hcju/isabelleps huggingface dataset:
phiArg_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'c set"   and phis :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'c list option"   and g :: "'a"   and v :: "'c"   and v' :: "'c" shows "CFG_SSA_wf_base.phiArg \<alpha>n defs phis g v v' \<equiv> \<exists>vs. CFG_SSA_wf_base.phi \<alpha>n defs phis g v = Some vs \<and> v' \<in> set vs"
strong_reduction_simulation_and_closures_1: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "strong_reduction_simulation Rel Cal" shows "strong_reduction_simulation (Rel\<^sup>=) Cal"
my_trancl_induct: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set"   and P :: "'b \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>+"   and "P a"   and "\<And>x y. \<lbrakk>(x, y) \<in> r; P x\<rbrakk> \<Longrightarrow> P y" shows "P b"
d_fixpoint: fixes x :: "'a" shows "(\<exists>y. x = d y) = (x = d x)"
in_these_eq: fixes x :: "'a"   and A :: "'a option set" shows "(x \<in> Option.these A) = (Some x \<in> A)"
map_index_congL: fixes xs :: "'b list"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<forall>p<length xs. f p (xs ! p) = xs ! p" shows "map_index f xs = xs"
delay_bisimulation_obs_locale: fixes t :: "'t" shows "delay_bisimulation_obs (r1 t) (r2 t) (bisim t) ta_bisim_bisim_syntax \<tau>move1 \<tau>move2"
assms: shows "\<not> the (rt1 ip) \<sqsubseteq> the (rt2 ip)"
set_restrict_spmf: fixes p :: "'d spmf"   and A :: "'d set" shows "set_spmf (p \<upharpoonleft> A) = set_spmf p \<inter> A"
valid_path_valid_swap: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "valid_path (\<lambda>x. (f x, g x))" shows "valid_path (prod.swap \<circ> (\<lambda>x. (f x, g x)))"
ZF2m: fixes T :: "(ZF, ZF) Axioms"   and A :: "ZF"   and e :: "(ZF, ZF) Expression"   and B :: "ZF" assumes "ZFAxioms T" shows "ZF2m T (m2ZF (TermEquivCl' T A e B)) = TermEquivCl' T A e B"
disc_market_pos_stock_axioms: shows "disc_market_pos_stock M F Mkt risk_free_asset"
inverts_components: fixes a :: "'c" assumes "C.ide a" shows "D.inverse_arrows (\<mu> a) (\<mu>'.map a)"
assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "min (min a b) c = min a (min b c)"
vec_rel_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'c vec"   and B :: "'d vec" shows "vec_rel R A B \<equiv> dim_vec A = dim_vec B \<and> (\<forall>i<dim_vec B. R (A $ i) (B $ i))"
assms: shows "ennreal w \<le> weight \<Gamma> x"
eventually_nhds_iff_sequentially: fixes "open" :: "'c set \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "class.first_countable_topology open" shows "eventually P (topological_space.nhds open a) = (\<forall>f. filterlim f (topological_space.nhds open a) sequentially \<longrightarrow> (\<forall>\<^sub>F n in sequentially. P (f n)))"
abs_fun_def: fixes a :: "'a"   and x :: "'b" shows "[a].x \<equiv> \<lambda>b. if b = a then nSome x else if b \<sharp> x then nSome ([(a, b)] \<bullet> x) else nNone"
tendsto_enn2erealI: fixes f :: "'a \<Rightarrow> ennreal"   and l :: "ennreal"   and F :: "'a filter" assumes "(f \<longlongrightarrow> l) F" shows "((\<lambda>i. enn2ereal (f i)) \<longlongrightarrow> enn2ereal l) F"
