hcju/isabelleps huggingface dataset:
finite_set_mset_mset_set: fixes A :: "'c set" assumes "finite A" shows "set_mset (mset_set A) = A"
strongly_preferred_def: fixes a :: "'b"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b" shows "a \<prec>[R] b \<equiv> R a b \<and> \<not> R b a"
if_then_Some_else_None_eq_1: fixes a :: "bool"   and b :: "'l"   and c :: "'l" shows "((if a then Some b else None) = Some c) = (a \<and> b = c)"
thin_quadrilaterals: fixes Gxy :: "'a set"   and x :: "'a"   and y :: "'a"   and Gyz :: "'a set"   and z :: "'a"   and Gzt :: "'a set"   and t :: "'a"   and Gxt :: "'a set"   and w :: "'a" assumes "geodesic_segment_between Gxy x y"   and "geodesic_segment_between Gyz y z"   and "geodesic_segment_between Gzt z t"   and "geodesic_segment_between Gxt x t"   and "w \<in> Gxt" shows "infdist w (Gxy \<union> Gyz \<union> Gzt) \<le> 8 * deltaG TYPE('a)"
B: shows "\<not> (\<exists>v\<in>sinks I D u (ys @ [y]). \<exists>w\<in>U. (v, w) \<in> I)"
powr_lower_bound: fixes l :: "real"   and x :: "real"   and u :: "real"   and z :: "real" assumes "0 < l"   and "l \<le> x"   and "x \<le> u" shows "min (l powr z) (u powr z) \<le> x powr z"
strict_prefix_def: fixes xs :: "'e list"   and ys :: "'e list" shows "strict_prefix xs ys = (prefix xs ys \<and> xs \<noteq> ys)"
bf_lit_def: fixes v :: "'c" shows "bf_lit v \<equiv> \<lambda>l. l v"
clock_sim: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e" assumes "Environment jkbp" shows "sim (PreEnvironmentJView.mkM envVal (Environment.clock_jview envObs) (PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal (Environment.clock_jview envObs))) (mkKripke (Environment.clock_sim jkbp envInit envAction envTrans envVal envObs ` PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal (Environment.clock_jview envObs)) (Environment.clock_simRels envObs) (Environment.clock_simVal envVal)) (Environment.clock_sim jkbp envInit envAction envTrans envVal envObs)"
computation_downpos: shows "rat_poly_times A (A - rat_poly_times (rat_poly_times B B) B) = rat_poly_inv (rat_poly_times B (B - rat_poly_times (rat_poly_times A A) A))"
full2: fixes \<sigma> :: "'tp"   and d :: "univ" shows "intTF \<sigma> d"
before_dist_fst_snd: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "before vs ram1 ram2"   and "distinct vs" shows "fst (splitAt ram2 (snd (splitAt ram1 vs))) = snd (splitAt ram1 (fst (splitAt ram2 vs)))"
simps_1: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "'a" assumes "Matrix_Tensor.mult id f" shows "mult.times f n [] = []"
assms_4: shows "\<not> formulaFalse (getF state) (elements (getM state))"
get: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and l :: "'l" assumes "s r = Some (\<sigma>, \<tau>, \<E> [Read (VE (Loc l))])"   and "l \<in> dom (\<sigma>;;\<tau>)" shows "revision_step r s (s(r \<mapsto> (\<sigma>, \<tau>, \<E> [VE (the ((\<sigma>;;\<tau>) l))])))"
outer_measure_of_mono: fixes A :: "'b set"   and B :: "'b set"   and M :: "'b measure" assumes "A \<subseteq> B" shows "outer_measure_of M A \<le> outer_measure_of M B"
assms_1: shows "image_mset f (mset_set A) = image_mset g (mset_set A)"
