hcju/isabelleps huggingface dataset:
i: shows "\<And>s. (PROP P s) \<Longrightarrow> (PROP Q s)"
simps_1: fixes uu :: "'b \<Rightarrow> bool"   and uv :: "bool" shows "has_disc_negated uu uv MatchAny = False"
arr_iff_in_hom: fixes f :: "'c" shows "C.arr f = \<guillemotleft>f : C.dom f \<rightarrow> C.cod f\<guillemotright>"
allE': fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<forall>x. P x" obtains "P x" and "\<forall>x. P x"
Suc_1: fixes lholed :: "Lholed"   and k :: "nat"   and LI :: "e list"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list"   and \<C>' :: "t_context" assumes "Lfilled i lholed (vs @ [$Br (i + k)]) LI"   and "k < length (label \<C>)"   and "label \<C> ! k = tcs"   and "length tcs = length vs"   and "\<S>\<bullet>\<C> \<turnstile> LI : ts _> ts'"   and "const_list vs" shows "\<S>\<bullet>\<C>' \<turnstile> vs : [] _> tcs"
kildall_dom_properties: shows "kildall r f step start \<in> list n A \<and> stables r step (kildall r f step start) \<and> start [\<sqsubseteq>\<^bsub>r\<^esub>] kildall r f step start \<and> (\<forall>ts\<in>list n A. start [\<sqsubseteq>\<^bsub>r\<^esub>] ts \<and> stables r step ts \<longrightarrow> kildall r f step start [\<sqsubseteq>\<^bsub>r\<^esub>] ts)"
apply_operations_def: fixes history :: "nat \<Rightarrow> ('c \<times> 'd) event list"   and interp :: "'d \<Rightarrow> 'e \<Rightarrow> 'e option"   and initial_state :: "'e"   and es :: "('c \<times> 'd) event list" assumes "network_with_ops history" shows "network_with_ops.apply_operations interp initial_state es \<equiv> happens_before.apply_operations (network_with_ops.interp_msg interp) (trivial_network.node_deliver_messages es) initial_state"
evaln_Callee: fixes G :: "prog"   and s :: "st"   and l :: "lname \<Rightarrow> val option"   and e :: "expr"   and n :: "nat"   and v :: "vals"   and s' :: "state" shows "G\<turnstile>Norm s \<midarrow>In1l (Callee l e)\<succ>\<midarrow>n\<rightarrow> (v, s') = False"
tdev_pdevs_of_list: fixes xs :: "'a list" shows "tdev (pdevs_of_list xs) = sum_list (map abs xs)"
b2: shows "\<exists>M. \<forall>n. norm (y n) < M"
cr_envT_stateTD: fixes s :: "'d"   and M :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m1 :: "('d, 'e) envT"   and m2 :: "('d, 'f) stateT" assumes "cr_envT_stateT s M m1 m2" shows "M (run_env m1 s) (run_state m2 s)"
noroot: fixes t :: "real" assumes "t \<in> {x<..r}" shows "f t \<noteq> 0"
Tarski_neutral_dimensionless_axioms: shows "Tarski_neutral_dimensionless Bet Cong"
ctr: fixes s1 :: "'c stream"   and s2 :: "'d stream" shows "szip s1 s2 = (shd s1, shd s2) ## szip (stl s1) (stl s2)"
ad_a: fixes x :: "'a" shows "a (domain_bdlo_monoid.d (1::'a) (\<cdot>) (\<sqinter>) \<top> x) = a x"
Abs_star_inverse: fixes y :: "(nat \<Rightarrow> 'b) set" assumes "y \<in> star" shows "Rep_star (Abs_star y) = y"
take_split: fixes n :: "nat"   and xs :: "'c list"   and m :: "nat" assumes "n \<le> length xs"   and "n \<le> m" shows "take n xs \<cdot> take (m - n) (drop n xs) = take m xs"
simps_1: fixes x :: "nat" shows "free_vars (CVar x) = {x}"
