author | paulson |
Tue, 23 Dec 2003 12:54:15 +0100 | |
changeset 14317 | 85125b18d38a |
parent 14310 | 1dd7439477dd |
child 14325 | 94ac3895822f |
permissions | -rw-r--r-- |
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9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
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(* Title: Real/RealOrd.thy |
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
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ID: $Id$ |
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
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Author: Jacques D. Fleuriot and Lawrence C. Paulson |
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Copyright: 1998 University of Cambridge |
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*) |
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header{*The Reals Form an Ordered Field, etc.*} |
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theory RealOrd = RealDef: |
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defs (overloaded) |
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real_abs_def: "abs (r::real) == (if 0 \<le> r then r else -r)" |
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HOL: installation of Ring_and_Field as the basis for Naturals and Reals
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subsection{*Properties of Less-Than Or Equals*} |
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lemma real_leI: "~(w < z) ==> z \<le> (w::real)" |
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by (unfold real_le_def, assumption) |
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lemma real_leD: "z\<le>w ==> ~(w<(z::real))" |
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by (unfold real_le_def, assumption) |
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lemma not_real_leE: "~ z \<le> w ==> w<(z::real)" |
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by (unfold real_le_def, blast) |
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lemma real_le_imp_less_or_eq: "!!(x::real). x \<le> y ==> x < y | x = y" |
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apply (unfold real_le_def) |
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apply (cut_tac real_linear) |
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apply (blast elim: real_less_irrefl real_less_asym) |
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done |
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lemma real_less_or_eq_imp_le: "z<w | z=w ==> z \<le>(w::real)" |
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apply (unfold real_le_def) |
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apply (cut_tac real_linear) |
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apply (fast elim: real_less_irrefl real_less_asym) |
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done |
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lemma real_le_less: "(x \<le> (y::real)) = (x < y | x=y)" |
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by (blast intro!: real_less_or_eq_imp_le dest!: real_le_imp_less_or_eq) |
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lemma real_le_refl: "w \<le> (w::real)" |
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by (simp add: real_le_less) |
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lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)" |
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apply (drule real_le_imp_less_or_eq) |
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apply (drule real_le_imp_less_or_eq) |
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apply (rule real_less_or_eq_imp_le) |
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apply (blast intro: real_less_trans) |
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done |
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lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)" |
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apply (drule real_le_imp_less_or_eq) |
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apply (drule real_le_imp_less_or_eq) |
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apply (fast elim: real_less_irrefl real_less_asym) |
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done |
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(* Axiom 'order_less_le' of class 'order': *) |
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lemma real_less_le: "((w::real) < z) = (w \<le> z & w ~= z)" |
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apply (simp add: real_le_def real_neq_iff) |
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apply (blast elim!: real_less_asym) |
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done |
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instance real :: order |
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by (intro_classes, |
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(assumption | |
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rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+) |
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(* Axiom 'linorder_linear' of class 'linorder': *) |
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lemma real_le_linear: "(z::real) \<le> w | w \<le> z" |
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apply (simp add: real_le_less) |
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apply (cut_tac real_linear, blast) |
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done |
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instance real :: linorder |
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by (intro_classes, rule real_le_linear) |
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subsection{*Theorems About the Ordering*} |
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lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)" |
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apply (auto simp add: real_of_preal_zero_less) |
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apply (cut_tac x = x in real_of_preal_trichotomy) |
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apply (blast elim!: real_less_irrefl real_of_preal_not_minus_gt_zero [THEN notE]) |
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done |
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lemma real_gt_preal_preal_Ex: "real_of_preal z < x ==> \<exists>y. x = real_of_preal y" |
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by (blast dest!: real_of_preal_zero_less [THEN real_less_trans] |
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intro: real_gt_zero_preal_Ex [THEN iffD1]) |
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lemma real_ge_preal_preal_Ex: "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y" |
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by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex) |
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lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x" |
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by (auto elim: order_le_imp_less_or_eq [THEN disjE] |
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intro: real_of_preal_zero_less [THEN [2] real_less_trans] |
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simp add: real_of_preal_zero_less) |
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lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x" |
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by (blast intro!: real_less_all_preal real_leI) |
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lemma real_of_preal_le_iff: "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)" |
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apply (auto intro!: preal_leI simp add: linorder_not_less [symmetric]) |
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done |
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subsection{*Monotonicity of Addition*} |
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lemma real_add_left_cancel: "((x::real) + y = x + z) = (y = z)" |
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apply safe |
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apply (drule_tac f = "%t. (-x) + t" in arg_cong) |
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apply (simp add: real_add_assoc [symmetric]) |
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done |
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lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y" |
