hcju/isabelleps huggingface dataset:
bs_clos_inv: fixes x :: "nat"   and e :: "exp"   and \<rho>'' :: "(nat \<times> bval) list"   and v1 :: "exp" assumes "bs_val (BClos x e \<rho>'') v1" obtains \<rho>' :: "(nat \<times> exp) list" where "v1 = ELam x (psubst ((x, EVar x) # \<rho>') e)" and "bs_env \<rho>'' \<rho>'" and "FV (psubst ((x, EVar x) # \<rho>') e) \<subseteq> {x}"
hom_boundary2: shows "\<exists>d. (\<forall>p X S. d p X S \<in> hom (relative_homology_group p X S) (homology_group (p - 1) (subtopology X S))) \<and> (\<forall>p X S c. singular_relcycle p X S c \<and> Suc 0 \<le> p \<longrightarrow> d (int p) X S (homologous_rel_set p X S c) = homologous_rel_set (p - Suc 0) (subtopology X S) {} (chain_boundary p c))"
DynProcModifyReturnNoAbr: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P dynCall init p return' c Q,A"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s\<in>P. \<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/UNIV \<^esub>{\<sigma>} Call (p s) (Modif \<sigma>),{}" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P dynCall init p return c Q,A"
intros_1: shows "local.S' (return_spmf (Inl None, (), Void)) (return_spmf (Void, None, Map.empty))"
resolution_classD1: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and S :: "'b set set" assumes "incidence_system point_set block_collection"   and "incidence_system.resolution_class point_set block_collection S" shows "partition_on point_set S"
vimage2p_def: fixes f :: "'a \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'e"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> 'c" shows "BNF_Def.vimage2p f g R = (\<lambda>x y. R (f x) (g y))"
admits_SNF_JNF_alt_def: shows "(\<forall>A. admits_SNF_JNF A) = (\<forall>A n. A \<in> carrier_mat n n \<and> isDiagonal_mat A \<longrightarrow> (\<exists>P Q. P \<in> carrier_mat n n \<and> Q \<in> carrier_mat n n \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat (P * A * Q)))"
index_valid_def: fixes as :: "('a \<times> 'b atom) set"   and s :: "('a, 'b) state" shows "index_valid as s = (\<forall>x b i. (Mapping.lookup (\<B>\<^sub>i\<^sub>l s) x = Some (i, b) \<longrightarrow> (i, Geq x b) \<in> as) \<and> (Mapping.lookup (\<B>\<^sub>i\<^sub>u s) x = Some (i, b) \<longrightarrow> (i, Leq x b) \<in> as))"
assms_1: shows "f \<in>\<^sub>\<circ> A ^\<^sub>\<times> n"
fun_relI: fixes A :: "('a \<times> 'c) set"   and f :: "'a \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'e"   and B :: "('d \<times> 'e) set" assumes "\<And>a a'. (a, a') \<in> A \<Longrightarrow> (f a, f' a') \<in> B" shows "(f, f') \<in> A \<rightarrow> B"
assms_1: fixes A :: "'a tensor" assumes "A \<in> set As" shows "dims A = ds"
fresh_owned: shows "dom (share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<S>) - dom \<S> \<subseteq> \<O>"
sub_vec: fixes nr :: "nat"   and v :: "'c list"   and sd :: "nat" assumes "vec nr v"   and "sd \<le> nr" shows "vec sd (sub_vec sd v)"
set_spies: fixes i :: "nat"   and j :: "nat"   and k :: "nat"   and A :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Says i j k A B X \<in> set evs" shows "X \<in> spies evs"
lsc_hull_of_convex_agrees_outside: fixes f :: "'b \<Rightarrow> ereal" assumes "Lower_Semicontinuous.convex_on UNIV f" shows "\<forall>x. x \<notin> closure (domain f) \<longrightarrow> f x = lsc_hull f x"
