author krauss Tue Nov 07 09:33:47 2006 +0100 (2006-11-07) changeset 21199 2d83f93c3580 parent 21198 48b8d8b8334d child 21200 2f6e276bfb93
* Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
Richer structures do not inherit from semiring_0 anymore, because
anihilation is a theorem there, not an axiom.

* Generalized axclass "recpower" to arbitrary monoid, not just
commutative semirings.
 src/HOL/Finite_Set.thy file | annotate | diff | revisions src/HOL/Hyperreal/Deriv.thy file | annotate | diff | revisions src/HOL/Hyperreal/StarClasses.thy file | annotate | diff | revisions src/HOL/Integ/NatBin.thy file | annotate | diff | revisions src/HOL/Power.thy file | annotate | diff | revisions src/HOL/Ring_and_Field.thy file | annotate | diff | revisions
```     1.1 --- a/src/HOL/Finite_Set.thy	Tue Nov 07 08:03:46 2006 +0100
1.2 +++ b/src/HOL/Finite_Set.thy	Tue Nov 07 09:33:47 2006 +0100
1.3 @@ -1645,7 +1645,7 @@
1.4  done
1.5
1.6
1.7 -lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::recpower)) = y^(card A)"
1.8 +lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{recpower, comm_monoid_mult})) = y^(card A)"
1.9    apply (erule finite_induct)
1.10    apply (auto simp add: power_Suc)
1.11    done
```
```     2.1 --- a/src/HOL/Hyperreal/Deriv.thy	Tue Nov 07 08:03:46 2006 +0100
2.2 +++ b/src/HOL/Hyperreal/Deriv.thy	Tue Nov 07 09:33:47 2006 +0100
2.3 @@ -241,7 +241,7 @@
2.4  done
2.5
2.6  lemma DERIV_power_Suc:
2.7 -  fixes f :: "real \<Rightarrow> 'a::{real_normed_algebra,recpower}"
2.8 +  fixes f :: "real \<Rightarrow> 'a::{real_normed_algebra,recpower,comm_monoid_mult}"
2.9    assumes f: "DERIV f x :> D"
2.10    shows "DERIV (\<lambda>x. f x ^ Suc n) x :> (of_nat n + 1) *# (D * f x ^ n)"
2.11  proof (induct n)
2.12 @@ -255,7 +255,7 @@
2.13  qed
2.14
2.15  lemma DERIV_power:
2.16 -  fixes f :: "real \<Rightarrow> 'a::{real_normed_algebra,recpower}"
2.17 +  fixes f :: "real \<Rightarrow> 'a::{real_normed_algebra,recpower,comm_monoid_mult}"
2.18    assumes f: "DERIV f x :> D"
2.19    shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n *# (D * f x ^ (n - Suc 0))"
2.20  by (cases "n", simp, simp add: DERIV_power_Suc f)
```
```     3.1 --- a/src/HOL/Hyperreal/StarClasses.thy	Tue Nov 07 08:03:46 2006 +0100
3.2 +++ b/src/HOL/Hyperreal/StarClasses.thy	Tue Nov 07 09:33:47 2006 +0100
3.3 @@ -361,7 +361,9 @@
3.4  apply (transfer, rule right_distrib)
3.5  done
3.6
3.7 -instance star :: (semiring_0) semiring_0 ..
3.8 +instance star :: (semiring_0) semiring_0
3.9 +by intro_classes (transfer, simp)+
3.10 +
3.11  instance star :: (semiring_0_cancel) semiring_0_cancel ..
3.12
3.13  instance star :: (comm_semiring) comm_semiring
3.14 @@ -417,7 +419,7 @@
3.15  done
3.16
3.17  instance star :: (pordered_comm_semiring) pordered_comm_semiring
3.18 -by (intro_classes, transfer, rule pordered_comm_semiring_class.mult_mono)
3.19 +by (intro_classes, transfer, rule mult_mono1_class.mult_mono)
3.20
3.21  instance star :: (pordered_cancel_comm_semiring) pordered_cancel_comm_semiring ..