is_empty_def: fixes A :: "'c set" shows "Set.is_empty A = (A = {})"
subformulas\<^sub>\<nu>_list_set: fixes \<phi> :: "'a ltln" shows "set (subformulas\<^sub>\<nu>_list \<phi>) = subformulas\<^sub>\<nu> \<phi>"
prec_st_def: fixes p :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" shows "prec_st p r = - (lft_pred_st (inpt_st r) until - lft_pred_st p)"
power_le_zero_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (power.power one times a n) zero = (0 < n \<and> (odd n \<and> less_eq a zero \<or> even n \<and> a = zero))"
update_cons_cons_env: fixes x :: "'c"   and y :: "'c"   and ys :: "'c list"   and opt :: "('a, 'b, 'c) env option"   and b :: "'b"   and es :: "'c \<Rightarrow> ('a, 'b, 'c) env option" shows "update (x # y # ys) opt (Env b es) = Env b (es(x := case es x of None \<Rightarrow> None | Some e \<Rightarrow> Some (update (y # ys) opt e)))"
abs_eq: shows "bot = Linear_Aggregation_Algebras.unit.Abs_unit 0"
intervalI: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and x :: "'b"   and b :: "'b" assumes "CLF cl f"   and "(a, x) \<in> order cl"   and "(x, b) \<in> order cl" shows "x \<in> interval (order cl) a b"
word_mult_def: fixes a :: "'a word"   and b :: "'a word" shows "a * b = word_of_int (uint a * uint b)"
AbstrLevels_S15_A93: fixes i :: "AbstrLevelsID" assumes "sS15 \<in> AbstrLevel i" shows "sA93 \<notin> AbstrLevel i"
init_NilE: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and b :: "bool"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>INIT C ([],b) \<leftarrow> unit,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "e' = unit \<and> s' = s"
results_gpv_GPV: fixes x :: "'d"   and \<Gamma> :: "('e, 'f) \<I>"   and gpv :: "('d, 'e, ('d, 'e, 'f) rpv) generat spmf" shows "(x \<in> results_gpv \<Gamma> (GPV gpv)) = (Pure x \<in> set_spmf gpv \<or> (\<exists>out c. IO out c \<in> set_spmf gpv \<and> (\<exists>input. input \<in> responses_\<I> \<Gamma> out \<and> x \<in> results_gpv \<Gamma> (c input))))"
STS: shows "closedin (top_of_set (S \<union> T)) S"
slg_rel_def_internal: fixes R :: "('c \<times> 'd) set" shows "slg_rel R \<equiv> (R \<rightarrow> \<langle>R\<rangle>list_set_rel) O br (\<lambda>succs. {(u, v). v \<in> succs u}) (\<lambda>_. True)"
perfect_sec_P2_def: fixes m1 :: "bool \<times> bool \<times> bool \<times> bool"   and m2 :: "bool \<times> bool" shows "OT_14.perfect_sec_P2 m1 m2 \<equiv> R2_14 m1 m2 = funct_OT_14 m1 m2 \<bind> (\<lambda>(s1, s2). S2_14 m2 s2)"
deg_monom: fixes a :: "int"   and n :: "nat" assumes "a \<noteq> \<zero>\<^bsub>R\<^esub>"   and "a \<in> carrier R" shows "deg R (monom (UP R) a n) = n"
assms: shows "wf_mformula \<sigma> j n R \<phi> \<phi>'"
part_def: fixes v :: "'b"   and \<R> :: "'b set set" shows "[v]\<^sub>\<R> \<equiv> THE R. R \<in> \<R> \<and> v \<in> R"
eval_ternary_simps_3_2: fixes T :: "ternaryvalue"   and x :: "ternaryformula" shows "(eval_ternary_And T (ternary_ternary_eval x) = TernaryTrue) = (ternary_ternary_eval x = TernaryTrue \<and> T = TernaryTrue)"