mat_kernelD_1: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'b Matrix.vec" assumes "A \<in> carrier_mat nr nc"   and "v \<in> mat_kernel A" shows "v \<in> carrier_vec nc"
path_Append: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and as :: "'a list"   and n'' :: "'b"   and as' :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n as n''"   and "CFG.path sourcenode targetnode valid_edge n'' as' n'" shows "CFG.path sourcenode targetnode valid_edge n (as @ as') n'"
rel_nondetT_simps: fixes M :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and m :: "'g"   and m' :: "'h" shows "rel_nondetT M (NondetT m) (NondetT m') = M m m'"
intros: fixes c :: "'form"   and r :: "'rule" assumes "c \<in> set (consequent r)" shows "natEff_Inst (r, c) c (f_antecedent r)"
dropWhile_append1: fixes x :: "'g"   and xs :: "'g list"   and P :: "'g \<Rightarrow> bool"   and ys :: "'g list" assumes "x \<in> set xs"   and "\<not> P x" shows "dropWhile P (xs @ ys) = dropWhile P xs @ ys"
power_one_right: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d" assumes "class.monoid_mult one times" shows "power.power one times a 1 = a"
ivl_zero_case: fixes l1 :: "real"   and u1 :: "real"   and l2 :: "real"   and u2 :: "real" assumes "l1 \<le> u1"   and "l2 \<le> u2" shows "l1 \<le> 0 \<and> 0 \<le> u1 \<and> l2 \<le> 0 \<and> 0 \<le> u2 \<or> l1 \<le> 0 \<and> 0 \<le> u1 \<and> 0 \<le> l2 \<or> l1 \<le> 0 \<and> 0 \<le> u1 \<and> u2 \<le> 0 \<or> 0 \<le> l1 \<and> l2 \<le> 0 \<and> 0 \<le> u2 \<or> u1 \<le> 0 \<and> l2 \<le> 0 \<and> 0 \<le> u2 \<or> u1 \<le> 0 \<and> u2 \<le> 0 \<or> u1 \<le> 0 \<and> 0 \<le> l2 \<or> 0 \<le> l1 \<and> u2 \<le> 0 \<or> 0 \<le> l1 \<and> 0 \<le> l2"
heap_is_wellformed_impl: shows "heap_is_wellformed = local.a_heap_is_wellformed"
holds_untilD: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and xs :: "'a stream"   and j :: "nat" assumes "(holds P until holds Q) xs"   and "\<forall>i\<le>j. \<not> Q (xs !! i)" shows "P (xs !! j)"
inp_dblk: shows "\<forall>p. inp (dblock s' p) \<in> allInput s' \<union> {NotAnInput}"
kop_galois: fixes f :: "'b \<Rightarrow> 'a set"   and g :: "'a \<Rightarrow> 'b set" shows "(op\<^sub>K f = g) = (op\<^sub>K g = f)"
rep_eq: fixes x :: "('a, 'b) oalist_ntm" shows "lookup (MP_oalist x) = OAlist_lookup_ntm x"
lens_indep_get: fixes x :: "'b \<Longrightarrow> 'c"   and y :: "'d \<Longrightarrow> 'c"   and \<sigma> :: "'c"   and v :: "'d" assumes "x \<bowtie> y" shows "get\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) = get\<^bsub>x\<^esub> \<sigma>"
assm: shows "i \<noteq> bot \<and> j \<noteq> bot \<and> maximum i + 1 = minimum j"
formalSpec: shows "Spec = new_configured_SecurityInvariant (sinvar_spec, nm_default m, nm_receiver_violation m, nP)"
Collect_conj_eq: fixes P :: "'f \<Rightarrow> bool"   and Q :: "'f \<Rightarrow> bool" shows "{x. P x \<and> Q x} = {x. P x} \<inter> {x. Q x}"
SPEC_rule_conjunct1: fixes A :: "'a nres"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "A \<le> SPEC (\<lambda>v. P v \<and> Q v)" shows "A \<le> SPEC P"
le_fin_length: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list" assumes "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2" shows "length w\<^sub>1 \<le> length w\<^sub>2"
simps_1: fixes uu :: "nat"   and uv :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatariths_aformerr uu [] uv = Some []"
