--- a/NEWS Fri Mar 30 15:25:47 2012 +0200
+++ b/NEWS Fri Mar 30 17:21:36 2012 +0200
@@ -162,6 +162,9 @@
mod_mult_distrib ~> mult_mod_left
mod_mult_distrib2 ~> mult_mod_right
+* Removed redundant theorems nat_mult_2 and nat_mult_2_right; use
+generic mult_2 and mult_2_right instead. INCOMPATIBILITY.
+
* More default pred/set conversions on a couple of relation operations
and predicates. Added powers of predicate relations.
Consolidation of some relation theorems:
--- a/src/HOL/Divides.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/Divides.thy Fri Mar 30 17:21:36 2012 +0200
@@ -1086,7 +1086,7 @@
by (simp add: numeral_2_eq_2 le_mod_geq)
lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
-by (simp add: nat_mult_2 [symmetric])
+by (simp add: mult_2 [symmetric])
lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
proof -
--- a/src/HOL/Enum.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/Enum.thy Fri Mar 30 17:21:36 2012 +0200
@@ -465,7 +465,7 @@
| "sublists (x#xs) = (let xss = sublists xs in map (Cons x) xss @ xss)"
lemma length_sublists:
- "length (sublists xs) = Suc (Suc (0\<Colon>nat)) ^ length xs"
+ "length (sublists xs) = 2 ^ length xs"
by (induct xs) (simp_all add: Let_def)
lemma sublists_powset:
@@ -484,9 +484,9 @@
shows "distinct (map set (sublists xs))"
proof (rule card_distinct)
have "finite (set xs)" by rule
- then have "card (Pow (set xs)) = Suc (Suc 0) ^ card (set xs)" by (rule card_Pow)
+ then have "card (Pow (set xs)) = 2 ^ card (set xs)" by (rule card_Pow)
with assms distinct_card [of xs]
- have "card (Pow (set xs)) = Suc (Suc 0) ^ length xs" by simp
+ have "card (Pow (set xs)) = 2 ^ length xs" by simp
then show "card (set (map set (sublists xs))) = length (map set (sublists xs))"
by (simp add: sublists_powset length_sublists)
qed
--- a/src/HOL/Finite_Set.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/Finite_Set.thy Fri Mar 30 17:21:36 2012 +0200
@@ -2212,12 +2212,13 @@
subsubsection {* Cardinality of the Powerset *}
-lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A" (* FIXME numeral 2 (!?) *)
+lemma card_Pow: "finite A ==> card (Pow A) = 2 ^ card A"
apply (induct rule: finite_induct)
apply (simp_all add: Pow_insert)
apply (subst card_Un_disjoint, blast)
apply (blast, blast)
apply (subgoal_tac "inj_on (insert x) (Pow F)")
+ apply (subst mult_2)
apply (simp add: card_image Pow_insert)
apply (unfold inj_on_def)
apply (blast elim!: equalityE)
--- a/src/HOL/Int.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/Int.thy Fri Mar 30 17:21:36 2012 +0200
@@ -8,7 +8,6 @@
theory Int
imports Equiv_Relations Wellfounded
uses
- ("Tools/numeral.ML")
("Tools/int_arith.ML")
begin
@@ -835,7 +834,6 @@
of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
of_int_0 of_int_1 of_int_add of_int_mult
-use "Tools/numeral.ML"
use "Tools/int_arith.ML"
declaration {* K Int_Arith.setup *}
@@ -844,16 +842,6 @@
"(m::'a::linordered_idom) = n") =
{* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}
-setup {*
- Reorient_Proc.add
- (fn Const (@{const_name numeral}, _) $ _ => true
- | Const (@{const_name neg_numeral}, _) $ _ => true
- | _ => false)
-*}
-
-simproc_setup reorient_numeral
- ("numeral w = x" | "neg_numeral w = y") = Reorient_Proc.proc
-
subsection{*Lemmas About Small Numerals*}
--- a/src/HOL/IsaMakefile Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/IsaMakefile Fri Mar 30 17:21:36 2012 +0200
@@ -1128,7 +1128,7 @@
Library/Code_Integer.thy Library/Code_Nat.thy \
Library/Code_Natural.thy Library/Efficient_Nat.thy \
Number_Theory/Primes.thy ex/Abstract_NAT.thy ex/Antiquote.thy \
