hcju/isabelleps huggingface dataset:
irreducibleE: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and p :: "'c" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.irreducible times one zero p" obtains "p \<noteq> zero" and "\<not> dvd.dvd times p one" and "\<And>a b. p = times a b \<Longrightarrow> dvd.dvd times a one \<or> dvd.dvd times b one"
fold_A_chamber_system_image_fixespointwise: fixes F :: "'a set" assumes "F \<in> ChamberComplex.chamber_system A" shows "fixespointwise fold_A (fold_A ` F)"
l2wi: fixes l :: "('b word \<times> 'b word) list" shows "wordinterval_to_set (l2wi l) = (\<Union>(i, j)\<in>set l. {i..j})"
type_wf_impl: shows "type_wf = ShadowRootClass.type_wf"
prime_elem_multiplicity_prod_distrib: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "zero \<notin> f ` A"   and "finite A" shows "factorial_semiring.multiplicity times one p (comm_monoid_mult.prod times one f A) = (\<Sum>x\<in>A. factorial_semiring.multiplicity times one p (f x))"
fgw: fixes w :: "'a" assumes "v \<noteq> w" shows "f w = g w"
Suc_n_not_le_n: fixes n :: "nat" shows "\<not> order.greater_eq n (Suc n)"
Av: shows "A *\<^sub>v v = k \<cdot>\<^sub>v v"
IH: fixes ys :: "'a list" assumes "x \<in> set xs" shows "\<exists>zs. remove_cycles xs x ys = x # zs \<and> x \<notin> set zs"
lsc_at_PInfty: fixes f :: "'b \<Rightarrow> ereal"   and x0 :: "'b" assumes "f x0 = \<infinity>" shows "lsc_at x0 f = isCont f x0"
tagged_division_ofD_3: fixes s :: "('c \<times> 'c set) set"   and i :: "'c set"   and x :: "'c"   and K :: "'c set" assumes "s tagged_division_of i"   and "(x, K) \<in> s" shows "K \<subseteq> i"
sgn_inverse: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.field_abs_sgn divide inverse times one plus zero minus uminus abs sgn" shows "sgn (inverse a) = inverse (sgn a)"
del_UnEdge_even: fixes v :: "'v"   and w :: "'w"   and v' :: "'v" assumes "(v, w, v') \<in> E"   and "finite E" shows "(v \<in> odd_nodes_set (del_unEdge v w v' G)) = even (degree v G)"
assms_2: shows "A \<in> Ob\<^bsub>AA\<^esub>"
unique_eigenvector_jnf: fixes A :: "'b mat"   and n_as :: "(nat \<times> 'b) list"   and lam :: "'b"   and v :: "'b Matrix.vec"   and w :: "'b Matrix.vec" assumes "jordan_nf A n_as"   and "order lam (char_poly A) = 1"   and "eigenvector A v lam"   and "eigenvector A w lam" shows "\<exists>a. v = a \<cdot>\<^sub>v w"
F_T: fixes s :: "'b event list"   and X :: "'b event set"   and P :: "'b process" assumes "(s, X) \<in> F P" shows "s \<in> T P"
connected_component_eq: fixes y :: "'c"   and S :: "'c set"   and x :: "'c" assumes "y \<in> connected_component_set S x" shows "connected_component_set S y = connected_component_set S x"
r_or_def: shows "r_or \<equiv> Cn 2 r_sign [r_mul]"
assms_2: shows "externals (ioa.asig A) = externals (ioa.asig B)"
map_simp_1: shows "D.map J.Zero = local.dom f0"
wf_action_translation_on_action_locD: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and E' :: "('a \<times> ('b, 'a) obs_event action) llist"   and A :: "nat set"   and f :: "nat \<Rightarrow> nat"   and a :: "nat"   and P :: "'c prog" assumes "wf_action_translation_on E E' A f"   and "a \<in> A" shows "action_loc P E a = action_loc P E' (f a)"
da_elim_cases_48: fixes Env :: "env"   and B :: "lname set"   and C :: "qtname"   and sig :: "sig"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1l (Methd C sig)\<guillemotright> A" obtains m :: "qtname \<times> methd" where "methd (prg Env) C sig = Some m" and "Env\<turnstile> B \<guillemotright>\<langle>Body (declclass m) (stmt (mbody (mthd m)))\<rangle>\<guillemotright> A"
