hcju/isabelleps huggingface dataset:
fold_congs_2: fixes r :: "('d, 'e, 'f) automaton_scheme"   and r' :: "('d, 'e, 'f) automaton_scheme"   and v' :: "('d \<times> 'e \<times> 'd) set"   and f :: "('d \<times> 'e \<times> 'd) set \<Rightarrow> ('d \<times> 'e \<times> 'd) set"   and f' :: "('d \<times> 'e \<times> 'd) set \<Rightarrow> ('d \<times> 'e \<times> 'd) set" assumes "r = r'"   and "automaton.trans r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "trans_update f r = trans_update f' r'"
circ_simulate_right: fixes z :: "'a"   and x :: "'a"   and y :: "'a"   and w :: "'a" shows "z * x \<le> y * z \<squnion> w \<longrightarrow> z * x\<^sup>\<circ> \<le> y\<^sup>\<circ> * (z \<squnion> w * x\<^sup>\<circ>)"
icrit_spec_component_crit: shows "punit.icrit_spec punit.component_crit"
le_infI: fixes x :: "'b multiset"   and a :: "'b multiset"   and b :: "'b multiset" assumes "x \<subseteq># a"   and "x \<subseteq># b" shows "x \<subseteq># a \<inter># b"
diagseq_holds: fixes k :: "nat" assumes "\<And>r s n. \<lbrakk>strict_mono r; convergent (\<lambda>na. f (s na) n)\<rbrakk> \<Longrightarrow> convergent (\<lambda>na. f ((s \<circ> r) na) n)" shows "convergent (\<lambda>n. f ((diagseq \<circ> (+) (Suc k)) n) k)"
lappend_ltakeWhile_ldropWhile: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b llist" shows "lappend (ltakeWhile P xs) (ldropWhile P xs) = xs"
right_idem: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "semilattice f" shows "f (f a b) b = f a b"
hsplit_def: fixes c :: "hf \<Rightarrow> hf \<Rightarrow> 'a" shows "hsplit c \<equiv> \<lambda>p. c (hfst p) (hsnd p)"
monom_var_in_vars: fixes l :: "linear_poly" assumes "is_monom l" shows "monom_var l \<in> vars l"
foldl_LTLAnd_prop_entailment: fixes S :: "'c ltl set"   and i :: "'c ltl"   and xs :: "'c ltl list" shows "S \<Turnstile>\<^sub>P foldl LTLAnd i xs = (S \<Turnstile>\<^sub>P i \<and> (\<forall>y\<in>set xs. S \<Turnstile>\<^sub>P y))"
vconn_implies_hpar_1: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<rightarrow> g\<guillemotright>" shows "src \<mu> = src f"
is_solution_in_cone: fixes t :: "real"   and x :: "real \<Rightarrow> 'a"   and Y :: "'a set" assumes "t \<in> T"   and "(x solves_ode f) {t0--t} Y"   and "x t0 = x0" shows "x t \<in> cball x0 (B * \<bar>t - t0\<bar>)"
Abs_measure_inverse: fixes y :: "'b set \<times> 'b set set \<times> ('b set \<Rightarrow> ennreal)" assumes "y \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>- A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>}" shows "Rep_measure (Abs_measure y) = y"
check_eqv_def: fixes r :: "nat rexp"   and s :: "nat rexp" shows "check_eqv r s = (let nr = norm r; ns = norm s; as = add_atoms nr (add_atoms ns []) in case closure as (nr, ns) of None \<Rightarrow> False | Some ([], x) \<Rightarrow> True | Some (a # list, x) \<Rightarrow> False)"
relcomppE: fixes r :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and s :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and a :: "'g"   and c :: "'i" assumes "(r OO s) a c" obtains b :: "'h" where "r a b" and "s b c"
mkCone_def: fixes p0 :: "'c"   and p1 :: "'c" shows "D.mkCone p0 p1 \<equiv> \<lambda>j. if j = J.MkIde False then p0 else if j = J.MkIde True then p1 else C.null"
Spectrum_def: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "Spectrum A = Collect (eigen_value A)"
