hcju/isabelleps huggingface dataset:
red_f_equiv: shows "empt_ord.Red_F_\<G> = Red_F_\<G>"
enat_of_option_simps_3: fixes n :: "nat"   and x :: "nat option" shows "(enat n = Dijkstra_Abstract.enat_of_option x) = (x = Some n)"
a3: fixes a :: "atom"   and b :: "atom" assumes "a \<in> S"   and "b \<notin> S"   and "sort_of a = sort_of b" shows "(a \<rightleftharpoons> b) \<bullet> x \<noteq> x"
Cfg''_2: shows "cfg1 \<turnstile> msg \<mapsto> cfg''"
borel_eq_closed: shows "borel = sigma UNIV (Collect closed)"
sys_reg_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "low_equal s1 s2" shows "sys_reg s1 = sys_reg s2"
ltl_prop_equiv_def: fixes \<phi> :: "'c ltl"   and \<psi> :: "'c ltl" shows "\<phi> \<equiv>\<^sub>P \<psi> \<equiv> \<forall>\<A>. \<A> \<Turnstile>\<^sub>P \<phi> = \<A> \<Turnstile>\<^sub>P \<psi>"
dim_eq_span: fixes S :: "('b, 'c) vec set"   and T :: "('b, 'c) vec set" assumes "S \<subseteq> T"   and "vec.dim T \<le> vec.dim S" shows "vec.span S = vec.span T"
liftable_set_extension_axioms: shows "liftable_set_extension {a, b, c} alts Fishb"
add_block_def: fixes bl :: "'a set"   and b :: "'a set" shows "add_block_sys.add_block bl b \<equiv> add_point_sys.add_block bl + {#b#}"
up_to_mono: fixes J :: "nat set"   and H :: "nat set" assumes "J \<subseteq> H"   and "finite H" shows "up_to J \<le> up_to H"
5: shows "w\<^sup>+ \<sqinter> top * prim_E w v e \<sqinter> - v = bot"
open_components: fixes u :: "'a set"   and S :: "'a set" assumes "open u"   and "S \<in> components u" shows "open S"
ground_clause: shows "is_ground_cls (subst_cls C \<sigma>)"
simps_2: fixes iface :: "iface"   and ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and m :: "'b common_primitive match_expr"   and rs :: "'b common_primitive rule list"   and allowed :: "'b wordinterval"   and denied1 :: "'b wordinterval" shows "no_spoofing_algorithm_executable iface ipassmt (Rule m Accept # rs) allowed denied1 = no_spoofing_algorithm_executable iface ipassmt rs (wordinterval_union allowed (get_exists_matching_src_ips_executable iface m)) denied1"
ideal_zero: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set" assumes "Ring R"   and "ideal R I" shows "\<zero>\<^bsub>R\<^esub> \<in> I"
map_Val_throw_False: fixes vs :: "val list"   and ws :: "val list"   and ex :: "expr"   and es :: "expr list" assumes "map Val vs = map Val ws @ throw ex # es" shows "False"
drop_def: shows "drop \<equiv> \<lambda>uu uua. rec_list (\<lambda>n. []) (\<lambda>x xs xsa n. case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> xsa m) uua uu"
subsetI: fixes A :: "'h set"   and B :: "'h set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B" shows "A \<subseteq> B"
app_insf2_LstSeq: fixes s :: "hf"   and k :: "hf"   and z :: "hf"   and k' :: "hf"   and y :: "hf" assumes "LstSeq s k z"   and "k' \<noteq> succ k" shows "app (insf s (succ k) y) k' = app s k'"
axioms_1: fixes C :: "('d, 'e, 'f) Category_scheme" assumes "Category C" shows "ExtCategory C"
same2: shows "(\<forall>p\<in>set (retrieve_polys qs (a # xa)). poly p x = 0) = (poly (qs ! a) x = 0 \<and> (\<forall>p\<in>set (retrieve_polys qs xa). poly p x = 0))"