encR: fixes S :: "'procS" shows "indRelRp (SourceTerm S) (TargetTerm (\<lbrakk>S\<rbrakk>))"
cod_dom: fixes f :: "'c" assumes "C.arr f" shows "C.cod (C.dom f) = C.dom f"
simE_2: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and \<alpha> :: "freeRes"   and Q' :: "pi" assumes "P \<leadsto><Rel> Q"   and "Q \<longmapsto> \<alpha> \<prec> Q'" shows "\<exists>P'. P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
butlast_zip: fixes xs :: "'d list"   and ys :: "'e list" assumes "length xs = length ys" shows "butlast (xs || ys) = butlast xs || butlast ys"
assert_sup: fixes p :: "'b Assertion"   and q :: "'b Assertion" shows "{\<cdot> p \<squnion> q } = {\<cdot> p } \<squnion> {\<cdot> q }"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "comm_monoid_mult_hom hom"   and "inj_monoid_mult_hom hom" shows "inj_comm_monoid_mult_hom hom"
AE_I: fixes M :: "'b measure"   and P :: "'b \<Rightarrow> bool"   and N :: "'b set" assumes "{x \<in> space M. \<not> P x} \<subseteq> N"   and "emeasure M N = 0"   and "N \<in> sets M" shows "almost_everywhere M P"
strong_induct: fixes x1 :: "(name \<times> ty) list"   and x2 :: "trm"   and x3 :: "ty"   and P :: "'a \<Rightarrow> (name \<times> ty) list \<Rightarrow> trm \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<turnstile> x2 : x3"   and "\<And>\<Gamma> x \<tau> y. \<lbrakk>valid \<Gamma>; (x, \<tau>) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P y \<Gamma> (Var x) \<tau>"   and "\<And>\<Gamma> s \<tau> \<sigma> t y. \<lbrakk>\<Gamma> \<turnstile> s : \<tau> \<rightarrow> \<sigma>; \<And>z. P z \<Gamma> s (\<tau> \<rightarrow> \<sigma>); \<Gamma> \<turnstile> t : \<tau>; \<And>z. P z \<Gamma> t \<tau>\<rbrakk> \<Longrightarrow> P y \<Gamma> (App s t) \<sigma>"   and "\<And>x \<Gamma> \<tau> t \<sigma> y. \<lbrakk>x \<sharp> y; x \<sharp> \<Gamma>; (x, \<tau>) # \<Gamma> \<turnstile> t : \<sigma>; \<And>z. P z ((x, \<tau>) # \<Gamma>) t \<sigma>\<rbrakk> \<Longrightarrow> P y \<Gamma> (\<Lambda> x . t) (\<tau> \<rightarrow> \<sigma>)"   and "\<And>\<Gamma> s \<sigma> y. \<lbrakk>\<Gamma> \<turnstile> s : \<sigma>; \<And>z. P z \<Gamma> s \<sigma>\<rbrakk> \<Longrightarrow> P y \<Gamma> [s] (T \<sigma>)"   and "\<And>x \<Gamma> s \<sigma> t \<tau> y. \<lbrakk>x \<sharp> y; x \<sharp> (\<Gamma>, s); \<Gamma> \<turnstile> s : T \<sigma>; \<And>z. P z \<Gamma> s (T \<sigma>); (x, \<sigma>) # \<Gamma> \<turnstile> t : T \<tau>; \<And>z. P z ((x, \<sigma>) # \<Gamma>) t (T \<tau>)\<rbrakk> \<Longrightarrow> P y \<Gamma> (s to x in t) (T \<tau>)" shows "P z x1 x2 x3"
assms_4: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"
s_le_p: fixes a :: "nat \<Rightarrow> nat"   and j :: "nat" assumes "a \<in> s" shows "a j \<le> p"
interval_not_empty: fixes a :: "'a"   and b :: "'a" assumes "trans r"   and "interval r a b \<noteq> {}" shows "(a, b) \<in> r"
