hcju/isabelleps huggingface dataset:
dg_Rel_is_arrD_1: fixes \<alpha> :: "V"   and A :: "V"   and B :: "V"   and S :: "V" assumes "S : A \<mapsto>\<^bsub>dg_Rel \<alpha>\<^esub> B" shows "arr_Rel \<alpha> S"
D: shows "map (inv p) (filter (\<lambda>x. x \<in> range p) xs') \<in> divergences P"
word_ctz_def: fixes w :: "'b word" shows "word_ctz w \<equiv> length (takeWhile Not (rev (to_bl w)))"
\<Z>_sgrp_basisI: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_vfsequence \<alpha> \<SS>"   and "vcard \<SS> = 2\<^sub>\<nat>"   and "binop (\<SS>\<lparr>\<A>\<rparr>) (\<SS>\<lparr>mbinop\<rparr>)" shows "\<Z>_sgrp_basis \<alpha> \<SS>"
spmf_rel_mono_strong: fixes A :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b spmf"   and g :: "'d spmf"   and B :: "'b \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_spmf A f g"   and "\<And>x y. \<lbrakk>x \<in> set_spmf f; y \<in> set_spmf g; A x y\<rbrakk> \<Longrightarrow> B x y" shows "rel_spmf B f g"
all_prime_nempty_g_one: fixes ps :: "nat list" assumes "all_prime ps"   and "ps \<noteq> []" shows "Suc 0 < (\<Prod>m\<in>#mset ps. m)"
sums_infsetsum_nat: fixes f :: "nat \<Rightarrow> 'b"   and A :: "nat set" assumes "f abs_summable_on A" shows "(\<lambda>n. if n \<in> A then f n else (0::'b)) sums infsetsum f A"
over_dim: shows "rat_poly.row_length (brickmat over) = 4"
assms_2: fixes x :: "'a"   and y :: "'a" assumes "a = Inl x"   and "b = Inl y"   and "\<not> P x y" shows "thesis"
simps_4: fixes t :: "ternaryvalue" shows "ternary_ternary_eval (TernaryValue t) = t"
pref_comp_eq: fixes u :: "'b list"   and v :: "'b list" assumes "u \<bowtie> v"   and "\<^bold>|u\<^bold>| = \<^bold>|v\<^bold>|" shows "u = v"
CompDiag_Ide_Diag: fixes t :: "'c local.term"   and a :: "'c local.term" assumes "Diag t"   and "Ide a"   and "Dom a = Cod t" shows "a \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> t = t"
exists_closure: obtains L :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) ring" where "algebraic_closure L (indexed_const ` carrier R)" and "indexed_const \<in> ring_hom R L"
valid_frac_def: fixes X :: "('a \<times> 'a) set" shows "valid_frac X \<equiv> \<exists>r\<in>R. \<exists>s\<in>S. r / s = X"
ge_kde1: fixes x :: "complex" assumes "x \<in> closed_segment (a + complex_of_real kde) (a + complex_of_real e)" shows "\<exists>y. x = a + complex_of_real y \<and> kde \<le> y"
sub: shows "\<Union> (fv ` set neg) \<subseteq> \<Union> (fv ` set pos)"
pred_mapping_map_upd: fixes Q :: "'a \<Rightarrow> bool"   and x :: "'a"   and P :: "'b \<Rightarrow> 'a option"   and p :: "'b" assumes "Q x"   and "pred_mapping Q P" shows "pred_mapping Q (P(p \<mapsto> x))"
simps_2: fixes n :: "nat" shows "next (Suc n) = nextsuffix \<circ> next n"
vrange_vunion: fixes r :: "V"   and s :: "V" shows "\<R>\<^sub>\<circ> (r \<union>\<^sub>\<circ> s) = \<R>\<^sub>\<circ> r \<union>\<^sub>\<circ> \<R>\<^sub>\<circ> s"
limp_path_def: fixes Z :: "'a list set"   and Y :: "'a list set" shows "limp_path Z Y = \<Union> {X. p_prod X Y \<subseteq> Z}"
assms: shows "(f ^ n) \<langle>$\<rangle> a = b"
