hcju/isabelleps huggingface dataset:
\<G>\<F>_implies_GF: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<forall>\<psi>\<in>\<G>\<F> \<phi> w. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>[\<F>\<G> \<phi> w]\<^sub>\<Sigma>\<^sub>1)"
rcos_ring_hom_ring: shows "ring_hom_ring R (R Quot I) ((+>) I)"
xints_of_def: fixes rect :: "'b rectangle list"   and "is" :: "nat set" shows "xints_of rect is = (\<lambda>i. IdxInterval (xint (rect ! i)) i) ` is"
dropWhile_append: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" shows "dropWhile P (xs @ ys) = (if \<forall>x\<in>set xs. P x then dropWhile P ys else dropWhile P xs @ ys)"
dg_GRPH_is_arrI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<^bsub>dg_GRPH \<alpha>\<^esub> \<BB>"
isomorphic_def: fixes a :: "'a"   and a' :: "'a" shows "L'.C.isomorphic a a' = (\<exists>f. L'.C.in_hom f a a' \<and> L'.C.iso f)"
NextMem_def: fixes c :: "('exp, 'id) MWLsCom"   and m :: "'id \<Rightarrow> 'val" shows "\<lbrakk>c\<rbrakk>(m) \<equiv> THE m'. \<exists>p \<alpha>. SRabbr c m \<alpha> p m'"
\<theta>_simps_4: shows "local.dom \<theta> = f \<star> w"
abelian_monoidE_5: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "abelian_monoid R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"
greatest_nrows: shows "Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A)) \<noteq> nrows A"
assms: shows "is \<lhd> input_sizes (deep_model_l rs)"
XY: shows "simple_distributed M (\<lambda>x. (X x, Y x)) (\<lambda>x. prob ((\<lambda>x. (X x, Y x)) -` {x} \<inter> space M))"
max_mult_distrib_left: fixes p :: "'a"   and x :: "'a"   and y :: "'a" shows "p * max x y = (if (0::'a) \<le> p then max (p * x) (p * y) else min (p * x) (p * y))"
simps_5: fixes t :: "('b, 'a) Term.term"   and t' :: "('b, 'a) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (delete\<langle>t,t'\<rangle>) = fv t \<union> fv t'"
assms_2: shows "A $ ma $ mod_type_class.from_nat (Suc k) \<noteq> (0::'a)"
h3: shows "M_mat_R (take_indices signs (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))) (take_indices subsets (rows_to_keep (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))))) = take_rows_from_matrix (take_cols_from_matrix (M_mat_R signs subsets) (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))) (rows_to_keep (take_cols_from_matrix (M_mat_R signs subsets) (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))))"
9: shows "r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> * (e\<^sup>T * e \<sqinter> g) = bot"
prepend: fixes vs :: "nat list"   and f :: "nat \<Rightarrow> nat option"   and x :: "nat" shows "(vs \<odot> f) x = (if x < length vs then Some (vs ! x) else f (x - length vs))"
simps_2: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and a :: "'e"   and x :: "'f"   and xs :: "'f list" shows "foldl f a (x # xs) = foldl f (f a x) xs"
landau_product_preprocess_8: fixes m :: "num"   and n :: "num" shows "num.Bit0 m + num.Bit0 n = num.Bit0 (m + n)"
opp_opp: fixes p :: "'b point" shows "opp (opp p) = p"
siterate_eq_scons: fixes f :: "'b \<Rightarrow> 'b"   and s :: "'b"   and a :: "'b"   and w :: "'b stream" shows "(siterate f s = a ## w) = (s = a \<and> siterate f (f s) = w)"
adjoint_equivalence_in_bicategory_axioms: shows "adjoint_equivalence_in_bicategory (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<tau>\<^sub>0 a) (map\<^sub>0 a) (unit a) (counit a)"