comm_monoid_add_hom_axioms: fixes d :: "nat" shows "comm_monoid_add_hom (\<lambda>x. monom x d)"
l7_16: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and P :: "'a"   and P' :: "'a"   and Q :: "'a"   and Q' :: "'a"   and R :: "'a"   and R' :: "'a"   and S :: "'a"   and S' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A P P'"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A Q Q'"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A R R'"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A S S'"   and "Cong P Q R S" shows "Cong P' Q' R' S'"
vector_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "vector x"   and "vector y" shows "vector (x \<sqinter> y)"
Cos: shows "((\<lambda>x. cos (c x)) expands_to Cos) basis"
monom_eq_const_iff: fixes c :: "'b"   and n :: "nat"   and d :: "'b" shows "(monom c n = [:d:]) = (c = d \<and> (c = (0::'b) \<or> n = 0))"
assms_1: shows "i cd\<^bsup>\<pi>\<^esup>\<rightarrow> l"
wI_1: shows "x = map \<pi> (enc (w, I))"
rfb_rbd_galois_var: fixes R :: "('d \<times> 'c) set"   and X :: "'d set"   and Y :: "'c set" shows "(bd\<^sub>\<R> R X \<subseteq> Y) = (X \<subseteq> fb\<^sub>\<R> R Y)"
small_trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<omega>[F](g)"   and "g \<in> \<omega>[F](h)" shows "f \<in> \<omega>[F](h)"
image_scaleR_ball: fixes c :: "real"   and a :: "'c"   and r :: "real" assumes "c \<noteq> 0" shows "(*\<^sub>R) c ` ball a r = ball (c *\<^sub>R a) (\<bar>c\<bar> *\<^sub>R r)"
wf_subtype: fixes \<Gamma> :: "binding list"   and P :: "POPLmarkRecord.type"   and Q :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> P <: Q" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f P \<and> \<Gamma> \<turnstile>\<^sub>w\<^sub>f Q"
heap_is_wellformed_disc_nodes_in_heap: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and node :: "(_) node_ptr" assumes "l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes"   and "heap_is_wellformed h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"   and "node \<in> set disc_nodes" shows "node |\<in>| node_ptr_kinds h"
sp_instr_op: fixes opinl :: "'opinl"   and op :: "'op"   and F :: "'a \<Rightarrow> 'b fundef option"   and \<Sigma> :: "Unboxed.type option list" assumes "\<DD>\<ee>\<II>\<nn>\<ll> opinl = op" shows "Subx.sp_instr F (Ubx.instr.IOp op) \<Sigma> = Subx.sp_instr F (Ubx.instr.IOpInl opinl) \<Sigma>"
If: fixes P :: "state \<Rightarrow> bool"   and b :: "state \<Rightarrow> bool"   and c1 :: "com"   and Q :: "state \<Rightarrow> bool"   and c2 :: "com" assumes "\<turnstile> {\<lambda>s. P s \<and> b s} c1 {Q}"   and "\<turnstile> {\<lambda>s. P s \<and> \<not> b s} c2 {Q}" shows "\<turnstile> {P} IF b THEN c1 ELSE c2 {Q}"
comp_same: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a" assumes "comparator comp" shows "comp x x = Eq"
demonic_def: fixes Q :: "'d \<Rightarrow> 'e"   and p :: "'e"   and s :: "'d" shows "[: Q :] p s = (Q s \<le> p)"
cindex_poly_def: fixes a :: "real"   and b :: "real"   and q :: "real poly"   and p :: "real poly" shows "cindex_poly a b q p \<equiv> \<Sum>x | poly p x = 0 \<and> a < x \<and> x < b. jump_poly q p x"
Q_\<pi>_consistent: fixes u :: "'v"   and d :: "nat"   and v :: "'v" assumes "Q u = enat d"   and "\<pi> u = Some v" shows "d = w {u, v}"
finite_Int: fixes F :: "'a set"   and G :: "'a set" assumes "finite F \<or> finite G" shows "finite (F \<inter> G)"
