hcju/isabelleps huggingface dataset:
assms_2: shows "(x, y) = split_n_m_x d m X"
pnet_init_in_net_tree_ips_in_net_ips: fixes s :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and p :: "net_tree"   and i :: "nat" assumes "s \<in> init (pnet np p)"   and "i \<in> net_tree_ips p" shows "i \<in> net_ips s"
af_subformulas\<^sub>\<nu>: fixes \<phi> :: "'b ltln"   and w :: "'b set list" shows "subformulas\<^sub>\<nu> (af \<phi> w) = subformulas\<^sub>\<nu> \<phi>"
exhaust: fixes y :: "('h, 'i, 'j) generat" obtains x1 :: "'h" where "y = Pure x1"   | x21 :: "'i" and x22 :: "'j" where "y = IO x21 x22"
simps_1: fixes a :: "'a" shows "local.plus_oracle_stop (Inl a) = map_spmf (apfst (map_option Inl)) (left s a)"
OO_Grp_alt: fixes A :: "'p set"   and f :: "'p \<Rightarrow> 'n"   and g :: "'p \<Rightarrow> 'o" shows "(Grp A f)\<inverse>\<inverse> OO Grp A g = (\<lambda>x y. \<exists>z. z \<in> A \<and> f z = x \<and> g z = y)"
3: shows "x\<^sup>T \<le> c * - c\<^sup>T"
HFun_Sigma_Eats: fixes H :: "fm set"   and r :: "tm"   and d :: "tm"   and d' :: "tm" assumes "H \<turnstile> HFun_Sigma r"   and "H \<turnstile> NotInDom d r"   and "H \<turnstile> OrdP d" shows "H \<turnstile> HFun_Sigma (Eats r (HPair d d'))"
splus_pprod_def: shows "splus_pprod \<equiv> pprod.splus"
exhaust: fixes y :: "pcount" obtains "y = a"   | "y = b"   | "y = g"
partition_filter1: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "fst (partition P xs) = filter P xs"
t2: shows "t2 \<in> existence_ivl t0 x0"
singular_chain_boundary_alt: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_chain (Suc p) X c" shows "singular_chain p X (chain_boundary (Suc p) c)"
cring_deg_mult: fixes q :: "nat \<Rightarrow> 'a"   and p :: "nat \<Rightarrow> 'a" assumes "q \<in> carrier P"   and "p \<in> carrier P"   and "q (deg R q) \<otimes> p (deg R p) \<noteq> \<zero>" shows "deg R (q \<otimes>\<^bsub>P\<^esub> p) = deg R p + deg R q"
r_both_4: fixes i :: "nat"   and f :: "recf"   and j :: "nat"   and g :: "recf"   and t :: "nat"   and x :: "nat"   and v :: "nat" assumes "\<And>x. eval r_phi [i, x] = eval f [x]"   and "\<And>x. eval r_phi [j, x] = eval g [x]"   and "eval r_result1 [t, i, x] \<down>= 0"   and "eval r_result1 [t, j, x] \<down>= Suc v" shows "eval r_both [t, i, j, x] \<down>= prod_encode (1, the (eval g [x]))"
sum_of_unit_vec_to_col: fixes A :: "complex mat"   and n :: "nat"   and i :: "nat"   and j :: "nat"   and c :: "complex" assumes "dim_col A = n"   and "i < n"   and "j < n" shows "col A i + c \<cdot>\<^sub>v col A j = col_fst (A * |unit_vec n i + c \<cdot>\<^sub>v unit_vec n j\<rangle>)"
real_alg_of_real_code: fixes x :: "real_alg" shows "real_alg_of_real (real_of x) = x"
paval'_paval: fixes a :: "aexp"   and ps :: "char list \<Rightarrow> int option" assumes "vars a \<subseteq> dom ps" shows "paval' a ps = Some (paval a ps)"
