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53 text {* chains of functions yield chains in the po range *} |
53 text {* chains of functions yield chains in the po range *} |
54 |
54 |
55 lemma ch2ch_fun: "chain S \<Longrightarrow> chain (\<lambda>i. S i x)" |
55 lemma ch2ch_fun: "chain S \<Longrightarrow> chain (\<lambda>i. S i x)" |
56 by (simp add: chain_def less_fun_def) |
56 by (simp add: chain_def less_fun_def) |
57 |
57 |
58 lemma ch2ch_fun_rev: "(\<And>x. chain (\<lambda>i. S i x)) \<Longrightarrow> chain S" |
58 lemma ch2ch_lambda: "(\<And>x. chain (\<lambda>i. S i x)) \<Longrightarrow> chain S" |
59 by (simp add: chain_def less_fun_def) |
59 by (simp add: chain_def less_fun_def) |
60 |
60 |
61 |
61 |
62 text {* upper bounds of function chains yield upper bound in the po range *} |
62 text {* upper bounds of function chains yield upper bound in the po range *} |
63 |
63 |