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apply (auto simp add: real_gt_zero_preal_Ex) |
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apply (rule_tac x = "y*ya" in exI) |
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apply (simp (no_asm_use) add: real_of_preal_mult) |
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done |
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lemma real_minus_add_distrib [simp]: "-(x + y) = (-x) + (- y :: real)" |
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apply (rule_tac z = x in eq_Abs_REAL) |
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apply (rule_tac z = y in eq_Abs_REAL) |
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apply (auto simp add: real_minus real_add) |
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done |
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(*Alternative definition for real_less*) |
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lemma real_less_add_positive_left_Ex: "R < S ==> \<exists>T::real. 0 < T & R + T = S" |
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apply (rule_tac x = R in real_of_preal_trichotomyE) |
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apply (rule_tac [!] x = S in real_of_preal_trichotomyE) |
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apply (auto dest!: preal_less_add_left_Ex simp add: real_of_preal_not_minus_gt_all real_of_preal_add real_of_preal_not_less_zero real_less_not_refl real_of_preal_not_minus_gt_zero) |
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apply (rule_tac x = "real_of_preal D" in exI) |
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apply (rule_tac [2] x = "real_of_preal m+real_of_preal ma" in exI) |
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apply (rule_tac [3] x = "real_of_preal D" in exI) |
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apply (auto simp add: real_of_preal_zero_less real_of_preal_sum_zero_less real_add_assoc) |
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apply (simp add: real_add_assoc [symmetric]) |
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done |
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lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))" |
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apply (drule real_less_add_positive_left_Ex) |
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apply (auto simp add: real_add_minus real_add_zero_right real_add_ac) |
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done |
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lemma real_lemma_change_eq_subj: "!!S::real. T = S + W ==> S = T + (-W)" |
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by (simp add: real_add_ac) |
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(* FIXME: long! *) |
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lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)" |
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apply (rule ccontr) |
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apply (drule real_leI [THEN real_le_imp_less_or_eq]) |
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apply (auto simp add: real_less_not_refl) |
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apply (drule real_less_add_positive_left_Ex, clarify, simp) |
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apply (drule real_lemma_change_eq_subj, auto) |
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apply (drule real_less_sum_gt_zero) |
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apply (auto elim: real_less_asym simp add: real_add_left_commute [of W] real_add_ac) |
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done |
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lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y" |
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apply (rule real_sum_gt_zero_less) |
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apply (drule real_less_sum_gt_zero [of x y]) |
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apply (drule real_mult_order, assumption) |
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apply (simp add: real_add_mult_distrib2) |
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done |
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lemma real_less_sum_gt_0_iff: "(0 < S + (-W::real)) = (W < S)" |
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by (blast intro: real_less_sum_gt_zero real_sum_gt_zero_less) |
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lemma real_less_eq_diff: "(x<y) = (x-y < (0::real))" |
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apply (unfold real_diff_def) |
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apply (subst real_minus_zero_less_iff [symmetric]) |
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apply (simp add: real_add_commute real_less_sum_gt_0_iff) |
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done |
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lemma real_less_eqI: "(x::real) - y = x' - y' ==> (x<y) = (x'<y')" |
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apply (subst real_less_eq_diff) |
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apply (rule_tac y1 = y in real_less_eq_diff [THEN ssubst], simp) |
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done |
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lemma real_le_eqI: "(x::real) - y = x' - y' ==> (y\<le>x) = (y'\<le>x')" |
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apply (drule real_less_eqI) |
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apply (simp add: real_le_def) |
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done |
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lemma real_add_left_mono: "x \<le> y ==> z + x \<le> z + (y::real)" |
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apply (rule real_le_eqI [THEN iffD1]) |
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prefer 2 apply assumption |
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apply (simp add: real_diff_def real_add_ac) |
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done |
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subsection{*The Reals Form an Ordered Field*} |
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instance real :: inverse .. |
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instance real :: ordered_field |
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proof |
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fix x y z :: real |
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show "(x + y) + z = x + (y + z)" by (rule real_add_assoc) |
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show "x + y = y + x" by (rule real_add_commute) |
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show "0 + x = x" by simp |
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show "- x + x = 0" by simp |
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show "x - y = x + (-y)" by (simp add: real_diff_def) |
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show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc) |
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show "x * y = y * x" by (rule real_mult_commute) |
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show "1 * x = x" by simp |
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show "(x + y) * z = x * z + y * z" by (simp add: real_add_mult_distrib) |
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show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one) |
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show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono) |
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show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2) |
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show "\<bar>x\<bar> = (if x < 0 then -x else x)" |
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by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le) |
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show "x \<noteq> 0 ==> inverse x * x = 1" by simp |
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show "y \<noteq> 0 ==> x / y = x * inverse y" by (simp add: real_divide_def) |
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qed |
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lemma real_zero_less_one: "0 < (1::real)" |
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by (rule Ring_and_Field.zero_less_one) |
