hcju/isabelleps huggingface dataset:
sint_ge: fixes x :: "'a Word.word" shows "- (2 ^ (LENGTH('a) - 1)) \<le> sint x"
padd_infinity_l: fixes a :: "'a"   and x :: "'a"   and y :: "'a"   and p :: "'a \<times> 'a \<times> 'a" shows "padd a (x, y, \<zero>) p = p"
esing_simps_2: fixes x' :: "'a"   and x :: "'a"   and n :: "'b" assumes "x' \<noteq> x" shows "(esing x\<cdot>n) x' = \<bottom>"
cp_OclAnd: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'a state \<times> 'a state" shows "(X and Y) \<tau> = ((\<lambda>_. X \<tau>) and (\<lambda>_. Y \<tau>)) \<tau>"
leibniz_rule: fixes U :: "'a set"   and a :: "'b"   and b :: "'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and fx :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'c"   and x0 :: "'a" assumes "\<And>x t. \<lbrakk>x \<in> U; t \<in> cbox a b\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x t) has_derivative blinfun_apply (fx x t)) (at x within U)"   and "\<And>x. x \<in> U \<Longrightarrow> f x integrable_on cbox a b"   and "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). fx x t)"   and "x0 \<in> U"   and "convex U" shows "((\<lambda>x. integral (cbox a b) (f x)) has_derivative blinfun_apply (integral (cbox a b) (fx x0))) (at x0 within U)"
power2_csqrt: fixes z :: "complex" shows "(csqrt z)\<^sup>2 = z"
A_authenticates_and_keydist_to_B: fixes servK :: "nat"   and T3 :: "nat"   and evs :: "event list"   and authK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and A :: "agent"   and Ta :: "nat"   and authTicket :: "msg" assumes "Crypt servK (Number T3) \<in> parts (knows Spy evs)"   and "Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "Key servK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "B \<noteq> Tgs"   and "evs \<in> kerbIV" shows "B Issues A with Crypt servK (Number T3) on evs"
the_gpv_def: fixes gpv :: "('a, 'b, 'c) gpv" shows "the_gpv gpv = (case gpv of GPV x \<Rightarrow> x)"
get_shadow_root_def: shows "get_shadow_root = (\<lambda>element_ptr. get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr shadow_root_opt)"
seqE: fixes g :: "'a2"   and f :: "'a2" assumes "A2.seq g f" obtains "A2.arr f" and "A2.arr g" and "A2.dom g = A2.cod f"
continuous_at_Sup_antimono: fixes f :: "'d \<Rightarrow> 'e"   and S :: "'d set" assumes "antimono f"   and "continuous (at_left (\<Squnion> S)) f"   and "S \<noteq> {}"   and "bdd_above S" shows "f (\<Squnion> S) = \<Sqinter> (f ` S)"
ZObis_finished_discr_R: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and c :: "('c, 'd, 'a) cmd"   and d :: "('c, 'd, 'a) cmd"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "proper c"   and "finished c"   and "(c, d) \<in> PL_Indis.ZObis aval tval cval indis" shows "PL_Indis.discr aval tval cval indis d"
