(* Author: Johannes Hoelzl <hoelzl@in.tum.de> 2009 *) theory Approximation_Ex imports Complex_Main "../Approximation" begin text ‹ Here are some examples how to use the approximation method. The approximation method has the following syntax: approximate "prec" (splitting: "x" = "depth" and "y" = "depth" ...)? (taylor: "z" = "derivates")? Here "prec" specifies the precision used in all computations, it is specified as number of bits to calculate. In the proposition to prove an arbitrary amount of variables can be used, but each one need to be bounded by an upper and lower bound. To specify the bounds either @{term "l⇩_{1}≤ x ∧ x ≤ u⇩_{1}"}, @{term "x ∈ { l⇩_{1}.. u⇩_{1}}"} or @{term "x = bnd"} can be used. Where the bound specification are again arithmetic formulas containing variables. They can be connected using either meta level or HOL equivalence. To use interval splitting add for each variable whos interval should be splitted to the "splitting:" parameter. The parameter specifies how often each interval should be divided, e.g. when x = 16 is specified, there will be @{term "65536 = 2^16"} intervals to be calculated. To use taylor series expansion specify the variable to derive. You need to specify the amount of derivations to compute. When using taylor series expansion only one variable can be used. › section "Compute some transcendental values" lemma "¦ ln 2 - 544531980202654583340825686620847 / 785593587443817081832229725798400 ¦ < (inverse (2^51) :: real)" by (approximation 60) lemma "¦ exp 1.626 - 5.083499996273 ¦ < (inverse 10 ^ 10 :: real)" by (approximation 40) lemma "¦ sqrt 2 - 1.4142135623730951 ¦ < inverse 10 ^ 16" by (approximation 60) lemma "¦ pi - 3.1415926535897932385 ¦ < inverse 10 ^ 18" by (approximation 70) lemma "¦ sin 100 + 0.50636564110975879 ¦ < (inverse 10 ^ 17 :: real)" by (approximation 70) section "Use variable ranges" lemma "0.5 ≤ x ∧ x ≤ 4.5 ⟹ ¦ arctan x - 0.91 ¦ < 0.455" by (approximation 10) lemma "x ∈ {0.5 .. 4.5} ⟶ ¦ arctan x - 0.91 ¦ < 0.455" by (approximation 10) lemma "0.49 ≤ x ∧ x ≤ 4.49 ⟹ ¦ arctan x - 0.91 ¦ < 0.455" by (approximation 20) lemma "1 / 2^1 ≤ x ∧ x ≤ 9 / 2^1 ⟹ ¦ arctan x - 0.91 ¦ < 0.455" by (approximation 10) lemma "3.2 ≤ (x::real) ∧ x ≤ 6.2 ⟹ sin x ≤ 0" by (approximation 10) lemma "3.2 ≤ (x::real) ∧ x ≤ 3.9 ⟹ real_of_int (ceiling x) ∈ {4 .. 4}" by (approximation 10) lemma defines "g ≡ 9.80665" and "v ≡ 128.61" and "d ≡ pi / 180" shows "g / v * tan (35 * d) ∈ { 3 * d .. 3.1 * d }" using assms by (approximation 20) lemma "x ∈ { 0 .. 1 :: real } ⟶ x⇧^{2}≤ x" by (approximation 30 splitting: x=1 taylor: x = 3) lemma "(n::real) ∈ {32 .. 62} ⟹ ⌈log 2 (2 * (⌊n⌋ div 2) + 1)⌉ = ⌈log 2 (⌊n⌋ + 1)⌉" unfolding eq_iff by (approximation 20) approximate 10 end