hcju/isabelleps huggingface dataset:
simps_8: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula" shows "Formula.sat \<sigma> V v i (formula.And \<phi> \<psi>) = (Formula.sat \<sigma> V v i \<phi> \<and> Formula.sat \<sigma> V v i \<psi>)"
iso_def: fixes f :: "'a2" shows "C2.iso f \<equiv> \<exists>g. C2.inverse_arrows f g"
add_comm: fixes x :: "'a"   and y :: "'a" assumes "D x y" shows "D y x \<and> x \<oplus> y = y \<oplus> x"
assms_2: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>"
cycle_free_diag: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat"   and i :: "nat" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "linordered_ab_monoid_add.cycle_free plus zero less_eq m n"   and "i \<le> n" shows "less_eq zero (m i i)"
assms_2: fixes i :: "nat"   and a :: "'a"   and b :: "'b" assumes "i < length bs"   and "P a b" shows "Q a (f i b)"
lapp_fin_fin_lemma: fixes r :: "'b llist"   and s :: "'b llist"   and A :: "'b set" assumes "r @@ s \<in> A\<^sup>\<star>" shows "r \<in> A\<^sup>\<star>"
simps: fixes a :: "'b list" shows "list_of_array (Array a) = a"
simps_5: fixes v :: "nat"   and Q1 :: "pol"   and j :: "nat"   and Q2 :: "pol" shows "isnorm (Pinj (Suc v) (PX Q1 j Q2)) = isnorm (PX Q1 j Q2)"
of_int_abs: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "ring_1.of_int one plus zero minus \<bar>x\<bar> = abs (ring_1.of_int one plus zero minus x)"
adjugate_eq: shows "adjugate = to_vec \<circ> Square_Matrix.adjugate \<circ> from_vec"
decode_c: shows "digit_decode (nths (dims T) (- rmodes)) (nths is (- rmodes)) < prod_list (nths (dims T) (- rmodes))"
mem_simps_9: fixes A :: "'e"   and C :: "'e set set" shows "(A \<in> \<Union> C) = (\<exists>X\<in>C. A \<in> X)"
basis_exists: fixes V :: "'a set" obtains B :: "'a set" where "B \<subseteq> V" and "cindependent B" and "V \<subseteq> cspan B" and "card B = cdim V"
closed_eucl_atLeastAtMost: fixes a :: "'a"   and b :: "'a" shows "closed {a..b}"
greaterThanLessThan_borel: fixes a :: "'a"   and b :: "'a" shows "{a<..<b} \<in> sets borel"
that: assumes "v11 f"   and "n \<in>\<^sub>\<circ> \<omega>"   and "\<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n"   and "\<R>\<^sub>\<circ> f = A" shows "thesis"
rt: shows "(if y ^ p \<le> n then (y, y ^ p = n) else root_newton_int_main y' n) = (x, b)"
fg: fixes x :: "'a"   and y :: "'a" assumes "R x y" shows "ord_spmf P (f x) (g y)"
mat_comp_allE: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b mat"   and B :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and i :: "nat"   and j :: "nat" assumes "mat_comp_all r A B"   and "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "i < nr"   and "j < nc" shows "r (A $$ (i, j)) (B $$ (i, j))"
homology_exactness_reduced_1: fixes X :: "'a topology"   and S :: "'a set"   and p :: "int" assumes "topspace X \<inter> S \<noteq> {}" shows "exact_seq ([reduced_homology_group (p - 1) (subtopology X S), relative_homology_group p X S, reduced_homology_group p X], [hom_boundary p X S, hom_induced p X {} X S id])"
ipurge_tr_rev_t_form_aux: fixes X :: "('c, 'd) ipurge_rec" shows "ipurge_tr_rev_t_form (ipurge_tr_rev_t_aux X)"
rel_sel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b dual"   and b :: "'c dual" shows "rel_dual R a b = R (\<partial>\<^sup>- a) (\<partial>\<^sup>- b)"