simps: fixes 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 a1 :: "char list \<Rightarrow> ty option"   and a2 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and a3 :: "expr list"   and a4 :: "ty list" shows "(P,a1,a2 \<turnstile> a3 [:] a4) = ((\<exists>E h. a1 = E \<and> a2 = h \<and> a3 = [] \<and> a4 = []) \<or> (\<exists>E h e T es Ts. a1 = E \<and> a2 = h \<and> a3 = e # es \<and> a4 = T # Ts \<and> P,E,h \<turnstile> e : T \<and> P,E,h \<turnstile> es [:] Ts))"
qFreshBinp_def: fixes xs :: "'varSort"   and x :: "'var"   and qbinp :: "'d \<Rightarrow> ('e, 'd, 'varSort, 'var, 'f) qAbs option" shows "qFreshBinp xs x qbinp \<equiv> liftAll (qFreshAbs xs x) qbinp"
tuple_in'_keys: fixes as :: "'a option list" assumes "as \<in> Mapping.keys tuple_in'" shows "as \<in> Mapping.keys tuple_in \<or> (\<exists>(t, X)\<in>set move. as \<in> {as \<in> X. valid_tuple tuple_since (t, as)})"
collapse: fixes inference :: "'b inference" shows "Infer (side_prems_of inference) (main_prem_of inference) (concl_of inference) = inference"
vec_normalize_def: fixes v :: "complex vec" shows "vec_normalize v = (if v = 0\<^sub>v (dim_vec v) then v else 1 / vec_norm v \<cdot>\<^sub>v v)"
w_def: shows "w = (1 - u) *\<^sub>R x + u *\<^sub>R y"
simple_path_endless: fixes c :: "real \<Rightarrow> 'a" assumes "simple_path c" shows "path_image c - {pathstart c, pathfinish c} = c ` {0<..<1}"
i_shrink_defs_1: fixes f :: "nat \<Rightarrow> 'd message_af"   and k :: "nat" shows "f \<div> k \<equiv> i_aggregate f k last_message"
naturality: fixes f :: "'a" assumes "A.arr f" shows "Map t (A.cod f) \<cdot>\<^sub>B Dom t f = Cod t f \<cdot>\<^sub>B Map t (A.dom f)"
ccpo: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lb :: "'a set \<Rightarrow> 'a" assumes "partial_function_definitions ord lb" shows "class.ccpo lb ord (mk_less ord)"
merge_val: fixes l :: "'c Tree"   and r :: "'c Tree" shows "val (merge l r) = val l \<or> val (merge l r) = val r"
ObjDf: shows "dom\<^bsub>C\<^esub> f \<in> obj\<^bsub>C\<^esub>"
dghm_id_ArrMap_vdomain: fixes \<CC> :: "V" shows "\<D>\<^sub>\<circ> (dghm_id \<CC>\<lparr>ArrMap\<rparr>) = \<CC>\<lparr>Arr\<rparr>"
i_Exec_Comp_Stream_Acc_Output__eq_NoMsg_iAll_finish_event_conv: fixes k :: "nat"   and event :: "nat \<Rightarrow> bool"   and t0 :: "nat"   and t :: "nat"   and s :: "nat \<Rightarrow> 'a message_af"   and output_fun :: "'b \<Rightarrow> 'a message_af"   and trans_fun :: "'c message_af \<Rightarrow> 'b \<Rightarrow> 'b"   and input :: "nat \<Rightarrow> 'c message_af"   and c :: "'b" assumes "Suc 0 < k"   and "\<And>t. event t = (t mod k = k - Suc 0)"   and "t0 = t * k"   and "s = output_fun \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) c" shows "(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = NoMsg) = (s t0 = NoMsg \<and> (\<circle> t' t0 [0\<dots>]. s t1 = NoMsg. t1 \<U> t2 [0\<dots>] \<oplus> t'. event t2 \<and> s t2 = NoMsg))"
order_iff_strict: fixes b :: "'e"   and a :: "'e" shows "(b \<le> a) = (b < a \<or> a = b)"
sum_upto_divisor_sum2: fixes f :: "nat \<Rightarrow> real"   and x :: "real" shows "sum_upto (\<lambda>n. \<Sum>d | d dvd n. f d) x = sum_upto (\<lambda>xa. sum_upto f (x / real xa)) x"
NDComm: fixes a :: "'b set set"   and b :: "'b set set" shows "netsDistinct a b = netsDistinct b a"
token_run_step': fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "x \<le> n" shows "semi_mojmir_def.token_run \<delta> q\<^sub>0 w x (Suc n) = \<delta> (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n) (w n)"