bit_eqI: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "\<And>n. power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero \<Longrightarrow> bit a n = bit b n" shows "a = b"
comp_parallel_wiring: fixes C1 :: "('i, 'm, 'q, 'u) converter"   and C2 :: "('j, 'n, 's, 'w) converter"   and C3 :: "('k, 'o, 'r, 'v) converter"   and C4 :: "('l, 'p, 't, 'x) converter" shows "((C1 |\<^sub>= C2) |\<^sub>= C3 |\<^sub>= C4) \<odot> parallel_wiring = parallel_wiring \<odot> ((C1 |\<^sub>= C3) |\<^sub>= C2 |\<^sub>= C4)"
MapsToOpOp: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and X :: "'d"   and Y :: "'d" assumes "f maps\<^bsub>Op C\<^esub> X to Y" shows "f maps\<^bsub>C\<^esub> Y to X"
lim: shows "(\<lambda>n. ereal (a n)) \<longlonglongrightarrow> ereal L"
assms_5: shows "decisionVars \<subseteq> vars (elements (getM state))"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and trm :: "'b \<Rightarrow> 'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "elementary_category_with_terminal_object C one trm" shows "elementary_cartesian_category C pr0 pr1 one trm"
of_int_eq_of_int_power_cancel_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and b :: "int"   and w :: "nat"   and x :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(power.power one times (ring_1.of_int one plus zero minus b) w = ring_1.of_int one plus zero minus x) = (b ^ w = x)"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" shows "a1 \<R>\<lbrakk>\<cdot>\<rbrakk>R a2 = (\<exists>S. S \<in>S a1 \<and> \<lbrakk>S\<rbrakk> \<in>T a2)"
power_le_zero_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (power.power one times a n) zero = (0 < n \<and> (odd n \<and> less_eq a zero \<or> even n \<and> a = zero))"
inv_o_cancel: fixes f :: "'b \<Rightarrow> 'c" assumes "inj f" shows "inv f \<circ> f = id"
greek_less_def: fixes r :: "('b \<times> 'b) set" shows "greek_less r = lfp (nest r)"
one_local_ring_at_def: shows "pi.one_local_ring_at \<equiv> local.one_rel"
zero_less_diff: fixes n :: "nat"   and m :: "nat" shows "(0 < n - m) = (m < n)"
cases: fixes r :: "'a simple_match" obtains iiface :: "iface" and oiface :: "iface" and src :: "'a word \<times> nat" and dst :: "'a word \<times> nat" and proto :: "protocol" and sports :: "16 word \<times> 16 word" and dports :: "16 word \<times> 16 word" where "r = \<lparr>iiface = iiface, oiface = oiface, src = src, dst = dst, proto = proto, sports = sports, dports = dports\<rparr>"
simps_1: shows "decomp_rm\<^sub>e\<^sub>s\<^sub>t [] = []"
poly_mod_type_axioms: shows "poly_mod_type TYPE('a) p"
lookup_update: fixes k :: "'a"   and v :: "'b"   and al :: "('a, 'b) alist" shows "DAList.lookup (DAList.update k v al) = DAList.lookup al(k \<mapsto> v)"
cl_inter: shows "closure (existence_ivl t0 x0 \<inter> {..t0}) \<inter> closure (existence_ivl t0 x0 \<inter> {t0..}) = {t0}"
np1: shows "num_params (tm_poly t1) \<le> length I"
cc_def: shows "cc \<equiv> \<lambda>x. if x \<in> T then c x else 0"
bind_gpv_sel': fixes r :: "('g, 'e, 'f) gpv"   and f :: "'g \<Rightarrow> ('d, 'e, 'f) gpv" shows "the_gpv (r \<bind> f) = the_gpv r \<bind> (\<lambda>generat. if is_Pure generat then the_gpv (f (generat.result generat)) else return_spmf (IO (generat.output generat) (\<lambda>input. continuation generat input \<bind> f)))"