par_nhop_quality_increases: fixes i :: "nat" shows "opaodv i \<langle>\<langle>\<^bsub>i\<^esub> qmsg \<Turnstile> (otherwith (=) {i} (orecvmsg (\<lambda>\<sigma> m. msg_fresh \<sigma> m \<and> msg_zhops m)), other quality_increases {i} \<rightarrow>) global (\<lambda>\<sigma>. \<forall>dip. let nhip = the (nhop (rt (\<sigma> i)) dip) in dip \<in> vD (rt (\<sigma> i)) \<inter> vD (rt (\<sigma> nhip)) \<and> nhip \<noteq> dip \<longrightarrow> rt (\<sigma> i) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> nhip))"
s_3: fixes x :: "'e"   and y :: "'a"   and i :: "nat" assumes "x \<in> space N"   and "y \<in> space M" shows "norm (s i (x, y)) \<le> 2 * norm (f x y)"
rec_disj_def: shows "rec_disj = Cn 2 rec_sg [Cn 2 rec_add [recf.id 2 0, recf.id 2 1]]"
connected_linear_image: fixes f :: "'b \<Rightarrow> 'c"   and s :: "'b set" assumes "linear f"   and "connected s" shows "connected (f ` s)"
member_filter: fixes x :: "'b"   and P :: "'b \<Rightarrow> bool"   and A :: "'b set" shows "(x \<in> Set.filter P A) = (x \<in> A \<and> P x)"
sum_cases: fixes x :: "'a + 'b" shows "(\<exists>y. x = Inl y) \<or> (\<exists>y. x = Inr y)"
d_const: fixes X :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'a" assumes "const X" shows "const (d\<^sub>x X)"
sum_list_length_split: fixes xs :: "real list"   and m :: "real" shows "length xs = length (list_neq xs m) + length (list_eq xs m)"
semantics_subset_vertices: fixes A :: "('d, 'e) labeled_graph"   and e :: "'d allegorical_term" assumes "graph A" shows ":A:\<lbrakk>e\<rbrakk> \<subseteq> vertices A \<times> vertices A"
llist_all2_rel: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and r :: "'e llist"   and s :: "'e llist" assumes "Quotient3 R Abs Rep" shows "llist_all2 R r s = (llist_all2 R r r \<and> llist_all2 R s s \<and> lmap Abs r = lmap Abs s)"
deliver_in_prefix_is_valid: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and i :: "nat"   and m :: "'a \<times> ('a, 'b) operation" assumes "xs prefix of i"   and "Deliver m \<in> set xs" shows "\<exists>state. valid_behaviours state m"
induct: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "\<And>j. \<lbrakk>j + 2 < n \<Longrightarrow> P (j + 1); \<And>x. \<lbrakk>j + 2 < n; x \<in> {j + 3..n - 1}\<rbrakk> \<Longrightarrow> P x\<rbrakk> \<Longrightarrow> P j" shows "P a0"
RPDi_Test_1: fixes i :: "nat" shows "RPDi i (MSkip 0) \<subseteq> {MSkip 0}"
less_iff_diff_less_0: fixes a :: "'a"   and b :: "'a" shows "(a < b) = (a - b < (0::'a))"
UTM_halt_lemma: fixes p :: "(action \<times> nat) list"   and rs :: "nat"   and args :: "nat list"   and i :: "nat"   and m :: "nat"   and k :: "nat" assumes "tm_wf0 p"   and "0 < rs"   and "args \<noteq> []"   and "{\<lambda>tp. tp = (Bk \<up> i, <args>)} p {\<lambda>tp. tp = (Bk \<up> m, Oc \<up> rs @ Bk \<up> k)}" shows "{\<lambda>tp. tp = ([], <code p # args>)} UTM {\<lambda>tp. \<exists>m n. tp = (Bk \<up> m, Oc \<up> rs @ Bk \<up> n)}"
path_pd_ipd0': fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat" assumes "is_path \<pi>"   and "\<pi> n \<noteq> local.return"   and "\<pi> n \<noteq> \<pi> 0"   and "\<pi> n pd\<rightarrow> \<pi> 0" obtains k :: "nat" where "k \<le> n" and "\<pi> k = ipd (\<pi> 0)"
thetaWhileW0_BisT: shows "thetaWhileW0 \<subseteq> BisT"
space_closed: shows "sigma_sets \<Omega> E \<subseteq> Pow \<Omega>"
open_point_def: fixes S :: "point set" shows "open S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
wqo_on_lists_over_finite_sets: shows "wqo_on subseq UNIV"
simps: fixes t :: "'t"   and a1 :: "'x2 \<times> 'm2"   and a2 :: "'x2 \<times> 'm2" shows "r2.silent_move t a1 a2 = (\<exists>s s' tl. a1 = s \<and> a2 = s' \<and> t \<turnstile> s -2-tl\<rightarrow> s' \<and> \<tau>move2 s tl s')"
assms: fixes i :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" assumes "i \<in> I" shows "flowsto i T CXS (f i)"
Busemann_function_inner: fixes z :: "'b"   and x :: "'b"   and y :: "'b" shows "Busemann_function_at (to_Gromov_completion z) x y = dist x z - dist y z"
acomplete_bal_list: fixes n :: "nat"   and xs :: "'a list" assumes "n \<le> length xs" shows "acomplete (bal_list n xs)"
simps_1: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b" shows "mapi f [] = []"
neq_h: shows "At = NeqUni (a, b, c) \<longrightarrow> (\<exists>w\<ge>x. w \<le> y \<and> a * w\<^sup>2 + b * w + c = 0)"
assms_3: shows "(\<xi>, p) \<in> reachable A I"
invertible_right_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "invertible x"   and "x \<in> M"   and "y \<in> M"   and "z \<in> M" shows "(y \<cdot> x = z \<cdot> x) = (y = z)"
AssertionStatEqTrans: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and \<Psi>'' :: "'b" assumes "\<Psi> \<simeq> \<Psi>'"   and "\<Psi>' \<simeq> \<Psi>''" shows "\<Psi> \<simeq> \<Psi>''"
get_root_node_si_def: fixes ptr :: "(_) object_ptr" shows "get_root_node_si ptr = Heap_Error_Monad.bind (get_ancestors_si ptr) (\<lambda>ancestors. return (last ancestors))"
integrable_fst': fixes f :: "'a \<times> 'b \<Rightarrow> 'c" assumes "integrable (M1 \<Otimes>\<^sub>M M2) f" shows "integrable M1 (\<lambda>x. LINT y|M2. f (x, y))"
accepting_pair\<^sub>G\<^sub>R_def: fixes \<delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>0 :: "'b"   and P :: "('b \<times> 'c \<times> 'b) set \<times> ('b \<times> 'c \<times> 'b) set set"   and w :: "nat \<Rightarrow> 'c" shows "accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 P w \<equiv> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> fst P = {} \<and> (\<forall>I\<in>snd P. limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> I \<noteq> {})"
simps_1: fixes b1 :: "bool"   and b2 :: "bool" shows "GExp_Lexorder.less_gexp_aux (Bc b1) (Bc b2) = (b1 < b2)"
sum_in_Rats: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<rat>" shows "sum f A \<in> \<rat>"
attractive_axioms: shows "attractive A (\<lambda>x y. y \<sqsubseteq> x)"
assms: shows "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"
conforms_StdXcpt: fixes xn :: "xname"   and s :: "st"   and G :: "prog"   and L :: "lname \<Rightarrow> ty option" shows "(Some (Xcpt (Std xn)), s)\<Colon>\<preceq>(G, L) = Norm s\<Colon>\<preceq>(G, L)"
simps_1: fixes s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" shows "preds [] s = True"
insert_image: fixes x :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "x \<in> A" shows "insert (f x) (f ` A) = f ` A"
LstSeqP_SeqAppendP_Eats: fixes H :: "fm set"   and f1 :: "tm"   and k1 :: "tm"   and f2 :: "tm"   and k2 :: "tm"   and g :: "tm"   and y1 :: "tm"   and y2 :: "tm"   and k :: "tm"   and z :: "tm" assumes "H \<turnstile> SeqAppendP f1 (SUCC k1) f2 (SUCC k2) g"   and "H \<turnstile> LstSeqP f1 k1 y1"   and "H \<turnstile> LstSeqP f2 k2 y2"   and "H \<turnstile> HaddP k1 k2 k" shows "H \<turnstile> LstSeqP (Eats g (HPair (SUCC (SUCC k)) z)) (SUCC (SUCC k)) z"
multi_nonempty_split: fixes M :: "'a multiset" assumes "M \<noteq> \<Empt>" shows "\<exists>A a. M = add_mset a A"
f_1_Ex1: shows "f_1 \<in> CofGroups.Ex1"
fps_exp_0: shows "fps_exp (0::'a) = 1"
bounded_iff: fixes b :: "'c"   and c :: "'c"   and a :: "'c" shows "(b + c \<le> a) = (b \<le> a \<and> c \<le> a)"
ResSet: fixes R :: "pi"   and S :: "pi"   and y :: "name" assumes "(R, S) \<in> Rel" shows "(<\<nu>y>R, <\<nu>y>S) \<in> Rel'"
cut_ge_mono: fixes t' :: "'a"   and t :: "'a"   and I :: "'a set" assumes "t' \<le> t" shows "I \<down>\<ge> t \<subseteq> I \<down>\<ge> t'"
distinct_1: fixes x11 :: "'f"   and x12 :: "'g"   and x13 :: "'h"   and x21 :: "'f"   and x22 :: "bool" shows "NewThread x11 x12 x13 \<noteq> ThreadExists x21 x22"
A: shows "\<nexists>i. t < max (te p i) (te q i)"
i_expand_nth_mult: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af"   and n :: "nat" assumes "0 < k" shows "(f \<odot> k) (n * k) = f n"
connected_arcs_empty: assumes "Digraph_Component.connected G"   and "arcs G = {}"   and "verts G \<noteq> {}" obtains v :: "'a" where "verts G = {v}"
assms_1: shows "s\<^sub>2 = (s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>;;\<tau>', \<E> [VE (CV Unit)])))(r'' := None)"
ereal_of_enat_le_iff: fixes m :: "enat"   and n :: "enat" shows "(ereal_of_enat m \<le> ereal_of_enat n) = (m \<le> n)"
wset_Suspend_okI: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and I :: "(('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set) set" assumes "s \<in> I"   and "\<And>t w. wset s t = \<lfloor>w\<rfloor> \<Longrightarrow> \<exists>s0\<in>I. \<exists>s1\<in>I. \<exists>ttas x x0 ta w' ln' ln''. s0 -t\<triangleright>ta\<rightarrow> s1 \<and> s1 -\<triangleright>ttas\<rightarrow>* s \<and> thr s0 t = \<lfloor>(x0, no_wait_locks)\<rfloor> \<and> t \<turnstile> \<langle>x0, shr s0\<rangle> -ta\<rightarrow> \<langle>x, shr s1\<rangle> \<and> Suspend w' \<in> set \<lbrace>ta\<rbrace>\<^bsub>w\<^esub> \<and> actions_ok s0 t ta \<and> thr s1 t = \<lfloor>(x, ln')\<rfloor> \<and> thr s t = \<lfloor>(x, ln'')\<rfloor>" shows "s \<in> wset_Suspend_ok I"
pderiv_power: fixes p :: "'a poly"   and n :: "nat" shows "pderiv (p ^ n) = Polynomial.smult (of_nat n) (p ^ (n - 1) * pderiv p)"
finite_surj: fixes A :: "'d set"   and B :: "'e set"   and f :: "'d \<Rightarrow> 'e" assumes "finite A"   and "B \<subseteq> f ` A" shows "finite B"
fps_inverse_fps_X_plus1: shows "inverse (1 + fps_X) = Abs_fps ((^) (- (1::'a)))"
simplicialcomplex_opp_half_apartment: shows "SimplicialComplex Y"
node_quality_increases: fixes i :: "nat"   and R :: "nat set" shows "\<langle>i : opaodv i \<langle>\<langle>\<^bsub>i\<^esub> qmsg : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg (\<lambda>_. rreq_rrep_sn) \<sigma>, other (\<lambda>_ _. True) {i} \<rightarrow>) globala (\<lambda>(\<sigma>, uu_, \<sigma>'). quality_increases (\<sigma> i) (\<sigma>' i))"
