src/HOL/Probability/Helly_Selection.thy
changeset 69861 62e47f06d22c
parent 69260 0a9688695a1b
equal deleted inserted replaced
69860:b58a575d211e 69861:62e47f06d22c
    86   then have nonempty: "{lim (?f n) |n. x < r n} \<noteq> {}" for x
    86   then have nonempty: "{lim (?f n) |n. x < r n} \<noteq> {}" for x
    87     by auto
    87     by auto
    88 
    88 
    89   define F where "F x = Inf {lim (?f n) |n. x < r n}" for x
    89   define F where "F x = Inf {lim (?f n) |n. x < r n}" for x
    90   have F_eq: "ereal (F x) = (INF n\<in>{n. x < r n}. ereal (lim (?f n)))" for x
    90   have F_eq: "ereal (F x) = (INF n\<in>{n. x < r n}. ereal (lim (?f n)))" for x
    91     unfolding F_def by (subst ereal_Inf'[OF bdd_below nonempty]) (simp add: setcompr_eq_image)
    91     unfolding F_def by (subst ereal_Inf'[OF bdd_below nonempty]) (simp add: setcompr_eq_image image_comp)
    92   have mono_F: "mono F"
    92   have mono_F: "mono F"
    93     using nonempty by (auto intro!: cInf_superset_mono simp: F_def bdd_below mono_def)
    93     using nonempty by (auto intro!: cInf_superset_mono simp: F_def bdd_below mono_def)
    94   moreover have "\<And>x. continuous (at_right x) F"
    94   moreover have "\<And>x. continuous (at_right x) F"
    95     unfolding continuous_within order_tendsto_iff eventually_at_right[OF less_add_one]
    95     unfolding continuous_within order_tendsto_iff eventually_at_right[OF less_add_one]
    96   proof safe
    96   proof safe