new material on matrices, etc., and consolidating duplicate results about of_nat
authorpaulson <lp15@cam.ac.uk>
Sun Feb 25 12:54:55 2018 +0000 (15 months ago)
changeset 67719bffb7482faaa
parent 67717 5a1b299fe4af
child 67720 b342f96e47b5
new material on matrices, etc., and consolidating duplicate results about of_nat
src/HOL/Analysis/Ball_Volume.thy
src/HOL/Analysis/Cartesian_Euclidean_Space.thy
src/HOL/Analysis/Great_Picard.thy
src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
src/HOL/HOL.thy
src/HOL/Library/Extended_Nonnegative_Real.thy
     1.1 --- a/src/HOL/Analysis/Ball_Volume.thy	Sat Feb 24 17:21:35 2018 +0100
     1.2 +++ b/src/HOL/Analysis/Ball_Volume.thy	Sun Feb 25 12:54:55 2018 +0000
     1.3 @@ -17,8 +17,11 @@
     1.4  definition unit_ball_vol :: "real \<Rightarrow> real" where
     1.5    "unit_ball_vol n = pi powr (n / 2) / Gamma (n / 2 + 1)"
     1.6  
     1.7 +lemma unit_ball_vol_pos [simp]: "n \<ge> 0 \<Longrightarrow> unit_ball_vol n > 0"
     1.8 +  by (force simp: unit_ball_vol_def intro: divide_nonneg_pos)
     1.9 +
    1.10  lemma unit_ball_vol_nonneg [simp]: "n \<ge> 0 \<Longrightarrow> unit_ball_vol n \<ge> 0"
    1.11 -  by (auto simp add: unit_ball_vol_def intro!: divide_nonneg_pos Gamma_real_pos)
    1.12 +  by (simp add: dual_order.strict_implies_order)
    1.13  
    1.14  text \<open>
    1.15    We first need the value of the following integral, which is at the core of
     2.1 --- a/src/HOL/Analysis/Cartesian_Euclidean_Space.thy	Sat Feb 24 17:21:35 2018 +0100
     2.2 +++ b/src/HOL/Analysis/Cartesian_Euclidean_Space.thy	Sun Feb 25 12:54:55 2018 +0000
     2.3 @@ -7,19 +7,6 @@
     2.4  lemma subspace_special_hyperplane: "subspace {x. x $ k = 0}"
     2.5    by (simp add: subspace_def)
     2.6  
     2.7 -lemma delta_mult_idempotent:
     2.8 -  "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)"
     2.9 -  by simp
    2.10 -
    2.11 -(*move up?*)
    2.12 -lemma sum_UNIV_sum:
    2.13 -  fixes g :: "'a::finite + 'b::finite \<Rightarrow> _"
    2.14 -  shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))"
    2.15 -  apply (subst UNIV_Plus_UNIV [symmetric])
    2.16 -  apply (subst sum.Plus)
    2.17 -  apply simp_all
    2.18 -  done
    2.19 -
    2.20  lemma sum_mult_product:
    2.21    "sum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
    2.22    unfolding sum_nat_group[of h B A, unfolded atLeast0LessThan, symmetric]
    2.23 @@ -632,7 +619,7 @@
    2.24      by simp
    2.25  qed
    2.26  
    2.27 -text\<open>Inverse matrices  (not necessarily square)\<close>
    2.28 +subsection\<open>Inverse matrices  (not necessarily square)\<close>
    2.29  
    2.30  definition
    2.31    "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
    2.32 @@ -728,9 +715,86 @@
    2.33    done
    2.34  
    2.35  
    2.36 -subsection \<open>lambda skolemization on cartesian products\<close>
    2.37 +subsection\<open>Some bounds on components etc. relative to operator norm.\<close>
    2.38 +
    2.39 +lemma norm_column_le_onorm:
    2.40 +  fixes A :: "real^'n^'m"
    2.41 +  shows "norm(column i A) \<le> onorm(( *v) A)"
    2.42 +proof -
    2.43 +  have bl: "bounded_linear (( *v) A)"
    2.44 +    by (simp add: linear_linear matrix_vector_mul_linear)
    2.45 +  have "norm (\<chi> j. A $ j $ i) \<le> norm (A *v axis i 1)"
    2.46 +    by (simp add: matrix_mult_dot cart_eq_inner_axis)
    2.47 +  also have "\<dots> \<le> onorm (( *v) A)"
    2.48 +    using onorm [OF bl, of "axis i 1"] by (auto simp: axis_in_Basis)
    2.49 +  finally have "norm (\<chi> j. A $ j $ i) \<le> onorm (( *v) A)" .
