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59 lines
1.8 KiB
Plaintext
59 lines
1.8 KiB
Plaintext
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theory HSV_chapter3 imports Complex_Main begin
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(* We use the following command to search Isabelle's library
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for theorems that contain a particular pattern. *)
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find_theorems "_ \<in> \<rat>"
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thm Rats_abs_nat_div_natE
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find_theorems "(_ / _)^_"
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thm power_divide
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(* A proof that the square root of 2 is irrational. *)
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theorem sqrt2_irrational: "sqrt 2 \<notin> \<rat>"
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proof auto
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assume "sqrt 2 \<in> \<rat>"
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then obtain m n where
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"n \<noteq> 0" and "\<bar>sqrt 2\<bar> = real m / real n" and "coprime m n"
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by (rule Rats_abs_nat_div_natE)
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hence "\<bar>sqrt 2\<bar>^2 = (real m / real n)^2" by auto
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hence "2 = (real m / real n)^2" by simp
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hence "2 = (real m)^2 / (real n)^2" unfolding power_divide by auto
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hence "2 * (real n)^2 = (real m)^2"
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by (simp add: nonzero_eq_divide_eq `n \<noteq> 0`)
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hence "real (2 * n^2) = (real m)^2" by auto
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hence *: "2 * n^2 = m^2"
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using of_nat_power_eq_of_nat_cancel_iff by blast
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hence "even (m^2)" by presburger
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hence "even m" by simp
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then obtain m' where "m = 2 * m'" by auto
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with * have "2 * n^2 = (2 * m')^2" by auto
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hence "2 * n^2 = 4 * m'^2" by simp
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hence "n^2 = 2 * m'^2" by simp
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hence "even (n^2)" by presburger
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hence "even n" by simp
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with `even m` and `coprime m n` show False by auto
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qed
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(* A proof that there is no greatest even number. *)
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theorem "\<forall>n::int. even n \<longrightarrow> (\<exists>m. even m \<and> m > n)"
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proof clarify
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fix n::int
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assume "even n"
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hence "even (n+2)" by simp
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moreover
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have "n < (n+2)" by simp
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ultimately
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show "\<exists>m. even m \<and> n < m" by blast
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qed
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(* The same proof in the procedural style. *)
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theorem "\<forall>n::int. even n \<longrightarrow> (\<exists>m. even m \<and> m > n)"
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apply clarify
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apply (rule_tac x="n+2" in exI)
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apply (rule conjI)
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apply simp
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apply (thin_tac "even n")
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apply simp
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done
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end
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