hcju/isabelleps huggingface dataset:
C1_differentiable_on_add: fixes f :: "real \<Rightarrow> 'a"   and S :: "real set"   and g :: "real \<Rightarrow> 'a" assumes "f C1_differentiable_on S"   and "g C1_differentiable_on S" shows "(\<lambda>x. f x + g x) C1_differentiable_on S"
match_def: fixes \<E> :: "char list \<Rightarrow> nat option"   and \<Gamma> :: "char list set" shows "match \<E> \<Gamma> \<equiv> dom \<E> = \<Gamma>"
assms: shows "\<C> \<turnstile> es @ [C v] : ts _> ts' @ [t]"
build_psimps_2: fixes ps :: "(real, 'a) vec list"   and k :: "'a"   and s :: "real"   and ks :: "'a list"   and l :: "(real, 'a) vec list"   and m :: "real"   and r :: "(real, 'a) vec list" assumes "1 < length ps"   and "(k, s) = widest_spread ks ps"   and "(l, m, r) = partition_by_median k ps"   and "build_dom (ks, l)"   and "build_dom (ks, r)" shows "build ks ps = Node k m (build ks l) (build ks r)"
laplace_exists_laplace_integrandI: fixes f :: "real \<Rightarrow> complex"   and s :: "complex" assumes "laplace_integrand f s integrable_on {0..}" obtains F :: "complex" where "(f has_laplace F) s"
get_writer_def: fixes get :: "('d \<Rightarrow> 'c) \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> ('a, 'b, 'c) writerT" shows "get_writer get f = WriterT (get (\<lambda>s. run_writer (f s)))"
isState_def: fixes state :: "('a, 'b, 'c) state" shows "isState state = (\<exists>x21 x22 x23. state = State x21 x22 x23)"
alphaAbs_qAbs_iff: fixes A :: "('a, 'b, 'c, 'd, 'e) qAbs"   and xs' :: "'c"   and x' :: "'d"   and X' :: "('a, 'b, 'c, 'd, 'e) qTerm" shows "(A $= qAbs xs' x' X') = (\<exists>x y X. A = qAbs xs' x X \<and> y \<notin> {x, x'} \<and> qAFresh xs' y X \<and> qAFresh xs' y X' \<and> X #[[y \<and> x]]_xs' #= X' #[[y \<and> x']]_xs')"
x': shows "x' \<equiv> getFr [x, yy] [t] []"
nat_eq_iff2: fixes m :: "nat"   and w :: "int" shows "(m = nat w) = (if 0 \<le> w then w = int m else m = 0)"
S_trms_instance_M: shows "has_all_wt_instances_of \<Gamma> (trms\<^sub>s\<^sub>s\<^sub>t S \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t S) (set M)"
atLeastAtMost_singleton_iff: fixes 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.order less_eq less" shows "(ord.atLeastAtMost less_eq a b = {c}) = (a = b \<and> b = c)"
divide_le_eq_numeral_1: fixes b :: "'a"   and c :: "'a"   and w :: "num" shows "(b / c \<le> numeral w) = (if (0::'a) < c then b \<le> numeral w * c else if c < (0::'a) then numeral w * c \<le> b else (0::'a) \<le> numeral w)"
bd: assumes "basis (set G)" shows "det (mat_of_rows n G) \<noteq> (0::'a)"
H_def: shows "rev.H \<equiv> pre_digraph.del_arc (G.del_arc rev.a') a'"
prime_power_iff: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and unit_factor :: "'d \<Rightarrow> 'd"   and normalize :: "'d \<Rightarrow> 'd"   and p :: "'d"   and n :: "nat" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalization_semidom.prime zero times one normalize (power.power one times p n) = (normalization_semidom.prime zero times one normalize p \<and> n = 1)"
SAT_def: fixes \<Phi> :: "('fsym, 'psym) lit list set" shows "M.SAT \<Phi> \<equiv> \<forall>\<xi>. M.wtE \<xi> \<longrightarrow> M.satPB \<xi> \<Phi>"
assms_2: shows "prefs_from_table_wf agents alts ys"
while: fixes P :: "'b set"   and b :: "'b set"   and c :: "'b com"   and X :: "'b set"   and Y :: "'b \<Rightarrow> nat" assumes "\<turnstile> (P \<inter> b) c P" shows "\<turnstile> P While b X Y c (P \<inter> - b)"
supp_def: fixes f :: "'b \<Rightarrow> real" shows "supp f = {x. f x \<noteq> 0}"
