--- a/src/HOL/Big_Operators.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Big_Operators.thy Fri Jan 14 15:44:47 2011 +0100
@@ -172,7 +172,7 @@
lemma (in comm_monoid_add) setsum_reindex_cong:
"[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|]
==> setsum h B = setsum g A"
- by (simp add: setsum_reindex cong: setsum_cong)
+ by (simp add: setsum_reindex)
lemma (in comm_monoid_add) setsum_0[simp]: "setsum (%i. 0) A = 0"
by (cases "finite A") (erule finite_induct, auto)
@@ -288,7 +288,7 @@
shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
proof -
interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
- from assms show ?thesis by (simp add: setsum_def fold_image_UN_disjoint cong: setsum_cong)
+ from assms show ?thesis by (simp add: setsum_def fold_image_UN_disjoint)
qed
text{*No need to assume that @{term C} is finite. If infinite, the rhs is
@@ -310,7 +310,7 @@
shows "(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"
proof -
interpret comm_monoid_mult "op +" 0 by (fact comm_monoid_mult)
- from assms show ?thesis by (simp add: setsum_def fold_image_Sigma split_def cong: setsum_cong)
+ from assms show ?thesis by (simp add: setsum_def fold_image_Sigma split_def)
qed
text{*Here we can eliminate the finiteness assumptions, by cases.*}
@@ -498,7 +498,7 @@
assumes "finite A" "A \<noteq> {}"
and "!!x. x:A \<Longrightarrow> f x < g x"
shows "setsum f A < setsum g A"
- using prems
+ using assms
proof (induct rule: finite_ne_induct)
case singleton thus ?case by simp
next
@@ -775,7 +775,7 @@
apply (subgoal_tac
"setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
apply (simp add: setsum_UN_disjoint del: setsum_constant)
-apply (simp cong: setsum_cong)
+apply simp
done
lemma card_Union_disjoint:
@@ -947,7 +947,7 @@
let ?f = "(\<lambda>k. if k=a then b k else 1)"
{assume a: "a \<notin> S"
hence "\<forall> k\<in> S. ?f k = 1" by simp
- hence ?thesis using a by (simp add: setprod_1 cong add: setprod_cong) }
+ hence ?thesis using a by (simp add: setprod_1) }
moreover
{assume a: "a \<in> S"
let ?A = "S - {a}"
@@ -959,7 +959,7 @@
have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
by simp
- then have ?thesis using a by (simp add: fA1 cong add: setprod_cong cong del: if_weak_cong)}
+ then have ?thesis using a by (simp add: fA1 cong: setprod_cong cong del: if_weak_cong)}
ultimately show ?thesis by blast
qed
@@ -975,7 +975,7 @@
"finite I ==> (ALL i:I. finite (A i)) ==>
(ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
-by(simp add: setprod_def fold_image_UN_disjoint cong: setprod_cong)
+ by (simp add: setprod_def fold_image_UN_disjoint)
lemma setprod_Union_disjoint:
"[| (ALL A:C. finite A);
@@ -990,7 +990,7 @@
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
(\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =
(\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
-by(simp add:setprod_def fold_image_Sigma split_def cong:setprod_cong)
+by(simp add:setprod_def fold_image_Sigma split_def)
text{*Here we can eliminate the finiteness assumptions, by cases.*}
lemma setprod_cartesian_product:
@@ -1332,7 +1332,7 @@
shows "sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B) = \<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B}"
using A proof (induct rule: finite_ne_induct)
case singleton thus ?case
- by (simp add: sup_Inf1_distrib [OF B] fold1_singleton_def [OF Inf_fin_def])
+ by (simp add: sup_Inf1_distrib [OF B])
next
interpret ab_semigroup_idem_mult inf
by (rule ab_semigroup_idem_mult_inf)
@@ -1347,7 +1347,7 @@
qed
have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
have "sup (\<Sqinter>\<^bsub>fin\<^esub>(insert x A)) (\<Sqinter>\<^bsub>fin\<^esub>B) = sup (inf x (\<Sqinter>\<^bsub>fin\<^esub>A)) (\<Sqinter>\<^bsub>fin\<^esub>B)"
- using insert by (simp add: fold1_insert_idem_def [OF Inf_fin_def])
+ using insert by simp
also have "\<dots> = inf (sup x (\<Sqinter>\<^bsub>fin\<^esub>B)) (sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B))" by(rule sup_inf_distrib2)
also have "\<dots> = inf (\<Sqinter>\<^bsub>fin\<^esub>{sup x b|b. b \<in> B}) (\<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B})"
using insert by(simp add:sup_Inf1_distrib[OF B])
@@ -1391,7 +1391,7 @@
interpret ab_semigroup_idem_mult sup
by (rule ab_semigroup_idem_mult_sup)
have "inf (\<Squnion>\<^bsub>fin\<^esub>(insert x A)) (\<Squnion>\<^bsub>fin\<^esub>B) = inf (sup x (\<Squnion>\<^bsub>fin\<^esub>A)) (\<Squnion>\<^bsub>fin\<^esub>B)"
- using insert by (simp add: fold1_insert_idem_def [OF Sup_fin_def])
+ using insert by simp
also have "\<dots> = sup (inf x (\<Squnion>\<^bsub>fin\<^esub>B)) (inf (\<Squnion>\<^bsub>fin\<^esub>A) (\<Squnion>\<^bsub>fin\<^esub>B))" by(rule inf_sup_distrib2)
also have "\<dots> = sup (\<Squnion>\<^bsub>fin\<^esub>{inf x b|b. b \<in> B}) (\<Squnion>\<^bsub>fin\<^esub>{inf a b|a b. a \<in> A \<and> b \<in> B})"
using insert by(simp add:inf_Sup1_distrib[OF B])
@@ -1551,15 +1551,15 @@
next
interpret ab_semigroup_idem_mult min
by (rule ab_semigroup_idem_mult_min)
- assume "A \<noteq> B"
+ assume neq: "A \<noteq> B"
have B: "B = A \<union> (B-A)" using `A \<subseteq> B` by blast
have "fold1 min B = fold1 min (A \<union> (B-A))" by(subst B)(rule refl)
also have "\<dots> = min (fold1 min A) (fold1 min (B-A))"
proof -
have "finite A" by(rule finite_subset[OF `A \<subseteq> B` `finite B`])
- moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`]) (* by(blast intro:finite_Diff prems) fails *)
- moreover have "(B-A) \<noteq> {}" using prems by blast
- moreover have "A Int (B-A) = {}" using prems by blast
+ moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`])
+ moreover have "(B-A) \<noteq> {}" using assms neq by blast
+ moreover have "A Int (B-A) = {}" using assms by blast
ultimately show ?thesis using `A \<noteq> {}` by (rule_tac fold1_Un)
qed
also have "\<dots> \<le> fold1 min A" by (simp add: min_le_iff_disj)
--- a/src/HOL/Deriv.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Deriv.thy Fri Jan 14 15:44:47 2011 +0100
@@ -355,7 +355,7 @@
lemma differentiableE [elim?]:
assumes "f differentiable x"
obtains df where "DERIV f x :> df"
- using prems unfolding differentiable_def ..