ext_inject: fixes inst :: "inst list"   and funcs :: "cl list"   and tab :: "cl option list list"   and mem :: "mem list"   and globs :: "global list"   and more :: "'a"   and inst' :: "inst list"   and funcs' :: "cl list"   and tab' :: "cl option list list"   and mem' :: "mem list"   and globs' :: "global list"   and more' :: "'a" shows "(\<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = \<lparr>inst = inst', funcs = funcs', tab = tab', mem = mem', globs = globs', \<dots> = more'\<rparr>) = (inst = inst' \<and> funcs = funcs' \<and> tab = tab' \<and> mem = mem' \<and> globs = globs' \<and> more = more')"
visible_methods_exist: fixes Cs :: "char list list"   and mthd :: "ty list \<times> ty \<times> char list list \<times> expr"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and M :: "char list" assumes "(Cs, mthd) \<in> MethodDefs P C M" shows "\<exists>Bs fs ms. class P (last Cs) = \<lfloor>(Bs, fs, ms)\<rfloor> \<and> map_of ms M = \<lfloor>mthd\<rfloor>"
integrable_const_bound: fixes f :: "real \<Rightarrow> 'b"   and B :: "real"   and n :: "nat" assumes "AE x in M n. norm (f x) \<le> B"   and "Mn.random_variable n borel f" shows "integrable (M n) f"
lim_1: fixes i :: "nat" shows "integrable (M1 \<Otimes>\<^sub>M M2) (s i)"
dom_map_of_conv_image_fst: fixes xys :: "('b \<times> 'c) list" shows "dom (map_of xys) = fst ` set xys"
charpo: shows "char_poly A = (\<Prod>a\<leftarrow>es. [:- a, 1:]) \<and> length es = n"
oalist_inv_raw_ConsD1: fixes x :: "'a \<times> 'c"   and xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw (x # xs)" shows "oalist_inv_raw xs"
BetSEq: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "BetS A B C" shows "Bet A B C \<and> A \<noteq> B \<and> A \<noteq> C \<and> B \<noteq> C"
quotient_map_def: fixes X :: "'a topology"   and X' :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "quotient_map X X' f = (f ` topspace X = topspace X' \<and> (\<forall>U\<subseteq>topspace X'. openin X {x \<in> topspace X. f x \<in> U} = openin X' U))"
coprime_power_left_iff: fixes a :: "'a"   and n :: "nat"   and b :: "'a" shows "coprime (a ^ n) b = (coprime a b \<or> n = 0)"
assms_3: shows "y \<^sup>@ k = u \<cdot> v \<cdot> w"
at_within_Icc_at_right: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.order_topology less_eq less open"   and "less a b" shows "topological_space.at_within open a (ord.atLeastAtMost less_eq a b) = topological_space.at_within open a (ord.greaterThan less a)"
lns_normI: fixes B :: "'b set"   and P :: "('b \<times> 'b) set" assumes "\<And>x e. \<lbrakk>x \<in> B; 0 < e\<rbrakk> \<Longrightarrow> \<exists>y\<in>B. norm (y - x) \<le> e \<and> y \<succeq>[P] x \<and> (x, y) \<notin> P" shows "local_nonsatiation B P"
partition_subset_imp_notin: fixes A :: "'b set"   and P :: "'b set set"   and X :: "'b set"   and X' :: "'b set" assumes "partition_on A P"   and "X \<in> P"   and "X' \<subset> X" shows "X' \<notin> P"
Sup_adjunct_Sup: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and W :: "'a set" assumes "S_inductive_set D"   and "f \<in> carrier D \<rightarrow> carrier D"   and "a \<in> carrier D"   and "\<forall>x\<in>carrier D. x \<preceq> f x"   and "W \<in> WWa D f a"   and "Algebra2.Sup D W \<notin> W" shows "Algebra2.Sup D (insert (Algebra2.Sup D W) W) = Algebra2.Sup D W"
ptrm_alpha_equiv_symp: shows "symp (\<approx>)"
