hcju/isabelleps huggingface dataset:
AndL: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "F, G, \<Gamma> \<Rightarrow> \<Delta>" shows "F \<^bold>\<and> G, \<Gamma> \<Rightarrow> \<Delta>"
sum_ideals: fixes I1 :: "'a set"   and I2 :: "'a set" assumes "ideal R I1"   and "ideal R I2" shows "ideal R (I1 \<minusplus> I2)"
B_mod_greater_zero: fixes p :: "nat"   and a :: "int"   and x :: "int" assumes "GAUSS p a"   and "x \<in> GAUSS.B p a" shows "0 < x mod int p"
mult_zero_circ: fixes x :: "'a" shows "top = top \<squnion> x \<sqinter> bot"
plus_right_mono: fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "z \<le> y" shows "x + z \<le> x + y"
less_imp_neq: fixes x :: "'b word"   and y :: "'b word" assumes "x <s y" shows "x \<noteq> y"
simps: fixes j :: "jump" shows "the_Jump (Jump j) = j"
WT_gpv_iff_outs_gpv: fixes \<I> :: "('y, 'z) \<I>"   and gpv :: "('aa, 'y, 'z) gpv" shows "\<I> \<turnstile>g gpv \<surd> = (outs_gpv \<I> gpv \<subseteq> outs_\<I> \<I>)"
proj_not_active_same: fixes n :: "nat"   and n' :: "enat"   and t :: "cnf llist"   and c :: "'id" assumes "enat n \<le> n'"   and "\<not> lfinite t \<or> n' - 1 < llength t"   and "\<not> (\<exists>k\<ge>n. enat k < n' \<and> enat k < llength t \<and> \<parallel>c\<parallel>\<^bsub>lnth t k\<^esub>)" shows "\<pi>\<^bsub>c\<^esub>ltake n' t = \<pi>\<^bsub>c\<^esub>ltake (enat n) t"
gcd_proj2_iff: fixes m :: "'a"   and n :: "'a" shows "(gcd m n = normalize n) = (n dvd m)"
Group_Ugp: fixes G :: "('e, 'f) Group_scheme" assumes "Ugp G" shows "Group G"
meet_def: fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y = (if \<exists>z. {x, y} >>| z then \<Sqinter>{x, y} else x)"
distinct_5: fixes x41 :: "('d, 'e, 'f) com"   and x42 :: "('d, 'e, 'f) com" shows "Skip \<noteq> Seq x41 x42"
rep_number_g0_exists: fixes B :: "'a set multiset"   and x :: "'a" assumes "0 < B rep x" obtains b :: "'a set" where "b \<in># B" and "x \<in> b"
rconf_In2_LVar: fixes va :: "Term.var"   and n :: "lname"   and G :: "prog"   and L :: "lenv"   and s :: "st"   and vf :: "vvar"   and T :: "ty" assumes "va = LVar n" shows "G,L,s\<turnstile>In2 va\<succ>In2 vf\<Colon>\<preceq>Inl T = (fst vf = the (locals s n) \<and> (locals s n \<noteq> None \<longrightarrow> G,s\<turnstile>fst vf\<Colon>\<preceq>T) \<and> s\<le>|snd vf\<preceq>T\<Colon>\<preceq>(G, L))"
characteristic_weak_formula_supp_aux: fixes P :: "'state" shows "supp (Abs_bset {distinguishing_weak_formula P Q |Q. \<not> P \<equiv>\<cdot> Q}) \<subseteq> supp P"
b: shows "((y' \<leftrightarrow> y) \<bullet> e')[y'::=x] = e'[y::=x]"
map_of_map': fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and xs :: "('b \<times> 'c) list"   and x :: "'b" shows "map_of (map (\<lambda>(k, v). (k, f k v)) xs) x = map_option (f x) (map_of xs x)"
K_def: shows "K = {Pi\<^sub>E UNIV X |X. (\<forall>i. X i \<in> B i) \<and> finite {i. X i \<noteq> UNIV}}"
ideal_subset: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and h :: "'c" assumes "Ring R"   and "ideal R I"   and "h \<in> I" shows "h \<in> carrier R"
orthocompl_span: fixes S :: "'a vec set"   and v :: "'a vec"   and n :: "nat"   and y :: "'a vec" assumes "\<And>x. x \<in> S \<Longrightarrow> v \<bullet> x = (0::'a)"   and "S \<subseteq> carrier_vec n"   and "v \<in> carrier_vec n"   and "y \<in> LinearCombinations.module.span class_ring (module_vec TYPE('a) n) S" shows "v \<bullet> y = (0::'a)"
upclosed_set_def: fixes X :: "'a set" shows "upclosed_set X = (X \<in> upsets)"
b: fixes nb :: "'node" assumes "nb \<in> phiDefNodes_aux g v (filter (\<lambda>n. (n, v) \<notin> Mapping.keys phis) (\<alpha>n g)) m"   and "\<forall>x\<in>set ms. nb \<notin> phiDefNodes_aux g v (filter (\<lambda>n. (n, v) \<notin> Mapping.keys phis) (\<alpha>n g)) x" shows "nb \<in> phiDefNodes_aux g v (filter (\<lambda>n. (n, v) \<notin> Mapping.keys (phis'_aux g v (set ms) phis)) (\<alpha>n g)) m"
simps: fixes solution_s :: "'b \<Rightarrow> 'c"   and asi :: "'d"   and tv :: "'c \<Rightarrow> 'a"   and ui :: "'e"   and s :: "'b" shows "solution_nsc solution_s ((asi, tv, ui), s) = tv (solution_s s)"
less_natpair_def: fixes A :: "natpair"   and B :: "natpair" shows "(A < B) = (case A of natpair.Pair (a, b) \<Rightarrow> case B of natpair.Pair (c, d) \<Rightarrow> if a = c then b < d else a < c)"
distinct_136: fixes x11 :: "Syntax.term"   and x5 :: "Syntax.term" shows "Prj1 x11 \<noteq> Inj1 x5"
isLam_eta_expand: fixes e :: "exp"   and n :: "nat" assumes "isLam e" shows "isLam (eta_expand n e)"
nMT_domMT: fixes p :: "((int \<times> int) set set, int) Combinators list"   and r :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "\<not> not_MT C p"   and "p \<noteq> []" shows "r \<notin> dom (C (list2FWpolicy p))"
countable_network_def: fixes \<Delta> :: "('c, 'd) network_scheme" shows "countable_network \<Delta> \<equiv> (countable \<^bold>E\<^bsub>\<Delta>\<^esub> \<and> source \<Delta> \<noteq> sink \<Delta>) \<and> (\<forall>e. e \<notin> \<^bold>E\<^bsub>\<Delta>\<^esub> \<longrightarrow> capacity \<Delta> e = 0) \<and> (\<forall>e. capacity \<Delta> e \<noteq> \<top>)"
oinitial_refine: shows "(Map.empty, {}) \<in> oGS_rel"
fps_X_power_unit_factor: fixes n :: "nat" shows "unit_factor (fps_X ^ n) = 1"
quotientI: fixes x :: "'a"   and A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "x \<in> A" shows "r `` {x} \<in> A // r"
b: fixes p :: "'b"   and q :: "'b" shows "((p, q) \<in> b) = (\<exists>t. p \<parallel> t \<and> t \<parallel> q)"
HeronConf_interp_stepwise_timedelayed_cases: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and \<delta>\<tau> :: "'c tag_const"   and K\<^sub>2 :: "clock"   and K\<^sub>3 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
either_above_arg2: fixes t' :: "'b ttree"   and t :: "'b ttree" shows "t' \<sqsubseteq> t \<oplus>\<oplus> t'"
inverts: fixes C :: "C"   and C' :: "C" shows "(C\<cdot>C \<sqsubseteq> C\<cdot>C') = (C \<sqsubseteq> C')"
assms: shows "X \<in> Sols f U S t\<^sub>0 s"
pi_approx_def: fixes n :: "nat" shows "pi_approx n = 16 * arctan_approx (3 * n) (1 / 5) - 4 * arctan_approx n (1 / 239)"
shiftr_0: fixes n :: "nat" shows "0 >> n = 0"
inverse_Quantity_ext_def: fixes x :: "('e, 'f, 'g) Quantity_scheme" shows "inverse x = \<lparr>mag = inverse (mag x), dim = inverse (Quantity.dim x), \<dots> = inverse (Quantity.more x)\<rparr>"
map_sum_o_inj_1: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c" shows "map_sum f g \<circ> Inl = Inl \<circ> f"
col_factor'_def: fixes c :: "nat" shows "col_factor' c = col_factor (digit_encode (nths (dims A) (- I)) c)"
borel_measurable_if_I: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "f \<in> borel_measurable (lebesgue_on S)"   and "S \<in> sets lebesgue" shows "(\<lambda>x. if x \<in> S then f x else (0::'b)) \<in> borel_measurable lebesgue"
invertibles_blinfun_def: shows "invertibles_blinfun = {w. \<exists>wi. w o\<^sub>L wi = 1\<^sub>L \<and> wi o\<^sub>L w = 1\<^sub>L}"
csa_in: shows "csa \<in> consistent_sign_vectors qs UNIV"
character_neq_1_exists: fixes x :: "('a \<Rightarrow> complex) \<Rightarrow> complex" assumes "x \<in> carrier (Characters (Characters G))"   and "x \<noteq> \<one>\<^bsub>Characters (Characters G)\<^esub>" obtains \<chi> :: "(('a \<Rightarrow> complex) \<Rightarrow> complex) \<Rightarrow> complex" where "character (Characters (Characters G)) \<chi>" and "\<chi> x \<noteq> 1"
arr_iff_in_hom: fixes f :: "'d" shows "Dop.arr f = Dop.in_hom f (Dop.dom f) (Dop.cod f)"
trivial_hom_iff: shows "(h ` carrier G = {\<one>\<^bsub>H\<^esub>}) = (kernel G H h = carrier G)"
inverse_inverse_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse (inverse a) = a"
bijective_regular: fixes x :: "'a" assumes "bijective x" shows "regular x"
invariant_ES: shows "invariant (ES \<union> {(Y, yrhs)})"
s_Roll_inv: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, Roll e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>" obtains e'a :: "Syntax.term" where "e' = Roll e'a" and "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'a\<ggreater>"
subformulas\<^sub>\<mu>_list_distinct: fixes \<phi> :: "'a ltln" shows "distinct (subformulas\<^sub>\<mu>_list \<phi>)"
unit_mult_left_cancel: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "is_unit a" shows "(a * b = a * c) = (b = c)"
emeasure_lub_spmf: fixes A :: "'a set" assumes "Y \<noteq> {}" shows "emeasure (measure_spmf local.lub_spmf) A = (\<Squnion>y\<in>Y. emeasure (measure_spmf y) A)"
erH_rHom: fixes A :: "('e, 'f) Ring_scheme"   and B :: "('e, 'g) Ring_scheme"   and Y :: "'e"   and h :: "'a \<Rightarrow> 'e" assumes "Idomain S"   and "PolynRg A B Y"   and "h \<in> rHom S B" shows "erH R S X A B Y h \<in> pHom R S X, A B Y"
product_integrable_prod: fixes I :: "'i set"   and f :: "'i \<Rightarrow> 'b \<Rightarrow> 'd" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> integrable (distr M borel (Y i)) (f i)" shows "integrable (Pi\<^sub>M I (\<lambda>i. distr M borel (Y i))) (\<lambda>x. \<Prod>i\<in>I. f i (x i))"
process_rule_2_traces: fixes xs :: "'e list"   and xs' :: "'e list"   and P :: "'e process" assumes "xs @ xs' \<in> traces P" shows "xs \<in> traces P"
ring_n_distinct_prime_divisors: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set" assumes "distinct_pds K n P" shows "Ring (Sr K {x \<in> carrier K. \<forall>j\<le>n. 0 \<le> (\<nu>\<^bsub>K P j\<^esub>) x})"
repFreePath_2: shows "distinct ((initial M2, initial M1) # states p (initial M2, initial M1))"
softSubst_def: fixes \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var" shows "softSubst \<phi> t x = exi x (cnj (eql (Var x) t) \<phi>)"
id_hom: fixes A :: "'a" assumes "A \<in> Ob" shows "Id A \<in> Hom A A"
set_mult_carrier_idem: fixes H :: "'a set" assumes "subgroup H G" shows "carrier G <#> H = carrier G"
neg: fixes x :: "'a" assumes "vectorspace H"   and "x \<in> H" shows "h (- x) = - h x"
simp_1: fixes res :: "(nat \<times> 'a mod_ring poly) list" shows "dist_degree_factorize_main v w d res = (if v = 1 then res else if degree v < d + d then (degree v, v) # res else let w = w ^ CARD('a) mod v; d = Suc d; gd = gcd (w - monom 1 1) v in if gd = 1 then dist_degree_factorize_main v w d res else let v' = v div gd in dist_degree_factorize_main v' (w mod v') d ((d, gd) # res))"
PREC_case: fixes f :: "nat list \<Rightarrow> nat"   and kf :: "nat"   and g :: "nat list \<Rightarrow> nat"   and kg :: "nat" assumes "\<And>l. f l < ack kf (sum_list l)"   and "\<And>l. g l < ack kg (sum_list l)" shows "\<exists>k. \<forall>l. PREC f g l < ack k (sum_list l)"
last_plus: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "m \<le> n" shows "sum g {m..n} = g n + sum g {m..<n}"
steps_def: shows "[\<leadsto>] = {(s, s') |s s'. \<exists>r. revision_step r s s'}"
unit_wfactors: fixes a :: "'a" assumes "a \<in> Units G" shows "wfactors G [] a"
Inv_def: fixes p :: "'b"   and c :: "'c"   and s :: "('a, 'b, 'c) SLin_state"   and s' :: "('a, 'b, 'c) SLin_state" shows "Inv p c s s' \<equiv> status s p = Ready \<and> s' = s\<lparr>pending := (pending s)(p := (p, c)), status := (status s)(p := Pending)\<rparr>"
more_arith_simps_3: fixes a :: "'b" shows "- a + a = (0::'b)"
mkIde_as_mkArr: fixes A :: "'s set" assumes "A \<subseteq> Univ"   and "|A| <o cardSuc (cmax |UNIV| natLeq)" shows "mkArr A A (\<lambda>x. x) = mkIde A"
total_on_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "total_on A r = (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r)"
ty_val_def: fixes \<Gamma> :: "String.literal \<Rightarrow> unit option"   and vars :: "String.literal \<Rightarrow> 32 Word.word option" shows "ty_val \<Gamma> vars \<equiv> dom \<Gamma> = dom vars"
\<delta>_statesI_1: fixes q :: "'a"   and l :: "'b"   and qs :: "'a list"   and \<delta> :: "('a, 'b) ta_rule set" assumes "q \<rightarrow> l qs \<in> \<delta>" shows "q \<in> \<delta>_states \<delta>"
simps_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a0 :: "'b"   and a1 :: "'b" assumes "binary_product_diagram C a0 a1" shows "binary_product_diagram.map C a0 a1 concrete_category.Null = partial_magma.null C"
language: fixes p :: "'c"   and A :: "('d, 'c) ngba"   and w :: "'d stream"   and r :: "'c stream" assumes "p \<in> initial A"   and "run A (w ||| r) p"   and "gen infs (accepting A) (p ## r)" shows "w \<in> language A"
term_okI: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and t :: "Core.term" assumes "term_ok' (sig \<Theta>) t"   and "typ_of t \<noteq> None" shows "term_ok \<Theta> t"
assms_6: fixes w :: "complex" assumes "w \<notin> S" shows "w \<in> outside (path_image \<gamma>)"
skip: fixes \<iota> :: "nat"   and m :: "'b \<Rightarrow> 'c"   and E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and BMap :: "'c \<Rightarrow> bool" shows "((skip\<^bsub>\<iota>\<^esub>, m), [], None, m) \<in> MWLs_semantics.MWLsSteps_det E BMap"
Abs: fixes xs :: "'varSort"   and s :: "'sort"   and X' :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and X :: "'a"   and x :: "'var" assumes "isInBar (xs, s)"   and "wls s X'"   and "gWls MOD s X"   and "phi s X" shows "phiAbs (xs, s) (gAbs MOD xs x X' X)"
case_1: fixes f1 :: "'k"   and f2 :: "'l \<Rightarrow> 'k" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
knows_Nil: fixes A :: "agent" shows "knows A [] = initState A"
simps_1: fixes m :: "nat" shows "m - 0 = m"
atoms_of_cnf_Un: fixes S :: "'b literal set set"   and T :: "'b literal set set" shows "atoms_of_cnf (S \<union> T) = atoms_of_cnf S \<union> atoms_of_cnf T"
pre_subdivFace_indexToVertexList: fixes g :: "graph"   and f :: "face"   and v :: "nat"   and e :: "nat list"   and i :: "nat" assumes "minGraphProps g"   and "f \<in> set (nonFinals g)"   and "v \<in> \<V> f"   and "e \<in> set (enumerator i |vertices f|)"   and "\<not> containsDuplicateEdge g f v e"   and "2 < i" shows "pre_subdivFace g f v (indexToVertexList f v e)"
case_2: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b msllist \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b msllist" shows "(case MSLCons x21 x22 of MSLNil \<Rightarrow> f1 | MSLCons x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
is_red_addsI: fixes f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and v :: "'x \<Rightarrow>\<^sub>0 nat"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "f \<in> F"   and "f \<noteq> 0"   and "v \<in> keys p"   and "local.punit.adds_term (lpp f) v" shows "punit.is_red F p"
assms_1: shows "n \<in> obs n' (PDG_BS S)"
card_word: shows "CARD('b word) = 2 ^ LENGTH('b)"
eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x = y" shows "domain_semigroup.ds_ord d (\<cdot>) x y"
FaultProp: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and c :: "('f, 'e, 'g) com"   and f :: "'g" shows "\<Gamma>\<turnstile> \<langle>c,Fault f\<rangle> \<Rightarrow> Fault f"
i1: fixes s :: "'a" assumes "PROP P s" shows "PROP Q s"
scale_minus_right: fixes a :: "'b"   and x :: "'b" shows "a * - x = - (a * x)"
post_meas0_bell10_fst: shows "post_meas0 2 |\<beta>\<^sub>1\<^sub>0\<rangle> 0 = |unit_vec 4 0\<rangle>"
scaleC: fixes r :: "complex"   and x :: "'a" shows "f (r *\<^sub>C x) = cnj r *\<^sub>C f x"
isSuccOrd_def: shows "r.isSuccOrd \<equiv> \<exists>j\<in>Field r. \<forall>i\<in>Field r. (i, j) \<in> r"
ker: shows "a_kernel R (RDirProd_list (map (\<lambda>i. R Quot I (Suc i)) [0..<Suc n])) (canonical_proj_ext (\<lambda>i. I (Suc i)) n) = (\<Inter>i\<le>n. I (Suc i))"
inject_2: fixes x2 :: "(val \<times> val) fset"   and y2 :: "(val \<times> val) fset" shows "(val.Fun x2 = val.Fun y2) = (x2 = y2)"
order_less_imp_not_eq2: fixes x :: "'b"   and y :: "'b" assumes "x < y" shows "(y = x) = False"
prenum_in_R2: shows "prenum \<in> \<R>\<^sup>2"
e_inverse: fixes e :: "'e \<rightarrow> 'f"   and p :: "'f \<rightarrow> 'e"   and x :: "'e" assumes "ep_pair e p" shows "p\<cdot>(e\<cdot>x) = x"
hyps: fixes j :: "nat" shows "j < length N \<longrightarrow> vec_mat_Tensor v N ! j = vec_vec_Tensor v (N ! j)"
uminus_literal_def: fixes L :: "'b literal" shows "- L = (if is_pos L then Neg else Pos) (atm_of L)"
delta_pf_support: fixes N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat" shows "support_set (delta_pf N der matur) \<subseteq> {stk}"
contlub_Mprefix: fixes A :: "'b set" shows "contlub (Mprefix A)"
equiv_exprel_iff: fixes x :: "freeExp"   and y :: "freeExp" shows "(exprel `` {x} = exprel `` {y}) = (x \<sim> y)"
I_lift_dnfeq_qe: fixes qe :: "atom list \<Rightarrow> atom fm"   and \<phi> :: "atom fm"   and xs :: "'b list" assumes "\<And>as. \<forall>a\<in>set as. depends\<^sub>d\<^sub>l\<^sub>o a \<Longrightarrow> qfree (qe as)"   and "\<And>as. \<forall>a\<in>set as. depends\<^sub>d\<^sub>l\<^sub>o a \<and> \<not> (case a of Less nat1 nat2 \<Rightarrow> False | Eq i j \<Rightarrow> i = 0 \<or> j = 0) \<Longrightarrow> \<forall>xs. DLO.I (qe as) xs = (\<exists>x. \<forall>a\<in>set as. I\<^sub>d\<^sub>l\<^sub>o a (x # xs))" shows "DLO.I (DLO\<^sub>e.lift_dnfeq_qe qe \<phi>) xs = DLO.I \<phi> xs"
invertibleE: fixes u :: "'a"   and thesis :: "bool" assumes "maxI.additive.invertible u"   and "\<And>v. \<lbrakk>addA u v = zeroA \<and> addA v u = zeroA; v \<in> A\<rbrakk> \<Longrightarrow> thesis"   and "u \<in> A" shows "thesis"
leq_refl: fixes x :: "'f llist" shows "lprefix x x"
cont_cfg_defined: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real) set" assumes "cfg \<in> MDP.valid_cfg"   and "s \<in> set_pmf (abst (action cfg))" shows "(abss (THE x. abss x = s \<and> x \<in> set_pmf (action cfg)) = s \<and> (THE x. abss x = s \<and> x \<in> set_pmf (action cfg)) \<in> set_pmf (action cfg)) \<and> (\<forall>y. abss y = s \<and> y \<in> set_pmf (action cfg) \<longrightarrow> y = (THE x. abss x = s \<and> x \<in> set_pmf (action cfg)))"
distinct_7: fixes x2 :: "'a"   and x3 :: "'b" shows "Abrupt x2 \<noteq> Fault x3"
continuous_on_iff: fixes s :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "continuous_on s f = (\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
open_continuous_vimage: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set" assumes "open s"   and "continuous_on s f"   and "open B" shows "open (f -` B \<inter> s)"
positively_invariant_trapG1: shows "g.positively_invariant trapG1"
upd_rows_transpose: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d set"   and p :: "'d \<Rightarrow> ('c, 'd) vec" shows "upd_rows (Square_Matrix.transpose M) i p = Square_Matrix.transpose (upd_cols M i p)"
cycle_permutes: fixes cs :: "'b list" shows "cycle_of_list cs permutes set cs"
f_aux: fixes i :: "nat" shows "flow \<Delta> (f_aux i)"
mat_of_rows_list_def: fixes nc :: "nat"   and rs :: "'a list list" shows "mat_of_rows_list nc rs = mat (length rs) nc (\<lambda>(i, j). rs ! i ! j)"
iMODb_mult_div_right_inj_on: fixes k :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" shows "inj_on (\<lambda>x. x div k) [ r, mod k * m, c ]"
lunit'_in_hom_2: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.ide f" shows "\<guillemotleft>S.lunit' f : f \<Rightarrow>\<^sub>S S.trg f \<star>\<^sub>S f\<guillemotright>"
emeasure_pair_measure_Times: fixes A :: "'c set"   and N :: "'c measure"   and B :: "'b set" assumes "A \<in> sets N"   and "B \<in> M2.events" shows "emeasure (N \<Otimes>\<^sub>M M2) (A \<times> B) = emeasure N A * emeasure M2 B"
YFtorFtor: fixes C :: "('d, 'e, 'f) LSCategory_scheme" assumes "LSCategory C" shows "Ftor YFtor C : C \<longrightarrow> CatExp (Op C) SET"
submonoid_invertible: fixes u :: "'a" assumes "sub.invertible u"   and "u \<in> G" shows "invertible u"
simps_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "monoidal_language C" shows "monoidal_language.Can C \<^bold>\<langle>f\<^bold>\<rangle> = partial_magma.ide C f"
simps_2: fixes x :: "char list"   and e :: "aexp"   and P :: "'a" shows "post (x ::= e {P}) = P"
simps_1: fixes x :: "'c"   and y :: "'d" shows "zip_option (Some x) (Some y) = Some (x, y)"
impl_blue_dfs_hash_def: shows "impl_blue_dfs_hash \<equiv> \<lambda>x. NDFS_SI_Statistics.start_nres \<bind> (\<lambda>xa. LIST_FOREACH' (RETURN (id (gi_V0 x))) (\<lambda>(xb, xc, xd). xd = NO_CYC) (\<lambda>xb (xc, xd, xe). if \<not> hs.memb xb xc then REC\<^sub>T (\<lambda>xf (xg, xh, xi, xj). let y' = hs.ins xj xg; y'a = hs.ins xj xi; y'b = bgi_F x xj in NDFS_SI_Statistics.vis_blue_nres \<bind> (\<lambda>xk. LIST_FOREACH' (RETURN (id (gi_E x xj))) (\<lambda>(xl, xm, xn, xo). xo = NO_CYC) (\<lambda>xl (xm, xn, xo, xp). if hs.memb xl xo \<and> (y'b \<or> bgi_F x xl) then RETURN (xm, xn, xo, init_wit_blue_early xj xl) else if \<not> hs.memb xl xm then xf (xm, xn, xo, xl) \<bind> (\<lambda>(xq, xr, xs, xt). RETURN (xq, xr, xs, prep_wit_blue xj xt)) else NDFS_SI_Statistics.match_blue_nres \<bind> (\<lambda>xq. RETURN (xm, xn, xo, xp))) (y', xh, y'a, NO_CYC) \<bind> (\<lambda>(xl, xm, xn, xo). (if xo = NO_CYC \<and> y'b then impl_red_dfs_hash (gi_E x) xn xm xj \<bind> (\<lambda>(xp, xq). RETURN (xp, init_wit_blue xj xq)) else RETURN (xm, xo)) \<bind> (\<lambda>(xp, xq). let y'c = hs.delete xj xn in RETURN (xl, xp, y'c, xq))))) (xc, xd, hs.empty (), xb) \<bind> (\<lambda>(xf, xg, xh, xi). RETURN (xf, xg, xi)) else RETURN (xc, xd, NO_CYC)) (hs.empty (), hs.empty (), NO_CYC) \<bind> (\<lambda>(xb, xc, xd). NDFS_SI_Statistics.stop_nres \<bind> (\<lambda>xe. RETURN (extract_res xd))))"
polynomial_pow_degree: fixes p :: "'a list list"   and n :: "nat" assumes "p \<in> carrier (poly_ring (poly_ring R))" shows "degree (p [^]\<^bsub>poly_ring (poly_ring R)\<^esub> n) = n * degree p"
wrap_Un: fixes M :: "'b set"   and x :: "'b" shows "wrap (M \<union> {x}) = {M \<union> {x}}"
DomainE: fixes a :: "'l"   and r :: "('l \<times> 'm) set" assumes "a \<in> Domain r" obtains b :: "'m" where "(a, b) \<in> r"
C: shows "Crypt (SesK SK) (PubKey C) \<in> parts (used s')"
b_ag_group: fixes A :: "('c, 'd) aGroup_scheme" assumes "aGroup A" shows "Group (b_ag A)"
HAStates_CompFun_SAs: fixes S :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "S \<in> HAStates A" shows "the (CompFun A S) \<subseteq> SAs A"
Gauss_Jordan_column_k_iarrays_PA_def: fixes A' :: "'d iarray iarray \<times> nat \<times> 'd iarray iarray"   and k :: "nat" shows "Gauss_Jordan_column_k_iarrays_PA A' k = (let P = fst A'; i = fst (snd A'); A = snd (snd A') in if vector_all_zero_from_index (i, column_iarray k A) \<or> i = nrows_iarray A then (P, i, A) else let Gauss = Gauss_Jordan_in_ij_iarrays_PA (P, A) i k in (fst Gauss, i + 1, snd Gauss))"
assms_1: shows "negligible (\<lambda>\<eta>. local.prf.advantage \<eta> (prf_adversary \<eta> (\<A> \<eta>)))"