match_def: fixes Pm :: "'a \<Rightarrow> ('b \<times> 'b) set"   and Pw :: "'b \<Rightarrow> ('a \<times> 'a) set"   and \<mu> :: "('a \<times> 'b) set" assumes "StableMarriage Pm Pw" shows "StableMarriage.match Pm Pw \<mu> = (inj_on fst \<mu> \<and> inj_on snd \<mu> \<and> \<mu> \<subseteq> (\<Union>m. {m} \<times> Field (Pm m)) \<inter> (\<Union>w. Field (Pw w) \<times> {w}))"
test_comp_mult1: fixes p :: "'a" assumes "test p" shows "!p \<cdot> p = (0::'a)"
intersect_normal: fixes K :: "'a set" assumes "finite (carrier G)"   and "K \<lhd> G" shows "composition_series (G\<lparr>carrier := K\<rparr>) (remdups_adj (map ((\<inter>) K) \<GG>))"
pos_divideR_less_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and c :: "real"   and b :: "'c"   and a :: "'c" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "0 < c" shows "less (scaleR (inverse c) b) a = less b (scaleR c a)"
a1: shows "low_equal s1 s2 \<and> t1 = add_data_cache s1 address w bm \<and> t2 = add_data_cache s2 address w bm"
orthocompl_span: fixes S :: "'a vec set"   and v :: "'a vec"   and y :: "'a vec" assumes "\<And>x. x \<in> S \<Longrightarrow> v \<bullet> x = (0::'a)"   and "S \<subseteq> carrier_vec n"   and "v \<in> carrier_vec n"   and "y \<in> local.span S" shows "v \<bullet> y = (0::'a)"
subgroup_mult_id: fixes H :: "'a set" assumes "subgroup H G" shows "H <#> H = H"
select_convs_2: fixes entries :: "'a list"   and getentry :: "'a \<Rightarrow> nat option"   and more :: "'b" shows "getentry \<lparr>entries = entries, getentry = getentry, \<dots> = more\<rparr> = getentry"
Suc_1: fixes i :: "nat"   and h :: "heap" assumes "k = j - i"   and "i \<le> j"   and "effect (for [i..<j] f) h h' u"   and "I i h"   and "\<And>k h h' r. \<lbrakk>i \<le> k; k < j; I k h; effect (f k) h h' r\<rbrakk> \<Longrightarrow> I (Suc k) h'" shows "I j h'"
sup_None_2: fixes x :: "'b option" shows "x \<squnion> None = x"
len_r: shows "length (map2 f xs ys) = length ys"
law_restrict_carrier: fixes R :: "('c, 'd) ring_scheme" shows "carrier (law_restrict R) = carrier R"
index_injective: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set"   and i :: "nat"   and j :: "nat" assumes "long_ch_by_ord2 f X"   and "finite X"   and "i < j"   and "j < card X" shows "f i \<noteq> f j"
msgs_unchanged_if_snapshotted_RecvMarker_for_other_is: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and i :: "nat"   and p :: "'p"   and q :: "'p"   and i' :: "nat" assumes "c \<turnstile> ev \<mapsto> c'"   and "ev = RecvMarker i p q"   and "ps c p \<noteq> None"   and "i' \<noteq> i" shows "msgs c i' = msgs c' i'"
rep_eq: fixes x :: "'d \<Rightarrow>\<^sub>F 'e" shows "domain x = dom (fmlookup x)"
remove_vertex_E': fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a"   and x :: "'a" assumes "Digraph G"   and "v \<rightarrow>\<^bsub>Digraph.remove_vertex G x\<^esub> w" shows "v \<rightarrow>\<^bsub>G\<^esub> w"
hom_closed: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (lsphere n) (equator n))" shows "hom_induced (int n) (lsphere n) (equator n) (nsphere n) (upper n) id x \<in> carrier (relative_homology_group (int n) (nsphere n) (upper n))"
arrI: fixes f :: "'a \<times> 'a" assumes "B.arr (fst f) \<and> B.arr (snd f) \<and> src\<^sub>B (fst f) = trg\<^sub>B (snd f)" shows "B.VV.arr f"
numgcdh0: fixes t :: "Ferrack.num"   and m :: "int" assumes "numgcdh t m = 0" shows "m = 0"
simple_action_conj_def: fixes a :: "simple_action"   and b :: "simple_action" shows "simple_action_conj a b \<equiv> if a = Accept \<and> b = Accept then Accept else Drop"
prems_1: shows "outstanding_refs (Not \<circ> is_volatile_Read\<^sub>s\<^sub>b) (x # xs) \<inter> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb = {}"
2: shows "[\<lbrace>x\<^sup>P,H\<rbrace> \<^bold>& \<^bold>\<not>\<lbrace>y\<^sup>P,H\<rbrace> \<^bold>\<or> \<^bold>\<not>\<lbrace>x\<^sup>P,H\<rbrace> \<^bold>& \<lbrace>y\<^sup>P,H\<rbrace> in v]"
neg_prod_lt: fixes c :: "'a"   and x :: "'a" assumes "c < (0::'a)" shows "c * x < (0::'a) \<equiv> (0::'a) < x"
constr_rep_1: fixes x1 :: "'d" shows "Rep_frame (\<langle>\<epsilon>, x1\<rangle>) = FAssert_Rep x1"
atLeast_Suc_atMost_Suc_shift: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "sum g {Suc m..Suc n} = sum (g \<circ> Suc) {m..n}"
dvd_1_left: fixes k :: "nat" shows "Suc 0 dvd k"
Suc: shows "make_vert_equiv k = mat1 (2 ^ k)"
intersecting_def: fixes v :: "'a set" shows "intersecting v = {i. V i \<inter> v \<noteq> {}}"
contractible_space_singleton: fixes X :: "'a topology"   and a :: "'a" assumes "topspace X = {a}" shows "contractible_space X"
zero_in_bigtheta_iff: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" shows "((\<lambda>_. 0::'b) \<in> \<Theta>[F](f)) = (\<forall>\<^sub>F x in F. f x = (0::'b))"
long_division_closed_2: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list"   and q :: "'c list" assumes "domain R"   and "subfield K R"   and "p \<in> carrier (K [X]\<^bsub>R\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>R\<^esub>)" shows "ring.pmod R p q \<in> carrier (K [X]\<^bsub>R\<^esub>)"
shift_degree': fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "deg R (shift (deg R f) f) = 0"
907: fixes x :: "'a" shows "- x \<sqinter> - x = - x"
ring_simprules_7: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<oplus>\<^bsub>R\<^esub> z = x \<oplus>\<^bsub>R\<^esub> (y \<oplus>\<^bsub>R\<^esub> z)"
Bang: fixes P :: "pi"   and V :: "Early_Semantics.residual" assumes "P \<parallel> !P \<longmapsto>\<^sub>e V" shows "!P \<longmapsto>\<^sub>e V"
rtrancl_into_rtrancl: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>*"
M_def: shows "M \<equiv> PM \<and> M_axioms"
mset_filter: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "mset (filter P xs) = filter_mset P (mset xs)"
hom_imp_img_group: fixes h :: "'a \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme" assumes "h \<in> hom G H" shows "Group.group (H\<lparr>carrier := h ` carrier G, one := h \<one>\<rparr>)"
surjective: fixes r :: "'b pState_scheme" shows "r = \<lparr>pid = pid r, vars = pState.vars r, pc = pc r, channels = pState.channels r, idx = pState.idx r, \<dots> = pState.more r\<rparr>"
div_exp_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide (divide a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m)) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) = divide a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) (m + n))"
geodesic_segment_topology_3: fixes G :: "'b set" assumes "geodesic_segment G" shows "path_connected G"
append_butlast_last_id: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "butlast xs ## last xs = xs"
vec_hom_zero_iff: fixes x :: "'a vec"   and n :: "nat" shows "(vec_hom x = 0\<^sub>v n) = (x = 0\<^sub>v n)"
approx_minus2: fixes a :: "'a star"   and b :: "'a star" assumes "- a \<approx> - b" shows "a \<approx> b"