ipcidr_union_set_def: fixes ips :: "('a word \<times> nat) set" shows "ipcidr_union_set ips \<equiv> \<Union>(base, len)\<in>ips. ipset_from_cidr base len"
deg_monom_minus: fixes a :: "'a" assumes "a \<in> carrier R"   and "carrier R \<noteq> {\<zero>}" shows "deg R (monom P \<one> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"
atLeastAtMost_iff: fixes i :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a"   and u :: "'a" shows "(i \<in> ord.atLeastAtMost less_eq l u) = (less_eq l i \<and> less_eq i u)"
rep_eq: fixes x :: "64 word" shows "Rep_uint64 (Abs_uint64' x) = x"
map_wordinterval_to_set_distinct: fixes xs :: "'a wordinterval list" assumes "distinct xs"   and "\<forall>x1\<in>set xs. \<forall>x2\<in>set xs. x1 \<noteq> x2 \<longrightarrow> wordinterval_to_set x1 \<inter> wordinterval_to_set x2 = {}"   and "\<forall>x\<in>set xs. \<not> wordinterval_empty x" shows "distinct (map wordinterval_to_set xs)"
natToVertexList_nth_Suc: fixes es :: "nat list"   and f :: "face"   and n :: "nat"   and v :: "nat" assumes "incrIndexList es |es| |vertices f|"   and "Suc n < |es|" shows "natToVertexList v f es ! Suc n = (if es ! n = es ! Suc n then None else Some f\<^bsup>es ! Suc n\<^esup> \<bullet> v)"
eventually_at_filter: fixes "open" :: "'b set \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and a :: "'b"   and s :: "'b set" assumes "class.topological_space open" shows "eventually P (topological_space.at_within open a s) = (\<forall>\<^sub>F x in topological_space.nhds open a. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x)"
AgentY_X_FWB: shows "\<lfloor>RightTo (Agent X) FWB\<rfloor>\<^sub>Y"
open_QuantT_def: fixes U :: "('a['b, 'c]) set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
FDOM_eff_subset_FDOM_valid_states_pair: fixes a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and s :: "('d, 'g) fmap" assumes "a \<in> PROB"   and "s \<in> valid_states PROB" shows "fmdom' (snd a) \<subseteq> fmdom' s"
preprm_inv_ext: fixes \<pi> :: "('b \<times> 'b) list"   and \<sigma> :: "('b \<times> 'b) list" assumes "\<pi> =p \<sigma>" shows "preprm_inv \<pi> =p preprm_inv \<sigma>"
akra_bazzi_exponent_def: fixes as :: "real list"   and bs :: "real list" shows "akra_bazzi_exponent as bs \<equiv> THE p. (\<Sum>i<length as. as ! i * bs ! i powr p) = 1"
reachable_conv': fixes u :: "'a"   and v :: "'a" assumes "u \<in> pverts (list_digraph G)" shows "(u \<rightarrow>\<^sup>*\<^bsub>with_proj (list_digraph G)\<^esub> v) = ((u, v) \<in> (parcs (list_digraph G))\<^sup>*)"
shift_left_inj: fixes xs :: "'c list"   and s1 :: "'c stream"   and s2 :: "'c stream" shows "(xs @- s1 = xs @- s2) = (s1 = s2)"
coeff_replace_coeff: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" assumes "f (0::'d) = (0::'c)" shows "coeff (replace_coeff f p) m = f (coeff p m)"
all_acquired_dropWhile: fixes P :: "'b memref \<Rightarrow> bool"   and sb :: "'b memref list" shows "all_acquired (dropWhile P sb) \<subseteq> all_acquired sb"
jnf: fixes A :: "real mat"   and n :: "nat"   and m :: "nat"   and lam :: "complex"   and n_as :: "(nat \<times> complex) list" assumes "spectral_radius_1_jnf_max A n m lam n_as" shows "jordan_nf (of_real_hom.mat_hom A) n_as"
