hcju/isabelleps huggingface dataset:
WTrt_elim_cases_14: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr"   and D :: "String.literal"   and F :: "String.literal"   and e2 :: "'addr expr"   and e3 :: "'addr expr"   and T :: "ty" assumes "P,E,h \<turnstile> e\<bullet>compareAndSwap(D\<bullet>F, e2, e3) : T" obtains U :: "ty" and C :: "String.literal" and T :: "ty" and fm :: "fmod" and T2 :: "ty" and T3 :: "ty" where "Ta__ = Boolean" and "P,E,h \<turnstile> e : U" and "class_type_of' U = \<lfloor>C\<rfloor>" and "P \<turnstile> C has F:T (fm) in D" and "volatile fm" and "P,E,h \<turnstile> e2 : T2__" and "P \<turnstile> T2__ \<le> T" and "P,E,h \<turnstile> e3 : T3__" and "P \<turnstile> T3__ \<le> T"   | T2 :: "ty" and T3 :: "ty" where "Ta__ = Boolean" and "P,E,h \<turnstile> e : NT" and "P,E,h \<turnstile> e2 : T2" and "P,E,h \<turnstile> e3 : T3"
Q'_def: shows "Q' \<equiv> Abs_poly (\<lambda>k. if n < k then 0 else (- 1) ^ (n - k) * e' (n - k))"
ra_in_s_set: fixes r :: "'a"   and a :: "'d"   and A :: "'d set"   and s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd" assumes "r \<in> carrier R"   and "a \<in> A" shows "r \<^sub>s\<cdot> a \<in> s_set R s A"
lang_dual_rexp_dual_of: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp" assumes "M2L_Equivalence_Checking.wf \<Sigma> n r" shows "lang_dual \<Sigma> n (rexp_dual_of r) = lang \<Sigma> n r"
compass_reflect_compass_2: shows "apply_cltn2 west meridian_reflect = east"
has_derivative_on_open: fixes t :: "real"   and s :: "'a" assumes "t \<in> T"   and "s \<in> S" obtains B :: "real set" where "t \<in> B" and "open B" and "B \<subseteq> T" and "D (\<lambda>\<tau>. \<phi> \<tau> s) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R f (\<phi> t s)) at t within B"
swap_swap: fixes i :: "'b"   and j :: "'b"   and x :: "'b" shows "swap i j (swap i j x) = x"
distinct_1: shows "EV_SIGNAL_PREP \<noteq> EV_SIGNAL_FINISH"
FreeUltrafilterNat_HFinite: fixes X :: "nat \<Rightarrow> 'b"   and u :: "real" assumes "\<forall>\<^sub>F n in \<U>. norm (X n) < u" shows "star_n X \<in> HFinite"
document_ptr_document_ptr_cast: fixes document_ptr :: "(_) document_ptr" shows "is_document_ptr_kind (cast document_ptr)"
Abs_lst_fcb2: fixes as :: "atom list"   and x :: "'e"   and bs :: "atom list"   and y :: "'e"   and f :: "atom list \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> 'f"   and c :: "'g" assumes "[as]lst. x = [bs]lst. y"   and "set as \<sharp>* f as x c"   and "set as \<sharp>* c"   and "set bs \<sharp>* c"   and "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> f as x c = f (p \<bullet> as) (p \<bullet> x) c"   and "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> f bs y c = f (p \<bullet> bs) (p \<bullet> y) c" shows "f as x c = f bs y c"
eff\<^sub>i_Load: fixes n :: "nat"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and ST :: "ty list"   and LT :: "ty err list" shows "eff\<^sub>i (Load n, P, ST, LT) = (ok_val (LT ! n) # ST, LT)"
trm_prm_apply_id: fixes M :: "'b trm" shows "\<epsilon> \<cdot> M = M"
vbrelation_vcomp: fixes r :: "V"   and s :: "V" assumes "vbrelation r"   and "vbrelation s" shows "vbrelation (r \<circ>\<^sub>\<circ> s)"
