hcju/isabelleps huggingface dataset:
elims: fixes x :: "'c \<Rightarrow> 'd"   and xa :: "'c list"   and y :: "'c list" assumes "remdups_gen x xa = y" obtains f :: "'c \<Rightarrow> 'd" where "x = f__" and "xa = []" and "y = []"   | f :: "'c \<Rightarrow> 'd" and xb :: "'c" and xs :: "'c list" where "x = f" and "xa = xb # xs" and "y = xb # remdups_gen f (filter (\<lambda>y. f xb \<noteq> f y) xs)"
isolate_variable_sparse_degree_eq_zero_iff: fixes p :: "'b mpoly"   and v :: "nat" shows "(isolate_variable_sparse p v (MPoly_Type.degree p v) = 0) = (p = 0)"
strand_sem_append'_1: fixes M :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S' :: "('fun, 'var) strand_step list" assumes "\<lbrakk>M; S\<rbrakk>\<^sub>c' \<theta>"   and "\<lbrakk>M \<union> ik\<^sub>s\<^sub>t S; S'\<rbrakk>\<^sub>c' \<theta>" shows "\<lbrakk>M; S @ S'\<rbrakk>\<^sub>c' \<theta>"
measure_translation: fixes a :: "'c"   and S :: "'c set" shows "Sigma_Algebra.measure lebesgue ((+) a ` S) = Sigma_Algebra.measure lebesgue S"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (write_reg w curr_win rd s1)) \<and> t2 = snd (fst (write_reg w curr_win rd s2))"
subc_subst_new: fixes c :: "'c"   and t :: "'c FOL_Fitting.term"   and s :: "'c FOL_Fitting.term"   and m :: "nat"   and p :: "('c, 'd) form" assumes "new_term c t" shows "subc c (s[t/m]) (p[t/m]) = subc c s p[t/m]"
is_NFlow: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and cf :: "nat \<times> nat \<Rightarrow> 'a" assumes "RGraph c s t cf" shows "NFlow c s t (Network.flow_of_cf c cf)"
streams_sets: fixes X :: "'c set"   and M :: "'c measure" assumes "X \<in> sets M" shows "streams X \<in> sets (stream_space M)"
homeomorphism_grouping_points_exists: fixes U :: "'b set"   and S :: "'b set"   and K :: "'b set"   and T :: "'b set" assumes "open U"   and "open S"   and "connected S"   and "U \<noteq> {}"   and "finite K"   and "K \<subseteq> S"   and "U \<subseteq> S"   and "S \<subseteq> T" obtains f :: "'b \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'b" where "homeomorphism T T f g" and "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S" and "bounded {x. \<not> (f x = x \<and> g x = x)}" and "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U"
subst_tm_def: fixes t :: "('a, 'b) tm"   and env :: "(name, ('a, 'b) tm) fmap" shows "subst t env = Lambda_Free_Term.subst (fmlookup_default env (Hd \<circ> hd.Var \<circ> of_name) \<circ> to_name) t"
older_seniors_Max_coboundedI: fixes a :: "nat"   and x :: "nat"   and n :: "nat" assumes "a \<in> older_seniors x n" shows "a \<le> Max (older_seniors x n)"
r: shows "\<integral>\<^sup>+ x. ennreal (max 0 (f x)) \<partial>lborel = ennreal r"
seqE: fixes g :: "'a1"   and f :: "'a1" assumes "C1.seq g f" obtains "C1.arr f" and "C1.arr g" and "C1.dom g = C1.cod f"
analz_into_parts: fixes c :: "msg"   and H :: "msg set" assumes "c \<in> analz H" shows "c \<in> parts H"
num_roots_le_deg: fixes d :: "nat" assumes "finite (carrier R)"   and "d \<noteq> 0" shows "card {x \<in> carrier R. x [^] d = \<one>} \<le> d"
