hcju/isabelleps huggingface dataset:
least: shows "\<forall>y\<in>col_space A. norm (b - p) \<le> norm (b - y)"
ofilter_def: fixes A :: "('a + 'c) set" shows "rt.ofilter A \<equiv> A \<subseteq> Field (r +o t) \<and> (\<forall>a\<in>A. rt.under a \<subseteq> A)"
start_2_goon: fixes lm :: "nat list"   and m :: "nat"   and b :: "cell list"   and list :: "cell list" assumes "lm \<noteq> []"   and "wprepare_loop_start m lm (b, Bk # list)" shows "(list = [] \<longrightarrow> wprepare_loop_goon m lm (Bk # b, [])) \<and> (list \<noteq> [] \<longrightarrow> wprepare_loop_goon m lm (Bk # b, list))"
YFtorNT'_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e" shows "YFtorNT' C f \<equiv> \<lparr>NTDom = Hom\<^bsub>C\<^esub>[\<midarrow>,dom\<^bsub>C\<^esub> f], NTCod = Hom\<^bsub>C\<^esub>[\<midarrow>,cod\<^bsub>C\<^esub> f], NatTransMap = \<lambda>B. Hom\<^bsub>C\<^esub>[B,f]\<rparr>"
K_block_design_axioms_def: fixes block_collection :: "'b set multiset"   and sizes :: "int set" shows "K_block_design_axioms block_collection sizes \<equiv> (\<forall>bl. bl \<in># block_collection \<longrightarrow> int (card bl) \<in> sizes) \<and> (\<forall>x. x \<in> sizes \<longrightarrow> 0 < x)"
adjacent_no_loop: fixes G :: "('a, 'b) graph"   and v :: "'a"   and v' :: "'a" assumes "valid_unMultigraph G"   and "valid_unMultigraph.adjacent G v v'" shows "v \<noteq> v'"
singleton_insert_inj_eq': fixes a :: "'b"   and A :: "'b set"   and b :: "'b" shows "(insert a A = {b}) = (a = b \<and> A \<subseteq> {b})"
assms_3: shows "\<bar>valof a + valof b\<bar> < threshold TYPE(('e, 'f) IEEE.float)"
p_2: shows "\<forall>x\<in>A. f x = f' (p x)"
Hausdorff_space_compact_separation: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "Hausdorff_space X"   and "compactin X S"   and "compactin X T"   and "disjnt S T" obtains U :: "'c set" and V :: "'c set" where "openin X U" and "openin X V" and "S \<subseteq> U" and "T \<subseteq> V" and "disjnt U V"
n_max: shows "\<forall>l\<in>C. nat_of_fatom (get_atom l) \<le> n"
Ia: shows "Inum (real_of_int i # bs) a = Inum (real_of_int i # bs) (CN 0 (fst (zsplit0 a)) (snd (zsplit0 a)))"
simps_2: fixes R :: "(ZF \<times> ZF) set"   and s :: "ZF"   and n :: "nat" shows "Ext_ZF_n R s (Suc n) = HOLZF.Sum (Repl (Ext_ZF_n R s n) (Ext_ZF R))"
nec_hintikka_scheme: fixes v :: "i"   and x :: "\<nu>"   and \<phi> :: "\<nu> \<Rightarrow> \<o>" shows "[x\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) \<^bold>\<equiv> \<^bold>\<A>\<phi> x \<^bold>& (\<^bold>\<forall>z. \<^bold>\<A>\<phi> z \<^bold>\<rightarrow> z \<^bold>= x) in v]"
in_progressS_elt: fixes r :: "Runs.var set option"   and S :: "Runs.var set"   and x :: "Runs.var" assumes "in_progressS r S"   and "x \<in> S" shows "in_progress r x"
sset_fromN: fixes n :: "nat" shows "sset (fromN n) = {n..}"
case_3: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and f3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and f4 :: "('b, 'c) Combinators \<Rightarrow> ('b, 'c) Combinators \<Rightarrow> 'a"   and x31 :: "'b"   and x32 :: "'b"   and x33 :: "'c" shows "(case AllowPortFromTo x31 x32 x33 of DenyAll \<Rightarrow> f1 | DenyAllFromTo x xa \<Rightarrow> f2 x xa | AllowPortFromTo x xa xb \<Rightarrow> f3 x xa xb | x \<oplus> xa \<Rightarrow> f4 x xa) = f3 x31 x32 x33"
set_enforce_option: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c option" shows "set_option (enforce_option P x) = {a \<in> set_option x. P a}"
has_integral_linear_cbox: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b"   and a :: "'a"   and b :: "'a"   and h :: "'b \<Rightarrow> 'c" assumes "(f has_integral y) (cbox a b)"   and "bounded_linear h" shows "(h \<circ> f has_integral h y) (cbox a b)"
False: shows "\<not> (poly p a = 0 \<or> poly p b = 0)"
poly_eqI: fixes p :: "'a poly"   and q :: "'a poly" assumes "\<And>n. poly.coeff p n = poly.coeff q n" shows "p = q"