c_tl_aux2: fixes u :: "nat" assumes "1 < c_len u" shows "c_tl u = c_pair (c_len u - 2) (c_snd (c_snd (u - 1))) + 1"
prems: shows "Suc p \<le> Suc (Suc n)"
surjective_sum: fixes f :: "'e + 'f \<Rightarrow> 'g" shows "case_sum (\<lambda>x. f (Inl x)) (\<lambda>y. f (Inr y)) = f"
msg_fresh_1: fixes \<sigma> :: "nat \<Rightarrow> state"   and hops :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool" shows "msg_fresh \<sigma> (Rreq hops dip dsn dsk oip osn sip handled) = (1 \<le> osn \<and> (sip \<noteq> oip \<longrightarrow> oip \<in> kD (rt (\<sigma> sip)) \<and> osn \<le> nsqn (rt (\<sigma> sip)) oip \<and> (nsqn (rt (\<sigma> sip)) oip = osn \<longrightarrow> the (dhops (rt (\<sigma> sip)) oip) \<le> hops \<or> the (flag (rt (\<sigma> sip)) oip) = Aodv_Basic.inv)))"
countable_insert: fixes A :: "'c set"   and a :: "'c" assumes "countable A" shows "countable (insert a A)"
elims_2: fixes x :: "'b hp" assumes "php x" obtains xa :: "'b" and hs :: "'b hp list" where "x = Hp xa hs" and "\<forall>h\<in>set hs. Ball (set_hp h) ((\<le>) xa) \<and> php h"
class_isSubobj: 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" assumes "is_class P C" shows "is_subobj P (C, [C])"
list_all2_Nil2: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list" shows "list_all2 P xs [] = (xs = [])"
individually_rational_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "Singleton_for_d.individually_rational_on ds X = (Singleton_for_d.CD_on ds X = X \<and> Singleton_for_d.CH X = X)"
box_def: fixes a :: "'c"   and b :: "'c" shows "box a b = {x. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"
bot_xor: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<bottom> \<oplus> x = x"
g_1: shows "g \<in> o(\<lambda>x. ln (b * x) powr - (e / 2 + 1))"
onode_bigger_than_next: fixes i :: "nat"   and R\<^sub>i :: "nat set" shows "\<langle>i : optoy i \<langle>\<langle>\<^bsub>i\<^esub> qmsg : R\<^sub>i\<rangle>\<^sub>o \<Turnstile> (otherwith nos_inc {i} (oarrivemsg msg_ok), other nos_inc {i} \<rightarrow>) global (\<lambda>\<sigma>. no (\<sigma> i) \<le> no (\<sigma> (nhid (\<sigma> i))))"
varsC_def: fixes c :: "('a, 'b) lit list" shows "varsC c = \<Union> (set (map varsL c))"
simps_1: shows "Eg2_var\<^sub>C_of_Eg1 control_var = control_var\<^sub>C"
prems: shows "x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (f # F)"
identity_arrow_of_spans_axioms: shows "identity_arrow_of_spans (\<cdot>) (r \<star> f)"
fact1: shows "(0, {#}) \<in> Mf ` Collect (factorization_m f)"
exhaust: fixes y :: "nat" obtains "y = 0"   | x2 :: "nat" where "y = Suc x2"
split_snoc_op: fixes pat :: "[:'r:]"   and xs :: "[:'r:]"   and x :: "'r" shows "split\<cdot>pat\<cdot>(xs :@ [:x:]) = op\<cdot>pat\<cdot>(split\<cdot>pat\<cdot>xs)\<cdot>x"
assms_1: shows "h \<in> set (f4_red_aux bs ps)"
normal_vector_def: fixes W :: "'a vec set" shows "normal_vector W = (let ws = SOME ws. set ws = W \<and> distinct ws; m = length ws; B = \<lambda>j. mat m m (\<lambda>(i, j'). ws ! i $ (if j' < j then j' else Suc j')) in vec n (\<lambda>j. (- (1::'a)) ^ (m + j) * det (B j)))"
homotopic_with_mono: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and Q :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" assumes "homotopic_with P X Y f g"   and "\<And>h. \<lbrakk>continuous_map X Y h; P h\<rbrakk> \<Longrightarrow> Q h" shows "homotopic_with Q X Y f g"