CAs0_def: shows "CAs0 = map CAs0f [0..<n]"
finite_less_Inf_iff: fixes X :: "'c set"   and a :: "'c" assumes "finite X"   and "X \<noteq> {}" shows "(a < Inf X) = (\<forall>x\<in>X. a < x)"
eval_Inf: fixes A :: "'a pred set" shows "eval (\<Sqinter> A) = \<Sqinter> (eval ` A)"
VUnion_in_VsetI: fixes A :: "V"   and i :: "V" assumes "A \<in>\<^sub>\<circ> Vset i" shows "\<Union>\<^sub>\<circ> A \<in>\<^sub>\<circ> Vset i"
remove_for_all_empty_children: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and h' :: "(_) heap" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h \<turnstile> forall_M Shadow_DOM.remove children \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r []"
trcl_empty_simp: fixes c :: "'c"   and c' :: "'c"   and t :: "('c \<times> 'd \<times> 'c) set" shows "((c, [], c') \<in> trcl t) = (c = c')"
ipurge_tr_aux_del_failures: fixes P :: "'b process"   and I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and ys :: "'b list"   and Z :: "'b set"   and zs :: "'b list"   and xs :: "'b list" assumes "secure P I D"   and "\<And>u z. \<lbrakk>u \<in> sinks_aux_less I D U ys; z \<in> Z \<union> set zs\<rbrakk> \<Longrightarrow> (u, D z) \<notin> I"   and "(xs @ ipurge_tr_aux I D U ys @ zs, Z) \<in> failures P"   and "xs @ ys \<in> traces P" shows "(xs @ ys @ zs, Z) \<in> failures P"
coeff_add: fixes p :: "'a mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and q :: "'a mpoly" shows "coeff p m + coeff q m = coeff (p + q) m"
A: shows "A = \<Union> (SN_rel_ext_step P Pw R Rw ` {top_s, top_ns, normal_s, normal_ns}) \<inter> M'"
that: fixes p' :: "('a, 'b, 'c, 'd) seqp" assumes "p = {l}receive(fmsg) . p'"   and "l' \<in> labels \<Gamma> qq" shows "thesis"
that: fixes env' :: "(name, 'a) fmap" assumes "match x t = Some env'"   and "P_env env' env" shows "thesis"
less_add_same_cancel2: fixes a :: "'a"   and b :: "'a" shows "(a < b + a) = ((0::'a) < b)"
zero_right_mult_decreasing: fixes x :: "'a" shows "x * bot \<le> x"
RETURN_ref_RETURND: fixes c :: "'g"   and R :: "('g \<times> 'h) set"   and a :: "'h" assumes "RETURN c \<le> \<Down> R (RETURN a)" shows "(c, a) \<in> R"
0: fixes f :: "'fsym" assumes "f \<in> fsyms T" shows "intF1 f = intF2 f"
norm_triangle_sub: 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"   and y :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm x \<le> norm y + norm (minus x y)"
distribute: shows "(c::'a::refinement_lattice)\<^sup>\<star> ; (d::'a::refinement_lattice) = d \<sqinter> c ; c\<^sup>\<star> ; d"
simps_3: fixes i :: "int"   and k :: "int"   and ks :: "int list" shows "inf\<^sub>- (Atom (Le i (k # ks))) = (if k < 0 then TrueF else if 0 < k then FalseF else Atom (Le i (0 # ks)))"
map2_val_compatD2: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and f :: "('a \<times> 'b) list \<Rightarrow> ('a \<times> 'c) list"   and zs :: "('a \<times> 'b) list" assumes "comparator comp"   and "comparator.map2_val_compat comp f"   and "comparator.oalist_inv_raw comp zs" shows "fst ` set (f zs) \<subseteq> fst ` set zs"
