src/HOL/Fun.thy
changeset 34209 c7f621786035
parent 34153 5da0f7abbe29
child 35115 446c5063e4fd
     1.1 --- a/src/HOL/Fun.thy	Wed Dec 30 01:08:33 2009 +0100
     1.2 +++ b/src/HOL/Fun.thy	Wed Dec 30 10:24:53 2009 +0100
     1.3 @@ -171,7 +171,7 @@
     1.4  by (simp add: surj_def) 
     1.5  
     1.6  lemma bij_id[simp]: "bij id"
     1.7 -by (simp add: bij_def inj_on_id surj_id) 
     1.8 +by (simp add: bij_def)
     1.9  
    1.10  lemma inj_onI:
    1.11      "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
    1.12 @@ -432,14 +432,14 @@
    1.13  by (rule ext, auto)
    1.14  
    1.15  lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
    1.16 -by(fastsimp simp:inj_on_def image_def)
    1.17 +by (fastsimp simp:inj_on_def image_def)
    1.18  
    1.19  lemma fun_upd_image:
    1.20       "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
    1.21  by auto
    1.22  
    1.23  lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
    1.24 -by(auto intro: ext)
    1.25 +by (auto intro: ext)
    1.26  
    1.27  
    1.28  subsection {* @{text override_on} *}
    1.29 @@ -496,7 +496,7 @@
    1.30    thus "inj_on f A" by simp 
    1.31  next
    1.32    assume "inj_on f A"
    1.33 -  with A show "inj_on (swap a b f) A" by(iprover intro: inj_on_imp_inj_on_swap)
    1.34 +  with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
    1.35  qed
    1.36  
    1.37  lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
    1.38 @@ -529,7 +529,7 @@
    1.39  lemma the_inv_into_f_f:
    1.40    "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
    1.41  apply (simp add: the_inv_into_def inj_on_def)
    1.42 -apply (blast intro: the_equality)
    1.43 +apply blast
    1.44  done
    1.45  
    1.46  lemma f_the_inv_into_f: