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diff --git a/tests/examplefiles/example.thy b/tests/examplefiles/example.thy new file mode 100644 index 00000000..abaa1af8 --- /dev/null +++ b/tests/examplefiles/example.thy @@ -0,0 +1,751 @@ +(* from Isabelle2013-2 src/HOL/Power.thy; BSD license *) + +(* Title: HOL/Power.thy + Author: Lawrence C Paulson, Cambridge University Computer Laboratory + Copyright 1997 University of Cambridge +*) + +header {* Exponentiation *} + +theory Power +imports Num +begin + +subsection {* Powers for Arbitrary Monoids *} + +class power = one + times +begin + +primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) where + power_0: "a ^ 0 = 1" + | power_Suc: "a ^ Suc n = a * a ^ n" + +notation (latex output) + power ("(_\<^bsup>_\<^esup>)" [1000] 1000) + +notation (HTML output) + power ("(_\<^bsup>_\<^esup>)" [1000] 1000) + +text {* Special syntax for squares. *} + +abbreviation (xsymbols) + power2 :: "'a \<Rightarrow> 'a" ("(_\<^sup>2)" [1000] 999) where + "x\<^sup>2 \<equiv> x ^ 2" + +notation (latex output) + power2 ("(_\<^sup>2)" [1000] 999) + +notation (HTML output) + power2 ("(_\<^sup>2)" [1000] 999) + +end + +context monoid_mult +begin + +subclass power . + +lemma power_one [simp]: + "1 ^ n = 1" + by (induct n) simp_all + +lemma power_one_right [simp]: + "a ^ 1 = a" + by simp + +lemma power_commutes: + "a ^ n * a = a * a ^ n" + by (induct n) (simp_all add: mult_assoc) + +lemma power_Suc2: + "a ^ Suc n = a ^ n * a" + by (simp add: power_commutes) + +lemma power_add: + "a ^ (m + n) = a ^ m * a ^ n" + by (induct m) (simp_all add: algebra_simps) + +lemma power_mult: + "a ^ (m * n) = (a ^ m) ^ n" + by (induct n) (simp_all add: power_add) + +lemma power2_eq_square: "a\<^sup>2 = a * a" + by (simp add: numeral_2_eq_2) + +lemma power3_eq_cube: "a ^ 3 = a * a * a" + by (simp add: numeral_3_eq_3 mult_assoc) + +lemma power_even_eq: + "a ^ (2 * n) = (a ^ n)\<^sup>2" + by (subst mult_commute) (simp add: power_mult) + +lemma power_odd_eq: + "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2" + by (simp add: power_even_eq) + +lemma power_numeral_even: + "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)" + unfolding numeral_Bit0 power_add Let_def .. + +lemma power_numeral_odd: + "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)" + unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right + unfolding power_Suc power_add Let_def mult_assoc .. + +lemma funpow_times_power: + "(times x ^^ f x) = times (x ^ f x)" +proof (induct "f x" arbitrary: f) + case 0 then show ?case by (simp add: fun_eq_iff) +next + case (Suc n) + def g \<equiv> "\<lambda>x. f x - 1" + with Suc have "n = g x" by simp + with Suc have "times x ^^ g x = times (x ^ g x)" by simp + moreover from Suc g_def have "f x = g x + 1" by simp + ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult_assoc) +qed + +end + +context comm_monoid_mult +begin + +lemma power_mult_distrib: + "(a * b) ^ n = (a ^ n) * (b ^ n)" + by (induct n) (simp_all add: mult_ac) + +end + +context semiring_numeral +begin + +lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k" + by (simp only: sqr_conv_mult numeral_mult) + +lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l" + by (induct l, simp_all only: numeral_class.numeral.simps pow.simps + numeral_sqr numeral_mult power_add power_one_right) + +lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)" + by (rule numeral_pow [symmetric]) + +end + +context semiring_1 +begin + +lemma of_nat_power: + "of_nat (m ^ n) = of_nat m ^ n" + by (induct n) (simp_all add: of_nat_mult) + +lemma power_zero_numeral [simp]: "(0::'a) ^ numeral k = 0" + by (simp add: numeral_eq_Suc) + +lemma zero_power2: "0\<^sup>2 = 0" (* delete? *) + by (rule power_zero_numeral) + +lemma one_power2: "1\<^sup>2 = 1" (* delete? *) + by (rule power_one) + +end + +context comm_semiring_1 +begin + +text {* The divides relation *} + +lemma le_imp_power_dvd: + assumes "m \<le> n" shows "a ^ m dvd a ^ n" +proof + have "a ^ n = a ^ (m + (n - m))" + using `m \<le> n` by simp + also have "\<dots> = a ^ m * a ^ (n - m)" + by (rule power_add) + finally show "a ^ n = a ^ m * a ^ (n - m)" . +qed + +lemma power_le_dvd: + "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b" + by (rule dvd_trans [OF le_imp_power_dvd]) + +lemma dvd_power_same: + "x dvd y \<Longrightarrow> x ^ n dvd y ^ n" + by (induct n) (auto simp add: mult_dvd_mono) + +lemma dvd_power_le: + "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m" + by (rule power_le_dvd [OF dvd_power_same]) + +lemma dvd_power [simp]: + assumes "n > (0::nat) \<or> x = 1" + shows "x dvd (x ^ n)" +using assms proof + assume "0 < n" + then have "x ^ n = x ^ Suc (n - 1)" by simp + then show "x dvd (x ^ n)" by simp +next + assume "x = 1" + then show "x dvd (x ^ n)" by simp +qed + +end + +context ring_1 +begin + +lemma power_minus: + "(- a) ^ n = (- 1) ^ n * a ^ n" +proof (induct n) + case 0 show ?case by simp +next + case (Suc n) then show ?case + by (simp del: power_Suc add: power_Suc2 mult_assoc) +qed + +lemma power_minus_Bit0: + "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)" + by (induct k, simp_all only: numeral_class.numeral.simps power_add + power_one_right mult_minus_left mult_minus_right minus_minus) + +lemma power_minus_Bit1: + "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))" + by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left) + +lemma power_neg_numeral_Bit0 [simp]: + "neg_numeral k ^ numeral (Num.Bit0 l) = numeral (Num.pow k (Num.Bit0 l))" + by (simp only: neg_numeral_def power_minus_Bit0 power_numeral) + +lemma power_neg_numeral_Bit1 [simp]: + "neg_numeral k ^ numeral (Num.Bit1 l) = neg_numeral (Num.pow k (Num.Bit1 l))" + by (simp only: neg_numeral_def power_minus_Bit1 power_numeral pow.simps) + +lemma power2_minus [simp]: + "(- a)\<^sup>2 = a\<^sup>2" + by (rule power_minus_Bit0) + +lemma power_minus1_even [simp]: + "-1 ^ (2*n) = 1" +proof (induct n) + case 0 show ?case by simp +next + case (Suc n) then show ?case by (simp add: power_add power2_eq_square) +qed + +lemma power_minus1_odd: + "-1 ^ Suc (2*n) = -1" + by simp + +lemma power_minus_even [simp]: + "(-a) ^ (2*n) = a ^ (2*n)" + by (simp add: power_minus [of a]) + +end + +context ring_1_no_zero_divisors +begin + +lemma field_power_not_zero: + "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0" + by (induct n) auto + +lemma zero_eq_power2 [simp]: + "a\<^sup>2 = 0 \<longleftrightarrow> a = 0" + unfolding power2_eq_square by simp + +lemma power2_eq_1_iff: + "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1" + unfolding power2_eq_square by (rule square_eq_1_iff) + +end + +context idom +begin + +lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y" + unfolding power2_eq_square by (rule square_eq_iff) + +end + +context division_ring +begin + +text {* FIXME reorient or rename to @{text nonzero_inverse_power} *} +lemma nonzero_power_inverse: + "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n" + by (induct n) + (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero) + +end + +context field +begin + +lemma nonzero_power_divide: + "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n" + by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse) + +end + + +subsection {* Exponentiation on ordered types *} + +context linordered_ring (* TODO: move *) +begin + +lemma sum_squares_ge_zero: + "0 \<le> x * x + y * y" + by (intro add_nonneg_nonneg zero_le_square) + +lemma not_sum_squares_lt_zero: + "\<not> x * x + y * y < 0" + by (simp add: not_less sum_squares_ge_zero) + +end + +context linordered_semidom +begin + +lemma zero_less_power [simp]: + "0 < a \<Longrightarrow> 0 < a ^ n" + by (induct n) (simp_all add: mult_pos_pos) + +lemma zero_le_power [simp]: + "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n" + by (induct n) (simp_all add: mult_nonneg_nonneg) + +lemma power_mono: + "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n" + by (induct n) (auto intro: mult_mono order_trans [of 0 a b]) + +lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n" + using power_mono [of 1 a n] by simp + +lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1" + using power_mono [of a 1 n] by simp + +lemma power_gt1_lemma: + assumes gt1: "1 < a" + shows "1 < a * a ^ n" +proof - + from gt1 have "0 \<le> a" + by (fact order_trans [OF zero_le_one less_imp_le]) + have "1 * 1 < a * 1" using gt1 by simp + also have "\<dots> \<le> a * a ^ n" using gt1 + by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le + zero_le_one order_refl) + finally show ?thesis by simp +qed + +lemma power_gt1: + "1 < a \<Longrightarrow> 1 < a ^ Suc n" + by (simp add: power_gt1_lemma) + +lemma one_less_power [simp]: + "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n" + by (cases n) (simp_all add: power_gt1_lemma) + +lemma power_le_imp_le_exp: + assumes gt1: "1 < a" + shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n" +proof (induct m arbitrary: n) + case 0 + show ?case by simp +next + case (Suc m) + show ?case + proof (cases n) + case 0 + with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp + with gt1 show ?thesis + by (force simp only: power_gt1_lemma + not_less [symmetric]) + next + case (Suc n) + with Suc.prems Suc.hyps show ?thesis + by (force dest: mult_left_le_imp_le + simp add: less_trans [OF zero_less_one gt1]) + qed +qed + +text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*} +lemma power_inject_exp [simp]: + "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n" + by (force simp add: order_antisym power_le_imp_le_exp) + +text{*Can relax the first premise to @{term "0<a"} in the case of the +natural numbers.*} +lemma power_less_imp_less_exp: + "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n" + by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"] + power_le_imp_le_exp) + +lemma power_strict_mono [rule_format]: + "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n" + by (induct n) + (auto simp add: mult_strict_mono le_less_trans [of 0 a b]) + +text{*Lemma for @{text power_strict_decreasing}*} +lemma power_Suc_less: + "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n" + by (induct n) + (auto simp add: mult_strict_left_mono) + +lemma power_strict_decreasing [rule_format]: + "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n" +proof (induct N) + case 0 then show ?case by simp +next + case (Suc N) then show ?case + apply (auto simp add: power_Suc_less less_Suc_eq) + apply (subgoal_tac "a * a^N < 1 * a^n") + apply simp + apply (rule mult_strict_mono) apply auto + done +qed + +text{*Proof resembles that of @{text power_strict_decreasing}*} +lemma power_decreasing [rule_format]: + "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n" +proof (induct N) + case 0 then show ?case by simp +next + case (Suc N) then show ?case + apply (auto simp add: le_Suc_eq) + apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp) + apply (rule mult_mono) apply auto + done +qed + +lemma power_Suc_less_one: + "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1" + using power_strict_decreasing [of 0 "Suc n" a] by simp + +text{*Proof again resembles that of @{text power_strict_decreasing}*} +lemma power_increasing [rule_format]: + "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N" +proof (induct N) + case 0 then show ?case by simp +next + case (Suc N) then show ?case + apply (auto simp add: le_Suc_eq) + apply (subgoal_tac "1 * a^n \<le> a * a^N", simp) + apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one]) + done +qed + +text{*Lemma for @{text power_strict_increasing}*} +lemma power_less_power_Suc: + "1 < a \<Longrightarrow> a ^ n < a * a ^ n" + by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one]) + +lemma power_strict_increasing [rule_format]: + "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N" +proof (induct N) + case 0 then show ?case by simp +next + case (Suc N) then show ?case + apply (auto simp add: power_less_power_Suc less_Suc_eq) + apply (subgoal_tac "1 * a^n < a * a^N", simp) + apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le) + done +qed + +lemma power_increasing_iff [simp]: + "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y" + by (blast intro: power_le_imp_le_exp power_increasing less_imp_le) + +lemma power_strict_increasing_iff [simp]: + "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y" +by (blast intro: power_less_imp_less_exp power_strict_increasing) + +lemma power_le_imp_le_base: + assumes le: "a ^ Suc n \<le> b ^ Suc n" + and ynonneg: "0 \<le> b" + shows "a \<le> b" +proof (rule ccontr) + assume "~ a \<le> b" + then have "b < a" by (simp only: linorder_not_le) + then have "b ^ Suc n < a ^ Suc n" + by (simp only: assms power_strict_mono) + from le and this show False + by (simp add: linorder_not_less [symmetric]) +qed + +lemma power_less_imp_less_base: + assumes less: "a ^ n < b ^ n" + assumes nonneg: "0 \<le> b" + shows "a < b" +proof (rule contrapos_pp [OF less]) + assume "~ a < b" + hence "b \<le> a" by (simp only: linorder_not_less) + hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono) + thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less) +qed + +lemma power_inject_base: + "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b" +by (blast intro: power_le_imp_le_base antisym eq_refl sym) + +lemma power_eq_imp_eq_base: + "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b" + by (cases n) (simp_all del: power_Suc, rule power_inject_base) + +lemma power2_le_imp_le: + "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y" + unfolding numeral_2_eq_2 by (rule power_le_imp_le_base) + +lemma power2_less_imp_less: + "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y" + by (rule power_less_imp_less_base) + +lemma power2_eq_imp_eq: + "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y" + unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp + +end + +context linordered_ring_strict +begin + +lemma sum_squares_eq_zero_iff: + "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0" + by (simp add: add_nonneg_eq_0_iff) + +lemma sum_squares_le_zero_iff: + "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0" + by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff) + +lemma sum_squares_gt_zero_iff: + "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0" + by (simp add: not_le [symmetric] sum_squares_le_zero_iff) + +end + +context linordered_idom +begin + +lemma power_abs: + "abs (a ^ n) = abs a ^ n" + by (induct n) (auto simp add: abs_mult) + +lemma abs_power_minus [simp]: + "abs ((-a) ^ n) = abs (a ^ n)" + by (simp add: power_abs) + +lemma zero_less_power_abs_iff [simp, no_atp]: + "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0" +proof (induct n) + case 0 show ?case by simp +next + case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff) +qed + +lemma zero_le_power_abs [simp]: + "0 \<le> abs a ^ n" + by (rule zero_le_power [OF abs_ge_zero]) + +lemma zero_le_power2 [simp]: + "0 \<le> a\<^sup>2" + by (simp add: power2_eq_square) + +lemma zero_less_power2 [simp]: + "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0" + by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff) + +lemma power2_less_0 [simp]: + "\<not> a\<^sup>2 < 0" + by (force simp add: power2_eq_square mult_less_0_iff) + +lemma abs_power2 [simp]: + "abs (a\<^sup>2) = a\<^sup>2" + by (simp add: power2_eq_square abs_mult abs_mult_self) + +lemma power2_abs [simp]: + "(abs a)\<^sup>2 = a\<^sup>2" + by (simp add: power2_eq_square abs_mult_self) + +lemma odd_power_less_zero: + "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0" +proof (induct n) + case 0 + then show ?case by simp +next + case (Suc n) + have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)" + by (simp add: mult_ac power_add power2_eq_square) + thus ?case + by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg) +qed + +lemma odd_0_le_power_imp_0_le: + "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a" + using odd_power_less_zero [of a n] + by (force simp add: linorder_not_less [symmetric]) + +lemma zero_le_even_power'[simp]: + "0 \<le> a ^ (2*n)" +proof (induct n) + case 0 + show ?case by simp +next + case (Suc n) + have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" + by (simp add: mult_ac power_add power2_eq_square) + thus ?case + by (simp add: Suc zero_le_mult_iff) +qed + +lemma sum_power2_ge_zero: + "0 \<le> x\<^sup>2 + y\<^sup>2" + by (intro add_nonneg_nonneg zero_le_power2) + +lemma not_sum_power2_lt_zero: + "\<not> x\<^sup>2 + y\<^sup>2 < 0" + unfolding not_less by (rule sum_power2_ge_zero) + +lemma sum_power2_eq_zero_iff: + "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0" + unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff) + +lemma sum_power2_le_zero_iff: + "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0" + by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero) + +lemma