lex_ord_pair_valI: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'c) list"   and aux :: "order option"   and k :: "'a"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> order option" assumes "tc.oalist_inv_raw xs"   and "tc.oalist_inv_raw ys"   and "aux \<noteq> Some Eq"   and "k \<in> fst ` set xs \<union> fst ` set ys"   and "aux = f k (lookup_pair_tc xs k) (lookup_pair_tc ys k)"   and "\<And>k'. \<lbrakk>k' \<in> fst ` set xs \<union> fst ` set ys; tc.lt k' k\<rbrakk> \<Longrightarrow> f k' (lookup_pair_tc xs k') (lookup_pair_tc ys k') = Some Eq" shows "lex_ord_pair_tc f xs ys = aux"
restrict_nonempty: fixes k :: "'c"   and ks :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd"   and l :: "'c"   and ls :: "'c list" shows "(k # ks) \<langle>f\<rangle> (l # ls) = [k] \<langle>f\<rangle> [l] \<oplus> [k] \<langle>f\<rangle> ls \<oplus> ks \<langle>f\<rangle> [l] \<oplus> ks \<langle>f\<rangle> ls"
natural_transformation_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" shows "natural_transformation A B F G \<tau> \<equiv> (category A \<and> category B) \<and> functor A B F \<and> functor A B G \<and> natural_transformation_axioms A B F G \<tau>"
dom_strip: fixes F :: "'f set"   and \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option" shows "dom (strip F \<Gamma>) = dom \<Gamma>"
SReal_approx_iff: fixes x :: "real star"   and y :: "real star" assumes "x \<in> \<real>"   and "y \<in> \<real>" shows "(x \<approx> y) = (x = y)"
intros_17: fixes c1 :: "f32"   and c2 :: "f32"   and fop :: "relop_f" shows "\<lparr>[$C ConstFloat32 c1, $C ConstFloat32 c2, $Relop_f T_f32 fop]\<rparr> \<leadsto> \<lparr>[$C ConstInt32 (wasm_bool (app_relop_f fop c1 c2))]\<rparr>"
program_trace_def: fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'b \<Rightarrow> 'b \<times> instr list \<Rightarrow> bool"   and c :: "nat \<Rightarrow> ('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and n :: "nat"   and k :: "nat" shows "program_trace program_step c n k \<equiv> \<forall>l<k. computation.concurrent_step direct_memop_step empty_storebuffer_step program_step (\<lambda>p p' is sb. sb) (c (n + l)) (c (n + Suc l))"
SeqHaddP_SUCC: fixes H :: "fm set"   and s :: "tm"   and j :: "tm"   and k :: "tm"   and y :: "tm"   and y' :: "name" assumes "H \<turnstile> SeqHaddP s j (SUCC k) y"   and "atom y' \<sharp> (s, j, k, y)" shows "H \<turnstile> SyntaxN.Ex y' (SeqHaddP s j k (Var y') AND y EQ SUCC (Var y'))"
nmp_eq: fixes x :: "i"   and y :: "i" assumes "M2" shows "PP x y = (P x y \<and> \<not> P y x)"
bernoulli_denom_odd: fixes n :: "nat" assumes "n \<noteq> 1"   and "odd n" shows "bernoulli_denom n = 1"
assms_3: shows "[M]\<^bsub>v,n\<^esub> \<noteq> {}"
3: shows "trms\<^sub>s\<^sub>t (a # A \<cdot>\<^sub>s\<^sub>t \<theta>) = (trms\<^sub>s\<^sub>t\<^sub>p a \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>) \<union> trms\<^sub>s\<^sub>t (A \<cdot>\<^sub>s\<^sub>t \<theta>)"
icds_1: shows "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
abs_e2: shows "\<bar>e2\<bar> \<le> snd (trunc_bound_pdevs p (add_pdevs (snd (fst X)) (snd (fst Y))))"
insert: fixes g :: "'c \<Rightarrow> 'b"   and x :: "'c"   and A :: "'c multiset" shows "\<Sum>\<^sub># (image_mset g (add_mset x A)) = g x + \<Sum>\<^sub># (image_mset g A)"