cgenideal_ideal: fixes R :: "('e, 'f) ring_scheme"   and a :: "'e" assumes "cring R"   and "a \<in> carrier R" shows "ideal (PIdl\<^bsub>R\<^esub> a) R"
omega_sum_length_less: fixes ms :: "nat list"   and ns :: "nat list" assumes "length ms < length ns"   and "normal ns" shows "omega_sum ms < omega_sum ns"
idem: fixes a :: "'a upred" shows "(a \<and> a) = a"
simps_4: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Dom (t \<^bold>\<cdot> u) = Dom u"
pre_convr: fixes p :: "('b, 'd) uexpr" shows "\<lceil>p\<rceil>\<^sub><\<^sup>- = \<lceil>p\<rceil>\<^sub>>"
rtranclp_trans: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "r\<^sup>*\<^sup>* x y"   and "r\<^sup>*\<^sup>* y z" shows "r\<^sup>*\<^sup>* x z"
hom_compose: fixes G :: "('c, 'd) monoid_scheme"   and h :: "'c \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme"   and i :: "'e \<Rightarrow> 'g"   and I :: "('g, 'h) monoid_scheme" assumes "Group.group G"   and "h \<in> hom G H"   and "i \<in> hom H I" shows "compose (carrier G) i h \<in> hom G I"
S_transform_formula_eqvt: fixes p :: "perm"   and x :: "('a, 'c, 'b) formula" shows "p \<bullet> S_transform_formula x = S_transform_formula (p \<bullet> x)"
minGraphProps12: fixes g :: "graph"   and f :: "face"   and a :: "nat"   and b :: "nat" assumes "minGraphProps g"   and "f \<in> \<F> g"   and "(a, b) \<in> \<E> f" shows "(b, a) \<notin> \<E> f"
zeroes: fixes f :: "(real, 'n) vec \<Rightarrow> (real, 'n) vec"   and i :: "'n" assumes "linear f"   and "\<And>x. f x $ i = 0" shows "P f"
induct: fixes P :: "'e prot_atom \<Rightarrow> bool"   and prot_atom :: "'e prot_atom" assumes "\<And>x. P (prot_atom.Atom x)"   and "P Value"   and "P SetType"   and "P AttackType"   and "P Bottom"   and "P OccursSecType" shows "P prot_atom"
alphaBoundOutputChain'': fixes xvec :: "name list"   and yvec :: "name list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" assumes "length xvec = length yvec"   and "yvec \<sharp>* N"   and "yvec \<sharp>* P"   and "yvec \<sharp>* xvec"   and "distinct yvec" shows "\<lparr>\<nu>*xvec\<rparr>N \<prec>' P = \<lparr>\<nu>*yvec\<rparr>([xvec yvec] \<bullet>\<^sub>v N) \<prec>' ([xvec yvec] \<bullet>\<^sub>v P)"
compute_butlast_\<ff>s_def: fixes s :: "'a list" shows "compute_butlast_\<ff>s s = (let \<ff>s = replicate (length s) 0; i = 0; j = 1 in WHILE\<^sub>T\<^bsup>I_out_cb (butlast s)\<^esup> (\<lambda>(b, i, j). j < length b) (\<lambda>(\<ff>s, i, j). ASSERT (j < length \<ff>s) \<bind> (\<lambda>_. WHILE\<^sub>T\<^bsup>I_in_cb (butlast s) j\<^esup> (\<lambda>i. 0 < i \<and> s ! (i - 1) \<noteq> s ! (j - 1)) (\<lambda>i. ASSERT (i - 1 < length \<ff>s) \<bind> (\<lambda>_. Let (\<ff>s ! (i - 1)) RETURN)) i \<bind> (\<lambda>i. let i = i + 1 in ASSERT (j < length \<ff>s) \<bind> (\<lambda>_. let \<ff>s = \<ff>s[j := i]; j = j + 1 in RETURN (\<ff>s, i, j))))) (\<ff>s, i, j) \<bind> (\<lambda>(\<ff>s, uu_, uu_). RETURN \<ff>s))"
length_seqs: fixes j0 :: "nat"   and j :: "nat"   and K :: "nat set" shows "length (seqs j0 j K) = Suc j"
full_def: fixes \<sigma> :: "'c" shows "full \<sigma> \<equiv> \<forall>d. intT \<sigma> d"
from_nat_to_nat_id: fixes x :: "'a" shows "mod_type_class.from_nat (mod_type_class.to_nat x) = x"
spanBC: shows "span (B \<union> C) = span (B \<union> {u})"
hyps_2: shows "m \<notin> keys (mapping_of p1)"
add_divide_eq_if_simps_5: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and z :: "'b"   and b :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "minus (divide a z) b = (if z = zero then uminus b else divide (minus a (times b z)) z)"
phiArg_def: fixes g :: "'g"   and v :: "'val"   and v' :: "'val" shows "step.phiArg g v v' \<equiv> \<exists>vs. step.phi g v = Some vs \<and> v' \<in> set vs"
hext_typeof_mono: fixes h :: "heap"   and h' :: "heap"   and v :: "val"   and T :: "ty" assumes "h \<unlhd> h'"   and "typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>" shows "typeof\<^bsub>h'\<^esub> v = \<lfloor>T\<rfloor>"
sorted_sort: fixes xs :: "'a list" shows "sorted (map f (local.sort xs))"
term_nchotomy: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set" assumes "good X" shows "(\<exists>xs x. X = Var xs x) \<or> (\<exists>delta inp binp. goodInp inp \<and> goodBinp binp \<and> X = Op delta inp binp)"
code: fixes x :: "gauss_int" shows "- x = Gauss_Int (- ReZ x) (- ImZ x)"
reachable_def: shows "G.reachable = G.reaches s\<^sub>0"
leq_antisym: fixes x :: "'a spmf"   and y :: "'a spmf" assumes "ord_spmf (=) x y"   and "ord_spmf (=) y x" shows "x = y"
sort_leq_trans: fixes x :: "String.literal set"   and y :: "String.literal set"   and z :: "String.literal set" assumes "sort_leq cs x y"   and "sort_leq cs y z" shows "sort_leq cs x z"
simps_2: fixes \<phi> :: "fmla" shows "max_depth (Neg \<phi>) = Suc (max_depth \<phi>)"
mult_monoms_alt_def: fixes x :: "char list list"   and y :: "char list list" shows "(RETURN \<circ>\<circ> mult_monoms) x y = REC\<^sub>T (\<lambda>f (p, q). case (p, q) of ([], xa) \<Rightarrow> RETURN q | (a # list, []) \<Rightarrow> RETURN p | (a # list, y # q) \<Rightarrow> if a = y then f (list, q) \<bind> (\<lambda>pq. RETURN (a # pq)) else if (a, y) \<in> var_order_rel then f (list, y # q) \<bind> (\<lambda>pq. RETURN (a # pq)) else f (a # list, q) \<bind> (\<lambda>pq. RETURN (y # pq))) (x, y)"
continuous_attains_inf: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "compact s"   and "s \<noteq> {}"   and "continuous_on s f" shows "\<exists>x\<in>s. \<forall>y\<in>s. f x \<le> f y"
cont: fixes z :: "real" assumes "\<bar>z - x\<bar> \<le> d" shows "isCont f z"
fbrelation_fconverse: shows "fbrelation (r\<inverse>\<^sub>\<bullet>)"
conic_sgn_def: fixes p :: "proj2" shows "conic_sgn p \<equiv> sgn (proj2_rep p \<bullet> (M *v proj2_rep p))"
init_pc_def: fixes pd :: "proc_decl" shows "init_pc pd \<equiv> \<lparr>command = body pd, state = \<lparr>local_state.variables = init_valuation (local_vars pd)\<rparr>\<rparr>"
sum_diff1: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and a :: "'c" assumes "finite A" shows "sum f (A - {a}) = (if a \<in> A then sum f A - f a else sum f A)"
eq_class_of_def: fixes S :: "('c, 'd) eq_object_scheme"   and x :: "'c" shows "class_of\<^bsub>S\<^esub> x = {y \<in> carrier S. x .=\<^bsub>S\<^esub> y}"
restricted_subformulas_subset': fixes \<phi> :: "'b ltln" shows "restricted_subformulas \<phi> \<subseteq> restricted_subformulas_inner \<phi>"