k_le: fixes k :: "nat" assumes "\<pi> k = x \<or> \<pi> k = y" shows "(LEAST xa. \<pi> xa = x \<or> \<pi> xa = y) \<le> k"
outputRenameSubject: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and K :: "'a" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<turnstile> M \<leftrightarrow> K"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* K" shows "\<Psi> \<rhd> P \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"
assms_4: fixes x :: "'node" assumes "x \<in> set ns" shows "v \<notin> oldDefs g x"
start: fixes x :: "'a"   and A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "x \<in> A" shows "x \<in> directed_towards A r"
zero_less_Fract_iff: fixes b :: "'b"   and a :: "'b" assumes "(0::'b) < b" shows "(0 < Fract a b) = ((0::'b) < a)"
vertices_conv_Union_edges: fixes f :: "face" shows "\<V> f = (\<Union>\<^bsub>(a, b)\<in>\<E> f\<^esub> {a})"
a_comm: fixes x :: "'a"   and y :: "'a" shows "ad x \<cdot> ad y = ad y \<cdot> ad x"
init_fin_simulation2: fixes t :: "'t"   and s1 :: "(status \<times> 'x1) \<times> 'm1"   and s2 :: "(status \<times> 'x2) \<times> 'm2"   and tl2 :: "('l, 't, status \<times> 'x2, 'm2, 'w, 'o action) thread_action"   and s2' :: "(status \<times> 'x2) \<times> 'm2" assumes "t \<turnstile> s1 \<approx>i s2"   and "r2.init_fin t s2 tl2 s2'"   and "\<not> r2.init_fin_\<tau>move s2 tl2 s2'" shows "\<exists>s1' s1'' tl1. (\<tau>trsys.silent_move (r1.init_fin t) r1.init_fin_\<tau>move)\<^sup>*\<^sup>* s1 s1' \<and> r1.init_fin t s1' tl1 s1'' \<and> \<not> r1.init_fin_\<tau>move s1' tl1 s1'' \<and> t \<turnstile> s1'' \<approx>i s2' \<and> ta_bisim init_fin_bisim tl1 tl2"
assms: shows "\<forall>\<^sub>F x in at_top. f x = g x"
rep_eq: fixes x :: "uint64"   and xa :: "nat" shows "Rep_uint64 (x >>> xa) = signed_drop_bit xa (Rep_uint64 x)"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "nat" assumes "stirling x xa = y" obtains "x = 0" and "xa = 0" and "y = 1"   | k :: "nat" where "x = 0" and "xa = Suc k__" and "y = 0"   | n :: "nat" where "x = Suc na__" and "xa = 0" and "y = 0"   | n :: "nat" and k :: "nat" where "x = Suc n" and "xa = Suc k" and "y = n * stirling n (Suc k) + stirling n k"
r: fixes r :: "(att \<times> char list, att, nat) env"   and x :: "operation"   and r' :: "(att \<times> char list, att, nat) env" assumes "r \<midarrow>x\<rightarrow> r'"   and "Q r"   and "P x" shows "Q r'"
invertible_inverse_invertible: fixes u :: "'a" assumes "invertible u"   and "u \<in> G" shows "invertible (local.inverse u)"
uniformly_continuous_on_def: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "uniformly_continuous_on s f = (\<forall>e>0. \<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
sylvester_mat_sub_0: fixes n :: "nat"   and q :: "'c poly" shows "sylvester_mat_sub 0 n 0 q = 0\<^sub>m n n"
basic_subtype_asym: fixes \<tau> :: "'b basic_type"   and \<sigma> :: "'b basic_type" assumes "\<tau> \<sqsubset>\<^sub>B \<sigma>"   and "\<sigma> \<sqsubset>\<^sub>B \<tau>" shows "False"
zero_order_1: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq zero x"
assms_5: shows "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"
assms: shows "S \<equiv> \<lambda>n. \<Sum>k | k \<in> {1..n} \<and> coprime k n. f k n"
valid_node_def: fixes n :: "'b LDCFG_node" shows "CFGExit_wf.valid_node n \<equiv> \<exists>a. lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a \<and> (n = src a \<or> n = trg a)"