set_map: fixes f :: "'d \<Rightarrow> 'c"   and v :: "'d stream" shows "sset (smap f v) = f ` sset v"
red_not_final_thread: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s -t\<triangleright>ta\<rightarrow> s'" shows "not_final_thread s t"
IntD1: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "c \<in> A \<inter> B" shows "c \<in> A"
cos_diff: fixes x :: "'a"   and y :: "'a" shows "cos (x - y) = cos x * cos y + sin x * sin y"
module_hom_sub: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "linear (*a) (*b) f"   and "linear (*a) (*b) g" shows "linear (*a) (*b) (\<lambda>x. f x - g x)"
assms_1: shows "A, n \<turnstile> (p # ps, a) # branch"
par_by_merge_seq_add: fixes P :: "('d, 'g) urel"   and M :: "(('d, 'g, 'h) mrg, 'f) urel"   and Q :: "('d, 'h) urel"   and R :: "('f, 'e) urel" shows "P \<parallel>\<^bsub>M\<^esub> Q ;; R = P \<parallel>\<^bsub>M ;; R\<^esub> Q"
inj_on_rev: fixes A :: "'a list set" shows "inj_on rev A"
lunit'_in_hom_2: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "\<guillemotleft>\<l>\<^sup>-\<^sup>1[f] : f \<Rightarrow> trg f \<star> f\<guillemotright>"
sigma_finite_measure_axioms: shows "sigma_finite_measure (count_space B)"
us: shows "us = getFrN (map snd txs) (map fst txs) [exi y \<phi>] (length txs)"
hom_decomp_shift_list: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and hU :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set"   and x :: "'x" assumes "hom_decomp ps"   and "hU \<in> set ps" shows "hom_decomp (shift_list hU x ps)"
list_length_hint1a: fixes l :: "'a list" assumes "l \<noteq> []" shows "0 < length l"
assms_1: shows "ip = ip' \<or> ip \<in> kD rt"
fun_pair_wf\<^sub>t\<^sub>r\<^sub>m: fixes t :: "('fun, 'b) Term.term"   and t' :: "('fun, 'b) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "wf\<^sub>t\<^sub>r\<^sub>m t'" shows "wf\<^sub>t\<^sub>r\<^sub>m (pair (t, t'))"
set_not_fresh: fixes x :: "'b"   and L :: "'b list" assumes "x \<in> set L" shows "\<not> atom x \<sharp> L"
emeasure_Collect_T: fixes P :: "('s \<times> 't) stream \<Rightarrow> bool"   and s :: "'s \<times> 't" assumes "Measurable.pred (stream_space (count_space UNIV)) P" shows "emeasure (T s) {x \<in> space (T s). P x} = \<integral>\<^sup>+ t. emeasure (T t) {x \<in> space (T t). P (t ## x)} \<partial>measure_pmf (K s)"
antisym_conv2: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "(\<not> x < y) = (x = y)"
\<phi>_less_iff: fixes \<gamma> :: "V"   and \<delta> :: "V" assumes "\<gamma> \<in> elts (tp D)"   and "\<delta> \<in> elts (tp D)" shows "(\<phi> \<gamma> < \<phi> \<delta>) = (\<gamma> < \<delta>)"
simps_2: fixes n1 :: "('a, 'b) Node" shows "FingerTreeStruc.is_measured_ftree (Single n1) = FingerTreeStruc.is_measured_node n1"
bar_cases: fixes x1ca :: "nat"   and xaa :: "nat"   and x3ba :: "'a" shows "bar x1ca xaa x3ba = (case xaa of 0 \<Rightarrow> 0 + x1ca | Suc x2ca \<Rightarrow> x2ca + x1ca)"
wf'_1: shows "case_nat REAL \<Gamma> \<turnstile>\<^sub>c f : REAL"
Integer_arith_strict_3: fixes x :: "Integer" shows "\<bottom> * x = \<bottom>"