dom_null: shows "Right_a.dom Right_a.null = Right_a.null"
lens_comp_indep_cong: fixes Z :: "'b \<Longrightarrow> 'c"   and X :: "'d \<Longrightarrow> 'b"   and Y :: "'e \<Longrightarrow> 'b" assumes "mwb_lens Z" shows "(X ;\<^sub>L Z \<bowtie> Y ;\<^sub>L Z) = (X \<bowtie> Y)"
Infty_neq_0_3: shows "- \<infinity> \<noteq> 0"
measure_pmf_conv_infsetsum: fixes p :: "'b pmf"   and A :: "'b set" shows "measure_pmf.prob p A = infsetsum (pmf p) A"
F_def: shows "F \<equiv> \<lambda>ltakeWhile xs. case xs of LNil \<Rightarrow> LNil | LCons x xs \<Rightarrow> if P x then LCons x (ltakeWhile xs) else LNil"
A: shows "((ereal \<circ> g \<circ> real_of_ereal) \<longlongrightarrow> A) (at_right a)"
prefix_distinct: fixes history :: "nat \<Rightarrow> 'c list"   and xs :: "'c list"   and i :: "nat" assumes "node_histories history"   and "node_histories.prefix_of_node_history history xs i" shows "distinct xs"
alw_alw: fixes \<phi> :: "'a stream \<Rightarrow> bool" shows "alw (alw \<phi>) = alw \<phi>"
v'_def: shows "v' \<equiv> \<lambda>i. if i \<le> n then THE c. c \<in> X \<and> v c = i else not_in_X"
assms_3: shows "\<iota>\<^sub>m x = Some q\<^sub>0"
id: shows "f v = ((+) (1::'a) ^^ k') (g v)"
rep_ltln\<^sub>P_def: shows "rep_ltln\<^sub>P \<equiv> quot_type.rep Rep_ltln\<^sub>P"
norm_le: fixes x :: "'a"   and y :: "'b" shows "norm (x ** y) \<le> norm x * norm y * K"
liftE_def: fixes f :: "'c \<Rightarrow> ('e \<times> 'c) set \<times> bool" shows "liftE f \<equiv> do r \<leftarrow> f; return (Inr r) od"
rep_plus: fixes r :: "'b \<Rightarrow>\<^sub>0 'a"   and s :: "'b \<Rightarrow>\<^sub>0 'a" shows "rep (r + s) = rep r + rep s"
witness_path: fixes c :: "'b \<Rightarrow> real"   and s :: "'a"   and v :: "'a"   and r :: "real" assumes "\<mu> c s v = ereal r" shows "\<exists>p. apath s p v \<and> \<mu> c s v = ereal (awalk_cost c p)"
seq_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b arrow_of_spans_data"   and \<mu> :: "'b arrow_of_spans_data" assumes "span_vertical_category C" shows "partial_magma.arr (span_vertical_category.vcomp C) (span_vertical_category.vcomp C \<nu> \<mu>) = (arrow_of_spans C \<mu> \<and> arrow_of_spans C \<nu> \<and> Dom \<nu> = Cod \<mu>)"
interpret_floatarith_eq_take_max_VarI: fixes ra :: "floatarith"   and ys :: "real list"   and zs :: "real list" assumes "take (max_Var_floatarith ra) ys = take (max_Var_floatarith ra) zs" shows "interpret_floatarith ra ys = interpret_floatarith ra zs"
vifintersection_vintersection: fixes I :: "V"   and J :: "V"   and f :: "V \<Rightarrow> V" assumes "I \<noteq> 0"   and "J \<noteq> 0" shows "\<Inter>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I) \<inter>\<^sub>\<circ> \<Inter>\<^sub>\<circ> (VLambda J f `\<^sub>\<circ> J) = \<Inter>\<^sub>\<circ> (VLambda (I \<union>\<^sub>\<circ> J) f `\<^sub>\<circ> (I \<union>\<^sub>\<circ> J))"
assms_2: assumes "0 \<le> x"   and "y \<le> 0" shows "P (lcm x (- y))"
stable_rank_tower: fixes j :: "nat"   and i :: "nat"   and x :: "nat"   and n :: "nat"   and y :: "nat" assumes "j \<le> i"   and "rank x n = Some j"   and "rank y n = Some i"   and "stable_rank y i" shows "stable_rank x j"