index_vec_of_list: fixes i :: "nat"   and xs :: "'b list" assumes "i < length xs" shows "vec_of_list xs $ i = xs ! i"
sumAssertCases: fixes \<Psi> :: "'b"   and \<phi> :: "'c"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rs :: "('a, 'b, 'c) residual" assumes "\<Psi> \<rhd> P \<oplus>\<^sub>\<phi> Q \<longmapsto> Rs"   and "\<Psi> \<turnstile> \<phi>" obtains "\<Psi> \<rhd> P \<longmapsto> Rs" and "guarded P"   | "\<Psi> \<rhd> Q \<longmapsto> Rs" and "guarded Q"
inj_on_subset: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" assumes "inj_on f A"   and "B \<subseteq> A" shows "inj_on f B"
lcs_comm: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and lcs :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'e"   and t :: "'e" assumes "class.lcs_powerprod plus minus zero lcs" shows "lcs s t = lcs t s"
d'_def: shows "d' = scalar_product (map of_bool x) is + 2 * d"
bind_lzero: fixes f :: "'g \<Rightarrow> 'f option" shows "None \<bind> f = None"
case_prod_Pair_iden: fixes p :: "'a \<times> 'b" shows "(case p of (x, y) \<Rightarrow> (x, y)) = p"
hom_eq_mult_hom_imp: fixes x :: "'a"   and y :: "'a"   and hz :: "'b" assumes "hom x = hom y * hz" shows "\<exists>z. hz = hom z \<and> x = y * z"
"8_3": shows "f T4 \<subseteq> wfvarsoccs\<^sub>s\<^sub>s\<^sub>t ((T1 @ T2) @ T3) \<union> X"
square_freeI: fixes p :: "'c poly" assumes "\<And>q. \<lbrakk>0 < degree q; q \<noteq> 0; q * q dvd p\<rbrakk> \<Longrightarrow> False"   and "p \<noteq> 0" shows "square_free p"
BisT_coind: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" assumes "sym theta"   and "theta \<subseteq> RetrT (theta \<union> BisT)" shows "theta \<subseteq> BisT"
assms_5: shows "a \<otimes> h [^] e = \<^bold>g [^] z"
simps_3: fixes x :: "'b"   and h :: "'b tree" shows "exec (Insert x) [h] = Pairing_Heap_Tree.insert x h"
norm_triangle_sub: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm x \<le> norm y + norm (minus x y)"
tabulate_toList: fixes ks :: "'c odlist"   and f :: "'c \<Rightarrow> 'd" shows "toList (tabulate ks f) = map (\<lambda>k. (k, f k)) (toList ks)"
ltln_Release_alterdef: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi> = w \<Turnstile>\<^sub>n G\<^sub>n \<psi> or\<^sub>n (\<psi> U\<^sub>n (\<phi> and\<^sub>n \<psi>))"
chambersubcomplex_image: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexMorphism X Y f" shows "ChamberComplex.ChamberSubcomplex Y (f \<turnstile> X)"
order_refl: fixes x :: "'k multiset" shows "x \<subseteq># x"
set_prefixes_eq: fixes xs :: "'b list" shows "set (prefixes xs) = {ys. prefix ys xs}"
find_first_correct_2: fixes x :: "'a"   and xs :: "'a list"   and i :: "nat" assumes "find_first x xs = Some i" shows "xs ! i = x"
hyps_for'_subset: fixes n :: "'node"   and p :: "'inPort" shows "hyps_for' n p \<subseteq> fset (outPorts n)"
VcompNml_Cod_Nml: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Nml V src trg t" shows "bicategorical_language.VcompNml V (bicategorical_language.Cod V src trg t) t = t"
pinf_3: fixes t :: "'b" shows "\<exists>z. \<forall>x>z. (x = t) = False"
assms_2: fixes s :: "'a" assumes "P s" shows "R s"