k2: fixes i :: "nat" assumes "i < k" shows "P ((A @ B) ! i)"
step: fixes p :: "('b, 'c, 'a, 'd) seqp"   and l :: "'d"   and \<xi> :: "'b"   and a :: "'c seq_action"   and \<xi>' :: "'b"   and q :: "('b, 'c, 'a, 'd) seqp"   and l' :: "'d"   and pp :: "('b, 'c, 'a, 'd) seqp" assumes "p \<in> cterms \<Gamma>"   and "l \<in> labels \<Gamma> p"   and "P (\<xi>, l)"   and "((\<xi>, p), a, \<xi>', q) \<in> seqp_sos \<Gamma>"   and "((\<xi>, p), a, \<xi>', q) \<in> automaton.trans A"   and "l' \<in> labels \<Gamma> q"   and "(\<xi>, pp) \<in> reachable A I"   and "p \<in> sterms \<Gamma> pp"   and "(\<xi>', q) \<in> reachable A I"   and "I a" shows "P (\<xi>', l')"
continuous_map_compose: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and X'' :: "'e topology"   and g :: "'d \<Rightarrow> 'e" assumes "continuous_map X X' f"   and "continuous_map X' X'' g" shows "continuous_map X X'' (g \<circ> f)"
zero: fixes V :: "'b set" assumes "vectorspace V" shows "(0::'b) \<in> V"
eval_def: fixes b :: "'a"   and c :: "'a" shows "eval b c \<equiv> SOME e. \<guillemotleft>e : local.prod (exp b c) b \<rightarrow> c\<guillemotright> \<and> (\<forall>a g. ide a \<and> \<guillemotleft>g : local.prod a b \<rightarrow> c\<guillemotright> \<longrightarrow> (\<exists>!f. \<guillemotleft>f : a \<rightarrow> exp b c\<guillemotright> \<and> g = C e (local.prod f b)))"
Times_same_infinite_bij_betw: fixes A :: "'b set" assumes "infinite A" shows "\<exists>f. bij_betw f (A \<times> A) A"
spanningE: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "spanning H G" obtains "subgraph H G \<and> verts G = verts H"
LeftDerives1_Derive: fixes \<alpha> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and \<gamma> :: "symbol list" assumes "LeftDerives1 \<alpha> i r \<gamma>" shows "Derive \<alpha> [(i, r)] = \<gamma>"
mfinalI: fixes s :: "('f, 'e) locks \<times> (('e \<Rightarrow> ('x \<times> 'f \<Rightarrow>f nat) option) \<times> 'g) \<times> ('e \<Rightarrow> 'h wait_set_status option) \<times> 'e set" assumes "\<And>t x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<Longrightarrow> final x \<and> ln = no_wait_locks \<and> wset s t = None" shows "mfinal s"
Model_def: fixes I :: "('a, 'b, 'c, 'd) Interpretation"   and Ax :: "('a, 'b) Axioms" shows "Model I Ax \<equiv> Interpretation I \<and> Axioms Ax \<and> Model_axioms I Ax"
lex_def: fixes r :: "('a \<times> 'a) set" shows "lex r = \<Union> (range (lexn r))"
perm_append2: fixes xs :: "'a list"   and ys :: "'a list"   and l :: "'a list" assumes "xs <~~> ys" shows "xs @ l <~~> ys @ l"
sel_1: fixes z :: "gauss_int" shows "ReZ (gauss_cnj z) = ReZ z"
Cons_1: fixes vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'b) list" assumes "p \<in> set (snd (sym_preproc_addnew gs vs fs v))" obtains "p \<in> set fs"   | g :: "'t \<Rightarrow>\<^sub>0 'b" and s :: "'a" where "g \<in> set gs" and "p = monom_mult (1::'b) s g"
ordertype_VWF_eq_iff: fixes \<alpha> :: "V"   and A :: "V set" assumes "Ord \<alpha>"   and "small A" shows "(tp A = \<alpha>) = (\<exists>f. bij_betw f A (elts \<alpha>) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. (f x < f y) = ((x, y) \<in> VWF)))"
map_of_reinsert_neq_None: fixes Ca :: "'a"   and D :: "'a"   and fs :: "('c \<times> 'b) list"   and F :: "'c" assumes "Ca \<noteq> D" shows "map_of (map (\<lambda>(F, y). ((F, Ca), y)) fs) (F, D) = None"