full_abstraction_and_trans_relation_contains_SRel_impl_TRel: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set"   and Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"   and "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel"   and "SRel = _Collect (S1, S2) ((SourceTerm S1, SourceTerm S2) \<in> Rel)"   and "trans (Rel \<union> _Collect (P, Q) (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T P \<and> S \<in>S Q))" shows "\<forall>S1 S2. ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel) = ((TargetTerm (\<lbrakk>S1\<rbrakk>), TargetTerm (\<lbrakk>S2\<rbrakk>)) \<in> Rel)"
simps_17: fixes x :: "nat"   and a :: "floatarith"   and n :: "nat"   and vs :: "real list" shows "isDERIV x (Power a (Suc n)) vs = isDERIV x a vs"
IH: shows "vars\<^sub>l\<^sub>s\<^sub>t A = (\<Union>l. vars\<^sub>l\<^sub>s\<^sub>t (proj l A))"
merge_token_succeeds: fixes x :: "nat"   and y :: "nat"   and i :: "nat" assumes "(x, y) \<in> local.merge i" shows "token_succeeds x = token_succeeds y"
natural_transformation_axioms: shows "natural_transformation A B F G \<tau>''.map"
assms_2: shows "pairwise (\<lambda>x y. disjnt (B x) (B y)) A"
equal_bit1_def: fixes x :: "'a bit1"   and y :: "'a bit1" shows "equal_class.equal x y = (Rep_bit1 x = Rep_bit1 y)"
poincare_between_0uv: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<noteq> 0\<^sub>h"   and "v \<noteq> 0\<^sub>h" shows "poincare_between 0\<^sub>h u v = (let u' = to_complex u; v' = to_complex v in arg u' = arg v' \<and> cmod u' \<le> cmod v')"
coboundedI2: fixes c :: "'b multiset"   and b :: "'b multiset"   and a :: "'b multiset" assumes "c \<subseteq># b" shows "c \<subseteq># a \<union># b"
is_natural_2: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b"   and f :: "'a" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "B (\<tau> (partial_magma.cod A f)) (F f) = \<tau> f"
rsp: shows "rel_fun (eq_onp (\<lambda>r. fst r \<le> snd r)) (=) snd snd"
simps_4: fixes t :: "'c local.term"   and u :: "'c local.term" shows "\<lbrace>t \<^bold>\<cdot> u\<rbrace> = \<lbrace>t\<rbrace> \<cdot> \<lbrace>u\<rbrace>"
Collect_bex_eq: fixes A :: "'b set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
tfr: shows "tfr\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>s\<^sub>t S)"
con_comp_futures: fixes P :: "'f process"   and Q :: "'g process"   and p :: "'f \<Rightarrow> 'e"   and q :: "'g \<Rightarrow> 'e"   and xs :: "'e list" shows "futures (P \<parallel> Q <p, q>) xs = {(ys, Y). (xs @ ys, Y) \<in> con_comp_failures P Q p q}"
hyps_2: shows "Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t ik\<^sub>e\<^sub>s\<^sub>t A \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t A"
amalgamation_V01_R1: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> V\<^sub>0\<^sub>1" shows "amalgamation (the (f 0)) (the (f 1)) \<in> \<R>"
comp_inv_arr: fixes f :: "'a"   and g :: "'a" assumes "C.inverse_arrows f g" shows "g \<cdot> f = C.dom f"
\<pi>0: shows "\<pi>' 0 = \<pi> l"
is_qfN_power: fixes n :: "nat"   and A :: "int"   and N :: "int" assumes "0 < n"   and "is_qfN A N" shows "is_qfN (A ^ n) N"
T1: fixes u :: "'a"   and \<omega> :: "'a" assumes "ide u"   and "\<guillemotleft>\<omega> : local.dom \<omega> \<Rightarrow> (k\<^sup>* \<star> h) \<star> u\<guillemotright>" shows "\<exists>w \<theta> \<nu>. ide w \<and> \<guillemotleft>\<theta> : p\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright> \<and> \<guillemotleft>\<nu> : local.dom \<omega> \<Rightarrow> p\<^sub>1 \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> (((k\<^sup>* \<star> h) \<star> \<theta>) \<cdot> \<a>[k\<^sup>* \<star> h, p\<^sub>0, w] \<cdot> (tab \<star> w)) \<cdot> \<nu> = \<omega>"
Atomic_Form_is_wf_dbfm: fixes x :: "hf"   and e :: "(name, hf) finfun" assumes "Atomic x" shows "\<exists>A. wf_dbfm A \<and> x = \<lbrakk>quot_dbfm A\<rbrakk>e"
assms_2: shows "(\<lambda>x. t x (g x)) \<in> \<Theta>[F](h)"
hX: fixes t :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real" assumes "healthy t" shows "healthy (wp (body ;; Embed t \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))"
DERIV_local_const: fixes f :: "real \<Rightarrow> real"   and l :: "real"   and x :: "real"   and d :: "real" assumes "(f has_real_derivative l) (at x)"   and "0 < d"   and "\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f x = f y" shows "l = 0"
comp_eq_id_dest: fixes a :: "'d \<Rightarrow> 'c"   and b :: "'b \<Rightarrow> 'd"   and c :: "'b \<Rightarrow> 'c"   and v :: "'b" assumes "a \<circ> b = id \<circ> c" shows "a (b v) = c v"
monom_add: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and n :: "nat" assumes "UP_ring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "monom (UP R) (a \<oplus>\<^bsub>R\<^esub> b) n = monom (UP R) a n \<oplus>\<^bsub>UP R\<^esub> monom (UP R) b n"
rel_conversep: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "stream_all2 R\<inverse>\<inverse> = (stream_all2 R)\<inverse>\<inverse>"
l_to_tree_order_axioms: shows "l_to_tree_order Shadow_DOM.to_tree_order"
row_add_iterate_A: shows "fst (row_add_iterate_PA (P, A) n i j) $ a $ b = P $ a $ b"
divideR_right: fixes r :: "real"   and y :: "'a"   and x :: "'a" assumes "r \<noteq> 0" shows "(y = x /\<^sub>R r) = (r *\<^sub>R y = x)"
one_step: fixes a :: "nat \<Rightarrow> nat"   and j :: "nat"   and p' :: "nat" assumes "a \<in> s"   and "j < n"   and "\<And>a'. \<lbrakk>a' \<in> s; a' \<noteq> a\<rbrakk> \<Longrightarrow> a' j = p'" shows "a j \<noteq> p'"
dis_introL: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" assumes "\<phi> \<pi>l" shows "dis \<phi> \<psi> \<pi>l"
ptensor_mat_dim_col: fixes m1 :: "'a mat"   and m2 :: "'a mat" shows "dim_col (st3.ptensor_mat m1 m2) = st3.d0"
N_uminus: fixes n :: "int" shows "N (- n) = N n"
a_closed: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "x \<oplus>\<^bsub>P\<^esub> y \<in> carrier P"
subst_fv_unfold_ground_img: fixes s :: "'c \<Rightarrow> ('d, 'c) Term.term"   and t :: "('d, 'c) Term.term" assumes "range_vars s = {}" shows "fv (t \<cdot> s) = fv t - subst_domain s"
check_def: fixes a :: "nat list"   and b :: "nat list" shows "check a b = filter (\<lambda>(x, y). a \<bullet> x = b \<bullet> y)"
conv_radius_leI': fixes f :: "nat \<Rightarrow> 'b"   and z :: "'b" assumes "\<not> summable (\<lambda>n. f n * z ^ n)" shows "conv_radius f \<le> ereal (norm z)"
Red1Try: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and v :: "'addr val"   and C :: "String.literal"   and V :: "nat"   and e2 :: "'addr expr1"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>try Val v catch(C V) e2,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Val v,s\<rangle>"