predicate2D: fixes P :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Q :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and x :: "'n"   and y :: "'o" assumes "P \<le> Q"   and "P x y" shows "Q x y"
subgraph_forest: fixes G :: "('a, 'b) graph"   and H :: "('a, 'b) graph" assumes "forest G"   and "subgraph H G" shows "forest H"
\<U>: shows "Ball \<U> (openin X) \<and> S \<union> T \<subseteq> \<Union> \<U>"
IH_1: fixes u :: "'a" assumes "linear t\<^sub>1" shows "case match t\<^sub>1 u of None \<Rightarrow> \<forall>env. subst (convert_term t\<^sub>1) env \<noteq> u | Some env \<Rightarrow> subst (convert_term t\<^sub>1) env = u"
finite_surj: fixes A :: "'b set"   and B :: "'c set"   and f :: "'b \<Rightarrow> 'c" assumes "finite A"   and "B \<subseteq> f ` A" shows "finite B"
isomorphic_implies_hpar_1: fixes f :: "'c"   and f' :: "'c" assumes "C.isomorphic f f'" shows "C.ide f"
eucl_of_list_aform_def: fixes xs :: "(real \<times> real pdevs) list" shows "eucl_of_list_aform xs = (eucl_of_list (map fst xs), sum_pdevs (\<lambda>i. pdevs_scaleR (snd (xs ! index Basis_list i)) i) Basis)"
trace_at_eq: fixes s :: "'b"   and \<omega> :: "(real \<times> 'b) stream"   and j :: "real" shows "trace_at s \<omega> j = (case sfirst (\<lambda>x. j < fst (shd x)) \<omega> of enat i \<Rightarrow> (s ## smap snd \<omega>) !! i)"
map_rsuml_plus_oracle: fixes oracle1 :: "'h \<Rightarrow> 'i \<Rightarrow> ('l \<times> 'h) spmf"   and oracle2 :: "'h \<Rightarrow> 'j \<Rightarrow> ('m \<times> 'h) spmf"   and oracle3 :: "'h \<Rightarrow> 'k \<Rightarrow> ('n \<times> 'h) spmf" shows "map_fun id (map_fun rsuml (map_spmf (map_prod lsumr id))) (oracle1 \<oplus>\<^sub>O oracle2 \<oplus>\<^sub>O oracle3) = (oracle1 \<oplus>\<^sub>O oracle2) \<oplus>\<^sub>O oracle3"
ccINF_lower2: fixes A :: "'c set"   and i :: "'c"   and f :: "'c \<Rightarrow> 'a"   and u :: "'a" assumes "countable A"   and "i \<in> A"   and "f i \<le> u" shows "Inf (f ` A) \<le> u"
monom_mult_def: fixes n :: "nat"   and f :: "'a poly" shows "monom_mult n f = Polynomial.monom (1::'a) n * f"
one: shows "R (arith_ops_record.one ff_ops) 1"
rcs_fixed2: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set" assumes "Group G"   and "G \<guillemotright> H " shows "\<forall>h\<in>H. H \<bullet>\<^bsub>G\<^esub> h = H"
neg_in_filterD: fixes A :: "'a"   and \<rho>' :: "'s"   and D :: "'a literal list" assumes "Neg A \<in># {#L \<cdot>l \<rho>'. L \<in># mset D#}" shows "\<exists>A0. Neg A0 \<in># mset D \<and> A = A0 \<cdot>a \<rho>'"
vars1'_def: shows "st1.vars1' = ind_in_set st1.vars0 ` vars1"
finprod_reindex: fixes f :: "'d \<Rightarrow> 'a"   and h :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "f \<in> h ` A \<rightarrow> carrier G"   and "inj_on h A" shows "finprod G f (h ` A) = (\<Otimes>x\<in>A. f (h x))"
take_cols_from_matrix_def: fixes matr :: "'a mat"   and indices_to_keep :: "nat list" shows "take_cols_from_matrix matr indices_to_keep = mat_of_cols (dim_row matr) (take_indices (cols matr) indices_to_keep)"
Int_absorb2: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subseteq> B" shows "A \<inter> B = A"