bind_fv2_Fv_fv: fixes x :: "variable"   and \<tau> :: "typ"   and lev :: "nat"   and t :: "Core.term" shows "fv (bind_fv2 (x, \<tau>) lev t) = fv t - {(x, \<tau>)}"
WTrt_binop_simps_12: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T :: "ty" shows "(P \<turnstile> T1\<guillemotleft>BinAnd\<guillemotright>T2 : T) = (T1 = Boolean \<and> T2 = Boolean \<and> T = Boolean \<or> T1 = Integer \<and> T2 = Integer \<and> T = Integer)"
valid_def: fixes C :: "Class"   and m :: "Method"   and l :: "int"   and A :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and B :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and I :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" shows " \<Turnstile> \<lbrace> A \<rbrace> C , m , l \<lbrace> B \<rbrace> I = (\<forall>M. mbody_is C m M \<longrightarrow> (\<forall>Mspec Minv Anno. MST\<down>(C, m) = Some (Mspec, Minv, Anno) \<longrightarrow> (\<forall>par code l0. M = (par, code, l0) \<longrightarrow> (\<forall>s0 s. MS M l0 (mkState s0) l s \<longrightarrow> A s0 s \<longrightarrow> (\<forall>h v. Opsem M l s h v \<longrightarrow> B s0 s (h, v)) \<and> (\<forall>ll r. (MS M l s ll r \<longrightarrow> (\<forall>Q. Anno\<down>ll = Some Q \<longrightarrow> Q s0 r)) \<and> (Reach M l s r \<longrightarrow> I s0 s (heap r)))))))"
step_elim_cases_11: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c1 :: "('b, 'a, 'c) com"   and c2 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (Catch c1 c2, s) \<rightarrow> u" obtains c\<^sub>1' :: "('b, 'a, 'c) com" and s' :: "('b, 'c) xstate" where "u = (Catch c\<^sub>1' c2, s')" and "\<Gamma>\<turnstile> (c1, s) \<rightarrow> (c\<^sub>1', s')"   | sa :: "'b" where "u = (c2, Normal sa)" and "c1 = Throw" and "s = Normal sa"   | "c1 = Skip" and "u = (Skip, s)"   | f :: "'c" where "s = Fault f__" and "u = (Skip, Fault f__)" and "redex c1 = Catch c1 c2"   | "s = Stuck" and "u = (Skip, Stuck)" and "redex c1 = Catch c1 c2"   | f :: "'b" where "s = Abrupt f" and "u = (Skip, Abrupt f)" and "redex c1 = Catch c1 c2"
simps_1: fixes l :: "nat"   and cl :: "nat" shows "prefixToLevel_aux [] l cl = []"
coinductive_eq_iff_lnth_eq: fixes a :: "'c llist"   and b :: "'c llist" shows "(a = b) = (llength a = llength b \<and> (\<forall>n. enat n < llength a \<longrightarrow> a $ n = b $ n))"
path_graph_empty: fixes u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" shows "path graph_empty u p v = (v = u \<and> p = [])"
mono_SUP: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'b \<Rightarrow> 'c"   and A :: "'d \<Rightarrow> 'b"   and I :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "order.mono less_eq f" shows "(\<Squnion>x\<in>I. f (A x)) \<le> f (Sup (A ` I))"
chg_map_new: fixes m :: "'d \<Rightarrow> 'c option"   and a :: "'d"   and f :: "'c \<Rightarrow> 'c" assumes "m a = None" shows "chg_map f a m = m"
t_def: shows "t \<equiv> \<langle>s, a, \<langle>u, b, r\<rangle>\<rangle>"
f_def: shows "f = (\<lambda>x. case quotient_of x of (pa, x) \<Rightarrow> pa * d div x)"
