--- a/src/HOL/Power.thy Fri Jan 09 01:28:24 2004 +0100
+++ b/src/HOL/Power.thy Fri Jan 09 10:46:18 2004 +0100
@@ -3,23 +3,451 @@
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1997 University of Cambridge
-The (overloaded) exponentiation operator, ^ :: [nat,nat]=>nat
-Also binomial coefficents
*)
-Power = Divides +
-consts
- binomial :: "[nat,nat] => nat" (infixl "choose" 65)
+header{*Exponentiation and Binomial Coefficients*}
+
+theory Power = Divides:
+
+subsection{*Powers for Arbitrary (Semi)Rings*}
+
+axclass ringpower \<subseteq> semiring, power
+ power_0 [simp]: "a ^ 0 = 1"
+ power_Suc: "a ^ (Suc n) = a * (a ^ n)"
+
+lemma power_0_Suc [simp]: "(0::'a::ringpower) ^ (Suc n) = 0"
+by (simp add: power_Suc)
+
+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::ringpower))"
+by (induct_tac "n", auto)
+
+lemma power_one [simp]: "1^n = (1::'a::ringpower)"
+apply (induct_tac "n")
+apply (auto simp add: power_Suc)
+done
+
+lemma power_one_right [simp]: "(a::'a::ringpower) ^ 1 = a"
+by (simp add: power_Suc)
+
+lemma power_add: "(a::'a::ringpower) ^ (m+n) = (a^m) * (a^n)"
+apply (induct_tac "n")
+apply (simp_all add: power_Suc mult_ac)
+done
+
+lemma power_mult: "(a::'a::ringpower) ^ (m*n) = (a^m) ^ n"
+apply (induct_tac "n")
+apply (simp_all add: power_Suc power_add)
+done
+
+lemma power_mult_distrib: "((a::'a::ringpower) * b) ^ n = (a^n) * (b^n)"
+apply (induct_tac "n")
+apply (auto simp add: power_Suc mult_ac)
+done
+
+lemma zero_less_power:
+ "0 < (a::'a::{ordered_semiring,ringpower}) ==> 0 < a^n"
+apply (induct_tac "n")
+apply (simp_all add: power_Suc zero_less_one mult_pos)
+done
+
+lemma zero_le_power:
+ "0 \<le> (a::'a::{ordered_semiring,ringpower}) ==> 0 \<le> a^n"
+apply (simp add: order_le_less)
+apply (erule disjE)
+apply (simp_all add: zero_less_power zero_less_one power_0_left)
+done
+
+lemma one_le_power:
+ "1 \<le> (a::'a::{ordered_semiring,ringpower}) ==> 1 \<le> a^n"
+apply (induct_tac "n")
+apply (simp_all add: power_Suc)
+apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
+apply (simp_all add: zero_le_one order_trans [OF zero_le_one])
+done
+
+lemma gt1_imp_ge0: "1 < a ==> 0 \<le> (a::'a::ordered_semiring)"
+ by (simp add: order_trans [OF zero_le_one order_less_imp_le])
+
+lemma power_gt1_lemma:
+ assumes gt1: "1 < (a::'a::{ordered_semiring,ringpower})"
+ shows "1 < a * a^n"
+proof -
+ have "1*1 < a * a^n"
+ proof (rule order_less_le_trans)
+ show "1*1 < a*1" by (simp add: gt1)
+ show "a*1 \<le> a * a^n"
+ by (simp only: mult_mono gt1 gt1_imp_ge0 one_le_power order_less_imp_le
+ zero_le_one order_refl)
+ qed
+ thus ?thesis by simp
+qed
+
+lemma power_gt1:
+ "1 < (a::'a::{ordered_semiring,ringpower}) ==> 1 < a ^ (Suc n)"
+by (simp add: power_gt1_lemma power_Suc)
+
+lemma power_le_imp_le_exp:
+ assumes gt1: "(1::'a::{ringpower,ordered_semiring}) < a"
+ shows "!!n. a^m \<le> a^n ==> m \<le> n"
+proof (induct "m")
+ case 0
+ show ?case by simp
+next
+ case (Suc m)
+ show ?case
+ proof (cases n)
+ case 0
+ from prems have "a * a^m \<le> 1" by (simp add: power_Suc)
+ with gt1 show ?thesis
+ by (force simp only: power_gt1_lemma
+ linorder_not_less [symmetric])
+ next
+ case (Suc n)
+ from prems show ?thesis
+ by (force dest: mult_left_le_imp_le
+ simp add: power_Suc order_less_trans [OF zero_less_one gt1])
+ qed
+qed
+
+text{*Surely we can strengthen this? It holds for 0<a<1 too.*}
+lemma power_inject_exp [simp]:
+ "1 < (a::'a::{ordered_semiring,ringpower}) ==> (a^m = a^n) = (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::{ringpower,ordered_semiring}) < a; a^m < a^n |] ==> m < n"
+by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"]