sum_group: fixes I :: "'f set"   and G :: "'f \<Rightarrow> ('g, 'h) monoid_scheme" assumes "\<And>i. i \<in> I \<Longrightarrow> Group.group (G i)" shows "Group.group (sum_group I G)"
abs_eq: fixes xa :: "'a"   and x :: "'a" shows "Fraction_Field.Fract xa x = abs_fract (if x = (0::'a) then (0::'a, 1::'a) else (xa, x))"
RPDs_MSkip: fixes n :: "nat" shows "RPDs (MSkip n) \<subseteq> MSkip ` {i. i \<le> n}"
lambdabinder1_def: shows "lambdabinder1 \<equiv> map_fun (map_fun id eval\<o>) make\<Pi>\<^sub>1 (\<lambda>\<phi> u s w. \<exists>x. \<nu>\<upsilon> x = u \<and> \<phi> x s w)"
add_strict_right_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ordered_cancel_ab_semigroup_add plus minus less_eq less"   and "less a b" shows "less (plus a c) (plus b c)"
option_rel_def: fixes R :: "('a \<times> 'b) set" shows "\<langle>R\<rangle>option_rel \<equiv> {(Some a, Some a') |a a'. (a, a') \<in> R} \<union> {(None, None)}"
exec_block: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and bdy :: "('e, 'd, 'f) com"   and init :: "'e \<Rightarrow> 'e"   and s :: "'e"   and t :: "'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and u :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Normal t"   and "\<Gamma>\<turnstile> \<langle>c s t,Normal (return s t)\<rangle> \<Rightarrow> u" shows "\<Gamma>\<turnstile> \<langle>block init bdy return c,Normal s\<rangle> \<Rightarrow> u"
iprefix_take_eq_iprefix_take_ex: fixes xs :: "'b list"   and f :: "nat \<Rightarrow> 'b" shows "(f \<Down> length xs = xs) = (\<exists>n. f \<Down> n = xs)"
monad_writerT: fixes return :: "'d \<times> 'e list \<Rightarrow> 'f"   and bind :: "'f \<Rightarrow> ('d \<times> 'e list \<Rightarrow> 'f) \<Rightarrow> 'f" assumes "monad return bind" shows "monad (return_writer return) (bind_writer return bind)"
open_Union: fixes K :: "'a set set" assumes "Ball K open" shows "open (\<Union> K)"
S\<^sub>i: fixes x :: "i"   and y :: "i" shows "E x \<^bold>\<and> E y \<^bold>\<leftarrow> E (x \<cdot> y)"
generate_ring_in_carrier: fixes H :: "'a set"   and h :: "'a" assumes "H \<subseteq> carrier R"   and "h \<in> generate_ring R H" shows "h \<in> carrier R"
bind_left_unit': fixes c :: "'f"   and P :: "'d \<Rightarrow> ('e \<times> 'd) option" shows "unit\<^sub>S\<^sub>E c ;- P = P"
independent_in_carrier: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list" assumes "ring R"   and "ring.independent R K Us" shows "set Us \<subseteq> carrier R"
init: shows "init Accessible_dfa \<in> states Accessible_dfa"
before_in_index1: fixes l :: "'c list"   and x :: "'c"   and y :: "'c" assumes "set l = {x, y}"   and "length l = 2"   and "x \<noteq> y" shows "(if x < y in l then 0 else 1) = index l x"
tendsto_Lambert_W': fixes f :: "'a \<Rightarrow> real"   and L :: "real"   and F :: "'a filter" assumes "(f \<longlongrightarrow> L) F"   and "(\<forall>\<^sub>F x in F. - exp (- 1) \<le> f x) \<or> - exp (- 1) < L"   and "L < 0" shows "((\<lambda>x. Lambert_W' (f x)) \<longlongrightarrow> Lambert_W' L) F"
axioms_6: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool" assumes "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf known_ptr known_ptrs" shows "l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs"
asprod_pos_mono: fixes w :: "int"   and x :: "ant"   and y :: "ant" assumes "0 < w" shows "(w *\<^sub>a x \<le> w *\<^sub>a y) = (x \<le> y)"