term_templateInduct: fixes rel :: "(('a, 'b, 'varSort, 'var, 'c) qTerm set \<times> ('a, 'b, 'varSort, 'var, 'c) qTerm set) set"   and phi :: "('a, 'b, 'varSort, 'var, 'c) qTerm set \<Rightarrow> bool"   and phiAbs :: "('a, 'b, 'varSort, 'var, 'c) qAbs set \<Rightarrow> bool"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set" assumes "\<And>X Y. \<lbrakk>good X; (X, Y) \<in> rel\<rbrakk> \<Longrightarrow> good Y \<and> skel Y = skel X"   and "\<And>xs x. phi (Var xs x)"   and "\<And>delta inp binp. \<lbrakk>goodInp inp; goodBinp binp; liftAll phi inp; liftAll phiAbs binp\<rbrakk> \<Longrightarrow> phi (Op delta inp binp)"   and "\<And>xs x X. \<lbrakk>good X; \<And>Y. (X, Y) \<in> rel \<Longrightarrow> phi Y\<rbrakk> \<Longrightarrow> phiAbs (Abs xs x X)" shows "(good X \<longrightarrow> phi X) \<and> (goodAbs A \<longrightarrow> phiAbs A)"
mat_cons1: fixes p :: "pat"   and env :: "v sem_env"   and s :: "'b state"   and v1 :: "v"   and env' :: "(char list \<times> v) list"   and ck :: "bool"   and e :: "exp0"   and bv :: "'b state \<times> (v, v) result"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v" assumes "allDistinct (pat_bindings p []) \<and> pmatch (c env) (refs s) p v1 [] = Match env' \<and> evaluate ck (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v env)) env) s e bv" shows "evaluate_match ck env s v1 ((p, e) # pes) err_v bv"
inject_1: fixes x2 :: "nat"   and x1 :: "NominalSequents.var list"   and y2 :: "nat"   and y1 :: "NominalSequents.var list" shows "(At x2 x1 = At y2 y1) = (x2 = y2 \<and> x1 = y1)"
is_red_union: fixes A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "is_red (A \<union> B) p = (is_red A p \<or> is_red B p)"
open_closed: fixes S :: "'a set" shows "open S = closed (- S)"
sorted_a: fixes x :: "'a extended"   and l :: "nat"   and y :: "'a extended"   and u :: "nat" assumes "idx s x = Inr l"   and "idx s y = Inr u" shows "(l \<le> u) = (a x \<le> a y)"
gcd_diff2_nat: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "gcd (n - m) n = gcd m n"
card_rows_with_1: shows "card {i \<in> example.rows_with_1. i < dim_row example.Aw'} = example.r ^ example.N_half"
inorder_insert: fixes t :: "'b tree23"   and a :: "'b" assumes "Sorted_Less.sorted (inorder t)" shows "inorder (insert a t) = ins_list a (inorder t)"
lens_indep_right_ext: fixes x :: "'d \<Longrightarrow> 'e"   and z :: "'f \<Longrightarrow> 'e"   and y :: "'g \<Longrightarrow> 'f" assumes "x \<bowtie> z" shows "x \<bowtie> y ;\<^sub>L z"
leadsTo_weaken_L: fixes F :: "'d program"   and A :: "'d set"   and A' :: "'d set"   and B :: "'d set" assumes "F \<in> A \<longmapsto> A'"   and "B \<subseteq> A" shows "F \<in> B \<longmapsto> A'"
I_rw: shows "I = {i. i < dim_row A \<and> i \<in> I}"
ev: shows "\<forall>\<^sub>F x in net. f x \<bullet> i = b"
topologyI: fixes T :: "'b set set" assumes "\<And>x y. \<lbrakk>x open\<^bsub>T\<^esub>; y open\<^bsub>T\<^esub>\<rbrakk> \<Longrightarrow> x \<inter> y open\<^bsub>T\<^esub>"   and "\<And>M. \<forall>m\<in>M. m open\<^bsub>T\<^esub> \<Longrightarrow> \<Union> M open\<^bsub>T\<^esub>" shows "topology T"
exhaust: fixes y :: "e" obtains x1 :: "b_e" where "y = $x1"   | "y = Trap"   | x3 :: "cl" where "y = Callcl x3"   | x41 :: "nat" and x42 :: "e list" and x43 :: "e list" where "y = Label x41 x42 x43"   | x51 :: "nat" and x52 :: "nat" and x53 :: "v list" and x54 :: "e list" where "y = Local x51 x52 x53 x54"
object_ptr_kinds_h_eq3: shows "object_ptr_kinds h = object_ptr_kinds h2"
