hcju/isabelleps huggingface dataset:
unfold2: fixes s1 :: "'d \<Rightarrow> ('c, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('c, 'd, 'e) F2"   and z :: "'e" shows "dtor2 (unfold2 s1 s2 z) = F2map id (unfold1 s1 s2) (unfold2 s1 s2) (s2 z)"
obtain_v1_v2_3: shows "disjnt (\<Union> S) vs"
solved_def: shows "solved = (\<forall>c. c |\<in>| conc_forms \<longrightarrow> (\<exists>\<Gamma> t. fst (tree.root t) = (\<Gamma> \<turnstile> c) \<and> \<Gamma> |\<subseteq>| ass_forms \<and> local.wf t \<and> tfinite t))"
abs_eq: fixes x :: "\<upsilon> \<Rightarrow> \<upsilon> \<Rightarrow> j \<Rightarrow> i \<Rightarrow> bool" shows "d\<^sub>2 (make\<Pi>\<^sub>2 x) = Some x"
upper_triangular_def: fixes A :: "'a mat" shows "upper_triangular A \<equiv> \<forall>i<dim_row A. \<forall>j<i. A $$ (i, j) = (0::'a)"
not_le_imp_less: fixes x :: "'b"   and y :: "'b" assumes "\<not> x \<le> y" shows "y < x"
invertible_left_inverse: fixes A :: "(('a, 'b) vec, 'b) vec" shows "invertible A = (\<exists>B. B ** A = mat (1::'a))"
r_2: shows "cball z r \<subseteq> s \<inter> g -` (- {0})"
mult_1s_3: fixes b :: "'a" shows "- Numeral1 * b = - b"
Greater_def: fixes \<theta> :: "('a, 'c) trm"   and \<theta>' :: "('a, 'c) trm" shows "Greater \<theta> \<theta>' = (Geq \<theta> \<theta>' && ! (Geq \<theta>' \<theta>))"
harm_less_ln: fixes m :: "nat" assumes "0 < m" shows "harm m < ln (real (2 * m + 1))"
between_inter_empty: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "pre_between vs ram1 ram2" shows "set (between vs ram1 ram2) \<inter> set (between vs ram2 ram1) = {}"
cobounded1: fixes a :: "'x"   and b :: "'x" shows "a \<le> sup a b"
simps: fixes x :: "'a"   and r :: "'a" shows "sqrtTest x r = ((0::'a) \<le> r \<and> r * r = x)"
i_take_length: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "length (f \<Down> n) = n"
vertical_composite_iso_inverse: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "natural_isomorphism A B F G \<tau>" shows "vertical_composite.map A B \<tau> (inverse_transformation.map A B G \<tau>) = F"
sub_of_denot_equivx: fixes E :: "char list \<Rightarrow> hterm"   and x :: "char list" shows "eval\<^sub>t E HFun (sub_of_denot E x) = E x"
mod_eq_0_iff_dvd: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "(modulo a b = zero) = dvd.dvd times b a"
simps_2: fixes l :: "'c tree"   and x :: "'c"   and r :: "'c tree" shows "mirror \<langle>l, x, r\<rangle> = \<langle>mirror r, x, mirror l\<rangle>"
orthogonal_transformation_scaleR: fixes f :: "'a \<Rightarrow> 'b"   and c :: "real"   and v :: "'a" assumes "orthogonal_transformation f" shows "f (c *\<^sub>R v) = c *\<^sub>R f v"
cong_gen_cong: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "local.cong (gen_cong R)"
compact_eq_Heine_Borel: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "class.topological_space open" shows "topological_space.compact open S = (\<forall>C. (\<forall>c\<in>C. open c) \<and> S \<subseteq> \<Union> C \<longrightarrow> (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union> D))"
iso_runit: fixes f :: "'a" assumes "C.ide f" shows "C.iso \<r>\<^sub>C[f]"
ext_inject: fixes static :: "bool"   and more :: "'a"   and static' :: "bool"   and more' :: "'a" shows "(\<lparr>static = static, \<dots> = more\<rparr> = \<lparr>static = static', \<dots> = more'\<rparr>) = (static = static' \<and> more = more')"
cinfinite_def: fixes r :: "('d \<times> 'd) set" shows "cinfinite r = infinite (Field r)"
dyn_def: fixes ts :: "traffic" shows "dyn ts \<equiv> fst (snd (snd (snd (Rep_traffic ts))))"
lookup_empty: fixes as :: "'d list" shows "lookup_tries (Trie None []) as = []"