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lemma real_add_less_mono: "[| R1 < S1; R2 < S2 |] ==> R1+R2 < S1+(S2::real)" |
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by (rule Ring_and_Field.add_strict_mono) |
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lemma real_add_le_mono: "[|i\<le>j; k\<le>l |] ==> i + k \<le> j + (l::real)" |
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by (rule Ring_and_Field.add_mono) |
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lemma real_le_minus_iff: "(-s \<le> -r) = ((r::real) \<le> s)" |
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by (rule Ring_and_Field.neg_le_iff_le) |
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lemma real_le_square [simp]: "(0::real) \<le> x*x" |
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by (rule Ring_and_Field.zero_le_square) |
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subsection{*Division Lemmas*} |
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(** Inverse of zero! Useful to simplify certain equations **) |
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lemma INVERSE_ZERO: "inverse 0 = (0::real)" |
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apply (unfold real_inverse_def) |
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apply (rule someI2) |
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apply (auto simp add: real_zero_not_eq_one) |
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done |
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lemma DIVISION_BY_ZERO [simp]: "a / (0::real) = 0" |
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by (simp add: real_divide_def INVERSE_ZERO) |
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instance real :: division_by_zero |
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proof |
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fix x :: real |
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show "inverse 0 = (0::real)" by (rule INVERSE_ZERO) |
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show "x/0 = 0" by (rule DIVISION_BY_ZERO) |
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qed |
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lemma real_mult_left_cancel: "(c::real) ~= 0 ==> (c*a=c*b) = (a=b)" |
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by auto |
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lemma real_mult_right_cancel: "(c::real) ~= 0 ==> (a*c=b*c) = (a=b)" |
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by auto |
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lemma real_mult_left_cancel_ccontr: "c*a ~= c*b ==> a ~= b" |
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by auto |
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lemma real_mult_right_cancel_ccontr: "a*c ~= b*c ==> a ~= b" |
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by auto |
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lemma real_inverse_not_zero: "x ~= 0 ==> inverse(x::real) ~= 0" |
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by (rule Ring_and_Field.nonzero_imp_inverse_nonzero) |
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lemma real_mult_not_zero: "[| x ~= 0; y ~= 0 |] ==> x * y ~= (0::real)" |
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by simp |
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lemma real_inverse_inverse: "inverse(inverse (x::real)) = x" |
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by (rule Ring_and_Field.inverse_inverse_eq) |
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lemma real_inverse_1: "inverse((1::real)) = (1::real)" |
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by (rule Ring_and_Field.inverse_1) |
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lemma real_minus_inverse: "inverse(-x) = -inverse(x::real)" |
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by (rule Ring_and_Field.inverse_minus_eq) |
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281 |
lemma real_inverse_distrib: "inverse(x*y) = inverse(x)*inverse(y::real)" |
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by (rule Ring_and_Field.inverse_mult_distrib) |
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283 |
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284 |
lemma real_add_divide_distrib: "(x+y)/(z::real) = x/z + y/z" |
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14293 | 285 |
by (rule Ring_and_Field.add_divide_distrib) |
14270 | 286 |
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287 |
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288 |
subsection{*More Lemmas*} |
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289 |
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290 |
lemma real_add_right_cancel: "(y + (x::real)= z + x) = (y = z)" |
|
291 |
by (rule Ring_and_Field.add_right_cancel) |
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292 |
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293 |
lemma real_add_less_mono1: "v < (w::real) ==> v + z < w + z" |
|
294 |
by (rule Ring_and_Field.add_strict_right_mono) |
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295 |
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296 |
lemma real_add_le_mono1: "v \<le> (w::real) ==> v + z \<le> w + z" |
|
297 |
by (rule Ring_and_Field.add_right_mono) |
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298 |
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299 |
lemma real_add_less_le_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::real)" |
|
300 |
apply (erule add_strict_right_mono [THEN order_less_le_trans]) |
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301 |
apply (erule add_left_mono) |
|
302 |
done |
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303 |
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304 |
lemma real_add_le_less_mono: "!!z z'::real. [| w'\<le>w; z'<z |] ==> w' + z' < w + z" |
|
305 |
apply (erule add_right_mono [THEN order_le_less_trans]) |
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306 |
apply (erule add_strict_left_mono) |
|
307 |
done |
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308 |
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309 |
lemma real_less_add_right_cancel: "!!(A::real). A + C < B + C ==> A < B" |
|
310 |
by (rule Ring_and_Field.add_less_imp_less_right) |
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311 |
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312 |
lemma real_less_add_left_cancel: "!!(A::real). C + A < C + B ==> A < B" |
|
313 |
by (rule Ring_and_Field.add_less_imp_less_left) |
|
314 |
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315 |
lemma real_le_add_right_cancel: "!!(A::real). A + C \<le> B + C ==> A \<le> B" |
|
316 |
by (rule Ring_and_Field.add_le_imp_le_right) |
|
317 |
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318 |
lemma real_le_add_left_cancel: "!!(A::real). C + A \<le> C + B ==> A \<le> B" |
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14317 | 319 |
by (rule Ring_and_Field.add_le_imp_le_left) |
14270 | 320 |
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14310 | 321 |
lemma real_add_right_cancel_less: "(v+z < w+z) = (v < (w::real))" |
14270 | 322 |
by (rule Ring_and_Field.add_less_cancel_right) |
323 |
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14310 | 324 |
lemma real_add_left_cancel_less: "(z+v < z+w) = (v < (w::real))" |
14270 | 325 |
by (rule Ring_and_Field.add_less_cancel_left) |
326 |
||
14310 | 327 |
lemma real_add_right_cancel_le: "(v+z \<le> w+z) = (v \<le> (w::real))" |
14270 | 328 |
by (rule Ring_and_Field.add_le_cancel_right) |
329 |
||
14310 | 330 |
lemma real_add_left_cancel_le: "(z+v \<le> z+w) = (v \<le> (w::real))" |
14270 | 331 |
by (rule Ring_and_Field.add_le_cancel_left) |
332 |
||
333 |
||
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|
334 |
subsection{*Inverse and Division*} |
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|
335 |
|
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|
336 |
lemma real_inverse_gt_0: "0 < x ==> 0 < inverse (x::real)" |
14277
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|
337 |
by (rule Ring_and_Field.positive_imp_inverse_positive) |
14265
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|
338 |
|
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|
339 |
lemma real_inverse_less_0: "x < 0 ==> inverse (x::real) < 0" |
14277
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|
340 |
by (rule Ring_and_Field.negative_imp_inverse_negative) |
14265
