hcju/isabelleps huggingface dataset:
is_equivalence_pseudofunctor: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" assumes "pseudofunctor_pseudonaturally_equivalent_to_equivalence_pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1" shows "equivalence_pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G"
simps_2: fixes a :: "'a"   and ts :: "'a FOL_Fitting.term list" shows "liftt (App a ts) = App a (liftts ts)"
tempD: fixes F :: "behavior \<Rightarrow> bool"   and sigma :: "behavior" assumes "\<turnstile> F" shows "F sigma"
Eq\<^sub>1I: fixes F :: "\<Pi>\<^sub>1"   and G :: "\<Pi>\<^sub>1"   and v :: "i" assumes "F = G" shows "[F \<^bold>=\<^sub>1 G in v]"
inner_axis: fixes x :: "('b, 'c) Finite_Cartesian_Product.vec"   and i :: "'c"   and y :: "'b" shows "x \<bullet> axis i y = x $h i \<bullet> y"
from_nat_into_to_nat_on: fixes A :: "'c set"   and a :: "'c" assumes "countable A"   and "a \<in> A" shows "from_nat_into A (to_nat_on A a) = a"
distinct_quicksort_acc: fixes ac :: "'a list"   and xs :: "'a list" shows "distinct (quicksort_acc ac xs) = distinct (ac @ xs)"
amin_def: fixes x :: "ant"   and y :: "ant" shows "amin x y = (if x \<le> y then x else y)"
cond_conj_distr: fixes P :: "'b upred"   and b :: "'b upred"   and Q :: "'b upred"   and S :: "'b upred" shows "(P \<and> Q \<triangleleft> b \<triangleright> S) = (P \<and> Q) \<triangleleft> b \<triangleright> (P \<and> S)"
hyperpow_one: fixes r :: "'a star" shows "r pow 1 = r"
openin_subtopology_diff_closed: fixes S :: "'c set"   and X :: "'c topology"   and T :: "'c set" assumes "S \<subseteq> topspace X"   and "closedin X T" shows "openin (subtopology X S) (S - T)"
degree_of_X_plus: fixes a :: "'a" assumes "a \<in> carrier R"   and "\<one> \<noteq> \<zero>" shows "deg R (X_poly_plus R a) = 1"
sum_indicator_eq_card: fixes A :: "'a set"   and B :: "'a set" assumes "finite A" shows "sum (chi B) A = card (A \<inter> B)"
cases: fixes a1 :: "'node"   and a2 :: "'var option"   and a3 :: "'edge list"   and a4 :: "'node" assumes "DynPDG_edge a1 a2 a3 a4" obtains n :: "'node" and n' :: "'node" and as :: "'edge list" where "a1 = n__" and "a2 = None" and "a3 = asa__" and "a4 = n'a__" and "n__ controls n'a__ via asa__"   | n :: "'node" and V :: "'var" and n' :: "'node" and as :: "'edge list" where "a1 = n" and "a2 = Some V" and "a3 = as" and "a4 = n'" and "n influences V in n' via as"
L: fixes u :: "nat \<Rightarrow> 'a" assumes "u \<longlonglongrightarrow> z" shows "limsup (\<lambda>n. ereal (dist x (u n) - dist y (u n))) = ereal (dist x z - dist y z)"
map_comp_None_iff: fixes m1 :: "'d \<Rightarrow> 'c option"   and m2 :: "'e \<Rightarrow> 'd option"   and k :: "'e" shows "((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None))"
le_bound_of: fixes f :: "'a \<Rightarrow> real"   and x :: "'a" assumes "Expectations.bounded f" shows "f x \<le> bound_of f"
d_3: shows "ennreal (1 - e) < emeasure M {x \<in> space M. \<forall>N. real (card {n \<in> {..<N}. \<exists>l\<in>{1..n}. u n ((Tinv ^^ n) x) - u (n - l) ((Tinv ^^ (n - l)) x) \<le> - delta l * real l}) \<le> e * real N}"
nonregular_event_induces_snapshot: fixes c :: "('a, 'b, 'c) configuration"   and p :: "'a"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration" assumes "\<not> ps c p \<noteq> None"   and "c \<turnstile> ev \<mapsto> c'"   and "occurs_on ev = p"   and "\<not> regular_event ev" shows "\<not> regular_event ev \<longrightarrow> ps c' p \<noteq> None"