k_def: shows "k \<equiv> \<lambda>p. frag_extend (\<lambda>f. chain_map (Suc p) f (subd p (frag_of (restrict id (standard_simplex p)))))"
Body1: fixes G :: "'b triple set"   and P :: "pname \<Rightarrow> 'b assn"   and Q :: "pname \<Rightarrow> 'b assn"   and Procs :: "pname set"   and pn :: "pname" assumes "G \<union> (\<lambda>p. {P p}. BODY p .{Q p}) ` Procs||-(\<lambda>p. {P p}. the (body p) .{Q p}) ` Procs"   and "pn \<in> Procs" shows "G|-{P pn}. BODY pn .{Q pn}"
card_partitions_atmost_k_parts: fixes n :: "nat"   and k :: "nat" shows "card {p. p partitions n \<and> sum p {..n} \<le> k} = Partition (n + k) k"
nth_take: fixes i :: "nat"   and n :: "nat"   and xs :: "'b list" assumes "i < n" shows "xs \<down> n ! i = xs ! i"
fref_def: fixes P :: "'f \<Rightarrow> bool"   and R :: "('d \<times> 'f) set"   and S :: "('e \<times> 'g) set" shows "[P]\<^sub>f R \<rightarrow> S \<equiv> {(f, g). \<forall>x y. P y \<and> (x, y) \<in> R \<longrightarrow> (f x, g y) \<in> S}"
augment_edge_rev_same: fixes v :: "nat"   and u :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'capacity"   and \<Delta> :: "'capacity" assumes "(v, u) \<in> E" shows "augment_edge f (u, v) \<Delta> (v, u) = f (v, u) - \<Delta>"
fls_shift_delta: fixes m :: "int"   and a :: "int"   and b :: "'a" shows "fls_shift m (Abs_fls (\<lambda>n. if n = a then b else (0::'a))) = Abs_fls (\<lambda>n. if n = a - m then b else (0::'a))"
s_UnauthI: fixes v :: "Syntax.term"   and \<pi> :: "Syntax.term list" assumes "value v" shows "\<lless>\<pi>, Unauth v\<ggreater> I\<rightarrow> \<lless>\<pi>, v\<ggreater>"
evaluate_clock_mono_1: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and v :: "v"   and pes :: "(pat \<times> exp0) list"   and v' :: "v"   and s' :: "'b state"   and r1 :: "(v, v) result" assumes "evaluate_match ck env s v pes v' (s', r1)" shows "clock s' \<le> clock s"
entails_def: fixes N1 :: "'f set"   and N2 :: "'f set" shows "(N1 \<Turnstile>Q N2) = (\<forall>q\<in>Q. entails_q q N1 N2)"
prime_nat_not_dvd: fixes p :: "nat"   and n :: "nat" assumes "prime p"   and "n < p"   and "n \<noteq> 1" shows "\<not> n dvd p"
component_connected: fixes g :: "'a ugraph"   and r :: "'a" shows "Undirected_Graph.connected (component_of g r)"
assms_1: shows "prefs_from_table_wf agents alts xss"
ring_def: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" shows "ring R addition multiplication zero unit \<equiv> abelian_group R addition zero \<and> Group_Theory.monoid R multiplication unit \<and> ring_axioms R addition multiplication"
axioms_1: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "total_ordered_set A less_eq" shows "total_quasi_ordered_set A less_eq"
convex_on_def: fixes S :: "'c set"   and f :: "'c \<Rightarrow> real" shows "convex_on S f = (\<forall>x\<in>S. \<forall>y\<in>S. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)"
index_smult_vec_2: fixes a :: "'a"   and v :: "'a Matrix.vec" shows "dim_vec (a \<cdot>\<^sub>v v) = dim_vec v"
merge_length1_nonempty: fixes as :: "nat list list"   and bs :: "nat list list"   and us :: "nat list list"   and vs :: "nat list list" assumes "merge as bs us vs"   and "as \<in> lists (- {[]})" shows "us \<in> lists (- {[]})"