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|
341 |
|
14268
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|
342 |
lemma real_mult_less_mono1: "[| (0::real) < z; x < y |] ==> x*z < y*z" |
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|
343 |
by (rule Ring_and_Field.mult_strict_right_mono) |
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|
344 |
|
14304 | 345 |
text{*The precondition could be weakened to @{term "0\<le>x"}*} |
14268
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|
346 |
lemma real_mult_less_mono: |
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|
347 |
"[| u<v; x<y; (0::real) < v; 0 < x |] ==> u*x < v* y" |
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|
348 |
by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le) |
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|
349 |
|
14266 | 350 |
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" |
14288 | 351 |
by (force elim: order_less_asym |
352 |
simp add: Ring_and_Field.mult_less_cancel_right) |
|
14265
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|
353 |
|
14268
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|
354 |
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)" |
14266 | 355 |
by (auto simp add: real_le_def) |
14265
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|
356 |
|
14268
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|
357 |
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)" |
14288 | 358 |
by (force elim: order_less_asym |
359 |
simp add: Ring_and_Field.mult_le_cancel_left) |
|
14265
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|
360 |
|
14304 | 361 |
text{*Only two uses?*} |
14266 | 362 |
lemma real_mult_less_mono': |
363 |
"[| x < y; r1 < r2; (0::real) \<le> r1; 0 \<le> x|] ==> r1 * x < r2 * y" |
|
14304 | 364 |
by (rule Ring_and_Field.mult_strict_mono') |
14265
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|
365 |
|
14268
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|
366 |
lemma real_inverse_less_swap: |
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|
367 |
"[| 0 < r; r < x |] ==> inverse x < inverse (r::real)" |
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|
368 |
by (rule Ring_and_Field.less_imp_inverse_less) |
14265
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|
369 |
|
14288 | 370 |
(*FIXME: remove the [iff], since the general theorem is already [simp]*) |
14266 | 371 |
lemma real_mult_is_0 [iff]: "(x*y = 0) = (x = 0 | y = (0::real))" |
14288 | 372 |
by (rule Ring_and_Field.mult_eq_0_iff) |
14265
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|
373 |
|
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|
374 |
lemma real_inverse_add: "[| x \<noteq> 0; y \<noteq> 0 |] |
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|
375 |
==> inverse x + inverse y = (x + y) * inverse (x * (y::real))" |
14288 | 376 |
by (simp add: Ring_and_Field.inverse_add mult_assoc) |
14265
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|
377 |
|
14288 | 378 |
text{*FIXME: delete or at least combine the next two lemmas*} |
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|
379 |
lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)" |
14270 | 380 |
apply (drule Ring_and_Field.equals_zero_I [THEN sym]) |
381 |
apply (cut_tac x = y in real_le_square) |
|
14266 | 382 |
apply (auto, drule real_le_anti_sym, auto) |
14265
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|
383 |
done |
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|
384 |
|
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|
385 |
lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)" |
14266 | 386 |
apply (rule_tac y = x in real_sum_squares_cancel) |
14265
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|
387 |
apply (simp add: real_add_commute) |
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|
388 |
done |
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|
389 |
|
14268
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|
390 |
|
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|
391 |
subsection{*Convenient Biconditionals for Products of Signs*} |
14265
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|
392 |
|
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|
393 |
lemma real_0_less_mult_iff: "((0::real) < x*y) = (0<x & 0<y | x<0 & y<0)" |
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|
394 |
by (rule Ring_and_Field.zero_less_mult_iff) |
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|
395 |
|
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|
396 |
lemma real_0_le_mult_iff: "((0::real)\<le>x*y) = (0\<le>x & 0\<le>y | x\<le>0 & y\<le>0)" |
14304 | 397 |
by (rule Ring_and_Field.zero_le_mult_iff) |
14265
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|
398 |
|
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|
399 |
lemma real_mult_less_0_iff: "(x*y < (0::real)) = (0<x & y<0 | x<0 & 0<y)" |
14304 | 400 |
by (rule Ring_and_Field.mult_less_0_iff) |
14265
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|
401 |
|
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|
402 |
lemma real_mult_le_0_iff: "(x*y \<le> (0::real)) = (0\<le>x & y\<le>0 | x\<le>0 & 0\<le>y)" |
14304 | 403 |
by (rule Ring_and_Field.mult_le_0_iff) |
14265
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|
404 |
|
14268
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|
405 |
subsection{*Hardly Used Theorems to be Deleted*} |
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|
406 |
|
14270 | 407 |
lemma real_add_less_mono2: "!!(A::real). A < B ==> C + A < C + B" |
408 |
by simp |
|
409 |
||
14268
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|
410 |
lemma real_add_order: "[| 0 < x; 0 < y |] ==> (0::real) < x + y" |
5cf13e80be0e
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|
411 |
apply (erule order_less_trans) |
14270 | 412 |
apply (drule real_add_less_mono2, simp) |
14268
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|
413 |
done |
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changeset
|
414 |
|
5cf13e80be0e
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|
415 |
lemma real_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::real) \<le> x + y" |
5cf13e80be0e
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parents:
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|
416 |
apply (drule order_le_imp_less_or_eq)+ |
5cf13e80be0e
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parents:
14266
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changeset
|
417 |
apply (auto intro: real_add_order order_less_imp_le) |
5cf13e80be0e
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parents:
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changeset
|
418 |
done |
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parents:
14266
diff
changeset
|
419 |
|
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parents:
14266
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|
420 |
|
14290 | 421 |
subsection{*An Embedding of the Naturals in the Reals*} |
422 |
||
423 |
lemma real_of_posnat_one: "real_of_posnat 0 = (1::real)" |
|
424 |
by (simp add: real_of_posnat_def pnat_one_iff [symmetric] |
|
425 |
real_of_preal_def symmetric real_one_def) |
|
426 |
||
427 |
lemma real_of_posnat_two: "real_of_posnat (Suc 0) = (1::real) + (1::real)" |
|
428 |
by (simp add: real_of_posnat_def real_of_preal_def real_one_def pnat_two_eq |
|
429 |
real_add |
|
430 |
prat_of_pnat_add [symmetric] preal_of_prat_add [symmetric] |
|
431 |
pnat_add_ac) |
|
432 |
||
433 |
lemma real_of_posnat_add: |
|
434 |
"real_of_posnat n1 + real_of_posnat n2 = real_of_posnat (n1 + n2) + (1::real)" |
|
435 |
apply (unfold real_of_posnat_def) |
|
436 |
apply (simp (no_asm_use) add: real_of_posnat_one [symmetric] real_of_posnat_def real_of_preal_add [symmetric] preal_of_prat_add [symmetric] prat_of_pnat_add [symmetric] pnat_of_nat_add) |
|
437 |
done |
|
438 |
||
439 |
lemma real_of_posnat_add_one: "real_of_posnat (n + 1) = real_of_posnat n + (1::real)" |
|
440 |
apply (rule_tac x1 = " (1::real) " in real_add_right_cancel [THEN iffD1]) |
|
441 |
apply (rule real_of_posnat_add [THEN subst]) |
|
442 |
apply (simp (no_asm_use) add: real_of_posnat_two real_add_assoc) |
|
443 |
done |
|
444 |
||
445 |
lemma real_of_posnat_Suc: "real_of_posnat (Suc n) = real_of_posnat n + (1::real)" |
|
446 |
by (subst real_of_posnat_add_one [symmetric], simp) |
|
447 |
||
448 |
lemma inj_real_of_posnat: "inj(real_of_posnat)" |
|
449 |
apply (rule inj_onI) |
|
450 |
apply (unfold real_of_posnat_def) |
|
451 |