sorted_mult_spvec_spmat: fixes c :: "(nat \<times> 'a) list"   and B :: "(nat \<times> (nat \<times> 'a) list) list"   and a :: "(nat \<times> 'a) list" assumes "sorted_spvec c"   and "sorted_spmat B" shows "sorted_spvec (mult_spvec_spmat c a B)"
strongBisimWeakBisim: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim>\<^sup>s Q" shows "P \<approx>\<^sup>s Q"
eta_lift: fixes s :: "dB"   and t :: "dB"   and i :: "nat" assumes "s \<rightarrow>\<^sub>\<eta> t" shows "lift s i \<rightarrow>\<^sub>\<eta> lift t i"
continuous_on_open_vimage: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "open s" shows "continuous_on s f = (\<forall>B. open B \<longrightarrow> open (f -` B \<inter> s))"
indexed_pmult_index_free: fixes P :: "'d multiset \<Rightarrow> 'a"   and j :: "'d"   and i :: "'d" assumes "index_free P j"   and "i \<noteq> j" shows "index_free (P \<Otimes> i) j"
mult_right_strict_mono: fixes n :: "nat" assumes "0 < n" shows "strict_mono (\<lambda>x. x * n)"
periodic_fun_simple'_def: fixes f :: "'a \<Rightarrow> 'b" shows "periodic_fun_simple' f \<equiv> \<forall>x. f (x + (1::'a)) = f x"
simps_9: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "not\<^sub>n (\<phi> R\<^sub>n \<psi>) = not\<^sub>n \<phi> U\<^sub>n not\<^sub>n \<psi>"
tfr\<^sub>s\<^sub>e\<^sub>t_def: fixes M :: "('fun, 'var) Term.term set" shows "tfr\<^sub>s\<^sub>e\<^sub>t M \<equiv> \<forall>s\<in>SMP M - range Var. \<forall>t\<in>SMP M - range Var. (\<exists>\<delta>. Unifier \<delta> s t) \<longrightarrow> \<Gamma> s = \<Gamma> t"
pre_zeta_def: fixes a :: "real"   and s :: "complex" shows "pre_zeta a s = pre_zeta_aux (nat (1 - \<lceil>Re s / 2\<rceil>)) a s"
unit_base: fixes v :: "state \<Rightarrow> unit" shows "basevars v"
IH2_4: fixes j :: "nat" assumes "hi < j"   and "j < length xs''" shows "xs''' ! j = xs'' ! j"
inducts: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and S :: "'a set" assumes "orthogonal \<Delta> f S" obtains "\<And>x y. \<lbrakk>edge \<Delta> x y; x \<in> S; y \<notin> S\<rbrakk> \<Longrightarrow> f (x, y) = capacity \<Delta> (x, y)" and "\<And>x y. \<lbrakk>edge \<Delta> x y; x \<notin> S; y \<in> S\<rbrakk> \<Longrightarrow> f (x, y) = 0"
cqt_basic_3: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and \<psi> :: "'a \<Rightarrow> \<o>" shows "[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<equiv> \<psi> \<alpha>) \<^bold>\<rightarrow> ((\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<forall>\<alpha>. \<psi> \<alpha>)) in v]"
Restrp_symmetric: shows "symmetric UNIV ((\<bowtie>) \<restriction> A)"
u: shows "u \<le> t \<and> u \<in>\<^sub>A frontier (c_pts c loc)"
unreach_on_path: fixes a :: "'a"   and Q :: "'a set"   and b :: "'a" assumes "a \<in> \<emptyset> Q b" shows "a \<in> Q"
nrows_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "nrows A = CARD('f)"
finite_primes_less: fixes p :: "nat" shows "finite {q. prime q \<and> q < p}"
fds_converges_remainder_iff: fixes m :: "nat"   and f :: "'b fds"   and s :: "'b" shows "fds_converges (fds_remainder m f) s = fds_converges f s"
1: fixes f :: "'a" assumes "hom'.arr f" shows "hom.arr (d\<^sub>1 \<star> f \<star> e\<^sub>0)"
set_quicksort: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "set (ord.quicksort less xs) = set xs"
pos: shows "0 < 1 + s * hb * (inverse (bs ! i) * ln x powr - (1 + e))"
\<pi>_1: shows "\<pi> permutes {0..<length xs}"
in_filtermap_iff: fixes f' :: "real \<Rightarrow> 'f"   and h' :: "real \<Rightarrow> real"   and F' :: "real filter"   and g' :: "real \<Rightarrow> 'f" shows "(f' \<in> L (filtermap h' F') g') = ((\<lambda>x. f' (h' x)) \<in> L' F' (\<lambda>x. g' (h' x)))"