filterlim_iff: fixes f :: "'b \<Rightarrow> 'c"   and F2 :: "'c filter"   and F1 :: "'b filter" shows "filterlim f F2 F1 = (\<forall>P. eventually P F2 \<longrightarrow> (\<forall>\<^sub>F x in F1. P (f x)))"
typ_of_Abs_body_typ': fixes Ts :: "typ list"   and T :: "typ"   and t :: "Core.term"   and ty :: "typ" assumes "typ_of1 Ts (Abs T t) = Some ty" shows "\<exists>rty. ty = T \<rightarrow> rty \<and> typ_of1 (T # Ts) t = Some rty"
openin_components_locally_connected: fixes S :: "'b set"   and c :: "'b set" assumes "locally connected S"   and "c \<in> components S" shows "openin (top_of_set S) c"
ex_infinite_decides_finite: fixes M :: "nat set"   and S :: "nat set"   and \<F> :: "nat set set" assumes "infinite M"   and "finite S" obtains N :: "nat set" where "N \<subseteq> M" and "infinite N" and "\<And>T. T \<subseteq> S \<Longrightarrow> decides \<F> T N"
gc1_def: shows "gc1 = gcd pR1 pI1"
InitPsi_def: shows "InitPsi \<equiv> pc1 = #a \<and> pc2 = #a \<and> x = #0 \<and> y = #0 \<and> sem = #1"
abs_ltln\<^sub>Q_def: shows "abs_ltln\<^sub>Q \<equiv> quot_type.abs (\<sim>\<^sub>Q) Abs_ltln\<^sub>Q"
simps: fixes a1 :: "nat \<Rightarrow> classDef option"   and a2 :: "nat list"   and a3 :: "nat list" shows "a1 \<turnstile>+ a2 <: a3 = ((\<exists>CT. a1 = CT \<and> a2 = [] \<and> a3 = []) \<or> (\<exists>CT C0 D0 Cs Ds. a1 = CT \<and> a2 = C0 # Cs \<and> a3 = D0 # Ds \<and> CT \<turnstile> C0 <: D0 \<and> CT \<turnstile>+ Cs <: Ds))"
ur3: shows "poly (real_of_int_poly p) (real_of_1 (p, l, r)) = 0"
simps_2: fixes x :: "quat"   and y :: "quat" shows "Im1 (x + y) = Im1 x + Im1 y"
ln2_sum: shows "ln 2 = ln (1 / 2 + 1) + ln (1 / 3 + 1)"
diff_strict_mono: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and d :: "'d"   and c :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less a b"   and "less d c" shows "less (minus a c) (minus b d)"
diff: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "Modules.additive f" shows "f (x - y) = f x - f y"
oexp_less_iff: fixes \<alpha> :: "V"   and \<beta> :: "V"   and \<gamma> :: "V" assumes "0 < \<alpha>"   and "Ord \<alpha>"   and "Ord \<beta>"   and "Ord \<gamma>" shows "(\<alpha> \<up> \<beta> < \<alpha> \<up> \<gamma>) = (\<beta> \<in> elts \<gamma> \<and> 1 < \<alpha>)"
simps_1: fixes P :: "'c \<Rightarrow> bool" shows "dropWhile P [] = []"
dghm_dghm_comp_dghm_id_left: shows "dghm_id \<BB> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF> = \<FF>"
map_eq_append_conv: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and ys :: "'c list"   and zs :: "'c list" shows "(map f xs = ys \<cdot> zs) = (\<exists>us vs. xs = us \<cdot> vs \<and> ys = map f us \<and> zs = map f vs)"
elims_2: fixes x :: "'c formula" assumes "is_nnf x" obtains F :: "'c formula" and G :: "'c formula" where "x = Fa__ \<^bold>\<and> G__" and "is_nnf Fa__ \<and> is_nnf G__"   | F :: "'c formula" and G :: "'c formula" where "x = F \<^bold>\<or> G" and "is_nnf F \<and> is_nnf G"   | v :: "'c" where "x = Atom va__" and "is_lit_plus (Atom va__)"   | "x = \<bottom>" and "is_lit_plus \<bottom>"   | v :: "'c formula" where "x = \<^bold>\<not> v" and "is_lit_plus (\<^bold>\<not> v)"
orthogonal_clauses_7: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b"   and a :: "'b"   and c :: "real" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner"   and "real_inner.orthogonal inner x a" shows "real_inner.orthogonal inner (scaleR c x) a"
pnop_n: shows "n \<in>\<^sub>\<circ> \<omega>"
image_set_diff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" assumes "inj f" shows "f ` (A - B) = f ` A - f ` B"