+ power_le_imp_le_exp)
+
+
+lemma power_mono:
+ "[|a \<le> b; (0::'a::{ringpower,ordered_semiring}) \<le> a|] ==> a^n \<le> b^n"
+apply (induct_tac "n")
+apply (simp_all add: power_Suc)
+apply (auto intro: mult_mono zero_le_power order_trans [of 0 a b])
+done
+
+lemma power_strict_mono [rule_format]:
+ "[|a < b; (0::'a::{ringpower,ordered_semiring}) \<le> a|]
+ ==> 0 < n --> a^n < b^n"
+apply (induct_tac "n")
+apply (auto simp add: mult_strict_mono zero_le_power power_Suc
+ order_le_less_trans [of 0 a b])
+done
+
+lemma power_eq_0_iff [simp]:
+ "(a^n = 0) = (a = (0::'a::{ordered_ring,ringpower}) & 0<n)"
+apply (induct_tac "n")
+apply (auto simp add: power_Suc zero_neq_one [THEN not_sym])
+done
+
+lemma field_power_eq_0_iff [simp]:
+ "(a^n = 0) = (a = (0::'a::{field,ringpower}) & 0<n)"
+apply (induct_tac "n")
+apply (auto simp add: power_Suc field_mult_eq_0_iff zero_neq_one[THEN not_sym])
+done
+
+lemma field_power_not_zero: "a \<noteq> (0::'a::{field,ringpower}) ==> a^n \<noteq> 0"
+by force
+
+text{*Perhaps these should be simprules.*}
+lemma power_inverse:
+ "inverse ((a::'a::{field,division_by_zero,ringpower}) ^ n) = (inverse a) ^ n"
+apply (induct_tac "n")
+apply (auto simp add: power_Suc inverse_mult_distrib)
+done
+
+lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_field,ringpower}) ^ n"
+apply (induct_tac "n")
+apply (auto simp add: power_Suc abs_mult)
+done
+
+lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{ring,ringpower}) ^ n"
+proof -
+ have "-a = (- 1) * a" by (simp add: minus_mult_left [symmetric])
+ thus ?thesis by (simp only: power_mult_distrib)
+qed
+
+text{*Lemma for @{text power_strict_decreasing}*}
+lemma power_Suc_less:
+ "[|(0::'a::{ordered_semiring,ringpower}) < a; a < 1|]
+ ==> a * a^n < a^n"
+apply (induct_tac n)
+apply (auto simp add: power_Suc mult_strict_left_mono)
+done
+
+lemma power_strict_decreasing:
+ "[|n < N; 0 < a; a < (1::'a::{ordered_semiring,ringpower})|]
+ ==> a^N < a^n"
+apply (erule rev_mp)
+apply (induct_tac "N")
+apply (auto simp add: power_Suc power_Suc_less less_Suc_eq)
+apply (rename_tac m)
+apply (subgoal_tac "a * a^m < 1 * a^n", simp)
+apply (rule mult_strict_mono)
+apply (auto simp add: zero_le_power zero_less_one order_less_imp_le)
+done
+
+text{*Proof resembles that of @{text power_strict_decreasing}*}
+lemma power_decreasing:
+ "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semiring,ringpower})|]
+ ==> a^N \<le> a^n"
+apply (erule rev_mp)
+apply (induct_tac "N")
+apply (auto simp add: power_Suc le_Suc_eq)
+apply (rename_tac m)
+apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)
+apply (rule mult_mono)
+apply (auto simp add: zero_le_power zero_le_one)
+done
+
+lemma power_Suc_less_one:
+ "[| 0 < a; a < (1::'a::{ordered_semiring,ringpower}) |] ==> a ^ Suc n < 1"
+apply (insert power_strict_decreasing [of 0 "Suc n" a], simp)
+done
+
+text{*Proof again resembles that of @{text power_strict_decreasing}*}
+lemma power_increasing:
+ "[|n \<le> N; (1::'a::{ordered_semiring,ringpower}) \<le> a|] ==> a^n \<le> a^N"
+apply (erule rev_mp)
+apply (induct_tac "N")
+apply (auto simp add: power_Suc le_Suc_eq)
+apply (rename_tac m)
+apply (subgoal_tac "1 * a^n \<le> a * a^m", simp)
+apply (rule mult_mono)
+apply (auto simp add: order_trans [OF zero_le_one] zero_le_power)
+done
+
+text{*Lemma for @{text power_strict_increasing}*}
+lemma power_less_power_Suc:
+ "(1::'a::{ordered_semiring,ringpower}) < a ==> a^n < a * a^n"
+apply (induct_tac n)
+apply (auto simp add: power_Suc mult_strict_left_mono order_less_trans [OF zero_less_one])
+done
+
+lemma power_strict_increasing:
+ "[|n < N; (1::'a::{ordered_semiring,ringpower}) < a|] ==> a^n < a^N"
+apply (erule rev_mp)
+apply (induct_tac "N")
+apply (auto simp add: power_less_power_Suc power_Suc less_Suc_eq)
+apply (rename_tac m)
+apply (subgoal_tac "1 * a^n < a * a^m", simp)
+apply (rule mult_strict_mono)
+apply (auto simp add: order_less_trans [OF zero_less_one] zero_le_power
+ order_less_imp_le)
+done
+
+lemma power_le_imp_le_base:
+ assumes le: "a ^ Suc n \<le> b ^ Suc n"
+ and xnonneg: "(0::'a::{ordered_semiring,ringpower}) \<le> a"
+ 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: prems power_strict_mono)
+ from le and this show "False"
+ by (simp add: linorder_not_less [symmetric])
+ qed
+
+lemma power_inject_base:
+ "[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
+ ==> a = (b::'a::{ordered_semiring,ringpower})"
+by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
+
+
+subsection{*Exponentiation for the Natural Numbers*}
primrec (power)
"p ^ 0 = 1"
"p ^ (Suc n) = (p::nat) * (p ^ n)"
+instance nat :: ringpower
+proof
+ fix z :: nat
+ fix n :: nat
+ show "z^0 = 1" by simp
+ show "z^(Suc n) = z * (z^n)" by simp
+qed
+
+lemma nat_one_le_power [simp]: "1 \<le> i ==> Suc 0 \<le> i^n"
+by (insert one_le_power [of i n], simp)
+
+lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
+apply (unfold dvd_def)
+apply (erule not_less_iff_le [THEN iffD2, THEN add_diff_inverse, THEN subst])
+apply (simp add: power_add)
+done
+
+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: "!!i::nat. [| 0 < i; i^m < i^n |] ==> m < n"
+apply (rule ccontr)
+apply (drule leI [THEN le_imp_power_dvd, THEN dvd_imp_le, THEN leD])
+apply (erule zero_less_power, auto)
+done
+
+lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"
+by (induct_tac "n", auto)
+
+lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
+apply (induct_tac "j")
+apply (simp_all add: le_Suc_eq)
+apply (blast dest!: dvd_mult_right)
+done
+
+lemma power_dvd_imp_le: "[|i^m dvd i^n; (1::nat) < i|] ==> m \<le> n"
+apply (rule power_le_imp_le_exp, assumption)
+apply (erule dvd_imp_le, simp)
+done
+
+
+subsection{*Binomial Coefficients*}
+
+text{*This development is based on the work of Andy Gordon and
+Florian Kammueller*}
+
+consts
+ binomial :: "[nat,nat] => nat" (infixl "choose" 65)
+
primrec
- binomial_0 "(0 choose k) = (if k = 0 then 1 else 0)"
+ binomial_0: "(0 choose k) = (if k = 0 then 1 else 0)"
+
+ binomial_Suc: "(Suc n choose k) =
+ (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"
+
+lemma binomial_n_0 [simp]: "(n choose 0) = 1"
+by (case_tac "n", simp_all)
+
+lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"
+by simp
+
+lemma binomial_Suc_Suc [simp]:
+ "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"
+by simp
+
+lemma binomial_eq_0 [rule_format]: "\<forall>k. n < k --> (n choose k) = 0"
+apply (induct_tac "n", auto)
+apply (erule allE)
+apply (erule mp, arith)
+done
+
+declare binomial_0 [simp del] binomial_Suc [simp del]
+
+lemma binomial_n_n [simp]: "(n choose n) = 1"
+apply (induct_tac "n")
+apply (simp_all add: binomial_eq_0)
+done
+
+lemma binomial_Suc_n [simp]: "(Suc n choose n) = Suc n"
+by (induct_tac "n", simp_all)
+
+lemma binomial_1 [simp]: "(n choose Suc 0) = n"
+by (induct_tac "n", simp_all)
+
+lemma zero_less_binomial [rule_format]: "k \<le> n --> 0 < (n choose k)"
+by (rule_tac m = n and n = k in diff_induct, simp_all)
- binomial_Suc "(Suc n choose k) =
- (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"
+lemma binomial_eq_0_iff: "(n choose k = 0) = (n<k)"
+apply (safe intro!: binomial_eq_0)
+apply (erule contrapos_pp)
+apply (simp add: zero_less_binomial)
+done
+