reindex: fixes h :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and g :: "'e \<Rightarrow> 'a" assumes "inj_on h A" shows "F g (h ` A) = F (g \<circ> h) A"
no_collision_react_initially: assumes "s\<^sub>o \<le> u_max"   and "u_max < other.s_stop" shows "no_collision_react {0..\<delta>}"
norm_matrix_vec_mult_le_transpose: fixes x :: "(real, 'b) vec"   and A :: "((real, 'b) vec, 'b) vec" assumes "\<parallel>x\<parallel> = 1" shows "\<parallel>A *v x\<parallel> \<le> sqrt (\<parallel>Finite_Cartesian_Product.transpose A ** A\<parallel>\<^sub>o\<^sub>p) * (\<parallel>x\<parallel>)"
ide_char: fixes a :: "'a" shows "hom.ide a = (hom.arr a \<and> ide a)"
subtype_correct: fixes t :: "'BExp set"   and P :: "'BExp set"   and t' :: "'BExp set" shows "t \<le>:\<^sub>P t' = (\<forall>mem. pred P mem \<longrightarrow> type_max t mem \<le> type_max t' mem)"
mset_set_set: fixes xs :: "'b list" assumes "distinct xs" shows "mset_set (set xs) = mset xs"
fds_exp_def: fixes f :: "'b fds" shows "fds_exp f = (let f' = fds (\<lambda>n. if n = 1 then 0::'b else fds_nth f n) in fds (\<lambda>n. exp (fds_nth f 1) * (\<Sum>k. fds_nth (f' ^ k) n /\<^sub>R fact k)))"
reach: shows "reachable (\<epsilon>(r \<mapsto> (\<epsilon>, \<epsilon>, e)))"
vAv: shows "inner_prod v (A *\<^sub>v v) = (\<Sum>i = 0..<m. \<Sum>j = 0..<m. conjugate (v $ i) * A $$ (i, j) * v $ j)"
insert_notmem: fixes a :: "'d"   and b :: "'d"   and B :: "'d set" assumes "a \<noteq> b"   and "a \<notin> B" shows "a \<notin> insert b B"
residuated_comp_closed: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "residuated f"   and "residuated g" shows "residuated (f \<circ> g)"
translation_map: fixes a :: "'a" assumes "a \<in> G" shows "(a)\<^sub>R \<in> G \<rightarrow>\<^sub>E G"
\<alpha>edges_sym: fixes M_empty :: "'a"   and M_update :: "'b \<Rightarrow> ('b \<times> nat) list \<Rightarrow> 'a \<Rightarrow> 'a"   and M_delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and M_lookup :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list option"   and M_invar :: "'a \<Rightarrow> bool"   and g :: "'a" assumes "wgraph_by_map M_empty M_update M_delete M_lookup M_invar"   and "wgraph_by_map.invar M_lookup M_invar g" shows "sym (wgraph_by_map.\<alpha>edges_aux M_lookup g)"
SemFalse: fixes b :: "BExpr"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and c2 :: "OBJ"   and n :: "nat"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and c1 :: "OBJ" assumes "\<not> evalB b (fst s)"   and " s , c2 \<rightarrow>\<^sub>n  t " shows " s , Iff b c1 c2 \<rightarrow>\<^sub>n + 1  t "
nth_list_update: fixes i :: "nat"   and xs :: "'d list"   and x :: "'d"   and j :: "nat" assumes "i < length xs" shows "xs[i := x] ! j = (if i = j then x else xs ! j)"
f11: fixes ss :: "symbol list"   and ssa :: "symbol list" shows "length ss = length (take (length ss) ssa) \<or> length (take (length ssa) ss) = length ssa"
assms: shows "addO n (exp\<omega> m) = addO n' (exp\<omega> m')"
mset_subset_eqD: fixes A :: "'d multiset"   and B :: "'d multiset"   and x :: "'d" assumes "A \<subseteq># B"   and "x \<in># A" shows "x \<in># B"
rreq_rrep_sn_any_step_invariant: fixes i :: "nat" shows "paodv i \<TTurnstile>\<^sub>A (recvmsg rreq_rrep_sn \<rightarrow>) onll \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(uu_, a, uu_). anycast rreq_rrep_sn a)"
coprime_cong_prod_nat: fixes A :: "'b set"   and m :: "'b \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" assumes "\<And>i j. \<lbrakk>i \<in> A; j \<in> A; i \<noteq> j\<rbrakk> \<Longrightarrow> coprime (m i) (m j)"   and "\<And>i. i \<in> A \<Longrightarrow> [x = y] (mod m i)" shows "[x = y] (mod prod m A)"