simps_2: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list" shows "\<lbrakk>M; send\<langle>t\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>c = (\<lambda>\<I>. M \<turnstile>\<^sub>c t \<cdot> \<I> \<and> \<lbrakk>M; S\<rbrakk>\<^sub>c \<I>)"
simps_2: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and P :: "'e \<Rightarrow> bool"   and U :: "'f set"   and x :: "'e"   and xs :: "'e list"   and X :: "'e set" shows "ipurge_ref_aux_foldr I D P U (x # xs) X = ipurge_ref_aux I D U ((if P x then [x] else []) @ ipurge_tr_aux_foldr I D P (sinks_aux I D U [x]) xs) (ipurge_ref_aux_foldr I D P (sinks_aux I D U [x]) xs X)"
scalar_product_append: fixes n :: "nat" shows "\<forall>xs ys zs ws. length zs = length ws \<and> length xs = length ys \<and> length xs = n \<longrightarrow> scalar_product (xs @ zs) (ys @ ws) = scalar_product xs ys + scalar_product zs ws"
mkCone_def: fixes p0 :: "'c"   and p1 :: "'c" shows "D.mkCone p0 p1 \<equiv> \<lambda>j. if j = J.AA then p0 else if j = J.BB then p1 else if j = J.AT then f0 \<cdot> p0 else if j = J.BT then f1 \<cdot> p1 else if j = J.TT then f0 \<cdot> p0 else C.null"
dom_uses_in_graph: fixes g :: "'g" shows "Mapping.keys (u_g g s1_uses g) \<subseteq> set (\<alpha>n g)"
index_mono_def: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd" shows "index_mono index key \<equiv> \<forall>x y n mi ma. {key x, key y} \<subseteq> {mi..ma} \<longrightarrow> key x \<le> key y \<longrightarrow> index key x n mi ma \<le> index key y n mi ma"
hom_pass2': fixes hs :: "'b heap list" assumes "no_Emptys hs" shows "hom (pass\<^sub>2' hs) = Pairing_Heap_Tree.pass\<^sub>2 (homs hs)"
start_heap_obs_not_Read: fixes ad :: "'a"   and al :: "addr_loc"   and v :: "'a val"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set" shows "ReadMem ad al v \<notin> set (heap_base.start_heap_obs empty_heap allocate)"
lr_of_tran_fbs_dropI: fixes rt :: "'b routing_rule list"   and fw :: "'b simple_rule list"   and p :: "('b, 'c) simple_packet_scheme"   and ifs :: "'d" assumes "valid_prefixes rt"   and "has_default_route rt"   and "no_oif_match fw"   and "has_default_policy fw"   and "simple_linux_router_nol12 rt fw p = None" shows "\<exists>r oif. generalized_sfw (lr_of_tran_fbs rt fw ifs) p = Some (r, oif, Drop)"
aboveS_decr: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "trans r"   and "antisym r"   and "(a, b) \<in> r" shows "aboveS r b \<subseteq> aboveS r a"
f: shows "q.eq_m f (Polynomial.smult (lead_coeff f) (Polynomial.smult ilc f))"
padic_mult_comm: fixes p :: "int"   and x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "prime p"   and "x \<in> carrier (padic_int p)"   and "y \<in> carrier (padic_int p)" shows "x \<otimes>\<^bsub>padic_int p\<^esub> y = y \<otimes>\<^bsub>padic_int p\<^esub> x"
wf_start_prog_Start_sees_start_method: fixes wf_md :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and P :: "jvm_prog"   and C :: "char list"   and M :: "char list" assumes "wf_prog wf_md P" shows "class_add P (start_class C M) \<turnstile> Start sees start_m, Static :  []\<rightarrow>Void = (1, 0, [Invokestatic C M 0, Return], []) in Start"