- ex/Arith_Examples.thy ex/Arithmetic_Series_Complex.thy ex/BT.thy \
+ ex/Arith_Examples.thy ex/BT.thy \
ex/BinEx.thy ex/Binary.thy ex/Birthday_Paradox.thy ex/CTL.thy \
ex/Case_Product.thy ex/Chinese.thy ex/Classical.thy \
ex/Code_Nat_examples.thy \
--- a/src/HOL/Library/Cardinality.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/Library/Cardinality.thy Fri Mar 30 17:21:36 2012 +0200
@@ -59,7 +59,7 @@
lemma card_set [simp]: "CARD('a set) = 2 ^ CARD('a::finite)"
unfolding Pow_UNIV [symmetric]
- by (simp only: card_Pow finite numeral_2_eq_2)
+ by (simp only: card_Pow finite)
lemma card_nat [simp]: "CARD(nat) = 0"
by (simp add: card_eq_0_iff)
--- a/src/HOL/Library/Code_Integer.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/Library/Code_Integer.thy Fri Mar 30 17:21:36 2012 +0200
@@ -22,7 +22,7 @@
proof -
have "2 = nat 2" by simp
show ?thesis
- apply (subst nat_mult_2 [symmetric])
+ apply (subst mult_2 [symmetric])
apply (auto simp add: Let_def divmod_int_mod_div not_le
nat_div_distrib nat_mult_distrib mult_div_cancel mod_2_not_eq_zero_eq_one_int)
apply (unfold `2 = nat 2`)
--- a/src/HOL/Library/Formal_Power_Series.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/Library/Formal_Power_Series.thy Fri Mar 30 17:21:36 2012 +0200
@@ -2781,7 +2781,7 @@
lemma binomial_Vandermonde_same: "setsum (\<lambda>k. (n choose k)^2) {0..n} = (2*n) choose n"
using binomial_Vandermonde[of n n n,symmetric]
- unfolding nat_mult_2 apply (simp add: power2_eq_square)
+ unfolding mult_2 apply (simp add: power2_eq_square)
apply (rule setsum_cong2)
by (auto intro: binomial_symmetric)
@@ -3139,7 +3139,7 @@
moreover
{assume on: "odd n"
from on obtain m where m: "n = 2*m + 1"
- unfolding odd_nat_equiv_def2 by (auto simp add: nat_mult_2)
+ unfolding odd_nat_equiv_def2 by (auto simp add: mult_2)
have "?l $n = ?r$n"
by (simp add: m fps_sin_def fps_cos_def power_mult_distrib
power_mult power_minus)}
--- a/src/HOL/Library/Target_Numeral.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/Library/Target_Numeral.thy Fri Mar 30 17:21:36 2012 +0200
@@ -385,7 +385,7 @@
by simp
then have "num_of_nat (nat (int_of k)) =
num_of_nat (nat (int_of k) div 2 + nat (int_of k) div 2 + nat (int_of k) mod 2)"
- by (simp add: nat_mult_2)
+ by (simp add: mult_2)
with ** have "num_of_nat (nat (int_of k)) =
num_of_nat (nat (int_of k) div 2 + nat (int_of k) div 2 + 1)"
by simp
@@ -395,7 +395,7 @@
by (auto simp add: num_of_int_def nat_of_def Let_def prod_case_beta
not_le Target_Numeral.int_eq_iff less_eq_int_def
nat_mult_distrib nat_div_distrib num_of_nat_One num_of_nat_plus_distrib
- nat_mult_2 aux add_One)
+ mult_2 [where 'a=nat] aux add_One)
qed
hide_const (open) int_of nat_of Pos Neg sub dup divmod_abs num_of_int
--- a/src/HOL/NSA/HSeries.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/NSA/HSeries.thy Fri Mar 30 17:21:36 2012 +0200
@@ -114,7 +114,7 @@
lemma sumhr_minus_one_realpow_zero [simp]:
"!!N. sumhr(0, N + N, %i. (-1) ^ (i+1)) = 0"
unfolding sumhr_app
-by transfer (simp del: power_Suc add: nat_mult_2 [symmetric])
+by transfer (simp del: power_Suc add: mult_2 [symmetric])
lemma sumhr_interval_const:
"(\<forall>n. m \<le> Suc n --> f n = r) & m \<le> na
--- a/src/HOL/Nat_Numeral.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/Nat_Numeral.thy Fri Mar 30 17:21:36 2012 +0200
@@ -61,17 +61,6 @@
lemmas zpower_numeral_even = power_numeral_even [where 'a=int]
lemmas zpower_numeral_odd = power_numeral_odd [where 'a=int]
-lemma nat_numeral_Bit0:
- "numeral (Num.Bit0 w) = (let n::nat = numeral w in n + n)"
- unfolding numeral_Bit0 Let_def ..