k_def: shows "k = index (take (Suc i) cs) z"
dghm_dghm_comp_dghm_id_right: shows "\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M dghm_id \<AA> = \<GG>"
islimpt_in_closure: fixes x :: "'b"   and S :: "'b set" shows "x islimpt S = (x \<in> closure (S - {x}))"
position_maxlength_cannotbe_augmented: fixes p :: "'a list"   and m :: "'a" assumes "enat (length p) = 2 * N" shows "\<not> position (p @ [m])"
map_resource_id1: fixes f :: "'e \<Rightarrow> 'd"   and res :: "('c, 'e) resource" shows "map_resource id f res = map_resource' f res"
remove1_filter_not: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" assumes "\<not> P x" shows "remove1 x (filter P xs) = filter P xs"
bisim: shows "rel_language letter_eq (Language1 (init1 r)) (Language2 (init2 s)) = (\<forall>a b. (\<exists>w1 w2. a = \<dd>s w1 (Language1 (init1 r)) \<and> b = \<dd>s w2 (Language2 (init2 s)) \<and> w1 \<in> lists (set alphabet1) \<and> w2 \<in> lists (set alphabet2) \<and> list_all2 letter_eq w1 w2) \<longrightarrow> \<oo> a = \<oo> b)"
hom_nat_pow: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier G" shows "h (x [^] n) = h x [^]\<^bsub>H\<^esub> n"
ccsqrt_def: fixes z :: "complex" shows "ccsqrt z = rcis (sqrt (cmod z)) (arg z / 2)"
opI: fixes a :: "'a" assumes "a \<in> S" shows "\<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and> (\<forall>x\<in>T. \<forall>y\<in>T. P x \<and> P y \<longrightarrow> R x y)"
StuckProp: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" shows "\<Gamma>\<turnstile> (c # cs, css, Stuck) \<rightarrow> (cs, css, Stuck)"
simps_4: fixes b :: "'a"   and c :: "('a, 'b) com" shows "noWhile (while b do c) = False"
select_convs_4: fixes name :: "nat"   and incoming :: "nat set"   and new :: "'b ltlr set"   and old :: "'b ltlr set"   and "next" :: "'b ltlr set"   and more :: "'c" shows "old \<lparr>name = name, incoming = incoming, new = new, old = old, next = next, \<dots> = more\<rparr> = old"
Input: fixes a :: "name"   and x :: "name"   and P :: "pi" shows "a<x>.P \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l P"
uminus_Abs_freeword_singleton: fixes s :: "'a"   and b :: "bool" shows "- Abs_freeword [(s, b)] = Abs_freeword [(s, \<not> b)]"
stream_defined: fixes xs :: "'b LList" shows "stream\<cdot>xs \<noteq> \<bottom>"
absorb1: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "a \<cdot> b = a"
less_half_sum: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a b" shows "less a (divide (plus a b) (plus one one))"
simps: fixes a1 :: "('a \<times> 'a) set"   and a2 :: "'a list" shows "sorted_on a1 a2 = ((\<exists>r. a1 = r \<and> a2 = []) \<or> (\<exists>x r xs. a1 = r \<and> a2 = x # xs \<and> x \<in> Field r \<and> (\<forall>y\<in>set xs. (x, y) \<in> r) \<and> sorted_on r xs))"
simps_2: fixes A :: "agent"   and ev :: "event"   and evs :: "event list" shows "knows A (ev # evs) = (case ev of Says A' B X \<Rightarrow> insert X (knows A evs) | Gets A' X \<Rightarrow> knows A evs | Notes A' X \<Rightarrow> if A' = A then insert X (knows A evs) else knows A evs)"
perp_col2_bis: 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"   and P :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C D"   and "Tarski_neutral_dimensionless.Col Bet C D P"   and "Tarski_neutral_dimensionless.Col Bet C D Q"   and "P \<noteq> Q" shows "Tarski_neutral_dimensionless.Perp Bet Cong A B P Q"
fps_X_power_iff: fixes n :: "nat" shows "fps_X ^ n = Abs_fps (\<lambda>m. if m = n then 1::'a else (0::'a))"