sim_action_simps_16: fixes obs :: "('a, 'b) obs_event action"   and ad :: "'a"   and al :: "addr_loc"   and v' :: "'a val" shows "obs \<approx> NormalAction (ReadMem ad al v') = (\<exists>v. obs = NormalAction (ReadMem ad al v))"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "String.literal \<Rightarrow> integer option \<Rightarrow> AST.expr option \<Rightarrow> 'a"   and f2 :: "String.literal \<Rightarrow> integer option \<Rightarrow> (integer \<times> AST.varType list) option \<Rightarrow> 'a"   and f3 :: "String.literal \<Rightarrow> integer \<Rightarrow> AST.expr option \<Rightarrow> 'a"   and f4 :: "String.literal \<Rightarrow> integer option \<Rightarrow> String.literal option \<Rightarrow> 'a"   and varDecl :: "AST.varDecl" shows "P (case varDecl of AST.VarDeclNum x xa xb \<Rightarrow> f1 x xa xb | AST.VarDeclChan x xa xb \<Rightarrow> f2 x xa xb | AST.VarDeclUnsigned x xa xb \<Rightarrow> f3 x xa xb | AST.VarDeclMType x xa xb \<Rightarrow> f4 x xa xb) = ((\<forall>x11 x12 x13. varDecl = AST.VarDeclNum x11 x12 x13 \<longrightarrow> P (f1 x11 x12 x13)) \<and> (\<forall>x21 x22 x23. varDecl = AST.VarDeclChan x21 x22 x23 \<longrightarrow> P (f2 x21 x22 x23)) \<and> (\<forall>x31 x32 x33. varDecl = AST.VarDeclUnsigned x31 x32 x33 \<longrightarrow> P (f3 x31 x32 x33)) \<and> (\<forall>x41 x42 x43. varDecl = AST.VarDeclMType x41 x42 x43 \<longrightarrow> P (f4 x41 x42 x43)))"
fbrelation_vintersectionI: fixes s :: "V" shows "fbrelation (r\<^sub>1 \<inter>\<^sub>\<circ> s)"
split_part: fixes P :: "bool"   and Q :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "(\<lambda>(a, b). P \<and> Q a b) = (\<lambda>ab. P \<and> (case ab of (x, xa) \<Rightarrow> Q x xa))"
Joints_imp_length_eq: fixes xs :: "'a list"   and XS :: "('a \<times> 'a pdevs) list" assumes "xs \<in> Joints XS" shows "length xs = length XS"
set_disconnected_nodes_impl: shows "set_disconnected_nodes = local.a_set_disconnected_nodes"
dom_closed: fixes f :: "'a" assumes "left b f" shows "left b (local.dom f)"
adjunction_in_bicategory_axioms: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g' ((\<phi> \<star> f) \<cdot> \<eta>) (\<epsilon> \<cdot> (f \<star> local.inv \<phi>))"
intros: fixes u :: "'d \<Rightarrow> 'e"   and A :: "('f \<times> ('d, 'e) cconstraint \<times> 'g \<times> 'd list \<times> 'f) set \<times> ('f \<Rightarrow> ('d, 'e) cconstraint)"   and l :: "'f"   and d :: "'e" assumes "u \<turnstile> inv_of A l"   and "u \<oplus> d \<turnstile> inv_of A l"   and "(0::'e) \<le> d" shows "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>\<^bsup>d\<^esup> \<langle>l, u \<oplus> d\<rangle>"
sum_minus_one: fixes y :: "nat" shows "(\<Sum>x = 1..y. (- 1) ^ (x + 1)) = (if odd y then 1 else 0)"
r_nf_u_def: shows "r_nf_u \<equiv> Cn 1 r_dec [Cn 1 r_result [r_pdec1, r_pdec12, r_pdec22]]"
lfp_trans_greatest: fixes T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and v :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>t P. \<lbrakk>le_trans (T t) t; \<And>P. sound P \<Longrightarrow> sound (t P)\<rbrakk> \<Longrightarrow> le_trans u t"   and "\<And>P. sound P \<Longrightarrow> sound (v P)"   and "le_trans (T v) v" shows "le_trans u (lfp_trans T)"