decode_regs_def: fixes n :: "nat" shows "decode_regs n = (\<lambda>i. let xs = nat_to_list n in if i < length xs then nat_to_list n ! i else 0)"
induct: fixes P :: "('f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option) \<Rightarrow> ('f, 'g, 'i) new_thread_action list \<Rightarrow> bool"   and a0 :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and a1 :: "('f, 'g, 'i) new_thread_action list" assumes "\<And>ts. P ts []"   and "\<And>ts ta tas. P (redT_updT ts ta) tas \<Longrightarrow> P ts (ta # tas)" shows "P a0 a1"
assms: shows "\<bar>x\<bar> < \<bar>y\<bar>"
spans_image: fixes V :: "'b set"   and B :: "'b set" assumes "V \<subseteq> local.span B" shows "f ` V \<subseteq> l.vs2.span (f ` B)"
butlast_prefix: fixes xs :: "'b list"   and ys :: "'b list" assumes "prefix xs ys" shows "prefix (butlast xs) (butlast ys)"
negated_euclidean_axiom': shows "\<exists>a b c. a \<in> unit_disc \<and> b \<in> unit_disc \<and> c \<in> unit_disc \<and> \<not> poincare_collinear {a, b, c} \<and> (\<nexists>x. x \<in> unit_disc \<and> poincare_distance a x = poincare_distance b x \<and> poincare_distance a x = poincare_distance c x)"
rem_implicit_pres_ops_path_to: fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and s :: "nat \<Rightarrow> nat option"   and \<pi>s :: "char list list"   and s' :: "nat \<Rightarrow> nat option" assumes "wf_ast_problem prob"   and "\<And>op. op \<in> set (ast_problem.ast\<delta> prob) \<Longrightarrow> consistent_pres_op op"   and "\<And>op. op \<in> set (ast_problem.ast\<delta> prob) \<Longrightarrow> is_standard_operator op"   and "s \<in> ast_problem.valid_states prob" shows "ast_problem.path_to (rem_implicit_pres_ops prob) s \<pi>s s' = ast_problem.path_to prob s \<pi>s s'"
C: fixes x :: "real" assumes "x \<in> {0..1 / 2}" shows "(1 - x) powr (b - 1) \<le> C"
PChain: shows "P \<parallel> !P \<Longrightarrow>\<^sub>\<tau> P'''"
hr_comp_def: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and R2 :: "('a \<times> 'c) set"   and a :: "'c"   and c :: "'b" shows "hr_comp R1 R2 a c \<equiv> \<exists>\<^sub>Ab. R1 b c * \<up> ((b, a) \<in> R2)"
Ogood_def: shows "Ogood = (\<Inter>N. On (N + 1))"
inner_minus_left: fixes x :: "'b"   and y :: "'b" shows "inner (- x) y = - inner x y"
KoenigLemma: fixes Tr :: "'a list set" assumes "infinite Tr"   and "\<forall>i. finite {t. \<exists>t'\<in>Tr. t = take i t'}" shows "\<exists>f. strict_mono f \<and> range f \<subseteq> {t. \<exists>t'\<in>Tr. t \<le> t'}"
simps_2: fixes n :: "nat"   and i :: "nat" shows "unit_vecs_last n (Suc i) = unit_vec n (n - Suc i) # unit_vecs_last n i"
nonneg_cexpr_comp: fixes V :: "nat set"   and t2 :: "pdf_type"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and t1 :: "pdf_type"   and f :: "cexpr" assumes "nonneg_cexpr (shift_var_set V) (case_nat t2 \<Gamma>) e"   and "case_nat t1 \<Gamma> \<turnstile>\<^sub>c f : t2"   and "free_vars f \<subseteq> shift_var_set V" shows "nonneg_cexpr (shift_var_set V) (case_nat t1 \<Gamma>) (e \<circ>\<^sub>c f)"
Union_Int_subset: fixes A :: "'a set set"   and B :: "'a set set" shows "\<Union> (A \<inter> B) \<subseteq> \<Union> A \<inter> \<Union> B"