prod_orD_def: fixes p1 :: "'e \<Rightarrow> 'g decision option"   and p2 :: "'f \<Rightarrow> 'h decision option" shows "p1 \<Otimes>\<^sub>\<or>\<^sub>D p2 = (\<lambda>(x, y). case p1 x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d1\<rfloor> \<Rightarrow> case p2 y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d2\<rfloor> \<Rightarrow> \<lfloor>allow (d1, d2)\<rfloor> | \<lfloor>deny d2\<rfloor> \<Rightarrow> \<lfloor>deny (d1, d2)\<rfloor> | \<lfloor>deny d1\<rfloor> \<Rightarrow> case p2 y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d2\<rfloor> \<Rightarrow> \<lfloor>deny (d1, d2)\<rfloor> | \<lfloor>deny d2\<rfloor> \<Rightarrow> \<lfloor>deny (d1, d2)\<rfloor>)"
pfun_upd_add: fixes f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun"   and x :: "'a"   and v :: "'b" shows "f + g(x \<mapsto> v)\<^sub>p = (f + g)(x \<mapsto> v)\<^sub>p"
pwd_spy_enc_sign_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "case pwd_spy_enc_sign n of (s, S, A, B, C, D) \<Rightarrow> pwd_spy_enc_sign_state n (S, A, B, C, D) \<subseteq> s \<and> {Key (Auth_ShaKey n)} \<in> key_sets (PriKey S) (crypts (Log -` spied s))"
Refinement_using_invariants: fixes R :: "('a \<times> 'b) set"   and I :: "'a set"   and J :: "'b set"   and Sa :: "('a, 'c) spec"   and Sc :: "('b, 'd) spec"   and pi :: "'d \<Rightarrow> 'c" assumes "{R \<inter> I \<times> J} TS.trans Sa, TS.trans Sc {> R}"   and "{I \<inter> Domain (R \<inter> UNIV \<times> J)} TS.trans Sa {> I}"   and "{J \<inter> Range (R \<inter> I \<times> UNIV)} TS.trans Sc {> J}"   and "init Sc \<subseteq> R `` init Sa"   and "init Sa \<subseteq> I"   and "init Sc \<subseteq> J"   and "obs_consistent (R \<inter> I \<times> J) pi Sa Sc" shows "refines (R \<inter> I \<times> J) pi Sa Sc"
vsv_vlrestriction_vinsert: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (xs \<restriction>\<^sup>l\<^sub>\<circ> k)" shows "(xs \<restriction>\<^sup>l\<^sub>\<circ> k) \<restriction>\<^sup>l\<^sub>\<circ> vinsert a A = vinsert \<langle>a, (xs \<restriction>\<^sup>l\<^sub>\<circ> k)\<lparr>a\<rparr>\<rangle> ((xs \<restriction>\<^sup>l\<^sub>\<circ> k) \<restriction>\<^sup>l\<^sub>\<circ> A)"
mem_affine_3_minus: fixes S :: "'c set"   and x :: "'c"   and y :: "'c"   and z :: "'c"   and v :: "real" assumes "affine S"   and "x \<in> S"   and "y \<in> S"   and "z \<in> S" shows "x + v *\<^sub>R (y - z) \<in> S"
Suc: fixes s :: "'s" shows "T s = T s \<bind> (\<lambda>\<omega>. distr (T ((s ## \<omega>) !! n)) (stream_space (count_space UNIV)) ((@-) (stake n \<omega>)))"
lastn_def: fixes n :: "nat"   and x :: "'b list" shows "lastn n x = drop (length x - n) x"
ip: shows "ip = rip \<and> Suc 0 \<le> hops \<or> ip \<noteq> rip \<and> ip \<in> kD rt"
strict_order_iff: fixes a :: "'b DBMEntry"   and b :: "'b DBMEntry" shows "(a \<prec> b) = (a = linordered_monoid.min a b \<and> a \<noteq> b)"
sin_vangle_zero_iff: fixes u :: "'b"   and v :: "'b" shows "(sin (vangle u v) = 0) = (vangle u v \<in> {0, pi})"
dom_clearjunk: fixes al :: "('c \<times> 'd) list" shows "fst ` set (clearjunk al) = fst ` set al"
healthy_intros_20: fixes p :: "'b \<Rightarrow> 'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>a s. a \<in> supp (p s) \<Longrightarrow> nearly_healthy (wlp (f a))"   and "\<And>s. sound (p s)"   and "\<And>s. sum (p s) (supp (p s)) \<le> 1" shows "nearly_healthy (wlp (SetPC f p))"
hyps: fixes p :: "'a node ref option"   and ys :: "'a list"   and q :: "'a node ref option" shows "<os_list xs p * os_list ys q> os_reverse_aux q p <os_list (rev xs @ ys)>"
