(2)[rule_format, OF span_0, of 0] by simp} @@ -4201,12 +4201,12 @@ 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) + 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 + from g show ?thesis unfolding linear_def using C apply clarsimp by blast qed @@ -4218,7 +4218,7 @@ proof(induct arbitrary: B rule: finite_induct[OF fA]) case 1 thus ?case by simp next - case (2 x s t) + case (2 x s t) thus ?case proof(induct rule: finite_induct[OF "2.prems"(1)]) case 1 then show ?case by simp @@ -4234,7 +4234,7 @@ qed qed -lemma card_subset_eq: assumes fB: "finite B" and AB: "A \ B" and +lemma card_subset_eq: assumes fB: "finite B" and AB: "A \ B" and c: "card A = card B" shows "A = B" proof- @@ -4245,27 +4245,27 @@ 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 + with AB show "A = B" by blast qed lemma subspace_isomorphism: - assumes s: "subspace (S:: (real ^'n) set)" and t: "subspace T" + 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 + 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 + 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 + 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) + 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 + 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) @@ -4277,9 +4277,9 @@ {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 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" + 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)]) @@ -4308,20 +4308,20 @@ qed lemma linear_eq_0: - assumes lf: "linear f" and SB: "S \ span B" and f0: "\x\B. f x = 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" + 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 +qed lemma linear_eq_stdbasis: assumes lf: "linear (f::'a::ring_1^'m \ 'a^'n)" and lg: "linear g" @@ -4329,7 +4329,7 @@ shows "f = g" proof- let ?U = "UNIV :: 'm set" - let ?I = "{basis i:: 'a^'m|i. i \ {1 .. dimindex ?U}}" + 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 @@ -4341,27 +4341,27 @@ (* Similar results for bilinear functions. *) lemma bilinear_eq: - assumes bf: "bilinear (f:: 'a::ring^'m \ 'a^'n \ 'a^'p)" + 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 + 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" + have "\x \ span B. \y\ span C. f x y = g x y" apply - apply (rule ballI) - apply (rule span_induct[of B ?P]) + 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 + 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]) @@ -4369,7 +4369,7 @@ qed lemma bilinear_eq_stdbasis: - assumes bf: "bilinear (f:: 'a::ring_1^'m \ 'a^'n \ 'a^'p)" + 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" @@ -4394,16 +4394,16 @@ 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) + 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 + then show ?thesis using h(1) by blast qed lemma linear_surjective_right_inverse: @@ -4411,18 +4411,18 @@ shows "\g. linear g \ f o g = id" proof- from linear_independent_extend[OF independent_stdbasis] - obtain h:: "real ^'n \ real ^'m" where + 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) + 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 + then show ?thesis using h(1) by blast qed lemma matrix_left_invertible_injective: @@ -4434,7 +4434,7 @@ 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 + 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" @@ -4454,25 +4454,25 @@ "(\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" + {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" + 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)] + 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 +qed lemma matrix_left_invertible_independent_columns: fixes A :: "real^'n^'m" @@ -4481,7 +4481,7 @@ 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" + {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" @@ -4493,11 +4493,11 @@ hence ?rhs by blast} moreover {assume H: ?rhs - {fix x assume x: "A *v x = 0" + {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 + ultimately show ?thesis unfolding matrix_left_invertible_ker by blast qed lemma matrix_right_invertible_independent_rows: @@ -4514,13 +4514,13 @@ have fU: "finite ?U" by simp have lhseq: "?lhs \ (\y. \(x::real^'m). setsum (\i. (x$i) *s column i A) ?U = y)" unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def - apply (subst eq_commute) .. + apply (subst eq_commute) .. have rhseq: "?rhs \ (\x. x \ span (columns A))" by blast {assume h: ?lhs - {fix x:: "real ^'n" + {fix x:: "real ^'n" from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m" where y: "setsum (\i. (y$i) *s column i A) ?U = x" by blast - have "x \ span (columns A)" + have "x \ span (columns A)" unfolding y[symmetric] apply (rule span_setsum[OF fU]) apply clarify @@ -4532,21 +4532,21 @@ moreover {assume h:?rhs let ?P = "\(y::real ^'n). \(x::real^'m). setsum (\i. (x$i) *s column i A) ?U = y" - {fix y have "?P y" + {fix y have "?P y" proof(rule span_induct_alt[of ?P "columns A"]) show "\x\real ^ 'm. setsum (\i. (x$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" + from y1 obtain i where i: "i \ ?U" "y1 = column i A" unfolding columns_def by blast - from y2 obtain x:: "real ^'m" where + 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$j))::real^'m" show "?P (c*s y1 + y2)" proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric]Cart_lambda_beta setsum_component cond_value_iff right_distrib cond_application_beta vector_component cong del: if_weak_cong, simp only: One_nat_def[symmetric]) - fix j + fix j have th: "\xa \ ?U. (if xa = i then (c + (x$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) @@ -4558,7 +4558,7 @@ by (simp add: setsum_addf) also have "\ = c * ((column i A)$j) + setsum (\xa. ((x$xa) * ((column xa A)$j))) ?U" unfolding setsum_delta[OF fU] - using i(1) by simp + using i(1) by simp finally show "setsum (\xa. if xa = i then (c + (x$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 @@ -4579,12 +4579,12 @@ (* 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" + 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" + 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)" @@ -4604,7 +4604,7 @@ (* And vice versa. *) -lemma surjective_iff_injective_gen: +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") @@ -4641,17 +4641,17 @@ qed lemma linear_surjective_imp_injective: - assumes lf: "linear (f::real ^'n => real ^'n)" and sf: "surj f" + 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" + 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)" + have fBi: "independent (f ` B)" apply (rule card_le_dim_spanning[of "f ` B" ?U]) apply blast using sf B(3) @@ -4676,12 +4676,12 @@ 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" + 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] + from th show ?thesis unfolding linear_injective_0[OF lf] using B(3) by blast qed @@ -4689,7 +4689,7 @@ lemma left_right_inverse_eq: assumes fg: "f o g = id" and gh: "g o h = id" - shows "f = h" + 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) @@ -4723,7 +4723,7 @@ {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 @@ -4735,13 +4735,13 @@ (* 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" + 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 + 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]) @@ -4750,13 +4750,13 @@ qed lemma right_inverse_linear: - assumes lf: "linear (f:: real ^'n \ real ^'n)" and gf: "f o g = id" + 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 + 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]) @@ -4767,7 +4767,7 @@ (* The same result in terms of square matrices. *) lemma matrix_left_right_inverse: - fixes A A' :: "real ^'n^'n" + 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" @@ -4779,7 +4779,7 @@ 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" + 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) @@ -4846,17 +4846,17 @@ 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 + 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 + unfolding infnorm_set_image bex_simps apply (subst th) - unfolding th1 + unfolding th1 unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma] - - unfolding infnorm_set_image ball_simps bex_simps + + unfolding infnorm_set_image ball_simps bex_simps apply (simp add: vector_add_component) apply (metis numseg_dimindex_nonempty th2) done @@ -4885,7 +4885,7 @@ apply (simp add: vector_component) done -lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)" +lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)" proof- have "y - x = - (x - y)" by simp then show ?thesis by (metis infnorm_neg) @@ -4896,7 +4896,7 @@ 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" + 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 . @@ -4911,11 +4911,11 @@ 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 + 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 + show ?thesis unfolding infnorm_def isUb_def setle_def unfolding infnorm_set_image ball_simps by auto qed @@ -4942,7 +4942,7 @@ 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)" + 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 @@ -4954,7 +4954,7 @@ (* 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_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 @@ -4968,20 +4968,20 @@ 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 + 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 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 id_def . + show ?thesis unfolding real_vector_norm_def id_def . qed (* Equality in Cauchy-Schwarz and triangle inequalities. *) @@ -5037,7 +5037,7 @@ {assume x: "x \ 0" and y: "y \ 0" hence "norm x \ 0" "norm y \ 0" by simp_all - hence n: "norm x > 0" "norm y > 0" + hence n: "norm x > 0" "norm y > 0" using norm_ge_zero[of x] norm_ge_zero[of y] by arith+ have th: "\(a::real) b c. a + b + c \ 0 ==> (a = b + c \ a^2 = (b + c)^2)" by algebra @@ -5058,7 +5058,7 @@ lemma collinear_empty: "collinear {}" by (simp add: collinear_def) -lemma collinear_sing: "collinear {(x::'a::ring_1^'n)}" +lemma collinear_sing: "collinear {(x::'a::ring_1^'n)}" apply (simp add: collinear_def) apply (rule exI[where x=0]) by simp @@ -5075,20 +5075,20 @@ 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 + {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 + 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" + from cx cy cx0 have "y = ?d *s x" by (simp add: vector_smult_assoc) hence ?rhs using x y by blast} moreover