3.22
```
```     4.1 --- a/src/HOL/Integ/NatBin.thy	Tue Nov 07 08:03:46 2006 +0100
4.2 +++ b/src/HOL/Integ/NatBin.thy	Tue Nov 07 09:33:47 2006 +0100
4.3 @@ -334,7 +334,7 @@
4.4  done
4.5
4.6  lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2"
4.7 -by (simp add: power_mult power_mult_distrib power2_eq_square)
4.8 +by (subst mult_commute) (simp add: power_mult)
4.9
4.10  lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"
```
```     5.1 --- a/src/HOL/Power.thy	Tue Nov 07 08:03:46 2006 +0100
5.2 +++ b/src/HOL/Power.thy	Tue Nov 07 09:33:47 2006 +0100
5.3 @@ -11,17 +11,17 @@
5.4  imports Divides
5.5  begin
5.6
5.7 -subsection{*Powers for Arbitrary Semirings*}
5.8 +subsection{*Powers for Arbitrary Monoids*}
5.9
5.10 -axclass recpower \<subseteq> comm_semiring_1_cancel, power
5.11 +axclass recpower \<subseteq> monoid_mult, power
5.12    power_0 [simp]: "a ^ 0       = 1"
5.13    power_Suc:      "a ^ (Suc n) = a * (a ^ n)"
5.14
5.15 -lemma power_0_Suc [simp]: "(0::'a::recpower) ^ (Suc n) = 0"
5.16 +lemma power_0_Suc [simp]: "(0::'a::{recpower,semiring_0}) ^ (Suc n) = 0"
5.18
5.19  text{*It looks plausible as a simprule, but its effect can be strange.*}
5.20 -lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::recpower))"
5.21 +lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::{recpower,semiring_0}))"
5.22  by (induct "n", auto)
5.23
5.24  lemma power_one [simp]: "1^n = (1::'a::recpower)"
5.25 @@ -32,17 +32,20 @@
5.26  lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
5.28
5.29 +lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n"
5.30 +by (induct "n") (simp_all add:power_Suc mult_assoc)
5.31 +
5.32  lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"
5.33 -apply (induct "n")
5.34 +apply (induct "m")
5.35  apply (simp_all add: power_Suc mult_ac)
5.36  done
5.37
5.38  lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"
5.39 -apply (induct "n")
5.40 +apply (induct "n")
5.42  done
5.43
5.44 -lemma power_mult_distrib: "((a::'a::recpower) * b) ^ n = (a^n) * (b^n)"
5.45 +lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)"
5.46  apply (induct "n")
5.47  apply (auto simp add: power_Suc mult_ac)
5.48  done
```
```     6.1 --- a/src/HOL/Ring_and_Field.thy	Tue Nov 07 08:03:46 2006 +0100
6.2 +++ b/src/HOL/Ring_and_Field.thy	Tue Nov 07 09:33:47 2006 +0100
6.3 @@ -27,9 +27,27 @@
6.4    left_distrib: "(a + b) * c = a * c + b * c"
6.5    right_distrib: "a * (b + c) = a * b + a * c"
6.6
6.7 -axclass semiring_0 \<subseteq> semiring, comm_monoid_add
6.8 +axclass mult_zero \<subseteq> times, zero
6.9 +  mult_zero_left [simp]: "0 * a = 0"
6.10 +  mult_zero_right [simp]: "a * 0 = 0"
6.11 +
6.12 +axclass semiring_0 \<subseteq> semiring, comm_monoid_add, mult_zero
6.13 +
6.15
6.16 -axclass semiring_0_cancel \<subseteq> semiring_0, cancel_ab_semigroup_add
6.17 +instance semiring_0_cancel \<subseteq> semiring_0
6.18 +proof
6.19 +  fix a :: 'a
6.20 +  have "0 * a + 0 * a = 0 * a + 0"
6.21 +    by (simp add: left_distrib [symmetric])
6.22 +  thus "0 * a = 0"
6.23 +    by (simp only: add_left_cancel)
6.24 +
6.25 +  have "a * 0 + a * 0 = a * 0 + 0"
6.26 +    by (simp add: right_distrib [symmetric])
6.27 +  thus "a * 0 = 0"
6.28 +    by (simp only: add_left_cancel)
6.29 +qed
6.30
6.31  axclass comm_semiring \<subseteq> ab_semigroup_add, ab_semigroup_mult
6.32    distrib: "(a + b) * c = a * c + b * c"
6.33 @@ -44,14 +62,16 @@
6.34    finally show "a * (b + c) = a * b + a * c" by blast
6.35  qed
6.36
6.37 -axclass comm_semiring_0 \<subseteq> comm_semiring, comm_monoid_add
6.38 +axclass comm_semiring_0 \<subseteq> comm_semiring, comm_monoid_add, mult_zero
6.39
6.40  instance comm_semiring_0 \<subseteq> semiring_0 ..