degree_poly_lift: fixes p :: "'a poly" shows "degree (poly_lift p) = degree p"
set_rev: fixes xs :: "'b list" shows "set (rev xs) = set xs"
boundedI: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subseteq># b"   and "a \<subseteq># c" shows "a \<subseteq># b \<inter># c"
affine_hull_nonempty_interior: fixes S :: "'c set" assumes "interior S \<noteq> {}" shows "affine hull S = UNIV"
add_comm: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "x \<union> y = y \<union> x"
shd_def: fixes stream :: "'b stream" shows "shd stream = (case stream of x1 ## x2 \<Rightarrow> x1)"
absolutely_continuous_def: fixes M :: "'d measure"   and N :: "'d measure" shows "absolutely_continuous M N = (null_sets M \<subseteq> null_sets N)"
ennreal_pmf_bind: fixes N :: "'i pmf"   and f :: "'i \<Rightarrow> 'h pmf"   and i :: "'h" shows "ennreal (pmf (N \<bind> f) i) = \<integral>\<^sup>+ x. ennreal (pmf (f x) i) \<partial>measure_pmf N"
wf_permute_alts: fixes \<sigma> :: "'alt \<Rightarrow> 'alt" assumes "\<sigma> permutes alts" shows "pref_profile_wf agents alts (permute_profile \<sigma> (permute_profile \<sigma> R))"
nvars_SNOCs: fixes k :: "nat"   and x :: "'x"   and \<AA> :: "'i" shows "#\<^sub>V ((SNOC x ^^ k) \<AA>) = #\<^sub>V \<AA>"
assms: fixes \<Psi> :: "'b" shows "\<Psi> \<rhd> P \<doteq> Q"
ceiling_log_eq_powr_iff: fixes x :: "real"   and b :: "real"   and k :: "nat" assumes "0 < x"   and "1 < b" shows "(\<lceil>log b x\<rceil> = int k + 1) = (b powr real k < x \<and> x \<le> b powr real (k + 1))"
\<b>_card_X: fixes X :: "'x set"   and m :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and i :: "nat" assumes "finite X"   and "exact_decomp X m ps"   and "Suc (card X) \<le> i" shows "\<b> ps i = \<a> ps"
rel_star_def: fixes X :: "('a \<times> 'a) set" shows "X\<^sup>* = \<Union> (range (rel_dioid.power X))"
hcomp_closed: fixes f :: "'d"   and g :: "'d" assumes "D.arr f \<and> Obj (src\<^sub>D f) \<and> Obj (trg\<^sub>D f)"   and "D.arr g \<and> Obj (src\<^sub>D g) \<and> Obj (trg\<^sub>D g)"   and "trg\<^sub>D f = src\<^sub>D g" shows "D.hseq g f \<and> Obj (src\<^sub>D (g \<star>\<^sub>D f)) \<and> Obj (trg\<^sub>D (g \<star>\<^sub>D f))"
simps_1: fixes \<mu> :: "'a" shows "E.Trg \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0"
span_finite: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'c set" assumes "module scale"   and "finite S" shows "module.span scale S = range (\<lambda>u. \<Sum>v\<in>S. scale (u v) v)"
assms_5: shows "safe_reach_upto n safe_free_flowing c\<^sub>0"
preserves_dom: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and f :: "'d" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "partial_magma.dom B (\<tau> f) = F (partial_magma.dom A f)"
Fourier_coefficient_def: shows "Fourier_coefficient \<equiv> orthonormal_coeff {- pi..pi} trigonometric_set"
att_orD2: fixes x1 :: "'a attree"   and x2 :: "'a attree list"   and x :: "'a set \<times> 'a set" assumes "\<turnstile>x1 # x2 \<oplus>\<^sub>\<or>\<^bsup>x\<^esup>" shows "\<turnstile>x2 \<oplus>\<^sub>\<or>\<^bsup>(fst x - fst (attack x1), snd x)\<^esup>"
top_uexpr_rep_eq: fixes b :: "'a" shows "\<lbrakk>\<top>\<rbrakk>\<^sub>e b = False"
inv_uptoI: fixes m :: "nat"   and p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat" assumes "\<And>i j. \<lbrakk>i < n; j < n; j < m\<rbrakk> \<Longrightarrow> p A i j" shows "local.inv_upto p A m"
