hcju/isabelleps huggingface dataset:
assms_4: fixes x3aa :: "com"   and \<rho> :: "complex mat" assumes "x3aa \<in> set x3a"   and "well_com x3aa"   and "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>" shows "positive (denote x3aa \<rho>) \<and> trace (denote x3aa \<rho>) \<le> trace \<rho> \<and> denote x3aa \<rho> \<in> carrier_mat d d"
Nml_Nmlize_3: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Arr V src trg t" shows "bicategorical_language.Trg trg (bicategorical_language.Nmlize V t) = bicategorical_language.Trg trg t"
bibdI: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int"   and grouping :: "int"   and index :: "int" assumes "t_design point_set block_collection u_block_size grouping index"   and "grouping = 2" shows "bibd point_set block_collection u_block_size index"
subseteq_qle_def: fixes M :: "'a list list set"   and N :: "'a list list set" shows "M \<subseteq>\<^sub>\<simeq> N \<equiv> \<forall>x\<in>M. x \<in>\<^sub>\<simeq> N"
ipresIGVar_termMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and MOD :: "('h, 'i, 'd, 'j, 'k, 'a, 'g, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "ipresIGVar h (FixSyn.termMOD Delta) MOD = FixSyn.ipresVar h MOD"
gt_rat_sign_change: fixes plr :: "int poly \<times> rat \<times> rat"   and y :: "rat" assumes "\<exists>!x. root_cond plr x"   and "poly_cond2 (poly_real_alg_1 plr)"   and "rai_lb plr \<le> y"   and "y \<le> rai_ub plr" shows "(sgn (ipoly (poly_real_alg_1 plr) y) = sgn (ipoly (poly_real_alg_1 plr) (rai_ub plr))) = (real_of_1 plr < real_of_rat y)"
subst_Op_simp: fixes Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and inp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and binp :: "'e \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qAbs set option"   and ys :: "'varSort"   and y :: "'var"   and delta :: "'f" assumes "good Y"   and "goodInp inp"   and "goodBinp binp" shows "Op delta inp binp #[Y / y]_ys = Op delta (inp %[Y / y]_ys) (binp %%[Y / y]_ys)"
is_sup_def: fixes x :: "'c"   and y :: "'c"   and sup :: "'c" shows "is_sup x y sup = (x \<sqsubseteq> sup \<and> y \<sqsubseteq> sup \<and> (\<forall>z. x \<sqsubseteq> z \<and> y \<sqsubseteq> z \<longrightarrow> sup \<sqsubseteq> z))"
to_nat_ex: fixes a :: "'a" shows "\<exists>!i. univ_list ! i = a \<and> i < length univ_list"
lr_coset_assoc: fixes g :: "'a"   and h :: "'a"   and P :: "'a set" assumes "g \<in> carrier G"   and "h \<in> carrier G"   and "P \<subseteq> carrier G" shows "g <# (P #> h) = g <# P #> h"
simps_8: fixes \<Sigma> :: "'b list"   and w :: "'b list"   and I :: "(nat + nat set) list"   and \<phi> :: "'b formula" assumes "formula \<Sigma>" shows "formula.satisfies (w, I) (FEXISTS \<phi>) = (\<exists>P\<subseteq>{0..length w - 1}. formula.satisfies (w, Inr P # I) \<phi>)"
id_eq_prop_prop_1: fixes v :: "i"   and F :: "\<Pi>\<^sub>1" shows "[F \<^bold>= F in v]"
9: shows "minarc (v * - v\<^sup>T \<sqinter> g) * t = bot"
prod_list_zero_iff: fixes xs :: "'a list" shows "(prod_list xs = (0::'a)) = ((0::'a) \<in> set xs)"
step_suc_sem: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and n :: "'a"   and \<sigma> :: "'b \<Rightarrow> 'c" shows "IFC_def.step prob (n, \<sigma>) = (IFC_def.suc prob n \<sigma>, IFC_def.sem prob n \<sigma>)"
syzygy_module_closed_plus: fixes s1 :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and s2 :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c" assumes "s1 \<in> pmdl.syzygy_module B"   and "s2 \<in> pmdl.syzygy_module B" shows "s1 + s2 \<in> pmdl.syzygy_module B"
rtrancl_subset_implies: fixes a :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set" assumes "a \<subseteq> b" shows "a \<subseteq> b\<^sup>*"