+ using assms unfolding differentiable_def ..
lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
by (simp add: differentiable_def)
@@ -408,7 +408,7 @@
assumes "f differentiable x"
assumes "g differentiable x"
shows "(\<lambda>x. f x - g x) differentiable x"
- unfolding diff_minus using prems by simp
+ unfolding diff_minus using assms by simp
lemma differentiable_mult [simp]:
assumes "f differentiable x"
@@ -437,13 +437,16 @@
assumes "f differentiable x"
assumes "g differentiable x" and "g x \<noteq> 0"
shows "(\<lambda>x. f x / g x) differentiable x"
- unfolding divide_inverse using prems by simp
+ unfolding divide_inverse using assms by simp
lemma differentiable_power [simp]:
fixes f :: "'a::{real_normed_field} \<Rightarrow> 'a"
assumes "f differentiable x"
shows "(\<lambda>x. f x ^ n) differentiable x"
- by (induct n, simp, simp add: prems)
+ apply (induct n)
+ apply simp
+ apply (simp add: assms)
+ done
subsection {* Nested Intervals and Bisection *}
@@ -1227,7 +1230,7 @@
assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"
shows "f a < f b"
proof (rule ccontr)
- assume "~ f a < f b"
+ assume f: "~ f a < f b"
have "EX l z. a < z & z < b & DERIV f z :> l
& f b - f a = (b - a) * l"
apply (rule MVT)
@@ -1236,13 +1239,12 @@
apply (metis DERIV_isCont)
apply (metis differentiableI less_le)
done
- then obtain l z where "a < z" and "z < b" and "DERIV f z :> l"
+ then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
and "f b - f a = (b - a) * l"
by auto
-
- from prems have "~(l > 0)"
+ with assms f have "~(l > 0)"
by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)
- with prems show False
+ with assms z show False
by (metis DERIV_unique less_le)
qed
@@ -1252,9 +1254,8 @@
"\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
shows "f a \<le> f b"
proof (rule ccontr, cases "a = b")
- assume "~ f a \<le> f b"
- assume "a = b"
- with prems show False by auto
+ assume "~ f a \<le> f b" and "a = b"
+ then show False by auto
next
assume A: "~ f a \<le> f b"
assume B: "a ~= b"
@@ -1266,13 +1267,13 @@
apply (metis DERIV_isCont)
apply (metis differentiableI less_le)
done
- then obtain l z where "a < z" and "z < b" and "DERIV f z :> l"
+ then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
and C: "f b - f a = (b - a) * l"
by auto
with A have "a < b" "f b < f a" by auto
with C have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
(metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono real_add_left_mono real_le_linear real_le_refl)
- with prems show False
+ with assms z show False
by (metis DERIV_unique order_less_imp_le)
qed
@@ -1509,14 +1510,14 @@
theorem GMVT:
fixes a b :: real
assumes alb: "a < b"
- and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
- and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
- and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
- and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
+ and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
+ and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
+ and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
+ and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
shows "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and> a < c \<and> c < b \<and> ((f b - f a) * g'c) = ((g b - g a) * f'c)"
proof -
let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
- from prems have "a < b" by simp
+ from assms have "a < b" by simp
moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
proof -
have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp
--- a/src/HOL/Divides.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Divides.thy Fri Jan 14 15:44:47 2011 +0100
@@ -681,8 +681,8 @@
ML {*
local
-structure CancelDivMod = CancelDivModFun(struct
-
+structure CancelDivMod = CancelDivModFun
+(
val div_name = @{const_name div};
val mod_name = @{const_name mod};
val mk_binop = HOLogic.mk_binop;
@@ -691,12 +691,9 @@
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
- val trans = trans;
-
val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
(@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac}))
-
-end)
+)
in
@@ -1352,15 +1349,16 @@
theorem posDivAlg_correct:
assumes "0 \<le> a" and "0 < b"
shows "divmod_int_rel a b (posDivAlg a b)"
-using prems apply (induct a b rule: posDivAlg.induct)
-apply auto
-apply (simp add: divmod_int_rel_def)
-apply (subst posDivAlg_eqn, simp add: right_distrib)
-apply (case_tac "a < b")
-apply simp_all
-apply (erule splitE)
-apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
-done
+ using assms
+ apply (induct a b rule: posDivAlg.induct)
+ apply auto
+ apply (simp add: divmod_int_rel_def)
+ apply (subst posDivAlg_eqn, simp add: right_distrib)
+ apply (case_tac "a < b")
+ apply simp_all
+ apply (erule splitE)
+ apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
+ done
subsubsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}
@@ -1381,15 +1379,16 @@
lemma negDivAlg_correct:
assumes "a < 0" and "b > 0"
shows "divmod_int_rel a b (negDivAlg a b)"
-using prems apply (induct a b rule: negDivAlg.induct)
-apply (auto simp add: linorder_not_le)
-apply (simp add: divmod_int_rel_def)
-apply (subst negDivAlg_eqn, assumption)
-apply (case_tac "a + b < (0\<Colon>int)")
-apply simp_all
-apply (erule splitE)
-apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
-done
+ using assms
+ apply (induct a b rule: negDivAlg.induct)
+ apply (auto simp add: linorder_not_le)
+ apply (simp add: divmod_int_rel_def)
+ apply (subst negDivAlg_eqn, assumption)
+ apply (case_tac "a + b < (0\<Colon>int)")
+ apply simp_all
+ apply (erule splitE)
+ apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
+ done
subsubsection{*Existence Shown by Proving the Division Algorithm to be Correct*}
@@ -1440,8 +1439,8 @@
ML {*
local
-structure CancelDivMod = CancelDivModFun(struct
-
+structure CancelDivMod = CancelDivModFun
+(
val div_name = @{const_name div};
val mod_name = @{const_name mod};
val mk_binop = HOLogic.mk_binop;
@@ -1450,12 +1449,9 @@
val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];
- val trans = trans;
-
val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
(@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))
-
-end)
+)
in
--- a/src/HOL/Fact.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Fact.thy Fri Jan 14 15:44:47 2011 +0100
@@ -12,12 +12,9 @@
begin
class fact =
-
-fixes
- fact :: "'a \<Rightarrow> 'a"
+ fixes fact :: "'a \<Rightarrow> 'a"
instantiation nat :: fact
-
begin
fun
@@ -26,7 +23,7 @@
fact_0_nat: "fact_nat 0 = Suc 0"
| fact_Suc: "fact_nat (Suc x) = Suc x * fact x"
-instance proof qed
+instance ..