compact_Times: fixes s :: "'b set"   and t :: "'c set" assumes "compact s"   and "compact t" shows "compact (s \<times> t)"
single_valuedD: fixes r :: "('b \<times> 'c) set"   and x :: "'b"   and y :: "'c"   and z :: "'c" assumes "single_valued r"   and "(x, y) \<in> r"   and "(x, z) \<in> r" shows "y = z"
dvd: shows "A * B * C div radical (A * B * C) dvd pderiv C * B - C * pderiv B"
\<chi>_def: shows "\<chi> = \<a>[k\<^sup>*, h, p\<^sub>0 \<star> w] \<cdot> \<a>[k\<^sup>* \<star> h, p\<^sub>0, w] \<cdot> (tab \<star> w)"
is_cont: shows "\<forall>i\<in>Basis. continuous_on s (\<lambda>x. f x \<bullet> i)"
super_def: fixes P :: "'a prog"   and C :: "char list" shows "super P C \<equiv> fst (the (class P C))"
hyps_2: assumes "rsquarefree p" shows "proots_count p s = card (proots_within p s)"
scale_zero_left: fixes x :: "'c" shows "0 *\<^sub>C x = (0::'c)"
assms: shows "coincide_on \<sigma> \<eta> (vars_of_lit l)"
wwf_mdecl_def: 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 C :: "char list" shows "wwf_mdecl P C \<equiv> \<lambda>(M, Ts, T, pns, body). length Ts = length pns \<and> distinct pns \<and> this \<notin> set pns \<and> fv body \<subseteq> {this} \<union> set pns"
range_split_def: fixes P :: "('c \<Rightarrow> 'b) \<times> ('c \<Rightarrow> 'b)"   and p :: "'a \<Rightarrow> 'c decision option" shows "P \<nabla> p = (\<lambda>x. case p x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow y\<rfloor> \<Rightarrow> \<lfloor>allow (fst P y)\<rfloor> | \<lfloor>deny y\<rfloor> \<Rightarrow> \<lfloor>deny (snd P y)\<rfloor>)"
nuc_prop2: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "nucleus f" shows "f (x \<cdot> y) = f (f x \<cdot> y)"
map_add_disj': fixes h\<^sub>1 :: "'c \<Rightarrow> 'd option"   and h\<^sub>2 :: "'c \<Rightarrow> 'd option"   and h\<^sub>0 :: "'c \<Rightarrow> 'd option" shows "h\<^sub>1 ++ h\<^sub>2 \<bottom> h\<^sub>0 = (h\<^sub>1 \<bottom> h\<^sub>0 \<and> h\<^sub>2 \<bottom> h\<^sub>0)"
2: shows "local.iso (\<a>[a, \<I> \<otimes> b, c] \<cdot> (\<a>[a, \<I>, b] \<otimes> c))"
that: shows "constant_term (denominator x) \<noteq> (0::'a)"
col_def: fixes A :: "'a mat"   and j :: "nat" shows "col A j = Matrix.vec (dim_row A) (\<lambda>i. A $$ (i, j))"
vec_uniqueness: fixes m :: "nat"   and v :: "'b list"   and n :: "nat" assumes "vec m v"   and "vec n v" shows "m = n"
right_mult_zero_mat: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n" shows "A * 0\<^sub>m n nc = 0\<^sub>m nr nc"
crit_pair_cbelow_distinct_component: fixes p :: "'t \<Rightarrow>\<^sub>0 'e"   and q :: "'t \<Rightarrow>\<^sub>0 'e"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and F :: "('t \<Rightarrow>\<^sub>0 'e) set" assumes "component_of_term (lt p) \<noteq> component_of_term (lt q)" shows "crit_pair_cbelow_on d m F p q"
simps_6: fixes r :: "mregex"   and \<phi>s :: "Formula.formula list" shows "from_mregex (MStar r) \<phi>s = regex.Star (from_mregex r \<phi>s)"
arr_MkArr: fixes F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" shows "arr (MkArr F G \<tau>) = natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G \<tau>"