xor_inverse: fixes x :: "'g" shows "x \<oplus> x = \<bottom>"
Int_subset_iff: fixes C :: "'a set"   and A :: "'a set"   and B :: "'a set" shows "order.greater_eq (A \<inter> B) C = (order.greater_eq A C \<and> order.greater_eq B C)"
bit_unsigned_of_bits_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and bs :: "bool list"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "bit (comm_semiring_0.horner_sum plus zero times (zero_neq_one.of_bool one zero) (numeral.numeral one plus (num.Bit0 num.One)) bs) n = nth_default False bs n"
ideD_1: fixes a :: "'b" assumes "\<phi>.B.ide a" shows "\<phi>.B.arr a"
altc_sample_option_def: fixes altc_sample :: "'d \<Rightarrow> ('e \<Rightarrow> 'm) \<Rightarrow> 'm"   and p :: "'d"   and f :: "'e \<Rightarrow> ('a, 'm) optionT" shows "local.altc_sample_option altc_sample p f = OptionT (altc_sample p (\<lambda>x. run_option (f x)))"
asymp_equivI'_const: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and c :: "'b"   and F :: "'a filter" assumes "((\<lambda>x. f x / g x) \<longlongrightarrow> c) F"   and "c \<noteq> (0::'b)" shows "f \<sim>[F] (\<lambda>x. c * g x)"
bottom_def: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "class.pcpo below" shows "pcpo.bottom below = (THE x. \<forall>y. below x y)"
test_def: fixes P :: "'b \<Rightarrow> bool" shows "\<questiondown>P? = (\<lambda>s. {x. x = s \<and> P x})"
sorted_insort_is_snoc: fixes xs :: "'b list"   and a :: "'b" assumes "sorted xs"   and "\<forall>x\<in>set xs. x \<le> a" shows "insort a xs = xs @ [a]"
map_SSup_def: fixes F :: "'d downset \<Rightarrow> 'e downset" shows "map_SSup F = SSup \<circ> F \<circ> ds"
assms_1: shows "g \<in> \<Theta>(\<lambda>x. real x powr p * ln (real x) powr p')"
ide_dom: fixes f :: "'d" assumes "D.arr f" shows "D.ide (D.dom f)"
r_one_unique: fixes a :: "'a"   and x :: "'a" assumes "a \<in> carrier G"   and "x \<in> carrier G"   and "a \<cdot> x = a" shows "x = \<one>"
param_DFS_def: fixes G :: "('c, 'd) graph_rec_scheme" shows "param_DFS G \<equiv> fb_graph G"
independent_mono: fixes A :: "('a, 'b) vec set"   and B :: "('a, 'b) vec set" assumes "vec.independent A"   and "B \<subseteq> A" shows "vec.independent B"
disjoint_insert_1: fixes B :: "'c set"   and a :: "'c"   and A :: "'c set" shows "(B \<inter> Set.insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
vec_simps_9: fixes x :: "'b"   and y :: "'b" shows "x \<bullet> y = (\<Sum>i<DIM('b). x $\<^sub>e i * y $\<^sub>e i)"
subst_psubst_compose: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y :: "'var" assumes "good X"   and "good Y"   and "goodEnv rho" shows "X #[rho] #[Y / y]_ys = X #[rho &[Y / y]_ys]"
decisive7_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and A :: "'a set"   and swf :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('a \<times> 'a) set"   and Is :: "'b set"   and j :: "'b" assumes "hasw [x, y, z] A"   and "iia swf A Is"   and "SWF swf A Is universal_domain"   and "weak_pareto swf A Is universal_domain"   and "semidecisive swf A Is {j} x y" shows "decisive swf A Is {j} y z"