tautology_selection: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "K .\<bottom>. \<phi> = {}" shows "\<gamma> K \<phi> = {K}"
ran_f: shows "\<R>\<^sub>\<circ> f = F"
single_def: fixes x :: "'b" shows "Predicate.single x = pred.Pred ((=) x)"
goodBinp_def: fixes binp :: "'i \<Rightarrow> ('j, 'k, 'varSort, 'var, 'l) qAbs set option" shows "goodBinp binp \<equiv> liftAll goodAbs binp \<and> |{i. binp i \<noteq> None}| <o |UNIV|"
incseq_le: fixes X :: "nat \<Rightarrow> 'e"   and L :: "'e"   and n :: "nat" assumes "incseq X"   and "X \<longlonglongrightarrow> L" shows "X n \<le> L"
act3: fixes X :: "'a set"   and p :: "'b" shows "\<alpha> (\<Squnion>X) p = \<Squnion>{\<alpha> x p |x. x \<in> X}"
IH_1: fixes n :: "nat"   and m :: "nat" shows "cut_bound n m F1 \<le> cut_bound' (max n m) F1"
double_diff: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "A \<subseteq> B"   and "B \<subseteq> C" shows "B - (C - A) = A"
bernoulli_stream_pseudo_prob_rec': fixes p :: "real"   and M :: "bool stream measure"   and n :: "nat"   and x :: "bool stream" assumes "infinite_coin_toss_space p M"   and "pseudo_proj_True n x = x" shows "Sigma_Algebra.measure M (pseudo_proj_True (Suc n) -` {x}) = Sigma_Algebra.measure M (pseudo_proj_True n -` {x}) * prob_component p x n"
product_part_in_factors: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "O x y"   and "P z (x \<otimes> y)" shows "P z x \<and> P z y"
choose_subspace_of_subspace: fixes n :: "nat"   and S :: "('c, 'd) vec set" assumes "n \<le> vec.dim S" obtains T :: "('c, 'd) vec set" where "vec.subspace T" and "T \<subseteq> vec.span S" and "vec.dim T = n"
cr_ord: shows "cr x ord \<in>\<^sub>i ci ord"
horizontal_composite: 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 \<sigma> :: "'a \<Rightarrow> 'b"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H :: "'b \<Rightarrow> 'd"   and K :: "'b \<Rightarrow> 'd"   and \<tau> :: "'b \<Rightarrow> 'd" assumes "natural_transformation A B F G \<sigma>"   and "natural_transformation B C H K \<tau>" shows "natural_transformation A C (H \<circ> F) (K \<circ> G) (\<tau> \<circ> \<sigma>)"
distr_M_component: fixes i :: "nat" shows "distr M (count_space UNIV) (\<lambda>f. f i) = measure_pmf (bernoulli_pmf p)"
vsv_vrange_vempty: assumes "\<R>\<^sub>\<circ> r = []\<^sub>\<circ>" shows "r = []\<^sub>\<circ>"
lunit_in_hom: fixes a :: "'d" assumes "D.ide a" shows "\<guillemotleft>\<l>\<^sub>D[a] : \<I>\<^sub>D \<otimes>\<^sub>D a \<rightarrow>\<^sub>D a\<guillemotright>"
bracket_compat: fixes x :: "nat itrm"   and n :: "nat"   and i :: "nat" assumes "set (opaque x) \<subseteq> {0..<n}"   and "i < n" shows "term_bracket i (unlift_vars n x) = map_option (unlift_vars n) (itrm_bracket i x)"
insert_sort_by_set: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a list" shows "set (insert_sort_by cmp l) = set l"
Operator_works: fixes t :: "nat"   and k :: "nat" assumes "valid_state_var (Operator t k)" shows "dimacs_to_var (var_to_dimacs (Operator t k)) = Operator t k"
image_mset_If_eq_notin: fixes C :: "'c"   and A :: "'c multiset"   and f :: "'e \<Rightarrow> 'd"   and a :: "'c \<Rightarrow> 'e"   and b :: "'c \<Rightarrow> 'e" assumes "C \<notin># A" shows "{#f (if x = C then a x else b x). x \<in># A#} = {#f (b x). x \<in># A#}"