-
-lemma nat_numeral_Bit1:
- "numeral (Num.Bit1 w) = (let n = numeral w in Suc (n + n))"
- unfolding numeral_Bit1 Let_def by simp
-
-lemmas eval_nat_numeral =
- nat_numeral_Bit0 nat_numeral_Bit1
-
lemmas nat_arith =
diff_nat_numeral
@@ -101,46 +90,14 @@
lemma Suc_diff_eq_diff_pred: "0 < n ==> Suc m - n = m - (n - Numeral1)"
by (simp split: nat_diff_split)
-text{*No longer required as a simprule because of the @{text inverse_fold}
- simproc*}
-lemma Suc_diff_numeral: "Suc m - (numeral v) = m - (numeral v - 1)"
- by (subst expand_Suc, simp only: diff_Suc_Suc)
-
lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
by (simp split: nat_diff_split)
-subsubsection{*For @{term nat_case} and @{term nat_rec}*}
-
-lemma nat_case_numeral [simp]:
- "nat_case a f (numeral v) = (let pv = nat (numeral v - 1) in f pv)"
- by (subst expand_Suc, simp only: nat.cases nat_numeral_diff_1 Let_def)
-
-lemma nat_case_add_eq_if [simp]:
- "nat_case a f ((numeral v) + n) = (let pv = nat (numeral v - 1) in f (pv + n))"
- by (subst expand_Suc, simp only: nat.cases nat_numeral_diff_1 Let_def add_Suc)
-
-lemma nat_rec_numeral [simp]:
- "nat_rec a f (numeral v) = (let pv = nat (numeral v - 1) in f pv (nat_rec a f pv))"
- by (subst expand_Suc, simp only: nat_rec_Suc nat_numeral_diff_1 Let_def)
-
-lemma nat_rec_add_eq_if [simp]:
- "nat_rec a f (numeral v + n) =
- (let pv = nat (numeral v - 1) in f (pv + n) (nat_rec a f (pv + n)))"
- by (subst expand_Suc, simp only: nat_rec_Suc nat_numeral_diff_1 Let_def add_Suc)
-
-
subsubsection{*Various Other Lemmas*}
text {*Evens and Odds, for Mutilated Chess Board*}
-text{*Lemmas for specialist use, NOT as default simprules*}
-lemma nat_mult_2: "2 * z = (z+z::nat)"
-by (rule mult_2) (* FIXME: duplicate *)
-
-lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
-by (rule mult_2_right) (* FIXME: duplicate *)
-
text{*Case analysis on @{term "n<2"}*}
lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
by (auto simp add: numeral_2_eq_2)
--- a/src/HOL/Num.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/Num.thy Fri Mar 30 17:21:36 2012 +0200
@@ -7,6 +7,8 @@
theory Num
imports Datatype
+uses
+ ("Tools/numeral.ML")
begin
subsection {* The @{text num} type *}
@@ -331,6 +333,9 @@
(@{const_syntax neg_numeral}, num_tr' "-")] end
*}
+use "Tools/numeral.ML"
+
+
subsection {* Class-specific numeral rules *}
text {*
@@ -507,7 +512,7 @@
lemma numeral_mult: "numeral (m * n) = numeral m * numeral n"
apply (induct n rule: num_induct)
apply (simp add: numeral_One)
- apply (simp add: mult_inc numeral_inc numeral_add numeral_inc right_distrib)
+ apply (simp add: mult_inc numeral_inc numeral_add right_distrib)
done
lemma numeral_times_numeral: "numeral m * numeral n = numeral (m * n)"
@@ -869,8 +874,7 @@
lemma numeral_eq_Suc: "numeral k = Suc (pred_numeral k)"
unfolding pred_numeral_def by simp
-lemma nat_number:
- "1 = Suc 0"
+lemma eval_nat_numeral:
"numeral One = Suc 0"
"numeral (Bit0 n) = Suc (numeral (BitM n))"
"numeral (Bit1 n) = Suc (numeral (Bit0 n))"
@@ -880,14 +884,14 @@
"pred_numeral Num.One = 0"