fequalityE: fixes A :: "'c fset"   and B :: "'c fset" assumes "A = B" obtains "A |\<subseteq>| B" and "B |\<subseteq>| A"
density_cong: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure"   and f' :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "f' \<in> borel_measurable M"   and "AE x in M. f x = f' x" shows "density M f = density M f'"
abel_semigroup_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "abel_semigroup f \<equiv> semigroup f \<and> abel_semigroup_axioms f"
effect_refI: fixes r :: "'b ref"   and h' :: "heap"   and v :: "'b"   and h :: "heap" assumes "(r, h') = Ref.alloc v h" shows "effect (ref v) h h' r"
bin_last_numeral_simps_2: shows "\<not> is_unit 2"
WAi_meas: shows "induced_map A --` WA \<in> sets M"
synth_monotone: fixes c :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "c \<in> synth G"   and "G \<subseteq> H" shows "c \<in> synth H"
fgeq: fixes z :: "complex" assumes "z \<in> S" shows "f z + \<i> * g z = exp (h z)"
\<phi>_inverts_ext: fixes y :: "'d"   and f :: "'c"   and x :: "'c" assumes "D.ide y"   and "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>" shows "arrow_to_functor.is_ext (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G (F y) (\<eta>.map y) x (\<phi> y f) f"
ifs_1: shows "\<not> p - Suc 0 \<le> lo"
wf_ipath_epath: fixes t :: "('state \<times> 'rule) tree"   and steps :: "('state \<times> 'rule) stream" assumes "local.wf t"   and "ipath t steps" shows "epath steps"
type_copy_Abs_o_Rep: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a" assumes "type_definition Rep Abs UNIV" shows "Abs \<circ> Rep = id"
assms_2: shows "y \<in> set_mset M - set_mset N"
HCP_adv_def: fixes I :: "('a \<times> 'b) spmf"   and domain :: "'a \<Rightarrow> 'c set"   and range :: "'a \<Rightarrow> 'c set"   and F :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and F\<^sub>i\<^sub>n\<^sub>v :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'a \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> (bool \<times> 'a \<times> bool \<times> bool \<Rightarrow> bool spmf) \<Rightarrow> 'c \<Rightarrow> bool spmf"   and \<sigma> :: "bool"   and b\<^sub>\<sigma> :: "bool"   and D :: "bool \<times> 'a \<times> bool \<times> bool \<Rightarrow> bool spmf" assumes "etp I domain range F F\<^sub>i\<^sub>n\<^sub>v" shows "ETP.etp.HCP_adv I range F\<^sub>i\<^sub>n\<^sub>v B A \<sigma> b\<^sub>\<sigma> D = \<bar>spmf (ETP.etp.HCP_game I range F\<^sub>i\<^sub>n\<^sub>v B A \<sigma> b\<^sub>\<sigma> D) True - 1 / 2\<bar>"
BSIA_implies_IA: fixes \<rho> :: "'e V_rec \<Rightarrow> 'e set" assumes "BSIA \<rho> \<V> Tr\<^bsub>(induceES SES)\<^esub>" shows "IA \<rho> \<V> Tr\<^bsub>(induceES SES)\<^esub>"
b: shows "\<forall>x\<in>{x \<in> s. norm x = norm a}. norm (f b) \<le> norm (f x)"
splus_zero: fixes pair_of_term :: "'d \<Rightarrow> 'e \<times> 'f"   and term_of_pair :: "'e \<times> 'f \<Rightarrow> 'd"   and v :: "'d" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.splus pair_of_term term_of_pair (0::'e) v = v"
only_one_path: shows "\<forall>P\<in>\<P>. P = Q"
nodes_sup_def: fixes G :: "nat graph_rec" assumes "cfg_doms G" shows "nodes_sup = inter_sorted_rev"
Abs_lst_fcb2: fixes as :: "atom list"   and x :: "'b"   and bs :: "atom list"   and y :: "'b"   and f :: "atom list \<Rightarrow> 'b \<Rightarrow> 'd \<Rightarrow> 'c"   and c :: "'d" assumes "[as]lst. x = [bs]lst. y"   and "set as \<sharp>* f as x c"   and "set as \<sharp>* c"   and "set bs \<sharp>* c"   and "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> f as x c = f (p \<bullet> as) (p \<bullet> x) c"   and "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> f bs y c = f (p \<bullet> bs) (p \<bullet> y) c" shows "f as x c = f bs y c"