of_bool_nth: fixes x :: "'b word"   and v :: "nat" shows "of_bool (x !! v) = (x >> v) AND 1"
map_def: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" shows "Set_Theory.map \<alpha> S T \<equiv> \<alpha> \<in> S \<rightarrow>\<^sub>E T"
pt_tyvrs_inst: shows "pt TYPE('a) TYPE(tyvrs)"
assms_4: fixes i :: "'b" assumes "i \<in> I'" shows "\<mu> i = \<mu>' i"
neg_numeral_less_zero: 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 n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (uminus (numeral.numeral one plus n)) zero"
connected_open_diff_countable: fixes S :: "'a set"   and T :: "'a set" assumes "2 \<le> DIM('a)"   and "open S"   and "connected S"   and "countable T" shows "connected (S - T)"
coboundedI2: fixes b :: "'a::semilattice_inf"   and c :: "'a::semilattice_inf"   and a :: "'a::semilattice_inf" assumes "(b::'a::semilattice_inf) \<le> (c::'a::semilattice_inf)" shows "(a::'a::semilattice_inf) \<sqinter> (b::'a::semilattice_inf) \<le> (c::'a::semilattice_inf)"
that: fixes p :: "(real \<times> real set) set" assumes "p tagged_division_of {a..b}"   and "g fine p" shows "thesis"
mod_mult_self4: 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 modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and b :: "'d"   and c :: "'d"   and a :: "'d" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus (times b c) a) b = modulo a b"
acute_col_perp__out: 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 A' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Acute Bet Cong A B C"   and "Tarski_neutral_dimensionless.Col Bet B C A'"   and "Tarski_neutral_dimensionless.Perp Bet Cong B C A A'" shows "Tarski_neutral_dimensionless.Out Bet B A' C"
cosh_gt_1: fixes x :: "real" assumes "0 < x" shows "1 < cosh x"
vrestr_intersec_in_space: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "A \<in> sets M"   and "B \<in> sets M" shows "A \<inter> f --` B = A \<inter> f -` B"
list_3_cases: fixes xs :: "'a list" obtains "xs = []"   | x :: "'a" where "xs = [x__]"   | x :: "'a" and y :: "'a" and ys :: "'a list" where "xs = x # y # ys"
2: shows "- (- - - - x \<squnion> - (- - x \<squnion> - z)) = - - - - - x \<sqinter> - - - z"
simps_2: fixes p :: "linear_poly"   and a :: "'a" shows "poly (GEQ_ns p a) = p"
iso_Ri: fixes x :: "'c"   and y :: "'b" shows "mono (Ri x y)"
servK_authentic: fixes authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "msg"   and evs :: "event list" assumes "Crypt authK \<lbrace>Key servK, Agent B, Ts\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "authK \<notin> range shrK"   and "evs \<in> kerbV" shows "\<exists>A ST. Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Ts\<rbrace>, ST\<rbrace> \<in> set evs"
main_G: shows "AE x in M. g x * q (real_cond_exp M F X x) + G x * (X x - real_cond_exp M F X x) \<le> g x * q (X x)"
assms_3: fixes t :: "'a \<Rightarrow>\<^sub>0 nat"   and p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and c :: "'b" assumes "t \<in> .[X]"   and "p \<in> P[X]"   and "c \<noteq> (0::'b)"   and "t \<notin> keys p"   and "P p" shows "P (monomial c t + p)"
inst_ok_nil: 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" shows "inst_ok \<Theta> []"
col_in_angle: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p" assumes "A \<noteq> B"   and "C \<noteq> B"   and "P \<noteq> B"   and "B Out A P \<or> B Out C P" shows "P InAngle A B C"
Labels_def: shows "Labels = fst \<circ> snd \<circ> snd \<circ> Rep_seqauto"
