# HG changeset patch # User huffman # Date 1235589961 28800 # Node ID c5497842ee35db6c1611cf4ecd4e11c666ec63c3 # Parent c6e184561159b8b6ba1a724ba037aeaf0cf6b1bd new theory of Archimedean fields diff -r c6e184561159 -r c5497842ee35 src/HOL/Archimedean_Field.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Archimedean_Field.thy Wed Feb 25 11:26:01 2009 -0800 @@ -0,0 +1,400 @@ +(* Title: Archimedean_Field.thy + Author: Brian Huffman +*) + +header {* Archimedean Fields, Floor and Ceiling Functions *} + +theory Archimedean_Field +imports Main +begin + +subsection {* Class of Archimedean fields *} + +text {* Archimedean fields have no infinite elements. *} + +class archimedean_field = ordered_field + number_ring + + assumes ex_le_of_int: "\z. x \ of_int z" + +lemma ex_less_of_int: + fixes x :: "'a::archimedean_field" shows "\z. x < of_int z" +proof - + from ex_le_of_int obtain z where "x \ of_int z" .. + then have "x < of_int (z + 1)" by simp + then show ?thesis .. +qed + +lemma ex_of_int_less: + fixes x :: "'a::archimedean_field" shows "\z. of_int z < x" +proof - + from ex_less_of_int obtain z where "- x < of_int z" .. + then have "of_int (- z) < x" by simp + then show ?thesis .. +qed + +lemma ex_less_of_nat: + fixes x :: "'a::archimedean_field" shows "\n. x < of_nat n" +proof - + obtain z where "x < of_int z" using ex_less_of_int .. + also have "\ \ of_int (int (nat z))" by simp + also have "\ = of_nat (nat z)" by (simp only: of_int_of_nat_eq) + finally show ?thesis .. +qed + +lemma ex_le_of_nat: + fixes x :: "'a::archimedean_field" shows "\n. x \ of_nat n" +proof - + obtain n where "x < of_nat n" using ex_less_of_nat .. + then have "x \ of_nat n" by simp + then show ?thesis .. +qed + +text {* Archimedean fields have no infinitesimal elements. *} + +lemma ex_inverse_of_nat_Suc_less: + fixes x :: "'a::archimedean_field" + assumes "0 < x" shows "\n. inverse (of_nat (Suc n)) < x" +proof - + from `0 < x` have "0 < inverse x" + by (rule positive_imp_inverse_positive) + obtain n where "inverse x < of_nat n" + using ex_less_of_nat .. + then obtain m where "inverse x < of_nat (Suc m)" + using `0 < inverse x` by (cases n) (simp_all del: of_nat_Suc) + then have "inverse (of_nat (Suc m)) < inverse (inverse x)" + using `0 < inverse x` by (rule less_imp_inverse_less) + then have "inverse (of_nat (Suc m)) < x" + using `0 < x` by (simp add: nonzero_inverse_inverse_eq) + then show ?thesis .. +qed + +lemma ex_inverse_of_nat_less: + fixes x :: "'a::archimedean_field" + assumes "0 < x" shows "\n>0. inverse (of_nat n) < x" + using ex_inverse_of_nat_Suc_less [OF `0 < x`] by auto + +lemma ex_less_of_nat_mult: + fixes x :: "'a::archimedean_field" + assumes "0 < x" shows "\n. y < of_nat n * x" +proof - + obtain n where "y / x < of_nat n" using ex_less_of_nat .. + with `0 < x` have "y < of_nat n * x" by (simp add: pos_divide_less_eq) + then show ?thesis .. +qed + + +subsection {* Existence and uniqueness of floor function *} + +lemma exists_least_lemma: + assumes "\ P 0" and "\n. P n" + shows "\n. \ P n \ P (Suc n)" +proof - + from `\n. P n` have "P (Least P)" by (rule LeastI_ex) + with `\ P 0` obtain n where "Least P = Suc n" + by (cases "Least P") auto + then have "n < Least P" by simp + then have "\ P n" by (rule not_less_Least) + then have "\ P n \ P (Suc n)" + using `P (Least P)` `Least P = Suc n` by simp + then show ?thesis .. +qed + +lemma floor_exists: + fixes x :: "'a::archimedean_field" + shows "\z. of_int z \ x \ x < of_int (z + 1)" +proof (cases) + assume "0 \ x" + then have "\ x < of_nat 0" by simp + then have "\n. \ x < of_nat n \ x < of_nat (Suc n)" + using ex_less_of_nat by (rule exists_least_lemma) + then obtain n where "\ x < of_nat n \ x < of_nat (Suc n)" .. + then have "of_int (int n) \ x \ x < of_int (int n + 1)" by simp + then show ?thesis .. +next + assume "\ 0 \ x" + then have "\ - x \ of_nat 0" by simp + then have "\n. \ - x \ of_nat n \ - x \ of_nat (Suc n)" + using ex_le_of_nat by (rule exists_least_lemma) + then obtain n where "\ - x \ of_nat n \ - x \ of_nat (Suc n)" .. + then have "of_int (- int n - 1) \ x \ x < of_int (- int n - 1 + 1)" by simp + then show ?thesis .. +qed + +lemma floor_exists1: + fixes x :: "'a::archimedean_field" + shows "\!z. of_int z \ x \ x < of_int (z + 1)" +proof (rule ex_ex1I) + show "\z. of_int z \ x \ x < of_int (z + 1)" + by (rule floor_exists) +next + fix y z assume + "of_int y \ x \ x < of_int (y + 1)" + "of_int z \ x \ x < of_int (z + 1)" + then have + "of_int y \ x" "x < of_int (y + 1)" + "of_int z \ x" "x < of_int (z + 1)" + by simp_all + from le_less_trans [OF `of_int y \ x` `x < of_int (z + 1)`] + le_less_trans [OF `of_int z \ x` `x < of_int (y + 1)`] + show "y = z" by (simp del: of_int_add) +qed + + +subsection {* Floor function *} + +definition + floor :: "'a::archimedean_field \ int" where + [code del]: "floor x = (THE z. of_int z \ x \ x < of_int (z + 1))" + +notation (xsymbols) + floor ("\_\") + +notation (HTML output) + floor ("\_\") + +lemma floor_correct: "of_int (floor x) \ x \ x < of_int (floor x + 1)" + unfolding floor_def using floor_exists1 by (rule theI') + +lemma floor_unique: "\of_int z \ x; x < of_int z + 1\ \ floor x = z" + using floor_correct [of x] floor_exists1 [of x] by auto + +lemma of_int_floor_le: "of_int (floor x) \ x" + using floor_correct .. + +lemma le_floor_iff: "z \ floor x \ of_int z \ x" +proof + assume "z \ floor x" + then have "(of_int z :: 'a) \ of_int (floor x)" by simp + also have "of_int (floor x) \ x" by (rule of_int_floor_le) + finally show "of_int z \ x" . +next + assume "of_int z \ x" + also have "x < of_int (floor x + 1)" using floor_correct .. + finally show "z \ floor x" by (simp del: of_int_add) +qed + +lemma floor_less_iff: "floor x < z \ x < of_int z" + by (simp add: not_le [symmetric] le_floor_iff) + +lemma less_floor_iff: "z < floor x \ of_int z + 1 \ x" + using le_floor_iff [of "z + 1" x] by auto + +lemma floor_le_iff: "floor x \ z \ x < of_int z + 1" + by (simp add: not_less [symmetric] less_floor_iff) + +lemma floor_mono: assumes "x \ y" shows "floor x \ floor y" +proof - + have "of_int (floor x) \ x" by (rule of_int_floor_le) + also note `x \ y` + finally show ?thesis by (simp add: le_floor_iff) +qed + +lemma floor_less_cancel: "floor x < floor y \ x < y" + by (auto simp add: not_le [symmetric] floor_mono) + +lemma floor_of_int [simp]: "floor (of_int z) = z" + by (rule floor_unique) simp_all + +lemma floor_of_nat [simp]: "floor (of_nat n) = int n" + using floor_of_int [of "of_nat n"] by simp + +text {* Floor with numerals *} + +lemma floor_zero [simp]: "floor 0 = 0" + using floor_of_int [of 0] by simp + +lemma floor_one [simp]: "floor 1 = 1" + using floor_of_int [of 1] by simp + +lemma floor_number_of [simp]: "floor (number_of v) = number_of v" + using floor_of_int [of "number_of v"] by simp + +lemma zero_le_floor [simp]: "0 \ floor x \ 0 \ x" + by (simp add: le_floor_iff) + +lemma one_le_floor [simp]: "1 \ floor x \ 1 \ x" + by (simp add: le_floor_iff) + +lemma number_of_le_floor [simp]: "number_of v \ floor x \ number_of v \ x" + by (simp add: le_floor_iff) + +lemma zero_less_floor [simp]: "0 < floor x \ 1 \ x" + by (simp add: less_floor_iff) + +lemma one_less_floor [simp]: "1 < floor x \ 2 \ x" + by (simp add: less_floor_iff) + +lemma number_of_less_floor [simp]: + "number_of v < floor x \ number_of v + 1 \ x" + by (simp add: less_floor_iff) + +lemma floor_le_zero [simp]: "floor x \ 0 \ x < 1" + by (simp add: floor_le_iff) + +lemma floor_le_one [simp]: "floor x \ 1 \ x < 2" + by (simp add: floor_le_iff) + +lemma floor_le_number_of [simp]: + "floor x \ number_of v \ x < number_of v + 1" + by (simp add: floor_le_iff) + +lemma floor_less_zero [simp]: "floor x < 0 \ x < 0" + by (simp add: floor_less_iff) + +lemma floor_less_one [simp]: "floor x < 1 \ x < 1" + by (simp add: floor_less_iff) + +lemma floor_less_number_of [simp]: + "floor x < number_of v \ x < number_of v" + by (simp add: floor_less_iff) + +text {* Addition and subtraction of integers *} + +lemma floor_add_of_int [simp]: "floor (x + of_int z) = floor x + z" + using floor_correct [of x] by (simp add: floor_unique) + +lemma floor_add_number_of [simp]: + "floor (x + number_of v) = floor x + number_of v" + using floor_add_of_int [of x "number_of v"] by simp + +lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1" + using floor_add_of_int [of x 1] by simp + +lemma floor_diff_of_int [simp]: "floor (x - of_int z) = floor x - z" + using floor_add_of_int [of x "- z"] by (simp add: algebra_simps) + +lemma floor_diff_number_of [simp]: + "floor (x - number_of v) = floor x - number_of v" + using floor_diff_of_int [of x "number_of v"] by simp + +lemma floor_diff_one [simp]: "floor (x - 1) = floor x - 1" + using floor_diff_of_int [of x 1] by simp + + +subsection {* Ceiling function *} + +definition + ceiling :: "'a::archimedean_field \ int" where + [code del]: "ceiling x = - floor (- x)" + +notation (xsymbols) + ceiling ("\_\") + +notation (HTML output) + ceiling ("\_\") + +lemma ceiling_correct: "of_int (ceiling x) - 1 < x \ x \ of_int (ceiling x)" + unfolding ceiling_def using floor_correct [of "- x"] by simp + +lemma ceiling_unique: "\of_int z - 1 < x; x \ of_int z\ \ ceiling x = z" + unfolding ceiling_def using floor_unique [of "- z" "- x"] by simp + +lemma le_of_int_ceiling: "x \ of_int (ceiling x)" + using ceiling_correct .. + +lemma ceiling_le_iff: "ceiling x \ z \ x \ of_int z" + unfolding ceiling_def using le_floor_iff [of "- z" "- x"] by auto + +lemma less_ceiling_iff: "z < ceiling x \ of_int z < x" + by (simp add: not_le [symmetric] ceiling_le_iff) + +lemma ceiling_less_iff: "ceiling x < z \ x \ of_int z - 1" + using ceiling_le_iff [of x "z - 1"] by simp + +lemma le_ceiling_iff: "z \ ceiling x \ of_int z - 1 < x" + by (simp add: not_less [symmetric] ceiling_less_iff) + +lemma ceiling_mono: "x \ y \ ceiling x \ ceiling y" + unfolding ceiling_def by (simp add: floor_mono) + +lemma ceiling_less_cancel: "ceiling x < ceiling y \ x < y" + by (auto simp add: not_le [symmetric] ceiling_mono) + +lemma ceiling_of_int [simp]: "ceiling (of_int z) = z" + by (rule ceiling_unique) simp_all + +lemma ceiling_of_nat [simp]: "ceiling (of_nat n) = int n" + using ceiling_of_int [of "of_nat n"] by simp + +text {* Ceiling with numerals *} + +lemma ceiling_zero [simp]: "ceiling 0 = 0" + using ceiling_of_int [of 0] by simp + +lemma ceiling_one [simp]: "ceiling 1 = 1" + using ceiling_of_int [of 1] by simp + +lemma ceiling_number_of [simp]: "ceiling (number_of v) = number_of v" + using ceiling_of_int [of "number_of v"] by simp + +lemma ceiling_le_zero [simp]: "ceiling x \ 0 \ x \ 0" + by (simp add: ceiling_le_iff) + +lemma ceiling_le_one [simp]: "ceiling x \ 1 \ x \ 1" + by (simp add: ceiling_le_iff) + +lemma ceiling_le_number_of [simp]: + "ceiling x \ number_of v \ x \ number_of v" + by (simp add: ceiling_le_iff) + +lemma ceiling_less_zero [simp]: "ceiling x < 0 \ x \ -1" + by (simp add: ceiling_less_iff) + +lemma ceiling_less_one [simp]: "ceiling x < 1 \ x \ 0" + by (simp add: ceiling_less_iff) + +lemma ceiling_less_number_of [simp]: + "ceiling x < number_of v \ x \ number_of v - 1" + by (simp add: ceiling_less_iff) + +lemma zero_le_ceiling [simp]: "0 \ ceiling x \ -1 < x" + by (simp add: le_ceiling_iff) + +lemma one_le_ceiling [simp]: "1 \ ceiling x \ 0 < x" + by (simp add: le_ceiling_iff) + +lemma number_of_le_ceiling [simp]: + "number_of v \ ceiling x\ number_of v - 1 < x" + by (simp add: le_ceiling_iff) + +lemma zero_less_ceiling [simp]: "0 < ceiling x \ 0 < x" + by (simp add: less_ceiling_iff) + +lemma one_less_ceiling [simp]: "1 < ceiling x \ 1 < x" + by (simp add: less_ceiling_iff) + +lemma number_of_less_ceiling [simp]: + "number_of v < ceiling x \ number_of v < x" + by (simp add: less_ceiling_iff) + +text {* Addition and subtraction of integers *} + +lemma ceiling_add_of_int [simp]: "ceiling (x + of_int z) = ceiling x + z" + using ceiling_correct [of x] by (simp add: ceiling_unique) + +lemma ceiling_add_number_of [simp]: + "ceiling (x + number_of v) = ceiling x + number_of v" + using ceiling_add_of_int [of x "number_of v"] by simp + +lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1" + using ceiling_add_of_int [of x 1] by simp + +lemma ceiling_diff_of_int [simp]: "ceiling (x - of_int z) = ceiling x - z" + using ceiling_add_of_int [of x "- z"] by (simp add: algebra_simps) + +lemma ceiling_diff_number_of [simp]: + "ceiling (x - number_of v) = ceiling x - number_of v" + using ceiling_diff_of_int [of x "number_of v"] by simp + +lemma ceiling_diff_one [simp]: "ceiling (x - 1) = ceiling x - 1" + using ceiling_diff_of_int [of x 1] by simp + + +subsection {* Negation *} + +lemma floor_minus [simp]: "floor (- x) = - ceiling x" + unfolding ceiling_def by simp + +lemma ceiling_minus [simp]: "ceiling (- x) = - floor x" + unfolding ceiling_def by simp + +end diff -r c6e184561159 -r c5497842ee35 src/HOL/IsaMakefile --- a/src/HOL/IsaMakefile Wed Feb 25 09:09:50 2009 -0800 +++ b/src/HOL/IsaMakefile Wed Feb 25 11:26:01 2009 -0800 @@ -267,6 +267,7 @@ @$(ISABELLE_TOOL) usedir -b -f main.ML -g true $(OUT)/Pure HOL-Main $(OUT)/HOL: ROOT.ML $(MAIN_DEPENDENCIES) \ + Archimedean_Field.thy \ Complex_Main.thy \ Complex.thy \ Deriv.thy \