closed_scale: fixes x :: "'b"   and a :: "'a" assumes "x \<in> U\<^sub>M" shows "a *s x \<in> U\<^sub>M"
lemma2_6_2_set: fixes S :: "'a set"   and T :: "'a set"   and r :: "('a \<times> 'a) set" assumes "S \<subseteq> T" shows "r \<down>s S \<subseteq> r \<down>s T"
exhaust: fixes y :: "('c, 'd, 'e) state" obtains x1 :: "'c" and x2 :: "'d" and x3 :: "'e list" where "y = State x1 x2 x3"
mult_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<le> b"   and "c \<le> d"   and "(0::'a) \<le> b"   and "(0::'a) \<le> c" shows "a * c \<le> b * d"
indexed_pmult_def: fixes P :: "'d multiset \<Rightarrow> 'c multiset \<Rightarrow> 'a"   and i :: "'d" shows "L.indexed_pmult P i = (\<lambda>m. if i \<in># m then P (m - {#i#}) else \<zero>\<^bsub>L\<^esub>)"
wlp_def: fixes \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wlp \<pi> c Q s \<equiv> \<forall>t. \<pi>: (c, s) \<Rightarrow> t \<longrightarrow> Q t"
class_add_widens: fixes P :: "jvm_prog"   and Ts :: "ty list"   and Ts' :: "ty list"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P \<turnstile> Ts [\<le>] Ts'"   and "\<not> is_class P C" shows "class_add P (C, cdec) \<turnstile> Ts [\<le>] Ts'"
Stirling_2_2: fixes n :: "nat" shows "Stirling (Suc (Suc n)) (Suc (Suc 0)) = 2 ^ Suc n - 1"
comparator_of_def: fixes x :: "'b"   and y :: "'b" shows "comparator_of x y = (if x < y then Lt else if x = y then Eq else Gt)"
inner_sum_Basis: fixes i :: "'c" assumes "i \<in> Basis" shows "One \<bullet> i = 1"
span_base: fixes a :: "'d"   and S :: "'d set" assumes "a \<in> S" shows "a \<in> cspan S"
s7_no_marker: shows "\<forall>cid. Marker \<notin> set (msgs s7 cid)"
finite_polytope_facets: fixes S :: "'b set" assumes "polytope S" shows "finite {T. T facet_of S}"
HEndPhase1_maxBalInp: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and b :: "nat"   and D :: "Disk set"   and v :: "InputsOrNi"   and p :: "Proc" assumes "HEndPhase1 s s' q"   and "b \<in> \<Union> (range Ballot)"   and "D \<in> MajoritySet"   and "maxBalInp s b v"   and "\<forall>d\<in>D. b \<le> bal (disk s d p) \<and> (\<forall>q. phase s q = 1 \<and> b \<le> mbal (dblock s q) \<and> hasRead s q d p \<longrightarrow> (\<exists>br\<in>blocksRead s q d. b \<le> bal (block br)))"   and "Inv1 s"   and "Inv2a s"   and "Inv2b s" shows "maxBalInp s' b v"
dg_prod_2_Dom_vdomain: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>" shows "\<D>\<^sub>\<circ> ((\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Dom\<rparr>) = (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Arr\<rparr>"
prv_inj_suc_var: shows "prv (imp (eql (suc (Var xx)) (suc (Var yy))) (eql (Var xx) (Var yy)))"
in_face_cycle_setD: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and y :: "'d"   and x :: "'d" assumes "digraph_map G M"   and "y \<in> pre_digraph_map.face_cycle_set M x"   and "x \<in> arcs G" shows "y \<in> arcs G"
boundl_impl_def: fixes a :: "nat list"   and b :: "nat list"   and x :: "nat list"   and y :: "nat list" shows "boundl_impl a b x y = (\<forall>i<length a. x ! i \<le> max_x_impl a b y i)"
geodesic_quasi_isometric_image: fixes lambda :: "real"   and C :: "real"   and f :: "'b \<Rightarrow> 'c"   and G :: "'b set"   and x :: "'b"   and y :: "'b" assumes "lambda C -quasi_isometry f"   and "geodesic_segment_between G x y" shows "hausdorff_distance (f ` G) {f x--f y} \<le> 92 * lambda\<^sup>2 * (C + deltaG TYPE('c))"