possible_steps_2_vend: fixes r :: "nat \<Rightarrow>f value option" assumes "\<exists>n. r $ 2 = Some (Num n) \<and> 100 \<le> n" shows "possible_steps drinks 1 r STR ''vend'' [] = {|(2, vend)|}"
conjugate_eq_aux: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "is_conjugation f g" shows "f (x \<cdot> - g y) \<le> f x \<cdot> - y"
legendre_aux_ge: fixes p :: "nat"   and k :: "real" assumes "prime p"   and "1 \<le> k" shows "nat \<lfloor>k / real p\<rfloor> \<le> legendre_aux k p"
Union_image_insert: fixes f :: "'d \<Rightarrow> 'c set"   and a :: "'d"   and B :: "'d set" shows "\<Union> (f ` insert a B) = f a \<union> \<Union> (f ` B)"
one_mult_sv: fixes v :: "complex \<times> complex" shows "1 *\<^sub>s\<^sub>v v = v"
corthogonal_matD: fixes A :: "'b mat"   and i :: "nat"   and j :: "nat" assumes "corthogonal_mat A"   and "i < dim_col A"   and "j < dim_col A" shows "(col A i \<bullet>c col A j = (0::'b)) = (i \<noteq> j)"
converged_cd_diverge: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and l :: "nat"   and m :: "nat"   and m' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "\<pi> 0 = \<pi>' 0"   and "\<nexists>l'. cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<pi>'\<^esup> l'"   and "l < m"   and "cs\<^bsup>\<pi>\<^esup> m = cs\<^bsup>\<pi>'\<^esup> m'" obtains k :: "nat" and k' :: "nat" where "cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'" and "l cd\<^bsup>\<pi>\<^esup>\<rightarrow> k" and "\<pi> (Suc k) \<noteq> \<pi>' (Suc k')"
arr_MkArr: fixes F :: "'c \<Rightarrow> 's"   and G :: "'c \<Rightarrow> 's"   and \<tau> :: "'c \<Rightarrow> 's" shows "Cop_S.arr (Cop_S.MkArr F G \<tau>) = natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) F G \<tau>"
ANR_openin: fixes T :: "'b set"   and S :: "'b set" assumes "ANR T"   and "openin (top_of_set T) S" shows "ANR S"
X_def: shows "X = [:0::'c, 1::'c:]"
colossless_gpv_lossless_spmfD: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv" assumes "colossless_gpv \<I> gpv" shows "lossless_spmf (the_gpv gpv)"
Base_declares_foo: shows "tprg\<turnstile> mdecl Base_foo declared_in Base"
wf_operators_2: shows "Ball (set ast\<delta>) wf_operator"
simps_6: fixes \<xi> :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> or\<^sub>n \<psi> = (\<xi> \<Turnstile>\<^sub>n \<phi> \<or> \<xi> \<Turnstile>\<^sub>n \<psi>)"
option_pred_o_map: fixes P :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" shows "pred_option P \<circ> map_option f = pred_option (P \<circ> f)"
strong_int_gr_induct_helper: fixes k :: "int"   and i :: "int"   and P :: "int \<Rightarrow> bool"   and j :: "int" assumes "k < i"   and "\<And>i. \<lbrakk>k < i; \<And>j. \<lbrakk>k < j; j < i\<rbrakk> \<Longrightarrow> P j\<rbrakk> \<Longrightarrow> P i"   and "k < j"   and "j < i" shows "P j"
WTs_implies_WTrts: 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 E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and Ts :: "ty list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" assumes "P,E \<turnstile> es [::] Ts" shows "P,E,h \<turnstile> es [:] Ts"
ring_irreducibleE_2: fixes r :: "'a" assumes "r \<in> carrier R"   and "ring_irreducible r" shows "irreducible R r"