apply (drule inj_real_of_preal [THEN injD]) |
|
452 |
apply (drule inj_preal_of_prat [THEN injD]) |
|
453 |
apply (drule inj_prat_of_pnat [THEN injD]) |
|
454 |
apply (erule inj_pnat_of_nat [THEN injD]) |
|
455 |
done |
|
456 |
||
457 |
lemma real_of_nat_zero [simp]: "real (0::nat) = 0" |
|
458 |
by (simp add: real_of_nat_def real_of_posnat_one) |
|
459 |
||
460 |
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)" |
|
461 |
by (simp add: real_of_nat_def real_of_posnat_two real_add_assoc) |
|
462 |
||
463 |
lemma real_of_nat_add [simp]: |
|
464 |
"real (m + n) = real (m::nat) + real n" |
|
465 |
apply (simp add: real_of_nat_def add_ac) |
|
466 |
apply (simp add: real_of_posnat_add add_assoc [symmetric]) |
|
467 |
apply (simp add: add_commute) |
|
468 |
apply (simp add: add_assoc [symmetric]) |
|
469 |
done |
|
470 |
||
471 |
(*Not for addsimps: often the LHS is used to represent a positive natural*) |
|
472 |
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)" |
|
473 |
by (simp add: real_of_nat_def real_of_posnat_Suc real_add_ac) |
|
474 |
||
475 |
lemma real_of_nat_less_iff [iff]: |
|
476 |
"(real (n::nat) < real m) = (n < m)" |
|
477 |
by (auto simp add: real_of_nat_def real_of_posnat_def) |
|
478 |
||
479 |
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)" |
|
480 |
by (simp add: linorder_not_less [symmetric]) |
|
481 |
||
482 |
lemma inj_real_of_nat: "inj (real :: nat => real)" |
|
483 |
apply (rule inj_onI) |
|
484 |
apply (auto intro!: inj_real_of_posnat [THEN injD] |
|
485 |
simp add: real_of_nat_def real_add_right_cancel) |
|
486 |
done |
|
487 |
||
488 |
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)" |
|
489 |
apply (induct_tac "n") |
|
490 |
apply (auto simp add: real_of_nat_Suc) |
|
491 |
apply (drule real_add_le_less_mono) |
|
492 |
apply (rule real_zero_less_one) |
|
493 |
apply (simp add: order_less_imp_le) |
|
494 |
done |
|
495 |
||
496 |
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n" |
|
497 |
apply (induct_tac "m") |
|
498 |
apply (auto simp add: real_of_nat_Suc real_add_mult_distrib real_add_commute) |
|
499 |
done |
|
500 |
||
501 |
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)" |
|
502 |
by (auto dest: inj_real_of_nat [THEN injD]) |
|
503 |
||
504 |
lemma real_of_nat_diff [rule_format]: |
|
505 |
"n \<le> m --> real (m - n) = real (m::nat) - real n" |
|
14293 | 506 |
apply (induct_tac "m", simp) |
14290 | 507 |
apply (simp add: real_diff_def Suc_diff_le le_Suc_eq real_of_nat_Suc add_ac) |
508 |
apply (simp add: add_left_commute [of _ "- 1"]) |
|
509 |
done |
|
510 |
||
511 |
lemma real_of_nat_zero_iff: "(real (n::nat) = 0) = (n = 0)" |
|
512 |
proof |
|
513 |
assume "real n = 0" |
|
514 |
have "real n = real (0::nat)" by simp |
|
515 |
then show "n = 0" by (simp only: real_of_nat_inject) |
|
516 |
next |
|
517 |
show "n = 0 \<Longrightarrow> real n = 0" by simp |
|
518 |
qed |
|
519 |
||
520 |
lemma real_of_nat_neg_int [simp]: "neg z ==> real (nat z) = 0" |
|
521 |
by (simp add: neg_nat real_of_nat_zero) |
|
522 |
||
523 |
||
14288 | 524 |
subsection{*Results About @{term real_of_posnat}: to be Deleted*} |
14268
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|
525 |
|
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parents:
14266
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changeset
|
526 |
lemma real_of_posnat_gt_zero: "0 < real_of_posnat n" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
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parents:
14266
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changeset
|
527 |
apply (unfold real_of_posnat_def) |
14270 | 528 |
apply (rule real_gt_zero_preal_Ex [THEN iffD2], blast) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
529 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
530 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
531 |
declare real_of_posnat_gt_zero [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
532 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
533 |
lemmas real_inv_real_of_posnat_gt_zero = real_of_posnat_gt_zero [THEN real_inverse_gt_0, standard] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
534 |
declare real_inv_real_of_posnat_gt_zero [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
535 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
536 |
lemmas real_of_posnat_ge_zero = real_of_posnat_gt_zero [THEN order_less_imp_le, standard] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
537 |
declare real_of_posnat_ge_zero [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
538 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
539 |
lemma real_of_posnat_not_eq_zero: "real_of_posnat n ~= 0" |
14270 | 540 |
by (rule real_of_posnat_gt_zero [THEN real_not_refl2, THEN not_sym]) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
541 |
declare real_of_posnat_not_eq_zero [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
542 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
543 |
declare real_of_posnat_not_eq_zero [THEN real_mult_inv_left, simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
544 |
declare real_of_posnat_not_eq_zero [THEN real_mult_inv_right, simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
545 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
546 |
lemma real_of_posnat_ge_one: "1 <= real_of_posnat n" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
547 |
apply (simp (no_asm) add: real_of_posnat_one [symmetric]) |
14293 | 548 |
apply (induct_tac "n", simp) |
14290 | 549 |
apply (simp add: real_of_posnat_Suc real_of_posnat_one order_less_imp_le) |
550 |
apply (rule add_le_cancel_right [THEN iffD1, of _ "- 1"]) |
|
14293 | 551 |
apply (simp add: add_assoc) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
552 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
553 |
declare real_of_posnat_ge_one [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
554 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
555 |
lemma real_of_posnat_real_inv_not_zero: "inverse(real_of_posnat n) ~= 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
556 |
apply (rule real_inverse_not_zero) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
557 |
apply (rule real_of_posnat_gt_zero [THEN real_not_refl2, THEN not_sym]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
558 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
559 |
declare real_of_posnat_real_inv_not_zero [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
560 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
561 |
lemma real_of_posnat_real_inv_inj: "inverse(real_of_posnat x) = inverse(real_of_posnat y) ==> x = y" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
562 |
apply (rule inj_real_of_posnat [THEN injD]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
563 |
apply (rule real_of_posnat_real_inv_not_zero |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
564 |
[THEN real_mult_left_cancel, THEN iffD1, of x]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
565 |
apply (simp add: real_mult_inv_left |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
566 |
real_of_posnat_gt_zero [THEN real_not_refl2, THEN not_sym]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
567 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
568 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
569 |
lemma real_mult_less_self: "0 < r ==> r*(1 + -inverse(real_of_posnat n)) < r" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
570 |
apply (simp (no_asm) add: real_add_mult_distrib2) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
571 |
apply (rule_tac C = "-r" in real_less_add_left_cancel) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
572 |
apply (auto intro: real_mult_order simp add: real_add_assoc [symmetric] real_minus_zero_less_iff2) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
573 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
574 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
575 |
lemma real_of_posnat_inv_Ex_iff: "(EX n. inverse(real_of_posnat n) < r) = (EX n. 1 < r * real_of_posnat n)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
576 |
apply safe |
14270 | 577 |
apply (drule_tac n1 = n in real_of_posnat_gt_zero [THEN real_mult_less_mono1]) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
578 |
apply (drule_tac [2] n2=n in real_of_posnat_gt_zero [THEN real_inverse_gt_0, THEN real_mult_less_mono1]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
579 |
apply (auto simp add: real_of_posnat_gt_zero [THEN real_not_refl2, THEN not_sym] real_mult_assoc) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