no_failure: fixes xs :: "'a list"   and x :: "'a"   and state :: "'b" assumes "op_history (xs @ [x])"   and "apply_operations xs initial_state = Some state" shows "\<langle>x\<rangle> state \<noteq> None"
simps_11: fixes c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "Language.normalize (Catch c\<^sub>1 c\<^sub>2) = Catch (Language.normalize c\<^sub>1) (Language.normalize c\<^sub>2)"
simps_2: fixes signals :: "signal \<Rightarrow> nat"   and more :: "'a" shows "a0n_state.more \<lparr>signals = signals, \<dots> = more\<rparr> = more"
SNF_PAQ: shows "Smith_normal_form_mat (P * A * Q)"
lin: shows "OF_priority_match OF_match_fields_safe oft = OF_match_linear OF_match_fields_safe oft"
lm101: fixes x :: "'c"   and X :: "'c set"   and Y :: "'c set" assumes "x \<in> X \<union> Y" shows "X <|| Y ,, x = chi X x"
FG_el_decomp: fixes c :: "('a, 'g) aezfun" assumes "c \<in> ActingGroup.group_ring" shows "\<exists>ras. set (map snd ras) \<subseteq> G \<and> c = (\<Sum>(r, a)\<leftarrow>ras. r \<delta>\<delta> a)"
trans: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "leq1 b a"   and "leq1 c b" shows "leq1 c a"
simps_1: fixes f :: "'c \<Rightarrow> ('c, 'd) Nested_Datatype.term"   and a :: "'c" shows "subst_term f (Var a) = f a"
h: shows "h = (\<lambda>(b, i). case f b of (d, fs) \<Rightarrow> (d ^ Suc i, map (\<lambda>f. (f, i)) fs))"
elims: fixes x :: "'a list list"   and y :: "'a list list" assumes "UP.normalize x = y" obtains "x = []" and "y = []"   | v :: "'a list" and va :: "'a list list" where "x = v # va" and "y = (if lead_coeff (v # va) \<noteq> \<zero>\<^bsub>poly_ring R\<^esub> then v # va else UP.normalize (tl (v # va)))"
induct: fixes P :: "16 word list \<Rightarrow> bool"   and a0 :: "16 word list" assumes "P []"   and "\<And>x xs. \<lbrakk>x = 0 \<Longrightarrow> P (dropWhile (\<lambda>x. x = 0) xs); x \<noteq> 0 \<Longrightarrow> P xs\<rbrakk> \<Longrightarrow> P (x # xs)" shows "P a0"
abs_eq: fixes xa :: "'a \<Rightarrow>\<^sub>0 'b"   and x :: "'a \<Rightarrow>\<^sub>0 'b" shows "PP xa - PP x = PP (xa - x)"
u: shows "u \<in> {0<..<exp (- 1)}"
coclop_idem_var: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "coclop f" shows "f (f x) = f x"
umformung_OPT': fixes qs :: "'c list"   and init :: "'c list"   and Strat :: "(nat \<times> nat list) list" assumes "set qs \<subseteq> set init"   and "distinct init"   and "length qs = length Strat"   and "\<And>x l. \<lbrakk>x < length Strat; l < length (snd (Strat ! x))\<rbrakk> \<Longrightarrow> Suc (snd (Strat ! x) ! l) < length init" shows "T\<^sub>p init qs Strat = (\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. ALGxy_det Strat qs init x y + ALG_Pxy Strat qs init x y)"
top_edge_is_compos_of_scal_and_g1: shows "(\<lambda>x. twoC (x, 1)) = (\<lambda>x. (x, g1 x)) \<circ> (\<lambda>x. a + (b - a) * x)"
Stuck: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com" shows "\<Gamma>\<turnstile>c \<down> Stuck"
reduced_GB_is_monomial_set_dgrad_p_set: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m"   and "is_monomial_set F" shows "is_monomial_set (reduced_GB F)"
that_2: shows "Rel (crel_vs R) (dp param) s"
enc_def: fixes \<eta> :: "nat"   and k :: "bool list"   and m :: "bool list" shows "enc \<eta> k m \<equiv> return_spmf (k [\<oplus>] m)"
polynomial_of_mset_def: fixes p :: "(char list multiset \<times> int) multiset" shows "polynomial_of_mset p = \<Sum>\<^sub># (mononom_of_vars `# p) 0"