    2.50 +  then show ?thesis
    2.51 +    unfolding column_def .
    2.52 +qed
    2.53 +
    2.54 +lemma matrix_component_le_onorm:
    2.55 +  fixes A :: "real^'n^'m"
    2.56 +  shows "\<bar>A $ i $ j\<bar> \<le> onorm(( *v) A)"
    2.57 +proof -
    2.58 +  have "\<bar>A $ i $ j\<bar> \<le> norm (\<chi> n. (A $ n $ j))"
    2.59 +    by (metis (full_types, lifting) component_le_norm_cart vec_lambda_beta)
    2.60 +  also have "\<dots> \<le> onorm (( *v) A)"
    2.61 +    by (metis (no_types) column_def norm_column_le_onorm)
    2.62 +  finally show ?thesis .
    2.63 +qed
    2.64 +
    2.65 +lemma component_le_onorm:
    2.66 +  fixes f :: "real^'m \<Rightarrow> real^'n"
    2.67 +  shows "linear f \<Longrightarrow> \<bar>matrix f $ i $ j\<bar> \<le> onorm f"
    2.68 +  by (metis matrix_component_le_onorm matrix_vector_mul)
    2.69  
    2.70 -(* FIXME: rename do choice_cart *)
    2.71 +lemma onorm_le_matrix_component_sum:
    2.72 +  fixes A :: "real^'n^'m"
    2.73 +  shows "onorm(( *v) A) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>)"
    2.74 +proof (rule onorm_le)
    2.75 +  fix x
    2.76 +  have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) $ i\<bar>)"
    2.77 +    by (rule norm_le_l1_cart)
    2.78 +  also have "\<dots> \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)"
    2.79 +  proof (rule sum_mono)
    2.80 +    fix i
    2.81 +    have "\<bar>(A *v x) $ i\<bar> \<le> \<bar>\<Sum>j\<in>UNIV. A $ i $ j * x $ j\<bar>"
    2.82 +      by (simp add: matrix_vector_mult_def)
    2.83 +    also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j * x $ j\<bar>)"
    2.84 +      by (rule sum_abs)
    2.85 +    also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)"
    2.86 +      by (rule sum_mono) (simp add: abs_mult component_le_norm_cart mult_left_mono)
    2.87 +    finally show "\<bar>(A *v x) $ i\<bar> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)" .
    2.88 +  qed
    2.89 +  finally show "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>) * norm x"
    2.90 +    by (simp add: sum_distrib_right)
    2.91 +qed
    2.92 +
    2.93 +lemma onorm_le_matrix_component:
    2.94 +  fixes A :: "real^'n^'m"
    2.95 +  assumes "\<And>i j. abs(A$i$j) \<le> B"
    2.96 +  shows "onorm(( *v) A) \<le> real (CARD('m)) * real (CARD('n)) * B"
    2.97 +proof (rule onorm_le)
    2.98 +  fix x :: "(real, 'n) vec"
    2.99 +  have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) $ i\<bar>)"
   2.100 +    by (rule norm_le_l1_cart)
   2.101 +  also have "\<dots> \<le> (\<Sum>i::'m \<in>UNIV. real (CARD('n)) * B * norm x)"
   2.102 +  proof (rule sum_mono)
   2.103 +    fix i
   2.104 +    have "\<bar>(A *v x) $ i\<bar> \<le> norm(A $ i) * norm x"
   2.105 +      by (simp add: matrix_mult_dot Cauchy_Schwarz_ineq2)
   2.106 +    also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>) * norm x"
   2.107 +      by (simp add: mult_right_mono norm_le_l1_cart)
   2.108 +    also have "\<dots> \<le> real (CARD('n)) * B * norm x"
   2.109 +      by (simp add: assms sum_bounded_above mult_right_mono)
   2.110 +    finally show "\<bar>(A *v x) $ i\<bar> \<le> real (CARD('n)) * B * norm x" .