diff_cancel: fixes a :: "'f" shows "a - a = (0::'f)"
assert_Sup: fixes X :: "'a set" shows "{. \<Squnion>X .} = \<Squnion>(assert ` X)"
eucl_lessThan: fixes a :: "real" shows "{x. eucl_less x a} = {..<a}"
le_infI1: fixes a :: "'c multiset"   and x :: "'c multiset"   and b :: "'c multiset" assumes "a \<subseteq># x" shows "a \<inter># b \<subseteq># x"
lstar_rec_isotone: fixes y :: "'a" shows "isotone (\<lambda>x. Rf x y)"
order_trans: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "ordc x y"   and "ordc y z" shows "ordc x z"
orthogonal_Gram_Schmidt_upt_k: fixes k :: "nat"   and A :: "(('a, 'b) vec, 'c) vec" assumes "k < ncols A" shows "pairwise orthogonal {column i (Gram_Schmidt_upt_k A k) |i. mod_type_class.to_nat i \<le> k}"
simps: fixes r :: "Vertex \<Rightarrow> Vertex"   and v :: "Vertex" shows "apply_rotation r v = r v"
mode_release_refinement_helper: fixes mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and SomeMode :: "Mode"   and x :: "Eg1.var" shows "mds\<^sub>A_of (mds\<^sub>C(SomeMode := {y \<in> mds\<^sub>C SomeMode. y \<noteq> Eg2_var\<^sub>C_of_Eg1 x})) = (mds\<^sub>A_of mds\<^sub>C)(SomeMode := {y \<in> mds\<^sub>A_of mds\<^sub>C SomeMode. y \<noteq> x})"
sel_2: fixes x1 :: "'a match_expr"   and x2 :: "action" shows "get_action (Rule x1 x2) = x2"
simps_4: fixes sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current :: "thread_id_t"   and obj :: "obj_id_t \<Rightarrow> obj_t"   and thread :: "thread_id_t \<Rightarrow> thread_t"   and more :: "'a" shows "obj \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr> = obj"
hyps_2: shows "terminal_path v\<^sub>2 v' pth"
simps_2: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])"
length_greater_0_conv: fixes xs :: "'a list" shows "xs \<in># length = (xs \<noteq> [])"
perp_not_eq_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C D" shows "C \<noteq> D"
insert_null_sets_iff: fixes a :: "'b"   and N :: "'b set" shows "(insert a N \<in> null_sets lebesgue) = (N \<in> null_sets lebesgue)"
find_mu_def: shows "find_mu \<equiv> (\<lambda>s. s\<lparr>m := 0, tortoise := x0\<rparr>) ;; while (\<lambda>s. hare s \<noteq> tortoise s) (\<lambda>s. s\<lparr>tortoise := f (tortoise s), hare := f (hare s), m := m s + 1\<rparr>)"
lexext_rev_singleton: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" shows "lexext_rev gt [y] [x] = gt y x"
roots1: fixes z :: "complex" assumes "z \<in> ball 0 (1 / R)" shows "poly q z \<noteq> 0"
assms: shows "A B Perp C D \<or> B A Perp C D \<or> A B Perp D C \<or> B A Perp D C \<or> C D Perp A B \<or> C D Perp B A \<or> D C Perp A B \<or> D C Perp B A"
amin_mult_pos: fixes z :: "int"   and x :: "ant"   and y :: "ant" assumes "0 < z" shows "amin (z *\<^sub>a x) (z *\<^sub>a y) = z *\<^sub>a amin x y"
tendsto_compose_filtermap: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e"   and T :: "'d"   and F :: "'c filter" shows "((g \<circ> f) \<longlongrightarrow> T) F = (g \<longlongrightarrow> T) (filtermap f F)"