sorted_rev_subset_len_lt: fixes a :: "'a list"   and b :: "'a list" assumes "Sorted_Less.sorted (rev a)"   and "Sorted_Less.sorted (rev b)"   and "set a \<subset> set b" shows "length a < length b"
reds: shows "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>1,b\<^sub>1\<rangle> \<rightarrow>* \<langle>e\<^sub>2,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"
res_pow_eq: fixes x :: "int"   and n :: "nat" shows "x [^] n = x ^ n mod int p"
P_def: shows "P \<equiv> \<Sum>j = 0..p. smult (b j) (Bernstein_Poly j p c d)"
append_eq_first_pref_spec: fixes s :: "'a list"   and t :: "'a list"   and r :: "'a list"   and x :: "'a" assumes "s @ t = r @ [x] \<and> t \<noteq> []" shows "s \<le> r"
DisjointPathsPlusOne_axioms: shows "DisjointPathsPlusOne G v0 v1 paths P_new"
simE2_2: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi"   and Q :: "pi"   and \<alpha> :: "freeRes"   and Q' :: "pi" assumes "\<And>R S. (R, S) \<in> Rel \<Longrightarrow> R \<leadsto><Rel> S"   and "eqvt Rel"   and "(P, Q) \<in> Rel"   and "Q \<Longrightarrow>\<^sup>^\<alpha> \<prec> Q'" shows "\<exists>P'. P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
Derivation_leftmost: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and a :: "symbol list"   and b :: "symbol list" assumes "D \<noteq> []"   and "Derivation a D b" shows "\<exists>i. leftmost i a"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Cod (t \<^bold>\<star> u) = Cod t \<^bold>\<star> Cod u"
supp_restrict0: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "order.greater_eq A (supp (restrict0 f A))"
assms_3: shows "h ` ({0..1} \<times> U) \<subseteq> S"
VAR_imp_eq: fixes m :: "nat"   and n :: "nat" assumes "VAR m \<sim> VAR n" shows "m = n"
is_run_def: fixes s :: "nat \<Rightarrow> 'c" shows "C.is_run s \<equiv> \<exists>l. C.is_lrun l s"
subgame_strategy_stays_in_subgame: fixes V' :: "'a set"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v :: "'a" assumes "ParityGame.strategy (G'.subgame V') p \<sigma>"   and "v \<in> (if p = Even then V0\<^bsub>G'.subgame V'\<^esub> else V\<^bsub>G'.subgame V'\<^esub> - V0\<^bsub>G'.subgame V'\<^esub>)"   and "\<not> \<not> (\<exists>w\<in>V\<^bsub>G'.subgame V'\<^esub>. v \<rightarrow>\<^bsub>G'.subgame V'\<^esub> w)" shows "\<sigma> v \<in> V'"
single_nz_val_dotP: fixes i :: "nat"   and n :: "nat"   and x :: "'a vec"   and v :: "'a" assumes "i < n"   and "dim_vec x = n" shows "single_nz_val n i v \<bullet> x = v * x $ i"
esssup_def: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'c" shows "esssup M f = (if f \<in> borel_measurable M then Limsup (ae_filter M) f else top)"
rem_implicit_pres_ops_valid_plan: 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 \<pi>s :: "char list list" 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" shows "ast_problem.valid_plan (rem_implicit_pres_ops prob) \<pi>s = ast_problem.valid_plan prob \<pi>s"
JN_constant: fixes I :: "'d set"   and c :: "'c program" shows "(\<Squnion>i\<in>I. c) = (if I = {} then \<bottom> else c)"
adj_mk_symmetric_eq: assumes "symmetric H" shows "parcs (mk_symmetric H) = arcs_ends H"
TheComplex_def: shows "\<Sigma> \<equiv> dual_order.PosetComplex \<P>"
subalgebra_sigma_finite: fixes M :: "'b measure"   and N :: "'b measure" assumes "prob_space M"   and "subalgebra M N" shows "sigma_finite_subalgebra M N"
