--- a/src/HOL/Algebra/FiniteProduct.thy Sun Nov 28 14:01:20 2010 +0100
+++ b/src/HOL/Algebra/FiniteProduct.thy Sun Nov 28 15:21:02 2010 +0100
@@ -130,7 +130,7 @@
apply (rule Suc_le_mono [THEN subst])
apply (simp add: card_Suc_Diff1)
apply (rule_tac A1 = "Aa - {xb}" in finite_imp_foldSetD [THEN exE])
- apply (blast intro: foldSetD_imp_finite finite_Diff)
+ apply (blast intro: foldSetD_imp_finite)
apply best
apply assumption
apply (frule (1) Diff1_foldSetD)
--- a/src/HOL/Big_Operators.thy Sun Nov 28 14:01:20 2010 +0100
+++ b/src/HOL/Big_Operators.thy Sun Nov 28 15:21:02 2010 +0100
@@ -707,7 +707,7 @@
proof -
have "A <+> B = Inl ` A \<union> Inr ` B" by auto
moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
- by(auto intro: finite_imageI)
+ by auto
moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
ultimately show ?thesis using fin by(simp add: setsum_Un_disjoint setsum_reindex)
--- a/src/HOL/Finite_Set.thy Sun Nov 28 14:01:20 2010 +0100
+++ b/src/HOL/Finite_Set.thy Sun Nov 28 15:21:02 2010 +0100
@@ -274,7 +274,7 @@
then show ?thesis by simp
qed
-lemma finite_Diff [simp]: "finite A ==> finite (A - B)"
+lemma finite_Diff [simp, intro]: "finite A ==> finite (A - B)"
by (rule Diff_subset [THEN finite_subset])
lemma finite_Diff2 [simp]:
@@ -303,7 +303,7 @@
text {* Image and Inverse Image over Finite Sets *}
-lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
+lemma finite_imageI[simp, intro]: "finite F ==> finite (h ` F)"
-- {* The image of a finite set is finite. *}
by (induct set: finite) simp_all
@@ -372,8 +372,9 @@
text {* The finite UNION of finite sets *}
-lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
- by (induct set: finite) simp_all
+lemma finite_UN_I[intro]:
+ "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
+by (induct set: finite) simp_all
text {*
Strengthen RHS to
@@ -385,7 +386,7 @@
lemma finite_UN [simp]:
"finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
-by (blast intro: finite_UN_I finite_subset)
+by (blast intro: finite_subset)
lemma finite_Collect_bex[simp]: "finite A \<Longrightarrow>
finite{x. EX y:A. Q x y} = (ALL y:A. finite{x. Q x y})"
@@ -428,9 +429,9 @@
text {* Sigma of finite sets *}
-lemma finite_SigmaI [simp]:
+lemma finite_SigmaI [simp, intro]:
"finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
- by (unfold Sigma_def) (blast intro!: finite_UN_I)
+ by (unfold Sigma_def) blast
lemma finite_cartesian_product: "[| finite A; finite B |] ==>
finite (A <*> B)"
@@ -2266,7 +2267,7 @@
apply (induct set: finite)
apply (simp_all add: Pow_insert)
apply (subst card_Un_disjoint, blast)
- apply (blast intro: finite_imageI, blast)
+ apply (blast, blast)
apply (subgoal_tac "inj_on (insert x) (Pow F)")
apply (simp add: card_image Pow_insert)
apply (unfold inj_on_def)
--- a/src/HOL/IMPP/Hoare.thy Sun Nov 28 14:01:20 2010 +0100
+++ b/src/HOL/IMPP/Hoare.thy Sun Nov 28 15:21:02 2010 +0100
@@ -367,7 +367,7 @@
apply (tactic {* ALLGOALS (clarsimp_tac @{clasimpset}) *})
apply (subgoal_tac "G = mgt_call ` U")
prefer 2
-apply (simp add: card_seteq finite_imageI)
+apply (simp add: card_seteq)
apply simp
apply (erule prems(3-)) (*MGF_lemma1*)
apply (rule ballI)
--- a/src/HOL/Library/Infinite_Set.thy Sun Nov 28 14:01:20 2010 +0100
+++ b/src/HOL/Library/Infinite_Set.thy Sun Nov 28 15:21:02 2010 +0100
@@ -339,7 +339,7 @@
shows "\<exists>y \<in> f`A. infinite (f -` {y})"
proof (rule ccontr)
assume "\<not> ?thesis"
- with img have "finite (UN y:f`A. f -` {y})" by (blast intro: finite_UN_I)
+ with img have "finite (UN y:f`A. f -` {y})" by blast
moreover have "A \<subseteq> (UN y:f`A. f -` {y})" by auto
moreover note dom
ultimately show False by (simp add: infinite_super)
--- a/src/HOL/List.thy Sun Nov 28 14:01:20 2010 +0100
+++ b/src/HOL/List.thy Sun Nov 28 15:21:02 2010 +0100
@@ -3625,7 +3625,7 @@
have "?S (Suc n) = (\<Union>x\<in>A. (\<lambda>xs. x#xs) ` ?S n)"
by (auto simp:length_Suc_conv)
then show ?case using `finite A`
- by (auto intro: finite_imageI Suc) (* FIXME metis? *)
+ by (auto intro: Suc) (* FIXME metis? *)
qed
lemma finite_lists_length_le:
--- a/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy Sun Nov 28 14:01:20 2010 +0100
+++ b/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy Sun Nov 28 15:21:02 2010 +0100
@@ -1300,7 +1300,7 @@