end
@@ -93,8 +90,7 @@
lemma fact_plus_one_int:
assumes "n >= 0"
shows "fact ((n::int) + 1) = (n + 1) * fact n"
-
- using prems unfolding fact_int_def
+ using assms unfolding fact_int_def
by (simp add: nat_add_distrib algebra_simps int_mult)
lemma fact_reduce_nat: "(n::nat) > 0 \<Longrightarrow> fact n = n * fact (n - 1)"
@@ -102,19 +98,19 @@
apply (erule ssubst)
apply (subst fact_Suc)
apply simp_all
-done
+ done
lemma fact_reduce_int: "(n::int) > 0 \<Longrightarrow> fact n = n * fact (n - 1)"
apply (subgoal_tac "n = (n - 1) + 1")
apply (erule ssubst)
apply (subst fact_plus_one_int)
apply simp_all
-done
+ done
lemma fact_nonzero_nat [simp]: "fact (n::nat) \<noteq> 0"
apply (induct n)
apply (auto simp add: fact_plus_one_nat)
-done
+ done
lemma fact_nonzero_int [simp]: "n >= 0 \<Longrightarrow> fact (n::int) ~= 0"
by (simp add: fact_int_def)
@@ -137,7 +133,7 @@
apply (erule ssubst)
apply (subst zle_int)
apply auto
-done
+ done
lemma dvd_fact_nat [rule_format]: "1 <= m \<longrightarrow> m <= n \<longrightarrow> m dvd fact (n::nat)"
apply (induct n)
@@ -147,7 +143,7 @@
apply (erule ssubst)
apply (rule dvd_triv_left)
apply auto
-done
+ done
lemma dvd_fact_int [rule_format]: "1 <= m \<longrightarrow> m <= n \<longrightarrow> m dvd fact (n::int)"
apply (case_tac "1 <= n")
@@ -155,7 +151,7 @@
apply (auto simp add: fact_plus_one_int)
apply (subgoal_tac "m = i + 1")
apply auto
-done
+ done
lemma interval_plus_one_nat: "(i::nat) <= j + 1 \<Longrightarrow>
{i..j+1} = {i..j} Un {j+1}"
--- a/src/HOL/Finite_Set.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Finite_Set.thy Fri Jan 14 15:44:47 2011 +0100
@@ -803,7 +803,7 @@
proof -
interpret fun_left_comm_idem inf by (fact fun_left_comm_idem_inf)
from `finite A` show ?thesis by (induct A arbitrary: B)
- (simp_all add: Inf_empty Inf_insert inf_commute fold_fun_comm)
+ (simp_all add: Inf_insert inf_commute fold_fun_comm)
qed
lemma sup_Sup_fold_sup:
@@ -812,7 +812,7 @@
proof -
interpret fun_left_comm_idem sup by (fact fun_left_comm_idem_sup)
from `finite A` show ?thesis by (induct A arbitrary: B)
- (simp_all add: Sup_empty Sup_insert sup_commute fold_fun_comm)
+ (simp_all add: Sup_insert sup_commute fold_fun_comm)
qed
lemma Inf_fold_inf:
@@ -833,7 +833,7 @@
interpret fun_left_comm_idem "\<lambda>A. inf (f A)" by (fact fun_left_comm_idem_apply)
from `finite A` show "?fold = ?inf"
by (induct A arbitrary: B)
- (simp_all add: INFI_def Inf_empty Inf_insert inf_left_commute)
+ (simp_all add: INFI_def Inf_insert inf_left_commute)
qed
lemma sup_SUPR_fold_sup:
@@ -844,7 +844,7 @@
interpret fun_left_comm_idem "\<lambda>A. sup (f A)" by (fact fun_left_comm_idem_apply)
from `finite A` show "?fold = ?sup"
by (induct A arbitrary: B)
- (simp_all add: SUPR_def Sup_empty Sup_insert sup_left_commute)
+ (simp_all add: SUPR_def Sup_insert sup_left_commute)
qed
lemma INFI_fold_inf:
@@ -1197,19 +1197,19 @@
by (auto simp add: nonempty_iff)
show ?thesis
proof cases
- assume "a = x"
- thus ?thesis
+ assume a: "a = x"
+ show ?thesis
proof cases
assume "A' = {}"
- with prems show ?thesis by simp
+ with A' a show ?thesis by simp
next
assume "A' \<noteq> {}"
- with prems show ?thesis
+ with A A' a show ?thesis
by (simp add: fold1_insert mult_assoc [symmetric])
qed
next
assume "a \<noteq> x"
- with prems show ?thesis
+ with A A' show ?thesis
by (simp add: insert_commute fold1_eq_fold)
qed
qed
--- a/src/HOL/GCD.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/GCD.thy Fri Jan 14 15:44:47 2011 +0100
@@ -36,11 +36,8 @@
subsection {* GCD and LCM definitions *}
class gcd = zero + one + dvd +
-
-fixes
- gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" and
- lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
-
+ fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
+ and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
begin
abbreviation
@@ -186,7 +183,7 @@
and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm (-x) y)"
and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm (-x) (-y))"
shows "P (lcm x y)"
-by (insert prems, auto simp add: lcm_neg1_int lcm_neg2_int, arith)
+ using assms by (auto simp add: lcm_neg1_int lcm_neg2_int) arith
lemma lcm_ge_0_int [simp]: "lcm (x::int) y >= 0"
by (simp add: lcm_int_def)
@@ -632,13 +629,12 @@
apply (subgoal_tac "b' = b div gcd a b")
apply (erule ssubst)
apply (rule div_gcd_coprime_nat)
- using prems
- apply force
+ using z apply force
apply (subst (1) b)
using z apply force
apply (subst (1) a)
using z apply force
-done
+ done
lemma gcd_coprime_int:
assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
@@ -650,13 +646,12 @@
apply (subgoal_tac "b' = b div gcd a b")
apply (erule ssubst)
apply (rule div_gcd_coprime_int)
- using prems
- apply force
+ using z apply force
apply (subst (1) b)
using z apply force
apply (subst (1) a)
using z apply force
-done
+ done
lemma coprime_mult_nat: assumes da: "coprime (d::nat) a" and db: "coprime d b"
shows "coprime d (a * b)"
@@ -1192,13 +1187,13 @@
by auto
moreover
{assume db: "d=b"
- from prems have ?thesis apply simp
+ with nz H have ?thesis apply simp
apply (rule exI[where x = b], simp)
apply (rule exI[where x = b])
by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
moreover
{assume db: "d < b"
- {assume "x=0" hence ?thesis using prems by simp }
+ {assume "x=0" hence ?thesis using nz H by simp }
moreover
{assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
from db have "d \<le> b - 1" by simp
--- a/src/HOL/Import/HOL4Compat.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Import/HOL4Compat.thy Fri Jan 14 15:44:47 2011 +0100
@@ -64,10 +64,10 @@
by simp
lemma sum_case_def: "(ALL f g x. sum_case f g (Inl x) = f x) & (ALL f g y. sum_case f g (Inr y) = g y)"
- by simp;
+ by simp
lemma one: "ALL v. v = ()"
- by simp;
+ by simp
lemma option_case_def: "(!u f. option_case u f None = u) & (!u f x. option_case u f (Some x) = f x)"
by simp
@@ -103,7 +103,7 @@
by (simp add: map_pair_def split_def)
lemma pair_case_def: "split = split"
- ..;
+ ..
lemma LESS_OR_EQ: "m <= (n::nat) = (m < n | m = n)"
by auto
@@ -128,12 +128,12 @@
lemma LESS_DEF: "m < n = (? P. (!n. P (Suc n) --> P n) & P m & ~P n)"
proof safe
- assume "m < n"
+ assume 1: "m < n"
def P == "%n. n <= m"
have "(!n. P (Suc n) \<longrightarrow> P n) & P m & ~P n"
proof (auto simp add: P_def)
assume "n <= m"
- from prems
+ with 1
show False
by auto
qed
@@ -187,7 +187,7 @@
show "m < n"
..
qed
-qed;
+qed
definition FUNPOW :: "('a => 'a) => nat => 'a => 'a" where
"FUNPOW f n == f ^^ n"
@@ -242,10 +242,10 @@
by auto
lemma num_case_def: "(!b f. nat_case b f 0 = b) & (!b f n. nat_case b f (Suc n) = f n)"
- by simp;
+ by simp
lemma divides_def: "(a::nat) dvd b = (? q. b = q * a)"
- by (auto simp add: dvd_def);
+ by (auto simp add: dvd_def)
lemma list_case_def: "(!v f. list_case v f [] = v) & (!v f a0 a1. list_case v f (a0#a1) = f a0 a1)"
by simp
@@ -263,21 +263,21 @@
(list_case v f M = list_case v' f' M')"
proof clarify
fix M M' v f
- assume "M' = [] \<longrightarrow> v = v'"
- and "!a0 a1. M' = a0 # a1 \<longrightarrow> f a0 a1 = f' a0 a1"
+ assume 1: "M' = [] \<longrightarrow> v = v'"
+ and 2: "!a0 a1. M' = a0 # a1 \<longrightarrow> f a0 a1 = f' a0 a1"
show "list_case v f M' = list_case v' f' M'"
proof (rule List.list.case_cong)
show "M' = M'"
..
