hcju/isabelleps huggingface dataset:
add_pairs_single_naive_def: fixes data :: "'g"   and flag :: "bool"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and h :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and ps :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "add_pairs_single_naive data flag gs bs h ps = ps @ map (\<lambda>g. (flag, h, g)) gs @ map (\<lambda>b. (flag, h, b)) bs"
prems_1: shows "\<forall>a\<in>\<O> \<union> all_acquired (x # sb). \<S>' a = \<S> a"
Rep_Store: fixes x :: "Store" shows "Rep_Store x \<in> Store"
ord_Zp_def: fixes p :: "int" assumes "padic_integers p" shows "padic_integers.ord_Zp p = padic_val p"
id_eq_prop_prop_5_b: fixes v :: "i"   and F :: "\<Pi>\<^sub>3"   and G :: "\<Pi>\<^sub>3" shows "[F \<^bold>= G \<^bold>\<rightarrow> G \<^bold>= F in v]"
domain_eq: shows "Domainp (pcr_nonzero (=)) = (\<lambda>x. x \<in> UNIV - {0::'b})"
farkas_assert_bound: fixes s :: "('i, 'a) state"   and as :: "('i \<times> 'a atom) set"   and ia :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<diamond> s"   and "index_valid as s"   and "\<U> (assert_bound ia s)" shows "\<exists>C. farkas_coefficients_atoms_tableau (snd ` insert ia as) (\<T> s) C"
index_correct_3: fixes idx_sf :: "('Q \<times> 'L, ('Q, 'L) ta_rule dlist) HashMap.hashmap" assumes "hta_idx_sf H = Some idx_sf" shows "hll_idx.is_index (\<lambda>r. (lhs r, rhsl r)) (ls.\<alpha> \<delta>) idx_sf"
homeomorphic_map_maps: fixes X :: "'d topology"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e" shows "homeomorphic_map X Y f = (\<exists>g. homeomorphic_maps X Y f g)"
set_l_is: shows "set (map (\<lambda>v. \<^bold>\<not> (Atom (Operator k (index (\<Pi>\<^sub>\<O>) op))) \<^bold>\<or> Atom (State k (index (\<Pi>\<^sub>\<V>) v))) (precondition_of op)) = {\<^bold>\<not> (Atom (Operator k (index (\<Pi>\<^sub>\<O>) op))) \<^bold>\<or> Atom (State k (index (\<Pi>\<^sub>\<V>) v)) |v. v \<in> set (precondition_of op)}"
restrict_def: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" shows "restrict f A = (\<lambda>x. if x \<in> A then f x else SOME y. True)"
finite_insert: fixes a :: "'a"   and A :: "'a set" shows "finite (insert a A) = finite A"
add_right_mono: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "a \<subseteq># b" shows "a + c \<subseteq># b + c"
ncoplanar_perm_22: 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 D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet D C A B"
conforms_RefTD: fixes G :: "prog"   and s :: "st"   and a' :: "val"   and t :: "ref_ty"   and x :: "abopt"   and L :: "lenv" assumes "G,s\<turnstile>a'\<Colon>\<preceq>RefT t"   and "a' \<noteq> Null"   and "(x, s)\<Colon>\<preceq>(G, L)" shows "\<exists>a obj. a' = Addr a \<and> globs s (Inl a) = Some obj \<and> G\<turnstile>obj_ty obj\<preceq>RefT t \<and> is_type G (obj_ty obj)"
is_lub_thelub: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'b"   and x :: "'b" assumes "class.cpo below"   and "po.chain below S"   and "po.is_ub below (range S) x" shows "below (po.lub below (range S)) x"
subset_image_iff: fixes B :: "'d set"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "(B \<subseteq> f ` A) = (\<exists>AA\<subseteq>A. B = f ` AA)"