arith_extra_simps_6: fixes a :: "'a" shows "a + (0::'a) = a"
nextElem_vFrom_suc1: fixes f :: "face"   and v :: "nat"   and i :: "nat"   and u :: "nat" assumes "distinct (vertices f)"   and "v \<in> \<V> f"   and "i < |vertices f|"   and "last (verticesFrom f v) \<noteq> u"   and "verticesFrom f v ! i = u" shows "f \<bullet> u = verticesFrom f v ! Suc i"
has_derivative_mult_right: fixes g :: "'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and x :: "'b" assumes "(g has_derivative g') F" shows "((\<lambda>xa. x * g xa) has_derivative (\<lambda>xa. x * g' xa)) F"
intT_proj: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<sigma> :: "'c"   and a :: "univ" assumes "ModelIkMcalc wtFsym wtPsym arOf resOf parOf \<Phi> infTp intT intF intP" shows "intT \<sigma> (ModelIkMcalc.proj intT \<sigma> a)"
continuous_map_iff_continuous2: fixes g :: "'c \<Rightarrow> 'd" shows "continuous_map euclidean euclidean g = continuous_on UNIV g"
card_UNIV_expr_def: shows "card_UNIV_class.card_UNIV = Phantom(expr) 0"
distinct_5: fixes x1 :: "nat" shows "At x1 \<noteq> Em"
simps_4: fixes \<xi> :: "'a ltln"   and \<xi>' :: "'a ltln" shows "suspendable (\<xi> or\<^sub>n \<xi>') = (suspendable \<xi> \<and> suspendable \<xi>')"
clI: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> carrier"   and "rank_of (insert x X) = rank_of X" shows "x \<in> cl X"
graph_union_idemp_2: fixes A :: "('a, 'b) labeled_graph"   and B :: "('a, 'b) labeled_graph" shows "graph_union A (graph_union A B) = graph_union A B"
inject_2: fixes x31 :: "('a, 'b) MWLfCom"   and x32 :: "('a, 'b) MWLfCom"   and y31 :: "('a, 'b) MWLfCom"   and y32 :: "('a, 'b) MWLfCom" shows "(x31;x32 = y31;y32) = (x31 = y31 \<and> x32 = y32)"
infsuff_def: fixes A :: "'b set"   and s :: "'b llist" shows "infsuff A s \<equiv> {r \<in> A\<^sup>\<omega>. s \<le> r}"
lax_aux: fixes X :: "'b set"   and x :: "'b"   and y :: "'b" assumes "\<forall>x. \<forall>y\<in>X. x \<rightarrow> y \<in> X"   and "\<forall>x. \<forall>y\<in>X. y \<leftarrow> x \<in> X" shows "\<Sqinter> {z \<in> X. x \<le> z} \<cdot> \<Sqinter> {z \<in> X. y \<le> z} \<le> \<Sqinter> {z \<in> X. x \<cdot> y \<le> z}"
Repl: fixes b :: "ZF"   and A :: "ZF"   and f :: "ZF \<Rightarrow> ZF" shows "b |\<in>| Repl A f = (\<exists>a. a |\<in>| A \<and> b = f a)"
right_total_Ex_transfer: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "right_total A" shows "((A ===> (=)) ===> (=)) (Bex (Collect (Domainp A))) Ex"
simps_17: fixes v :: "'a local.term" shows "Diag \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[v\<^bold>] = False"
borel_measurable_lebesgue_integral: fixes f :: "'a \<Rightarrow> val \<Rightarrow> 'b"   and N :: "'a measure"   and i :: "nat" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M stock_measure (\<Gamma> i))" shows "(\<lambda>x. integral\<^sup>L (stock_measure (\<Gamma> i)) (f x)) \<in> borel_measurable N"
backlinks_in_nodes_G: fixes G :: "'b list_graph" assumes "fst ` set (edgesL G) \<subseteq> set (nodesL G)"   and "snd ` set (edgesL G) \<subseteq> set (nodesL G)" shows "fst ` set (edgesL (FiniteListGraph.undirected G)) \<subseteq> set (nodesL (FiniteListGraph.undirected G)) \<and> snd ` set (edgesL (FiniteListGraph.undirected G)) \<subseteq> set (nodesL (FiniteListGraph.undirected G))"