prod_rev_ffact_nat': fixes n :: "nat"   and m :: "nat" assumes "n \<le> m" shows "\<Prod>{m - n + 1..m} = ffact n m"
assms: shows "(as, x) \<approx>res (=) supp p (as', x')"
safetyI: fixes A :: "'b set"   and P :: "'b llist set" assumes "\<And>t. \<lbrakk>t \<in> A\<^sup>\<infinity>; \<forall>r\<in>finpref A t. \<exists>s\<in>A\<^sup>\<infinity>. r @@ s \<in> P\<rbrakk> \<Longrightarrow> t \<in> P" shows "safety A P"
INF_image: fixes Inf :: "'d set \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> 'e"   and A :: "'f set" shows "Inf (g ` f ` A) = Inf ((g \<circ> f) ` A)"
enumerate_regs_empty_reg_unconstrained: fixes a :: "vname aexp" assumes "enumerate_regs a = {}" shows "\<forall>r. \<not> aexp_constrains a (V (R r))"
his_determ: fixes n' :: "nat"   and nid' :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid"   and nid'' :: "'nid" assumes "(n', nid') \<in> his t n nid"   and "(n', nid'') \<in> his t n nid" shows "nid' = nid''"
\<sigma>1s: shows "set (labels \<sigma>1') \<subseteq> r \<down>s {fst \<beta>_step}"
fds_exp_ln: fixes l :: "'a"   and f :: "'a fds" assumes "exp l = fds_nth f (Suc 0)" shows "fds_exp (fds_ln l f) = f"
collinear_3_imp_in_affine_hull: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "collinear {a, b, c}"   and "a \<noteq> b" shows "c \<in> affine hull {a, b}"
arr_iff_in_hom: fixes f :: "'a" shows "C.arr f = \<guillemotleft>f : C.dom f \<rightarrow>\<^sub>C C.cod f\<guillemotright>"
continuous_on_tendsto_compose: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c"   and l :: "'c"   and F :: "'e filter" assumes "continuous_on s f"   and "(g \<longlongrightarrow> l) F"   and "l \<in> s"   and "\<forall>\<^sub>F x in F. g x \<in> s" shows "((\<lambda>x. f (g x)) \<longlongrightarrow> f l) F"
Rep_G_order: fixes a :: "'a G"   and b :: "'a G" shows "(a \<le> b) = (Rep_G a \<le> Rep_G b)"
mset_ordering_singleton: fixes E1 :: "'b multiset"   and a :: "'b"   and r :: "('b \<times> 'b) set" assumes "\<forall>x. x \<in># E1 \<longrightarrow> (x, a) \<in> r" shows "(E1, {#a#}) \<in> mult r"
sep_disj_commute: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and sep_disj :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.pre_sep_algebra plus zero sep_disj" shows "sep_disj x y = sep_disj y x"
check_Root_CTRL: shows "(Root_CTRL_States, Root_CTRL_Init, Root_CTRL_Labels, Root_CTRL_Delta) \<in> seqauto"
intro: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "ext_irrefl_before_trans ext"   and "ext_hd_or_tl ext" shows "ext_wf_bounded ext"
formulaEntailsClause_def: fixes formula :: "Literal list list"   and clause :: "Literal list" shows "formulaEntailsClause formula clause \<equiv> \<forall>valuation. model valuation formula \<longrightarrow> model valuation clause"
rel_cases: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "('d, 'f) nondetT"   and b :: "('e, 'g) nondetT" assumes "rel_nondetT' R1 R2 a b" obtains xa :: "'f" and ya :: "'g" where "a = NondetT xa" and "b = NondetT ya" and "R2 xa ya"
npemul_correct: fixes xs :: "'a list"   and e1 :: "pexpr"   and e2 :: "pexpr" assumes "in_carrier xs" shows "peval xs (npemul e1 e2) = peval xs (PExpr2 (PMul e1 e2))"
polyadd_commute: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p"
outside: shows "x \<notin> S \<or> s x \<noteq> 0"
vrestr_compose: fixes g :: "'a \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "g \<in> M \<rightarrow>\<^sub>M M" shows "(\<lambda>x. f (g x)) --` A = g --` f --` A"