6.41
6.42 -axclass comm_semiring_0_cancel \<subseteq> comm_semiring_0, cancel_ab_semigroup_add
6.44
6.45  instance comm_semiring_0_cancel \<subseteq> semiring_0_cancel ..
6.46
6.47 +instance comm_semiring_0_cancel \<subseteq> comm_semiring_0 ..
6.48 +
6.49  axclass zero_neq_one \<subseteq> zero, one
6.50    zero_neq_one [simp]: "0 \<noteq> 1"
6.51
6.52 @@ -64,31 +84,37 @@
6.53  axclass no_zero_divisors \<subseteq> zero, times
6.54    no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
6.55
6.56 -axclass semiring_1_cancel \<subseteq> semiring_1, cancel_ab_semigroup_add
6.58
6.59  instance semiring_1_cancel \<subseteq> semiring_0_cancel ..
6.60
6.61 -axclass comm_semiring_1_cancel \<subseteq> comm_semiring_1, cancel_ab_semigroup_add (* previously semiring *)
6.62 +instance semiring_1_cancel \<subseteq> semiring_1 ..
6.63 +
6.64 +axclass comm_semiring_1_cancel \<subseteq>
6.67
6.68  instance comm_semiring_1_cancel \<subseteq> semiring_1_cancel ..
6.69
6.70  instance comm_semiring_1_cancel \<subseteq> comm_semiring_0_cancel ..
6.71
6.72 +instance comm_semiring_1_cancel \<subseteq> comm_semiring_1 ..
6.73 +
6.74  axclass ring \<subseteq> semiring, ab_group_add
6.75
6.76  instance ring \<subseteq> semiring_0_cancel ..
6.77
6.78 -axclass comm_ring \<subseteq> comm_semiring_0, ab_group_add
6.79 +axclass comm_ring \<subseteq> comm_semiring, ab_group_add
6.80
6.81  instance comm_ring \<subseteq> ring ..
6.82
6.83  instance comm_ring \<subseteq> comm_semiring_0_cancel ..
6.84
6.85 -axclass ring_1 \<subseteq> ring, semiring_1
6.86 +axclass ring_1 \<subseteq> ring, zero_neq_one, monoid_mult
6.87
6.88  instance ring_1 \<subseteq> semiring_1_cancel ..
6.89
6.90 -axclass comm_ring_1 \<subseteq> comm_ring, comm_semiring_1 (* previously ring *)
6.91 +axclass comm_ring_1 \<subseteq> comm_ring, zero_neq_one, comm_monoid_mult (* previously ring *)
6.92
6.93  instance comm_ring_1 \<subseteq> ring_1 ..