ANR_eq_absolute_neighbourhood_extensor: fixes S :: "'a set" shows "ANR S = (\<forall>f U T. continuous_on T f \<longrightarrow> f ` T \<subseteq> S \<longrightarrow> closedin (top_of_set U) T \<longrightarrow> (\<exists>V g. T \<subseteq> V \<and> openin (top_of_set U) V \<and> continuous_on V g \<and> g ` V \<subseteq> S \<and> (\<forall>x\<in>T. g x = f x)))"
pullback_topology_def: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and T :: "'e topology" shows "pullback_topology A f T = topology (\<lambda>S. \<exists>U. openin T U \<and> S = f -` U \<inter> A)"
"4_6": fixes uv :: "poscheckvariant"   and t :: "('d, 'c) Term.term"   and t' :: "('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<langle>uv: t \<in> t'\<rangle> = fv t \<union> fv t'"
ide_char: fixes f :: "('a, 'a set) Maps.arr" shows "Maps.ide f = (f \<noteq> Maps.Null \<and> Maps.Dom f \<in> Collect obj \<and> Maps.Cod f = Maps.Dom f \<and> Maps.Map f = \<lbrakk>Maps.Dom f\<rbrakk>)"
iT_Plus_neg_icard: fixes I :: "nat set"   and k :: "nat" shows "icard (I \<oplus>- k) = icard (I \<down>\<ge> k)"
sum_nonzero_representation_eq: fixes basis :: "'b set"   and v :: "'b" assumes "independent basis"   and "v \<in> span basis" shows "(\<Sum>b | representation basis v b \<noteq> (0::'a). representation basis v b *s b) = v"
that: fixes n :: "nat" assumes "q \<in> dgrad_p_set d n"   and "G \<subseteq> dgrad_p_set d n"   and "insert q (G - {p}) \<subseteq> dgrad_p_set d n" shows "thesis"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> [$C v, $e] : ts _> ts'"
gen_dfs_le_gen_dfsT: fixes gds :: "('b, 'a) gen_dfs_struct"   and V0 :: "'b set" shows "gen_dfs_defs.gen_dfs gds V0 \<le> gen_dfs_defs.gen_dfsT gds V0"
ext_inject: fixes nodes :: "'c set"   and edges :: "('c \<times> 'd \<times> 'c) set"   and more :: "'e"   and nodes' :: "'c set"   and edges' :: "('c \<times> 'd \<times> 'c) set"   and more' :: "'e" shows "(\<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = \<lparr>nodes = nodes', edges = edges', \<dots> = more'\<rparr>) = (nodes = nodes' \<and> edges = edges' \<and> more = more')"
prems: shows "{x1_}unicast(x2_, x3_) . p1_ \<triangleright> p2_ \<in> A"
add_to_nat_def: fixes x :: "'a"   and y :: "'a" shows "x + y = mod_type_class.from_nat (mod_type_class.to_nat x + mod_type_class.to_nat y)"
fifo_gap_relabel_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and Q :: "nat list"   and u :: "nat" shows "fifo_gap_relabel f l Q u \<equiv> assert (u \<in> V - {s, t}) \<bind> (\<lambda>_. assert (Height_Bounded_Labeling c s t f l) \<bind> (\<lambda>_. let Q = Q @ [u] in assert (relabel_precond f l u) \<bind> (\<lambda>_. assert (l u < 2 * card V \<and> relabel_effect f l u u < 2 * card V) \<bind> (\<lambda>_. let l = gap_relabel_effect f l u in return (l, Q)))))"
cosh_dist_unique: fixes a :: "proj2"   and b :: "proj2"   and c :: "proj2"   and p :: "proj2" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "c \<in> hyp2"   and "p \<in> S"   and "B\<^sub>\<real> cart2_pt a cart2_pt b cart2_pt p"   and "B\<^sub>\<real> cart2_pt a cart2_pt c cart2_pt p"   and "cosh_dist a b = cosh_dist a c" shows "b = c"
doesnt_modify_def: fixes c :: "'Com"   and x :: "'Var" shows "doesnt_modify c x = (\<forall>mds mem c' mds' mem'. \<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle> \<longrightarrow> mem x = mem' x)"