CallUndefined: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and p :: "'b"   and s :: "'a" assumes "\<Gamma> p = None" shows "\<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow> Stuck"
OrdP_max_imp: fixes j :: "name"   and x :: "tm"   and k :: "name" assumes "atom j \<sharp> x"   and "atom k \<sharp> (x, j)" shows "{OrdP x, Neg (x EQ Zero)} \<turnstile> SyntaxN.Ex j (Var j IN x AND All2 k x (Var k SUBS Var j))"
homeomorphic_trans: fixes S :: "'c set"   and T :: "'d set"   and U :: "'e set" assumes "S homeomorphic T"   and "T homeomorphic U" shows "S homeomorphic U"
nrename_fresh_interesting1: fixes z :: "name"   and M :: "trm"   and x :: "name"   and y :: "name" assumes "z \<sharp> M[x\<turnstile>n>y]"   and "z \<sharp> (x, y)" shows "z \<sharp> M"
tendstoD: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and l :: "'b"   and F :: "'c filter"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "filterlim f (topological_space.nhds open l) F"   and "0 < e" shows "\<forall>\<^sub>F x in F. dist (f x) l < e"
asymp_equiv_add_left: fixes h :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "h \<in> o[F](g)" shows "((\<lambda>x. h x + f x) \<sim>[F] g) = (f \<sim>[F] g)"
implies_mp_eq: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> (- x \<squnion> y) = x \<sqinter> y"
ent_disjI1': fixes A :: "assn"   and B :: "assn"   and C :: "assn" assumes "A \<Longrightarrow>\<^sub>A B" shows "A \<Longrightarrow>\<^sub>A B \<or>\<^sub>A C"
kleisli_comm_cong: fixes x :: "'c \<Rightarrow> 'c option"   and y :: "'c \<Rightarrow> 'c option"   and z :: "'c \<Rightarrow> 'c option" assumes "x \<rhd> y = y \<rhd> x" shows "z \<rhd> x \<rhd> y = z \<rhd> y \<rhd> x"
sum_eq: fixes f :: "'c \<Rightarrow> ('a, 'b) vec"   and S :: "'c set" shows "sum f S = (\<chi>i. \<Sum>x\<in>S. f x $ i)"
True_then: fixes P :: "bool" assumes "True \<longrightarrow> P" shows "P"
cosine_law_vector: fixes u :: "'b"   and v :: "'b" shows "(norm (u - v))\<^sup>2 = (norm u)\<^sup>2 + (norm v)\<^sup>2 - 2 * norm u * norm v * cos (vangle u v)"
max2_def: fixes a :: "'a Bonk_Schramm_extension_unfolded"   and b :: "'a Bonk_Schramm_extension_unfolded" shows "wo.max2 a b \<equiv> if (a, b) \<in> Bonk_Schramm_extension_unfolded_wo then b else a"
convergent_imp_bounded: fixes S :: "nat \<Rightarrow> 'b"   and l :: "'b" assumes "S \<longlonglongrightarrow> l" shows "bounded (range S)"
Plus_def: fixes A :: "'b set"   and B :: "'c set" shows "A <+> B = Inl ` A \<union> Inr ` B"
distinct_9: fixes x1 :: "'e"   and x6 :: "nat" shows "broadcast x1 \<noteq> deliver x6"
sup_measure_F_mono': fixes J :: "'a measure set"   and I :: "'a measure set" assumes "finite J"   and "finite I" shows "sup_measure.F id I \<le> sup_measure.F id (I \<union> J)"
heterogenous_same_type_cblinfun_def: fixes x :: "'a itself"   and y :: "'b itself" shows "heterogenous_same_type_cblinfun x y = (unitary heterogenous_cblinfun_id \<and> unitary heterogenous_cblinfun_id)"
lm005: fixes f :: "'a \<Rightarrow> 'b" shows "Graph f = graph UNIV f"