T5: fixes w :: "i"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>" shows "[\<psi> \<^bold>\<rightarrow> \<chi> in w] = (\<not> [\<psi> in w] \<or> [\<chi> in w])"
set_map_\<alpha>_cong: fixes \<alpha>' :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> M \<Longrightarrow> \<alpha>' x = \<eta> x"   and "\<alpha>' \<in> extensional M" shows "Set_Theory.map \<alpha>' M M'"
sum_list_mono_lt: fixes xs :: "real list"   and m :: "real" assumes "xs \<noteq> [] \<and> (\<forall>x\<in>set xs. x < m)" shows "sum_list xs < m * real (length xs)"
is_extensional: fixes f :: "'c \<times> 'c" assumes "\<not> C.VV.arr f" shows "G.H\<^sub>DoFF.map f = D.null"
Preflow_axioms: shows "Preflow c s t (augment (augmentingFlow p))"
asm: shows "\<xi> \<Turnstile>\<^sub>c \<phi> W\<^sub>c \<psi>"
SystemCEFml1_def: shows "SystemCEFml1 = Geq (Differential (Var vid1)) (Differential (Const 0))"
weak_respection_of_barbs_and_closures_1: fixes Rel :: "('b \<times> 'b) set"   and CWB :: "('b, 'c) calculusWithBarbs" assumes "rel_weakly_respects_barbs Rel CWB" shows "rel_weakly_respects_barbs (Rel\<^sup>=) CWB"
A: shows "(r \<star> \<r>[f]) \<cdot> \<a>[r, f, trg f\<^sub>a] \<cdot> (\<rho> \<star> trg f\<^sub>a) = ((r \<star> \<l>[f] \<cdot> (\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, f\<^sub>a, f]) \<cdot> \<a>[r, f, f\<^sub>a \<star> f] \<cdot> (\<rho> \<star> f\<^sub>a \<star> f)) \<cdot> \<a>[g, f\<^sub>a, f] \<cdot> (\<nu> \<star> f) \<cdot> \<rho> \<cdot> \<r>[g]"
simps_7: fixes opt :: "atom fm \<Rightarrow> atom fm"   and step :: "nat \<Rightarrow> nat \<Rightarrow> atom list \<Rightarrow> atom fm list \<Rightarrow> atom fm"   and a :: "atom" shows "QE_dnf opt step (fm.Atom a) = simp_atom a"
phi_diff_aux: fixes xs :: "'b list"   and ys :: "'b list"   and x :: "'b" shows "int (card (Inv xs ys \<union> {(y, x) |y. y < x in xs \<and> y < x in ys} - {(x, y) |y. x < y in xs \<and> y < x in ys})) = int (card (Inv xs ys) + card (before x xs \<inter> before x ys)) - int (card (after x xs \<inter> before x ys))"
subst_apply_term_empty: fixes t :: "('a, 'b) Term.term" shows "subst_atm_abbrev t Var = t"
set_insert: fixes x :: "'b list"   and s :: "'b trie" assumes "True" shows "Trie_Fun.set (Trie_Fun.insert x s) = Trie_Fun.set s \<union> {x}"
small_step_fragmentsLS: fixes s :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j"   and \<alpha> :: "('g, 'i) seq_label"   and s' :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j" assumes "s \<rightarrow>\<^bsub>\<alpha>\<^esub> s'" shows "fragmentsLS s' \<subseteq> fragmentsLS s"
eq_2: shows "H \<turnstile> x2 EQ x2'"
has_fps_expansion_cos: fixes c :: "'a" shows "(\<lambda>x. cos (c * x)) has_fps_expansion fps_cos c"
insert_3: shows "dom (F(x \<mapsto> y)) = dom F2"
inverse_aform: fixes e :: "nat \<Rightarrow> real"   and p :: "nat"   and X :: "real \<times> real pdevs"   and Y :: "(real \<times> real pdevs) \<times> real" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "inverse_aform p X = Some Y" shows "inverse (aform_val e X) \<in> aform_err e Y"
meld_def: fixes bq1 :: "('c, 'd) SkewBinomialTree list"   and bq2 :: "('c, 'd) SkewBinomialTree list" shows "meld bq1 bq2 = meldUniq (uniqify bq1) (uniqify bq2)"