P1: shows "P1 \<in> carrier_mat (m - 1) (m - 1)"
round_robin_reschedule_invar_None: fixes final :: "'c \<Rightarrow> bool"   and r :: "('b, 'a, 'c, 'd, 'e, 'f) semantics"   and convert_RA :: "'b \<Rightarrow>f nat \<Rightarrow> 'f list"   and t0 :: "'a"   and queue :: "'a list"   and n0 :: "nat"   and s :: "('b, 'a) locks \<times> (('a \<Rightarrow> ('c \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('a \<Rightarrow> 'e wait_set_status option) \<times> 'a set"   and t :: "'a"   and \<sigma>' :: "'a list \<times> nat" assumes "multithreaded_base.round_robin_reschedule final r convert_RA t0 queue n0 s = \<lfloor>(t, None, \<sigma>')\<rfloor>"   and "Round_Robin.round_robin_invar (queue, n0) (dom (thr s))"   and "t0 \<in> set queue" shows "Round_Robin.round_robin_invar \<sigma>' (dom (thr s))"
A_NO_SPAWN: fixes s :: "'a list"   and ce :: "'a list multiset"   and s' :: "'a list" assumes "((s, ce), e, s', ce) \<in> trss fg"   and "c = {#s#} + ce"   and "c' = {#s'#} + ce" shows "P"
ys: shows "ys = LCons y0 (LCons y1 ys')"
implmt1_def: fixes G :: "prog" shows "implmt1 G = {(C, I). C \<noteq> Object \<and> (? c:class G C: I \<in> set (superIfs c))}"
sbh_step_preserves_write_sops_bound: fixes "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and sb' :: "'a memref list"   and m' :: "nat \<Rightarrow> nat"   and \<D>' :: "bool"   and \<O>' :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option"   and t :: "nat" assumes "(is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is', \<theta>', sb', m', \<D>', \<O>', \<R>', \<S>')"   and "\<forall>i\<in>\<Union> (fst ` store_sops is). i < t"   and "\<forall>i\<in>\<Union> (fst ` write_sops sb). i < t" shows "\<forall>i\<in>\<Union> (fst ` write_sops sb'). i < t"
splitTree_invpres_1: fixes s :: "('a, 'b) FingerTreeStruc"   and p :: "'b \<Rightarrow> bool"   and i :: "'b"   and l :: "('a, 'b) FingerTreeStruc"   and e :: "'a"   and a :: "'b"   and r :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.ft_invar s"   and "\<not> p i"   and "p (i + FingerTreeStruc.annot s)"   and "FingerTreeStruc.splitTree p i s = (l, (e, a), r)" shows "FingerTreeStruc.ft_invar l"
compat_append_right: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and zs :: "'b list" assumes "lexext_rev gt ys xs" shows "lexext_rev gt (ys @ zs) (xs @ zs)"
cINF_lower: fixes f :: "'d \<Rightarrow> 'c multiset"   and A :: "'d set"   and x :: "'d" assumes "subset_mset.bdd_below (f ` A)"   and "x \<in> A" shows "Inf (f ` A) \<subseteq># f x"
diagvector_mult_vector: fixes x :: "'b \<Rightarrow> 'a"   and y :: "('a, 'b) Finite_Cartesian_Product.vec" shows "diagvector x *v y = (\<chi>i. x i * y $h i)"
bind_option_def: fixes return :: "'e option \<Rightarrow> 'f"   and bind :: "'f \<Rightarrow> ('e option \<Rightarrow> 'f) \<Rightarrow> 'f" shows "bind_option return bind \<equiv> rec_optionT (\<lambda>x f. OptionT (bind x (\<lambda>x. case x of None \<Rightarrow> return None | Some y \<Rightarrow> run_option (f y))))"
COMB_init_def: fixes h :: "nat list"   and init :: "nat list" shows "COMB_init h init = Sum_pmf (8 / 10) (fst BIT init) (fst (Partial_Cost_Model.embed (rTS h)) init)"