next
assume "M' = []"
- with prems
+ with 1 2
show "v = v'"
by auto
next
fix a0 a1
assume "M' = a0 # a1"
- with prems
+ with 1 2
show "f a0 a1 = f' a0 a1"
by auto
qed
@@ -302,14 +302,14 @@
by auto
next
fix fn1 fn2
- assume "ALL h t. fn1 (h # t) = f (fn1 t) h t"
- assume "ALL h t. fn2 (h # t) = f (fn2 t) h t"
- assume "fn2 [] = fn1 []"
+ assume 1: "ALL h t. fn1 (h # t) = f (fn1 t) h t"
+ assume 2: "ALL h t. fn2 (h # t) = f (fn2 t) h t"
+ assume 3: "fn2 [] = fn1 []"
show "fn1 = fn2"
proof
fix xs
show "fn1 xs = fn2 xs"
- by (induct xs,simp_all add: prems)
+ by (induct xs) (simp_all add: 1 2 3)
qed
qed
@@ -411,7 +411,7 @@
by (simp add: Let_def)
lemma REVERSE: "(rev [] = []) & (!h t. rev (h#t) = (rev t) @ [h])"
- by simp;
+ by simp
lemma REAL_SUP_ALLPOS: "\<lbrakk> ALL x. P (x::real) \<longrightarrow> 0 < x ; EX x. P x; EX z. ALL x. P x \<longrightarrow> x < z \<rbrakk> \<Longrightarrow> EX s. ALL y. (EX x. P x & y < x) = (y < s)"
proof safe
@@ -424,12 +424,11 @@
show "ALL x : Collect P. 0 < x"
proof safe
fix x
- assume "P x"
+ assume P: "P x"
from allx
have "P x \<longrightarrow> 0 < x"
..
- thus "0 < x"
- by (simp add: prems)
+ with P show "0 < x" by simp
qed
next
from px
@@ -461,7 +460,7 @@
by simp
lemma REAL_LT_TOTAL: "((x::real) = y) | x < y | y < x"
- by auto;
+ by auto
lemma [hol4rew]: "real (0::nat) = 0"
by simp
--- a/src/HOL/Import/HOL4Setup.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Import/HOL4Setup.thy Fri Jan 14 15:44:47 2011 +0100
@@ -90,11 +90,11 @@
have ed: "TYPE_DEFINITION P Rep"
proof (auto simp add: TYPE_DEFINITION)
fix x y
- assume "Rep x = Rep y"
+ assume b: "Rep x = Rep y"
from td have "x = Abs (Rep x)"
by auto
also have "Abs (Rep x) = Abs (Rep y)"
- by (simp add: prems)
+ by (simp add: b)
also from td have "Abs (Rep y) = y"
by auto
finally show "x = y" .
--- a/src/HOL/Lim.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Lim.thy Fri Jan 14 15:44:47 2011 +0100
@@ -653,7 +653,7 @@
moreover have "\<forall>n. ?F n \<noteq> a"
by (rule allI) (rule F1)
- moreover from prems have "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by simp
+ moreover note `\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L`
ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp
moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
--- a/src/HOL/Ln.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Ln.thy Fri Jan 14 15:44:47 2011 +0100
@@ -71,7 +71,7 @@
qed
moreover have "x ^ (Suc n + 2) <= x ^ (n + 2)"
apply (simp add: mult_compare_simps)
- apply (simp add: prems)
+ apply (simp add: assms)
apply (subgoal_tac "0 <= x * (x * x^n)")
apply force
apply (rule mult_nonneg_nonneg, rule a)+
@@ -91,7 +91,7 @@
by simp
also have "... <= 1 / 2 * (x ^ 2 / 2 * (1 / 2) ^ n)"
apply (rule mult_left_mono)
- apply (rule prems)
+ apply (rule c)
apply simp
done
also have "... = x ^ 2 / 2 * (1 / 2 * (1 / 2) ^ n)"
@@ -129,7 +129,7 @@
have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <=
suminf (%n. (x^2/2) * ((1/2)^n))"
apply (rule summable_le)
- apply (auto simp only: aux1 prems)
+ apply (auto simp only: aux1 a b)
apply (rule exp_tail_after_first_two_terms_summable)
by (rule sums_summable, rule aux2)
also have "... = x^2"
@@ -155,14 +155,14 @@
apply (rule divide_left_mono)
apply (auto simp add: exp_ge_add_one_self_aux)
apply (rule add_nonneg_nonneg)
- apply (insert prems, auto)
+ using a apply auto
apply (rule mult_pos_pos)
apply auto
apply (rule add_pos_nonneg)
apply auto
done
also from a have "... <= 1 + x"
- by(simp add:field_simps zero_compare_simps)
+ by (simp add: field_simps zero_compare_simps)
finally show ?thesis .
qed
@@ -192,14 +192,14 @@
finally have "(1 - x) * (1 + x + x ^ 2) <= 1" .
moreover have "0 < 1 + x + x^2"
apply (rule add_pos_nonneg)
- apply (insert a, auto)
+ using a apply auto
done
ultimately have "1 - x <= 1 / (1 + x + x^2)"
by (elim mult_imp_le_div_pos)
also have "... <= 1 / exp x"
apply (rule divide_left_mono)
apply (rule exp_bound, rule a)
- apply (insert prems, auto)
+ using a b apply auto
apply (rule mult_pos_pos)
apply (rule add_pos_nonneg)
apply auto
@@ -256,10 +256,10 @@
also have "- (x / (1 - x)) = -x / (1 - x)"
by auto
finally have d: "- x / (1 - x) <= ln (1 - x)" .
- have "0 < 1 - x" using prems by simp
+ have "0 < 1 - x" using a b by simp
hence e: "-x - 2 * x^2 <= - x / (1 - x)"
- using mult_right_le_one_le[of "x*x" "2*x"] prems
- by(simp add:field_simps power2_eq_square)
+ using mult_right_le_one_le[of "x*x" "2*x"] a b
+ by (simp add:field_simps power2_eq_square)
from e d show "- x - 2 * x^2 <= ln (1 - x)"
by (rule order_trans)
qed
@@ -292,7 +292,7 @@
"0 <= x ==> x <= 1 ==> abs(ln (1 + x) - x) <= x^2"
proof -
assume x: "0 <= x"
- assume "x <= 1"
+ assume x1: "x <= 1"
from x have "ln (1 + x) <= x"
by (rule ln_add_one_self_le_self)
then have "ln (1 + x) - x <= 0"
@@ -303,7 +303,7 @@
by simp
also have "... <= x^2"
proof -
- from prems have "x - x^2 <= ln (1 + x)"
+ from x x1 have "x - x^2 <= ln (1 + x)"
by (intro ln_one_plus_pos_lower_bound)
thus ?thesis
by simp
@@ -314,19 +314,19 @@
lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
"-(1 / 2) <= x ==> x <= 0 ==> abs(ln (1 + x) - x) <= 2 * x^2"
proof -
- assume "-(1 / 2) <= x"
- assume "x <= 0"
+ assume a: "-(1 / 2) <= x"
+ assume b: "x <= 0"
have "abs(ln (1 + x) - x) = x - ln(1 - (-x))"
apply (subst abs_of_nonpos)
apply simp
apply (rule ln_add_one_self_le_self2)
- apply (insert prems, auto)
+ using a apply auto
done
also have "... <= 2 * x^2"
apply (subgoal_tac "- (-x) - 2 * (-x)^2 <= ln (1 - (-x))")
apply (simp add: algebra_simps)
apply (rule ln_one_minus_pos_lower_bound)
- apply (insert prems, auto)
+ using a b apply auto
done
finally show ?thesis .