sup_ty_opt_trans: fixes P :: "'a prog"   and a :: "ty err"   and b :: "ty err"   and c :: "ty err" assumes "P \<turnstile> a \<le>\<^sub>\<top> b"   and "P \<turnstile> b \<le>\<^sub>\<top> c" shows "P \<turnstile> a \<le>\<^sub>\<top> c"
colours_distinct_4: fixes r :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state" assumes "white r s" shows "\<not> black r s"
transpose_move_matrix: fixes A :: "'b matrix"   and x :: "int"   and y :: "int" shows "transpose_matrix (move_matrix A x y) = move_matrix (transpose_matrix A) y x"
pivot_fun_monoton: fixes A :: "'b mat"   and f :: "nat \<Rightarrow> nat"   and nr :: "nat"   and i :: "nat"   and k :: "nat" assumes "pivot_fun A f (dim_col A)"   and "dim_row A = nr"   and "i < nr" shows "k < nr \<and> i < k \<longrightarrow> f i \<le> f k"
assms: shows "[a, c]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<circ>\<^sub>\<bullet> s"
rel_sel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b tree"   and b :: "'c tree" shows "rel_tree R a b = ((a = Leaf) = (b = Leaf) \<and> (a \<noteq> Leaf \<longrightarrow> b \<noteq> Leaf \<longrightarrow> l a = l b \<and> r a = r b \<and> rel_option R (val a) (val b) \<and> rel_tree R (lchild a) (lchild b) \<and> rel_tree R (rchild a) (rchild b)))"
psubst_Var_simp2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option"   and xs :: "'varSort"   and x :: "'var"   and X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set" assumes "goodEnv rho"   and "rho xs x = Some X" shows "Var xs x #[rho] = X"
reachable_step: fixes s :: "'c"   and A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool"   and a :: "'d"   and s' :: "'c" assumes "s \<in> reachable A I"   and "(s, a, s') \<in> automaton.trans A"   and "I a" shows "s' \<in> reachable A I"
bet_cop2__cop: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and U :: "'p"   and W :: "'p"   and V :: "'p" assumes "Coplanar A B C U"   and "Coplanar A B C W"   and "Bet U V W" shows "Coplanar A B C V"
Units_closed: fixes x :: "nat" assumes "x \<in> Units R" shows "x \<in> carrier R"
is_SDG_path_valid_SDG_node_2: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n is-ns\<rightarrow>\<^sub>d* n'" shows "valid_SDG_node n'"
prems_2: shows "(([u], ce), w, r', ce') \<in> trcl (trss fg)"
add: fixes x :: "complex"   and y :: "complex"   and xa :: "'a" shows "(x + y) *\<^sub>C xa = x *\<^sub>C xa + y *\<^sub>C xa"
wf_subcls_nCls': fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C\<^sub>0 :: "char list"   and cd :: "'a cdecl" assumes "wf_prog wf_md P"   and "\<not> is_class P C\<^sub>0"   and "cd \<in> set P" shows "(fst cd, C\<^sub>0) \<notin> (subcls1 P)\<^sup>*"
Ensures_def: fixes A :: "'d set"   and B :: "'d set" shows "A Ensures B \<equiv> {F. F \<in> reachable F \<inter> A ensures B}"
bipartite_E: fixes \<Gamma> :: "('a, 'b) web_scheme"   and x :: "'a"   and y :: "'a" assumes "countable_bipartite_web \<Gamma>"   and "edge \<Gamma> x y" shows "x \<in> A \<Gamma> \<and> y \<in> B \<Gamma>"
some_eq_trivial: fixes x :: "'b" shows "(SOME y. y = x) = x"
Int_space_eq2: fixes x :: "'b set"   and M :: "'b measure" assumes "x \<in> sets M" shows "x \<inter> space M = x"