substitute: shows "integral (cbox (pathstart \<gamma> \<bullet> i) (pathfinish \<gamma> \<bullet> i)) (\<lambda>f_var. F (f_var *\<^sub>R i + g f_var) \<bullet> i) = integral (cbox 0 1) (\<lambda>x. F (\<gamma> x) \<bullet> i * vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1}))"
complex_of_real_integrable_eq: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real" shows "complex_integrable M (\<lambda>x. complex_of_real (f x)) = integrable M f"
recurse_def: fixes f :: "'d \<Rightarrow> ('e, 'f) state" shows "recurse f = un_B \<circ> traverse_rev (B \<circ> f)"
prj_simps_1: shows "C.seq doms.\<mu>\<nu>.prj\<^sub>1 \<p>\<^sub>1[doms.\<nu>.leg0 \<cdot> doms.\<mu>\<nu>.prj\<^sub>0, doms.\<pi>.leg1]"
intros: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Bs :: "base list"   and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and D :: "char list" assumes "class P C = \<lfloor>(Bs, rest)\<rfloor>"   and "Repeats D \<in> set Bs" shows "subclsRp P C D"
fundamental_lemma: fixes bad1 :: "'a \<Rightarrow> bool"   and bad2 :: "'b \<Rightarrow> bool"   and A :: "'a \<Rightarrow> bool"   and B :: "'b \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'b spmf" assumes "rel_spmf (\<lambda>x y. bad1 x = bad2 y \<and> (\<not> bad2 y \<longrightarrow> A x = B y)) p q" shows "\<bar>Sigma_Algebra.measure (measure_spmf p) {x. A x} - Sigma_Algebra.measure (measure_spmf q) {y. B y}\<bar> \<le> Sigma_Algebra.measure (measure_spmf p) {x. bad1 x}"
homeomorphic_to_disc: assumes "S \<noteq> {}"   and "S = UNIV \<or> (\<exists>f g. f holomorphic_on S \<and> g holomorphic_on ball 0 1 \<and> (\<forall>z\<in>S. f z \<in> ball 0 1 \<and> g (f z) = z) \<and> (\<forall>z\<in>ball 0 1. g z \<in> S \<and> f (g z) = z))" shows "S homeomorphic ball 0 1"
isomorphic_relative_homology_by_contractible: fixes X :: "'b topology"   and S :: "'b set"   and p :: "int" assumes "contractible_space (subtopology X S)"   and "topspace X \<inter> S \<noteq> {}" shows "relative_homology_group p X S \<cong> reduced_homology_group p X"
lens_get_put_quasi_commute: fixes Y :: "'d \<Longrightarrow> 'e"   and X :: "'f \<Longrightarrow> 'e"   and s :: "'e"   and v :: "'f" assumes "vwb_lens Y"   and "X \<subseteq>\<^sub>L Y" shows "get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> s v) = put\<^bsub>X /\<^sub>L Y\<^esub> (get\<^bsub>Y\<^esub> s) v"
transaction_strand_subsets_10: fixes T :: "('e, 'f, 'g, 'h) prot_transaction" shows "set (unlabel (transaction_send T)) \<subseteq> set (unlabel (transaction_strand T))"
Ind: fixes j :: "name"   and i :: "name"   and A :: "fm"   and H :: "fm set" assumes "atom j \<sharp> (i, A)"   and "H \<turnstile> A(i::=Zero)"   and "H \<turnstile> SyntaxN.All i (SyntaxN.All j (A IMP A(i::=Var j) IMP A(i::=Eats (Var i) (Var j))))" shows "H \<turnstile> A"
set_integral_discrete_difference: fixes X :: "'a set"   and A :: "'a set"   and B :: "'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" assumes "countable X"   and "A - B \<union> (B - A) \<subseteq> X"   and "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"   and "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M" shows "set_lebesgue_integral M A f = set_lebesgue_integral M B f"
dining_transitions_def: fixes n :: "nat" shows "Tr\<^sub>D n = (\<lambda>(ps, fs). Tr\<^sub>P n ps \<inter> Tr\<^sub>F n fs)"
const_poly_dvd_iff: fixes c :: "'b"   and p :: "'b poly" shows "([:c:] dvd p) = (\<forall>n. c dvd coeff p n)"
rreq_def: shows "rreq \<equiv> \<lambda>(hops, dip, dsn, dsk, oip, osn, sip). Rreq hops dip dsn dsk oip osn sip"