qed
@@ -343,9 +343,9 @@
lemma ln_x_over_x_mono: "exp 1 <= x ==> x <= y ==> (ln y / y) <= (ln x / x)"
proof -
- assume "exp 1 <= x" and "x <= y"
+ assume x: "exp 1 <= x" "x <= y"
have a: "0 < x" and b: "0 < y"
- apply (insert prems)
+ apply (insert x)
apply (subgoal_tac "0 < exp (1::real)")
apply arith
apply auto
@@ -361,12 +361,12 @@
done
also have "y / x = (x + (y - x)) / x"
by simp
- also have "... = 1 + (y - x) / x" using a prems by(simp add:field_simps)
+ also have "... = 1 + (y - x) / x" using x a by (simp add: field_simps)
also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
apply (rule mult_left_mono)
apply (rule ln_add_one_self_le_self)
apply (rule divide_nonneg_pos)
- apply (insert prems a, simp_all)
+ using x a apply simp_all
done
also have "... = y - x" using a by simp
also have "... = (y - x) * ln (exp 1)" by simp
@@ -375,16 +375,16 @@
apply (subst ln_le_cancel_iff)
apply force
apply (rule a)
- apply (rule prems)
- apply (insert prems, simp)
+ apply (rule x)
+ using x apply simp
done
also have "... = y * ln x - x * ln x"
by (rule left_diff_distrib)
finally have "x * ln y <= y * ln x"
by arith
- then have "ln y <= (y * ln x) / x" using a by(simp add:field_simps)
- also have "... = y * (ln x / x)" by simp
- finally show ?thesis using b by(simp add:field_simps)
+ then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps)
+ also have "... = y * (ln x / x)" by simp
+ finally show ?thesis using b by (simp add: field_simps)
qed
end
--- a/src/HOL/Log.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Log.thy Fri Jan 14 15:44:47 2011 +0100
@@ -251,10 +251,11 @@
apply (erule order_less_imp_le)
done
-lemma ln_powr_bound2: "1 < x ==> 0 < a ==> (ln x) powr a <= (a powr a) * x"
+lemma ln_powr_bound2:
+ assumes "1 < x" and "0 < a"
+ shows "(ln x) powr a <= (a powr a) * x"
proof -
- assume "1 < x" and "0 < a"
- then have "ln x <= (x powr (1 / a)) / (1 / a)"
+ from assms have "ln x <= (x powr (1 / a)) / (1 / a)"
apply (intro ln_powr_bound)
apply (erule order_less_imp_le)
apply (rule divide_pos_pos)
@@ -264,14 +265,14 @@
by simp
finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a"
apply (intro powr_mono2)
- apply (rule order_less_imp_le, rule prems)
+ apply (rule order_less_imp_le, rule assms)
apply (rule ln_gt_zero)
- apply (rule prems)
+ apply (rule assms)
apply assumption
done
also have "... = (a powr a) * ((x powr (1 / a)) powr a)"
apply (rule powr_mult)
- apply (rule prems)
+ apply (rule assms)
apply (rule powr_gt_zero)
done
also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
@@ -279,35 +280,37 @@
also have "... = x"
apply simp
apply (subgoal_tac "a ~= 0")
- apply (insert prems, auto)
+ using assms apply auto
done
finally show ?thesis .
qed
-lemma LIMSEQ_neg_powr: "0 < s ==> (%x. (real x) powr - s) ----> 0"
+lemma LIMSEQ_neg_powr:
+ assumes s: "0 < s"
+ shows "(%x. (real x) powr - s) ----> 0"
apply (unfold LIMSEQ_iff)
apply clarsimp
apply (rule_tac x = "natfloor(r powr (1 / - s)) + 1" in exI)
apply clarify
- proof -
- fix r fix n
- assume "0 < s" and "0 < r" and "natfloor (r powr (1 / - s)) + 1 <= n"
- have "r powr (1 / - s) < real(natfloor(r powr (1 / - s))) + 1"
- by (rule real_natfloor_add_one_gt)
- also have "... = real(natfloor(r powr (1 / -s)) + 1)"
- by simp
- also have "... <= real n"
- apply (subst real_of_nat_le_iff)
- apply (rule prems)
- done
- finally have "r powr (1 / - s) < real n".
- then have "real n powr (- s) < (r powr (1 / - s)) powr - s"
- apply (intro powr_less_mono2_neg)
- apply (auto simp add: prems)
- done
- also have "... = r"
- by (simp add: powr_powr prems less_imp_neq [THEN not_sym])
- finally show "real n powr - s < r" .
- qed
+proof -
+ fix r fix n
+ assume r: "0 < r" and n: "natfloor (r powr (1 / - s)) + 1 <= n"
+ have "r powr (1 / - s) < real(natfloor(r powr (1 / - s))) + 1"
+ by (rule real_natfloor_add_one_gt)
+ also have "... = real(natfloor(r powr (1 / -s)) + 1)"
+ by simp
+ also have "... <= real n"
+ apply (subst real_of_nat_le_iff)
+ apply (rule n)
+ done
+ finally have "r powr (1 / - s) < real n".
+ then have "real n powr (- s) < (r powr (1 / - s)) powr - s"
+ apply (intro powr_less_mono2_neg)
+ apply (auto simp add: s)
+ done
+ also have "... = r"
+ by (simp add: powr_powr s r less_imp_neq [THEN not_sym])
+ finally show "real n powr - s < r" .