resource_of_oracle_extend_state_oracle: fixes "oracle" :: "'q \<Rightarrow> 'n \<Rightarrow> ('o \<times> 'q) spmf"   and s' :: "'p"   and s :: "'q" shows "RES \<dagger>oracle (s', s) = RES oracle s"
pltl_seq_sim: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and \<phi> :: "'b pltl"   and \<tau> :: "nat \<Rightarrow> 'b" assumes "\<sigma> \<simeq>atoms_pltl \<phi>\<simeq> \<tau>" shows "\<sigma> \<Turnstile>\<^sub>p \<phi> = \<tau> \<Turnstile>\<^sub>p \<phi>"
known_ptr_defs: fixes ptr :: "(_) object_ptr" shows "ElementClass.known_ptr ptr = (NodeClass.known_ptr ptr \<or> is_element_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)"
ns_mul_ext_Id_eq: fixes R :: "('a \<times> 'a) set" shows "ns_mul_ext Id R = (s_mul_ext Id R)\<^sup>="
g: shows "\<guillemotleft>g : R a' \<Rightarrow> R a\<guillemotright>"
comp_def: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'e" shows "f \<circ> g = (\<lambda>x. f (g x))"
x0_2: shows "a21 * fst x0 + a22 * snd x0 = b2"
fls_base_factor_def: fixes f :: "'a fls" shows "fls_base_factor f = fls_shift (fls_subdegree f) f"
negGorbit_list_V: fixes gs :: "'g list"   and T :: "'a \<Rightarrow> 'w"   and as :: "'a list" assumes "set gs \<subseteq> H"   and "T ` set as \<subseteq> W" shows "set (concat (HRep.negGorbit_list gs T as)) \<subseteq> W"
has_derivative_minus: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "(f has_derivative f') F" shows "((\<lambda>x. - f x) has_derivative (\<lambda>x. - f' x)) F"
convex_same_rel_interior_closure_straddle: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "convex T" shows "(rel_interior S = rel_interior T) = (rel_interior S \<subseteq> T \<and> T \<subseteq> closure S)"
select_convs_2: fixes counter :: "nat"   and discovered :: "'c \<Rightarrow> nat option"   and finished :: "'c \<Rightarrow> nat option"   and pending :: "('c \<times> 'c) set"   and stack :: "'c list"   and tree_edges :: "('c \<times> 'c) set"   and back_edges :: "('c \<times> 'c) set"   and cross_edges :: "('c \<times> 'c) set"   and sccs :: "'c set set"   and lowlink :: "'c \<Rightarrow> nat option"   and tj_stack :: "'c list"   and more :: "'d" shows "lowlink \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, sccs = sccs, lowlink = lowlink, tj_stack = tj_stack, \<dots> = more\<rparr> = lowlink"
conn1: fixes pid :: "'pid"   and k :: "cnf" assumes "\<parallel>pid\<parallel>\<^bsub>k\<^esub>" shows "pbsb (\<sigma>\<^bsub>pid\<^esub>k) = (\<Union>sid\<in>{sid. \<parallel>sid\<parallel>\<^bsub>k\<^esub>}. {sbsb (\<sigma>\<^bsub>sid\<^esub>k)})"
sym_allrel: fixes A :: "'b set" shows "sym (A \<times> A)"
OES1: fixes \<Gamma> :: "'b formula list"   and \<AA> :: "'b formula"   and \<Theta> :: "'b formula list"   and \<BB> :: "'b formula" assumes "\<Gamma> \<Rightarrow> \<AA> # \<Theta>" shows "\<Gamma> \<Rightarrow> \<AA> \<^bold>\<or> \<BB> # \<Theta>"
homotopic_paths_imp_pathstart: fixes s :: "'c set"   and p :: "real \<Rightarrow> 'c"   and q :: "real \<Rightarrow> 'c" assumes "homotopic_paths s p q" shows "pathstart p = pathstart q"