contains_propagates_fdisj: fixes n :: "nat"   and g :: "form"   and h :: "form" assumes "infinite (deriv s)"   and "contains f n (0, FDisj g h)" shows "\<exists>y. contains f (n + y) (0, g) \<and> contains f (n + y) (0, h)"
simps_1: shows "combine \<langle>\<rangle> \<langle>\<rangle> = \<langle>\<rangle>"
seq_inf_distrib: 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\<^sub>0 :: "'b::refinement_lattice"   and d\<^sub>1 :: "'b::refinement_lattice" assumes "seq_finite_conjunctive (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) (c::'b::refinement_lattice) ((d\<^sub>0::'b::refinement_lattice) \<sqinter> (d\<^sub>1::'b::refinement_lattice)) = seq c d\<^sub>0 \<sqinter> seq c d\<^sub>1"
keq: fixes x :: "real \<times> 'a" assumes "x \<in> B" shows "k x = h' x"
sconst_streams: fixes x :: "'d"   and A :: "'d set" assumes "x \<in> A" shows "sconst x \<in> streams A"
is_Zp_cauchy_imp_res_eventually_const: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and n :: "nat" assumes "is_Zp_cauchy s" obtains N :: "nat" and r :: "int" where "r \<in> carrier (residue_ring (p ^ n))" and "\<And>m. N < m \<Longrightarrow> s m n = r"
that: assumes "\<And>i. p i \<noteq> 0 \<Longrightarrow> 1 \<le> i \<and> i \<le> n"   and "(\<Sum>i\<le>n. p i * i) = n" shows "thesis"
atLeastLessThan_iff: fixes i :: "'a"   and l :: "'a"   and u :: "'a" shows "(i \<in> {l..<u}) = (l \<le> i \<and> i < u)"
prop_unfold_equiv_helper_2: fixes X :: "'b ltln set" assumes "finite (abs_ltln\<^sub>P ` X)" shows "card (abs_ltln\<^sub>Q ` X) \<le> card (abs_ltln\<^sub>P ` X)"
branch_inv_Branching: fixes b :: "bool list"   and l :: "tree"   and r :: "tree" shows "branch b (Branching l r) = (\<exists>a b'. b = a # b' \<and> (a \<longrightarrow> branch b' l) \<and> (\<not> a \<longrightarrow> branch b' r))"
impl_of_cases: fixes y :: "('b \<times> 'c) list" assumes "y \<in> {xs. (distinct \<circ> map fst) xs}" obtains x :: "('b, 'c) alist" where "y = alist.impl_of x"
mult2_alt_def: fixes b :: "bool"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" shows "mult2_alt b ns s = {(X, Y). \<exists>X1 X2 Y1 Y2. X = X1 + X2 \<and> Y = Y1 + Y2 \<and> (X1, Y1) \<in> multpw ns \<and> (b \<or> Y2 \<noteq> {#}) \<and> (\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s))}"
uint8_test_bit_def: fixes x :: "uint8"   and n :: "integer" shows "uint8_test_bit x n = (if n < 0 \<or> 7 < n then undefined (!!) x n else x !! nat_of_integer n)"
swap_Var_simp: fixes ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and xs :: "'varSort"   and x :: "'var" shows "Var xs x #[y1 \<and> y2]_ys = Var xs (x @xs[y1 \<and> y2]_ys)"
\<H>_def: shows "\<H> \<equiv> \<G> \<union> {D. \<exists>C\<in>\<F> - \<G>. D face_of C \<and> aff_dim D < aff_dim T}"
analytically_valid_def: fixes s :: "'a set"   and F :: "'a \<Rightarrow> 'b"   and i :: "'a" shows "analytically_valid s F i \<equiv> (\<forall>a\<in>s. partially_vector_differentiable F i a) \<and> continuous_on s F \<and> integrable lborel (\<lambda>p. partial_vector_derivative F i p * indicator s p) \<and> (\<lambda>x. integral UNIV (\<lambda>y. partial_vector_derivative F i (y *\<^sub>R i + x *\<^sub>R \<Sum> (Basis - {i})) * indicator s (y *\<^sub>R i + x *\<^sub>R \<Sum> (Basis - {i})))) \<in> borel_measurable lborel"