induct: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and x1 :: "'c rule list"   and x2 :: "state"   and x3 :: "state"   and P :: "'c rule list \<Rightarrow> state \<Rightarrow> state \<Rightarrow> bool" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>x1, x2\<rangle> \<Rightarrow>\<^sub>s x3"   and "\<And>t. P [] t t"   and "\<And>m rs. matches \<gamma> m p \<Longrightarrow> P (Rule m Accept # rs) Undecided (Decision FinalAllow)"   and "\<And>m rs. matches \<gamma> m p \<Longrightarrow> P (Rule m Drop # rs) Undecided (Decision FinalDeny)"   and "\<And>m rs. matches \<gamma> m p \<Longrightarrow> P (Rule m Reject # rs) Undecided (Decision FinalDeny)"   and "\<And>m rs t. \<lbrakk>matches \<gamma> m p; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>s t; P rs Undecided t\<rbrakk> \<Longrightarrow> P (Rule m Log # rs) Undecided t"   and "\<And>m rs t. \<lbrakk>matches \<gamma> m p; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>s t; P rs Undecided t\<rbrakk> \<Longrightarrow> P (Rule m Empty # rs) Undecided t"   and "\<And>m rs t a. \<lbrakk>\<not> matches \<gamma> m p; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>s t; P rs Undecided t\<rbrakk> \<Longrightarrow> P (Rule m a # rs) Undecided t"   and "\<And>m chain rs\<^sub>1 m' rs\<^sub>2 rrs t. \<lbrakk>matches \<gamma> m p; \<Gamma> chain = Some (rs\<^sub>1 @ Rule m' Return # rs\<^sub>2); matches \<gamma> m' p; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow>\<^sub>s Undecided; P rs\<^sub>1 Undecided Undecided; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rrs, Undecided\<rangle> \<Rightarrow>\<^sub>s t; P rrs Undecided t\<rbrakk> \<Longrightarrow> P (Rule m (Call chain) # rrs) Undecided t"   and "\<And>m chain rs X rrs. \<lbrakk>matches \<gamma> m p; \<Gamma> chain = Some rs; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>s Decision X; P rs Undecided (Decision X)\<rbrakk> \<Longrightarrow> P (Rule m (Call chain) # rrs) Undecided (Decision X)"   and "\<And>m chain rs rrs t. \<lbrakk>matches \<gamma> m p; \<Gamma> chain = Some rs; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>s Undecided; P rs Undecided Undecided; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rrs, Undecided\<rangle> \<Rightarrow>\<^sub>s t; P rrs Undecided t\<rbrakk> \<Longrightarrow> P (Rule m (Call chain) # rrs) Undecided t" shows "P x1 x2 x3"
owner_document: shows "h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr) \<rightarrow>\<^sub>r owner_document"
vsv_vconverse: shows "vsv ((\<FF>\<lparr>ArrMap\<rparr>)\<inverse>\<^sub>\<circ>)"
simps_1: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" shows "foldr f [] = id"
carA: shows "gso ` {0..<i} \<subseteq> carrier_vec n"
inverse_transformation_axioms: shows "inverse_transformation (\<cdot>) (\<cdot>) (\<lambda>f. tensor f unity) local.map \<rho>"
f_Exec_Stream_length: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and c :: "'d" shows "length (f_Exec_Comp_Stream trans_fun xs c) = length xs"
in_borel_measurable_borel: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" shows "(f \<in> borel_measurable M) = (\<forall>S\<in>sets borel. f -` S \<inter> space M \<in> sets M)"
rtos_rs_Tr1: fixes r :: "nat"   and s :: "nat" assumes "0 < r"   and "0 < s" shows "rtos r s (r * s) = r * s"
better_T_TApp: fixes \<Gamma> :: "binding list"   and t\<^sub>1 :: "trm"   and X :: "tyvrs"   and T12 :: "ty"   and T11 :: "ty"   and T2 :: "ty" assumes "\<Gamma> \<turnstile> t\<^sub>1 : (\<forall>X<:T11. T12)"   and "\<Gamma>\<turnstile>T2<:T11" shows "\<Gamma> \<turnstile> t\<^sub>1 \<cdot>\<^sub>\<tau> T2 : T12[X \<mapsto> T2]\<^sub>\<tau>"
simps_2: fixes r0 :: "nat"   and r1 :: "nat"   and m :: "(nat \<times> nat) convnet" shows "output_size (Conv (r0, r1) m) = r0"