+qed
end
--- a/src/HOL/Map.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Map.thy Fri Jan 14 15:44:47 2011 +0100
@@ -111,7 +111,7 @@
assumes "m(a\<mapsto>x) = n(a\<mapsto>y)"
shows "x = y"
proof -
- from prems have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp
+ from assms have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp
then show ?thesis by simp
qed
--- a/src/HOL/Matrix/ComputeFloat.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Matrix/ComputeFloat.thy Fri Jan 14 15:44:47 2011 +0100
@@ -57,7 +57,7 @@
show ?case by simp
next
case (Suc m)
- show ?case by (auto simp add: algebra_simps pow2_add1 prems)
+ show ?case by (auto simp add: algebra_simps pow2_add1 1 Suc)
qed
next
case (2 n)
@@ -88,7 +88,7 @@
apply (subst pow2_neg[of "int m - a + 1"])
apply (subst pow2_neg[of "int m + 1"])
apply auto
- apply (insert prems)
+ apply (insert Suc)
apply (auto simp add: algebra_simps)
done
qed
@@ -147,8 +147,8 @@
assumes "real_is_int a" "real_is_int b"
shows "(int_of_real (a*b)) = (int_of_real a) * (int_of_real b)"
proof -
- from prems have a: "?! (a'::int). real a' = a" by (rule_tac real_is_int_rep, auto)
- from prems have b: "?! (b'::int). real b' = b" by (rule_tac real_is_int_rep, auto)
+ from assms have a: "?! (a'::int). real a' = a" by (rule_tac real_is_int_rep, auto)
+ from assms have b: "?! (b'::int). real b' = b" by (rule_tac real_is_int_rep, auto)
from a obtain a'::int where a':"a = real a'" by auto
from b obtain b'::int where b':"b = real b'" by auto
have r: "real a' * real b' = real (a' * b')" by auto
@@ -286,16 +286,16 @@
show ?case
proof cases
assume u: "u \<noteq> 0 \<and> even u"
- with prems have ind: "float (u div 2, v + 1) = float (norm_float (u div 2) (v + 1))" by auto
+ with 1 have ind: "float (u div 2, v + 1) = float (norm_float (u div 2) (v + 1))" by auto
with u have "float (u,v) = float (u div 2, v+1)" by (simp add: float_transfer_even)
then show ?thesis
apply (subst norm_float.simps)
apply (simp add: ind)
done
next
- assume "~(u \<noteq> 0 \<and> even u)"
- then show ?thesis
- by (simp add: prems float_def)
+ assume nu: "~(u \<noteq> 0 \<and> even u)"
+ show ?thesis
+ by (simp add: nu float_def)
qed
qed
}
--- a/src/HOL/Matrix/LP.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Matrix/LP.thy Fri Jan 14 15:44:47 2011 +0100
@@ -12,7 +12,7 @@
"c <= d"
shows "a <= b + d"
apply (rule_tac order_trans[where y = "b+c"])
- apply (simp_all add: prems)
+ apply (simp_all add: assms)
done
lemma linprog_dual_estimate:
@@ -26,8 +26,8 @@
shows
"c * x \<le> y * b' + (y * \<delta>A + abs (y * A' - c') + \<delta>c) * r"
proof -
- from prems have 1: "y * b <= y * b'" by (simp add: mult_left_mono)
- from prems have 2: "y * (A * x) <= y * b" by (simp add: mult_left_mono)
+ from assms have 1: "y * b <= y * b'" by (simp add: mult_left_mono)
+ from assms have 2: "y * (A * x) <= y * b" by (simp add: mult_left_mono)
have 3: "y * (A * x) = c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x" by (simp add: algebra_simps)
from 1 2 3 have 4: "c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x <= y * b'" by simp
have 5: "c * x <= y * b' + abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"
@@ -44,23 +44,23 @@
have 11: "abs (c'-c) = abs (c-c')"
by (subst 10, subst abs_minus_cancel, simp)
have 12: "(abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x"
- by (simp add: 11 prems mult_right_mono)
+ by (simp add: 11 assms mult_right_mono)
have 13: "(abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x <= (abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * abs x"
- by (simp add: prems mult_right_mono mult_left_mono)
+ by (simp add: assms mult_right_mono mult_left_mono)
have r: "(abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * abs x <= (abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * r"
apply (rule mult_left_mono)
- apply (simp add: prems)
+ apply (simp add: assms)
apply (rule_tac add_mono[of "0::'a" _ "0", simplified])+
apply (rule mult_left_mono[of "0" "\<delta>A", simplified])
apply (simp_all)
- apply (rule order_trans[where y="abs (A-A')"], simp_all add: prems)
- apply (rule order_trans[where y="abs (c-c')"], simp_all add: prems)
+ apply (rule order_trans[where y="abs (A-A')"], simp_all add: assms)
+ apply (rule order_trans[where y="abs (c-c')"], simp_all add: assms)
done
from 6 7 8 9 12 13 r have 14:" abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <=(abs y * \<delta>A + abs (y*A'-c') + \<delta>c) * r"
by (simp)
show ?thesis
apply (rule le_add_right_mono[of _ _ "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"])
- apply (simp_all only: 5 14[simplified abs_of_nonneg[of y, simplified prems]])
+ apply (simp_all only: 5 14[simplified abs_of_nonneg[of y, simplified assms]])
done
qed
@@ -73,10 +73,10 @@
have "0 <= A - A1"
proof -
have 1: "A - A1 = A + (- A1)" by simp
- show ?thesis by (simp only: 1 add_right_mono[of A1 A "-A1", simplified, simplified prems])
+ show ?thesis by (simp only: 1 add_right_mono[of A1 A "-A1", simplified, simplified assms])
qed
then have "abs (A-A1) = A-A1" by (rule abs_of_nonneg)
- with prems show "abs (A-A1) <= (A2-A1)" by simp
+ with assms show "abs (A-A1) <= (A2-A1)" by simp
qed
lemma mult_le_prts:
@@ -95,31 +95,31 @@
then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
by (simp add: algebra_simps)
moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
- by (simp_all add: prems mult_mono)
+ by (simp_all add: assms mult_mono)
moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
proof -
have "pprt a * nprt b <= pprt a * nprt b2"
- by (simp add: mult_left_mono prems)
+ by (simp add: mult_left_mono assms)
moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
- by (simp add: mult_right_mono_neg prems)
+ by (simp add: mult_right_mono_neg assms)
ultimately show ?thesis
by simp
qed
moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
proof -
have "nprt a * pprt b <= nprt a2 * pprt b"
- by (simp add: mult_right_mono prems)
+ by (simp add: mult_right_mono assms)
moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
- by (simp add: mult_left_mono_neg prems)
+ by (simp add: mult_left_mono_neg assms)
ultimately show ?thesis
by simp
qed
moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
proof -
have "nprt a * nprt b <= nprt a * nprt b1"
- by (simp add: mult_left_mono_neg prems)
+ by (simp add: mult_left_mono_neg assms)
moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
- by (simp add: mult_right_mono_neg prems)
+ by (simp add: mult_right_mono_neg assms)
ultimately show ?thesis
by simp
qed
@@ -141,19 +141,19 @@
"c * x \<le> y * b + (let s1 = c1 - y * A2; s2 = c2 - y * A1 in pprt s2 * pprt r2 + pprt s1 * nprt r2 + nprt s2 * pprt r1 + nprt s1 * nprt r1)"
(is "_ <= _ + ?C")
proof -
- from prems have "y * (A * x) <= y * b" by (simp add: mult_left_mono)
+ from assms have "y * (A * x) <= y * b" by (simp add: mult_left_mono)
moreover have "y * (A * x) = c * x + (y * A - c) * x" by (simp add: algebra_simps)
ultimately have "c * x + (y * A - c) * x <= y * b" by simp
then have "c * x <= y * b - (y * A - c) * x" by (simp add: le_diff_eq)
then have cx: "c * x <= y * b + (c - y * A) * x" by (simp add: algebra_simps)
have s2: "c - y * A <= c2 - y * A1"
- by (simp add: diff_minus prems add_mono mult_left_mono)
+ by (simp add: diff_minus assms add_mono mult_left_mono)
have s1: "c1 - y * A2 <= c - y * A"
- by (simp add: diff_minus prems add_mono mult_left_mono)
+ by (simp add: diff_minus assms add_mono mult_left_mono)
have prts: "(c - y * A) * x <= ?C"
apply (simp add: Let_def)
apply (rule mult_le_prts)
- apply (simp_all add: prems s1 s2)
+ apply (simp_all add: assms s1 s2)
done
then have "y * b + (c - y * A) * x <= y * b + ?C"
by simp
--- a/src/HOL/Nominal/Nominal.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Nominal/Nominal.thy Fri Jan 14 15:44:47 2011 +0100
@@ -785,7 +785,7 @@
hence "((UNIV::'x set) - A) \<noteq> ({}::'x set)" by (force simp only:)
then obtain c::"'x" where "c\<in>((UNIV::'x set) - A)" by force
then have "c\<notin>A" by simp
- then show ?thesis using prems by simp
+ then show ?thesis ..