+lemma zero_less_binomial_iff: "(0 < n choose k) = (k\<le>n)"
+by (simp add: linorder_not_less [symmetric] binomial_eq_0_iff [symmetric])
+
+(*Might be more useful if re-oriented*)
+lemma Suc_times_binomial_eq [rule_format]:
+ "\<forall>k. k \<le> n --> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
+apply (induct_tac "n")
+apply (simp add: binomial_0, clarify)
+apply (case_tac "k")
+apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq
+ binomial_eq_0)
+done
+
+text{*This is the well-known version, but it's harder to use because of the
+ need to reason about division.*}
+lemma binomial_Suc_Suc_eq_times:
+ "k \<le> n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
+by (simp add: Suc_times_binomial_eq div_mult_self_is_m zero_less_Suc
+ del: mult_Suc mult_Suc_right)
+
+text{*Another version, with -1 instead of Suc.*}
+lemma times_binomial_minus1_eq:
+ "[|k \<le> n; 0<k|] ==> (n choose k) * k = n * ((n - 1) choose (k - 1))"
+apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq)
+apply (simp split add: nat_diff_split, auto)
+done
+
+text{*ML bindings for the general exponentiation theorems*}
+ML
+{*
+val power_0 = thm"power_0";
+val power_Suc = thm"power_Suc";
+val power_0_Suc = thm"power_0_Suc";
+val power_0_left = thm"power_0_left";
+val power_one = thm"power_one";
+val power_one_right = thm"power_one_right";
+val power_add = thm"power_add";
+val power_mult = thm"power_mult";
+val power_mult_distrib = thm"power_mult_distrib";
+val zero_less_power = thm"zero_less_power";
+val zero_le_power = thm"zero_le_power";
+val one_le_power = thm"one_le_power";
+val gt1_imp_ge0 = thm"gt1_imp_ge0";
+val power_gt1_lemma = thm"power_gt1_lemma";
+val power_gt1 = thm"power_gt1";
+val power_le_imp_le_exp = thm"power_le_imp_le_exp";
+val power_inject_exp = thm"power_inject_exp";
+val power_less_imp_less_exp = thm"power_less_imp_less_exp";
+val power_mono = thm"power_mono";
+val power_strict_mono = thm"power_strict_mono";
+val power_eq_0_iff = thm"power_eq_0_iff";
+val field_power_eq_0_iff = thm"field_power_eq_0_iff";
+val field_power_not_zero = thm"field_power_not_zero";
+val power_inverse = thm"power_inverse";
+val power_abs = thm"power_abs";
+val power_minus = thm"power_minus";
+val power_Suc_less = thm"power_Suc_less";
+val power_strict_decreasing = thm"power_strict_decreasing";
+val power_decreasing = thm"power_decreasing";
+val power_Suc_less_one = thm"power_Suc_less_one";
+val power_increasing = thm"power_increasing";
+val power_strict_increasing = thm"power_strict_increasing";
+val power_le_imp_le_base = thm"power_le_imp_le_base";
+val power_inject_base = thm"power_inject_base";
+*}
+
+text{*ML bindings for the remaining theorems*}
+ML
+{*
+val nat_one_le_power = thm"nat_one_le_power";
+val le_imp_power_dvd = thm"le_imp_power_dvd";
+val nat_power_less_imp_less = thm"nat_power_less_imp_less";
+val nat_zero_less_power_iff = thm"nat_zero_less_power_iff";
+val power_le_dvd = thm"power_le_dvd";
+val power_dvd_imp_le = thm"power_dvd_imp_le";
+val binomial_n_0 = thm"binomial_n_0";
+val binomial_0_Suc = thm"binomial_0_Suc";
+val binomial_Suc_Suc = thm"binomial_Suc_Suc";
+val binomial_eq_0 = thm"binomial_eq_0";
+val binomial_n_n = thm"binomial_n_n";
+val binomial_Suc_n = thm"binomial_Suc_n";
+val binomial_1 = thm"binomial_1";
+val zero_less_binomial = thm"zero_less_binomial";
+val binomial_eq_0_iff = thm"binomial_eq_0_iff";
+val zero_less_binomial_iff = thm"zero_less_binomial_iff";
+val Suc_times_binomial_eq = thm"Suc_times_binomial_eq";
+val binomial_Suc_Suc_eq_times = thm"binomial_Suc_Suc_eq_times";
+val times_binomial_minus1_eq = thm"times_binomial_minus1_eq";
+*}
end