assms_5: fixes xs :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ys :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" shows "set (comb xs ys) = set xs \<union> set ys"
whisker_right: fixes f :: "'a"   and \<nu> :: "'a"   and \<mu> :: "'a" assumes "ide f"   and "seq \<nu> \<mu>" shows "\<nu> \<cdot> \<mu> \<star> f = (\<nu> \<star> f) \<cdot> (\<mu> \<star> f)"
path_connectedin_subtopology: fixes X :: "'b topology"   and S :: "'b set"   and T :: "'b set" shows "path_connectedin (subtopology X S) T = (path_connectedin X T \<and> T \<subseteq> S)"
analz_auth_data_out: fixes x :: "rat"   and H :: "msg set"   and y :: "rat" assumes "Pri_AgrK x \<notin> analz H"   and "Pri_AgrK y \<notin> analz H" shows "analz (insert (Auth_Data x y) H) = insert (Auth_Data x y) (analz H)"
nth_takefill: fixes m :: "nat"   and n :: "nat"   and fill :: "'a"   and l :: "'a list" assumes "m < n" shows "takefill fill n l ! m = (if m < length l then l ! m else fill)"
Ide_implies_Can: fixes t :: "'c local.term" assumes "Ide t" shows "Can t"
complex_homo_coords_eq_sym: fixes z1 :: "complex_homo_coords"   and z2 :: "complex_homo_coords" assumes "z1 \<approx> z2" shows "z2 \<approx> z1"
cone_empty: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "cone (h, {}) = range (\<lambda>c. c \<cdot> h)"
lossless2: fixes s2 :: "'s2"   and x :: "'a" assumes "bad2 s2"   and "I2 s2"   and "x \<in> outs_\<I> \<I>" shows "lossless_spmf (oracle2 s2 x)"
negative_resolvent_is_resolvent: shows "less_restrictive resolvent ordered_negative_resolvent"
comp_ide_arr: fixes b :: "(bool, 'arr list) concrete_category.arr"   and f :: "(bool, 'arr list) concrete_category.arr" assumes "ide b"   and "seq b f" shows "b \<cdot> f = f"
set_tree_union: fixes t2 :: "('a \<times> 'b) tree"   and t1 :: "('a \<times> 'b) tree" assumes "Tree2.bst t2" shows "Tree2.set_tree (local.union t1 t2) = Tree2.set_tree t1 \<union> Tree2.set_tree t2"
inst: shows "wtc c pc (wtl (take pc ins) c 0 s\<^sub>0) = wti c pc (\<tau>s ! pc)"
simps_9: fixes f :: "'c \<Rightarrow> 'b"   and x2 :: "'c" shows "map_option f \<lfloor>x2\<rfloor> = \<lfloor>f x2\<rfloor>"
CL_def: fixes cl :: "'b potype" shows "CL cl \<equiv> cl \<in> CompleteLattice"
mult_mm_cancel_l: fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det M \<noteq> 0"   and "M *\<^sub>m\<^sub>m A = M *\<^sub>m\<^sub>m B" shows "A = B"
surjective: fixes r :: "('d, 'e) edge_scheme" shows "r = \<lparr>src = src r, tgt = tgt r, \<dots> = edge.more r\<rparr>"
expand_inv_impl: fixes \<phi> :: "'b ltlr"   and n :: "('b, 'c) node_scheme"   and ns :: "('b, 'd) node_scheme set"   and \<psi> :: "'b ltlr"   and ns' :: "('b, 'd) node_scheme set"   and n' :: "('b, 'c) node_scheme"   and nm :: "nat" assumes "expand_inv \<phi> (n, ns)"   and "\<psi> \<in> new n"   and "old_next_pair ` ns \<subseteq> old_next_pair ` ns'"   and "expand_inv_result \<phi> ns'"   and "n' = n\<lparr>new := new n - {\<psi>}, old := {\<psi>} \<union> old n\<rparr> \<or> n' = n\<lparr>new := new1 \<psi> \<union> (new n - {\<psi>}), old := {\<psi>} \<union> old n, next := next1 \<psi> \<union> next n\<rparr> \<or> n' = n\<lparr>name := nm, new := new2 \<psi> \<union> (new n - {\<psi>}), old := {\<psi>} \<union> old n\<rparr>" shows "((n', ns'), n, ns) \<in> expand_ord \<phi> \<and> expand_inv \<phi> (n', ns')"