set_times_mono4: fixes a :: "'a"   and C :: "'a set"   and D :: "'a set" assumes "a \<in> C" shows "a *o D \<subseteq> D * C"
enter_subresultant_prs: fixes G2 :: "'b poly"   and G1 :: "'b poly" assumes "length (coeffs G2) \<le> length (coeffs G1)"   and "G2 \<noteq> 0" shows "\<exists>F n d f k b. subresultant_prs_locale2 F n d f k b G1 G2"
elims_2: fixes x :: "'d mat \<times> 'd mat \<times> 'd mat"   and xa :: "(('d, 'e) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec \<times> (('d, 'f) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec \<times> (('d, 'f) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" assumes "HMA_M3 x xa" obtains P :: "'d mat" and A :: "'d mat" and Q :: "'d mat" and P' :: "(('d, 'e) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" and A' :: "(('d, 'f) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" and Q' :: "(('d, 'f) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" where "x = (P, A, Q)" and "xa = (P', A', Q')" and "Mod_Type_Connect.HMA_M P P' \<and> Mod_Type_Connect.HMA_M A A' \<and> Mod_Type_Connect.HMA_M Q Q'"
lhss_def: fixes ES :: "('b \<times> 'c) set" shows "lhss ES \<equiv> {uu_. \<exists>Y yrhs. uu_ = Y \<and> (Y, yrhs) \<in> ES}"
wf_rexp_of'': fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n \<phi>" shows "M2L_Equivalence_Checking.wf enum_class.enum n (rexp_of'' n \<phi>)"
gf_app: assumes "1\<^sub>\<nat> \<in>\<^sub>\<circ> 2\<^sub>\<nat>" shows "gf\<lparr>1\<^sub>\<nat>\<rparr> \<in>\<^sub>\<circ> (if 1\<^sub>\<nat> = []\<^sub>\<circ> \<Rightarrow> \<AA> | otherwise \<Rightarrow> \<BB>)\<lparr>Arr\<rparr>"
relevant_entries_def: fixes P :: "'a prog"   and i :: "instr"   and pc :: "nat" shows "relevant_entries P i pc = filter (is_relevant_entry P i pc)"
structure_is_trivial: shows "T\<^sub>DoFF.map = FoT\<^sub>C.map"
execlass_les_def: fixes cs :: "('a \<times> 'a) list"   and c1 :: "'a"   and c2 :: "'a" shows "execlass_les cs c1 c2 = (List.member cs (c1, c2) \<and> \<not> List.member cs (c2, c1))"
fimage_iff: fixes z :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c fset" shows "(z |\<in>| f |`| A) = (\<exists>x|\<in>|A. z = f x)"
Confidentiality_Tgs_bis: fixes A :: "agent"   and Ka :: "nat"   and authK :: "nat"   and Ta :: "nat"   and authTicket :: "msg"   and evs :: "event list"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg" assumes "Says Kas A (Crypt Ka \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace>) \<in> set evs"   and "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "\<not> expiredAK Ta evs"   and "\<not> expiredSK Ts evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV" shows "Key servK \<notin> analz (knows Spy evs)"
integral_spike_set: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "negligible {x \<in> S - T. f x \<noteq> (0::'b)}"   and "negligible {x \<in> T - S. f x \<noteq> (0::'b)}" shows "integral S f = integral T f"
strict_weak_trans: fixes a :: "'b pmf"   and b :: "'b pmf"   and c :: "'b pmf" assumes "a \<prec>[Pareto (SD \<circ> R)] b"   and "Pareto (SD \<circ> R) b c" shows "a \<prec>[Pareto (SD \<circ> R)] c"
simps_1: fixes x11 :: "agent"   and x12 :: "agent"   and y11 :: "agent"   and y12 :: "agent" shows "(sharK x11 x12 = sharK y11 y12) = (x11 = y11 \<and> x12 = y12)"