that: fixes a :: "'a"   and b :: "'a" assumes "x = a * b"   and "a \<in> A"   and "b \<in> B" shows "thesis"
I_def: fixes s :: "'a" shows "I s = map_pmf (Pair s) (pmf_of_set \<O>\<^sub>s)"
A: shows "(\<exists>d e. K = d * e \<and> IntAgrK (S (Card n, n, run)) = Some d \<and> ExtAgrK (S (Card n, n, run)) = Some e) \<or> (\<exists>b. Pri_AgrK b \<in> A \<and> ExtMapK (S (User m, n, run)) = Some b)"
ring_r_one: fixes R :: "('h, 'i) Ring_scheme"   and x :: "'h" assumes "Ring R"   and "x \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> 1\<^sub>r\<^bsub>R\<^esub> = x"
Gateway_L4: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and t :: "nat" assumes "Gateway req dt a stop lose d ack i vc"   and "msg (Suc 0) req"   and "msg (Suc 0) a"   and "msg (Suc 0) stop"   and "ts lose"   and "ack (t + d) = [sending_data]"   and "a (Suc t) = [sc_ack]"   and "\<forall>k\<le>Suc d. lose (t + k) = [False]" shows "vc (Suc (t + d)) = [vc_com]"
transversal_segment_def: fixes a :: "'a"   and b :: "'a" shows "transversal_segment a b = (a \<noteq> b \<and> {a--b} \<subseteq> X \<and> (\<forall>z\<in>{a--b}. f z \<bullet> rot (a - b) \<noteq> 0))"
set_pair_pmf: fixes A :: "'c pmf"   and B :: "'d pmf" shows "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
match_fvi_trm_Some: fixes f :: "nat \<Rightarrow> event_data option"   and ts :: "Formula.trm list"   and xs :: "event_data list"   and t :: "Formula.trm"   and x :: "nat" assumes "Some f = match ts xs"   and "t \<in> set ts"   and "x \<in> fv_trm t" shows "f x \<noteq> None"
isos_compose: fixes f :: "'d"   and f' :: "'d" assumes "D.iso f"   and "D.iso f'"   and "D.seq f' f" shows "D.iso (f' \<cdot>\<^sub>D f)"
select_convs_1: fixes inpt :: "Proc \<Rightarrow> InputsOrNi"   and outpt :: "Proc \<Rightarrow> InputsOrNi"   and disk :: "Disk \<Rightarrow> Proc \<Rightarrow> DiskBlock"   and dblock :: "Proc \<Rightarrow> DiskBlock"   and phase :: "Proc \<Rightarrow> nat"   and disksWritten :: "Proc \<Rightarrow> Disk set"   and blocksRead :: "Proc \<Rightarrow> Disk \<Rightarrow> BlockProc set"   and allInput :: "InputsOrNi set"   and chosen :: "InputsOrNi"   and more :: "'a" shows "inpt \<lparr>inpt = inpt, outpt = outpt, disk = disk, dblock = dblock, phase = phase, disksWritten = disksWritten, blocksRead = blocksRead, allInput = allInput, chosen = chosen, \<dots> = more\<rparr> = inpt"
continuous_cblinfun_componentwiseI1: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C\<^sub>L 'c" assumes "\<And>i. i \<in> CBasis \<Longrightarrow> continuous F (\<lambda>x. cblinfun_apply (f x) i)" shows "continuous F f"
Abs_formula_inverse: fixes t\<^sub>\<alpha> :: "('b, 'c, 'd, 'e) Tree\<^sub>\<alpha>" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "Rep_formula (Abs_formula t\<^sub>\<alpha>) = t\<^sub>\<alpha>"
base: shows "U (min (Monitor.progress \<sigma> P \<phi> j) (Monitor.progress \<sigma> P \<psi> j)) z"
writes_singleton2: fixes f :: "('x, 'y, 'z) prog"   and h :: "'x"   and h' :: "'x" shows "writes {f} f h h'"