eval_Runit_Ide: fixes t :: "'a local.term" assumes "Ide t" shows "\<lbrace>\<^bold>\<r>\<^bold>[t\<^bold>]\<rbrace> = \<r>[\<lbrace>t\<rbrace>]"
prod_mset_prime_factorization_weak: fixes x :: "'b poly" assumes "x \<noteq> 0" shows "smult (inverse (lead_coeff (\<Prod>\<^sub># (field_poly.prime_factorization x)))) (\<Prod>\<^sub># (field_poly.prime_factorization x)) = smult (inverse (lead_coeff x)) x"
linearB: shows "\<exists>es. char_poly B = (\<Prod>a\<leftarrow>es. [:- a, 1::'a:])"
magQ_plus: fixes x :: "'a['b, 'c]"   and y :: "'a['b, 'c]" shows "\<lbrakk>x + y\<rbrakk>\<^sub>Q = \<lbrakk>x\<rbrakk>\<^sub>Q + \<lbrakk>y\<rbrakk>\<^sub>Q"
a_known_ptr_def: fixes ptr :: "(_) object_ptr" shows "l_known_ptr\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t.a_known_ptr ptr = (DocumentClass.known_ptr ptr \<or> (case cast ptr of None \<Rightarrow> False | Some x \<Rightarrow> is_shadow_root_ptr x))"
TESL_interpretation_stepwise_cons_morph: fixes \<phi> :: "'b TESL_atomic"   and n :: "nat"   and \<Phi> :: "'b TESL_atomic list" shows "\<lbrakk> \<phi> \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> \<inter> \<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> = \<lbrakk>\<lbrakk> \<phi> # \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup>"
innerRes: shows "Some (r, m) = foldr (div_int_poly_step q) (coeffs p) (Some (0, 0))"
bind_assoc: fixes f :: "('l, 'm, 'p) prog"   and g :: "'p \<Rightarrow> ('l, 'm, 'o) prog"   and h :: "'o \<Rightarrow> ('l, 'm, 'n) prog" shows "Heap_Error_Monad.bind (Heap_Error_Monad.bind f g) h = Heap_Error_Monad.bind f (\<lambda>x. Heap_Error_Monad.bind (g x) h)"
disjnt_sym: fixes A :: "'a set"   and B :: "'a set" assumes "disjnt A B" shows "disjnt B A"
assms: shows "run_step d i (s, vs, es) = (s', vs', RSBreak n res)"
Int_open: fixes T :: "'b set set"   and x :: "'b set"   and y :: "'b set" assumes "topology T"   and "x open\<^bsub>T\<^esub>"   and "y open\<^bsub>T\<^esub>" shows "x \<inter> y open\<^bsub>T\<^esub>"
greaterThanAtMost_empty: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a"   and k :: "'a" assumes "class.order less_eq less"   and "less_eq l k" shows "ord.greaterThanAtMost less_eq less k l = {}"
t_fourtimes_append_pre: fixes ires :: "cell list"   and rs :: "nat"   and n :: "nat"   and stp :: "nat"   and ln :: "nat"   and rn :: "nat" assumes "steps0 (Suc 0, Bk # Bk # ires, Oc \<up> Suc rs @ Bk \<up> n) t_fourtimes stp = (Suc t_fourtimes_len, Bk \<up> ln @ Bk # Bk # ires, Oc \<up> Suc (4 * rs) @ Bk \<up> rn)" shows "steps0 (Suc 0 + length (t_wcode_main_first_part @ shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, Bk # Bk # ires, Oc \<up> Suc rs @ Bk \<up> n) ((t_wcode_main_first_part @ shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) @ shift t_fourtimes (length (t_wcode_main_first_part @ shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2) @ [(L, 1), (L, 1)]) stp = (Suc t_fourtimes_len + length (t_wcode_main_first_part @ shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, Bk \<up> ln @ Bk # Bk # ires, Oc \<up> Suc (4 * rs) @ Bk \<up> rn)"
count_violations_def: fixes t :: "('a, 'b, 'c) event list" shows "count_violations t = (\<Sum>i = 0..<length t. if postrecording_event t i then card {j \<in> {i + 1..<length t}. prerecording_event t j} else 0)"