assms_3: fixes as :: "('i \<times> 'a atom) set"   and bs :: "('i \<times> 'a atom) set"   and t :: "('i, 'a) state" assumes "as \<subseteq> bs"   and "P as t" shows "P bs t"
path_valid_node_1: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>* n'" shows "valid_node n"
igSubstIGVar1_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gSubstGVar1 MOD" shows "igSubstIGVar1 (fromMOD MOD)"
reduce_root_bal: fixes t :: "'b btree" shows "bal (reduce_root t) = bal t"
P'''Trans: shows "P''' \<longmapsto> a<\<nu>x> \<prec> P''"
Subset_Zero_sf: fixes i :: "name" shows "Sigma_fm (Var i SUBS Zero)"
simps: fixes abort :: "bool"   and x :: "'a \<times> real"   and xs :: "('a \<times> real) msllist"   and p :: "real" shows "powr_ms_aux abort (MSLCons x xs) p = scale_shift_ms_aux (powr_expansion abort (fst x) p, snd x * p) (powser_ms_aux (gbinomial_series abort p) (scale_shift_ms_aux (inverse (fst x), - snd x) xs))"
static_mhd_simp: fixes m :: "ref_ty \<times> mhead" shows "static (mhd m) = is_static m"
IH: assumes "pairwise coprime (f ` A)"   and "inj_on f A" shows "totient (prod f A) = (\<Prod>a\<in>A. totient (f a))"
Inv_preserves_Can_3: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Can C t" shows "monoidal_language.Cod C (monoidal_language.Inv C t) = monoidal_language.Dom C t"
select_convs_1: fixes iiface :: "iface"   and oiface :: "iface"   and src :: "'a word \<times> nat"   and dst :: "'a word \<times> nat"   and proto :: "protocol"   and sports :: "16 word \<times> 16 word"   and dports :: "16 word \<times> 16 word"   and more :: "'b" shows "iiface \<lparr>iiface = iiface, oiface = oiface, src = src, dst = dst, proto = proto, sports = sports, dports = dports, \<dots> = more\<rparr> = iiface"
\<tau>move2xcp: fixes pc :: "nat"   and e :: "'addr expr1"   and P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and xcp :: "'addr" assumes "pc < length (compE2 e)" shows "\<tau>move2 P h stk e pc \<lfloor>xcp\<rfloor>"
h: fixes n :: "nat" shows "(g n has_integral h n) (cbox a b)"
random_permutation_def: fixes \<sigma> :: "'b \<Rightarrow> 'a option"   and x :: "'b" shows "rp.random_permutation \<sigma> x = (case \<sigma> x of None \<Rightarrow> spmf_of_set (A - ran \<sigma>) \<bind> (\<lambda>y. return_spmf (y, \<sigma>(x \<mapsto> y))) | Some y \<Rightarrow> return_spmf (y, \<sigma>))"
VInter_vsubset: fixes A :: "V"   and B :: "V" assumes "\<And>x. x \<in>\<^sub>\<circ> A \<Longrightarrow> x \<subseteq>\<^sub>\<circ> B"   and "A \<noteq> 0" shows "\<Inter>\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> B"
bigtheta_trans2: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)"   and "g \<in> o[F](h)" shows "f \<in> o[F](h)"
fill_offs_enum_index: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and i :: "nat"   and n :: "nat"   and j :: "nat"   and x :: "'c" assumes "index_less index key"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}"   and "i < n"   and "0 < offs_num n xs index key mi ma i"   and "j \<in> {offs (enum xs index key n mi ma) 0 ! i..<offs_next (offs (enum xs index key n mi ma) 0) (length xs) xs index key mi ma i}"   and "fill xs (offs (enum xs index key n mi ma) 0) index key (length xs) mi ma ! j = Some x" shows "index key x n mi ma = i"