canonical_subs_def: fixes n :: "nat"   and I :: "nat set"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b" shows "canonical_subs n I m = (\<forall>i j k. i \<le> n \<and> k \<le> n \<and> j \<in> I \<longrightarrow> m i k \<le> m i j + m j k)"
simps_7: fixes n :: "nat"   and r :: "'a atom rexp" shows "local.wf n (Star r) = local.wf n r"
ereal_minus_less_iff: fixes x :: "ereal"   and y :: "ereal"   and z :: "ereal" shows "(x - y < z) = (y \<noteq> - \<infinity> \<and> (y = \<infinity> \<longrightarrow> x \<noteq> \<infinity> \<and> z \<noteq> - \<infinity>) \<and> (y \<noteq> \<infinity> \<longrightarrow> x < z + y))"
Compositionality_Fork: fixes c :: "('exp, 'id) MWLfCom"   and c' :: "('exp, 'id) MWLfCom"   and d :: "'d"   and V :: "('exp, 'id) MWLfCom list"   and V' :: "('exp, 'id) MWLfCom list" assumes "[c] \<approx>\<^bsub>d\<^esub> [c']"   and "V \<approx>\<^bsub>d\<^esub> V'" shows "[fork c V] \<approx>\<^bsub>d\<^esub> [fork c' V']"
simps_17: fixes a :: "floatarith"   and n :: "nat" shows "max_Var_floatarith (a ^\<^sub>e n) = max_Var_floatarith a"
reachable_def: shows "reachable \<equiv> step\<^sup>*\<^sup>* s\<^sub>0"
mlex_less: fixes f :: "'b \<Rightarrow> nat"   and x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'b) set" assumes "f x < f y" shows "(x, y) \<in> f <*mlex*> R"
finBD: shows "finite ((\<lambda>v. v @\<^sub>v 0\<^sub>v m) ` baseB \<union> (@\<^sub>v) (0\<^sub>v n) ` baseD)"
distinct_101: fixes x81 :: "'a ltln"   and x82 :: "'a ltln"   and x101 :: "'a ltln"   and x102 :: "'a ltln" shows "x81 U\<^sub>n x82 \<noteq> x101 W\<^sub>n x102"
antisym_conv3: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and y :: "'e"   and x :: "'e" assumes "class.linorder less_eq less"   and "\<not> less y x" shows "(\<not> less x y) = (x = y)"
seq_nil_left: fixes a :: "'a::refinement_lattice" shows "nil ; (a::'a::refinement_lattice) = a"
discr0_MtransC: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "discr0 c"   and "mustT c s"   and "(c, s) \<rightarrow>*c (c', s')" shows "discr0 c'"
real_of_int_floor_add_one_ge: fixes r :: "real" shows "r \<le> real_of_int \<lfloor>r\<rfloor> + 1"
encode_initial_state_valid: fixes sv :: "sat_plan_variable"   and Prob :: "'a strips_problem" assumes "sv \<in> atoms (\<Phi>\<^sub>I Prob)" shows "valid_state_var sv"
dghm_comp_is_tm_dghm: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<CC>"
actions_ok'_iff: fixes s :: "('h, 'i) locks \<times> (('i \<Rightarrow> ('x \<times> 'h \<Rightarrow>f nat) option) \<times> 'j) \<times> ('i \<Rightarrow> 'k wait_set_status option) \<times> 'i set"   and t :: "'i"   and ta :: "('h, 'i, 'l, 'j, 'k, 'm) thread_action" shows "actions_ok' s t ta = (lock_ok_las' (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub> \<and> thread_oks (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub> \<and> cond_action_oks' s t \<lbrace>ta\<rbrace>\<^bsub>c\<^esub> \<and> wset_actions_ok (wset s) t \<lbrace>ta\<rbrace>\<^bsub>w\<^esub> \<and> interrupt_actions_ok' (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>)"
finite_mult_not_eq_zero_sumI: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite {a. f a \<noteq> (0::'c)}"   and "finite {b. g b \<noteq> (0::'c)}" shows "finite {a + b |a b. f a * g b \<noteq> (0::'c)}"
partitions_sorted: fixes js :: "'b list" assumes "\<And>xs ys x y. \<lbrakk>js = xs @ ys; x \<in> set xs; y \<in> set ys\<rbrakk> \<Longrightarrow> x \<le> y" shows "sorted js"
