--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Examples/Sqrt.thy Sat Jun 05 12:57:52 2021 +0200
@@ -0,0 +1,104 @@
+(* Title: HOL/Examples/Sqrt.thy
+ Author: Makarius
+ Author: Tobias Nipkow, TU Muenchen
+*)
+
+section \<open>Square roots of primes are irrational\<close>
+
+theory Sqrt
+ imports Complex_Main "HOL-Computational_Algebra.Primes"
+begin
+
+text \<open>
+ The square root of any prime number (including 2) is irrational.
+\<close>
+
+theorem sqrt_prime_irrational:
+ fixes p :: nat
+ assumes "prime p"
+ shows "sqrt p \<notin> \<rat>"
+proof
+ from \<open>prime p\<close> have p: "p > 1" by (rule prime_gt_1_nat)
+ assume "sqrt p \<in> \<rat>"
+ then obtain m n :: nat
+ where n: "n \<noteq> 0"
+ and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
+ and "coprime m n" by (rule Rats_abs_nat_div_natE)
+ have eq: "m\<^sup>2 = p * n\<^sup>2"
+ proof -
+ from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp
+ then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2" by (simp add: power_mult_distrib)
+ also have "(sqrt p)\<^sup>2 = p" by simp
+ also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp
+ finally show ?thesis by linarith
+ qed
+ have "p dvd m \<and> p dvd n"
+ proof
+ from eq have "p dvd m\<^sup>2" ..
+ with \<open>prime p\<close> show "p dvd m" by (rule prime_dvd_power)
+ then obtain k where "m = p * k" ..
+ with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by algebra
+ with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
+ then have "p dvd n\<^sup>2" ..
+ with \<open>prime p\<close> show "p dvd n" by (rule prime_dvd_power)
+ qed
+ then have "p dvd gcd m n" by simp
+ with \<open>coprime m n\<close> have "p = 1" by simp
+ with p show False by simp
+qed
+
+corollary sqrt_2_not_rat: "sqrt 2 \<notin> \<rat>"
+ using sqrt_prime_irrational [of 2] by simp
+
+text \<open>
+ Here is an alternative version of the main proof, using mostly linear
+ forward-reasoning. While this results in less top-down structure, it is
+ probably closer to proofs seen in mathematics.
+\<close>
+
+theorem
+ fixes p :: nat
+ assumes "prime p"
+ shows "sqrt p \<notin> \<rat>"
+proof
+ from \<open>prime p\<close> have p: "p > 1" by (rule prime_gt_1_nat)
+ assume "sqrt p \<in> \<rat>"
+ then obtain m n :: nat
+ where n: "n \<noteq> 0"
+ and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
+ and "coprime m n" by (rule Rats_abs_nat_div_natE)
+ from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp
+ then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2" by (auto simp add: power2_eq_square)
+ also have "(sqrt p)\<^sup>2 = p" by simp
+ also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp
+ finally have eq: "m\<^sup>2 = p * n\<^sup>2" by linarith
+ then have "p dvd m\<^sup>2" ..
+ with \<open>prime p\<close> have dvd_m: "p dvd m" by (rule prime_dvd_power)
+ then obtain k where "m = p * k" ..
+ with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by algebra
+ with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
+ then have "p dvd n\<^sup>2" ..
+ with \<open>prime p\<close> have "p dvd n" by (rule prime_dvd_power)
+ with dvd_m have "p dvd gcd m n" by (rule gcd_greatest)
+ with \<open>coprime m n\<close> have "p = 1" by simp
+ with p show False by simp
+qed
+
+
+text \<open>
+ Another old chestnut, which is a consequence of the irrationality of
+ \<^term>\<open>sqrt 2\<close>.
+\<close>
+
+lemma "\<exists>a b::real. a \<notin> \<rat> \<and> b \<notin> \<rat> \<and> a powr b \<in> \<rat>" (is "\<exists>a b. ?P a b")
+proof (cases "sqrt 2 powr sqrt 2 \<in> \<rat>")
+ case True
+ with sqrt_2_not_rat have "?P (sqrt 2) (sqrt 2)" by simp
+ then show ?thesis by blast
+next
+ case False
+ with sqrt_2_not_rat powr_powr have "?P (sqrt 2 powr sqrt 2) (sqrt 2)" by simp
+ then show ?thesis by blast
+qed
+
+end
--- a/src/HOL/ROOT Sat Jun 05 12:45:00 2021 +0200
+++ b/src/HOL/ROOT Sat Jun 05 12:57:52 2021 +0200
@@ -20,7 +20,7 @@
Notable Examples in Isabelle/HOL.
"
sessions
- "HOL-Library"
+ "HOL-Computational_Algebra"
theories
Adhoc_Overloading_Examples
Ackermann
@@ -36,6 +36,7 @@
Peirce
Records
Seq
+ Sqrt
document_files
"root.bib"
"root.tex"
@@ -706,7 +707,6 @@
Sketch_and_Explore
Sorting_Algorithms_Examples
Specifications_with_bundle_mixins
- Sqrt
Sqrt_Script
Sudoku
Sum_of_Powers
--- a/src/HOL/ex/Sqrt.thy Sat Jun 05 12:45:00 2021 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,104 +0,0 @@
-(* Title: HOL/ex/Sqrt.thy
- Author: Makarius
- Author: Tobias Nipkow, TU Muenchen
-*)
-
-section \<open>Square roots of primes are irrational\<close>
-
-theory Sqrt
- imports Complex_Main "HOL-Computational_Algebra.Primes"
-begin
-
-text \<open>
- The square root of any prime number (including 2) is irrational.