comp_sub: fixes i :: "nat" shows "component_of_term ` Keys (f i) \<subseteq> K"
fimage_fimage: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and A :: "'f fset" shows "f |`| g |`| A = (\<lambda>x. f (g x)) |`| A"
assms_1: shows "p \<circ>\<^sub>p q = 0"
scan_last: fixes a :: "'d"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list" shows "last (a # scan f xs a) = fold f xs a"
verts_Gu: fixes G :: "nat set \<times> nat set set"   and u :: "nat" shows "uverts (G -- u) = uverts G - {u}"
opt_bst_simps_1: fixes j :: "int"   and i :: "int"   and w :: "int \<Rightarrow> int \<Rightarrow> nat" assumes "j < i" shows "Optimal_BST.opt_bst w i j = \<langle>\<rangle>"
convergent_norm: fixes f :: "nat \<Rightarrow> 'b" assumes "convergent f" shows "convergent (\<lambda>n. norm (f n))"
wide_subdg_obj_eq: shows "\<AA>\<lparr>Obj\<rparr> = \<BB>\<lparr>Obj\<rparr>"
low_eq_def: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b" assumes "sifum_security_init dma \<C>_vars eval" shows "sifum_security_init.low_eq dma mem\<^sub>1 mem\<^sub>2 \<equiv> \<forall>x. dma mem\<^sub>1 x = Low \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"
no_active_nodes: shows "\<forall>v\<in>V - {s, t}. sum f (incoming v) \<le> sum f (outgoing v)"
iMOD_Suc_pred_insert_conv: fixes r :: "nat"   and m :: "nat" shows "insert r [ r + m, mod m ] = [ r, mod m ]"
enumdom_empty_domain: fixes s :: "nat \<Rightarrow> nat option"   and i :: "nat"   and y :: "nat" assumes "total_cons s"   and "\<And>x. \<phi> i x \<up>" shows "eval r_enumdom [i, y] \<up>"
plus_backward_simulation_def: fixes step1 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and step2 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and match :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "plus_backward_simulation step1 step2 match \<equiv> \<forall>s1 s2 s2'. match s1 s2 \<longrightarrow> step2 s2 s2' \<longrightarrow> (\<exists>s1'. step1\<^sup>+\<^sup>+ s1 s1' \<and> match s1' s2')"
lookup_uminus: fixes f :: "'b \<Rightarrow>\<^sub>0 'a"   and k :: "'b" shows "lookup (- f) k = - lookup f k"
assms_2: fixes e' :: "'a \<times> 'a"   and es :: "('a \<times> 'a) list" assumes "e' \<noteq> e"   and "e' \<noteq> (snd e, fst e)"   and "P e es" shows "P e (e' # es)"
proper_uprod_simps: fixes x :: "'b"   and y :: "'b" shows "proper_uprod (Upair x y) = (x \<noteq> y)"
assms_4: shows "\<forall>\<^sub>F x in F. g x \<le> f x"
prefix_Par_Int3: fixes a :: "'a"   and C :: "'a set"   and b :: "'a"   and P :: "'a process"   and Q :: "'a process" assumes "a \<in> C"   and "b \<notin> C" shows "(a \<rightarrow> P \<lbrakk>C\<rbrakk> b \<rightarrow> Q) = (b \<rightarrow> (a \<rightarrow> P \<lbrakk>C\<rbrakk> Q))"
in_Above_under: fixes a :: "'a"   and r :: "('a \<times> 'a) set" assumes "a \<in> Field r" shows "a \<in> Above r (under r a)"
\<F>\<G>_suffix: fixes \<phi> :: "'b ltln"   and i :: "nat"   and w :: "nat \<Rightarrow> 'b set" shows "\<F>\<G> \<phi> (suffix i w) = \<F>\<G> \<phi> w"