   2.111 +  qed
   2.112 +  also have "\<dots> \<le> CARD('m) * real (CARD('n)) * B * norm x"
   2.113 +    by simp
   2.114 +  finally show "norm (A *v x) \<le> CARD('m) * real (CARD('n)) * B * norm x" .
   2.115 +qed
   2.116 +
   2.117 +subsection \<open>lambda skolemization on cartesian products\<close>
   2.118  
   2.119  lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
   2.120     (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
   2.121 @@ -751,6 +815,32 @@
   2.122    ultimately show ?thesis by metis
   2.123  qed
   2.124  
   2.125 +lemma rational_approximation:
   2.126 +  assumes "e > 0"
   2.127 +  obtains r::real where "r \<in> \<rat>" "\<bar>r - x\<bar> < e"
   2.128 +  using Rats_dense_in_real [of "x - e/2" "x + e/2"] assms by auto
   2.129 +
   2.130 +lemma matrix_rational_approximation:
   2.131 +  fixes A :: "real^'n^'m"
   2.132 +  assumes "e > 0"
   2.133 +  obtains B where "\<And>i j. B$i$j \<in> \<rat>" "onorm(\<lambda>x. (A - B) *v x) < e"
   2.134 +proof -
   2.135 +  have "\<forall>i j. \<exists>q \<in> \<rat>. \<bar>q - A $ i $ j\<bar> < e / (2 * CARD('m) * CARD('n))"
   2.136 +    using assms by (force intro: rational_approximation [of "e / (2 * CARD('m) * CARD('n))"])
   2.137 +  then obtain B where B: "\<And>i j. B$i$j \<in> \<rat>" and Bclo: "\<And>i j. \<bar>B$i$j - A $ i $ j\<bar> < e / (2 * CARD('m) * CARD('n))"
   2.138 +    by (auto simp: lambda_skolem Bex_def)
   2.139 +  show ?thesis
   2.140 +  proof
   2.141 +    have "onorm (( *v) (A - B)) \<le> real CARD('m) * real CARD('n) *
   2.142 +    (e / (2 * real CARD('m) * real CARD('n)))"
   2.143 +      apply (rule onorm_le_matrix_component)
   2.144 +      using Bclo by (simp add: abs_minus_commute less_imp_le)
   2.145 +    also have "\<dots> < e"
   2.146 +      using \<open>0 < e\<close> by (simp add: divide_simps)
   2.147 +    finally show "onorm (( *v) (A - B)) < e" .