Max_in: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'g set" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Max less_eq A \<in> A"
justifying: fixes aa :: "nat"   and na :: "nat" assumes "enat aa < llength (justifying_exec (J na))"   and "action_obs (justifying_exec (J na)) aa = NormalAction (ReadMem ad al v)"   and "0 < na" shows "\<exists>T. P \<turnstile> ad@al : T \<and> P \<turnstile> v :\<le> T"
oInv_eq_0: fixes a :: "ordinal" assumes "a \<le> F 0" shows "oInv F a = 0"
prems_2: shows "\<rho> \<in> carrier_mat d d"
empty_imp_negligible: fixes S :: "'a set" assumes "S = {}" shows "negligible S"
set_nths_subset: fixes xs :: "'g list"   and I :: "nat set" shows "set (nths xs I) \<subseteq> set xs"
dom_m_def: fixes N :: "('e, 'f) fmap" shows "dom_m N = mset_fset (fmdom N)"
relative_homology_group_restrict: fixes p :: "int"   and X :: "'b topology"   and S :: "'b set" shows "relative_homology_group p X (topspace X \<inter> S) = relative_homology_group p X S"
degree_map_poly_hom: fixes p :: "'a poly" shows "degree (map_poly (\<lambda>a. [:a:]) p) = degree p"
lt_set_iff: fixes t :: "'t"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "(t \<in> lt_set F) = (\<exists>f\<in>F. f \<noteq> 0 \<and> lt f = t)"
iso_hcomp\<^sub>R\<^sub>W\<^sub>C_1: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "local.iso \<mu>"   and "local.iso \<nu>"   and "sources \<nu> \<inter> targets \<mu> \<noteq> {}" shows "local.iso (\<nu> \<star> \<mu>)"
lt_take: fixes n1 :: "nat"   and n2 :: "nat"   and al :: "'a list" assumes "n1 < n2"   and "n2 \<le> length al" shows "take n1 al < take n2 al"
power_divide: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and n :: "nat" assumes "class.field divide inverse times one plus zero minus uminus" shows "power.power one times (divide a b) n = divide (power.power one times a n) (power.power one times b n)"
dim_fs: fixes k :: "nat" assumes "k < length fs" shows "dim_vec (fs ! k) = n"
round_robin_reschedule_induct: fixes t0 :: "'a"   and queue :: "'a list"   and P :: "'a list \<Rightarrow> bool" assumes "t0 \<in> set queue"   and "\<And>queue. P (t0 # queue)"   and "\<And>queue t. \<lbrakk>t \<noteq> t0; t0 \<in> set queue; P (queue @ [t])\<rbrakk> \<Longrightarrow> P (t # queue)" shows "P queue"
exec_to_execn: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" shows "\<exists>n. \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"
wf_fun_pair_eqs_ineqs_map: fixes X :: "'var set"   and A :: "('fun, 'var) strand_step list"   and Di :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term" assumes "wf\<^sub>s\<^sub>t X A"   and "Di \<in> set (subseqs D)"   and "\<forall>(t, t')\<in>set D. fv t \<union> fv t' \<subseteq> X" shows "wf\<^sub>s\<^sub>t X (map (\<lambda>d. \<langle>check: pair (t, s) \<doteq> pair d\<rangle>\<^sub>s\<^sub>t) Di @ map (\<lambda>d. \<forall>[]\<langle>\<or>\<noteq>: [(pair (t, s), pair d)]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) D) @ A)"
wf_trm_subst_rangeI: fixes \<delta> :: "'b \<Rightarrow> ('fun, 'b) Term.term" assumes "\<And>x. wf\<^sub>t\<^sub>r\<^sub>m (\<delta> x)" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
and_num_2097152_128: shows "2097152 AND 128 = 0"
bnds_cos_lower: fixes xl :: "float"   and x :: "real"   and xu :: "float"   and y :: "real"   and prec :: "nat" assumes "real_of_float xl \<le> x"   and "x \<le> real_of_float xu"   and "cos x \<le> y" shows "real_of_float (fst (bnds_cos prec xl xu)) \<le> y"
