[STATEMENT]
lemma lin_independent_length_le_dim :
  "set us \<subseteq> V \<Longrightarrow> lin_independent us \<Longrightarrow> length us \<le> dim V"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>set us \<subseteq> V; R_lin_independent us\<rbrakk> \<Longrightarrow> length us \<le> dim_R V
[PROOF STEP]
using basis_ex max_lin_independent_set_in_Span dim_eq_size_basis
[PROOF STATE]
proof (prove)
using this:
\<exists>vs. set vs \<subseteq> V \<and> V = RSpan vs \<and> R_lin_independent vs
\<lbrakk>set ?vs \<subseteq> V; set ?us \<subseteq> RSpan ?vs; R_lin_independent ?us\<rbrakk> \<Longrightarrow> length ?us \<le> length ?vs
set ?vs \<subseteq> V \<and> V = RSpan ?vs \<and> R_lin_independent ?vs \<Longrightarrow> length ?vs = dim_R V

goal (1 subgoal):
 1. \<lbrakk>set us \<subseteq> V; R_lin_independent us\<rbrakk> \<Longrightarrow> length us \<le> dim_R V
[PROOF STEP]
by    force


[STATEMENT]
lemma image_of_maximum:
  fixes f::"'a::order \<Rightarrow> 'b::conditionally_complete_lattice"
  assumes "mono f"
    and "\<And>x. x:M \<Longrightarrow> x\<le>m"
    and "m:M"
  shows "(SUP x\<in>M. f x) = f m"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. Sup (f ` M) = f m
[PROOF STEP]
by (smt (verit, ccfv_threshold) assms(1) assms(2) assms(3) cSup_eq_maximum imageE imageI monoD)


[STATEMENT]
lemma acyclic_irreflexive_star_antisymmetric:
  "acyclic x \<longleftrightarrow> irreflexive x \<and> antisymmetric (x\<^sup>\<star>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. pd_kleene_allegory_class.acyclic x = (irreflexive x \<and> antisymmetric (x\<^sup>\<star>))
[PROOF STEP]
by (metis acyclic_star_inf_conv_iff conv_star_commute order.trans reflexive_inf_closed star.circ_mult_increasing star.circ_reflexive order.antisym)