qed
text {* there always exists a fresh name for an object with finite support *}
--- a/src/HOL/Power.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Power.thy Fri Jan 14 15:44:47 2011 +0100
@@ -297,7 +297,7 @@
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)
+ by (simp only: assms power_strict_mono)
from le and this show False
by (simp add: linorder_not_less [symmetric])
qed
--- a/src/HOL/Predicate.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Predicate.thy Fri Jan 14 15:44:47 2011 +0100
@@ -93,10 +93,10 @@
subsubsection {* Top and bottom elements *}
lemma bot1E [no_atp, elim!]: "bot x \<Longrightarrow> P"
- by (simp add: bot_fun_def bot_bool_def)
+ by (simp add: bot_fun_def)
lemma bot2E [elim!]: "bot x y \<Longrightarrow> P"
- by (simp add: bot_fun_def bot_bool_def)
+ by (simp add: bot_fun_def)
lemma bot_empty_eq: "bot = (\<lambda>x. x \<in> {})"
by (auto simp add: fun_eq_iff)
@@ -105,64 +105,64 @@
by (auto simp add: fun_eq_iff)
lemma top1I [intro!]: "top x"
- by (simp add: top_fun_def top_bool_def)
+ by (simp add: top_fun_def)
lemma top2I [intro!]: "top x y"
- by (simp add: top_fun_def top_bool_def)
+ by (simp add: top_fun_def)
subsubsection {* Binary intersection *}
lemma inf1I [intro!]: "A x ==> B x ==> inf A B x"
- by (simp add: inf_fun_def inf_bool_def)
+ by (simp add: inf_fun_def)
lemma inf2I [intro!]: "A x y ==> B x y ==> inf A B x y"
- by (simp add: inf_fun_def inf_bool_def)
+ by (simp add: inf_fun_def)
lemma inf1E [elim!]: "inf A B x ==> (A x ==> B x ==> P) ==> P"
- by (simp add: inf_fun_def inf_bool_def)
+ by (simp add: inf_fun_def)
lemma inf2E [elim!]: "inf A B x y ==> (A x y ==> B x y ==> P) ==> P"
- by (simp add: inf_fun_def inf_bool_def)
+ by (simp add: inf_fun_def)
lemma inf1D1: "inf A B x ==> A x"
- by (simp add: inf_fun_def inf_bool_def)
+ by (simp add: inf_fun_def)
lemma inf2D1: "inf A B x y ==> A x y"
- by (simp add: inf_fun_def inf_bool_def)
+ by (simp add: inf_fun_def)
lemma inf1D2: "inf A B x ==> B x"
- by (simp add: inf_fun_def inf_bool_def)
+ by (simp add: inf_fun_def)
lemma inf2D2: "inf A B x y ==> B x y"
- by (simp add: inf_fun_def inf_bool_def)
+ by (simp add: inf_fun_def)
lemma inf_Int_eq: "inf (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
- by (simp add: inf_fun_def inf_bool_def mem_def)
+ by (simp add: inf_fun_def mem_def)
lemma inf_Int_eq2 [pred_set_conv]: "inf (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)"
- by (simp add: inf_fun_def inf_bool_def mem_def)
+ by (simp add: inf_fun_def mem_def)
subsubsection {* Binary union *}
lemma sup1I1 [elim?]: "A x \<Longrightarrow> sup A B x"
- by (simp add: sup_fun_def sup_bool_def)
+ by (simp add: sup_fun_def)
lemma sup2I1 [elim?]: "A x y \<Longrightarrow> sup A B x y"
- by (simp add: sup_fun_def sup_bool_def)
+ by (simp add: sup_fun_def)
lemma sup1I2 [elim?]: "B x \<Longrightarrow> sup A B x"
- by (simp add: sup_fun_def sup_bool_def)
+ by (simp add: sup_fun_def)
lemma sup2I2 [elim?]: "B x y \<Longrightarrow> sup A B x y"
- by (simp add: sup_fun_def sup_bool_def)
+ by (simp add: sup_fun_def)
lemma sup1E [elim!]: "sup A B x ==> (A x ==> P) ==> (B x ==> P) ==> P"
- by (simp add: sup_fun_def sup_bool_def) iprover
+ by (simp add: sup_fun_def) iprover
lemma sup2E [elim!]: "sup A B x y ==> (A x y ==> P) ==> (B x y ==> P) ==> P"
- by (simp add: sup_fun_def sup_bool_def) iprover
+ by (simp add: sup_fun_def) iprover
text {*
\medskip Classical introduction rule: no commitment to @{text A} vs
@@ -170,16 +170,16 @@
*}
lemma sup1CI [intro!]: "(~ B x ==> A x) ==> sup A B x"
- by (auto simp add: sup_fun_def sup_bool_def)
+ by (auto simp add: sup_fun_def)
lemma sup2CI [intro!]: "(~ B x y ==> A x y) ==> sup A B x y"
- by (auto simp add: sup_fun_def sup_bool_def)
+ by (auto simp add: sup_fun_def)
lemma sup_Un_eq: "sup (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
- by (simp add: sup_fun_def sup_bool_def mem_def)
+ by (simp add: sup_fun_def mem_def)
lemma sup_Un_eq2 [pred_set_conv]: "sup (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)"
- by (simp add: sup_fun_def sup_bool_def mem_def)
+ by (simp add: sup_fun_def mem_def)
subsubsection {* Intersections of families *}
@@ -257,7 +257,7 @@
lemma pred_comp_rel_comp_eq [pred_set_conv]:
"((\<lambda>x y. (x, y) \<in> r) OO (\<lambda>x y. (x, y) \<in> s)) = (\<lambda>x y. (x, y) \<in> r O s)"
- by (auto simp add: fun_eq_iff elim: pred_compE)
+ by (auto simp add: fun_eq_iff)
subsubsection {* Converse *}
@@ -292,12 +292,10 @@
elim: pred_compE dest: conversepD)
lemma converse_meet: "(inf r s)^--1 = inf r^--1 s^--1"
- by (simp add: inf_fun_def inf_bool_def)
- (iprover intro: conversepI ext dest: conversepD)
+ by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)
lemma converse_join: "(sup r s)^--1 = sup r^--1 s^--1"
- by (simp add: sup_fun_def sup_bool_def)
- (iprover intro: conversepI ext dest: conversepD)
+ by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)
lemma conversep_noteq [simp]: "(op ~=)^--1 = op ~="
by (auto simp add: fun_eq_iff)
@@ -756,7 +754,7 @@
apply (rule ext)
apply (simp add: unit_eq)
done
- from this prems show ?thesis by blast
+ from this assms show ?thesis by blast
qed
lemma unit_pred_cases:
--- a/src/HOL/RComplete.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/RComplete.thy Fri Jan 14 15:44:47 2011 +0100
@@ -517,10 +517,10 @@
apply simp
done
-lemma natfloor_div_nat: "1 <= x ==> y > 0 ==>
- natfloor (x / real y) = natfloor x div y"
+lemma natfloor_div_nat:
+ assumes "1 <= x" and "y > 0"
+ shows "natfloor (x / real y) = natfloor x div y"
proof -
- assume "1 <= (x::real)" and "(y::nat) > 0"
have "natfloor x = (natfloor x) div y * y + (natfloor x) mod y"
by simp
then have a: "real(natfloor x) = real ((natfloor x) div y) * real y +
@@ -535,8 +535,7 @@
by simp
also have "... = real((natfloor x) div y) + real((natfloor x) mod y) /
real y + (x - real(natfloor x)) / real y"
- by (auto simp add: algebra_simps add_divide_distrib
- diff_divide_distrib prems)
+ by (auto simp add: algebra_simps add_divide_distrib diff_divide_distrib)
finally have "natfloor (x / real y) = natfloor(...)" by simp