content_cbox_cases: fixes a :: "'a"   and b :: "'a" shows "content (cbox a b) = (if \<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i then \<Prod>i\<in>Basis. b \<bullet> i - a \<bullet> i else 0)"
inv_def: fixes f :: "'a" shows "local.inv f = (SOME g. inverse_arrows f g)"
sho: shows "p \<in> SHOs (round (rho n) q) q"
same_uptoI: fixes j :: "nat"   and A :: "'a mat"   and B :: "'a mat" assumes "\<And>i' j'. \<lbrakk>i' < n; j' < j\<rbrakk> \<Longrightarrow> A $$ (i', j') = B $$ (i', j')" shows "local.same_upto j A B"
zminus_zadd_cancel: fixes z :: "int"   and w :: "int" shows "- z + (z + w) = w"
assms_5: shows "v\<^sup>T = r\<^sup>T * t\<^sup>\<star>"
dH_indistinguishable_def: fixes d :: "'d"   and H :: "('d \<times> 'exp) set"   and e1 :: "'exp"   and e2 :: "'exp" shows "e1 \<equiv>\<^bsub>d,H\<^esub> e2 \<equiv> \<forall>m m'. m \<sim>\<^bsub>d,H\<^esub> m' \<longrightarrow> E e1 m = E e2 m'"
a: fixes b :: "'a set" assumes "b \<in># remove1_mset bl \<B>"   and "ps \<subseteq> b" shows "ps \<subseteq> b - bl"
discI: fixes nat :: "nat"   and x2 :: "nat" assumes "nat = Suc x2" shows "nat \<noteq> 0"
execn_merge_guards_to_execn_Normal: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>merge_guards c,Normal s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> t"
distinct_21: fixes x21 :: "POPLmarkRecord.type"   and x22 :: "trm"   and x6 :: "(char list \<times> trm) list" shows "(\<lambda>:x21. x22) \<noteq> Rcd x6"
Q''Chain: shows "Q''[x::=u] \<Longrightarrow>\<^sub>\<tau> Q'"
plen_mono: fixes \<pi> :: "'a prefix"   and \<pi>' :: "'a prefix" assumes "\<pi> \<le> \<pi>'" shows "plen \<pi> \<le> plen \<pi>'"
assms_2: fixes x :: "'fun" shows "rel_option (\<lambda>ar fd. arity fd = ar) (\<A> x) (Finca_get F1 x)"
sr: shows "complex_of_real (HMA_Connect.spectral_radius A) = complex_of_real ev"
parPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and R :: "pi" assumes "P \<leadsto>[Rel] Q"   and "(P, Q) \<in> Rel"   and "\<And>S T U. (S, T) \<in> Rel \<Longrightarrow> (S \<parallel> U, T \<parallel> U) \<in> Rel'"   and "\<And>S T x. (S, T) \<in> Rel' \<Longrightarrow> (<\<nu>x>S, <\<nu>x>T) \<in> Rel'" shows "P \<parallel> R \<leadsto>[Rel'] Q \<parallel> R"
lift_assn_def: fixes R :: "('c \<times> 'd) set"   and \<Phi> :: "'d \<Rightarrow> bool"   and s :: "'c" shows "Autoref_Misc.lift_assn R \<Phi> s \<equiv> \<exists>s'. (s, s') \<in> R \<and> \<Phi> s'"
Id: shows "id\<^bsub>CatExp (Op C) SET\<^esub> (Hom\<^bsub>C\<^esub>[\<midarrow>,X]) \<in> mor\<^bsub>CatExp (Op C) SET\<^esub>"
\<R>_to_\<R>\<^sub>3: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> Sec option"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>u\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   and "\<turnstile> \<Gamma> {c} \<Gamma>'" shows "\<langle>c\<^sub>1 ;; c, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>'\<^esub> \<langle>c\<^sub>2 ;; c, mds, mem\<^sub>2\<rangle>"
index_map_mat_3: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d mat" shows "dim_col (map_mat f A) = dim_col A"
quasi_inverse_antipar_2: fixes f :: "'a" assumes "equivalence_map f" shows "trg f\<^sup>~ = src f"
starfun_shift_one: fixes f :: "nat \<Rightarrow> 'a"   and N :: "nat star" shows "(*f* (\<lambda>n. f (Suc n))) N = (*f* f) (N + 1)"
K': shows "K' \<equiv> {(x, y). rel_set T x y} `` K"
continuous_right_inverse_imp_quotient_map: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set"   and g :: "'b \<Rightarrow> 'a"   and U :: "'b set" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "continuous_on T g"   and "g ` T \<subseteq> S"   and "\<And>y. y \<in> T \<Longrightarrow> f (g y) = y"   and "U \<subseteq> T" shows "openin (top_of_set S) (S \<inter> f -` U) = openin (top_of_set T) U"