sigma_algebra_iff2: fixes \<Omega> :: "'c set"   and M :: "'c set set" shows "sigma_algebra \<Omega> M = (M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>s\<in>M. \<Omega> - s \<in> M) \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> \<Union> (range A) \<in> M))"
rep_eq: fixes x :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "'a set['c]"   and xb :: "'b set['c]" shows "rel_bset x xa xb = rel_set x (set_bset xa) (set_bset xb)"
Suc_2: shows "swapidseq_ext S (Suc n) p"
lunit_ide_eq: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "\<l>[f] = \<lparr>Chn = \<p>\<^sub>0[C.cod (Leg1 (Dom f)), Leg1 (Dom f)], Dom = \<lparr>Leg0 = Leg0 (Dom f) \<cdot> \<p>\<^sub>0[C.cod (Leg1 (Dom f)), Leg1 (Dom f)], Leg1 = \<p>\<^sub>1[C.cod (Leg1 (Dom f)), Leg1 (Dom f)]\<rparr>, Cod = Cod f\<rparr>"
open_ball: fixes x :: "'b"   and e :: "real" shows "open (ball x e)"
top_le_conv: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'c"   and x :: "'c" assumes "order r"   and "Semilat.top r T" shows "(T \<sqsubseteq>\<^bsub>r\<^esub> x) = (x = T)"
univ_commute: fixes P :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "equivP P \<phi>"   and "f respectsP \<phi>"   and "P x" shows "Equiv_Relation2.univ f (proj \<phi> x) = f x"
multp_iff: fixes R :: "('k \<times> 'k) set"   and P :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and N :: "'k multiset"   and M :: "'k multiset" assumes "irrefl R"   and "trans R"   and "\<And>x y. P x y = ((x, y) \<in> R)" shows "multp P N M = ((N, M) \<in> mult R)"
dg_prod_2_is_arrE: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and ab :: "V"   and cd :: "V"   and gf :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "gf : ab \<mapsto>\<^bsub>\<AA> \<times>\<^sub>D\<^sub>G \<BB>\<^esub> cd" obtains g :: "V" and f :: "V" and a :: "V" and b :: "V" and c :: "V" and d :: "V" where "gf = [g, f]\<^sub>\<circ>" and "ab = [a, b]\<^sub>\<circ>" and "cd = [c, d]\<^sub>\<circ>" and "g : a \<mapsto>\<^bsub>\<AA>\<^esub> c" and "f : b \<mapsto>\<^bsub>\<BB>\<^esub> d"
simps: fixes v :: "'a Point"   and s :: "'a" shows "mkCone v s = \<lparr>vertex = v, slope = s\<rparr>"
Cons_1: fixes thesis :: "bool" assumes "\<And>ys zs. xs = ys @ x # zs \<Longrightarrow> thesis"   and "x \<in> set xs" shows "thesis"
Dirichlet_L_Re_gt_1_nonzero: fixes s :: "complex" assumes "1 < Re s" shows "Dirichlet_L n \<chi> s \<noteq> 0"
assms: shows "filter is_mw_fM (sys_mem_store_buffers gc s) = []"
finite_has_maximal: fixes A :: "'b set" assumes "finite A"   and "A \<noteq> {}" shows "\<exists>m\<in>A. \<forall>b\<in>A. m \<le> b \<longrightarrow> m = b"
tape_of_nl_cons: fixes m :: "nat"   and lm :: "nat list" shows "<m # lm> = (if lm = [] then Oc \<up> Suc m else Oc \<up> Suc m @ Bk # <lm>)"
AndL1_in_SNa: fixes M :: "trm"   and x :: "name"   and y :: "name" assumes "SNa M" shows "SNa (AndL1 x.M y)"
vproductD_2: fixes f :: "V"   and I :: "V"   and A :: "V \<Rightarrow> V" assumes "f \<in>\<^sub>\<circ> vproduct I A" shows "\<D>\<^sub>\<circ> f = I"