out: fixes x :: "'v" assumes "x \<in> \<^bold>V"   and "x \<noteq> source \<Delta>" shows "d_OUT (capacity \<Delta>) x < \<top>"
L_def: fixes F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "O[F](g) = {f. \<exists>c>0. \<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)}"
hyps_2: shows "P \<turnstile> \<langle>es2,s2,b2\<rangle> [\<rightarrow>]* \<langle>es'',s'',b''\<rangle>"
lookup_operator'_def: shows "lookup_operator' \<equiv> \<lambda>(D, I, G, \<delta>) name. find (\<lambda>(n, uu_, uu_, uu_). n = name) \<delta>"
Blacks_def: fixes M :: "node list" shows "Blacks M \<equiv> {i. i < length M \<and> M ! i = Black}"
exec_n_split: fixes P :: "instr list"   and c :: "instr list"   and P' :: "instr list"   and i :: "int"   and s :: "(char list \<Rightarrow> int) \<times> int list"   and n :: "nat"   and j :: "int"   and s' :: "(char list \<Rightarrow> int) \<times> int list" assumes "P @ c @ P' \<turnstile> (size P + i, s) \<rightarrow>^n (j, s')"   and "0 \<le> i"   and "i < size c"   and "j \<notin> {size P..<size P + size c}" shows "\<exists>s'' i' k m. c \<turnstile> (i, s) \<rightarrow>^k (i', s'') \<and> i' \<in> exits c \<and> P @ c @ P' \<turnstile> (size P + i', s'') \<rightarrow>^m (j, s') \<and> n = k + m"
rtranclp_reflclp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "R\<^sup>=\<^sup>=\<^sup>*\<^sup>* = R\<^sup>*\<^sup>*"
le_iff_plus_unchanged2: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "Semilat A r f"   and "x \<in> A"   and "y \<in> A" shows "(x \<sqsubseteq>\<^bsub>r\<^esub> y) = (y \<squnion>\<^bsub>f\<^esub> x = y)"
simps_2: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" shows "filter_option P (Some x) = (if P x then Some x else None)"
eq1': shows "(l - eval_dioph ks (map (\<lambda>x. x mod 2) xs)) mod 2 = 0"
Int_absorb1: fixes B :: "'b set"   and A :: "'b set" assumes "order.greater_eq A B" shows "A \<inter> B = B"
liftPoly_sub: fixes z :: "nat"   and a :: "'a mpoly"   and b :: "'a mpoly" shows "liftPoly 0 z (a - b) = liftPoly 0 z a - liftPoly 0 z b"
cong_mid2__cong: fixes M :: "'p"   and A :: "'p"   and B :: "'p"   and M' :: "'p"   and A' :: "'p"   and B' :: "'p" assumes "M Midpoint A B"   and "M' Midpoint A' B'"   and "Cong A M A' M'" shows "Cong A B A' B'"
picard_lindeloef_axioms: shows "picard_lindeloef f T S t\<^sub>0"
lincomb_singles: fixes smult :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'b"   and m :: "'a" shows "scalar_mult.lincomb smult [a] [m] = smult a m"
lfinite_LConsI: fixes xs :: "'c llist"   and x :: "'c" assumes "lfinite xs" shows "lfinite (LCons x xs)"
throw_def2: fixes a' :: "val"   and x :: "abopt" shows "throw a' x = abrupt_if True (Some (Xcpt (Loc (the_Addr a')))) (np a' x)"
summands_valid: fixes R :: "('a, 'b) ring_scheme"   and M :: "('a, 'c, 'd) module_scheme"   and A :: "'c set"   and a :: "'c \<Rightarrow> 'a" assumes "Module.module R M"   and "A \<subseteq> carrier M"   and "a \<in> A \<rightarrow> carrier R" shows "\<forall>v\<in>A. a v \<odot>\<^bsub>M\<^esub> v \<in> carrier M"
snd: fixes P :: "'a trm"   and A :: "'a trm"   and B :: "'a trm" assumes "P \<noteq> SimplyTyped.Pair A B"   and "NF P" shows "NF (Snd P)"