lift_to_metric_completion_isometry: fixes f :: "'b \<Rightarrow> 'c" assumes "isometry_on UNIV f" shows "\<exists>g. isometry_on UNIV g \<and> range g = closure (range f) \<and> f = g \<circ> to_metric_completion \<and> (\<forall>x\<in>range to_metric_completion. g x = f (inv to_metric_completion x))"
inv: fixes i :: "'a" assumes "i \<in> J" shows "f' (f i) = i"
BLP_m_def: shows "BLP_m \<equiv> new_configured_list_SecurityInvariant SINVAR_LIB_BLPtrusted \<lparr>node_properties = BLP_security_levels\<rparr> ''Two has confidential information''"
approx_def: fixes x :: "'b star"   and y :: "'b star" shows "(x \<approx> y) = (x - y \<in> Infinitesimal)"
arr_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool"   and f :: "'b" assumes "subcategory C Arr" shows "partial_magma.arr (subcategory.comp C Arr) f = Arr f"
cop: shows "comm_monoid_mult_class.coprime (lead_coeff f) p"
termFSbImorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'g"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'h"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'g, 'h, 'i) model_scheme" shows "termFSbImorph h hA MOD \<equiv> ipresWlsAll h hA MOD \<and> ipresCons h hA MOD \<and> ipresFreshAll h hA MOD \<and> ipresSubstAll h hA MOD"
that: fixes d' :: "('test, 'atom) com"   and t' :: "'state" assumes "(d, t) \<rightarrow>*c (d', t')"   and "s' \<approx> t'"   and "c' \<approx>wT d'" shows "thesis"
simps_8: fixes a :: "MIR.num" shows "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if 0 < v then T else F | _ \<Rightarrow> Gt (reducecoeff a'))"
injectionsUnionCommute: fixes x :: "'c"   and X :: "'c set"   and Y :: "'d set" assumes "x \<notin> X" shows "(\<Union>f\<in>injections X Y. {f \<union> {(x, y)} |y. y \<in> Y - Range f}) = injections ({x} \<union> X) Y"
rsp: shows "eq_onp (\<lambda>s. length s \<le> MAX_STR) [] []"
IH_1: shows "((l, send\<langle>t\<rangle>) \<in> set A) = ((l, receive\<langle>t\<rangle>) \<in> set (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
to_poly_add: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "to_polynomial R (a \<oplus> b) = to_polynomial R a \<oplus>\<^bsub>P\<^esub> to_polynomial R b"
components_are_iso: fixes a :: "'a" assumes "C.ide a" shows "D.iso (G a)"
star_unfoldl: fixes x :: "'a" shows "(1::'a) + x \<odot> x\<^sup>\<star> \<le> x\<^sup>\<star>"
P_Var: fixes T :: "POPLmarkRecord.type" shows "\<turnstile> PVar T : T \<Rightarrow> [VarB T]"
above_l: shows "\<forall>n. l \<le> (\<Sum>i<2 * n + 1. (- 1) ^ i * a i)"
cil_\<alpha>n_cons2: fixes w :: "('a, 'b) label list list"   and wa :: "('a, 'b) label list list"   and fg :: "('d, 'a, 'e, 'c, 'f) flowgraph_rec_scheme"   and wb :: "('a, 'b) label list list"   and e :: "('a, 'b) label list" assumes "w \<in> wa \<otimes>\<^bsub>\<alpha>n fg\<^esub> wb"   and "fst (\<alpha>n fg e) \<inter> mon_ww fg wa = {}" shows "e # w \<in> wa \<otimes>\<^bsub>\<alpha>n fg\<^esub> e # wb"
smult_one: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme"   and x :: "'g" assumes "module R M"   and "x \<in> carrier M" shows "\<one>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> x = x"