elims: fixes x :: "'a list"   and y :: "'a list list" assumes "suffixes x = y" obtains "x = []" and "y = []"   | xa :: "'a" and xs :: "'a list" where "x = xa # xs" and "y = suffixes xs @ [xa # xs]"
equiv_domain_and_range_permutation: fixes A :: "'c set"   and B :: "'d set" shows "equiv (A \<rightarrow>\<^sub>E B) (domain_and_range_permutation A B)"
g_def: fixes mrk :: "'node set"   and lbl :: "'a \<Rightarrow> 'index"   and ptr :: "'a \<Rightarrow> 'node"   and x :: "'a" shows "g mrk lbl ptr x \<equiv> ptr x \<noteq> nil \<and> ptr x \<notin> mrk \<and> lbl x = none"
min_dep_imp_ex1_supset_circuit: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> carrier"   and "indep X"   and "\<not> indep (insert x X)" shows "\<exists>!C. circuit C \<and> C \<subseteq> insert x X"
cInf_singleton: fixes x :: "'b multiset" shows "\<Sqinter> {x} = x"
lappend_ltake_ldrop: fixes n :: "enat"   and xs :: "'a llist" shows "ltake n xs @\<^sub>l ldrop n xs = xs"
assms_2: fixes i :: "'a" assumes "i \<in> Basis" shows "dist (x \<bullet> i) (y \<bullet> i) \<le> r / sqrt (real DIM('a))"
alpha_Tree_induct': fixes P :: "('a, 'b, 'c, 'd) Tree \<Rightarrow> ('a, 'b, 'c, 'd) Tree \<Rightarrow> bool"   and a0 :: "('a, 'b, 'c, 'd) Tree"   and a1 :: "('a, 'b, 'c, 'd) Tree" assumes "\<And>tset1 tset2. (\<And>a b. \<lbrakk>a \<in> set_bset tset1; b \<in> set_bset tset2\<rbrakk> \<Longrightarrow> P a b) \<Longrightarrow> P (tConj tset1) (tConj tset2)"   and "\<And>t1 t2. P t1 t2 \<Longrightarrow> P (tNot t1) (tNot t2)"   and "\<And>f1 \<phi>1 f2 \<phi>2. P (tPred f1 \<phi>1) (tPred f2 \<phi>2)"   and "\<And>f1 \<alpha>1 t1 f2 \<alpha>2 t2. \<lbrakk>\<And>x. P (x \<bullet> t1) t2; \<And>x a b. P ((a \<rightleftharpoons> b) \<bullet> t1) t1; \<And>x a b. P ((a \<rightleftharpoons> b) \<bullet> t2) t2\<rbrakk> \<Longrightarrow> P (tAct f1 \<alpha>1 t1) (tAct f2 \<alpha>2 t2)"   and "\<And>v va. P (tNot v) (tConj va)"   and "\<And>v va vb. P (tNot v) (tPred va vb)"   and "\<And>v va vb vc. P (tNot v) (tAct va vb vc)"   and "\<And>v va vb. P (tPred v va) (tConj vb)"   and "\<And>v va vb. P (tPred v va) (tNot vb)"   and "\<And>v va vb vc vd. P (tPred v va) (tAct vb vc vd)"   and "\<And>v va vb vc. P (tAct v va vb) (tConj vc)"   and "\<And>v va vb vc. P (tAct v va vb) (tNot vc)"   and "\<And>v va vb vc vd. P (tAct v va vb) (tPred vc vd)"   and "\<And>va v. P (tConj va) (tNot v)"   and "\<And>va vb v. P (tPred va vb) (tNot v)"   and "\<And>va vb vc v. P (tAct va vb vc) (tNot v)"   and "\<And>vb v va. P (tConj vb) (tPred v va)"   and "\<And>vb v va. P (tNot vb) (tPred v va)"   and "\<And>vb vc vd v va. P (tAct vb vc vd) (tPred v va)"   and "\<And>vc v va vb. P (tConj vc) (tAct v va vb)"   and "\<And>vc v va vb. P (tNot vc) (tAct v va vb)"   and "\<And>vc vd v va vb. P (tPred vc vd) (tAct v va vb)" shows "P a0 a1"
key_seed_gen_def: shows "key_seed_gen = key_gen \<bind> (\<lambda>k. seed_gen \<bind> (\<lambda>s. return_spmf (k, s)))"
insert_is_Un: fixes a :: "'a"   and A :: "'a set" shows "Set.insert a A = {a} \<union> A"