coeff_eq_0: fixes p :: "'a poly"   and n :: "nat" assumes "degree p < n" shows "coeff p n = (0::'a)"
filterlim_at: fixes f :: "'a \<Rightarrow> 'b"   and b :: "'b"   and s :: "'b set"   and F :: "'a filter" shows "filterlim f (at b within s) F = ((\<forall>\<^sub>F x in F. f x \<in> s \<and> f x \<noteq> b) \<and> (f \<longlongrightarrow> b) F)"
exec_gpv_ap: fixes callee :: "'g \<Rightarrow> 'h \<Rightarrow> ('i \<times> 'g) spmf"   and f :: "('j \<Rightarrow> 'f, 'h, 'i) gpv"   and x :: "('j, 'h, 'i) gpv"   and \<sigma> :: "'g" shows "exec_gpv callee (f \<diamondop> x) \<sigma> = exec_gpv callee f \<sigma> \<bind> (\<lambda>(f', \<sigma>'). return_spmf (\<lambda>(x', \<sigma>''). (f' x', \<sigma>'')) \<diamondop> exec_gpv callee x \<sigma>')"
cont: fixes x :: "'a" assumes "x \<in> X" shows "continuous_on T (\<lambda>t. f t x)"
dg_finite_pdigraph_dg_singleton: fixes j :: "V" assumes "j \<in>\<^sub>\<circ> Vset \<alpha>" shows "finite_pdigraph \<alpha> (ZFC_in_HOL.set {j}) (\<lambda>i. \<CC>)"
assms_3: shows "set (p\<^sub>2 # ps) = PS\<^sub>L \<union> PS\<^sub>R"
ab_semigroup_add_ow: shows "class.ab_semigroup_add = ab_semigroup_add_ow UNIV"
complex_quadratic_equation_two_roots: fixes a :: "complex"   and \<xi> :: "complex"   and b :: "complex"   and c :: "complex" assumes "a \<noteq> 0"   and "a * \<xi>\<^sup>2 + b * \<xi> + c = 0" shows "\<xi> = (- b + ccsqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<or> \<xi> = (- b - ccsqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)"
v_2: shows "[v]\<^sub>\<R> = region X I r"
open_real_lessThan: fixes a :: "real" shows "open {..<a}"
lm034: fixes P :: "('c \<times> 'd) set"   and Q :: "('c \<times> 'd) set" assumes "P +< Q = Q" shows "Domain P \<subseteq> Domain Q"
var_expr_inj_term: fixes t :: "Term.var"   and w :: "expr" shows "\<langle>t\<rangle> \<noteq> \<langle>w\<rangle>"
arg_inv_pi: fixes z :: "complex" assumes "z \<noteq> 0"   and "arg z = pi" shows "arg (1 / z) = pi"
atLeastatMost_empty: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'d"   and a :: "'d" assumes "class.order less_eq less"   and "less b a" shows "ord.atLeastAtMost less_eq a b = {}"
intros: fixes X :: "'a set"   and Y :: "'a set"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "finite X"   and "finite Y"   and "Y \<noteq> {x. \<bottom> x}"   and "\<And>x. x \<in> X \<Longrightarrow> Bex Y (R x)" shows "max_extp R X Y"
wadjust_loop_erase_nonempty_7: fixes m :: "nat"   and rs :: "nat"   and c :: "cell list"   and b :: "cell list" assumes "wadjust_goon_left_moving m rs (c, b)" shows "c \<noteq> []"
lookup_plus: fixes A :: "'m tensor"   and B :: "'m tensor"   and "is" :: "nat list" assumes "dims A = dims B"   and "is \<lhd> dims A" shows "lookup (A + B) is = lookup A is + lookup B is"
unlabel_def: fixes S :: "('f \<times> 'e) list" shows "unlabel S \<equiv> map snd S"
cases: fixes a1 :: "nat"   and a2 :: "nat"   and rel :: "(nat \<times> nat) set"   and n :: "nat" assumes "(a1, a2) \<in> Clos rel n" obtains x :: "nat" and y :: "nat" where "a1 = xa__" and "a2 = ya__" and "(xa__, ya__) \<in> rel"   | x :: "nat" and z :: "nat" and y :: "nat" where "a1 = x" and "a2 = y" and "(x, z) \<in> Clos rel n" and "(z, y) \<in> Clos rel n" and "z < n"