path_independent_onD: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and B :: "'a set"   and C :: "'a set" assumes "path_independent_on A f"   and "B \<subseteq> A"   and "C \<subseteq> A" shows "f (B \<union> C) = f (B \<union> f C)"
closure_closure: fixes S :: "'a set" shows "closure (closure S) = closure S"
phis_finite: fixes g :: "'g" shows "finite (dom (phis g))"
less_funE: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "less_fun f g" obtains k :: "'c" where "f k < g k" and "\<And>k'. k' < k \<Longrightarrow> f k' = g k'"
no_\<tau>move1_\<tau>s_to_no_\<tau>move2: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and s1 :: "'g"   and s2 :: "'i" assumes "delay_bisimulation_diverge trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"   and "bisim s1 s2"   and "\<And>s1'. \<not> \<tau>trsys.silent_move trsys1 \<tau>move1 s1 s1'" shows "\<exists>s2'. (\<tau>trsys.silent_move trsys2 \<tau>move2)\<^sup>*\<^sup>* s2 s2' \<and> (\<forall>s2''. \<not> \<tau>trsys.silent_move trsys2 \<tau>move2 s2' s2'') \<and> bisim s1 s2'"
simps_24: fixes g1 :: "'a gexp"   and g2 :: "'a gexp"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_gexp_aux (Nor g1 g2) (Gt v va) = False"
meas_M: shows "(\<rightarrow>\<^sub>M) M = (\<rightarrow>\<^sub>M) borel"
hr_comp_pure: fixes R :: "('b \<times> 'c) set"   and S :: "('c \<times> 'a) set" shows "hr_comp (pure R) S = pure (R O S)"
nret1: shows "\<pi>\<^sub>1 l \<noteq> local.return"
finter_finsert_left: fixes a :: "'a"   and B :: "'a fset"   and C :: "'a fset" shows "finsert a B |\<inter>| C = (if a |\<in>| C then finsert a (B |\<inter>| C) else B |\<inter>| C)"
runit'_in_hom_2: fixes f :: "'a" assumes "ide f" shows "\<guillemotleft>\<r>\<^sup>-\<^sup>1[f] : f \<Rightarrow> f \<star> src f\<guillemotright>"
normalize_poly_spec_alt_def: fixes p :: "int mpoly" shows "normalize_poly_spec p = SPEC (\<lambda>r. r - p \<in> More_Modules.ideal polynomial_bool \<and> vars r \<subseteq> vars p)"
disj_def: fixes \<gamma> :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool" shows "\<gamma> \<or>\<^sup>c \<gamma>' \<equiv> \<lambda>t n. \<gamma> t n \<or> \<gamma>' t n"
yun_monic_factorization: fixes hom :: "'a \<Rightarrow> 'b"   and f :: "'a poly" assumes "field_hom_0' hom"   and "monic f" shows "yun_gcd.yun_monic_factorization gcd (map_poly hom f) = map (\<lambda>(f, i). (map_poly hom f, i)) (yun_gcd.yun_monic_factorization gcd f)"
Skip: fixes s :: "loc \<Rightarrow> nat"   and eval :: "((exp \<times> (loc \<Rightarrow> nat)) \<times> nat \<times> (loc \<Rightarrow> nat)) set" shows "(SKIP, s) -[eval]-> s"
num_params_tm_sub_le: fixes t1 :: "taylor_model"   and X :: "nat"   and t2 :: "taylor_model" assumes "num_params (tm_poly t1) \<le> X"   and "num_params (tm_poly t2) \<le> X" shows "num_params (tm_poly (tm_sub t1 t2)) \<le> X"
set_awalk_verts: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "G.awalk u p v" shows "set (G.awalk_verts u p) = {u} \<union> set (map (tail G) p) \<union> set (map (head G) p)"
ideal_inv1_closed: fixes I :: "'a set"   and x :: "'a" assumes "ideal R I"   and "x \<in> I" shows "-\<^sub>a x \<in> I"
