hcju/isabelleps huggingface dataset:
bot_1: fixes x :: "'a" assumes "\<not> x \<le> bound" shows "ord bot (f x)"
distinct_3: shows "ccode.Gt \<noteq> ccode.Eq"
finfun_update_update_code: fixes f :: "('a, 'b) finfun"   and a :: "'a"   and b :: "'b"   and a' :: "'a"   and b' :: "'b" shows "finfun_update (finfun_update_code f a b) a' b' = (if a = a' then finfun_update f a b' else finfun_update_code (finfun_update f a' b') a b)"
verit_sko_forall': fixes P :: "'b \<Rightarrow> bool"   and A :: "bool" assumes "P (SOME x. \<not> P x) = A" shows "(\<forall>x. P x) = A"
unbounded_inner: shows "\<not> bounded {x. B \<le> i \<bullet> x}"
wf_hull_rel_relcomp: fixes R :: "('a \<times> 'a) set" assumes "wf R"   and "eqvt R" shows "wf (hull_rel O R)"
abs_fun_eq1: fixes a :: "'b"   and x :: "'c"   and y :: "'c" shows "([a].x = [a].y) = (x = y)"
con_comp_consistent_maps: fixes D :: "'e \<Rightarrow> 'f"   and E :: "'g \<Rightarrow> 'f"   and p :: "'e \<Rightarrow> 'h"   and q :: "'g \<Rightarrow> 'h" assumes "consistent_maps D E p q" shows "con_comp_map D E p q = con_comp_map E D q p"
assms_3: shows "op \<in> set (map (strips_op_to_sasp \<Psi>) ops')"
P_iter_def: fixes ode_ops :: "ode_ops" shows "P_iter ode_ops \<equiv> \<lambda>uu uua uuaa. rec_nat (\<lambda>X0 h X. let _ = trace_set (ST ''P_iter failed (0)'') (Some X) in RETURN None) (\<lambda>i ia X0 h X. ASSERT (0 \<le> h) \<bind> (\<lambda>_. op_ivl_rep_of_set X \<bind> (\<lambda>(l, u). ASSERT (l \<le> u) \<bind> (\<lambda>_. mk_safe ode_ops ({l..u} ::: appr_rel) \<bind> (\<lambda>ivl. Picard_step_ivl ode_ops X0 0 h ivl \<bind> (\<lambda>X'. case X' of None \<Rightarrow> let _ = trace_set (ST ''P_iter failed (Picard_step)'') (Some X) in RETURN None | Some X' \<Rightarrow> op_ivl_rep_of_set X' \<bind> (\<lambda>(l', u'). do_widening_spec i \<bind> (\<lambda>do_widening. let l' = inf l' l - (if do_widening then \<bar>l' - l\<bar> else (0::'c)); u' = sup u' u + (if do_widening then \<bar>u' - u\<bar> else (0::'c)) in ASSERT (l' \<le> u') \<bind> (\<lambda>_. safe_set ode_ops {l'..u'} \<bind> (\<lambda>s. if s then mk_safe ode_ops {l'..u'} \<bind> (\<lambda>ivl'. if l \<le> l' \<and> u' \<le> u then RETURN (Some ivl) else ia X0 h ivl') else let _ = trace_set (ST ''P_iter failed (unsafe interval after widening)'') (Some X) in RETURN None)))))))))) uuaa uu uua"
surj_hom_imp_solvable: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h"   and "h ` carrier G = carrier H"   and "solvable G" shows "solvable H"
predicate_sum_decompose: fixes X :: "'c set"   and x :: "'c"   and A :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d \<Rightarrow> bool" assumes "finite X"   and "x \<in> X"   and "A (f x)"   and "\<forall>Z. B (sum f Z)"   and "\<And>x Z. \<lbrakk>A (f x); B (sum f Z)\<rbrakk> \<Longrightarrow> A (f x + sum f Z)"   and "\<And>x Z. \<lbrakk>B (f x); A (sum f Z)\<rbrakk> \<Longrightarrow> A (f x + sum f Z)" shows "A (sum f X)"
finite_dom_finite_map_graph: fixes f :: "'a \<Rightarrow> 'b option" assumes "finite (dom f)" shows "finite (map_graph f)"
assms_3: fixes x :: "'a" assumes "x \<in> I" shows "0 \<le> f x"
quotient_is_ring: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "ideal I R" shows "ring (R Quot I)"
S1: shows "f x (b_least2 f x y) \<noteq> 0"
get_pow_exp: fixes z :: "'b list"   and t :: "'b list"   and m :: "nat" assumes "z <p t" shows "m = \<^bold>|t \<^sup>@ m \<cdot> z\<^bold>| div \<^bold>|t\<^bold>|"
d3: shows "degree (map_poly M (u * map_poly M h)) = degree (u * map_poly M h)"
assms: shows "list_all (finite \<circ> DBA.nodes) AA"
horizontal_homs_axioms: shows "horizontal_homs (\<cdot>\<^sub>D) src\<^sub>D trg\<^sub>D"
shiftr_slice: fixes w :: "'a word"   and n :: "nat" shows "w >> n = slice n w"
reduced_word_forI_length': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and as :: "'b list"   and A :: "'b set"   and a :: "'b" assumes "class.monoid_add plus zero"   and "as \<in> lists A"   and "monoid_add.sum_list plus zero as = a"   and "order.greater_eq (monoid_add.word_length plus zero A a) (length as)" shows "monoid_add.reduced_word_for plus zero A a as"
finite_UNIV_finite_2_def: shows "finite_UNIV = Phantom(Enum.finite_2) True"
simps: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and g :: "'a"   and a :: "'b list" shows "graph_path_base.path \<alpha>n invar inEdges' g a = ((\<exists>n. a = [n] \<and> n \<in> set (\<alpha>n g) \<and> invar g) \<or> (\<exists>ns n'. a = n' # ns \<and> graph_path_base.path \<alpha>n invar inEdges' g ns \<and> n' \<in> set (graph_path_base.predecessors inEdges' g (hd ns))))"
add_neg_numeral_special_8: shows "- (1::'a) + (1::'a) = (0::'a)"
RModule_axioms_def: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set" shows "RModule_axioms R smult M \<equiv> ((\<forall>r m. r \<in> R \<longrightarrow> m \<in> M \<longrightarrow> smult r m \<in> M) \<and> (\<forall>r m n. r \<in> R \<longrightarrow> m \<in> M \<longrightarrow> n \<in> M \<longrightarrow> smult r (m + n) = smult r m + smult r n)) \<and> (\<forall>r s m. r \<in> R \<longrightarrow> s \<in> R \<longrightarrow> m \<in> M \<longrightarrow> smult (r + s) m = smult r m + smult s m) \<and> (\<forall>r s m. r \<in> R \<longrightarrow> s \<in> R \<longrightarrow> m \<in> M \<longrightarrow> smult r (smult s m) = smult (r * s) m) \<and> (\<forall>m. m \<in> M \<longrightarrow> smult (1::'a) m = m)"
xfer_fair_run: fixes as\<^sub>0 :: "'b"   and alstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and cs\<^sub>0 :: "'e"   and clstep :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and cs :: "nat \<Rightarrow> 'e" assumes "simulation as\<^sub>0 alstep cs\<^sub>0 clstep R"   and "df_system.is_fair_run cs\<^sub>0 clstep cs" obtains as :: "nat \<Rightarrow> 'b" where "df_system.is_fair_run as\<^sub>0 alstep as" and "\<forall>i. R (as i) (cs i)"
M: shows "CU.Model TE_wtFsym wtPsym (length \<circ> TE_arOf) (length \<circ> parOf) TE_\<Phi> D eintF eintP"
inv_char: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "x \<oplus> y = \<zero>"   and "y \<oplus> x = \<zero>" shows "inv\<^bsub>add_monoid R\<^esub> x = y"
cod_simps_3: shows "trg.cod.leg1 = trg.leg1"
intro: fixes R :: "('c, 'd) ring_scheme" assumes "abelian_group R"   and "Group.monoid R"   and "ring_axioms R" shows "ring R"
not_eq_order_implies_strict: fixes a :: "'b list"   and b :: "'b list" assumes "a \<noteq> b"   and "prefix a b" shows "strict_prefix a b"
join_le_iff: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and join :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Top :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.SL_top le join Top" shows "le (join x y) z = (le x z \<and> le y z)"
prv_eqv_imp_transi_rev: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi> \<in> fmla"   and "prv (eqv \<phi>1 \<phi>2)"   and "prv (imp \<phi>1 \<chi>)" shows "prv (imp \<phi>2 \<chi>)"
divide_less_eq_numeral_1: fixes b :: "'c"   and c :: "'c"   and w :: "num" shows "(b / c < numeral w) = (if (0::'c) < c then b < numeral w * c else if c < (0::'c) then numeral w * c < b else (0::'c) < numeral w)"
trimonoD_flatf_ge: fixes B :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c" assumes "trimono B" shows "monotone flatf_ge flatf_ge B"
enat_cocase_inf: fixes a :: "'c"   and f :: "enat \<Rightarrow> 'c" shows "(case \<infinity> of 0 \<Rightarrow> a | eSuc b \<Rightarrow> f b) = f \<infinity>"
mult_cancel_left: fixes f1 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g1 :: "'b \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "f1 \<in> \<Theta>[F](g1)"   and "\<forall>\<^sub>F x in F. g1 x \<noteq> (0::'c)" shows "((\<lambda>x. f1 x * f2 x) \<in> O[F](\<lambda>x. g1 x * g2 x)) = (f2 \<in> O[F](g2))"
map_le_refl: fixes f :: "'a \<Rightarrow> 'b option" shows "f \<subseteq>\<^sub>m f"
non_volatile_owned_or_read_only_pending_write_antimono: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and xs :: "'b memref list" assumes "non_volatile_owned_or_read_only True \<S> \<O> xs" shows "non_volatile_owned_or_read_only False \<S> \<O> xs"
gcd_coprime_exists: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "gcd a b \<noteq> zero" shows "\<exists>a' b'. a = times a' (gcd a b) \<and> b = times b' (gcd a b) \<and> algebraic_semidom.coprime times one a' b'"
quasi_isometric_equiv_rel_3: fixes X :: "'b set"   and Y :: "'c set" assumes "quasi_isometric X Y" shows "quasi_isometric Y X"
simps_1: fixes p :: "fm"   and q :: "fm" shows "d_\<delta> (And p q) = (\<lambda>d. d_\<delta> p d \<and> d_\<delta> q d)"
Exit_empty: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g" assumes "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" shows "Def Exit = {} \<and> Use Exit = {}"
conjunctiveI: fixes a :: "'a" assumes "\<And>b c. a * (b \<sqinter> c) = a * b \<sqinter> (a * c)" shows "a \<in> conjunctive.conjunctive (\<sqinter>) (\<sqinter>) (*)"
subcls: shows "G\<turnstile>C\<prec>\<^sub>C declclass old"
basic_subtype_x_Enum: fixes \<tau> :: "'a basic_type"   and \<E> :: "('a, String.literal) phantom" assumes "\<tau> \<sqsubset>\<^sub>B Enum \<E>" obtains "\<tau> = OclVoid"
size_if_complete: fixes kdt :: "'b kdt" assumes "KD_Tree.complete kdt" shows "KD_Tree.size_kdt kdt = 2 ^ height kdt"
fps_mult_subdegree_ge: fixes f :: "'b fps"   and g :: "'b fps" assumes "f * g \<noteq> 0" shows "subdegree f + subdegree g \<le> subdegree (f * g)"
aux_lemma1: fixes Ia :: "'a set"   and i :: "'a" assumes "ideal Ia R"   and "i \<in> Ia" shows "Ia +> i = \<zero>\<^bsub>R Quot Ia\<^esub>"
prod_prime_factors: fixes n :: "nat" assumes "0 < n" shows "f n = (\<Prod>p\<in>prime_factors n. f (p ^ multiplicity p n))"
simps: fixes v :: "nat"   and p :: "real mpoly" shows "to_list v p = map (\<lambda>x. (isolate_variable_sparse p v x, x)) [0..<MPoly_Type.degree p v + 1]"
ordinal_of_nat_plus: fixes x :: "nat"   and y :: "nat" shows "ordinal_of_nat x + ordinal_of_nat y = ordinal_of_nat (x + y)"
simps: fixes a :: "'k"   and f :: "'k \<times> 'k \<Rightarrow> ennreal" shows "(a \<in> MFMC_Network.SINK f) = (\<exists>x. a = x \<and> d_OUT f x = 0)"
TC_equiv_tranclp: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" shows "TC R a b = R\<^sup>+\<^sup>+ a b"
ereal_decseq_approx: fixes a :: "ereal"   and b :: "ereal" assumes "a < b" obtains X :: "nat \<Rightarrow> real" where "decseq X" and "\<And>i. a < ereal (X i)" and "\<And>i. ereal (X i) < b" and "(\<lambda>x. ereal (X x)) \<longlonglongrightarrow> a"
lt_le_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "(lt p \<preceq>\<^sub>t v) = (\<forall>u. v \<prec>\<^sub>t u \<longrightarrow> lookup p u = (0::'c))"
one_dim_mat_mult_is_scale: fixes B :: "'c mat"   and n :: "nat"   and a :: "'c" assumes "B \<in> carrier_mat 1 n" shows "mat 1 1 (\<lambda>k. a) * B = a \<cdot>\<^sub>m B"
False: shows "\<not> (n = 0 \<or> n = 1)"
add_divide_eq_if_simps_4: fixes a :: "'a"   and b :: "'a"   and z :: "'a" shows "a - b / z = (if z = (0::'a) then a else (a * z - b) / z)"
ext_inject: fixes mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and more :: "'d"   and mult' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one' :: "'c"   and more' :: "'d" shows "(\<lparr>monoid.mult = mult, one = one, \<dots> = more\<rparr> = \<lparr>monoid.mult = mult', one = one', \<dots> = more'\<rparr>) = (mult = mult' \<and> one = one' \<and> more = more')"
sum_sum_distl_fun: fixes h :: "'c \<Rightarrow> 'c set"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. finite (h x)"   and "finite X" shows "\<Sum> ((\<lambda>x. f x \<cdot> \<Sum> (g ` h x)) ` X) = \<Sum> {\<Sum> {f x \<cdot> g y |y. y \<in> h x} |x. x \<in> X}"
is_moebius_def: fixes f :: "complex_homo \<Rightarrow> complex_homo" shows "is_moebius f = (\<exists>M. f = moebius_pt M)"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat" assumes "\<And>i n. (\<not> n \<le> i \<Longrightarrow> P (Suc i) n) \<Longrightarrow> P i n" shows "P a0 a1"
Abs_MonoTran_inverse: fixes y :: "'b \<Rightarrow> 'b" assumes "y \<in> {f. mono f}" shows "Rep_MonoTran (Abs_MonoTran y) = y"
subgraph_restrict: fixes G :: "('e, 'f) labeled_graph" shows "subgraph G (restrict G) = graph G"
quotient_of_nonzero_1: fixes r :: "rat" shows "0 < snd (quotient_of r)"
dense_ge: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and z :: "'a"   and y :: "'a" assumes "class.dense_linorder less_eq less"   and "\<And>x. less z x \<Longrightarrow> less_eq y x" shows "less_eq y z"
1: shows "deg R g \<le> deg (R\<lparr>carrier := S\<rparr>) g"
trivial_limit_eq: fixes net :: "'b filter" shows "(net = bot) = (\<forall>P. eventually P net)"
ccSup_upper: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set"   and x :: "'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A"   and "x \<in> A" shows "less_eq x (Sup A)"
commit_correct: fixes \<eta> :: "nat" shows "chaum_ped_\<Sigma>_commit.abstract_com.correct \<eta>"
simps_1: fixes e :: "'a"   and a :: "'b" shows "FingerTreeStruc.nodeToDigit (Tip e a) = One (Tip e a)"
subspace_orthogonal_comp: fixes W :: "'a set" shows "subspace (W\<^sup>\<bottom>)"
attract: fixes x :: "'a set"   and y :: "'a set"   and z :: "'a set" assumes "x \<sqsubseteq>\<^sup>s y"   and "y \<sqsubseteq>\<^sup>s x"   and "y \<sqsubseteq>\<^sup>s z"   and "x \<in> Q"   and "y \<in> Q"   and "z \<in> Q" shows "x \<sqsubseteq>\<^sup>s z"
less_eq_element_ptr_def: fixes x :: "(_) element_ptr"   and y :: "(_) element_ptr" shows "x \<le> y \<equiv> case x of Ref i \<Rightarrow> case y of Ref j \<Rightarrow> i \<le> j | element_ptr.Ext x \<Rightarrow> True | element_ptr.Ext i \<Rightarrow> case y of Ref x \<Rightarrow> False | element_ptr.Ext j \<Rightarrow> i \<le> j"
set_shadow_root_get_disconnected_nodes: fixes set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) document_ptr" assumes "l_set_shadow_root_get_disconnected_nodes set_shadow_root_locs get_disconnected_nodes_locs" shows "\<forall>w\<in>set_shadow_root_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
mult_mono': fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e"   and d :: "'e" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq c d"   and "less_eq zero a"   and "less_eq zero c" shows "less_eq (times a c) (times b d)"
lookup_oalist_of_list: fixes xs :: "('a \<times> 'b) list"   and ko :: "'a nat_term_order" assumes "distinct (map fst xs)" shows "OAlist_lookup_ntm (OAlist_ntm (xs, ko)) = lookup_dflt xs"
in_keys_poly_substE: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "t \<in> keys (poly_subst f p)" obtains s :: "'d \<Rightarrow>\<^sub>0 nat" where "s \<in> keys p" and "t \<in> keys (subst_pp f s)"
that: fixes r :: "real"   and g :: "complex \<Rightarrow> complex" assumes "0 < r"   and "g holomorphic_on ball \<xi> r"   and "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = ((w - \<xi>) * g w) ^ n"   and "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0" shows "thesis"
intros_4: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs"   and Q :: "ccs" assumes "P \<longmapsto>\<alpha> \<prec> P'" shows "P \<parallel> Q \<longmapsto>\<alpha> \<prec> P' \<parallel> Q"
deriv: fixes x :: "real" assumes "a \<le> x" shows "(g has_real_derivative g' x) (at x)"
trivialpair_in_USdB: fixes SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> ('b \<Rightarrow> 'c)) set"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e" shows "([], []) \<in> Strong_Security.USdB SR DA d"
add_cancel_left_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(plus b a = a) = (b = zero)"
assms_1: shows "v \<leadsto>xs\<leadsto> w"
is_class_supclass: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "char list"   and D :: "char list" assumes "wf_prog wf_md P"   and "P \<turnstile> C \<preceq>\<^sup>* D"   and "is_class P C" shows "is_class P D"
nat_discrete_filtration: fixes p :: "real"   and M :: "bool stream measure" assumes "infinite_coin_toss_space p M" shows "Filtration.filtration M (infinite_coin_toss_space.nat_filtration M)"
has_int: shows "((\<lambda>x. c / max b \<bar>x - a\<bar> ^ n) has_integral A - B) {l'..r'}"
cases: fixes a :: "'e"   and r :: "('e \<times> 'f) set" assumes "a \<in> Domain r" obtains aa :: "'e" and b :: "'f" where "a = aa" and "(aa, b) \<in> r"
Conf_def: shows "Conf \<equiv> fst \<circ> snd \<circ> Rep_status"
card_insert_disjoint: fixes A :: "'c set"   and x :: "'c" assumes "finite A"   and "x \<notin> A" shows "card (insert x A) = Suc (card A)"
readCArray_def: fixes array :: "nat \<Rightarrow> 'b"   and index :: "nat" shows "readCArray array index \<equiv> array index"
rw: fixes k :: "'n" shows "(\<Sum>ka\<in>UNIV. P $h i $h ka * A $h ka $h k) = P $h i $h k * A $h k $h k"
assms_2: assumes "\<not> c" shows "effect e h h' r"
mult_onel: fixes x :: "'c \<times> 'd" shows "times_prod' one_prod x = x"
hyps: fixes c :: "nat"   and ys :: "'a list"   and a :: "nat" assumes "take c xs = take c ys"   and "a \<le> c" shows "take a xs = take a ys"
geq0: shows "dterm_sem I (f1 fid2 vid1) (sol 0, b) < dterm_sem I (f1 fid1 vid1) (sol 0, b)"
pre_mk_ps_chain: fixes \<Pi> :: "(node \<rightharpoonup>\<^sub>f topbot, node + node fset \<times> command_gadget \<times> node fset) chain"   and \<pi> :: "(node + node fset \<times> command_gadget \<times> node fset) list" shows "pre (mk_ps_chain \<Pi> \<pi>) = pre \<Pi>"
atom_map_def_var: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a" assumes "class.bounded_lattice inf less_eq less sup bot top" shows "bounded_lattice.atom_map less_eq less bot x = ord.downset less_eq x \<inter> Collect (bounded_lattice.atom less bot)"
equivp_symp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "equivp R"   and "R x y" shows "R y x"
envalid_def2: fixes n :: "nat"   and P :: "assn"   and e :: "expr"   and Q :: "vassn" shows "\<Turnstile>n:\<^sub>e (P, e, Q) \<equiv> \<forall>s v t. s -e\<succ>v-n\<rightarrow> t \<longrightarrow> P s \<longrightarrow> Q v t"
inorder_insert: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (local.insert x t) = ins_list x (Tree2.inorder t)"
subst_atm_mset_lists_Cons: fixes C :: "'a multiset"   and Cs :: "'a multiset list"   and \<sigma> :: "'s"   and \<sigma>s :: "'s list" shows "(C # Cs) \<cdot>\<cdot>aml (\<sigma> # \<sigma>s) = C \<cdot>am \<sigma> # Cs \<cdot>\<cdot>aml \<sigma>s"
finite_elements: fixes A :: "'c set"   and P :: "'c set set" assumes "finite A"   and "partition_on A P" shows "finite P"
BrnFT_def: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('a, 'b, 'c) cmd list" shows "BrnFT cl dl \<equiv> \<Union>n\<in>theFTOne cl dl. {brnL cl n..<brnL cl n + brn (cl ! n)}"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and f2 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> bool) \<Rightarrow> 'a"   and f3 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> 'c option) list \<Rightarrow> 'a"   and f4 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> ('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and edge_kind :: "('b, 'c, 'd, 'e) edge_kind" shows "P (case edge_kind of \<Up>x \<Rightarrow> f1 x | (x)\<^sub>\<surd> \<Rightarrow> f2 x | x:xa\<hookrightarrow>\<^bsub>xb\<^esub>xc \<Rightarrow> f3 x xa xb xc | x\<hookleftarrow>\<^bsub>xa\<^esub>xb \<Rightarrow> f4 x xa xb) = (\<not> ((\<exists>x1. edge_kind = \<Up>x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x2. edge_kind = (x2)\<^sub>\<surd> \<and> \<not> P (f2 x2)) \<or> (\<exists>x31 x32 x33 x34. edge_kind = x31:x32\<hookrightarrow>\<^bsub>x33\<^esub>x34 \<and> \<not> P (f3 x31 x32 x33 x34)) \<or> (\<exists>x41 x42 x43. edge_kind = x41\<hookleftarrow>\<^bsub>x42\<^esub>x43 \<and> \<not> P (f4 x41 x42 x43))))"
reduce_holds: fixes s :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "strict_mono s" shows "P n (s \<circ> reduce s n)"
f_s_fin: shows "\<integral>\<^sup>+ x. ennreal (norm (f x - s i x)) \<partial>M < \<infinity>"
wf_PLUS: fixes wf_atom :: "nat \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat"   and xs :: "'c rexp list" shows "alphabet.wf wf_atom n (PLUS xs) = (\<forall>r\<in>set xs. alphabet.wf wf_atom n r)"
ring_iso_memE_2: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "h \<in> ring_iso R S"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
prv_pls_zer: fixes t :: "'trm" assumes "t \<in> atrm" shows "prv (eql (pls t zer) t)"
\<tau>move2_IfElseD: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e :: "'addr expr1"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and pc' :: "nat"   and xcp' :: "'addr option" assumes "\<tau>move2 P h stk (if (e) e1 else e2) (Suc (Suc (length (compE2 e) + length (compE2 e1) + pc'))) xcp'" shows "\<tau>move2 P h stk e2 pc' xcp'"
fMax_finsert: fixes x :: "'a"   and A :: "'a fset" shows "fMax (finsert x A) = (if A = {||} then x else max x (fMax A))"
oalist_inv_list_of_oalist: fixes x :: "'x" shows "oalist_inv (list_of_oalistx x)"