disc_filtr_prob_space_def: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure" shows "disc_filtr_prob_space M F \<equiv> prob_space M \<and> disc_filtr_prob_space_axioms M F"
ext_inject: fixes getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a"   and getSATFlag' :: "ExtendedBool"   and getF' :: "Literal list list"   and getM' :: "(Literal \<times> bool) list"   and getConflictFlag' :: "bool"   and getConflictClause' :: "nat"   and getQ' :: "Literal list"   and getReason' :: "Literal \<Rightarrow> nat option"   and getWatch1' :: "nat \<Rightarrow> Literal option"   and getWatch2' :: "nat \<Rightarrow> Literal option"   and getWatchList' :: "Literal \<Rightarrow> nat list"   and getC' :: "Literal list"   and getCl' :: "Literal"   and getCll' :: "Literal"   and getCn' :: "nat"   and more' :: "'a" shows "(\<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = \<lparr>getSATFlag = getSATFlag', getF = getF', getM = getM', getConflictFlag = getConflictFlag', getConflictClause = getConflictClause', getQ = getQ', getReason = getReason', getWatch1 = getWatch1', getWatch2 = getWatch2', getWatchList = getWatchList', getC = getC', getCl = getCl', getCll = getCll', getCn = getCn', \<dots> = more'\<rparr>) = (getSATFlag = getSATFlag' \<and> getF = getF' \<and> getM = getM' \<and> getConflictFlag = getConflictFlag' \<and> getConflictClause = getConflictClause' \<and> getQ = getQ' \<and> getReason = getReason' \<and> getWatch1 = getWatch1' \<and> getWatch2 = getWatch2' \<and> getWatchList = getWatchList' \<and> getC = getC' \<and> getCl = getCl' \<and> getCll = getCll' \<and> getCn = getCn' \<and> more = more')"
maintained_then_fin_maintained: fixes R :: "('e, 'f) labeled_graph \<times> ('e, 'f) labeled_graph"   and G :: "('e, 'g) labeled_graph" assumes "maintained R G" shows "fin_maintained R G"
sum_expand_lemma: fixes n :: "nat" shows "(\<Sum>d = 1..n. (- 1) ^ (d + 1) * psi (n div d)) = (\<Sum>d = 1..n. (if odd (n div d) then 1 else 0) * mangoldt d)"
assms_5: fixes x :: "'b"   and y :: "'b" assumes "g x"   and "g y" shows "g' x = g' y"
P\<^sub>1_lower_bound_card: fixes P :: "'a set set"   and P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and S :: "'a set"   and L :: "'a set" assumes "bp P"   and "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 (S \<union> L)"   and "bij_exists P\<^sub>1 (\<Union> (P\<^sub>2 \<union> wrap B\<^sub>2))" shows "card P\<^sub>1 + 1 \<le> card P"
ideal_field_cases: fixes B :: "'c set" obtains "ideal B = {0::'c}"   | "ideal B = UNIV"
bigo_plus_subset2: fixes A :: "('c \<Rightarrow> 'd) set"   and f :: "'c \<Rightarrow> 'd"   and B :: "('c \<Rightarrow> 'd) set" assumes "A \<subseteq> O(f)"   and "B \<subseteq> O(f)" shows "A + B \<subseteq> O(f)"
summable_geometric_iff: fixes c :: "'c" shows "summable ((^) c) = (norm c < 1)"
embed_inj_on: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and f :: "'d \<Rightarrow> 'e" assumes "Well_order r"   and "embed r r' f" shows "inj_on f (Field r)"