down: fixes Y :: "V set"   and x :: "V" assumes "Y \<subseteq> elts x" shows "small Y"
expectation_left_gpv: fixes fail :: "ennreal"   and \<I> :: "('a, 'c) \<I>"   and \<I>' :: "('b, 'd) \<I>"   and f :: "'e \<Rightarrow> ennreal"   and gpv :: "('e, 'a, 'c) gpv" shows "expectation_gpv fail (\<I> \<oplus>\<^sub>\<I> \<I>') f (left_gpv gpv) = expectation_gpv fail \<I> f gpv"
coll_add_trans: fixes x :: "real \<times> real"   and y :: "real \<times> real"   and z :: "real \<times> real" assumes "coll 0 x (y + z)"   and "coll 0 y z"   and "x \<noteq> 0"   and "y \<noteq> 0"   and "z \<noteq> 0"   and "y + z \<noteq> 0" shows "coll 0 x z"
conflictFlagIsDecreasedByConflict_1: fixes stateA :: "State"   and stateB :: "State" assumes "appliedConflict stateA stateB" shows "getM stateA = getM stateB"
trg_dom: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "trg (local.dom \<mu>) = trg \<mu>"
map_simp: fixes f :: "'a arrow_of_spans_data" assumes "V.arr f" shows "V.map f = f"
f_join_nth: fixes n :: "nat"   and xs :: "'b list"   and I :: "nat set" assumes "n < length (xs \<Join>\<^sub> I)" shows "xs \<Join>\<^sub> I ! n = xs ! (I \<rightarrow> n)"
inter_subset_eq_union: fixes A :: "'b multiset"   and B :: "'b multiset" shows "A \<inter># B \<subseteq># A + B"
iwlsFSw_iterAll_termFSwImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSw MOD" shows "termFSwImorph (iter MOD) (iterAbs MOD) MOD"
exhaust_sel: fixes prod :: "'b \<times> 'c" obtains "prod = (fst prod, snd prod)"
disjoint_add_mset_2: fixes A :: "'b multiset"   and b :: "'b"   and B :: "'b multiset" shows "({#} = A \<inter># add_mset b B) = (b \<notin># A \<and> {#} = A \<inter># B)"
filtermap_exp_at_top: shows "filtermap exp at_top = at_top"
compute_bitlen: fixes x :: "int" shows "bitlen x = (if 0 < x then bitlen (x div 2) + 1 else 0)"
ladder_i_L'_thm: fixes index :: "nat" assumes "index < length L" shows "ladder_i L' index + length \<delta> = ladder_i L index"
distr_marginal2: fixes S :: "'d measure"   and T :: "'e measure"   and X :: "'a \<Rightarrow> 'd"   and Y :: "'a \<Rightarrow> 'e"   and Pxy :: "'d \<times> 'e \<Rightarrow> ennreal" assumes "sigma_finite_measure S"   and "sigma_finite_measure T"   and "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" shows "distributed M T Y (\<lambda>y. \<integral>\<^sup>+ x. Pxy (x, y) \<partial>S)"
Union_WWa_sub_carrier: fixes g :: "'a \<Rightarrow> 'a"   and a :: "'a" shows "\<Union> (WWa D g a) \<subseteq> carrier D"
summable_complex_powr_iff: fixes s :: "complex" assumes "Re s < - 1" shows "summable (\<lambda>n. exp (complex_of_real (ln (real n)) * s))"
it_to_it_fold': fixes it :: "'c \<Rightarrow> ('d list \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'd list \<Rightarrow> 'd list) \<Rightarrow> 'd list \<Rightarrow> 'd list"   and it' :: "'c \<Rightarrow> ('e \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'c" assumes "proper_it' it it'" shows "it_to_it (it s) = it' s"
bar3: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and xs :: "((int \<times> int) set set, int) Combinators list"   and xa :: "((int \<times> int) set set, int) Combinators" assumes "x \<in> dom (C (list2FWpolicy (xs @ [xa])))" shows "x \<in> dom (C (list2FWpolicy xs)) \<or> x \<in> dom (C xa)"
mkM_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'e Trace set" assumes "PreEnvironmentJView jkbp jview" shows "PreEnvironmentJView.mkM envVal jview T \<equiv> \<lparr>worlds = T, relations = \<lambda>a. rel_ext (\<lambda>(t, t'). {t, t'} \<subseteq> T \<and> jview a t = jview a t'), valuation = envVal \<circ> tLast\<rparr>"