580 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
581 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
582 |
lemma real_of_posnat_inv_iff: "(inverse(real_of_posnat n) < r) = (1 < r * real_of_posnat n)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
583 |
apply safe |
14270 | 584 |
apply (drule_tac n1 = n in real_of_posnat_gt_zero [THEN real_mult_less_mono1]) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
585 |
apply (drule_tac [2] n2=n in real_of_posnat_gt_zero [THEN real_inverse_gt_0, THEN real_mult_less_mono1]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
586 |
apply (auto simp add: real_mult_assoc) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
587 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
588 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
589 |
lemma real_mult_le_le_mono1: "[| (0::real) <=z; x<=y |] ==> z*x<=z*y" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
590 |
by (rule Ring_and_Field.mult_left_mono) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
591 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
592 |
lemma real_mult_le_le_mono2: "[| (0::real)<=z; x<=y |] ==> x*z<=y*z" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
593 |
by (rule Ring_and_Field.mult_right_mono) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
594 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
595 |
lemma real_of_posnat_inv_le_iff: "(inverse(real_of_posnat n) <= r) = (1 <= r * real_of_posnat n)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
596 |
apply safe |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
597 |
apply (drule_tac n2=n in real_of_posnat_gt_zero [THEN order_less_imp_le, THEN real_mult_le_le_mono1]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
598 |
apply (drule_tac [2] n3=n in real_of_posnat_gt_zero [THEN real_inverse_gt_0, THEN order_less_imp_le, THEN real_mult_le_le_mono1]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
599 |
apply (auto simp add: real_mult_ac) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
600 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
601 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
602 |
lemma real_of_posnat_less_iff: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
603 |
"(real_of_posnat n < real_of_posnat m) = (n < m)" |
14270 | 604 |
apply (unfold real_of_posnat_def, auto) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
605 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
606 |
declare real_of_posnat_less_iff [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
607 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
608 |
lemma real_of_posnat_le_iff: "(real_of_posnat n <= real_of_posnat m) = (n <= m)" |
14270 | 609 |
by (auto dest: inj_real_of_posnat [THEN injD] simp add: real_le_less le_eq_less_or_eq) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
610 |
declare real_of_posnat_le_iff [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
611 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
612 |
lemma real_mult_less_cancel3: "[| (0::real)<z; x*z<y*z |] ==> x<y" |
14270 | 613 |
by auto |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
614 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
615 |
lemma real_mult_less_cancel4: "[| (0::real)<z; z*x<z*y |] ==> x<y" |
14288 | 616 |
by (force elim: order_less_asym |
617 |
simp add: Ring_and_Field.mult_less_cancel_left) |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
618 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
619 |
lemma real_of_posnat_less_inv_iff: "0 < u ==> (u < inverse (real_of_posnat n)) = (real_of_posnat n < inverse(u))" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
620 |
apply safe |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
621 |
apply (rule_tac n2=n in real_of_posnat_gt_zero [THEN real_inverse_gt_0, THEN real_mult_less_cancel3]) |
14270 | 622 |
apply (rule_tac [2] x1 = u in real_inverse_gt_0 [THEN real_mult_less_cancel3]) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
623 |
apply (auto simp add: real_not_refl2 [THEN not_sym]) |
14270 | 624 |
apply (rule_tac z = u in real_mult_less_cancel4) |
625 |
apply (rule_tac [3] n1 = n in real_of_posnat_gt_zero [THEN real_mult_less_cancel4]) |
|
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
626 |
apply (auto simp add: real_not_refl2 [THEN not_sym] real_mult_assoc [symmetric]) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
627 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
628 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
629 |
lemma real_of_posnat_inv_eq_iff: "0 < u ==> (u = inverse(real_of_posnat n)) = (real_of_posnat n = inverse u)" |
14270 | 630 |
by auto |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
631 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
632 |
lemma real_add_one_minus_inv_ge_zero: "0 <= 1 + -inverse(real_of_posnat n)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
633 |
apply (rule_tac C = "inverse (real_of_posnat n) " in real_le_add_right_cancel) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
634 |
apply (simp (no_asm) add: real_add_assoc real_of_posnat_inv_le_iff) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
635 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
636 |
|
14288 | 637 |
(*Used just below and in HahnBanach/Aux.thy*) |
638 |
lemma real_mult_le_less_mono1: "[| (0::real) \<le> z; x < y |] ==> x*z \<le> y*z" |
|
639 |
apply (rule real_less_or_eq_imp_le) |
|
640 |
apply (drule order_le_imp_less_or_eq) |
|
641 |
apply (auto intro: real_mult_less_mono1) |
|
642 |
done |
|
643 |
||
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
644 |
lemma real_mult_add_one_minus_ge_zero: "0 < r ==> 0 <= r*(1 + -inverse(real_of_posnat n))" |
14270 | 645 |
by (drule real_add_one_minus_inv_ge_zero [THEN real_mult_le_less_mono1], auto) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
646 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
647 |
lemma real_inverse_unique: "x*y = (1::real) ==> y = inverse x" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
648 |
apply (case_tac "x ~= 0") |
14270 | 649 |
apply (rule_tac c1 = x in real_mult_left_cancel [THEN iffD1], auto) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
650 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
651 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
652 |
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x" |
14270 | 653 |
by (auto dest: real_inverse_less_swap) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
654 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
655 |
lemma real_of_nat_gt_zero_cancel_iff: "(0 < real (n::nat)) = (0 < n)" |
14270 | 656 |
by (rule real_of_nat_less_iff [THEN subst], auto) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
657 |
declare real_of_nat_gt_zero_cancel_iff [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
658 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
659 |
lemma real_of_nat_le_zero_cancel_iff: "(real (n::nat) <= 0) = (n = 0)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
660 |
apply (rule real_of_nat_zero [THEN subst]) |
14270 | 661 |
apply (subst real_of_nat_le_iff, auto) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
662 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
663 |
declare real_of_nat_le_zero_cancel_iff [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
664 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
665 |
lemma not_real_of_nat_less_zero: "~ real (n::nat) < 0" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
666 |
apply (simp (no_asm) add: real_le_def [symmetric] real_of_nat_ge_zero) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
667 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
668 |
declare not_real_of_nat_less_zero [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
669 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
670 |
lemma real_of_nat_ge_zero_cancel_iff: |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
671 |
"(0 <= real (n::nat)) = (0 <= n)" |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
672 |
apply (unfold real_le_def le_def) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
673 |
apply (simp (no_asm)) |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
674 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
675 |
declare real_of_nat_ge_zero_cancel_iff [simp] |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
676 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
677 |