sum_power2_gt_zero_iff: + "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0" + unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff) + +end + + +subsection {* Miscellaneous rules *} + +lemma power_eq_if: "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))" + unfolding One_nat_def by (cases m) simp_all + +lemma power2_sum: + fixes x y :: "'a::comm_semiring_1" + shows "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y" + by (simp add: algebra_simps power2_eq_square mult_2_right) + +lemma power2_diff: + fixes x y :: "'a::comm_ring_1" + shows "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y" + by (simp add: ring_distribs power2_eq_square mult_2) (rule mult_commute) + +lemma power_0_Suc [simp]: + "(0::'a::{power, semiring_0}) ^ Suc n = 0" + by simp + +text{*It looks plausible as a simprule, but its effect can be strange.*} +lemma power_0_left: + "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))" + by (induct n) simp_all + +lemma power_eq_0_iff [simp]: + "a ^ n = 0 \<longleftrightarrow> + a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0" + by (induct n) + (auto simp add: no_zero_divisors elim: contrapos_pp) + +lemma (in field) power_diff: + assumes nz: "a \<noteq> 0" + shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n" + by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero) + +text{*Perhaps these should be simprules.*} +lemma power_inverse: + fixes a :: "'a::division_ring_inverse_zero" + shows "inverse (a ^ n) = inverse a ^ n" +apply (cases "a = 0") +apply (simp add: power_0_left) +apply (simp add: nonzero_power_inverse) +done (* TODO: reorient or rename to inverse_power *) + +lemma power_one_over: + "1 / (a::'a::{field_inverse_zero, power}) ^ n = (1 / a) ^ n" + by (simp add: divide_inverse) (rule power_inverse) + +lemma power_divide: + "(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ n" +apply (cases "b = 0") +apply (simp add: power_0_left) +apply (rule nonzero_power_divide) +apply assumption +done + +text {* Simprules for comparisons where common factors can be cancelled. *} + +lemmas zero_compare_simps = + add_strict_increasing add_strict_increasing2 add_increasing + zero_le_mult_iff zero_le_divide_iff + zero_less_mult_iff zero_less_divide_iff + mult_le_0_iff divide_le_0_iff + mult_less_0_iff divide_less_0_iff + zero_le_power2 power2_less_0 + + +subsection {* Exponentiation for the Natural Numbers *} + +lemma nat_one_le_power [simp]: + "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n" + by (rule one_le_power [of i n, unfolded One_nat_def]) + +lemma nat_zero_less_power_iff [simp]: + "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0" + by (induct n) auto + +lemma nat_power_eq_Suc_0_iff [simp]: + "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0" + by (induct m) auto + +lemma power_Suc_0 [simp]: + "Suc 0 ^ n = Suc 0" + by simp + +text{*Valid for the naturals, but what if @{text"0<i<1"}? +Premises cannot be weakened: consider the case where @{term "i=0"}, +@{term "m=1"} and @{term "n=0"}.*} +lemma nat_power_less_imp_less: + assumes nonneg: "0 < (i\<Colon>nat)" + assumes less: "i ^ m < i ^ n" + shows "m < n" +proof (cases "i = 1") + case True with less power_one [where 'a = nat] show ?thesis by simp +next + case False with nonneg have "1 < i" by auto + from power_strict_increasing_iff [OF this] less show ?thesis .. +qed + +lemma power_dvd_imp_le: + "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n" + apply (rule power_le_imp_le_exp, assumption) + apply (erule dvd_imp_le, simp) + done + +lemma power2_nat_le_eq_le: + fixes m n :: nat + shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n" + by (auto intro: power2_le_imp_le power_mono) + +lemma power2_nat_le_imp_le: + fixes m n :: nat + assumes "m\<^sup>2 \<le> n" + shows "m \<le> n" + using assms by (cases m) (simp_all add: power2_eq_square) + + + +subsection {* Code generator tweak *} + +lemma power_power_power [code]: + "power = power.power (1::'a::{power}) (op *)" + unfolding power_def power.power_def .. + +declare power.power.simps [code] + +code_identifier + code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith + +end + |