divide_geq1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "rat"   and a :: "'a" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat"   and "less_eq b (scaleRat c a)"   and "c < 0" shows "less_eq a (scaleRat (1 / c) b)"
open_UNIV: fixes "open" :: "'a set \<Rightarrow> bool" assumes "class.topological_space open" shows "open UNIV"
a: shows "(x * e1 + (q - s1)) mod q = (x * y + x * q - x * b + (q - (x * y + x * (q - b) + (q - r)) mod q)) mod q"
right_ideal_explicit: fixes S :: "'a set" shows "right_ideal_generated S = {y. \<exists>f U. finite U \<and> U \<subseteq> S \<and> (\<Sum>i\<in>U. i * f i) = y}"
zero_less_abs_iff: fixes abs :: "'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less zero (abs a) = (a \<noteq> zero)"
iT_Plus_neg_not_empty_iff: fixes I :: "nat set"   and k :: "nat" shows "(I \<oplus>- k \<noteq> {}) = (\<exists>x\<in>I. k \<le> x)"
intros_9: fixes X :: "msg"   and H :: "msg set"   and Y :: "msg" assumes "X \<in> synth H"   and "Y \<in> synth H" shows "Enc X Y \<in> synth H"
Abs_seqauto_cases: fixes x :: "('d, 'e, 'f) seqauto" obtains y :: "'d set \<times> 'd \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) set \<times> ('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set" where "x = Abs_seqauto y" and "y \<in> seqauto"
bot_unique: fixes a :: "'a" shows "(a \<le> (0::'a)) = (a = (0::'a))"
Dag_root_not_in_subdag_l: fixes l :: "ref \<Rightarrow> ref"   and p :: "ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag" assumes "Dag (l p) l r t" shows "p \<notin> set_of t"
quorum_of_def: fixes slices :: "'a \<Rightarrow> 'a set set"   and W :: "'a set"   and p :: "'a"   and Q :: "'a set" assumes "stellar slices W" shows "stellar.quorum_of slices W p Q \<equiv> stellar.quorum slices W Q \<and> (p \<notin> W \<or> (\<exists>Sl\<in>slices p. Sl \<subseteq> Q))"
in_Reals_iff_Re: fixes q :: "quat" shows "(q \<in> \<real>) = (quat_of_real (quat.Re q) = q)"
safe: fixes loc1 :: "'loc"   and t :: "'t"   and s :: "'sum"   and loc2 :: "'loc" assumes "0 < zcount (c_pts c0 loc1) t"   and "s \<in>\<^sub>A path_summary loc1 loc2" shows "\<exists>t'\<le>results_in t s. t' \<in>\<^sub>A frontier (c_imp c0 loc2)"
in_production_imp_produces: fixes A :: "'a"   and C :: "'a literal multiset" assumes "A \<in> production C" shows "production C = {A}"
mat_ge_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b mat"   and B :: "'b mat" shows "ord.mat_ge less_eq A B = (\<forall>i<dim_row A. \<forall>j<dim_col A. less_eq (B $$ (i, j)) (A $$ (i, j)))"
k: shows "\<gamma> \<in> elts (\<omega> \<up> ord_of_nat n * ord_of_nat k)"
Im_eq_0: fixes z :: "complex" assumes "\<bar>Re z\<bar> = cmod z" shows "is_real z"
Zero_SubsetE: fixes H :: "fm set"   and A :: "fm"   and X :: "tm" assumes "H \<turnstile> A" shows "insert (Zero SUBS X) H \<turnstile> A"
PFMT_2: shows "ch = {#[entry fg q]#} + che"
simps_1: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list" shows "noDenyAll (x # xs) = (\<not> member DenyAll x \<and> noDenyAll xs)"
e: shows "exi x \<phi> = exi y (subst \<phi> (Var y) x)"
complete_def: fixes CC :: "'b set set"   and A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "CC-complete A r \<equiv> \<forall>X\<subseteq>A. X \<in> CC \<longrightarrow> Ex (extreme_bound A r X)"