Tw_ord_injTr: fixes E :: "('e, 'f) Order_scheme"   and a :: "'a"   and b :: "'a" assumes "Worder E"   and "\<forall>a\<in>carrier D. \<exists>b\<in>carrier E. ord_equiv (Iod D (segment D a)) (Iod E (segment E b))"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "a \<prec> b" shows "(Tw\<^bsub>D,E\<^esub>) a \<prec>\<^bsub>E\<^esub> (Tw\<^bsub>D,E\<^esub>) b"
inv_in_span: fixes U :: "'a vec set"   and a :: "'a vec" assumes "U \<subseteq> carrier_vec n"   and "a \<in> local.span U" shows "- a \<in> local.span U"
cases: fixes a :: "('b com option list \<times> 'b) list" assumes "a \<in> par_cptn" obtains P :: "'b com option list" and s :: "'b" where "a = [(P__, sa__)]"   | P :: "'b com option list" and t :: "'b" and xs :: "('b com option list \<times> 'b) list" and s :: "'b" where "a = (Pa__, sb__) # (Pa__, t__) # xsa__" and "(Pa__, t__) # xsa__ \<in> par_cptn"   | P :: "'b com option list" and s :: "'b" and Q :: "'b com option list" and t :: "'b" and xs :: "('b com option list \<times> 'b) list" where "a = (P, s) # (Q, t) # xs" and "(P, s) -pc\<rightarrow> (Q, t)" and "(Q, t) # xs \<in> par_cptn"
ffd_def: shows "fd\<^sub>\<F> = bd\<^sub>\<F> \<circ> op\<^sub>K"
midpoint_cong: fixes B :: "'p"   and A :: "'p"   and C :: "'p" assumes "B Midpoint A C" shows "Cong A B B C"
indets_monomial_zero: fixes c :: "'d" shows "indets (monomial c 0) = {}"
sound_modes: shows "locally_sound_mode_use \<langle>c, mds, mem\<rangle>"
pivot_funD_1: fixes A :: "'c mat"   and nr :: "nat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat"   and i :: "nat" assumes "dim_row A = nr"   and "pivot_fun A f nc"   and "i < nr" shows "f i \<le> nc"
filters_def: shows "filters = {X. X \<noteq> {} \<and> upclosed_set X \<and> filtered X}"
integrableI_nn_integral_finite: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure"   and x :: "real" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = ennreal x" shows "integrable M f"
fbspec: fixes A :: "'b fset"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "fBall A P"   and "x |\<in>| A" shows "P x"
comm_groupE_1: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and y :: "'d" assumes "comm_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"
assms_2: shows "f \<in> space M \<rightarrow> \<Omega>"
simps: fixes n :: "nat"   and uu :: "nat"   and uv :: "pratt_tree list" shows "pratt_tree_number (Pratt_Node (n, uu, uv)) = n"
carrier_subgroup_generated_subgroup: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "subgroup H G" shows "carrier (subgroup_generated G H) = H"
atLeast_borel: fixes a :: "'b" shows "{a..} \<in> sets borel"
max_PInf2: fixes x :: "ereal" shows "max x \<infinity> = \<infinity>"
lagrange_basis_poly_def: fixes xs :: "'b list"   and xj :: "'b" shows "lagrange_basis_poly xs xj \<equiv> let ys = filter (\<lambda>x. x \<noteq> xj) xs in \<Prod>xi\<leftarrow>ys. smult (inverse (xj - xi)) [:- xi, 1::'b:]"
nxtAct_active: fixes active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b"   and t :: "nat \<Rightarrow> cnf"   and i :: "nat" assumes "active c (t i)" shows "dynamic_component.nxtAct active c t i = i"
existsM1inM: fixes M :: "'a set set" assumes "M \<in> calM // RelM" shows "\<exists>M1. M1 \<in> M"
tranclp_converse: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "r\<inverse>\<inverse>\<^sup>+\<^sup>+ = r\<^sup>+\<^sup>+\<inverse>\<inverse>"
assms_3: shows "Sigma_Algebra.measure M (space M - s) = Sigma_Algebra.measure M (space M - t)"
0: shows "P \<in> partitionsUniverse \<and> R \<subseteq> UNIV \<times> P"