index_of_length: fixes x :: "'c"   and y :: "'c list" shows "index_of x y \<le> length y"
d_surjective_iff: fixes x :: "'a" shows "test x = (\<exists>y. d y = x)"
af_U: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl"   and x :: "'a set"   and xs :: "'a set list" shows "af (\<phi> U \<psi>) (x # xs) = af (\<phi> U \<psi>) xs and af \<phi> (x # xs) or af \<psi> (x # xs)"
path_connectedin_subtopology: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "path_connectedin (subtopology X S) T = (path_connectedin X T \<and> T \<subseteq> S)"
mem_carrier: fixes H :: "'f set"   and G :: "('f, 'g) monoid_scheme"   and x :: "'f" assumes "subgroup H G"   and "x \<in> H" shows "x \<in> carrier G"
2: shows "t \<in> F \<and> u \<in> G \<and> y = t \<squnion> u"
seqE: fixes g :: "'s"   and f :: "'s" assumes "seq g f" obtains "arr f" and "arr g" and "local.dom g = cod f"
sum_div_reduce: fixes d :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> complex" assumes "d dvd k"   and "0 < d" shows "(\<Sum>n | n \<in> {1..k} \<and> d dvd n. f n) = (\<Sum>c = 1..k div d. f (c * d))"
invertible_mat_zero: fixes A :: "'b mat" assumes "A \<in> carrier_mat 0 0" shows "invertible_mat A"
terminates_Normal_elim_cases_11: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile>Catch c1 c2 \<down> Normal s" obtains "\<Gamma>\<turnstile>c1 \<down> Normal s" and "\<forall>s'. \<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow> Abrupt s' \<longrightarrow> \<Gamma>\<turnstile>c2 \<down> Normal s'"
u_4: assumes "g \<noteq> h" shows "is_Abs g \<and> is_Abs h \<and> (the_Abs g, the_Abs h) \<in> set TI"
trace_in_unique: fixes y1 :: "'a"   and t :: "real"   and x :: "'a"   and \<omega> :: "(real \<times> 'a) stream"   and y2 :: "'a" assumes "trace_in {y1} t x \<omega>"   and "trace_in {y2} t x \<omega>" shows "y1 = y2"
llist_all2_coinduct: fixes X :: "'c llist \<Rightarrow> 'd llist \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "X xs ys"   and "\<And>xs ys. X xs ys \<Longrightarrow> lnull xs = lnull ys"   and "\<And>xs ys. \<lbrakk>X xs ys; \<not> lnull xs; \<not> lnull ys\<rbrakk> \<Longrightarrow> P (lhd xs) (lhd ys) \<and> (X (ltl xs) (ltl ys) \<or> llist_all2 P (ltl xs) (ltl ys))" shows "llist_all2 P xs ys"
Diff_subset: fixes A :: "'b set"   and B :: "'b set" shows "A - B \<subseteq> A"
wfactors_ee_cong_l: fixes G :: "('c, 'd) monoid_scheme"   and as :: "'c list"   and bs :: "'c list"   and b :: "'c" assumes "comm_monoid_cancel G"   and "essentially_equal G as bs"   and "wfactors G bs b"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "wfactors G as b"
proj_def: fixes r :: "('e \<times> 'd) set"   and x :: "'e" shows "Equiv_Relations.proj r x = r `` {x}"
SUP_least: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'd"   and u :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) u" shows "less_eq (Sup (f ` A)) u"
a_comm: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.antidomain_near_semiring antidomain_op plus times one zero less_eq less" shows "times (antidomain_op x) (antidomain_op y) = times (antidomain_op y) (antidomain_op x)"
is_interval_tree_def: fixes t :: "interval_tree" shows "is_interval_tree t = (tree_sorted t \<and> tree_max_inv t \<and> tree_interval_inv t)"
cInf_lower: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> X"   and "bdd_below X" shows "Inf X \<le> x"