trace_callee_eqI: fixes A :: "'a set"   and callee1 :: "'c \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'c) spmf"   and p :: "'c spmf"   and callee2 :: "'d \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'd) spmf"   and q :: "'d spmf" assumes "\<And>xs x. \<lbrakk>set xs \<subseteq> A \<times> UNIV; x \<in> A\<rbrakk> \<Longrightarrow> trace_callee callee1 p xs x = trace_callee callee2 q xs x" shows "trace_callee_eq callee1 callee2 A p q"
is_tdghmD_7: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<D>\<^sub>\<circ> (\<NN>\<lparr>NTMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>"
orderI: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "a = a \<inter># b" shows "a \<subseteq># b"
Cons: fixes x :: "'b"   and xs :: "'b list" shows "prod_list (x # xs) = x * prod_list xs"
bot_unique: fixes a :: "'a" shows "(a \<le> \<bottom>) = (a = \<bottom>)"
positively_responsiveD: fixes scf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set"   and P :: "'c \<Rightarrow> ('d \<times> 'd) set"   and P' :: "'c \<Rightarrow> ('d \<times> 'd) set"   and x :: "'d"   and y :: "'d" assumes "positively_responsive scf A Is"   and "profile A Is P"   and "profile A Is P'"   and "x \<in> A"   and "y \<in> A"   and "\<And>i. \<lbrakk>i \<in> Is; x \<^bsub>(P i)\<^esub>\<prec> y\<rbrakk> \<Longrightarrow> x \<^bsub>(P' i)\<^esub>\<prec> y"   and "\<And>i. \<lbrakk>i \<in> Is; x \<^bsub>(P i)\<^esub>\<approx> y\<rbrakk> \<Longrightarrow> x \<^bsub>(P' i)\<^esub>\<preceq> y"   and "\<exists>k\<in>Is. x \<^bsub>(P k)\<^esub>\<approx> y \<and> x \<^bsub>(P' k)\<^esub>\<prec> y \<or> y \<^bsub>(P k)\<^esub>\<prec> x \<and> x \<^bsub>(P' k)\<^esub>\<preceq> y"   and "x \<^bsub>(scf P)\<^esub>\<preceq> y" shows "x \<^bsub>(scf P')\<^esub>\<prec> y"
module_hom_def: fixes R :: "('d, 'e) ring_scheme"   and M :: "('d, 'b, 'f) module_scheme"   and N :: "('d, 'c, 'g) module_scheme" shows "module_hom R M N = {f \<in> carrier M \<rightarrow> carrier N. (\<forall>m1 m2. m1 \<in> carrier M \<and> m2 \<in> carrier M \<longrightarrow> f (m1 \<oplus>\<^bsub>M\<^esub> m2) = f m1 \<oplus>\<^bsub>N\<^esub> f m2) \<and> (\<forall>r m. r \<in> carrier R \<and> m \<in> carrier M \<longrightarrow> f (r \<odot>\<^bsub>M\<^esub> m) = r \<odot>\<^bsub>N\<^esub> f m)}"
dist_circle_segment: fixes radius :: "real"   and \<alpha> :: "real"   and \<beta> :: "real"   and center :: "(real, 2) vec" assumes "0 \<le> radius"   and "0 \<le> \<alpha>"   and "\<alpha> \<le> \<beta>"   and "\<beta> \<le> 2 * pi" shows "dist (center + radius *\<^sub>R of_radiant \<alpha>) (center + radius *\<^sub>R of_radiant \<beta>) = 2 * radius * sin ((\<beta> - \<alpha>) / 2)"
refine: fixes n :: "nat"   and Xi :: "real list"   and X :: "'b"   and Mi :: "nat"   and M :: "nat" assumes "DIM_precond TYPE('b) n"   and "(Xi, X) \<in> lv_rel"   and "(Mi, M) \<in> nat_rel" shows "(max_coord_lv_rel n Mi Xi, max_coord M X) \<in> lv_rel"
comp_null_1: fixes f :: "'b" shows "D.null \<cdot>\<^sub>D f = D.null"
select_convs_3: fixes newOID :: "CTypeId \<Rightarrow> nat"   and newAID :: "Arraytype \<Rightarrow> nat"   and vals :: "Location \<Rightarrow> Value"   and more :: "'a" shows "vals \<lparr>newOID = newOID, newAID = newAID, vals = vals, \<dots> = more\<rparr> = vals"
bin_rest_trunc_i: fixes n :: "nat"   and bin :: "int" shows "take_bit n (bin div 2) = take_bit (Suc n) bin div 2"
ts_c_def_1: shows "ts'' = ts_c @ t1s"
IP2: fixes x :: "'d"   and it :: "'d set"   and \<sigma> :: "'e"   and z :: "'e nres" assumes "x \<in> it"   and "it \<subseteq> S"   and "I it \<sigma>"   and "SPEC (\<lambda>f'. c \<sigma> f' \<le> SPEC (I it)) \<le> z" shows "f x \<le> z"