fund_circuit_in_def: fixes \<E> :: "'a set"   and x :: "'a"   and B :: "'a set" shows "\<E>.fund_circuit_in \<E> x B \<equiv> matroid.fund_circuit \<E> (\<E>.indep_in \<E>) x B"
path_Exit_source_1: fixes as :: "'edge list"   and n' :: "'node" assumes "(_Exit_) -as\<rightarrow>* n'" shows "n' = (_Exit_)"
upper_dim_axiom_def: shows "UpperDimAxiom \<equiv> \<forall>A B C P Q. P \<noteq> Q \<and> Cong A P A Q \<and> Cong B P B Q \<and> Cong C P C Q \<longrightarrow> Bet A B C \<or> Bet B C A \<or> Bet C A B"
Fun_canonical: fixes v :: "trm"   and T\<^sub>1 :: "POPLmarkRecord.type"   and T\<^sub>2 :: "POPLmarkRecord.type" assumes "[] \<turnstile> v : T\<^sub>1 \<rightarrow> T\<^sub>2"   and "v \<in> value" shows "\<exists>t S. v = (\<lambda>:S. t)"
induced_automorphism_sym: shows "OpposedThinChamberComplexFoldings.induced_automorphism X g f = \
"
map_of_SomeD: fixes xs :: "('c \<times> 'b) list"   and k :: "'c"   and y :: "'b" assumes "map_of xs k = \<lfloor>y\<rfloor>" shows "(k, y) \<in> set xs"
continuous_def: fixes F :: "'d filter"   and f :: "'d \<Rightarrow> 'e" shows "continuous F f = (f \<longlongrightarrow> f (netlimit F)) F"
bijective_def: fixes R :: "('c \<times> 'd) set" shows "bijective R \<equiv> (\<forall>x y z. (x, y) \<in> R \<and> (x, z) \<in> R \<longrightarrow> y = z) \<and> (\<forall>x y z. (x, z) \<in> R \<and> (y, z) \<in> R \<longrightarrow> x = y)"
list_of_set_set_def: fixes s :: "'a set" shows "list_of_set_set s = {l. set l = s \<and> distinct l}"
assms: fixes i :: "nat" assumes "i \<le> x" shows "f1 ! i = f2 ! i"
mS: shows "Sigma_Algebra.measure lebesgue S \<le> 2 * e * (2 * m) ^ (CARD('n) - 1)"
assms_2: shows "f \<cdot> g = f \<cdot> g'"
extend_measure_def: fixes \<Omega> :: "'c set"   and I :: "'d set"   and G :: "'d \<Rightarrow> 'c set"   and \<mu> :: "'d \<Rightarrow> ennreal" shows "extend_measure \<Omega> I G \<mu> = (if (\<exists>\<mu>''. (\<forall>i\<in>I. \<mu>'' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G ` I)) \<mu>'') \<and> \<not> (\<forall>i\<in>I. \<mu> i = 0) then measure_of \<Omega> (G ` I) (SOME \<mu>''. (\<forall>i\<in>I. \<mu>'' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G ` I)) \<mu>'') else sigma \<Omega> (G ` I))"
Bot_map_not_empty: fixes B :: "'f \<times> 'l" assumes "B \<in> Bot_FL" shows "\<G>_F_L B \<noteq> {}"
semilattice_order_axioms: shows "semilattice_order sup (\<lambda>x y. y \<le> x) (\<lambda>x y. y < x)"
density_cong: fixes f :: "'d \<Rightarrow> ennreal"   and M :: "'d measure"   and f' :: "'d \<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'"
S_hyp2_S_cart2_append1: fixes p :: "proj2"   and q :: "proj2"   and a :: "proj2"   and l :: "proj2_line" assumes "p \<noteq> q"   and "p \<in> S"   and "q \<in> S"   and "a \<in> hyp2"   and "proj2_incident p l"   and "proj2_incident q l"   and "proj2_incident a l" shows "\<exists>k>0. k < 1 \<and> cart2_append1 a = k *\<^sub>R cart2_append1 q + (1 - k) *\<^sub>R cart2_append1 p"
vdomain_vcomp: fixes r :: "V"   and s :: "V" shows "\<D>\<^sub>\<circ> (r \<circ>\<^sub>\<circ> s) = s -`\<^sub>\<circ> \<D>\<^sub>\<circ> r"
l_inv_ex: fixes x :: "'c" assumes "x \<in> carrier G2" shows "\<exists>y\<in>carrier G2. y \<otimes>\<^bsub>G2\<^esub> x = \<one>\<^bsub>G2\<^esub>"
equiv'_def: shows "equiv' \<equiv> \<lambda>a b. a \<sim> b \<and> PA a \<and> PB b"