concrete_category_axioms: shows "concrete_category (Collect B.obj) Hom B.iso_class (\<lambda>_ _ _. Comp)"
boundedByMonoD: fixes m :: "nat"   and B :: "(nat \<times> 'b) set"   and M :: "nat" assumes "boundedBy m B"   and "m < M" shows "boundedBy M B"
fbox_fdia_demorgan: fixes R :: "('e \<times> 'f) set"   and f :: "'f \<Rightarrow> 'd"   and x :: "'e" shows "|R] f x = -  |R\<rangle> (\<lambda>y. - f y) x"
eq_0_min_height: fixes kdt :: "'b kdt" shows "(0 = min_height kdt) = (\<exists>p. kdt = Leaf p)"
u: fixes v :: "real" assumes "0 < v"   and "v \<le> u" shows "norm (v *\<^sub>R (z - x) - (y - x)) < norm (y - x)"
simps_1: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and ao :: "'a option" shows "proper_intrvl.Compl_set_less_eq_aux less proper_interval ao [] [] = (\<not> proper_interval ao None)"
mem_ball_imp_mem_cball: fixes x :: "'c"   and y :: "'c"   and e :: "real" assumes "x \<in> ball y e" shows "x \<in> cball y e"
assms_2: shows "\<turnstile>\<^sub>! (p \<^bold>\<longleftrightarrow>\<^sub>! q)"
assms_4: shows "topspace X \<inter> T \<noteq> {}"
enc_respects_barb_set_iff_source_target_rel: fixes Barbs :: "'barbs set" shows "((\<forall>S x. x \<in> Barbs \<and> ((\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<down><TWB>x)) \<longrightarrow> x \<in> Barbs \<and> ((\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<down><TWB>x))) \<and> (\<forall>S x. x \<in> Barbs \<and> ((\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<down><TWB>x)) \<longrightarrow> x \<in> Barbs \<and> ((\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<down><TWB>x)))) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_respects_barb_set Rel (STCalWB SWB TWB) Barbs)"
distinct_1: fixes x1 :: "('c, 'd) Term.term"   and x2 :: "('c, 'd) Term.term" shows "send\<langle>x1\<rangle>\<^sub>s\<^sub>t \<noteq> receive\<langle>x2\<rangle>\<^sub>s\<^sub>t"
word_and_not: fixes x :: "'a word" shows "x && ~~ x = 0"
G_monotonic: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool" assumes "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl" shows "MonotProblem (ProblemIkTpartG.GE_wtFsym wtFsym resOf protCl) (ProblemIkTpartG.GE_wtPsym wtPsym prot protFw) (ProblemIkTpartG.GE_arOf arOf) (ProblemIkTpartG.GE_resOf resOf) (ProblemIkTpartG.GE_parOf parOf) (ProblemIkTpartG.gPB wtFsym arOf resOf \<Phi> prot protFw protCl)"
\<psi>_stream: fixes xs :: "('s \<times> ('c \<Rightarrow> real)) stream"   and x :: "'s \<times> ('c \<Rightarrow> real)" assumes "\<And>x y. \<lbrakk>x \<in> S; x ~ y\<rbrakk> \<Longrightarrow> x \<in> S"   and "\<And>x y. \<lbrakk>x \<in> S; x ~ y\<rbrakk> \<Longrightarrow> x ~ y"   and "pred_stream (\<lambda>s. s \<in> S) xs"   and "x \<in> S" shows "pred_stream (\<lambda>s. \<psi> (reps (abss s)) = \<psi> s) (x ## xs)"
is_OK_returns_result_I: fixes h :: "'x"   and f :: "('x, 'y, 'z) prog"   and y :: "'z" assumes "h \<turnstile> f \<rightarrow>\<^sub>r y" shows "h \<turnstile> ok f"
simply_connected_imp_winding_number_zero: fixes S :: "complex set"   and g :: "real \<Rightarrow> complex"   and z :: "complex" assumes "simply_connected S"   and "path g"   and "path_image g \<subseteq> S"   and "pathfinish g = pathstart g"   and "z \<notin> S" shows "winding_number g z = 0"
snd_sylow_axioms: shows "snd_sylow (G\<lparr>carrier := P\<rparr>) p a k"
S_def: shows "S \<equiv> trace_rest_closure rest1 rest2 IA IU p q"