also have "... = natfloor(real((natfloor x) mod y) /
real y + (x - real(natfloor x)) / real y + real((natfloor x) div y))"
@@ -547,11 +546,11 @@
apply (rule add_nonneg_nonneg)
apply (rule divide_nonneg_pos)
apply simp
- apply (simp add: prems)
+ apply (simp add: assms)
apply (rule divide_nonneg_pos)
apply (simp add: algebra_simps)
apply (rule real_natfloor_le)
- apply (insert prems, auto)
+ using assms apply auto
done
also have "natfloor(real((natfloor x) mod y) /
real y + (x - real(natfloor x)) / real y) = 0"
@@ -560,13 +559,13 @@
apply (rule add_nonneg_nonneg)
apply (rule divide_nonneg_pos)
apply force
- apply (force simp add: prems)
+ apply (force simp add: assms)
apply (rule divide_nonneg_pos)
apply (simp add: algebra_simps)
apply (rule real_natfloor_le)
- apply (auto simp add: prems)
- apply (insert prems, arith)
- apply (simp add: add_divide_distrib [THEN sym])
+ apply (auto simp add: assms)
+ using assms apply arith
+ using assms apply (simp add: add_divide_distrib [THEN sym])
apply (subgoal_tac "real y = real y - 1 + 1")
apply (erule ssubst)
apply (rule add_le_less_mono)
--- a/src/HOL/RealDef.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/RealDef.thy Fri Jan 14 15:44:47 2011 +0100
@@ -1200,7 +1200,7 @@
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) =
real (x div d) + (real (x mod d)) / (real d)"
proof -
- assume "d ~= 0"
+ assume d: "d ~= 0"
have "x = (x div d) * d + x mod d"
by auto
then have "real x = real (x div d) * real d + real(x mod d)"
@@ -1208,7 +1208,7 @@
then have "real x / real d = ... / real d"
by simp
then show ?thesis
- by (auto simp add: add_divide_distrib algebra_simps prems)
+ by (auto simp add: add_divide_distrib algebra_simps d)
qed
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
@@ -1353,7 +1353,7 @@
lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) =
real (x div d) + (real (x mod d)) / (real d)"
proof -
- assume "0 < d"
+ assume d: "0 < d"
have "x = (x div d) * d + x mod d"
by auto
then have "real x = real (x div d) * real d + real(x mod d)"
@@ -1361,7 +1361,7 @@
then have "real x / real d = \<dots> / real d"
by simp
then show ?thesis
- by (auto simp add: add_divide_distrib algebra_simps prems)
+ by (auto simp add: add_divide_distrib algebra_simps d)
qed
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==>
--- a/src/HOL/Transcendental.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Transcendental.thy Fri Jan 14 15:44:47 2011 +0100
@@ -164,7 +164,7 @@
{
have "?s 0 = 0" by auto
have Suc_m1: "\<And> n. Suc n - 1 = n" by auto
- { fix B T E have "(if \<not> B then T else E) = (if B then E else T)" by auto } note if_eq = this
+ have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto
have "?s sums y" using sums_if'[OF `f sums y`] .
from this[unfolded sums_def, THEN LIMSEQ_Suc]
@@ -348,7 +348,7 @@
fixes z :: "'a :: {monoid_mult,comm_ring}" shows
"(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
(\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
-by(auto simp add: algebra_simps power_add [symmetric] cong: strong_setsum_cong)
+by(auto simp add: algebra_simps power_add [symmetric])
lemma sumr_diff_mult_const2:
"setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
@@ -1849,7 +1849,7 @@
lemma sin_less_zero:
assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
proof -
- have "0 < sin (- x)" using prems by (simp only: sin_gt_zero2)
+ have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
thus ?thesis by simp
qed
@@ -2107,7 +2107,7 @@
lemma tan_less_zero:
assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
proof -
- have "0 < tan (- x)" using prems by (simp only: tan_gt_zero)
+ have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
thus ?thesis by simp
qed
--- a/src/HOL/Word/Word.thy Fri Jan 14 15:43:04 2011 +0100
+++ b/src/HOL/Word/Word.thy Fri Jan 14 15:44:47 2011 +0100
@@ -2171,7 +2171,7 @@
lemma word_of_int_power_hom:
"word_of_int a ^ n = (word_of_int (a ^ n) :: 'a :: len word)"
- by (induct n) (simp_all add : word_of_int_hom_syms power_Suc)
+ by (induct n) (simp_all add: word_of_int_hom_syms)
lemma word_arith_power_alt:
"a ^ n = (word_of_int (uint a ^ n) :: 'a :: len word)"
@@ -2367,7 +2367,7 @@
using word_of_int_Ex [where x=x]
word_of_int_Ex [where x=y]
word_of_int_Ex [where x=z]
- by (auto simp: bwsimps bbw_ao_dist simp del: bin_ops_comm)
+ by (auto simp: bwsimps bbw_ao_dist)
lemma word_oa_dist:
fixes x :: "'a::len0 word"
@@ -2375,7 +2375,7 @@
using word_of_int_Ex [where x=x]
word_of_int_Ex [where x=y]
word_of_int_Ex [where x=z]
- by (auto simp: bwsimps bbw_oa_dist simp del: bin_ops_comm)
+ by (auto simp: bwsimps bbw_oa_dist)
lemma word_add_not [simp]:
fixes x :: "'a::len0 word"
@@ -2571,7 +2571,7 @@
fixes w :: "'a::len0 word"
assumes "m ~= n"
shows "set_bit (set_bit w m x) n y = set_bit (set_bit w n y) m x"
- by (rule word_eqI) (clarsimp simp add : test_bit_set_gen word_size prems)
+ by (rule word_eqI) (clarsimp simp add: test_bit_set_gen word_size assms)
lemma test_bit_no':
fixes w :: "'a::len0 word"
@@ -2623,7 +2623,7 @@
done
lemma word_msb_n1: "msb (-1::'a::len word)"
- unfolding word_msb_alt word_msb_alt to_bl_n1 by simp
+ unfolding word_msb_alt to_bl_n1 by simp
declare word_set_set_same [simp] word_set_nth [simp]
test_bit_no [simp] word_set_no [simp] nth_0 [simp]
@@ -3047,7 +3047,7 @@
lemma shiftl_t2n: "shiftl (w :: 'a :: len word) n = 2 ^ n * w"
unfolding shiftl_def
- by (induct n) (auto simp: shiftl1_2t power_Suc)
+ by (induct n) (auto simp: shiftl1_2t)
lemma shiftr1_bintr [simp]:
"(shiftr1 (number_of w) :: 'a :: len0 word) =
@@ -3940,12 +3940,12 @@
apply (clarsimp simp: word_size)+
apply (rule trans)
apply (rule test_bit_rcat [OF refl refl])
- apply (simp add : word_size msrevs)
+ apply (simp add: word_size msrevs)
apply (subst nth_rev)
apply arith
- apply (simp add : le0 [THEN [2] xtr7, THEN diff_Suc_less])
+ apply (simp add: le0 [THEN [2] xtr7, THEN diff_Suc_less])
apply safe
- apply (simp add : diff_mult_distrib)
+ apply (simp add: diff_mult_distrib)
apply (rule mpl_lem)
apply (cases "size ws")
apply simp_all