l1_step2_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and KE :: "msg" shows "l1_step2 Rb A B KE \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> Rb \<notin> dom (progress s) \<and> guessed_frame Rb xpkE = Some KE \<and> (can_signal s A B \<longrightarrow> (\<exists>Ra. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> in_progress (progress s Ra) xpkE \<and> guessed_frame Ra xpkE = Some KE)) \<and> (Rb = test \<longrightarrow> NonceF (Rb $ sk) \<notin> synth (analz (ik s))) \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xpkE, xsk}), secret := {x. x = NonceF (Rb $ sk) \<and> Rb = test} \<union> secret s, l1_state.signals := if can_signal s A B then addSignal (l1_state.signals s) (Running A B \<langle>KE, NonceF (Rb $ sk)\<rangle>) else l1_state.signals s\<rparr>}"
pdevs_val_sum_list: fixes c :: "real"   and X :: "'a pdevs" shows "pdevs_val (\<lambda>_. c) X = c *\<^sub>R sum_list (map snd (list_of_pdevs X))"
VC_def: fixes ct :: "(char list \<times> nat \<times> unit list) list"   and P :: "'a" shows "Case_Labeling.VC ct P \<equiv> P"
someI2_ex: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" assumes "\<exists>a. P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (SOME x. P x)"
outs_stop_\<I>: fixes \<I> :: "('h, 'i) \<I>" shows "outs_\<I> (stop_\<I> \<I>) = outs_\<I> \<I>"
int_transE_nv2T: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and pol :: "'c \<Rightarrow> 'b \<Rightarrow> epol"   and grdOf :: "('a, 'b) lit list \<Rightarrow> ('a, 'b) lit \<Rightarrow> Preliminaries.var \<Rightarrow> ('a, 'b) lit"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and T :: "'a trm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "ModelIkPolMcalc2C wtFsym wtPsym arOf resOf parOf \<Phi> infTp pol grdOf intT intF intP"   and "Signature.wt wtFsym arOf resOf T"   and "Struct.wtE (ModelIkPolMcalc2C.I_intT infTp intT) \<xi>"   and "infTp (Signature.tpOf resOf T) \<or> (\<forall>x\<in>ProblemIkPol.nv2T T. Signature.tpOfV x \<noteq> Signature.tpOf resOf T)" shows "Struct.int intF (ModelIkPolMcalc2C.transE intT \<xi>) T = Struct.int (ModelIkPolMcalc2C.I_intF arOf intT intF) \<xi> T"
monomial_decompI: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "\<And>h U. (h, U) \<in> set ps \<Longrightarrow> is_monomial h"   and "\<And>h U. (h, U) \<in> set ps \<Longrightarrow> lcf h = (1::'b)" shows "monomial_decomp ps"
empty_neutral_1: fixes x :: "'f multiset" shows "{#} + x = x"
k_def: shows "k \<equiv> \<lambda>x. C x g"
NPadmit_Test_simps: fixes \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm"   and p :: "('c + 'b, 'e, 'd) formula" shows "NPadmit \<sigma> (? p) = NFadmit \<sigma> p"
deletion': fixes ss :: "'w list" assumes "ss \<in> lists S"   and "\<not> distinct (lconjseq ss)" shows "\<exists>a b as bs cs. ss = as @ [a] @ bs @ [b] @ cs \<and> sum_list ss = sum_list (as @ bs @ cs)"
rep_eq: fixes x :: "('b, 'a) pp" shows "deg_pp x = deg_pm (pp.mapping_of x)"
simps_2: fixes l :: "linear_poly"   and r :: "'a" shows "zero_satisfies (GEQ_ns l r) = (r \<le> (0::'a))"
le_0_eq: fixes n :: "nat" shows "(n \<le> zero_class.zero) = (n = zero_class.zero)"
character_mult: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex"   and \<chi>' :: "'c \<Rightarrow> complex" assumes "character G \<chi>"   and "character G \<chi>'" shows "character G (\<lambda>x. \<chi> x * \<chi>' x)"
rcut1: fixes A :: "fm"   and B :: "fm"   and H :: "fm set"   and C :: "fm" assumes "{A} \<turnstile> B"   and "insert B H \<turnstile> C" shows "insert A H \<turnstile> C"
simps_7: fixes n :: "nat" shows "norm_instr (Inca.instr.ICJump n) = Std.instr.ICJump n"
cut_of_sep_def: fixes \<Delta> :: "('a, 'b) network_scheme"   and S :: "('a \<times> 'a) set" shows "cut_of_sep \<Delta> S = RF\<^sup>N\<^bsub>\<Delta>\<^esub> (fst ` \<E>\<^bsub>web_of_network \<Delta>\<^esub> S)"