y_prop: shows "r < y \<and> (\<forall>(d, e, f)\<in>set les. \<forall>x\<in>{r<..y}. d * x\<^sup>2 + e * x + f < 0)"
mono_neutral_cong_right: fixes T :: "'b set"   and S :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (1::'c)"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "prod g T = prod h S"
R_def: shows "R = (\<lambda>s. case s of Inl (m, m') \<Rightarrow> 0 < m \<and> m' = m\<^sup>2 * D + 1 \<and> (\<forall>y\<in>{0<..<m}. \<not> is_square (y\<^sup>2 * D + 1)) | Inr x \<Rightarrow> Q x)"
fresh_list_cons: fixes a :: "'a"   and x :: "'b"   and xs :: "'b list" shows "a \<sharp> (x # xs) = (a \<sharp> x \<and> a \<sharp> xs)"
summable_comparison_test': fixes g :: "nat \<Rightarrow> real"   and N :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "summable g"   and "\<And>n. N \<le> n \<Longrightarrow> norm (f n) \<le> g n" shows "summable f"
pt_def: shows "pt TYPE('a) TYPE('b) \<equiv> (\<forall>x. [] \<bullet> x = x) \<and> (\<forall>pi1 pi2 x. (pi1 @ pi2) \<bullet> x = pi1 \<bullet> pi2 \<bullet> x) \<and> (\<forall>pi1 pi2 x.  pi1 \<triangleq> pi2  \<longrightarrow> pi1 \<bullet> x = pi2 \<bullet> x)"
span_Un: fixes U :: "'a vec set"   and U' :: "'a vec set"   and W :: "'a vec set"   and W' :: "'a vec set" assumes "U \<subseteq> carrier_vec n"   and "U' \<subseteq> carrier_vec n"   and "local.span U = local.span U'"   and "W \<subseteq> carrier_vec n"   and "W' \<subseteq> carrier_vec n"   and "local.span W = local.span W'" shows "local.span (U \<union> W) = local.span (U' \<union> W')"
Sup_flat_complete_lattice_def: fixes A :: "'a flat_complete_lattice set" shows "Sup A = (if A = {} \<or> A = {flat_complete_lattice.Bot} then flat_complete_lattice.Bot else if \<exists>v. A - {flat_complete_lattice.Bot} = {flat_complete_lattice.Value v} then flat_complete_lattice.Value (THE v. A - {flat_complete_lattice.Bot} = {flat_complete_lattice.Value v}) else flat_complete_lattice.Top)"
Func_Pfunc: fixes A :: "'a set"   and B :: "'b set" shows "Func_option A B \<subseteq> Pfunc A B"
real_poly_minus: fixes p :: "'a poly"   and q :: "'a poly" assumes "set (coeffs p) \<subseteq> \<real>"   and "set (coeffs q) \<subseteq> \<real>" shows "set (coeffs (p - q)) \<subseteq> \<real>"
cast_RefT2: fixes G :: "prog"   and S :: "ty"   and R :: "ref_ty" assumes "G\<turnstile>S\<preceq>? RefT R" shows "\<exists>t. S = RefT t"
det_linear_row_finsum: fixes S :: "'b set"   and c :: "nat \<Rightarrow> 'c vec"   and n :: "nat"   and k :: "nat"   and a :: "nat \<Rightarrow> 'b \<Rightarrow> 'c vec" assumes "finite S"   and "c \<in> {0..<n} \<rightarrow> carrier_vec n"   and "k < n"   and "a k \<in> S \<rightarrow> carrier_vec n" shows "det (mat\<^sub>r n n (\<lambda>i. if i = k then finsum_vec TYPE('c) n (a i) S else c i)) = (\<Sum>j\<in>S. det (mat\<^sub>r n n (\<lambda>i. if i = k then a i j else c i)))"
tail_invar_def: fixes bq :: "('e, 'f) SkewBinomialTree list" shows "tail_invar bq = (queue_invar bq \<and> rank_invar bq)"
both_comm: fixes t :: "'a ttree"   and t' :: "'a ttree" shows "t \<otimes>\<otimes> t' = t' \<otimes>\<otimes> t"