get_shadow_root_eq_h: fixes ptr' :: "(_) element_ptr"   and shadow_root_opt :: "(_) shadow_root_ptr option" assumes "ptr \<noteq> ptr'" shows "h \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_opt = h2 \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_opt"
IH: fixes i :: "nat" assumes "sorted xs"   and "i < length xs" shows "{#x \<in># mset xs. x < xs ! i#} \<subseteq># mset (take i xs)"
division_of_trivial: fixes s :: "'a set set" shows "(s division_of {}) = (s = {})"
WT_converter_parallel_converter: fixes \<I>1 :: "('i, 'j) \<I>"   and \<I> :: "('k, 'l) \<I>"   and conv1 :: "('i, 'j, 'k, 'l) converter"   and \<I>2 :: "('m, 'n) \<I>"   and conv2 :: "('m, 'n, 'k, 'l) converter" assumes "\<I>1, \<I> \<turnstile>\<^sub>C conv1 \<surd>"   and "\<I>2, \<I> \<turnstile>\<^sub>C conv2 \<surd>" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>2, \<I> \<turnstile>\<^sub>C conv1 |\<^sub>\<propto> conv2 \<surd>"
error_free_check_field_access_eq: fixes G :: "prog"   and accC :: "qtname"   and statDeclC :: "qtname"   and fn :: "vname"   and stat :: "bool"   and a :: "val"   and s :: "state" assumes "error_free (check_field_access G accC statDeclC fn stat a s)" shows "check_field_access G accC statDeclC fn stat a s = s"
Skip_WCFG_edge_Exit: fixes prog :: "cmd"   and l :: "nat" assumes "labels prog l Skip" shows "prog \<turnstile> (_ l _) -\<Up>id\<rightarrow> (_Exit_)"
top_assn_def: shows "top \<equiv> Abs_assn in_range"
Basic: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'e \<Rightarrow> 'e"   and s :: "'e" shows "\<Gamma>\<turnstile>Basic f \<down> Normal s"
passive_subset_def: fixes M :: "('a literal multiset \<times> label) set" shows "FL.passive_subset M = {CL \<in> M. snd CL \<noteq> Old}"
IN_j_finite: fixes x :: "'v option" shows "d_IN j x \<noteq> \<top>"
simps_1: fixes n :: "nat"   and \<phi> :: "Formula.formula" shows "minit0 n (formula.Neg \<phi>) = (if fv \<phi> = {} then MNeg (minit0 n \<phi>) else MRel empty_table)"
vunion_vinsert_commutativity_left: fixes a :: "V"   and A :: "V"   and B :: "V" shows "vinsert a A \<union>\<^sub>\<circ> B = A \<union>\<^sub>\<circ> vinsert a B"
Dom_in_Obj: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" shows "Dom f \<in> {False, True}"
finite_measure_mono: fixes A :: "nat set set set"   and B :: "nat set set set" assumes "A \<subseteq> B"   and "B \<in> E.events" shows "E.prob A \<le> E.prob B"
simps_2: fixes n\<^sub>s :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid" assumes "\<not> (\<exists>n'<n\<^sub>s. \<not> Option.is_none (devBC t n nid n'))"   and "Option.is_none (devBC t n nid n\<^sub>s)" shows "devExt t n nid n\<^sub>s 0 = []"
enum_set_get_one_enum_Some: fixes S :: "'b set"   and x :: "'b" assumes "enum_set_get_one enum_class.enum S = Some x" shows "x \<in> S"
in_set_member: fixes x :: "'a"   and xs :: "'a list" shows "(x \<in> list.set xs) = List.member xs x"
strict_iff_order: fixes b :: "'b multiset"   and a :: "'b multiset" shows "(b \<subset># a) = (b \<subseteq># a \<and> a \<noteq> b)"
P_const_const: fixes name :: "name" shows "P (const name) (const name)"
subint: fixes C :: "qtname"   and I :: "qtname"   and G :: "prog"   and J :: "qtname" assumes "G\<turnstile>C\<leadsto>1I"   and "G\<turnstile>I\<preceq>I J" shows "G\<turnstile>C\<leadsto>J"