Lxy_def: fixes xs :: "'b list"   and S :: "'b set" shows "Lxy xs S = filter (\<lambda>z. z \<in> S) xs"
invimage_vcomp: fixes r :: "V"   and s :: "V"   and x :: "V" shows "(r \<circ>\<^sub>\<circ> s) -`\<^sub>\<circ> x = s -`\<^sub>\<circ> r -`\<^sub>\<circ> x"
dw_closed_rtrancl: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and a :: "'b"   and b :: "'b" assumes "relation.dw_closed r A"   and "a \<in> A"   and "r\<^sup>*\<^sup>* a b" shows "b \<in> A"
less_LiminfD: fixes y :: "'c"   and F :: "'d filter"   and f :: "'d \<Rightarrow> 'c" assumes "y < Liminf F f" shows "\<forall>\<^sub>F x in F. y < f x"
induct: fixes P :: "bool list \<times> bool list \<Rightarrow> (nat, nat) atomic \<Rightarrow> bool"   and a0 :: "bool list \<times> bool list"   and a1 :: "(nat, nat) atomic" assumes "\<And>bs1 bs2 m. P (bs1, bs2) (Fo m)"   and "\<And>bs1 bs2 m. P (bs1, bs2) (Z m)"   and "\<And>bs1 bs2 m1 m2. P (bs1, bs2) (Less m1 m2)"   and "\<And>bs1 bs2 m M. P (bs1, bs2) (In m M)" shows "P a0 a1"
sumxlim: shows "\<forall>x. (\<lambda>i. X x * indpre i x) sums (X x * indicat_real (Y -` (A \<inter> range Y) \<inter> space M) x)"
sf_eq_arg: fixes x :: "nat" assumes "sf x = x" shows "x \<le> 1"
tab_def: shows "tab \<equiv> T0.trnr\<^sub>\<eta> (tab\<^sub>1 r) (rep r)"
IInit_def: fixes s :: "Istate" shows "IInit s = (range (iinput s) \<subseteq> Inputs \<and> ioutput s = (\<lambda>p. NotAnInput) \<and> ichosen s = NotAnInput \<and> iallInput s = range (iinput s))"
un_MSLCons1_def: fixes msllist :: "'a msllist" shows "un_MSLCons1 msllist = (case msllist of MSLCons x21 x22 \<Rightarrow> x21)"
sum_representation_eq: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and basis :: "'e set"   and v :: "'e"   and B :: "'e set" assumes "module scale"   and "\<not> module.dependent scale basis"   and "v \<in> module.span scale basis"   and "finite B"   and "basis \<subseteq> B" shows "(\<Sum>b\<in>B. scale (module.representation scale basis v b) b) = v"
intros_3: fixes A\<^sub>1 :: "char list set"   and c\<^sub>1 :: "com"   and A\<^sub>2 :: "char list set"   and c\<^sub>2 :: "com"   and A\<^sub>3 :: "char list set" assumes "D A\<^sub>1 c\<^sub>1 A\<^sub>2"   and "D A\<^sub>2 c\<^sub>2 A\<^sub>3" shows "D A\<^sub>1 (c\<^sub>1;; c\<^sub>2) A\<^sub>3"
ps_get_op_def: fixes am :: "nat \<Rightarrow> nat list"   and u :: "nat" shows "ps_get_op am u \<equiv> assert (u \<in> V) \<bind> (\<lambda>_. return (am u))"
wf_idecl_hidings: fixes G :: "prog"   and I :: "qtname"   and i :: "iface" assumes "wf_idecl G (I, i)" shows "\<lambda>s. set_option (table_of (imethods i) s) hidings Un_tables (DeclConcepts.imethds G ` set (isuperIfs i)) entails \<lambda>new old. G\<turnstile>resTy new\<preceq>resTy old"
S_def: shows "S = {(x, y). token_succeeds x \<and> token_succeeds y}"
mult_numeral_left_semiring_numeral: fixes v :: "num"   and w :: "num"   and z :: "'c" shows "numeral v * (numeral w * z) = numeral (v * w) * z"
fls_shift_plus: fixes m :: "int"   and f :: "'a fls"   and g :: "'a fls" shows "fls_shift m (f + g) = fls_shift m f + fls_shift m g"
is_PUSH_project_conv: shows "is_PUSH \<circ> project_operation = (\<lambda>x. is_SAT_PUSH' x \<or> is_NONSAT_PUSH' x)"