is_strongly_dom_component_safe_impl: shows "is_strongly_dom_component_safe = local.a_is_strongly_dom_component_safe"
simps_2: fixes v1 :: "'c \<times> nat"   and e :: "('c \<times> nat) edge"   and es :: "('c \<times> nat) edge list"   and v2 :: "'c \<times> nat"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" shows "SubRel.ces v1 (e # es) v2 subs = ((v1 = src e \<or> (v1, src e) \<in> subs\<^sup>+) \<and> SubRel.ces (tgt e) es v2 subs)"
division_contains: fixes s :: "'a set set"   and i :: "'a set" assumes "s division_of i" shows "\<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
TUA: shows "TUadmit \<sigma> \<theta> UNIV"
zdvd1_eq: fixes x :: "int" shows "is_unit x = (\<bar>x\<bar> = 1)"
has_integral_diff: fixes f :: "'c \<Rightarrow> 'd"   and k :: "'d"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and l :: "'d" assumes "(f has_integral k) S"   and "(g has_integral l) S" shows "((\<lambda>x. f x - g x) has_integral k - l) S"
MSB_def: fixes Is :: "nat fset list" shows "MSB Is \<equiv> if \<forall>P\<in>set Is. P = {||} then 0 else Suc (Max (\<Union> (fset ` set Is)))"
le_oInv: fixes F :: "ordinal \<Rightarrow> ordinal"   and z :: "ordinal"   and a :: "ordinal" assumes "normal F"   and "F z \<le> a" shows "z \<le> oInv F a"
B: shows "ipurge_tr_rev_aux I D (sources_aux I D U [z]) (xs @ ys @ zs) = ipurge_tr_rev_aux I D (sources_aux I D U [z]) (xs @ ipurge_tr_aux I D (D ` set ys) zs)"
plan_action_enabled_def: fixes P :: "ast_problem"   and \<pi> :: "plan_action"   and M :: "object atom formula set" shows "ast_problem.plan_action_enabled P \<pi> M = (ast_problem.wf_plan_action P \<pi> \<and> M \<^sup>c\<TTurnstile>\<^sub>= ground_action.precondition (ast_problem.resolve_instantiate P \<pi>))"
prems_3: fixes k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "(k, \<gamma>) \<in> set (a # l)" shows "valid_path \<gamma>"
sum_list_scf_list: fixes ts :: "('f, 'a) Term.term list"   and f :: "nat \<Rightarrow> nat" assumes "\<And>i. i < length ts \<Longrightarrow> 0 < f i" shows "sum_list (map weight ts) \<le> sum_list (map weight (scf_list f ts))"
strong_induct: fixes x :: "fm"   and P :: "'a \<Rightarrow> fm \<Rightarrow> bool"   and c :: "'a" assumes "se_fm x"   and "\<And>t u c. P c (t IN u)"   and "\<And>A B c. \<lbrakk>se_fm A; \<And>c. P c A; se_fm B; \<And>c. P c B\<rbrakk> \<Longrightarrow> P c (A OR B)"   and "\<And>A B c. \<lbrakk>se_fm A; \<And>c. P c A; se_fm B; \<And>c. P c B\<rbrakk> \<Longrightarrow> P c (A AND B)"   and "\<And>A i c. \<lbrakk>{atom i} \<sharp>* c; se_fm A; \<And>c. P c A\<rbrakk> \<Longrightarrow> P c (SyntaxN.Ex i A)"   and "\<And>A i t c. \<lbrakk>{atom i} \<sharp>* c; se_fm A; \<And>c. P c A; atom i \<sharp> t\<rbrakk> \<Longrightarrow> P c (All2 i t A)" shows "P c x"
less_eq_multiset_def: fixes M' :: "'a multiset"   and M :: "'a multiset" shows "(M' \<le> M) = (M' < M \<or> M' = M)"
update_convs_1: fixes output_iface' :: "char list \<Rightarrow> char list"   and output_iface :: "char list"   and next_hop :: "'b word option"   and more :: "'c" shows "output_iface_update output_iface' \<lparr>output_iface = output_iface, next_hop = next_hop, \<dots> = more\<rparr> = \<lparr>output_iface = output_iface' output_iface, next_hop = next_hop, \<dots> = more\<rparr>"