ide_char: fixes f :: "('obj, 'obj) concrete_category.arr" shows "ide f = (f \<noteq> Null \<and> Dom f \<in> Obj \<and> Cod f = Dom f \<and> Map f = Dom f)"
trace_and_start_determines_end: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and c' :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration" assumes "trace c t c'"   and "trace c t d'" shows "c' = d'"
less_dag_def: fixes s :: "dag"   and t :: "dag" shows "(s < t) = subdag t s"
A: shows "xs \<cong> {ipurge_tr_rev I D u xs, rev (ipurge_tr (I\<inverse>) D u (rev xs)), \<lambda>y ys. D y \<in> sinks (I\<inverse>) D u (rev ys @ [y])}"
uniformly_Cauchy_on_def: fixes X :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" shows "uniformly_Cauchy_on X f = (\<forall>e>0. \<exists>M. \<forall>x\<in>X. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e)"
f_one_to_one_on_index_subset_1: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set"   and i :: "nat"   and j :: "nat"   and Y :: "'a set"   and y :: "'a" assumes "[f[f 0 ..]X]"   and "0 \<le> i"   and "i < j"   and "Y = f ` {i..j}"   and "y \<in> Y" shows "\<exists>!k. k \<in> {i..j} \<and> f k = y"
InvariantWatchesEl_def: fixes formula :: "Literal list list"   and watch1 :: "nat \<Rightarrow> Literal option"   and watch2 :: "nat \<Rightarrow> Literal option" shows "InvariantWatchesEl formula watch1 watch2 \<equiv> \<forall>clause. 0 \<le> clause \<and> clause < length formula \<longrightarrow> (\<exists>w1 w2. watch1 clause = Some w1 \<and> watch2 clause = Some w2 \<and> w1 el formula ! clause \<and> w2 el formula ! clause)"
iffs: fixes ext :: "real_int"   and lan :: "nat_int"   and own :: "cars"   and more :: "'a"   and ext' :: "real_int"   and lan' :: "nat_int"   and own' :: "cars"   and more' :: "'a" shows "(\<lparr>ext = ext, lan = lan, own = own, \<dots> = more\<rparr> = \<lparr>ext = ext', lan = lan', own = own', \<dots> = more'\<rparr>) = (ext = ext' \<and> lan = lan' \<and> own = own' \<and> more = more')"
l_form: shows "l dvd N \<and> l dvd d \<and> \<not> is_unit l"
axioms_5: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'a \<times> 'c \<Rightarrow> 'd"   and G :: "'a \<times> 'c \<Rightarrow> 'd"   and \<tau> :: "'a \<times> 'c \<Rightarrow> 'd" assumes "binary_functor_transformation A1 A2 B F G \<tau>" shows "binary_functor A1 A2 B F"
hrelation_shift: fixes f :: "hf"   and delta :: "hf" shows "hrelation (shift f delta)"
IsConfSet_Status: fixes ST :: "('d, 'e, 'f) status" shows "IsConfSet (SAs (HA ST)) (CompFun (HA ST)) (Conf ST)"
homeomorphic_imp_continuous_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "homeomorphic_map X Y f" shows "continuous_map X Y f"
one_le_floor: fixes x :: "'a" shows "(1 \<le> \<lfloor>x\<rfloor>) = ((1::'a) \<le> x)"
C: shows "bs2 = [] \<or> as2 \<noteq> [] \<and> hd as2 < hd bs2"
Im_sin_nonneg2: fixes z :: "complex" assumes "Re z = pi"   and "Im z \<le> 0" shows "0 \<le> Im (sin z)"
bpp_def: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" shows "a \<^sub>f+ b = f a b"
measurable_fst: fixes M1 :: "'c measure"   and M2 :: "'d measure" shows "fst \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M M1"