"pred_numeral (Num.Bit0 k) = numeral (Num.BitM k)"
"pred_numeral (Num.Bit1 k) = numeral (Num.Bit0 k)"
- unfolding pred_numeral_def nat_number
+ unfolding pred_numeral_def eval_nat_numeral
by (simp_all only: diff_Suc_Suc diff_0)
lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
- by (simp add: nat_number(2-4))
+ by (simp add: eval_nat_numeral)
lemma numeral_3_eq_3: "3 = Suc (Suc (Suc 0))"
- by (simp add: nat_number(2-4))
+ by (simp add: eval_nat_numeral)
lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
by (simp only: numeral_One One_nat_def)
@@ -919,6 +923,12 @@
lemma le_Suc_numeral [simp]: "Suc n \<le> numeral k \<longleftrightarrow> n \<le> pred_numeral k"
by (simp add: numeral_eq_Suc)
+lemma diff_Suc_numeral [simp]: "Suc n - numeral k = n - pred_numeral k"
+ by (simp add: numeral_eq_Suc)
+
+lemma diff_numeral_Suc [simp]: "numeral k - Suc n = pred_numeral k - n"
+ by (simp add: numeral_eq_Suc)
+
lemma max_Suc_numeral [simp]:
"max (Suc n) (numeral k) = Suc (max n (pred_numeral k))"
by (simp add: numeral_eq_Suc)
@@ -935,6 +945,26 @@
"min (numeral k) (Suc n) = Suc (min (pred_numeral k) n)"
by (simp add: numeral_eq_Suc)
+text {* For @{term nat_case} and @{term nat_rec}. *}
+
+lemma nat_case_numeral [simp]:
+ "nat_case a f (numeral v) = (let pv = pred_numeral v in f pv)"
+ by (simp add: numeral_eq_Suc)
+
+lemma nat_case_add_eq_if [simp]:
+ "nat_case a f ((numeral v) + n) = (let pv = pred_numeral v in f (pv + n))"
+ by (simp add: numeral_eq_Suc)
+
+lemma nat_rec_numeral [simp]:
+ "nat_rec a f (numeral v) =
+ (let pv = pred_numeral v in f pv (nat_rec a f pv))"
+ by (simp add: numeral_eq_Suc Let_def)
+
+lemma nat_rec_add_eq_if [simp]:
+ "nat_rec a f (numeral v + n) =
+ (let pv = pred_numeral v in f (pv + n) (nat_rec a f (pv + n)))"
+ by (simp add: numeral_eq_Suc Let_def)
+
subsection {* Numeral equations as default simplification rules *}
@@ -950,10 +980,6 @@
subsection {* Setting up simprocs *}
-lemma numeral_reorient:
- "(numeral w = x) = (x = numeral w)"
- by auto
-
lemma mult_numeral_1: "Numeral1 * a = (a::'a::semiring_numeral)"
by simp
@@ -974,6 +1000,16 @@
mult_numeral_1 mult_numeral_1_right
mult_minus1 mult_minus1_right
+setup {*
+ Reorient_Proc.add
+ (fn Const (@{const_name numeral}, _) $ _ => true
+ | Const (@{const_name neg_numeral}, _) $ _ => true
+ | _ => false)
+*}
+
+simproc_setup reorient_numeral
+ ("numeral w = x" | "neg_numeral w = y") = Reorient_Proc.proc
+
subsubsection {* Simplification of arithmetic operations on integer constants. *}
--- a/src/HOL/Parity.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/Parity.thy Fri Mar 30 17:21:36 2012 +0200
@@ -45,11 +45,20 @@
lemma odd_1_nat [simp]: "odd (1::nat)" by presburger
+lemma even_numeral_int [simp]: "even (numeral (Num.Bit0 k) :: int)"
+ unfolding even_def by simp
+
+lemma odd_numeral_int [simp]: "odd (numeral (Num.Bit1 k) :: int)"
+ unfolding even_def by simp
+
(* TODO: proper simp rules for Num.Bit0, Num.Bit1 *)
-declare even_def[of "numeral v", simp] for v
declare even_def[of "neg_numeral v", simp] for v