ordertype_eq_0_iff: fixes X :: "'a set"   and r :: "('a \<times> 'a) set" assumes "small X"   and "wf r" shows "(ordertype X r = 0) = (X = {})"
split: fixes P :: "'e \<Rightarrow> bool"   and f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'e"   and f3 :: "'g \<Rightarrow> 'f \<Rightarrow> 'e"   and step :: "('g, 'f) step" shows "P (case step of Done \<Rightarrow> f1 | Skip x \<Rightarrow> f2 x | Yield x xa \<Rightarrow> f3 x xa) = ((step = Done \<longrightarrow> P f1) \<and> (\<forall>x2. step = Skip x2 \<longrightarrow> P (f2 x2)) \<and> (\<forall>x31 x32. step = Yield x31 x32 \<longrightarrow> P (f3 x31 x32)))"
fmdom'_add: fixes m :: "('d, 'e) fmap"   and n :: "('d, 'e) fmap" shows "fmdom' (m ++\<^sub>f n) = fmdom' m \<union> fmdom' n"
set_child_nodes_get_tag_name_is_l_set_child_nodes_get_tag_name: shows "l_set_child_nodes_get_tag_name ShadowRootClass.type_wf Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs get_tag_name get_tag_name_locs"
inj_on_push_forward_inclusion: fixes p :: "'a" assumes "p \<in> sub.carrier" shows "inj_on inclusion.push_forward (sub.tangent_space p)"
scale_minus_left: fixes a :: "rat"   and x :: "'b" shows "- a *R x = - (a *R x)"
assms: shows "a \<noteq> (0::'a) \<or> b \<noteq> 0"
q'_def: shows "q' \<equiv> bridge k j xs w w' (\<^bold>\<not> p)"
sig_red_single_sing_reg_cases: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> nat"   and term_of_pair :: "'d \<times> nat \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and fs :: "('d \<Rightarrow>\<^sub>0 'e) list"   and top_tail :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'e"   and q :: "'c \<Rightarrow>\<^sub>0 'e"   and f :: "'c \<Rightarrow>\<^sub>0 'e"   and t :: "'d" assumes "qpm_inf_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "qpm_inf_term.sig_red_single pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs ord_term top_tail p q f t = (qpm_inf_term.sig_red_single pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs (=) top_tail p q f t \<or> qpm_inf_term.sig_red_single pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs ord_term_strict top_tail p q f t)"
cases: fixes x :: "lit uprod" obtains l1 :: "lit" and l2 :: "lit" where "x = Upair l1 l2"
abs: fixes s :: "Core.term"   and t :: "Core.term"   and T :: "typ" assumes "s \<rightarrow>\<^sub>\<beta> t" shows "Abs T s \<rightarrow>\<^sub>\<beta> Abs T t"
pr_coincidence: shows "\<iota> = \<pp>\<^sub>1[\<one>, \<one>]"
measurable_on_restrict: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "f measurable_on UNIV"   and "S \<in> sets lebesgue" shows "(\<lambda>x. if x \<in> S then f x else (0::'b)) measurable_on UNIV"
Z_mono: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m" shows "Z m \<subseteq> Z n"
le_degree: fixes p :: "'c poly"   and n :: "nat" assumes "coeff p n \<noteq> (0::'c)" shows "n \<le> degree p"
supp_fin_insert: fixes X :: "'a set"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "fs TYPE('a) TYPE('b)"   and "finite X" shows "supp (insert x X) = supp x \<union> supp X"
pres_typeD2: fixes step :: "nat \<Rightarrow> 'c \<Rightarrow> (nat \<times> 'c) list"   and n :: "nat"   and A :: "'c set"   and s :: "'c"   and p :: "nat" assumes "pres_type step n A"   and "s \<in> A"   and "p < n" shows "snd ` set (step p s) \<subseteq> A"
rv_mon_conv_sfis: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and f :: "'b \<Rightarrow> real" assumes "measure_space M"   and "f \<in> rv M"   and "nonnegative f" shows "\<exists>u x. u\<up>f \<and> (\<forall>n. x n \<in> sfis (u n) M)"
trans_of_def: shows "PTA.trans_of \<equiv> fst"
cast_inversions_1: fixes x :: "('a, 'b, 'c) unboxed"   and d :: "'a" assumes "cast_Dyn x = Some d" shows "x = OpDyn d"
assms_3: fixes x :: "'a" assumes "x \<in> set xs" shows "(algebraic over K) x"