shows "finite {abs((x::'a::abs ^'n)$i) |i. i\<in> (UNIV :: 'n set)}"
and "{abs(x$i) |i. i\<in> (UNIV :: 'n::finite set)} \<noteq> {}"
unfolding infnorm_set_image_cart
- by (auto intro: finite_imageI)
+ by auto
lemma component_le_infnorm_cart:
shows "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
--- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Sun Nov 28 14:01:20 2010 +0100
+++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Sun Nov 28 15:21:02 2010 +0100
@@ -2929,7 +2929,7 @@
using sf B(3)
unfolding span_linear_image[OF lf] surj_def subset_eq image_iff
apply blast
- using fB apply (blast intro: finite_imageI)
+ using fB apply blast
unfolding d[symmetric]
apply (rule card_image_le)
apply (rule fB)
@@ -3035,7 +3035,7 @@
shows "finite {abs((x::'a::euclidean_space)$$i) |i. i<DIM('a)}"
and "{abs(x$$i) |i. i<DIM('a::euclidean_space)} \<noteq> {}"
unfolding infnorm_set_image
- by (auto intro: finite_imageI)
+ by auto
lemma infnorm_pos_le: "0 \<le> infnorm (x::'a::euclidean_space)"
unfolding infnorm_def
--- a/src/HOL/Old_Number_Theory/Euler.thy Sun Nov 28 14:01:20 2010 +0100
+++ b/src/HOL/Old_Number_Theory/Euler.thy Sun Nov 28 15:21:02 2010 +0100
@@ -94,7 +94,7 @@
subsection {* Properties of SetS *}
lemma SetS_finite: "2 < p ==> finite (SetS a p)"
- by (auto simp add: SetS_def SRStar_finite [of p] finite_imageI)
+ by (auto simp add: SetS_def SRStar_finite [of p])
lemma SetS_elems_finite: "\<forall>X \<in> SetS a p. finite X"
by (auto simp add: SetS_def MultInvPair_def)
--- a/src/HOL/Old_Number_Theory/EulerFermat.thy Sun Nov 28 14:01:20 2010 +0100
+++ b/src/HOL/Old_Number_Theory/EulerFermat.thy Sun Nov 28 15:21:02 2010 +0100
@@ -257,8 +257,7 @@
apply (subst setprod_insert)
apply (rule_tac [2] Bnor_prod_power_aux)
apply (unfold inj_on_def)
- apply (simp_all add: zmult_ac Bnor_fin finite_imageI
- Bnor_mem_zle_swap)
+ apply (simp_all add: zmult_ac Bnor_fin Bnor_mem_zle_swap)
done
--- a/src/HOL/Old_Number_Theory/Finite2.thy Sun Nov 28 14:01:20 2010 +0100
+++ b/src/HOL/Old_Number_Theory/Finite2.thy Sun Nov 28 15:21:02 2010 +0100
@@ -55,7 +55,7 @@
subsection {* Cardinality of explicit finite sets *}
lemma finite_surjI: "[| B \<subseteq> f ` A; finite A |] ==> finite B"
- by (simp add: finite_subset finite_imageI)
+by (simp add: finite_subset)
lemma bdd_nat_set_l_finite: "finite {y::nat . y < x}"
by (rule bounded_nat_set_is_finite) blast
--- a/src/HOL/Old_Number_Theory/Gauss.thy Sun Nov 28 14:01:20 2010 +0100
+++ b/src/HOL/Old_Number_Theory/Gauss.thy Sun Nov 28 15:21:02 2010 +0100
@@ -73,22 +73,22 @@
done
lemma finite_B: "finite (B)"
- by (auto simp add: B_def finite_A finite_imageI)
+by (auto simp add: B_def finite_A)
lemma finite_C: "finite (C)"
- by (auto simp add: C_def finite_B finite_imageI)
+by (auto simp add: C_def finite_B)
lemma finite_D: "finite (D)"
- by (auto simp add: D_def finite_Int finite_C)
+by (auto simp add: D_def finite_Int finite_C)
lemma finite_E: "finite (E)"
- by (auto simp add: E_def finite_Int finite_C)
+by (auto simp add: E_def finite_Int finite_C)
lemma finite_F: "finite (F)"
- by (auto simp add: F_def finite_E finite_imageI)
+by (auto simp add: F_def finite_E)
lemma C_eq: "C = D \<union> E"
- by (auto simp add: C_def D_def E_def)
+by (auto simp add: C_def D_def E_def)
lemma A_card_eq: "card A = nat ((p - 1) div 2)"
apply (auto simp add: A_def)
--- a/src/HOL/Probability/Lebesgue_Integration.thy Sun Nov 28 14:01:20 2010 +0100
+++ b/src/HOL/Probability/Lebesgue_Integration.thy Sun Nov 28 15:21:02 2010 +0100
@@ -511,7 +511,7 @@
(\<lambda>(x,y). f -` {x} \<inter> g -` {y} \<inter> space M) ` (f`space M \<times> g`space M)"
by auto
hence "finite (?p ` (A \<inter> space M))"
- by (rule finite_subset) (auto intro: finite_SigmaI finite_imageI) }
+ by (rule finite_subset) auto }
note this[intro, simp]
{ fix x assume "x \<in> space M"
--- a/src/HOL/ex/While_Combinator_Example.thy Sun Nov 28 14:01:20 2010 +0100
+++ b/src/HOL/ex/While_Combinator_Example.thy Sun Nov 28 15:21:02 2010 +0100
@@ -28,7 +28,7 @@
apply (fastsimp intro!: lfp_lowerbound)
apply (blast intro: wf_finite_psubset Int_lower2 [THEN [2] wf_subset])
apply (clarsimp simp add: finite_psubset_def order_less_le)
-apply (blast intro!: finite_Diff dest: monoD)
+apply (blast dest: monoD)
done