4: fixes x :: "nat" assumes "x < length qs" shows "finite (set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) (take x qs)))"
compl_struct_disjoint: fixes compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   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 data :: "nat \<times> 'g" assumes "compl_struct compl"   and "sps \<noteq> []"   and "set sps \<subseteq> set ps" shows "fst ` set (fst (compl gs bs (ps -- sps) sps data)) \<inter> fst ` (set gs \<union> set bs) = {}"
HDomain_Incl_graph_aux_def: shows "HDomain_Incl_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>HDomain_Incl_aux x r d y z. x1 = (r, d) \<and> x2 = All2 x d (SyntaxN.Ex y (SyntaxN.Ex z (Var z IN r AND Var z EQ HPair (Var x) (Var y)))) \<and> atom x \<sharp> (r, d, y, z) \<and> atom y \<sharp> (r, d, z) \<and> atom z \<sharp> (r, d))"
add_mset_eq_add_mset_ne: fixes a :: "'a"   and b :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" assumes "a \<noteq> b" shows "(add_mset a A = add_mset b B) = (a \<in># B \<and> b \<in># A \<and> A = add_mset b (remove1_mset a B))"
upd_incoming__elem: fixes nd :: "('d, 'e) node_scheme"   and n :: "('d, 'f) node_scheme"   and ns :: "('d, 'e) node_scheme set" assumes "nd \<in> upd_incoming n ns" shows "nd \<in> ns \<or> (\<exists>nd'\<in>ns. nd = nd'\<lparr>incoming := incoming n \<union> incoming nd'\<rparr> \<and> old nd' = old n \<and> next nd' = next n)"
move: shows "\<tau>Exec_mover ci P t body h (stk, loc, pc, xcp) (stk', loc', pc', xcp')"
no_Emptys_pass1: fixes hs :: "'b heap list" assumes "no_Emptys hs" shows "no_Emptys (pass\<^sub>1' hs)"
rd_assign_is_CSP: fixes s :: "'d" shows "is_CSP_process (R (true \<turnstile> \<lambda>(A, A'). ref A' = ref A \<and> tr A' = tr A \<and> \<not> wait A' \<and> alpha_rp.more A' = s))"
new_psubst_image: fixes c :: "'d"   and p :: "('d, 'e) form"   and d :: "'f"   and f :: "'d \<Rightarrow> 'f" assumes "new c p"   and "d \<notin> f ` params p" shows "new d (psubst (f(c := d)) p)"
integral: fixes x :: "real" assumes "2 < x" shows "(r has_integral f x - \<MM> x / ln x - (ln (ln x) - ln (ln 2))) {2..x}"
step_preserves_le: fixes S :: "(char list \<Rightarrow> int) set"   and cs :: "(char list \<Rightarrow> int) set acom"   and S' :: "(char list \<Rightarrow> int) set"   and cs' :: "(char list \<Rightarrow> int) set acom" assumes "step S cs = cs"   and "S' \<subseteq> S"   and "cs' \<le> cs" shows "step S' cs' \<le> cs"
not_eq_order_implies_strict: fixes a :: "nat list"   and b :: "nat list" assumes "a \<noteq> b"   and "a \<le>\<^sub>v b" shows "a <\<^sub>v b"
inv_equality: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x. g (f x) = x"   and "\<And>y. f (g y) = y" shows "inv f = g"
vconverse_vconverse: fixes r :: "V" assumes "vbrelation r" shows "(r\<inverse>\<^sub>\<circ>)\<inverse>\<^sub>\<circ> = r"
member_rec_2: fixes y :: "'b" shows "List.member [] y = False"
distinct_delete: fixes xs :: "['b]"   and a :: "'b" assumes "distinct xs"   and "\<forall>x\<in>set xs. eq\<cdot>a\<cdot>x \<noteq> \<bottom>" shows "distinct (delete\<cdot>a\<cdot>xs)"