lemma real_of_nat_num_if: "real n = (if n=0 then 0 else 1 + real ((n::nat) - 1))" |
14293 | 678 |
apply (case_tac "n", simp) |
14290 | 679 |
apply (simp add: real_of_nat_Suc add_commute) |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
680 |
done |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
681 |
|
14304 | 682 |
lemma real_mult_self_sum_ge_zero: "(0::real) \<le> x*x + y*y" |
683 |
proof - |
|
684 |
have "0 + 0 \<le> x*x + y*y" by (blast intro: add_mono zero_le_square) |
|
685 |
thus ?thesis by simp |
|
686 |
qed |
|
687 |
||
688 |
declare real_mult_self_sum_ge_zero [simp] |
|
14290 | 689 |
|
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
690 |
ML |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
691 |
{* |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
692 |
val real_abs_def = thm "real_abs_def"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
693 |
|
14270 | 694 |
val real_less_eq_diff = thm "real_less_eq_diff"; |
695 |
||
696 |
val real_add_right_cancel = thm"real_add_right_cancel"; |
|
697 |
val real_mult_congruent2_lemma = thm"real_mult_congruent2_lemma"; |
|
698 |
val real_mult_congruent2 = thm"real_mult_congruent2"; |
|
699 |
val real_mult = thm"real_mult"; |
|
700 |
val real_mult_commute = thm"real_mult_commute"; |
|
701 |
val real_mult_assoc = thm"real_mult_assoc"; |
|
702 |
val real_mult_left_commute = thm"real_mult_left_commute"; |
|
703 |
val real_mult_1 = thm"real_mult_1"; |
|
704 |
val real_mult_1_right = thm"real_mult_1_right"; |
|
705 |
val real_mult_0 = thm"real_mult_0"; |
|
706 |
val real_mult_0_right = thm"real_mult_0_right"; |
|
707 |
val real_mult_minus_eq1 = thm"real_mult_minus_eq1"; |
|
708 |
val real_minus_mult_eq1 = thm"real_minus_mult_eq1"; |
|
709 |
val real_mult_minus_eq2 = thm"real_mult_minus_eq2"; |
|
710 |
val real_minus_mult_eq2 = thm"real_minus_mult_eq2"; |
|
711 |
val real_mult_minus_1 = thm"real_mult_minus_1"; |
|
712 |
val real_mult_minus_1_right = thm"real_mult_minus_1_right"; |
|
713 |
val real_minus_mult_cancel = thm"real_minus_mult_cancel"; |
|
714 |
val real_minus_mult_commute = thm"real_minus_mult_commute"; |
|
715 |
val real_add_assoc_cong = thm"real_add_assoc_cong"; |
|
716 |
val real_add_mult_distrib = thm"real_add_mult_distrib"; |
|
717 |
val real_add_mult_distrib2 = thm"real_add_mult_distrib2"; |
|
718 |
val real_diff_mult_distrib = thm"real_diff_mult_distrib"; |
|
719 |
val real_diff_mult_distrib2 = thm"real_diff_mult_distrib2"; |
|
720 |
val real_zero_not_eq_one = thm"real_zero_not_eq_one"; |
|
721 |
val real_zero_iff = thm"real_zero_iff"; |
|
722 |
val preal_le_linear = thm"preal_le_linear"; |
|
723 |
val real_mult_inv_right_ex = thm"real_mult_inv_right_ex"; |
|
724 |
val real_mult_inv_left_ex = thm"real_mult_inv_left_ex"; |
|
725 |
val real_mult_inv_left = thm"real_mult_inv_left"; |
|
726 |
val real_mult_inv_right = thm"real_mult_inv_right"; |
|
727 |
val preal_lemma_eq_rev_sum = thm"preal_lemma_eq_rev_sum"; |
|
728 |
val preal_add_left_commute_cancel = thm"preal_add_left_commute_cancel"; |
|
729 |
val preal_lemma_for_not_refl = thm"preal_lemma_for_not_refl"; |
|
730 |
val real_less_not_refl = thm"real_less_not_refl"; |
|
731 |
val real_less_irrefl = thm"real_less_irrefl"; |
|
732 |
val real_not_refl2 = thm"real_not_refl2"; |
|
733 |
val preal_lemma_trans = thm"preal_lemma_trans"; |
|
734 |
val real_less_trans = thm"real_less_trans"; |
|
735 |
val real_less_not_sym = thm"real_less_not_sym"; |
|
736 |
val real_less_asym = thm"real_less_asym"; |
|
737 |
val real_of_preal_add = thm"real_of_preal_add"; |
|
738 |
val real_of_preal_mult = thm"real_of_preal_mult"; |
|
739 |
val real_of_preal_ExI = thm"real_of_preal_ExI"; |
|
740 |
val real_of_preal_ExD = thm"real_of_preal_ExD"; |
|
741 |
val real_of_preal_iff = thm"real_of_preal_iff"; |
|
742 |
val real_of_preal_trichotomy = thm"real_of_preal_trichotomy"; |
|
743 |
val real_of_preal_trichotomyE = thm"real_of_preal_trichotomyE"; |
|
744 |
val real_of_preal_lessD = thm"real_of_preal_lessD"; |
|
745 |
val real_of_preal_lessI = thm"real_of_preal_lessI"; |
|
746 |
val real_of_preal_less_iff1 = thm"real_of_preal_less_iff1"; |
|
747 |
val real_of_preal_minus_less_self = thm"real_of_preal_minus_less_self"; |
|
748 |
val real_of_preal_minus_less_zero = thm"real_of_preal_minus_less_zero"; |
|
749 |
val real_of_preal_not_minus_gt_zero = thm"real_of_preal_not_minus_gt_zero"; |
|
750 |
val real_of_preal_zero_less = thm"real_of_preal_zero_less"; |
|
751 |
val real_of_preal_not_less_zero = thm"real_of_preal_not_less_zero"; |
|
752 |
val real_minus_minus_zero_less = thm"real_minus_minus_zero_less"; |
|
753 |
val real_of_preal_sum_zero_less = thm"real_of_preal_sum_zero_less"; |
|
754 |
val real_of_preal_minus_less_all = thm"real_of_preal_minus_less_all"; |
|
755 |
val real_of_preal_not_minus_gt_all = thm"real_of_preal_not_minus_gt_all"; |
|
756 |
val real_of_preal_minus_less_rev1 = thm"real_of_preal_minus_less_rev1"; |
|
757 |
val real_of_preal_minus_less_rev2 = thm"real_of_preal_minus_less_rev2"; |
|
758 |
val real_of_preal_minus_less_rev_iff = thm"real_of_preal_minus_less_rev_iff"; |
|
759 |
val real_linear = thm"real_linear"; |
|
760 |
val real_neq_iff = thm"real_neq_iff"; |
|
761 |
val real_linear_less2 = thm"real_linear_less2"; |
|
762 |
val real_leI = thm"real_leI"; |
|
763 |
val real_leD = thm"real_leD"; |
|
764 |
val not_real_leE = thm"not_real_leE"; |
|
765 |
val real_le_imp_less_or_eq = thm"real_le_imp_less_or_eq"; |
|
766 |
val real_less_or_eq_imp_le = thm"real_less_or_eq_imp_le"; |
|
767 |
val real_le_less = thm"real_le_less"; |
|
768 |
val real_le_refl = thm"real_le_refl"; |
|
769 |
val real_le_linear = thm"real_le_linear"; |
|
770 |
val real_le_trans = thm"real_le_trans"; |
|
771 |
val real_le_anti_sym = thm"real_le_anti_sym"; |
|
772 |
val real_less_le = thm"real_less_le"; |
|
773 |
val real_minus_zero_less_iff = thm"real_minus_zero_less_iff"; |
|
774 |
val real_minus_zero_less_iff2 = thm"real_minus_zero_less_iff2"; |
|
775 |
val real_less_add_positive_left_Ex = thm"real_less_add_positive_left_Ex"; |
|
776 |
val real_less_sum_gt_zero = thm"real_less_sum_gt_zero"; |
|
777 |
val real_sum_gt_zero_less = thm"real_sum_gt_zero_less"; |
|
778 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
779 |
val real_gt_zero_preal_Ex = thm "real_gt_zero_preal_Ex"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
780 |
val real_gt_preal_preal_Ex = thm "real_gt_preal_preal_Ex"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
781 |
val real_ge_preal_preal_Ex = thm "real_ge_preal_preal_Ex"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
782 |
val real_less_all_preal = thm "real_less_all_preal"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
783 |
val real_less_all_real2 = thm "real_less_all_real2"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
784 |
val real_of_preal_le_iff = thm "real_of_preal_le_iff"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
785 |
val real_mult_order = thm "real_mult_order"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
786 |
val real_zero_less_one = thm "real_zero_less_one"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
787 |
val real_add_right_cancel_less = thm "real_add_right_cancel_less"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
788 |
val real_add_left_cancel_less = thm "real_add_left_cancel_less"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
789 |
val real_add_right_cancel_le = thm "real_add_right_cancel_le"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
790 |
val real_add_left_cancel_le = thm "real_add_left_cancel_le"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
791 |
val real_add_less_mono1 = thm "real_add_less_mono1"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
792 |
val real_add_le_mono1 = thm "real_add_le_mono1"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
793 |
val real_add_less_le_mono = thm "real_add_less_le_mono"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
794 |
val real_add_le_less_mono = thm "real_add_le_less_mono"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
795 |
val real_add_less_mono2 = thm "real_add_less_mono2"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
796 |
val real_less_add_right_cancel = thm "real_less_add_right_cancel"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
797 |
val real_less_add_left_cancel = thm "real_less_add_left_cancel"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
798 |
val real_le_add_right_cancel = thm "real_le_add_right_cancel"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
799 |