transpose_matrix_add: fixes A :: "'b matrix"   and B :: "'b matrix" shows "transpose_matrix (A + B) = transpose_matrix A + transpose_matrix B"
valid_history_axioms: shows "valid_history program_step ts\<^sub>s\<^sub>b"
disjoint_Aux2: fixes c :: "('c, 'd) sequent"   and A :: "('c, 'd) form"   and B :: "('c, 'd) form"   and S :: "('c, 'd) sequent" assumes "ModalSequents.mset c = \<LM> A  \<RM>"   and "A \<noteq> B"   and "ModalSequents.mset (extend S c) = \<LM> B  \<RM>" shows "False"
preflect_preflect: fixes t0 :: "'b"   and t :: "'b" shows "preflect t0 (preflect t0 t) = t"
AE_E: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b measure" assumes "almost_everywhere M P" obtains N :: "'b set" where "{x \<in> space M. \<not> P x} \<subseteq> N" and "emeasure M N = 0" and "N \<in> sets M"
cod_inv: fixes f :: "'a \<times> 'a" assumes "B.VV.iso f" shows "B.VV.cod (B.VV.inv f) = B.VV.dom f"
map_of_alt_def: fixes xs :: "('a \<times> 'b) list" shows "map_of xs = hash_of_assoc_map xs"
wfp_on_imp_irreflp_on: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "wfp_on P A" shows "irreflp_on P A"
cut_le_empty: fixes t :: "'a" shows "{} \<down>\<le> t = {}"
bind_rpv_Done: fixes f :: "('b, 'c, 'a) rpv" shows "Generative_Probabilistic_Value.Done \<bind> f = f"
assms_1: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[r], Undecided\<rangle> \<Rightarrow> t"
\<phi>'_def: shows "\<phi>' \<equiv> absp \<phi>"
trg_inv: fixes \<mu> :: "'c" assumes "C.iso \<mu>" shows "trg\<^sub>C (C.inv \<mu>) = trg\<^sub>C \<mu>"
left_unique_pcr_spmf_gpv: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "left_unique A" shows "left_unique (pcr_spmf_gpv A)"
map\<^sub>0_def: fixes a :: "'d" shows "F.map\<^sub>0 a \<equiv> src\<^sub>C (F a)"
maxpositions_def: fixes l :: "'b list" shows "maxpositions l = filterpositions2 ((\<le>) (Max (set l))) l"
set_disconnected_nodes_def: shows "set_disconnected_nodes = (\<lambda>document_ptr. put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes_update)"
set_eqD2: fixes R :: "('d, 'e) eq_object_scheme"   and A :: "'d set"   and A' :: "'d set"   and a' :: "'d" assumes "A {.=}\<^bsub>R\<^esub> A'"   and "a' \<in> A'" shows "\<exists>a\<in>A. a' .=\<^bsub>R\<^esub> a"
rangeE: fixes b :: "'k"   and f :: "'l \<Rightarrow> 'k" assumes "b \<in> range f" obtains x :: "'l" where "b = f x"
l2_inv4_def: shows "l2_inv4 \<equiv> {s. \<forall>Ra A B gny. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<longrightarrow> in_progress (progress s Ra) xsk \<longrightarrow> guessed_frame Ra xgny = Some gny \<longrightarrow> guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx)))}"
finite_sysI: assumes "finite \<V>\<^sup>+" shows "finite_incidence_system \<V>\<^sup>+ \<B>\<^sup>+"
Sup_upper2: fixes u :: "'a"   and A :: "'a set"   and v :: "'a" assumes "u \<in> A"   and "v \<le> u" shows "v \<le> \<Squnion> A"
c: fixes c :: "'b" shows "f (K$ c) = cnst c"
module_pair_ow_def: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>M\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>M\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" shows "module_pair_ow U\<^sub>M\<^sub>_\<^sub>1 plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 plus\<^sub>M\<^sub>_\<^sub>2 zero\<^sub>M\<^sub>_\<^sub>2 minus\<^sub>M\<^sub>_\<^sub>2 uminus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2 \<equiv> module_ow U\<^sub>M\<^sub>_\<^sub>1 plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 \<and> module_ow U\<^sub>M\<^sub>_\<^sub>2 plus\<^sub>M\<^sub>_\<^sub>2 zero\<^sub>M\<^sub>_\<^sub>2 minus\<^sub>M\<^sub>_\<^sub>2 uminus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2"