assms_2: fixes x :: "'a" shows "(V has_derivative V' x) (at x)"
mem_assign_refinement_helper_var: fixes mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat"   and x :: "Eg1.var"   and y :: "Eg1.var" shows "mem\<^sub>A_of (mem\<^sub>C(Eg2_var\<^sub>C_of_Eg1 x := mem\<^sub>C (Eg2_var\<^sub>C_of_Eg1 y))) = (mem\<^sub>A_of mem\<^sub>C)(x := mem\<^sub>A_of mem\<^sub>C y)"
equality: fixes r :: "'b score_scheme"   and r' :: "'b score_scheme" assumes "Score r = Score r'"   and "S1 r = S1 r'"   and "S2 r = S2 r'"   and "score.more r = score.more r'" shows "r = r'"
join_fmap_fmap: fixes f :: "'f \<rightarrow> 'd"   and xss :: "'f\<cdot>'e\<cdot>'e" shows "join\<cdot>(fmap\<cdot>(fmap\<cdot>f)\<cdot>xss) = fmap\<cdot>f\<cdot>(join\<cdot>xss)"
psi_u_2: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> U" shows "\<exists>i. psi_u i = f"
assign_def: fixes x :: "'d"   and e :: "('c, 'd) vec \<Rightarrow> 'c" shows "x ::= e = (\<lambda>s. skip (HS_VC_Spartan.vec_upd s x (e s)))"
mupdate_def: fixes i :: "int"   and a :: "'b"   and t :: "(int \<times> 'b) Tree" shows "mupdate i a t \<equiv> binsert fst (i, a) t"
rw_rat_def: fixes p :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and q :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" shows "rw_rat p q = (let u = punit.lt (snd q) \<oplus> fst p; v = punit.lt (snd p) \<oplus> fst q in u \<prec>\<^sub>t v \<or> u = v \<and> fst p \<preceq>\<^sub>t fst q)"
transp_def: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool" shows "transp r = (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
integrable_on_cmult_iff: fixes c :: "real"   and f :: "'a \<Rightarrow> real"   and S :: "'a set" assumes "c \<noteq> 0" shows "((\<lambda>x. c * f x) integrable_on S) = (f integrable_on S)"
B: shows "B = 1 / 2 \<cdot>\<^sub>m (A + adjoint A)"
cnf_BigOr: fixes x :: "'b formula list"   and y :: "'b formula list" shows "cnf \<^bold>\<Or>x @ y = {f \<union> g |f g. f \<in> cnf \<^bold>\<Or>x \<and> g \<in> cnf \<^bold>\<Or>y}"
assms: shows "{x \<in> space M. \<not> P x} \<in> sets M"
Aleph_succ: fixes x :: "V" shows "\<aleph>ZFC_in_HOL.succ x = csucc (\<aleph>x)"
eqpoll_connected_components_Euclidean_complements: fixes n :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and T :: "(nat \<Rightarrow> real) set" assumes "closedin (Euclidean_space n) S"   and "closedin (Euclidean_space n) T"   and "subtopology (Euclidean_space n) S homeomorphic_space subtopology (Euclidean_space n) T" shows "connected_components_of (subtopology (Euclidean_space n) (topspace (Euclidean_space n) - S)) \<approx> connected_components_of (subtopology (Euclidean_space n) (topspace (Euclidean_space n) - T))"
object_ptr_kinds_eq2: shows "|h \<turnstile> object_ptr_kinds_M|\<^sub>r = |h' \<turnstile> object_ptr_kinds_M|\<^sub>r"
someI: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "P x" shows "P (Eps P)"
axioms_2: fixes G :: "('c, 'd) graph_rec_scheme"   and s :: "'c tarjan_state" assumes "Tarjan_invar G s" shows "DFS_invar G (Tarjan_def.tarjan_params G) s"
y: shows "eval r_enumdom [i, x] \<down>= y"