emeasure_Un_Int: fixes A :: "'b set"   and M :: "'b measure"   and B :: "'b set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "emeasure M A + emeasure M B = emeasure M (A \<union> B) + emeasure M (A \<inter> B)"
simps_1: fixes R :: "('a \<times> 'a) set" shows "R ^^ 0 = Id"
ordinal_utility_axioms: shows "ordinal_utility (\<P> outcomes) relation (\<lambda>p. measure_pmf.expectation p u)"
iteratei_correct: fixes S :: "'m2" assumes "m2.invar S" shows "map_iterator (m2.iteratei S) (m2.\<alpha> S)"
less_add2: fixes j :: "nat"   and i :: "nat" assumes "0 < j" shows "i < j + i"
flow_vderiv_on_left: fixes x0 :: "'a" shows "(flow t0 x0 has_vderiv_on (\<lambda>x. f x (flow t0 x0 x))) (existence_ivl t0 x0 \<inter> {..t0})"
s\<^sub>1_r: shows "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Rjoin (VE (Rid r''))])"
set_sort: fixes f :: "'d \<Rightarrow> 'e Word.word"   and xs :: "'d list" shows "set (signed.sort_key f xs) = set xs"
sdrop_sappend_same: fixes xs :: "[:'r:]"   and ys :: "[:'r:]" assumes "xs \<noteq> \<bottom>" shows "sdrop\<cdot>(slength\<cdot>xs)\<cdot>(xs :@ ys) = ys"
one_integer_code: shows "1 = Code_Numeral.Pos num.One"
strategy_attracts_irrelevant_override: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and W :: "'a set"   and \<sigma>' :: "'a \<Rightarrow> 'a" assumes "strategy_attracts p \<sigma> A W"   and "strategy p \<sigma>"   and "strategy p \<sigma>'" shows "strategy_attracts p (override_on \<sigma>' \<sigma> (A - W)) A W"
map_id: fixes t :: "'d pmf" shows "map_pmf id t = t"
assms_2: shows "d_IN f x \<noteq> \<top>"
that: fixes A :: "'a set"   and B :: "'b set" assumes "open A"   and "open B"   and "x \<in> A \<times> B"   and "A \<times> B \<subseteq> S" shows "thesis"
respection_of_binary_predicates_and_closures_4: fixes Rel :: "('a \<times> 'a) set"   and Pred :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "rel_respects_binary_pred Rel Pred" shows "rel_respects_binary_pred (symcl (Rel\<^sup>=)) Pred"
cong: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c" shows "comm_monoid_mult.prod times one \<equiv> comm_monoid_mult.prod times one"
subalgebra_def: fixes M :: "'b measure"   and F :: "'b measure" shows "subalgebra M F = (space F = space M \<and> sets F \<subseteq> sets M)"
simplex_likeD_iso: fixes P :: "'a set" assumes "simplex_like P" shows "\<exists>f A. OrderingSetIso (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater P f \<and> f ` P = Pow A"
elims: fixes x :: "nat"   and y :: "complex mat" assumes "exexH_k x = y" obtains k :: "nat" where "x = k" and "y = tensor_P (exH_k k) (1\<^sub>m K)"
usubstappp_det: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<alpha> :: "game"   and V :: "variable set" assumes "snd (usubstappp \<sigma> U \<alpha>) \<noteq> undefg"   and "snd (usubstappp \<sigma> V \<alpha>) \<noteq> undefg" shows "snd (usubstappp \<sigma> U \<alpha>) = snd (usubstappp \<sigma> V \<alpha>)"
col_space_eq': fixes A :: "(('a, 'c) vec, 'b) vec" shows "col_space A = range ((*v) A)"