1.1 --- a/src/HOL/BNF_FP_Base.thy Wed Apr 23 10:23:26 2014 +0200
1.2 +++ b/src/HOL/BNF_FP_Base.thy Wed Apr 23 10:23:26 2014 +0200
1.3 @@ -89,34 +89,34 @@
1.4 lemma spec2: "\<forall>x y. P x y \<Longrightarrow> P x y"
1.5 by blast
1.6
1.7 -lemma rewriteR_comp_comp: "\<lbrakk>g o h = r\<rbrakk> \<Longrightarrow> f o g o h = f o r"
1.8 +lemma rewriteR_comp_comp: "\<lbrakk>g \<circ> h = r\<rbrakk> \<Longrightarrow> f \<circ> g \<circ> h = f \<circ> r"
1.9 unfolding comp_def fun_eq_iff by auto
1.10
1.11 -lemma rewriteR_comp_comp2: "\<lbrakk>g o h = r1 o r2; f o r1 = l\<rbrakk> \<Longrightarrow> f o g o h = l o r2"
1.12 +lemma rewriteR_comp_comp2: "\<lbrakk>g \<circ> h = r1 \<circ> r2; f \<circ> r1 = l\<rbrakk> \<Longrightarrow> f \<circ> g \<circ> h = l \<circ> r2"
1.13 unfolding comp_def fun_eq_iff by auto
1.14
1.15 -lemma rewriteL_comp_comp: "\<lbrakk>f o g = l\<rbrakk> \<Longrightarrow> f o (g o h) = l o h"
1.16 +lemma rewriteL_comp_comp: "\<lbrakk>f \<circ> g = l\<rbrakk> \<Longrightarrow> f \<circ> (g \<circ> h) = l \<circ> h"
1.17 unfolding comp_def fun_eq_iff by auto
1.18
1.19 -lemma rewriteL_comp_comp2: "\<lbrakk>f o g = l1 o l2; l2 o h = r\<rbrakk> \<Longrightarrow> f o (g o h) = l1 o r"
1.20 +lemma rewriteL_comp_comp2: "\<lbrakk>f \<circ> g = l1 \<circ> l2; l2 \<circ> h = r\<rbrakk> \<Longrightarrow> f \<circ> (g \<circ> h) = l1 \<circ> r"
1.21 unfolding comp_def fun_eq_iff by auto
1.22
1.23 -lemma convol_o: "<f, g> o h = <f o h, g o h>"
1.24 +lemma convol_o: "<f, g> \<circ> h = <f \<circ> h, g \<circ> h>"
1.25 unfolding convol_def by auto
1.26
1.27 -lemma map_prod_o_convol: "map_prod h1 h2 o <f, g> = <h1 o f, h2 o g>"
1.28 +lemma map_prod_o_convol: "map_prod h1 h2 \<circ> <f, g> = <h1 \<circ> f, h2 \<circ> g>"
1.29 unfolding convol_def by auto
1.30
1.31 lemma map_prod_o_convol_id: "(map_prod f id \<circ> <id , g>) x = <id \<circ> f , g> x"
1.32 unfolding map_prod_o_convol id_comp comp_id ..
1.33
1.34 -lemma o_case_sum: "h o case_sum f g = case_sum (h o f) (h o g)"
1.35 +lemma o_case_sum: "h \<circ> case_sum f g = case_sum (h \<circ> f) (h \<circ> g)"
1.36 unfolding comp_def by (auto split: sum.splits)
1.37
1.38 -lemma case_sum_o_map_sum: "case_sum f g o map_sum h1 h2 = case_sum (f o h1) (g o h2)"
1.39 +lemma case_sum_o_map_sum: "case_sum f g \<circ> map_sum h1 h2 = case_sum (f \<circ> h1) (g \<circ> h2)"
1.40 unfolding comp_def by (auto split: sum.splits)
1.41
1.42 -lemma case_sum_o_map_sum_id: "(case_sum id g o map_sum f id) x = case_sum (f o id) g x"
1.43 +lemma case_sum_o_map_sum_id: "(case_sum id g \<circ> map_sum f id) x = case_sum (f \<circ> id) g x"
1.44 unfolding case_sum_o_map_sum id_comp comp_id ..
1.45
1.46 lemma rel_fun_def_butlast:
1.47 @@ -144,7 +144,7 @@
1.48
1.49 lemma
1.50 assumes "type_definition Rep Abs UNIV"
1.51 - shows type_copy_Rep_o_Abs: "Rep \<circ> Abs = id" and type_copy_Abs_o_Rep: "Abs o Rep = id"
1.52 + shows type_copy_Rep_o_Abs: "Rep \<circ> Abs = id" and type_copy_Abs_o_Rep: "Abs \<circ> Rep = id"
1.53 unfolding fun_eq_iff comp_apply id_apply
1.54 type_definition.Abs_inverse[OF assms UNIV_I] type_definition.Rep_inverse[OF assms] by simp_all
1.55
1.56 @@ -152,7 +152,7 @@
1.57 assumes "type_definition Rep Abs UNIV"
1.58 "type_definition Rep' Abs' UNIV"
1.59 "type_definition Rep'' Abs'' UNIV"
1.60 - shows "Abs' o M o Rep'' = (Abs' o M1 o Rep) o (Abs o M2 o Rep'') \<Longrightarrow> M1 o M2 = M"
1.61 + shows "Abs' \<circ> M \<circ> Rep'' = (Abs' \<circ> M1 \<circ> Rep) \<circ> (Abs \<circ> M2 \<circ> Rep'') \<Longrightarrow> M1 \<circ> M2 = M"
1.62 by (rule sym) (auto simp: fun_eq_iff type_definition.Abs_inject[OF assms(2) UNIV_I UNIV_I]
1.63 type_definition.Abs_inverse[OF assms(1) UNIV_I]
1.64 type_definition.Abs_inverse[OF assms(3) UNIV_I] dest: spec[of _ "Abs'' x" for x])
1.65 @@ -160,7 +160,7 @@
1.66 lemma vimage2p_id: "vimage2p id id R = R"
1.67 unfolding vimage2p_def by auto
1.68
1.69 -lemma vimage2p_comp: "vimage2p (f1 o f2) (g1 o g2) = vimage2p f2 g2 o vimage2p f1 g1"
1.70 +lemma vimage2p_comp: "vimage2p (f1 \<circ> f2) (g1 \<circ> g2) = vimage2p f2 g2 \<circ> vimage2p f1 g1"
1.71 unfolding fun_eq_iff vimage2p_def o_apply by simp
1.72
1.73 ML_file "Tools/BNF/bnf_fp_util.ML"