Rep_Pg_inject: fixes x :: "Pg"   and y :: "Pg" shows "(Rep_Pg x = Rep_Pg y) = (x = y)"
Sigma_mset_Int_distrib1: fixes I :: "'b multiset"   and J :: "'b multiset"   and C :: "'b \<Rightarrow> 'c multiset" shows "Sigma_mset (I \<inter># J) C = Sigma_mset I C \<inter># Sigma_mset J C"
infimum_def: fixes A :: "'a set" shows "infimum A = {a \<in> lower_bound A. \<forall>x\<in>lower_bound A. x \<le> a}"
mat_times_vec_eqI: fixes A :: "'a mat"   and x :: "'a vec"   and c :: "'a vec" assumes "dim_col A\<^sup>T = dim_vec x"   and "dim_row A\<^sup>T = dim_vec c"   and "\<And>i. i < dim_vec c \<Longrightarrow> (A\<^sup>T *\<^sub>v x) $ i = c $ i" shows "[x\<^sub>v*A]=c"
trans_source_target_relation_impl_full_abstraction: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set" assumes "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel \<and> (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel"   and "trans Rel" shows "\<forall>S1 S2. ((S1, S2) \<in> _Collect (S1, S2) ((SourceTerm S1, SourceTerm S2) \<in> Rel)) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> _Collect (T1, T2) ((TargetTerm T1, TargetTerm T2) \<in> Rel))"
case_eq_if: fixes f :: "'b \<Rightarrow> 'b stream \<Rightarrow> 'a"   and stream :: "'b stream" shows "(case stream of x ## xa \<Rightarrow> f x xa) = f (shd stream) (stl stream)"
g_orbitalD: fixes s' :: "'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> bool"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and s :: "'b" assumes "s' \<in> x\<acute>=f & G on U S @ t\<^sub>0 s" obtains X :: "real \<Rightarrow> 'b" and t :: "real" where "X \<in> Sols f U S t\<^sub>0 s" and "X t = s'" and "t \<in> U s" and "\<P> X (down (U s) t) \<subseteq> {s. G s}"
abupd_def2: fixes f :: "abopt \<Rightarrow> abopt"   and x :: "abopt"   and s :: "st" shows "abupd f (x, s) = (f x, s)"
hom_zero: shows "to_fract (0::'b) = 0"
univ_second_countable: obtains \<B> :: "'c set set" where "countable \<B>" and "\<And>C. C \<in> \<B> \<Longrightarrow> open C" and "\<And>S. open S \<Longrightarrow> \<exists>U\<subseteq>\<B>. S = \<Union> U"
rtrancl_last_visit: fixes q :: "'a"   and q' :: "'a"   and R :: "('a \<times> 'a) set"   and S :: "'a set" assumes "(q, q') \<in> R\<^sup>*" obtains "(q, q') \<in> (R - UNIV \<times> S)\<^sup>*"   | qt :: "'a" where "qt \<in> S" and "(q, qt) \<in> R\<^sup>+" and "(qt, q') \<in> (R - UNIV \<times> S)\<^sup>*"
a1: shows "s' = user_reg_mod d w r s \<and> get_S (cpu_reg_val PSR s) = 0"
cos_periodic_pi2: fixes x :: "real" shows "cos (pi + x) = - cos x"
power_even_eq: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (2 * n) = power.power one times (power.power one times a n) 2"
compl_sup_top: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<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 x :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "sup (uminus x) x = top"
not_one_less_zero: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less one zero"
f_f1: shows "f = (\<lambda>u. f1 u u)"
eSum_singleTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> ant"   and j :: "nat" shows "(\<forall>j\<le>n. f j \<in> Z\<^sub>\<infinity>) \<and> j \<le> n \<and> (\<forall>l\<in>{h. h \<le> n} - {j}. f l = 0) \<longrightarrow> ASum f n = f j"