vmatch'_rel: fixes t\<^sub>1 :: "value"   and t\<^sub>2 :: "value"   and p :: "pat" assumes "Q t\<^sub>1 t\<^sub>2" shows "rel_option (fmrel Q) (vmatch' p t\<^sub>1) (vmatch' p t\<^sub>2)"
assms: shows "set (Polynomial.coeffs p) \<subseteq> range (\<lambda>x. Fraction_Field.Fract x (1::'a))"
complete_sets_sandwich_fmeasurable: fixes A :: "'c set"   and M :: "'c measure"   and C :: "'c set"   and B :: "'c set" assumes "A \<in> fmeasurable (completion M)"   and "C \<in> fmeasurable (completion M)"   and "A \<subseteq> B"   and "B \<subseteq> C"   and "Sigma_Algebra.measure (completion M) A = Sigma_Algebra.measure (completion M) C" shows "B \<in> fmeasurable (completion M)"
src_hcomp: fixes \<mu> :: "'b"   and \<nu> :: "'b" assumes "hseq \<mu> \<nu>" shows "src (\<mu> \<star> \<nu>) = src \<nu>"
disjoint_alt_simp2: fixes A :: "'c set"   and B :: "'c set" shows "(A - B \<noteq> A) = (A \<inter> B \<noteq> {})"
keys_monom_mult: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and c :: "'d"   and t :: "'b"   and p :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair"   and "c \<noteq> (0::'d)" shows "keys (term_powerprod.monom_mult pair_of_term term_of_pair c t p) = term_powerprod.splus pair_of_term term_of_pair t ` keys p"
ell2_norm_0: fixes x :: "'c \<Rightarrow> complex" assumes "has_ell2_norm x" shows "(ell2_norm x = 0) = (x = (\<lambda>_. 0))"
llist_all2_lnthD2: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist"   and n :: "nat" assumes "llist_all2 P xs ys"   and "enat n < llength ys" shows "P (lnth xs n) (lnth ys n)"
i'_in: shows "\<iota>' \<in> Inf_from N"
reduced_word_letter_set: fixes w :: "'w"   and ss :: "'w list" assumes "S_reduced_for w ss" shows "reduced_letter_set S w = set ss"
IH: assumes "\<And>s. s \<in> F \<Longrightarrow> closed s"   and "\<And>s. s \<in> F \<Longrightarrow> continuous_on s f" shows "continuous_on (\<Union> F) f"
analz_shrK_Decrypt: fixes A :: "agent"   and X :: "msg"   and H :: "msg set" assumes "Crypt (shrK A) X \<in> analz H"   and "Key (shrK A) \<in> analz H" shows "X \<in> analz H"
frontier_Int_subset: fixes S :: "'b set"   and T :: "'b set" shows "frontier (S \<inter> T) \<subseteq> frontier S \<union> frontier T"
inf_fun_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "inf f g = (\<lambda>x. inf (f x) (g x))"
collapse_6: fixes ltl :: "'b ltl" assumes "is_LTLGlobal ltl" shows "G theG ltl = ltl"
hyps: assumes "0 < degree p" shows "((\<lambda>r. Re (winding_number (poly p \<circ> part_circlepath z0 r 0 pi) 0)) \<longlongrightarrow> real (degree p) / 2) at_top"
monoid_def: fixes f :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and z :: "'b::type" shows "monoid (f::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (z::'b::type) \<equiv> semigroup f \<and> monoid_axioms f z"
bij_betw_subset: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and A' :: "'c set"   and B :: "'b set"   and B' :: "'c set" assumes "bij_betw f A A'"   and "order.greater_eq A B"   and "f ` B = B'" shows "bij_betw f B B'"
sds_lowering_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf"   and agents' :: "'a set"   and alts' :: "'b set" shows "sds_lowering agents alts sds agents' alts' \<equiv> ex_post_efficient_sds agents alts sds \<and> sds_lowering_axioms agents alts agents' alts'"
simps_5: fixes t :: "'a local.term" shows "Arr \<^bold>\<l>\<^bold>[t\<^bold>] = Arr t"
linear: fixes x :: "'t"   and y :: "'t" shows "le_of_nat_term_order cmp_term x y \<or> le_of_nat_term_order cmp_term y x"
inf_associative: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- x \<sqinter> (- y \<sqinter> - z) = - x \<sqinter> - y \<sqinter> - z"