   2.148 +  qed (use B in auto)
   2.149 +qed
   2.150 +
   2.151  lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
   2.152    unfolding inner_simps scalar_mult_eq_scaleR by auto
   2.153  
   2.154 @@ -1139,7 +1229,7 @@
   2.155      and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
   2.156    using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
   2.157  
   2.158 -lemma inter_interval_cart:
   2.159 +lemma Int_interval_cart:
   2.160    fixes a :: "real^'n"
   2.161    shows "cbox a b \<inter> cbox c d =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
   2.162    unfolding Int_interval
   2.163 @@ -1225,6 +1315,11 @@
   2.164      by (auto simp: axis_eq_axis inj_on_def *)
   2.165  qed
   2.166  
   2.167 +lemma dim_subset_UNIV_cart:
   2.168 +  fixes S :: "(real^'n) set"
   2.169 +  shows "dim S \<le> CARD('n)"
   2.170 +  by (metis dim_subset_UNIV DIM_cart DIM_real mult.right_neutral)
   2.171 +
   2.172  lemma affinity_inverses:
   2.173    assumes m0: "m \<noteq> (0::'a::field)"
   2.174    shows "(\<lambda>x. m *s x + c) \<circ> (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
   2.175 @@ -1430,13 +1525,7 @@
   2.176    unfolding vector_def by simp_all
   2.177  
   2.178  lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
   2.179 -  apply auto
   2.180 -  apply (erule_tac x="v$1" in allE)
   2.181 -  apply (subgoal_tac "vector [v$1] = v")
   2.182 -  apply simp
   2.183 -  apply (vector vector_def)
   2.184 -  apply simp
   2.185 -  done
   2.186 +  by (metis vector_1 vector_one)
   2.187  
   2.188  lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
   2.189    apply auto
     3.1 --- a/src/HOL/Analysis/Great_Picard.thy	Sat Feb 24 17:21:35 2018 +0100
     3.2 +++ b/src/HOL/Analysis/Great_Picard.thy	Sun Feb 25 12:54:55 2018 +0000
     3.3 @@ -46,7 +46,7 @@
     3.4    have "(n-1)^2 \<le> n^2 - 1"
     3.5      using assms by (simp add: algebra_simps power2_eq_square)
     3.6    then have "real (n - 1) \<le> sqrt (real (n\<^sup>2 - 1))"
     3.7 -    by (metis Extended_Nonnegative_Real.of_nat_le_iff of_nat_power real_le_rsqrt)
     3.8 +    by (metis of_nat_le_iff of_nat_power real_le_rsqrt)
     3.9    then have "n-1 \<le> sqrt(real n ^ 2 - 1)"
    3.10      by (simp add: Suc_leI assms of_nat_diff)
    3.11    then show ?thesis
     4.1 --- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Sat Feb 24 17:21:35 2018 +0100
     4.2 +++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Sun Feb 25 12:54:55 2018 +0000
     4.3 @@ -46,6 +46,14 @@
     4.4  lemma content_cbox_if: "content (cbox a b) = (if cbox a b = {} then 0 else \<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
     4.5    by (simp add: content_cbox')
     4.6  
     4.7 +lemma content_cbox_cart:
     4.8 +   "cbox a b \<noteq> {} \<Longrightarrow> content(cbox a b) = prod (\<lambda>i. b$i - a$i) UNIV"
     4.9 +  by (simp add: content_cbox_if Basis_vec_def cart_eq_inner_axis axis_eq_axis prod.UNION_disjoint)
    4.10 +
    4.11 +lemma content_cbox_if_cart:
    4.12 +   "content(cbox a b) = (if cbox a b = {} then 0 else prod (\<lambda>i. b$i - a$i) UNIV)"
    4.13 +  by (simp add: content_cbox_cart)
    4.14 +
    4.15  lemma content_division_of:
    4.16    assumes "K \<in> \<D>" "\<D> division_of S"
    4.17    shows "content K = (\<Prod>i \<in> Basis. interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)"
    4.18 @@ -357,6 +365,9 @@
    4.19    unfolding integral_def