replacement: fixes A :: "'c set"   and B :: "'c set" assumes "finite A"   and "finite B"   and "B \<subseteq> carrier V"   and "lin_indpt A"   and "A \<subseteq> span B" shows "\<exists>C. finite C \<and> C \<subseteq> carrier V \<and> C \<subseteq> span B \<and> C \<inter> A = {} \<and> int (card C) \<le> int (card B) - int (card A) \<and> span (A \<union> C) = span B"
nonzero_mod_p: fixes p :: "nat"   and a :: "int"   and x :: "int" assumes "GAUSS p a"   and "0 < x"   and "x < int p" shows "[x \<noteq> 0] (mod int p)"
every_eqclass_has_transition: fixes s :: "'b list"   and c :: "'b"   and X :: "'b list set"   and A :: "'b list set" assumes "s @ [c] \<in> X"   and "X \<in> UNIV // \<approx>A" obtains Y :: "'b list set" where "Y \<in> UNIV // \<approx>A" and "Y \<cdot> {[c]} \<subseteq> X" and "s \<in> Y"
span_mono: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> S"   and "B \<subseteq> S"   and "A \<subseteq> B" shows "??.Real_Vector_Spaces.span A \<subseteq> ??.Real_Vector_Spaces.span B"
rquot_prop: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "D x z \<and> y = x \<oplus> z" shows "z = rquot y x"
set_fmap_minus_iff: fixes F :: "'g \<Rightarrow> 'h option"   and x :: "'g \<times> 'h" shows "set_fmap (F -- x) = set_fmap F - {x}"
semiring_normalization_rules_34: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * (y + z) = x * y + x * z"
transitions_consD: fixes root :: "(att \<times> char list, att, nat) env"   and x :: "operation"   and xs :: "operation list"   and root'' :: "(att \<times> char list, att, nat) env"   and root' :: "(att \<times> char list, att, nat) env" assumes "root \<Midarrow>(x # xs)\<Rightarrow> root''"   and "root \<midarrow>x\<rightarrow> root'" shows "root' \<Midarrow>xs\<Rightarrow> root''"
iso_is_section: fixes f :: "'a arrow_of_spans_data" assumes "V.iso f" shows "V.section f"
seq_F'F: shows "A_B.seq (fst Fg') (fst Fg)"
pow_carrier_mod: fixes g :: "'a"   and k :: "nat" assumes "g \<in> carrier G" shows "g [^] (k mod order G) = g [^] k"
M: fixes m :: "'a set set" assumes "m \<in> M" shows "m \<subseteq> Pow \<Omega>"
rep_eq: fixes x :: "'a regular"   and xa :: "'a regular" shows "Rep_regular (x - xa) = Rep_regular x \<sqinter> - Rep_regular xa"
ht_at_phl_export1: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "\<lparr>\<tt>\<tt> x \<cdot> \<tt>\<tt> y\<rparr>z\<lparr>z \<cdot> w\<rparr>" shows "\<lparr>\<tt>\<tt> x\<rparr>\<tt>\<tt> y \<cdot> z\<lparr>\<tt>\<tt> y \<cdot> z \<cdot> w\<rparr>"
STCalWB_hasBarbST: fixes P :: "('procS, 'procT) Proc"   and a :: "'barbs" shows "P\<down><STCalWB SWB TWB>a = ((\<exists>S. S \<in>S P \<and> S\<down><SWB>a) \<or> (\<exists>T. T \<in>T P \<and> T\<down><TWB>a))"
X_Y_closed: fixes x :: "('s \<times> ('c \<Rightarrow> real) set) stream"   and y :: "('s \<times> ('c \<Rightarrow> real)) stream" assumes "P x"   and "smap abss y = x"   and "x \<in> X" shows "y \<in> Y \<and> Q y"
fineD: fixes d :: "'a \<Rightarrow> 'b set"   and s :: "('a \<times> 'b set) set"   and x :: "'a"   and k :: "'b set" assumes "d fine s"   and "(x, k) \<in> s" shows "k \<subseteq> d x"
left_imp_right: assumes "pvalid p x" shows "pvalid_left p x"
simps: fixes h :: "'d" shows "the_state (Some h) = h"
deadends_def: fixes p :: "Player" shows "deadends p \<equiv> {v \<in> VV p. deadend v}"
fps_lr_inverse_one_one_2: shows "fps_right_inverse 1 (1::'a) = 1"
chains_def: fixes A :: "'b set set" shows "chains A = {C. C \<subseteq> A \<and> chain\<^sub>\<subseteq> C}"
mod_by_0: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "modulo a zero = a"