take_bit_add: fixes divide :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and modulo :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and euclidean_size :: "'k \<Rightarrow> nat"   and division_segment :: "'k \<Rightarrow> 'k"   and bit :: "'k \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and drop_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and take_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and n :: "nat"   and a :: "'k"   and b :: "'k" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "take_bit n (plus (take_bit n a) (take_bit n b)) = take_bit n (plus a b)"
eventually_comp_filtermap: fixes P :: "'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "eventually (P \<circ> f) F = eventually P (filtermap f F)"
even_signed_take_bit_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and m :: "nat"   and a :: "'c" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not m a) = dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a"
cl_mono: fixes X :: "'a set"   and Y :: "'a set" assumes "X \<subseteq> Y"   and "Y \<subseteq> carrier" shows "cl X \<subseteq> cl Y"
uminus_ms_aux: fixes basis :: "(real \<Rightarrow> real) list"   and F :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" assumes "basis_wf basis"   and "is_expansion_aux F f basis" shows "is_expansion_aux (uminus_ms_aux F) (\<lambda>x. - f x) basis"
norm_not_less_zero: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "\<not> norm x < 0"
ccompatible4: fixes c :: "real"   and k :: "'d \<Rightarrow> nat"   and x :: "'d"   and X :: "'d set" assumes "c \<le> real (k x)"   and "c \<in> \<nat>"   and "x \<in> X" shows "ccompatible {region X I r |I r. valid_region X k I r} (GT x c)"
DFT_upper: fixes m :: "nat"   and i :: "nat"   and a :: "nat \<Rightarrow> complex" assumes "0 < m"   and "m \<le> i" shows "DFT (2 * m) a i = DFT m (\<lambda>i. a (2 * i)) (i - m) - FFT.root (2 * m) ^ (i - m) * DFT m (\<lambda>i. a (2 * i + 1)) (i - m)"
eheight_rbst: fixes A :: "'a set" assumes "finite A" shows "eheight_rbst (card A) = map_pmf eheight (random_bst A)"
Mg: shows "set_borel_measurable borel {a..b} g"
partial_equivalence_axioms: shows "partial_equivalence A (\<sim>)"
sorted_wrt_alls: fixes B :: "nat"   and xs :: "nat list" shows "sorted_wrt (<\<^sub>r\<^sub>l\<^sub>e\<^sub>x) (map fst (alls B xs))"
eq_mod_dvd_minus: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<noteq> (0::'a)"   and "a mod c = b mod c" shows "c dvd a - b"
numbound0_I: fixes a :: "Cooper.num"   and b :: "int"   and bs :: "int list"   and b' :: "int" assumes "numbound0 a" shows "Inum (b # bs) a = Inum (b' # bs) a"
simps: fixes a1 :: "('Var, 'AExp, 'BExp) Stmt"   and a2 :: "Mode \<Rightarrow> 'Var set"   and a3 :: "'Var \<Rightarrow> 'Val"   and a4 :: "('Var, 'AExp, 'BExp) Stmt"   and a5 :: "Mode \<Rightarrow> 'Var set"   and a6 :: "'Var \<Rightarrow> 'Val"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option" shows "\<langle>a1, a2, a3\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>'\<^esub> \<langle>a4, a5, a6\<rangle> = (\<exists>\<Gamma> c mds mem\<^sub>1 mem\<^sub>2. a1 = c \<and> a2 = mds \<and> a3 = mem\<^sub>1 \<and> a4 = c \<and> a5 = mds \<and> a6 = mem\<^sub>2 \<and> \<turnstile> \<Gamma> {c} \<Gamma>' \<and> mds_consistent mds \<Gamma> \<and> mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2)"