bind_dens_ctxt_measure_cong: fixes \<rho> :: "nat \<Rightarrow> val"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> 'a measure"   and g :: "(nat \<Rightarrow> val) \<Rightarrow> 'a measure"   and N :: "'a measure" assumes "\<And>\<sigma>. (\<And>x. x \<in> V' \<Longrightarrow> \<sigma> x = \<rho> x) \<Longrightarrow> f \<sigma> = g \<sigma>"   and "\<rho> \<in> space (state_measure V' \<Gamma>)"   and "f \<in> state_measure (V \<union> V') \<Gamma> \<rightarrow>\<^sub>M subprob_algebra N"   and "g \<in> state_measure (V \<union> V') \<Gamma> \<rightarrow>\<^sub>M subprob_algebra N" shows "dens_ctxt_measure \<Y> \<rho> \<bind> f = dens_ctxt_measure \<Y> \<rho> \<bind> g"
Lambert_W_times_exp_self': fixes x :: "real" assumes "- exp (- 1) \<le> x" shows "exp (Lambert_W x) * Lambert_W x = x"
sum_Un: fixes A :: "'d set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "finite A"   and "finite B" shows "sum f (A \<union> B) = sum f A + sum f B - sum f (A \<inter> B)"
delete_notin_dom: fixes k :: "'c"   and al :: "('c \<times> 'd) list" shows "k \<notin> fst ` set (delete k al)"
qSwapBinp_def: fixes xs :: "'i"   and x :: "'j"   and y :: "'j"   and qbinp :: "'f \<Rightarrow> ('g, 'h, 'i, 'j, 'k) qAbs option" shows "qSwapBinp xs x y qbinp \<equiv> lift (qSwapAbs xs x y) qbinp"
Cf: shows "cod\<^bsub>C\<^esub> f = Y"
case_1: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "char list \<Rightarrow> 'a" shows "(case Void of Void \<Rightarrow> f1 | Boolean \<Rightarrow> f2 | Integer \<Rightarrow> f3 | NT \<Rightarrow> f4 | Class x \<Rightarrow> f5 x) = f1"
has_vector_derivative_eq_rhs: fixes f :: "real \<Rightarrow> 'e"   and X :: "'e"   and F :: "real filter"   and Y :: "'e" assumes "(f has_vector_derivative X) F"   and "X = Y" shows "(f has_vector_derivative Y) F"
numeral_Bit1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit1 n) = plus (plus (numeral.numeral one plus n) (numeral.numeral one plus n)) one"
if_mono_cong: fixes b :: "bool"   and x :: "'b"   and x' :: "'b"   and y :: "'b"   and y' :: "'b" assumes "b \<Longrightarrow> x \<le> x'"   and "\<not> b \<Longrightarrow> y \<le> y'" shows "(if b then x else y) \<le> (if b then x' else y')"
induct: fixes P :: "real poly \<Rightarrow> bool"   and a0 :: "real poly" assumes "\<And>p. (p \<noteq> 0 \<Longrightarrow> P (pderiv p)) \<Longrightarrow> P p" shows "P a0"
fin1: shows "finite (supp (f (as \<inter> supp x) x c))"
surjective: fixes r :: "('c, 'd) tarjan_state_scheme" shows "r = \<lparr>counter = counter r, discovered = discovered r, finished = finished r, pending = pending r, stack = stack r, tree_edges = tree_edges r, back_edges = back_edges r, cross_edges = cross_edges r, sccs = sccs r, lowlink = lowlink r, tj_stack = tj_stack r, \<dots> = tarjan_state.more r\<rparr>"