simps_85: fixes f1 :: "'a"   and f2 :: "'b basic_type \<Rightarrow> 'a"   and f3 :: "'b basic_type \<Rightarrow> 'a"   and f4 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f5 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f6 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f7 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f8 :: "'b OCL_Types.type \<Rightarrow> 'a"   and f9 :: "String.literal \<rightharpoonup>\<^sub>f 'b OCL_Types.type \<Rightarrow> 'a"   and x5 :: "'b OCL_Types.type" shows "(case Set x5 of OclSuper \<Rightarrow> f1 | x[1] \<Rightarrow> f2 x | x[?] \<Rightarrow> f3 x | Collection x \<Rightarrow> f4 x | Set x \<Rightarrow> f5 x | OrderedSet x \<Rightarrow> f6 x | Bag x \<Rightarrow> f7 x | Sequence x \<Rightarrow> f8 x | Tuple x \<Rightarrow> f9 x) = f5 x5"
K_subset: fixes i :: "int"   and j :: "int" assumes "i \<le> j" shows "K i j \<in> {i..j}"
cols_def: fixes sort1 :: "('b list \<Rightarrow> 'b) \<Rightarrow> 'b list list \<Rightarrow> 'b list list"   and xss :: "'c list list"   and n :: "nat" assumes "Radix_Sort sort1" shows "cols xss n = (\<forall>xs\<in>set xss. length xs = n)"
length_bl_of_nth: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "length (bl_of_nth n f) = n"
dist_les_2: fixes discr :: "bool discr"   and cfun' :: "Value \<rightarrow> Value" shows "B\<cdot>discr \<notsqsubseteq> Fn\<cdot>cfun'"
depth_swapLeaves_neither: fixes t :: "'a tree"   and c :: "'a"   and a :: "'a"   and b :: "'a"   and w\<^sub>a :: "nat"   and w\<^sub>b :: "nat" assumes "consistent t"   and "c \<noteq> a"   and "c \<noteq> b" shows "depth (swapLeaves t w\<^sub>a a w\<^sub>b b) c = depth t c"
abrupt_if_not_None: fixes y :: "abopt"   and c :: "bool"   and x :: "abopt" assumes "y \<noteq> None" shows "abrupt_if c x y = y"
H_face_cycle_succ_2: shows "H.face_cycle_succ vw = rev_H uw"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F F F"
Liminf_compose_continuous_antimono: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'a" assumes "continuous_on UNIV f"   and "antimono f"   and "F \<noteq> bot" shows "Liminf F (\<lambda>n. f (g n)) = f (Limsup F g)"
call_list_lengths_def: fixes p :: "lambda" shows "call_list_lengths p = {0, 1, 2, 3} \<union> (\<lambda>c. case c of App x xa xb \<Rightarrow> length xb | call.Let nat list call \<Rightarrow> 0) ` calls p"
finite_finite_vimage_IntI: fixes F :: "'a set"   and h :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "finite F"   and "\<And>y. y \<in> F \<Longrightarrow> finite (h -` {y} \<inter> A)" shows "finite (h -` F \<inter> A)"
return_def: fixes result :: "'c" shows "return result = Prog (\<lambda>h. Inr (result, h))"
has_lock_acquire_locks_implies_has_lock: fixes l :: "('d \<times> nat) option"   and t :: "'d"   and n :: "nat"   and t' :: "'d" assumes "has_lock (acquire_locks l t n) t'"   and "t \<noteq> t'" shows "has_lock l t'"
sheaf_spec_on_open_is_comm_ring: fixes U :: "'a set set" assumes "is_zariski_open U" shows "comm_ring (\<O> U) (add_sheaf_spec U) (mult_sheaf_spec U) (zero_sheaf_spec U) (one_sheaf_spec U)"
SLN_right: fixes P :: "assn" shows "P * Automation.SLN = P"
clock_repRels_def: shows "clock_repRels \<equiv> \<lambda>a. rel_ext (\<lambda>(s, s'). envObs a s = envObs a s')"
assoc: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "semigroup f" shows "f (f a b) c = f a (f b c)"
greaterThan_iff: fixes i :: "'d"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and k :: "'d" shows "(i \<in> ord.greaterThan less k) = less k i"