only_owner_enter_normal: fixes s :: "event list"   and r :: "room"   and k :: "key"   and g :: "guest" assumes "hotel s"   and "safe\<^sub>0 s r"   and "(k, roomk s r) \<in> cards s g" shows "owns s r = \<lfloor>g\<rfloor>"
map_disj_common: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option"   and p :: "'c"   and v :: "'d"   and v' :: "'d" assumes "h\<^sub>0 \<bottom> h\<^sub>1"   and "h\<^sub>0 p = Some v"   and "h\<^sub>1 p = Some v'" shows "False"
add_assoc: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd"   and c :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<oplus> b \<oplus> c = a \<oplus> (b \<oplus> c)"
closure_of_alt: fixes X :: "'c topology"   and S :: "'c set" shows "X closure_of S = topspace X \<inter> S \<union> X derived_set_of S"
bisimilar_simulation_step: fixes P :: "'state"   and Q :: "'state"   and \<alpha> :: "'act"   and P' :: "'state" assumes "P \<sim>\<cdot> Q"   and "bn \<alpha> \<sharp>* Q"   and "P \<rightarrow> \<langle>\<alpha>,P'\<rangle>" obtains Q' :: "'state" where "Q \<rightarrow> \<langle>\<alpha>,Q'\<rangle>" and "P' \<sim>\<cdot> Q'"
single_valued_lfp: fixes f :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" assumes "cont f"   and "\<And>r. single_valued r \<Longrightarrow> single_valued (f r)" shows "single_valued (lfp f)"
e0_2: shows "\<forall>y. norm (y - x) < e0 \<longrightarrow> norm (f y - f x - f' (y - x)) \<le> 1 / (2 * B) * norm (y - x)"
self_codomain_iff_ide: fixes a :: "'a arrow_of_spans_data" shows "(a \<in> V.codomains a) = V.ide a"
hby: shows "\<forall>l\<in>C. \<forall>m\<in>C. \<bar>f l - f m\<bar> \<le> y"
simps: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "argmax f A = {x \<in> A. f x = Max (f ` A)}"
SUP_id_eq: fixes Sup :: "'b set \<Rightarrow> 'b"   and A :: "'b set" shows "Sup (id ` A) = Sup A"
ih: assumes "as \<noteq> bs" shows "as \<parallel> bs"
are_all_operators_non_interfering_set_contains_no_distinct_interfering_operator_pairs: fixes ops :: "'b strips_operator list"   and op\<^sub>1 :: "'b strips_operator"   and op\<^sub>2 :: "'b strips_operator" assumes "are_all_operators_non_interfering ops"   and "are_operators_interfering op\<^sub>1 op\<^sub>2"   and "op\<^sub>1 \<noteq> op\<^sub>2" shows "op\<^sub>1 \<notin> set ops \<or> op\<^sub>2 \<notin> set ops"
multlist_perm_cong: fixes as :: "'a list"   and bs :: "'a list" assumes "as <~~> bs"   and "set as \<subseteq> carrier G" shows "foldr (\<otimes>) as \<one> = foldr (\<otimes>) bs \<one>"
6: shows "w\<^sub>2 = w\<^sub>2 @ v\<^sub>2"
gamma2_differentiable: shows "\<forall>x\<in>{0..1}. \<gamma> differentiable at x"
preprocess_part_1_def: fixes l :: "('a \<times> QDelta ns_constraint) list" shows "preprocess_part_1 l \<equiv> let start = start_fresh_variable l; is = preprocess' l start in (Tableau is, Atoms is, UnsatIndices is)"
span_explicit: fixes b :: "'a set" shows "More_Modules.ideal b = {\<Sum>a\<in>t. r a * a |t r. finite t \<and> t \<subseteq> b}"
simps_13: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc"   and \<psi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c \<phi> W\<^sub>c \<psi> = (\<forall>i. suffix i \<xi> \<Turnstile>\<^sub>c \<phi> \<or> (\<exists>j\<le>i. suffix j \<xi> \<Turnstile>\<^sub>c \<psi>))"