finite_dimensional_vector_space_on_with_def: fixes S :: "'c set"   and pls :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mns :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and um :: "'c \<Rightarrow> 'c"   and zero :: "'c"   and scl :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and basis :: "'c set" shows "finite_dimensional_vector_space_on_with S pls mns um zero scl basis = (vector_space_on_with S pls mns um zero scl \<and> finite basis \<and> \<not> dependent_with pls zero scl basis \<and> span_with pls zero scl basis = S)"
designI: assumes "\<And>b. b \<in># blocks_image B \<pi> \<Longrightarrow> b \<noteq> {}"   and "blocks_image B \<pi> \<noteq> {#}"   and "\<pi> ` V \<noteq> {}" shows "design (\<pi> ` V) (blocks_image B \<pi>)"
dec_nemp': fixes u :: "'b list"   and G :: "'b list set" assumes "u \<noteq> \<epsilon>"   and "u \<in> \<langle>G\<rangle>" shows "Dec G u \<noteq> \<epsilon>"
dom_m_fmdrop: fixes C :: "'a"   and N :: "('a, 'b) fmap" shows "dom_m (fmdrop C N) = remove1_mset C (dom_m N)"
ipresIGSubstAbs_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'h"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'varSort, 'l, 'm, 'var, 'h, 'i, 'n) model_scheme" shows "ipresIGSubstAbs h hA termMOD MOD = ipresSubstAbs h hA MOD"
handshake_invL_def: fixes m :: "'g" shows "mut_m.handshake_invL m = (\<lambda>s. (atS (mutator m) hs_noop_locs s \<longrightarrow> sys_hs_type s\<down> = ht_NOOP) \<and> (atS (mutator m) hs_get_roots_locs s \<longrightarrow> sys_hs_type s\<down> = ht_GetRoots) \<and> (atS (mutator m) hs_get_work_locs s \<longrightarrow> sys_hs_type s\<down> = ht_GetWork) \<and> (atS (mutator m) ht_loaded_locs s \<longrightarrow> mutator_hs_pending (s\<down> (mutator m)) \<longrightarrow> hs_type (s\<down> (mutator m)) = sys_hs_type s\<down>) \<and> (atS (mutator m) hs_pending_loaded_locs s \<longrightarrow> mutator_hs_pending (s\<down> (mutator m)) \<longrightarrow> sys_hs_pending m s\<down>) \<and> (atS (mutator m) hs_pending_locs s \<longrightarrow> mutator_hs_pending (s\<down> (mutator m))) \<and> (atS (mutator m) no_pending_mutations_locs s \<longrightarrow> filter (\<lambda>s. is_mw_Mutate s \<or> is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s\<down>) = []))"
islimpt_EMPTY: fixes x :: "'b" shows "\<not> x islimpt {}"
whileAnnoFix: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P guards gs (whileAnnoFix b J V (\<lambda>Z. Seq (c Z) (guards gs Skip))) Q,A"
mon_e_simps_2: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and p :: "'c" shows "mon_e fg (LCall p) = mon fg p"
a: shows "wf (r1 <*lex*> r2 <*lex*> r3)"
simps_3: fixes \<phi>\<^sub>1 :: "'c fm"   and \<phi>\<^sub>2 :: "'c fm" shows "qfree (Or \<phi>\<^sub>1 \<phi>\<^sub>2) = (qfree \<phi>\<^sub>1 \<and> qfree \<phi>\<^sub>2)"
measurable_If_set: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and M' :: "'d measure"   and g :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "f \<in> M \<rightarrow>\<^sub>M M'"   and "g \<in> M \<rightarrow>\<^sub>M M'"   and "A \<inter> space M \<in> sets M" shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> M \<rightarrow>\<^sub>M M'"
le_TC_0_iff: fixes x :: "V" shows "(x \<sqsubseteq> 0) = (x = 0)"
less_int_def: shows "(<) \<equiv> (Rep_Integ ---> Rep_Integ ---> id) (\<lambda>(x, y) (u, v). x + v < u + y)"