    4.20    by (rule some_equality) (auto intro: has_integral_unique)
    4.21  
    4.22 +lemma has_integral_iff: "(f has_integral i) S \<longleftrightarrow> (f integrable_on S \<and> integral S f = i)"
    4.23 +  by blast
    4.24 +
    4.25  lemma eq_integralD: "integral k f = y \<Longrightarrow> (f has_integral y) k \<or> ~ f integrable_on k \<and> y=0"
    4.26    unfolding integral_def integrable_on_def
    4.27    apply (erule subst)
    4.28 @@ -3005,7 +3016,6 @@
    4.29    shows "f integrable_on {c..d}"
    4.30    by (metis assms box_real(2) integrable_subinterval)
    4.31  
    4.32 -
    4.33  subsection \<open>Combining adjacent intervals in 1 dimension.\<close>
    4.34  
    4.35  lemma has_integral_combine:
    4.36 @@ -4673,6 +4683,46 @@
    4.37      using as by auto
    4.38  qed
    4.39  
    4.40 +subsection\<open>Integrals on set differences\<close>
    4.41 +
    4.42 +lemma has_integral_setdiff:
    4.43 +  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
    4.44 +  assumes S: "(f has_integral i) S" and T: "(f has_integral j) T"
    4.45 +    and neg: "negligible (T - S)"
    4.46 +  shows "(f has_integral (i - j)) (S - T)"
    4.47 +proof -
    4.48 +  show ?thesis
    4.49 +    unfolding has_integral_restrict_UNIV [symmetric, of f]
    4.50 +  proof (rule has_integral_spike [OF neg])
    4.51 +    have eq: "(\<lambda>x. (if x \<in> S then f x else 0) - (if x \<in> T then f x else 0)) =
    4.52 +            (\<lambda>x. if x \<in> T - S then - f x else if x \<in> S - T then f x else 0)"
    4.53 +      by (force simp add: )
    4.54 +    have "((\<lambda>x. if x \<in> S then f x else 0) has_integral i) UNIV"
    4.55 +      "((\<lambda>x. if x \<in> T then f x else 0) has_integral j) UNIV"
    4.56 +      using S T has_integral_restrict_UNIV by auto
    4.57 +    from has_integral_diff [OF this]
    4.58 +    show "((\<lambda>x. if x \<in> T - S then - f x else if x \<in> S - T then f x else 0)
    4.59 +                   has_integral i-j) UNIV"
    4.60 +      by (simp add: eq)
    4.61 +  qed force
    4.62 +qed
    4.63 +
    4.64 +lemma integral_setdiff:
    4.65 +  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
    4.66 +  assumes "f integrable_on S" "f integrable_on T" "negligible(T - S)"
    4.67 + shows "integral (S - T) f = integral S f - integral T f"
    4.68 +  by (rule integral_unique) (simp add: assms has_integral_setdiff integrable_integral)
    4.69 +
    4.70 +lemma integrable_setdiff:
    4.71 +  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
    4.72 +  assumes "(f has_integral i) S" "(f has_integral j) T" "negligible (T - S)"
    4.73 +  shows "f integrable_on (S - T)"
    4.74 +  using has_integral_setdiff [OF assms]
    4.75 +  by (simp add: has_integral_iff)
    4.76 +
    4.77 +lemma negligible_setdiff [simp]: "T \<subseteq> S \<Longrightarrow> negligible (T - S)"
    4.78 +  by (metis Diff_eq_empty_iff negligible_empty)
    4.79 +
    4.80  lemma negligible_on_intervals: "negligible s \<longleftrightarrow> (\<forall>a b. negligible(s \<inter> cbox a b))" (is "?l \<longleftrightarrow> ?r")
    4.81  proof
    4.82    assume ?r
    4.83 @@ -4757,9 +4807,7 @@
    4.84  lemma has_integral_closure:
    4.85    fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
    4.86    shows "negligible(frontier S) \<Longrightarrow> (f has_integral y) (closure S) \<longleftrightarrow> (f has_integral y) S"
    4.87 -  apply (rule has_integral_spike_set_eq)