val real_le_add_left_cancel = thm "real_le_add_left_cancel"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
800 |
val real_add_order = thm "real_add_order"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
801 |
val real_le_add_order = thm "real_le_add_order"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
802 |
val real_add_less_mono = thm "real_add_less_mono"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
803 |
val real_add_le_mono = thm "real_add_le_mono"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
804 |
val real_le_minus_iff = thm "real_le_minus_iff"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
805 |
val real_le_square = thm "real_le_square"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
806 |
val real_mult_less_mono1 = thm "real_mult_less_mono1"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
807 |
val real_mult_less_mono2 = thm "real_mult_less_mono2"; |
14290 | 808 |
|
809 |
val real_inverse_gt_0 = thm "real_inverse_gt_0"; |
|
810 |
val real_inverse_less_0 = thm "real_inverse_less_0"; |
|
811 |
val real_mult_less_iff1 = thm "real_mult_less_iff1"; |
|
812 |
val real_mult_le_cancel_iff1 = thm "real_mult_le_cancel_iff1"; |
|
813 |
val real_mult_le_cancel_iff2 = thm "real_mult_le_cancel_iff2"; |
|
814 |
val real_mult_less_mono = thm "real_mult_less_mono"; |
|
815 |
val real_mult_less_mono' = thm "real_mult_less_mono'"; |
|
816 |
val real_inverse_less_swap = thm "real_inverse_less_swap"; |
|
817 |
val real_mult_is_0 = thm "real_mult_is_0"; |
|
818 |
val real_inverse_add = thm "real_inverse_add"; |
|
819 |
val real_sum_squares_cancel = thm "real_sum_squares_cancel"; |
|
820 |
val real_sum_squares_cancel2 = thm "real_sum_squares_cancel2"; |
|
821 |
val real_0_less_mult_iff = thm "real_0_less_mult_iff"; |
|
822 |
val real_0_le_mult_iff = thm "real_0_le_mult_iff"; |
|
823 |
val real_mult_less_0_iff = thm "real_mult_less_0_iff"; |
|
824 |
val real_mult_le_0_iff = thm "real_mult_le_0_iff"; |
|
825 |
||
826 |
val INVERSE_ZERO = thm"INVERSE_ZERO"; |
|
827 |
val DIVISION_BY_ZERO = thm"DIVISION_BY_ZERO"; |
|
828 |
val real_mult_left_cancel = thm"real_mult_left_cancel"; |
|
829 |
val real_mult_right_cancel = thm"real_mult_right_cancel"; |
|
830 |
val real_mult_left_cancel_ccontr = thm"real_mult_left_cancel_ccontr"; |
|
831 |
val real_mult_right_cancel_ccontr = thm"real_mult_right_cancel_ccontr"; |
|
832 |
val real_inverse_not_zero = thm"real_inverse_not_zero"; |
|
833 |
val real_mult_not_zero = thm"real_mult_not_zero"; |
|
834 |
val real_inverse_inverse = thm"real_inverse_inverse"; |
|
835 |
val real_inverse_1 = thm"real_inverse_1"; |
|
836 |
val real_minus_inverse = thm"real_minus_inverse"; |
|
837 |
val real_inverse_distrib = thm"real_inverse_distrib"; |
|
838 |
val real_add_divide_distrib = thm"real_add_divide_distrib"; |
|
839 |
||
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
840 |
val real_of_posnat_one = thm "real_of_posnat_one"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
841 |
val real_of_posnat_two = thm "real_of_posnat_two"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
842 |
val real_of_posnat_add = thm "real_of_posnat_add"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
843 |
val real_of_posnat_add_one = thm "real_of_posnat_add_one"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
844 |
val real_of_posnat_Suc = thm "real_of_posnat_Suc"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
845 |
val inj_real_of_posnat = thm "inj_real_of_posnat"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
846 |
val real_of_nat_zero = thm "real_of_nat_zero"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
847 |
val real_of_nat_one = thm "real_of_nat_one"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
848 |
val real_of_nat_add = thm "real_of_nat_add"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
849 |
val real_of_nat_Suc = thm "real_of_nat_Suc"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
850 |
val real_of_nat_less_iff = thm "real_of_nat_less_iff"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
851 |
val real_of_nat_le_iff = thm "real_of_nat_le_iff"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
852 |
val inj_real_of_nat = thm "inj_real_of_nat"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
853 |
val real_of_nat_ge_zero = thm "real_of_nat_ge_zero"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
854 |
val real_of_nat_mult = thm "real_of_nat_mult"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
855 |
val real_of_nat_inject = thm "real_of_nat_inject"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
856 |
val real_of_nat_diff = thm "real_of_nat_diff"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
857 |
val real_of_nat_zero_iff = thm "real_of_nat_zero_iff"; |
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
858 |
val real_of_nat_neg_int = thm "real_of_nat_neg_int"; |
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
859 |
|
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
860 |
val real_of_posnat_gt_zero = thm "real_of_posnat_gt_zero"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
861 |
val real_inv_real_of_posnat_gt_zero = thm "real_inv_real_of_posnat_gt_zero"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
862 |
val real_of_posnat_ge_zero = thm "real_of_posnat_ge_zero"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
863 |
val real_of_posnat_not_eq_zero = thm "real_of_posnat_not_eq_zero"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
864 |
val real_of_posnat_ge_one = thm "real_of_posnat_ge_one"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
865 |
val real_of_posnat_real_inv_not_zero = thm "real_of_posnat_real_inv_not_zero"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
866 |
val real_of_posnat_real_inv_inj = thm "real_of_posnat_real_inv_inj"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
867 |
val real_mult_less_self = thm "real_mult_less_self"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
868 |
val real_of_posnat_inv_Ex_iff = thm "real_of_posnat_inv_Ex_iff"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
869 |
val real_of_posnat_inv_iff = thm "real_of_posnat_inv_iff"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
870 |
val real_mult_le_le_mono1 = thm "real_mult_le_le_mono1"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
871 |
val real_mult_le_le_mono2 = thm "real_mult_le_le_mono2"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
872 |
val real_of_posnat_inv_le_iff = thm "real_of_posnat_inv_le_iff"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
873 |
val real_of_posnat_less_iff = thm "real_of_posnat_less_iff"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
874 |
val real_of_posnat_le_iff = thm "real_of_posnat_le_iff"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
875 |
val real_of_posnat_less_inv_iff = thm "real_of_posnat_less_inv_iff"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
876 |
val real_of_posnat_inv_eq_iff = thm "real_of_posnat_inv_eq_iff"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
877 |
val real_add_one_minus_inv_ge_zero = thm "real_add_one_minus_inv_ge_zero"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
878 |
val real_mult_add_one_minus_ge_zero = thm "real_mult_add_one_minus_ge_zero"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
879 |
val real_inverse_unique = thm "real_inverse_unique"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
880 |
val real_inverse_gt_one = thm "real_inverse_gt_one"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
881 |
val real_of_nat_gt_zero_cancel_iff = thm "real_of_nat_gt_zero_cancel_iff"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
882 |
val real_of_nat_le_zero_cancel_iff = thm "real_of_nat_le_zero_cancel_iff"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
883 |
val not_real_of_nat_less_zero = thm "not_real_of_nat_less_zero"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
884 |
val real_of_nat_ge_zero_cancel_iff = thm "real_of_nat_ge_zero_cancel_iff"; |
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14266
diff
changeset
|
885 |
val real_of_nat_num_if = thm "real_of_nat_num_if"; |
14269 | 886 |
|
14290 | 887 |
val real_minus_add_distrib = thm"real_minus_add_distrib"; |
888 |
val real_add_left_cancel = thm"real_add_left_cancel"; |
|
14304 | 889 |
val real_mult_self_sum_ge_zero = thm "real_mult_self_sum_ge_zero"; |
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
9043
diff
changeset
|
890 |
*} |
9043
ca761fe227d8
First round of changes, towards installation of simprocs
paulson
parents:
9013
diff
changeset
|
891 |
|
7334 | 892 |
end |