uniform_limit_ge_const: fixes I :: "'a filter"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and X :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "I \<noteq> bot"   and "uniform_limit X f g I"   and "\<forall>\<^sub>F i in I. \<forall>x\<in>X. h x \<le> f i x"   and "x \<in> X" shows "h x \<le> g x"
assms_3: shows "\<gamma> (Trans s a ou s') = \<gamma> (Trans s1 a1 ou1 s1')"
step_op_inl_hit: fixes F :: "'fenv_inca"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and opinl :: "'opinl"   and ar :: "nat"   and \<Sigma> :: "'dyn list"   and x :: "'dyn"   and H :: "'henv"   and st :: "('fun, 'dyn) frame list" assumes "Finca_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = Inca.instr.IOpInl opinl"   and "\<AA>\<rr>\<ii>\<tt>\<yy> (\<DD>\<ee>\<II>\<nn>\<ll> opinl) = ar"   and "ar \<le> length \<Sigma>"   and "\<II>\<ss>\<II>\<nn>\<ll> opinl (take ar \<Sigma>)"   and "\<II>\<nn>\<ll>\<OO>\<pp> opinl (take ar \<Sigma>) = x" shows "Sinca.step (State F H (Frame f pc \<Sigma> # st)) (State F H (Frame f (Suc pc) (x # drop ar \<Sigma>) # st))"
nat_set_cset_partial_bij: obtains f :: "nat set cset \<Rightarrow> nat set" where "bij_betw f {A. A \<noteq> {}\<^sub>c} UNIV"
sup_inf_absorb: fixes x :: "'b upred"   and y :: "'b upred" shows "(x \<or> x \<and> y) = x"
consequence_relation_family_axioms: shows "consequence_relation_family Bot_FL Q (\<lambda>q N1 N2. entails_q q (\<Union> (\<G>_F_L_q q ` N1)) (\<Union> (\<G>_F_L_q q ` N2)))"
assms_1: shows "Ftor F : A \<longrightarrow> B"
imp_pre_def: fixes P1 :: "'e \<Rightarrow> bool"   and P2 :: "'e \<Rightarrow> bool" shows "imp_pre P1 P2 \<equiv> \<lambda>x. P1 x \<longrightarrow> P2 x"
M_ge_0: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool" assumes "grover_state n f" shows "0 < grover_state.M n f"
assms: shows "(s, ss1 @ ss2) \<in> conv ars"
2: fixes n :: "nat" assumes "odd n" shows "\<exists>i. odd i \<and> - (real n * pi) = real_of_int i * pi"
bisimulation_axioms: shows "bisimulation trsys1 trsys2 bisim tlsim"
that: fixes a :: "'a"   and b :: "real" assumes "a \<in> S"   and "a \<noteq> (0::'a)"   and "0 < b"   and "\<And>x. x \<in> S \<Longrightarrow> b < a \<bullet> x" shows "thesis"
ev: shows "\<forall>\<^sub>F x in at_top. 1 < b x"
bounded_linear_zero: shows "bounded_linear (\<lambda>x. 0::'d)"
transfer_FOREACHc_nres: fixes iterate :: "('c dres \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c dres \<Rightarrow> 'c dres) \<Rightarrow> 'c dres \<Rightarrow> 'c dres"   and s :: "'d set"   and fi :: "'d \<Rightarrow> 'c \<Rightarrow> 'c dres"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and c :: "'c \<Rightarrow> bool"   and \<sigma> :: "'c" assumes "set_iterator iterate s"   and "\<And>x \<sigma>. nres_of (fi x \<sigma>) \<le> f x \<sigma>" shows "nres_of (iterate (case_dres False False c) (\<lambda>x s. s \<bind> fi x) (dRETURN \<sigma>)) \<le> FOREACH\<^sub>C s c f \<sigma>"
prv_expl: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "bprv fls" shows "bprv \<phi>"
dvd_mult_left: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times (times a b) c" shows "dvd.dvd times a c"
long_division_add_2: fixes K :: "'a set"   and a :: "'a list"   and b :: "'a list"   and q :: "'a list" assumes "subfield K R"   and "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X])"   and "q \<in> carrier (K [X])" shows "a \<oplus>\<^bsub>K [X]\<^esub> b pmod q = a pmod q \<oplus>\<^bsub>K [X]\<^esub> (b pmod q)"
sumZero: fixes P :: "pi" shows "P \<oplus> \<zero> \<sim> P"
euclidean_size_mult: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "euclidean_size (times a b) = euclidean_size a * euclidean_size b"
partitions_eq_implies_permutes: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and f' :: "'c \<Rightarrow> 'd" assumes "f \<in> A \<rightarrow>\<^sub>E B"   and "f' \<in> A \<rightarrow>\<^sub>E B"   and "finite B"   and "(\<lambda>b. {x \<in> A. f x = b}) ` B - {{}} = (\<lambda>b. {x \<in> A. f' x = b}) ` B - {{}}" shows "\<exists>p. p permutes B \<and> (\<forall>x\<in>A. f x = p (f' x))"