    4.88 -  apply (auto simp: Un_Diff closure_Un_frontier negligible_diff)
    4.89 -  by (simp add: Diff_eq closure_Un_frontier)
    4.90 +  by (rule has_integral_spike_set_eq) (simp add: Un_Diff closure_Un_frontier negligible_diff)
    4.91  
    4.92  lemma has_integral_open_interval:
    4.93    fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
    4.94 @@ -5344,9 +5392,6 @@
    4.95  
    4.96  subsection \<open>Also tagged divisions\<close>
    4.97  
    4.98 -lemma has_integral_iff: "(f has_integral i) S \<longleftrightarrow> (f integrable_on S \<and> integral S f = i)"
    4.99 -  by blast
   4.100 -
   4.101  lemma has_integral_combine_tagged_division:
   4.102    fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   4.103    assumes "p tagged_division_of S"
     5.1 --- a/src/HOL/HOL.thy	Sat Feb 24 17:21:35 2018 +0100
     5.2 +++ b/src/HOL/HOL.thy	Sun Feb 25 12:54:55 2018 +0000
     5.3 @@ -1404,6 +1404,9 @@
     5.4  lemma ex_if_distrib: "(\<exists>x. if x = a then P x else Q x) \<longleftrightarrow> P a \<or> (\<exists>x. x\<noteq>a \<and> Q x)"
     5.5    by auto
     5.6  
     5.7 +lemma if_if_eq_conj: "(if P then if Q then x else y else y) = (if P \<and> Q then x else y)"
     5.8 +  by simp
     5.9 +
    5.10  text \<open>As a simplification rule, it replaces all function equalities by
    5.11    first-order equalities.\<close>
    5.12  lemma fun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
     6.1 --- a/src/HOL/Library/Extended_Nonnegative_Real.thy	Sat Feb 24 17:21:35 2018 +0100
     6.2 +++ b/src/HOL/Library/Extended_Nonnegative_Real.thy	Sun Feb 25 12:54:55 2018 +0000
     6.3 @@ -158,17 +158,6 @@
     6.4    using tendsto_cmult_ereal[of c f "f x" "at x within A" for x]
     6.5    by (auto simp: continuous_on_def simp del: tendsto_cmult_ereal)
     6.6  
     6.7 -context linordered_nonzero_semiring
     6.8 -begin
     6.9 -
    6.10 -lemma of_nat_nonneg [simp]: "0 \<le> of_nat n"
    6.11 -  by (induct n) simp_all
    6.12 -
    6.13 -lemma of_nat_mono[simp]: "i \<le> j \<Longrightarrow> of_nat i \<le> of_nat j"
    6.14 -  by (auto simp add: le_iff_add intro!: add_increasing2)
    6.15 -
    6.16 -end
    6.17 -
    6.18  lemma real_of_nat_Sup:
    6.19    assumes "A \<noteq> {}" "bdd_above A"
    6.20    shows "of_nat (Sup A) = (SUP a:A. of_nat a :: real)"
    6.21 @@ -181,21 +170,6 @@
    6.22      by (intro cSUP_upper bdd_above_image_mono assms) (auto simp: mono_def)
    6.23  qed
    6.24  
    6.25 -\<comment> \<open>These generalise their counterparts in \<open>Nat.linordered_semidom_class\<close>\<close>
    6.26 -lemma of_nat_less[simp]:
    6.27 -  "m < n \<Longrightarrow> of_nat m < (of_nat n::'a::{linordered_nonzero_semiring, semiring_char_0})"
    6.28 -  by (auto simp: less_le)
    6.29 -
    6.30 -lemma of_nat_le_iff[simp]:
    6.31 -  "of_nat m \<le> (of_nat n::'a::{linordered_nonzero_semiring, semiring_char_0}) \<longleftrightarrow> m \<le> n"
    6.32 -proof (safe intro!: of_nat_mono)
    6.33 -  assume "of_nat m \<le> (of_nat n::'a)" then show "m \<le> n"
    6.34 -  proof (intro leI notI)
    6.35 -    assume "n < m" from less_le_trans[OF of_nat_less[OF this] \<open>of_nat m \<le> of_nat n\<close>] show False
    6.36 -      by blast
    6.37 -  qed
    6.38 -qed
    6.39 -
    6.40  lemma (in complete_lattice) SUP_sup_const1:
    6.41    "I \<noteq> {} \<Longrightarrow> (SUP i:I. sup c (f i)) = sup c (SUP i:I. f i)"
    6.42    using SUP_sup_distrib[of "\<lambda>_. c" I f] by simp