\<mu>_small_row_def: fixes i :: "nat"   and fs :: "int vec list"   and j :: "nat" shows "\<mu>_small_row i fs j = (\<forall>j'\<ge>j. j' < i \<longrightarrow> \<bar>gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j'\<bar> \<le> inverse 2)"
in_multiset_in_set: fixes x :: "'c"   and xs :: "'c list" shows "(x \<in># mset xs) = (x \<in> set xs)"
left_sided_localp: fixes y :: "'b \<times> 'c"   and x :: "'b \<times> 'c" assumes "times_prod' top_prod y = y" shows "less_eq_prod (times_prod' x y) y"
set: fixes x1 :: "'d"   and x2 :: "int" shows "set_label (x1-:x2) = {x1}"
prep: fixes vs :: "'b list"   and bs :: "'b list"   and p :: "fm" shows "Ifm vs bs (prep p) = Ifm vs bs p"
Rep_unit2: fixes x :: "unit2" shows "Rep_unit2 x \<in> {0..1}"
assms_5: shows "InvariantClCurrentLevel (getCl state) (getM state)"
iface_rec_induct': fixes P :: "prog \<Rightarrow> qtname \<Rightarrow> bool"   and a0 :: "prog"   and a1 :: "qtname" assumes "\<And>G I f. (\<And>x2 x. \<lbrakk>iface G I = Some x2; ws_prog G; x \<in> set (isuperIfs x2)\<rbrakk> \<Longrightarrow> P G x) \<Longrightarrow> P G I" shows "P a0 a1"
vs''_1: shows "((Suc 0, vs), 0, vs'') \<in> (step M)\<^sup>*"
lists_fcs_def: shows "lists_fcs \<equiv> orbits_list (lists_fc_succ xss)"
waryI: fixes s :: "('s, 'v) tm" shows "wary s"
snd_lens_plus: fixes x :: "'b \<Longrightarrow> 'c"   and y :: "'d \<Longrightarrow> 'c" assumes "wb_lens x"   and "x \<bowtie> y" shows "snd\<^sub>L ;\<^sub>L (x +\<^sub>L y) = y"
in_hom_char: fixes f :: "'b"   and a :: "'b"   and b :: "'b" shows "hhom\<^sub>D.in_hom f a b = (hhom\<^sub>D.arr a \<and> hhom\<^sub>D.arr b \<and> hhom\<^sub>D.arr f \<and> \<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>)"
kH00_eq_Hi0: shows "div_op (H $$ (i, 0)) (H $$ (0, 0)) * H $$ (0, 0) = H $$ (i, 0)"
insert_into_member_def: fixes new_el :: "'b"   and Sets :: "'b set set"   and S :: "'b set" shows "insert_into_member new_el Sets S = insert (S \<union> {new_el}) (Sets - {S})"
strategyproofness_1: shows "pmf (sds R10) a + (pmf (sds R10) c + pmf (sds R10) d) \<le> pmf (sds R12) d + (pmf (sds R12) b + pmf (sds R12) a)"
fiter_induct_eq: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "finite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "(seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (finite_iteration.fiter seq (nil::'b::refinement_lattice) (c::'b::refinement_lattice)) (d::'b::refinement_lattice) = gfp (\<lambda>x::'b::refinement_lattice. seq c x \<sqinter> d)"
D_2: shows "D \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta> = B"
take_subseq_def: fixes s :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> nat" shows "take_subseq s f = (\<lambda>k. s (f k))"
EASSERT_simps_1: shows "EASSERT True = ERETURN ()"
collinear_3_imp_in_affine_hull: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "collinear {a, b, c}"   and "a \<noteq> b" shows "c \<in> affine hull {a, b}"
rtrancl_beta_AppR: fixes t :: "Core.term"   and t' :: "Core.term"   and s :: "Core.term" assumes "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'" shows "s $ t \<rightarrow>\<^sub>\<beta>\<^sup>* s $ t'"
simps_3: fixes P :: "'a pred"   and Q :: "'a pred"   and xq :: "'a seq" shows "adjunct P (Join Q xq) = Join Q (adjunct P xq)"
simps_4: fixes vb :: "nat"   and va :: "(val \<times> val) fset" shows "VNat vb \<squnion> VFun va = None"
ubound_def: fixes x :: "real" shows "ubound x = max 0 x"
simps_30: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x51 :: "nat"   and x52 :: "nat"   and x53 :: "nat" shows "(case Pkt x51 x52 x53 of Rreq x xa xb xc xd xe xf \<Rightarrow> f1 x xa xb xc xd xe xf | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f5 x51 x52 x53"
more_arith_simps_5: fixes a :: "'b" shows "(1::'b) * a = a"