author | paulson |
Thu, 01 Jan 2004 10:06:32 +0100 | |
changeset 14334 | 6137d24eef79 |
parent 14329 | ff3210fe968f |
child 14352 | a8b1a44d8264 |
permissions | -rw-r--r-- |
10722 | 1 |
theory RealArith = RealArith0 |
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files ("real_arith.ML"): |
031a5a051bb4
Converting more of the "real" development to Isar scripts
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031a5a051bb4
Converting more of the "real" development to Isar scripts
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use "real_arith.ML" |
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setup real_arith_setup |
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subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*} |
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text{*Needed in this non-standard form by Hyperreal/Transcendental*} |
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lemma real_0_le_divide_iff: |
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"((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))" |
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by (simp add: real_divide_def zero_le_mult_iff, auto) |
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lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" |
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by arith |
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lemma real_add_eq_0_iff [iff]: "(x+y = (0::real)) = (y = -x)" |
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by auto |
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lemma real_add_less_0_iff [iff]: "(x+y < (0::real)) = (y < -x)" |
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by auto |
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lemma real_0_less_add_iff [iff]: "((0::real) < x+y) = (-x < y)" |
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by auto |
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lemma real_add_le_0_iff [iff]: "(x+y \<le> (0::real)) = (y \<le> -x)" |
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by auto |
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lemma real_0_le_add_iff [iff]: "((0::real) \<le> x+y) = (-x \<le> y)" |
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by auto |
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(** Simprules combining x-y and 0 (needed??) **) |
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lemma real_0_less_diff_iff [iff]: "((0::real) < x-y) = (y < x)" |
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by auto |
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lemma real_0_le_diff_iff [iff]: "((0::real) \<le> x-y) = (y \<le> x)" |
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by auto |
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(* |
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FIXME: we should have this, as for type int, but many proofs would break. |
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It replaces x+-y by x-y. |
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Addsimps [symmetric real_diff_def] |
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*) |
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subsubsection{*Division By @{term "-1"}*} |
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lemma real_divide_minus1 [simp]: "x/-1 = -(x::real)" |
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by simp |
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lemma real_minus1_divide [simp]: "-1/(x::real) = - (1/x)" |
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by (simp add: real_divide_def inverse_minus_eq) |
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lemma real_lbound_gt_zero: |
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"[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2" |
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apply (rule_tac x = " (min d1 d2) /2" in exI) |
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apply (simp add: min_def) |
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done |
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(*** Density of the Reals ***) |
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text{*Similar results are proved in @{text Ring_and_Field}*} |
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lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)" |
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by auto |
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lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y" |
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by auto |
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lemma real_dense: "x < y ==> \<exists>r::real. x < r & r < y" |
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by (rule Ring_and_Field.dense) |
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subsection{*Absolute Value Function for the Reals*} |
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lemma abs_nat_number_of [simp]: |
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"abs (number_of v :: real) = |
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(if neg (number_of v) then number_of (bin_minus v) |
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else number_of v)" |
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by (simp add: real_abs_def bin_arith_simps minus_real_number_of |
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le_real_number_of_eq_not_less less_real_number_of real_of_int_le_iff) |
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(*---------------------------------------------------------------------------- |
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Properties of the absolute value function over the reals |
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(adapted version of previously proved theorems about abs) |
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----------------------------------------------------------------------------*) |
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text{*FIXME: these should go!*} |
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lemma abs_eqI1: "(0::real)\<le>x ==> abs x = x" |
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by (unfold real_abs_def, simp) |
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lemma abs_eqI2: "(0::real) < x ==> abs x = x" |
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by (unfold real_abs_def, simp) |
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lemma abs_minus_eqI2: "x < (0::real) ==> abs x = -x" |
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by (unfold real_abs_def real_le_def, simp) |
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lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))" |
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by (unfold real_abs_def, simp) |
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lemma abs_minus_one [simp]: "abs (-1) = (1::real)" |
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by (unfold real_abs_def, simp) |
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lemma abs_interval_iff: "(abs x < r) = (-r < x & x < (r::real))" |
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by (force simp add: Ring_and_Field.abs_less_iff) |
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lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))" |
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by (force simp add: Ring_and_Field.abs_le_iff) |
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lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)" |
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by (unfold real_abs_def, auto) |
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lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)" |
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by (auto intro: abs_eqI1 simp add: real_of_nat_ge_zero) |
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lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x" |
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apply (simp add: linorder_not_less) |
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apply (auto intro: abs_ge_self [THEN order_trans]) |
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done |
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text{*Used only in Hyperreal/Lim.ML*} |
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lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)" |
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apply (simp add: real_add_assoc) |
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apply (rule_tac a1 = y in add_left_commute [THEN ssubst]) |
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apply (rule real_add_assoc [THEN subst]) |
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apply (rule abs_triangle_ineq) |
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done |
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ML |
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{* |
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val real_0_le_divide_iff = thm"real_0_le_divide_iff"; |
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val real_add_minus_iff = thm"real_add_minus_iff"; |
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val real_add_eq_0_iff = thm"real_add_eq_0_iff"; |
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val real_add_less_0_iff = thm"real_add_less_0_iff"; |
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val real_0_less_add_iff = thm"real_0_less_add_iff"; |
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val real_add_le_0_iff = thm"real_add_le_0_iff"; |
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val real_0_le_add_iff = thm"real_0_le_add_iff"; |
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val real_0_less_diff_iff = thm"real_0_less_diff_iff"; |
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val real_0_le_diff_iff = thm"real_0_le_diff_iff"; |
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val real_divide_minus1 = thm"real_divide_minus1"; |
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val real_minus1_divide = thm"real_minus1_divide"; |
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val real_lbound_gt_zero = thm"real_lbound_gt_zero"; |
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val real_less_half_sum = thm"real_less_half_sum"; |
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val real_gt_half_sum = thm"real_gt_half_sum"; |
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val real_dense = thm"real_dense"; |
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val abs_nat_number_of = thm"abs_nat_number_of"; |
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val abs_split = thm"abs_split"; |
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val abs_zero = thm"abs_zero"; |
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val abs_eqI1 = thm"abs_eqI1"; |
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val abs_eqI2 = thm"abs_eqI2"; |
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val abs_minus_eqI2 = thm"abs_minus_eqI2"; |
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val abs_ge_zero = thm"abs_ge_zero"; |
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val abs_idempotent = thm"abs_idempotent"; |
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val abs_zero_iff = thm"abs_zero_iff"; |
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val abs_ge_self = thm"abs_ge_self"; |
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val abs_ge_minus_self = thm"abs_ge_minus_self"; |
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val abs_mult = thm"abs_mult"; |
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val abs_inverse = thm"abs_inverse"; |
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val abs_triangle_ineq = thm"abs_triangle_ineq"; |
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val abs_minus_cancel = thm"abs_minus_cancel"; |
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val abs_minus_add_cancel = thm"abs_minus_add_cancel"; |
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val abs_minus_one = thm"abs_minus_one"; |
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val abs_interval_iff = thm"abs_interval_iff"; |
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val abs_le_interval_iff = thm"abs_le_interval_iff"; |
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val abs_add_one_gt_zero = thm"abs_add_one_gt_zero"; |
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val abs_le_zero_iff = thm"abs_le_zero_iff"; |
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val abs_real_of_nat_cancel = thm"abs_real_of_nat_cancel"; |
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val abs_add_one_not_less_self = thm"abs_add_one_not_less_self"; |
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val abs_sum_triangle_ineq = thm"abs_sum_triangle_ineq"; |
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val abs_mult_less = thm"abs_mult_less"; |
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*} |
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end |