add_abs_int: fixes x :: "nat"   and y :: "nat"   and u :: "nat"   and v :: "nat" shows "abs_int (x, y) + abs_int (u, v) = abs_int (x + u, y + v)"
fls_lr_inverse_one_2: fixes y :: "'a" shows "fls_right_inverse 1 y = fls_const y"
Rf_subs_Rf_diff_Rf: fixes N :: "'b literal multiset set" shows "standard_redundancy_criterion.Rf N \<subseteq> standard_redundancy_criterion.Rf (N - standard_redundancy_criterion.Rf N)"
inext_cut_le_conv: fixes n :: "nat"   and I :: "nat set"   and t :: "nat" assumes "inext n I \<le> t" shows "inext n (I \<down>\<le> t) = inext n I"
get_disconnected_nodes_new_shadow_root_different_pointers: fixes new_shadow_root_ptr :: "(_) shadow_root_ptr"   and ptr' :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr \<noteq> ptr'"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "r \<in> get_disconnected_nodes_locs ptr'" shows "r h h'"
no_data_corruption: fixes i :: "'a \<times> nat"   and s :: "'a \<times> nat"   and m :: "('a \<times> nat, 'b) message" assumes "event_at i (Receive s m)" shows "event_at s (Send m)"
fmdrop_fmupd_same: fixes x :: "'c"   and y :: "'d"   and m :: "('c, 'd) fmap" shows "fmdrop x (m(x $$:= y)) = fmdrop x m"
lerch_lemma_real: fixes h :: "real \<Rightarrow> real"   and u :: "real" assumes "continuous_on {0..1} h"   and "\<And>n. ((\<lambda>u. u ^ n * h u) has_integral 0) {0..1}"   and "0 \<le> u"   and "u \<le> 1" shows "h u = 0"
indoc_Cons: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and v :: "'a list" assumes "traces ind" shows "traces.indoc ind a (b # v) = (a = b \<or> a \<noteq> b \<and> ind a b \<and> traces.indoc ind a v)"
abs_eq: fixes x :: "('b, 'c, 'd) Tree" shows "Not\<^sub>\<alpha> (abs_Tree\<^sub>\<alpha> x) = abs_Tree\<^sub>\<alpha> (tNot x)"
prems: shows "P,E \<turnstile> e # es [::] Ts'"
finprod_empty: fixes G :: "('e, 'f) monoid_scheme"   and f :: "'g \<Rightarrow> 'e" assumes "Group.comm_monoid G" shows "finprod G f {} = \<one>\<^bsub>G\<^esub>"
simps_2: fixes l :: "nat"   and i :: "nat"   and j :: "nat" shows "\<sigma>s (Suc l) i j = (\<mu>' (Suc l) (Suc l) * \<sigma>s l i j + \<mu>' i (Suc l) * \<mu>' j (Suc l)) div \<mu>' l l"
Conj_def: fixes xset :: "('e, 'f, 'g) formula set['e]" shows "Conj xset = Abs_formula (Conj\<^sub>\<alpha> (map_bset Rep_formula xset))"
simps_9: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case Rh of Lh \<Rightarrow> f1 | Bal \<Rightarrow> f2 | Rh \<Rightarrow> f3) = f3"
suc_ofilter_in: fixes A :: "'a set"   and b :: "'a" assumes "local.ofilter A"   and "local.AboveS A \<noteq> {}"   and "(b, suc A) \<in> r"   and "b \<noteq> suc A" shows "b \<in> A"
le_var_order_rel: shows "(\<le>) = (\<lambda>x y. x = y \<or> (x, y) \<in> var_order_rel)"
smooth_on_mult: fixes k :: "enat"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (\<lambda>x. f x * g x)"
finalE: fixes e :: "expr" assumes "final e" obtains v :: "val" where "e = Val v"   | r :: "nat \<times> char list list" where "e = Throw r"
antisym_conv3: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "class.linorder less_eq less"   and "\<not> less y x" shows "(\<not> less x y) = (x = y)"
int_relprime_odd_power_divisors: fixes n :: "nat"   and a :: "int"   and b :: "int"   and c :: "int" assumes "odd n"   and "a * b = c ^ n"   and "coprime a b" shows "\<exists>k. a = k ^ n"
axioms_9: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd"   and \<eta> :: "'d \<Rightarrow> 'd"   and \<epsilon> :: "'c \<Rightarrow> 'c" assumes "unit_counit_adjunction C D F G \<eta> \<epsilon>" shows "natural_transformation D D (identity_functor.map D) (G \<circ> F) \<eta>"