single_addition_posTr: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "int"   and m :: "int" assumes "commute_bpp f (aug_pm_set z i {a})"   and "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z"   and "0 < n"   and "0 < m" shows "(THE x. x \<in> add_set f {a} (nat (n - 1))) \<^sub>f+ (THE x. x \<in> add_set f {a} (nat (m - 1))) = (THE x. x \<in> add_set f {a} (nat (n + m - 1)))"
I_and: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm"   and xs :: "int list" shows "Z.I (and \<phi>\<^sub>1 \<phi>\<^sub>2) xs = Z.I (And \<phi>\<^sub>1 \<phi>\<^sub>2) xs"
not_finite_existsD: fixes P :: "'d \<Rightarrow> bool" assumes "infinite {a. P a}" shows "\<exists>a. P a"
homom4Tr1: fixes G :: "('e, 'f) Group_scheme"   and N :: "'e set"   and H :: "'e set" assumes "Group G"   and "G \<triangleright> N"   and "G \<guillemotright> H " shows "Group (\<natural>\<^bsub>G\<^esub>H \<diamondop>\<^bsub>G\<^esub> N / N)"
mset_wfactorsEx: fixes Cs :: "'a set multiset" assumes "\<And>X. X \<in># Cs \<Longrightarrow> \<exists>x. (x \<in> carrier G \<and> irreducible G x) \<and> X = assocs G x" shows "\<exists>c cs. c \<in> carrier G \<and> set cs \<subseteq> carrier G \<and> wfactors G cs c \<and> fmset G cs = Cs"
Init_simps_1: fixes y :: "behavior \<Rightarrow> bool" shows "Init y \<equiv> y"
semiattractive_axioms: shows "semiattractive Q (\<lambda>x y. y \<sqsubseteq>\<^sup>s x)"
assms_3: shows "A \<subseteq>\<^sub>\<circ> B"
of_int_less_iff: 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 w :: "int"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z) = (w < z)"
reindex_map_def: fixes H :: "('a, 'b, 'c) hashedTa_scheme" shows "reindex_map H \<equiv> \<lambda>q. the (hm.lookup q (hh_map_to_nat.map_to_nat (hta_states H)))"
sqUn_lem: fixes A :: "'b set option"   and A' :: "'b set option"   and B :: "'b set option" assumes "A \<sqsubseteq> A'" shows "A \<squnion> B \<sqsubseteq> A' \<squnion> B"
assms_2: shows "A \<noteq> 0 \<or> B \<noteq> 0"
wp_arraycpy_eq: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and a :: "char list"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wp \<pi> (x[] ::= a) Q s = Q (s(x := s a))"
idom_tOp_nonzeros: fixes R :: "('d, 'e) Ring_scheme"   and x :: "'d"   and y :: "'d" assumes "Idomain R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "x \<noteq> \<zero>\<^bsub>R\<^esub>"   and "y \<noteq> \<zero>\<^bsub>R\<^esub>" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<noteq> \<zero>\<^bsub>R\<^esub>"
f_borel_Suc_preimage: fixes f :: "bool stream \<Rightarrow> 'b"   and n :: "nat"   and N :: "'b measure"   and x :: "bool stream" assumes "f \<in> F n \<rightarrow>\<^sub>M N"   and "set_discriminating n f N" shows "range (pseudo_proj_True (Suc n)) \<inter> f -` {f x} = pseudo_proj_True n ` f -` {f x} \<union> pseudo_proj_False n ` f -` {f x}"
lens_override_idem: fixes X :: "'e \<Longrightarrow> 'f"   and S :: "'f" assumes "vwb_lens X" shows "S \<triangleleft>\<^bsub>X\<^esub> S = S"
phils_phil: fixes n :: "nat"   and e :: "dining_event"   and s :: "nat list" assumes "0 < n"   and "e \<in> Tr\<^sub>P n s" shows "phil e < n"
assms_2: shows "e \<le> v ; - v\<^sup>T"