language_state_for_input_alt_def: fixes M :: "('d, 'e, 'f) FSM"   and q :: "'f"   and xs :: "'d list" shows "language_state_for_input M q xs = LS\<^sub>i\<^sub>n M q {xs}"
assms_3: shows "\<forall>\<^sub>F x in at_top. f x \<le> g x"
border_rev_conv: fixes x :: "'b list"   and w :: "'b list" shows "rev x \<le>b rev w = x \<le>b w"
geometric_sum: fixes x :: "'b"   and n :: "nat" assumes "x \<noteq> (1::'b)" shows "sum ((^) x) {..<n} = (x ^ n - (1::'b)) / (x - (1::'b))"
assms_3: assumes "C = add_mset s (C - {#s#})"   and "A = B + (C - {#s#})" shows "P"
simps_1: fixes uu :: "int" shows "isProc (DI uu) = False"
impl_one_bl: fixes a :: "'a" shows "a l\<rightarrow> a = (1::'a)"
P_xs_ia: shows "vec_to_iarray A !! (rev [0..<IArray.length (vec_to_iarray A)] ! ia) \<noteq> (0::'b)"
y_D_yp: shows "y = D *v map_vector complex_of_real yp"
Tn_meas: fixes n :: "nat" shows "T ^^ n \<in> M \<rightarrow>\<^sub>M M"
hyps_insert: fixes p :: "'b pl"   and v :: "'b"   and t :: "'b set" shows "hyps p (insert v t) \<subseteq> insert #v (hyps p t - {#v -> false})"
trms\<^sub>s\<^sub>t_cons: fixes x :: "('c, 'd) strand_step"   and S :: "('c, 'd) strand_step list" shows "trms\<^sub>s\<^sub>t (x # S) = trms\<^sub>s\<^sub>t\<^sub>p x \<union> trms\<^sub>s\<^sub>t S"
in_range_dist_union: fixes h :: "heap"   and as :: "nat set"   and as' :: "nat set" shows "in_range (h, as \<union> as') = (in_range (h, as) \<and> in_range (h, as'))"
simps_2: fixes Q :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool"   and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret"   and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" shows "pred (Q)\<^sub>\<surd> (cf # cfs) = Q (fst cf)"
inj_idom_divide_hom_axioms: shows "inj_idom_divide_hom hom"
Cons_1: assumes "Ind {a} (set v)" shows "[a] @ v =\<^sub>F v @ [a]"
ipresIGCons_termMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('b, 'c, 'd, 'i, 'f, 'a, 'g, 'h, 'j) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.ipresIGCons Delta h hA (FixSyn.termMOD Delta) MOD = FixSyn.ipresCons Delta h hA MOD"
basis_col_space_eq_basis_row_space_transpose: fixes A :: "(('b, 'd) vec, 'c) vec" shows "basis_col_space A = basis_row_space (Finite_Cartesian_Product.transpose A)"
g_def: fixes i :: "nat" shows "g i = isolate_variable_sparse p var i * sq ^ (i div 2) * Const (1 - real (i mod 2))"
bot_in_downset: fixes x :: "'a" shows "bot \<in> \<down>x"
Bin1OpThrow1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and bop :: "bop"   and e\<^sub>2 :: "'addr expr1"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>Throw a \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
RR_a_def: shows "RR_a \<equiv> {: p, t, left, right, atom, mrk \<leadsto> p', t', lnk, lbl, mrk' . lnk none = left \<and> lnk some = right \<and> lbl = (\<lambda>x. if atom x then some else none) \<and> p' = p \<and> t' = t \<and> mrk' = mrk :}"
src_cod: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "src\<^sub>B (B.cod \<mu>) = src\<^sub>B \<mu>"
DList_set_def: shows "DList_set = Collect \<circ> DList_Set.member"
Int_commute: fixes A :: "'g set"   and B :: "'g set" shows "A \<inter> B = B \<inter> A"