-declare even_nat_def[of "numeral v", simp] for v
+lemma even_numeral_nat [simp]: "even (numeral (Num.Bit0 k) :: nat)"
+ unfolding even_nat_def by simp
+
+lemma odd_numeral_nat [simp]: "odd (numeral (Num.Bit1 k) :: nat)"
+ unfolding even_nat_def by simp
subsection {* Even and odd are mutually exclusive *}
@@ -349,10 +358,6 @@
text {* Simplify, when the exponent is a numeral *}
-lemma power_0_left_numeral [simp]:
- "0 ^ numeral w = (0::'a::{power,semiring_0})"
-by (simp add: power_0_left)
-
lemmas zero_le_power_eq_numeral [simp] =
zero_le_power_eq [of _ "numeral w"] for w
--- a/src/HOL/Power.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/Power.thy Fri Mar 30 17:21:36 2012 +0200
@@ -185,7 +185,7 @@
lemma power_minus_Bit1:
"(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
- by (simp only: nat_number(4) power_Suc power_minus_Bit0 mult_minus_left)
+ 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))"
--- a/src/HOL/SetInterval.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/SetInterval.thy Fri Mar 30 17:21:36 2012 +0200
@@ -1282,19 +1282,21 @@
subsection {* The formula for arithmetic sums *}
-lemma gauss_sum: (* FIXME: rephrase in terms of "2" *)
- "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
+lemma gauss_sum:
+ "(2::'a::comm_semiring_1)*(\<Sum>i\<in>{1..n}. of_nat i) =
of_nat n*((of_nat n)+1)"
proof (induct n)
case 0
show ?case by simp
next
case (Suc n)
- then show ?case by (simp add: algebra_simps del: one_add_one) (* FIXME *)
+ then show ?case
+ by (simp add: algebra_simps add: one_add_one [symmetric] del: one_add_one)
+ (* FIXME: make numeral cancellation simprocs work for semirings *)
qed
theorem arith_series_general:
- "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
+ "(2::'a::comm_semiring_1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
of_nat n * (a + (a + of_nat(n - 1)*d))"
proof cases
assume ngt1: "n > 1"
@@ -1307,26 +1309,27 @@
also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
unfolding One_nat_def
by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)
- also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"
- by (simp add: left_distrib right_distrib del: one_add_one)
+ also have "2*\<dots> = 2*?n*a + d*2*(\<Sum>i\<in>{1..<n}. ?I i)"
+ by (simp add: algebra_simps)
also from ngt1 have "{1..<n} = {1..n - 1}"
by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
also from ngt1
- have "(1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..n - 1}. ?I i) = ((1+1)*?n*a + d*?I (n - 1)*?I n)"
+ have "2*?n*a + d*2*(\<Sum>i\<in>{1..n - 1}. ?I i) = (2*?n*a + d*?I (n - 1)*?I n)"
by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)
(simp add: mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
- finally show ?thesis by (simp add: algebra_simps del: one_add_one)
+ finally show ?thesis
+ unfolding mult_2 by (simp add: algebra_simps)
next
assume "\<not>(n > 1)"
hence "n = 1 \<or> n = 0" by auto
- thus ?thesis by (auto simp: algebra_simps mult_2_right)
+ thus ?thesis by (auto simp: mult_2)
qed
lemma arith_series_nat:
- "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
+ "(2::nat) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
proof -
have
- "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
+ "2 * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
of_nat(n) * (a + (a + of_nat(n - 1)*d))"
by (rule arith_series_general)
thus ?thesis
@@ -1334,15 +1337,8 @@
qed
lemma arith_series_int:
- "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
- of_nat n * (a + (a + of_nat(n - 1)*d))"
-proof -
- have
- "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
- of_nat(n) * (a + (a + of_nat(n - 1)*d))"
- by (rule arith_series_general)
- thus ?thesis by simp
-qed
+ "2 * (\<Sum>i\<in>{..<n}. a + int i * d) = int n * (a + (a + int(n - 1)*d))"
+ by (fact arith_series_general) (* FIXME: duplicate *)
lemma sum_diff_distrib:
fixes P::"nat\<Rightarrow>nat"
--- a/src/HOL/Word/Bit_Int.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/Word/Bit_Int.thy Fri Mar 30 17:21:36 2012 +0200
@@ -501,8 +501,8 @@
lemma bin_sc_numeral [simp]:
"bin_sc (numeral k) b w =
- bin_sc (numeral k - 1) b (bin_rest w) BIT bin_last w"
- by (subst expand_Suc, rule bin_sc.Suc)
+ bin_sc (pred_numeral k) b (bin_rest w) BIT bin_last w"
+ by (simp add: numeral_eq_Suc)
subsection {* Splitting and concatenation *}
--- a/src/HOL/Word/Bit_Representation.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/Word/Bit_Representation.thy Fri Mar 30 17:21:36 2012 +0200
@@ -229,8 +229,8 @@
by (cases n) auto
lemma bin_nth_numeral:
- "bin_rest x = y \<Longrightarrow> bin_nth x (numeral n) = bin_nth y (numeral n - 1)"
- by (subst expand_Suc, simp only: bin_nth.simps)
+ "bin_rest x = y \<Longrightarrow> bin_nth x (numeral n) = bin_nth y (pred_numeral n)"
+ by (simp add: numeral_eq_Suc)
lemmas bin_nth_numeral_simps [simp] =
bin_nth_numeral [OF bin_rest_numeral_simps(2)]
@@ -543,35 +543,35 @@
lemma bintrunc_numeral:
"bintrunc (numeral k) x =
- bintrunc (numeral k - 1) (bin_rest x) BIT bin_last x"
- by (subst expand_Suc, rule bintrunc.simps)
+ bintrunc (pred_numeral k) (bin_rest x) BIT bin_last x"
+ by (simp add: numeral_eq_Suc)
lemma sbintrunc_numeral:
"sbintrunc (numeral k) x =
- sbintrunc (numeral k - 1) (bin_rest x) BIT bin_last x"
- by (subst expand_Suc, rule sbintrunc.simps)
+ sbintrunc (pred_numeral k) (bin_rest x) BIT bin_last x"
+ by (simp add: numeral_eq_Suc)
lemma bintrunc_numeral_simps [simp]:
"bintrunc (numeral k) (numeral (Num.Bit0 w)) =
- bintrunc (numeral k - 1) (numeral w) BIT 0"
+ bintrunc (pred_numeral k) (numeral w) BIT 0"
"bintrunc (numeral k) (numeral (Num.Bit1 w)) =
- bintrunc (numeral k - 1) (numeral w) BIT 1"
+ bintrunc (pred_numeral k) (numeral w) BIT 1"
"bintrunc (numeral k) (neg_numeral (Num.Bit0 w)) =
- bintrunc (numeral k - 1) (neg_numeral w) BIT 0"
+ bintrunc (pred_numeral k) (neg_numeral w) BIT 0"
"bintrunc (numeral k) (neg_numeral (Num.Bit1 w)) =
- bintrunc (numeral k - 1) (neg_numeral (w + Num.One)) BIT 1"
+ bintrunc (pred_numeral k) (neg_numeral (w + Num.One)) BIT 1"
"bintrunc (numeral k) 1 = 1"
by (simp_all add: bintrunc_numeral)