6.94
6.95 @@ -115,22 +141,6 @@
6.96  instance field \<subseteq> division_ring
6.97  by (intro_classes, erule field_left_inverse, erule field_right_inverse)
6.98
6.99 -lemma mult_zero_left [simp]: "0 * a = (0::'a::semiring_0_cancel)"
6.100 -proof -
6.101 -  have "0*a + 0*a = 0*a + 0"
6.102 -    by (simp add: left_distrib [symmetric])
6.103 -  thus ?thesis
6.104 -    by (simp only: add_left_cancel)
6.105 -qed
6.106 -
6.107 -lemma mult_zero_right [simp]: "a * 0 = (0::'a::semiring_0_cancel)"
6.108 -proof -
6.109 -  have "a*0 + a*0 = a*0 + 0"
6.110 -    by (simp add: right_distrib [symmetric])
6.111 -  thus ?thesis
6.112 -    by (simp only: add_left_cancel)
6.113 -qed
6.114 -
6.115  lemma field_mult_eq_0_iff [simp]:
6.116    "(a*b = (0::'a::division_ring)) = (a = 0 | b = 0)"
6.117  proof cases
6.118 @@ -182,15 +192,22 @@
6.119  by (simp add: left_distrib diff_minus
6.120                minus_mult_left [symmetric] minus_mult_right [symmetric])
6.121
6.122 -axclass pordered_semiring \<subseteq> semiring_0, pordered_ab_semigroup_add
6.123 +axclass mult_mono \<subseteq> times, zero, ord
6.124    mult_left_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b"
6.125    mult_right_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> a * c <= b * c"
6.126
6.127 -axclass pordered_cancel_semiring \<subseteq> pordered_semiring, cancel_ab_semigroup_add
6.128 +axclass pordered_semiring \<subseteq> mult_mono, semiring_0, pordered_ab_semigroup_add
6.129 +
6.130 +axclass pordered_cancel_semiring \<subseteq>
6.134
6.135  instance pordered_cancel_semiring \<subseteq> semiring_0_cancel ..
6.136
6.137 -axclass ordered_semiring_strict \<subseteq> semiring_0, ordered_cancel_ab_semigroup_add
6.138 +instance pordered_cancel_semiring \<subseteq> pordered_semiring ..
6.139 +
6.141    mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
6.142    mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
6.143
6.144 @@ -204,18 +221,30 @@
6.146  done
6.147
6.148 -axclass pordered_comm_semiring \<subseteq> comm_semiring_0, pordered_ab_semigroup_add
6.149 +axclass mult_mono1 \<subseteq> times, zero, ord
6.150    mult_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b"
6.151
6.152 -axclass pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring, cancel_ab_semigroup_add
6.153 +axclass pordered_comm_semiring \<subseteq> comm_semiring_0, pordered_ab_semigroup_add, mult_mono1
6.154
6.155 +axclass pordered_cancel_comm_semiring \<subseteq>
6.157 +
6.158  instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring ..
6.159
6.160  axclass ordered_comm_semiring_strict \<subseteq> comm_semiring_0, ordered_cancel_ab_semigroup_add
6.161    mult_strict_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
6.162
6.163  instance pordered_comm_semiring \<subseteq> pordered_semiring
6.164 -by (intro_classes, insert mult_mono, simp_all add: mult_commute, blast+)
6.165 +proof
6.166 +  fix a b c :: 'a
6.167 +  assume A: "a <= b" "0 <= c"
6.168 +  with mult_mono show "c * a <= c * b" .
6.169 +
6.170 +  from mult_commute have "a * c = c * a" ..
6.171 +  also from mult_mono A have "\<dots> <= c * b" .
6.172 +  also from mult_commute have "c * b = b * c" ..
6.173 +  finally show "a * c <= b * c" .
6.174 +qed
6.175
6.176  instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring ..
6.177
6.178 @@ -229,12 +258,10 @@
6.179  apply (auto simp add: mult_strict_left_mono order_le_less)
6.180  done
6.181
6.182 -axclass pordered_ring \<subseteq> ring, pordered_semiring
6.183 +axclass pordered_ring \<subseteq> ring, pordered_cancel_semiring
6.184
6.185  instance pordered_ring \<subseteq> pordered_ab_group_add ..
6.186
6.187 -instance pordered_ring \<subseteq> pordered_cancel_semiring ..
6.188 -
6.189  axclass lordered_ring \<subseteq> pordered_ring, lordered_ab_group_abs
6.190
6.191  instance lordered_ring \<subseteq> lordered_ab_group_meet ..
```