--- a/src/HOL/Power.thy Thu Jun 24 17:52:02 2004 +0200
+++ b/src/HOL/Power.thy Thu Jun 24 17:52:55 2004 +0200
@@ -11,55 +11,55 @@
subsection{*Powers for Arbitrary (Semi)Rings*}
-axclass ringpower \<subseteq> comm_semiring_1_cancel, power
- power_0 [simp]: "a ^ 0 = 1"
- power_Suc: "a ^ (Suc n) = a * (a ^ n)"
+axclass recpower \<subseteq> comm_semiring_1_cancel, 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"
+lemma power_0_Suc [simp]: "(0::'a::recpower) ^ (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))"
+lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::recpower))"
by (induct_tac "n", auto)
-lemma power_one [simp]: "1^n = (1::'a::ringpower)"
+lemma power_one [simp]: "1^n = (1::'a::recpower)"
apply (induct_tac "n")
apply (auto simp add: power_Suc)
done
-lemma power_one_right [simp]: "(a::'a::ringpower) ^ 1 = a"
+lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
by (simp add: power_Suc)
-lemma power_add: "(a::'a::ringpower) ^ (m+n) = (a^m) * (a^n)"
+lemma power_add: "(a::'a::recpower) ^ (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"
+lemma power_mult: "(a::'a::recpower) ^ (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)"
+lemma power_mult_distrib: "((a::'a::recpower) * 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_semidom,ringpower}) ==> 0 < a^n"
+ "0 < (a::'a::{ordered_semidom,recpower}) ==> 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_semidom,ringpower}) ==> 0 \<le> a^n"
+ "0 \<le> (a::'a::{ordered_semidom,recpower}) ==> 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_semidom,ringpower}) ==> 1 \<le> a^n"
+ "1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n"
apply (induct_tac "n")
apply (simp_all add: power_Suc)
apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
@@ -70,7 +70,7 @@
by (simp add: order_trans [OF zero_le_one order_less_imp_le])
lemma power_gt1_lemma:
- assumes gt1: "1 < (a::'a::{ordered_semidom,ringpower})"
+ assumes gt1: "1 < (a::'a::{ordered_semidom,recpower})"
shows "1 < a * a^n"
proof -
have "1*1 < a*1" using gt1 by simp
@@ -81,11 +81,11 @@
qed
lemma power_gt1:
- "1 < (a::'a::{ordered_semidom,ringpower}) ==> 1 < a ^ (Suc n)"
+ "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"
by (simp add: power_gt1_lemma power_Suc)
lemma power_le_imp_le_exp:
- assumes gt1: "(1::'a::{ringpower,ordered_semidom}) < a"
+ assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a"
shows "!!n. a^m \<le> a^n ==> m \<le> n"
proof (induct m)
case 0
@@ -109,26 +109,26 @@
text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
lemma power_inject_exp [simp]:
- "1 < (a::'a::{ordered_semidom,ringpower}) ==> (a^m = a^n) = (m=n)"
+ "1 < (a::'a::{ordered_semidom,recpower}) ==> (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_semidom}) < a; a^m < a^n |] ==> m < n"
+ "[| (1::'a::{recpower,ordered_semidom}) < 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_semidom}) \<le> a|] ==> a^n \<le> b^n"
+ "[|a \<le> b; (0::'a::{recpower,ordered_semidom}) \<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_semidom}) \<le> a|]
+ "[|a < b; (0::'a::{recpower,ordered_semidom}) \<le> a|]
==> 0 < n --> a^n < b^n"
apply (induct_tac "n")
apply (auto simp add: mult_strict_mono zero_le_power power_Suc
@@ -136,51 +136,51 @@
done
lemma power_eq_0_iff [simp]:
- "(a^n = 0) = (a = (0::'a::{ordered_idom,ringpower}) & 0<n)"
+ "(a^n = 0) = (a = (0::'a::{ordered_idom,recpower}) & 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)"
+ "(a^n = 0) = (a = (0::'a::{field,recpower}) & 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"
+lemma field_power_not_zero: "a \<noteq> (0::'a::{field,recpower}) ==> a^n \<noteq> 0"
by force
lemma nonzero_power_inverse:
- "a \<noteq> 0 ==> inverse ((a::'a::{field,ringpower}) ^ n) = (inverse a) ^ n"
+ "a \<noteq> 0 ==> inverse ((a::'a::{field,recpower}) ^ n) = (inverse a) ^ n"
apply (induct_tac "n")
apply (auto simp add: power_Suc nonzero_inverse_mult_distrib mult_commute)
done
text{*Perhaps these should be simprules.*}
lemma power_inverse:
- "inverse ((a::'a::{field,division_by_zero,ringpower}) ^ n) = (inverse a) ^ n"
+ "inverse ((a::'a::{field,division_by_zero,recpower}) ^ n) = (inverse a) ^ n"
apply (induct_tac "n")
apply (auto simp add: power_Suc inverse_mult_distrib)
done
lemma nonzero_power_divide:
- "b \<noteq> 0 ==> (a/b) ^ n = ((a::'a::{field,ringpower}) ^ n) / (b ^ n)"
+ "b \<noteq> 0 ==> (a/b) ^ n = ((a::'a::{field,recpower}) ^ n) / (b ^ n)"
by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
lemma power_divide:
- "(a/b) ^ n = ((a::'a::{field,division_by_zero,ringpower}) ^ n / b ^ n)"
+ "(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)"
apply (case_tac "b=0", simp add: power_0_left)
apply (rule nonzero_power_divide)
apply assumption
done
-lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,ringpower}) ^ n"
+lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n"
apply (induct_tac "n")
apply (auto simp add: power_Suc abs_mult)
done
lemma zero_less_power_abs_iff [simp]:
- "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,ringpower}) | n=0)"
+ "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower}) | n=0)"
proof (induct "n")
case 0
show ?case by (simp add: zero_less_one)
@@ -190,12 +190,12 @@
qed
lemma zero_le_power_abs [simp]:
- "(0::'a::{ordered_idom,ringpower}) \<le> (abs a)^n"
+ "(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n"
apply (induct_tac "n")
apply (auto simp add: zero_le_one zero_le_power)
done
-lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{comm_ring_1,ringpower}) ^ n"
+lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{comm_ring_1,recpower}) ^ n"
proof -
have "-a = (- 1) * a" by (simp add: minus_mult_left [symmetric])
thus ?thesis by (simp only: power_mult_distrib)
@@ -203,14 +203,14 @@
text{*Lemma for @{text power_strict_decreasing}*}
lemma power_Suc_less:
- "[|(0::'a::{ordered_semidom,ringpower}) < a; a < 1|]
+ "[|(0::'a::{ordered_semidom,recpower}) < 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_semidom,ringpower})|]
+ "[|n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})|]
==> a^N < a^n"
apply (erule rev_mp)
apply (induct_tac "N")
@@ -223,7 +223,7 @@
text{*Proof resembles that of @{text power_strict_decreasing}*}
lemma power_decreasing:
- "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,ringpower})|]
+ "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,recpower})|]
==> a^N \<le> a^n"
apply (erule rev_mp)
apply (induct_tac "N")
@@ -235,13 +235,13 @@
done
lemma power_Suc_less_one:
- "[| 0 < a; a < (1::'a::{ordered_semidom,ringpower}) |] ==> a ^ Suc n < 1"
+ "[| 0 < a; a < (1::'a::{ordered_semidom,recpower}) |] ==> 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_semidom,ringpower}) \<le> a|] ==> a^n \<le> a^N"
+ "[|n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a|] ==> a^n \<le> a^N"
apply (erule rev_mp)
apply (induct_tac "N")
apply (auto simp add: power_Suc le_Suc_eq)
@@ -253,13 +253,13 @@
text{*Lemma for @{text power_strict_increasing}*}
lemma power_less_power_Suc:
- "(1::'a::{ordered_semidom,ringpower}) < a ==> a^n < a * a^n"
+ "(1::'a::{ordered_semidom,recpower}) < 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_semidom,ringpower}) < a|] ==> a^n < a^N"
+ "[|n < N; (1::'a::{ordered_semidom,recpower}) < 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)
@@ -272,7 +272,7 @@
lemma power_le_imp_le_base:
assumes le: "a ^ Suc n \<le> b ^ Suc n"
- and xnonneg: "(0::'a::{ordered_semidom,ringpower}) \<le> a"
+ and xnonneg: "(0::'a::{ordered_semidom,recpower}) \<le> a"
and ynonneg: "0 \<le> b"
shows "a \<le> b"
proof (rule ccontr)
@@ -286,7 +286,7 @@
lemma power_inject_base:
"[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
- ==> a = (b::'a::{ordered_semidom,ringpower})"
+ ==> a = (b::'a::{ordered_semidom,recpower})"
by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
@@ -296,7 +296,7 @@
"p ^ 0 = 1"
"p ^ (Suc n) = (p::nat) * (p ^ n)"
-instance nat :: ringpower
+instance nat :: recpower
proof
fix z n :: nat
show "z^0 = 1" by simp