trace_callee_eq'_run_gpv: fixes callee1 :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and callee2 :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf"   and A :: "'b set"   and s1 :: "'a"   and s2 :: "'d"   and I1 :: "'a \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>"   and I2 :: "'d \<Rightarrow> bool"   and gpv :: "('e, 'b, 'c) gpv" assumes "A \<turnstile>\<^sub>C callee1(s1) \<approx> callee2(s2)"   and "callee_invariant_on callee1 I1 \<I>"   and "callee_invariant_on callee2 I2 \<I>"   and "\<I> \<turnstile>g gpv \<surd>"   and "outs_gpv \<I> gpv \<subseteq> A"   and "I1 s1"   and "I2 s2" shows "run_gpv callee1 gpv s1 = run_gpv callee2 gpv s2"
Exh_tr: fixes t :: "tr" shows "t = \<bottom> \<or> t = TT \<or> t = FF"
intro: fixes z :: "'a"   and U :: "'a set"   and inverse :: "'a \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "z \<in> U"   and "\<And>a. a \<in> U \<Longrightarrow> inverse a \<in> U"   and "\<And>a. a \<in> U \<Longrightarrow> f z a = a"   and "\<And>a. a \<in> U \<Longrightarrow> f (inverse a) a = z" shows "group_ow_axioms U f z inverse"
split_length: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and ts :: "('b btree \<times> 'b) list"   and x :: "'b"   and ls :: "('b btree \<times> 'b) list"   and rs :: "('b btree \<times> 'b) list" assumes "split split"   and "split ts x = (ls, rs)" shows "length ls + length rs = length ts"
A: fixes c :: "'b"   and f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "c \<noteq> (0::'b)" shows "(\<lambda>x. c * f x) \<in> L F f"
list_all2_reorder_left_invariance: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and xs' :: "'a list" assumes "list_all2 R xs ys"   and "mset xs' = mset xs" shows "\<exists>ys'. list_all2 R xs' ys' \<and> mset ys' = mset ys"
rmigneq: fixes s :: "bdd"   and ni1 :: "nat"   and n1 :: "nat ifex"   and ni2 :: "nat"   and n2 :: "nat ifex" assumes "bdd_sane s"   and "Rmi_g ni1 n1 s"   and "Rmi_g ni2 n2 s"   and "ni1 \<noteq> ni2" shows "n1 \<noteq> n2"
e2lenls_lenls: fixes f :: "recf"   and xs :: "nat list"   and ls :: "nat list"   and ss :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" shows "e2lenls (encode_config ((f, xs, ls) # ss, rv)) = length ls"
less: shows "integral\<^sup>N (count_space (\<^bold>E `` X)) (weight \<Gamma>) < integral\<^sup>N (count_space X) (weight \<Gamma>)"
zero_vec_def: fixes n :: "nat" shows "0\<^sub>v n \<equiv> Matrix.vec n (\<lambda>i. 0::'b)"
functor_axioms: shows "functor J (\<cdot>\<^sub>B) (F \<circ> D)"
path_conforms_with_strategy_prefix: fixes p :: "Player"   and P :: "'a llist"   and \<sigma> :: "'a \<Rightarrow> 'a"   and P' :: "'a llist" assumes "G'.path_conforms_with_strategy p P \<sigma>"   and "lprefix P' P" shows "G'.path_conforms_with_strategy p P' \<sigma>"
divides_ff_refl: fixes x :: "'b fract" shows "divides_ff x x"