lemma sbintrunc_numeral_simps [simp]:
"sbintrunc (numeral k) (numeral (Num.Bit0 w)) =
- sbintrunc (numeral k - 1) (numeral w) BIT 0"
+ sbintrunc (pred_numeral k) (numeral w) BIT 0"
"sbintrunc (numeral k) (numeral (Num.Bit1 w)) =
- sbintrunc (numeral k - 1) (numeral w) BIT 1"
+ sbintrunc (pred_numeral k) (numeral w) BIT 1"
"sbintrunc (numeral k) (neg_numeral (Num.Bit0 w)) =
- sbintrunc (numeral k - 1) (neg_numeral w) BIT 0"
+ sbintrunc (pred_numeral k) (neg_numeral w) BIT 0"
"sbintrunc (numeral k) (neg_numeral (Num.Bit1 w)) =
- sbintrunc (numeral k - 1) (neg_numeral (w + Num.One)) BIT 1"
+ sbintrunc (pred_numeral k) (neg_numeral (w + Num.One)) BIT 1"
"sbintrunc (numeral k) 1 = 1"
by (simp_all add: sbintrunc_numeral)
@@ -754,12 +754,12 @@
by (cases n) simp_all
lemma funpow_numeral [simp]:
- "f ^^ numeral k = f \<circ> f ^^ (numeral k - 1)"
- by (subst expand_Suc, rule funpow.simps)
+ "f ^^ numeral k = f \<circ> f ^^ (pred_numeral k)"
+ by (simp add: numeral_eq_Suc)
lemma replicate_numeral [simp]: (* TODO: move to List.thy *)
- "replicate (numeral k) x = x # replicate (numeral k - 1) x"
- by (subst expand_Suc, rule replicate_Suc)
+ "replicate (numeral k) x = x # replicate (pred_numeral k) x"
+ by (simp add: numeral_eq_Suc)
lemmas power_minus_simp =
trans [OF gen_minus [where f = "power f"] power_Suc] for f
--- a/src/HOL/Word/Bool_List_Representation.thy Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/Word/Bool_List_Representation.thy Fri Mar 30 17:21:36 2012 +0200
@@ -633,12 +633,12 @@
takefill_minus [symmetric, THEN trans]]
lemma takefill_numeral_Nil [simp]:
- "takefill fill (numeral k) [] = fill # takefill fill (numeral k - 1) []"
- by (subst expand_Suc, rule takefill_Suc_Nil)
+ "takefill fill (numeral k) [] = fill # takefill fill (pred_numeral k) []"
+ by (simp add: numeral_eq_Suc)
lemma takefill_numeral_Cons [simp]:
- "takefill fill (numeral k) (x # xs) = x # takefill fill (numeral k - 1) xs"
- by (subst expand_Suc, rule takefill_Suc_Cons)
+ "takefill fill (numeral k) (x # xs) = x # takefill fill (pred_numeral k) xs"
+ by (simp add: numeral_eq_Suc)
(* links with function bl_to_bin *)
--- a/src/HOL/ex/Arithmetic_Series_Complex.thy Fri Mar 30 15:25:47 2012 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,23 +0,0 @@
-(* Title: HOL/ex/Arithmetic_Series_Complex.thy
- Author: Benjamin Porter, 2006
-*)
-
-
-header {* Arithmetic Series for Reals *}
-
-theory Arithmetic_Series_Complex
-imports Complex_Main
-begin
-
-lemma arith_series_real:
- "(2::real) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
- of_nat n * (a + (a + of_nat(n - 1)*d))"
-proof -
- have
- "((1::real) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat(i)*d) =
- of_nat(n) * (a + (a + of_nat(n - 1)*d))"
- by (rule arith_series_general)
- thus ?thesis by simp
-qed
-
-end
--- a/src/HOL/ex/ROOT.ML Fri Mar 30 15:25:47 2012 +0200
+++ b/src/HOL/ex/ROOT.ML Fri Mar 30 17:21:36 2012 +0200
@@ -57,7 +57,6 @@
"Sqrt",
"Sqrt_Script",
"Transfer_Ex",
- "Arithmetic_Series_Complex",
"HarmonicSeries",
"Refute_Examples",
"Landau",