-\<close>
-
-theorem sqrt_prime_irrational:
- fixes p :: nat
- assumes "prime p"
- shows "sqrt p \<notin> \<rat>"
-proof
- from \<open>prime p\<close> have p: "p > 1" by (rule prime_gt_1_nat)
- assume "sqrt p \<in> \<rat>"
- then obtain m n :: nat
- where n: "n \<noteq> 0"
- and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
- and "coprime m n" by (rule Rats_abs_nat_div_natE)
- have eq: "m\<^sup>2 = p * n\<^sup>2"
- proof -
- from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp
- then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2" by (simp add: power_mult_distrib)
- also have "(sqrt p)\<^sup>2 = p" by simp
- also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp
- finally show ?thesis by linarith
- qed
- have "p dvd m \<and> p dvd n"
- proof
- from eq have "p dvd m\<^sup>2" ..
- with \<open>prime p\<close> show "p dvd m" by (rule prime_dvd_power)
- then obtain k where "m = p * k" ..
- with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by algebra
- with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
- then have "p dvd n\<^sup>2" ..
- with \<open>prime p\<close> show "p dvd n" by (rule prime_dvd_power)
- qed
- then have "p dvd gcd m n" by simp
- with \<open>coprime m n\<close> have "p = 1" by simp
- with p show False by simp
-qed
-
-corollary sqrt_2_not_rat: "sqrt 2 \<notin> \<rat>"
- using sqrt_prime_irrational [of 2] by simp
-
-text \<open>
- Here is an alternative version of the main proof, using mostly linear
- forward-reasoning. While this results in less top-down structure, it is
- probably closer to proofs seen in mathematics.
-\<close>
-
-theorem
- fixes p :: nat
- assumes "prime p"
- shows "sqrt p \<notin> \<rat>"
-proof
- from \<open>prime p\<close> have p: "p > 1" by (rule prime_gt_1_nat)
- assume "sqrt p \<in> \<rat>"
- then obtain m n :: nat
- where n: "n \<noteq> 0"
- and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
- and "coprime m n" by (rule Rats_abs_nat_div_natE)
- from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp
- then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2" by (auto simp add: power2_eq_square)
- also have "(sqrt p)\<^sup>2 = p" by simp
- also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp
- finally have eq: "m\<^sup>2 = p * n\<^sup>2" by linarith
- then have "p dvd m\<^sup>2" ..
- with \<open>prime p\<close> have dvd_m: "p dvd m" by (rule prime_dvd_power)
- then obtain k where "m = p * k" ..
- with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by algebra
- with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
- then have "p dvd n\<^sup>2" ..
- with \<open>prime p\<close> have "p dvd n" by (rule prime_dvd_power)
- with dvd_m have "p dvd gcd m n" by (rule gcd_greatest)
- with \<open>coprime m n\<close> have "p = 1" by simp
- with p show False by simp
-qed
-
-
-text \<open>
- Another old chestnut, which is a consequence of the irrationality of
- \<^term>\<open>sqrt 2\<close>.
-\<close>
-
-lemma "\<exists>a b::real. a \<notin> \<rat> \<and> b \<notin> \<rat> \<and> a powr b \<in> \<rat>" (is "\<exists>a b. ?P a b")
-proof (cases "sqrt 2 powr sqrt 2 \<in> \<rat>")
- case True
- with sqrt_2_not_rat have "?P (sqrt 2) (sqrt 2)" by simp
- then show ?thesis by blast
-next
- case False
- with sqrt_2_not_rat powr_powr have "?P (sqrt 2 powr sqrt 2) (sqrt 2)" by simp
- then show ?thesis by blast
-qed
-
-end
--- a/src/HOL/ex/Sqrt_Script.thy Sat Jun 05 12:45:00 2021 +0200
+++ b/src/HOL/ex/Sqrt_Script.thy Sat Jun 05 12:57:52 2021 +0200
@@ -5,14 +5,14 @@
section \<open>Square roots of primes are irrational (script version)\<close>
-theory Sqrt_Script
-imports Complex_Main "HOL-Computational_Algebra.Primes"
-begin
+text \<open>
+ Contrast this linear Isabelle/Isar script with the more mathematical version
+ in \<^file>\<open>~~/src/HOL/Examples/Sqrt.thy\<close> by Makarius Wenzel.
+\<close>
-text \<open>
- \medskip Contrast this linear Isabelle/Isar script with Markus
- Wenzel's more mathematical version.
-\<close>
+theory Sqrt_Script
+ imports Complex_Main "HOL-Computational_Algebra.Primes"
+begin
subsection \<open>Preliminaries\<close>