simplicial_subdivision_of_cell_complex_lowdim: fixes \<M> :: "'b set set"   and d :: "int" assumes "finite \<M>"   and "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"   and "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1"   and "\<And>C. C \<in> \<M> \<Longrightarrow> aff_dim C \<le> d" obtains \<T> :: "'b set set" where "simplicial_complex \<T>" and "\<And>K. K \<in> \<T> \<Longrightarrow> aff_dim K \<le> d" and "\<Union> \<T> = \<Union> \<M>" and "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union> F" and "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> K \<subseteq> C"
map_poly_inj_idom_divide_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "map_poly_inj_idom_divide_hom hom \<equiv> inj_idom_divide_hom hom"
coeff_lp_monom: fixes c :: "rat"   and v :: "nat"   and v' :: "nat" shows "Abstract_Linear_Poly.coeff (lp_monom c v) v' = (if v = v' then c else 0)"
termMOD_def: shows "termMOD \<equiv> \<lparr>igWls = wls, igWlsAbs = wlsAbs, igVar = Var, igAbs = Abs, igOp = Op, igFresh = fresh, igFreshAbs = freshAbs, igSwap = swap, igSwapAbs = swapAbs, igSubst = subst, igSubstAbs = substAbs\<rparr>"
no_nonce_OR1_OR2: fixes A :: "agent"   and NA :: "msg"   and B :: "agent"   and evs :: "event list"   and NA' :: "msg"   and A' :: "agent" assumes "Crypt (shrK A) \<lbrace>NA, Agent A, Agent B\<rbrace> \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> otway" shows "Crypt (shrK A) \<lbrace>NA', NA, Agent A', Agent A\<rbrace> \<notin> parts (knows Spy evs)"
case_1: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Pos of Pos \<Rightarrow> f1 | Neg \<Rightarrow> f2) = f1"
pow_distrib1: shows "order \<G> * c * x1 - c * x1 = (order \<G> - 1) * c * x1"
FGModuleHomSet_is_Gmaps_in_VectorSpaceHomSet: fixes smult' :: "('f, 'g) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and W :: "'a set" assumes "FGModule G smult' W" shows "FGModuleHomSet G (\<cdot>) V smult' W = VectorSpaceHomSet (\<sharp>\<cdot>) V (aezfun_scalar_mult.fsmult smult') W \<inter> {T. \<forall>g\<in>G. \<forall>v\<in>V. T (g *\<cdot> v) = aezfun_scalar_mult.Gmult smult' g (T v)}"
fds_const_converges: fixes c :: "'b"   and s :: "'b" shows "fds_converges (fds_const c) s"
k_adj: fixes v :: "'v" assumes "v \<in> V" shows "card {n. adjacent v n} = k"
abs_of_pos: fixes abs :: "'d \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less zero a" shows "abs a = a"
mrconc: fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> 'f option"   and x :: "'e"   and p :: "'d list"   and a :: "'d"   and b :: "'d" assumes "applied_rule_rev C x p = \<lfloor>a\<rfloor>" shows "applied_rule_rev C x (b # p) = \<lfloor>a\<rfloor>"
nonpos_Reals_of_real_iff: fixes r :: "real" shows "(of_real r \<in> \<real>\<^sub>\<le>\<^sub>0) = (r \<le> 0)"
simps: fixes g :: "('e, 'f) generator"   and s :: "'f" shows "unstream g s = (case generator g s of Done \<Rightarrow> [] | Skip x \<Rightarrow> unstream g x | Yield x s' \<Rightarrow> x # unstream g s')"
r_shift_prim: fixes f :: "recf"   and a :: "nat" assumes "prim_recfn a f" shows "prim_recfn (Suc a) (r_shift f)"
Abs_hf_inverse: fixes y :: "nat" assumes "y \<in> UNIV" shows "Rep_hf (Abs_hf y) = y"
rep_eq: fixes x :: "'a list" shows "Rep_vec (vec_of_list x) = (length x, mk_vec (length x) ((!) x))"
lexordp_append_rightI: fixes ys :: "'a list list"   and xs :: "'a list list" assumes "ys \<noteq> \<epsilon>" shows "ord.lexordp (\<lambda>x y. y <lex x) xs (xs \<cdot> ys)"
norm_stack_def: fixes \<Sigma> :: "('dyn, 'ubx1, 'ubx2) unboxed list" shows "norm_stack \<Sigma> \<equiv> map Subx.norm_unboxed \<Sigma>"
bi_totalI: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "left_total R"   and "right_total R" shows "bi_total R"
nodes_target: fixes p :: "nat \<times> 'a"   and A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set"   and r :: "(nat \<times> 'a) list" assumes "p \<in> gnodes A w V"   and "gpath A w V r p" shows "gtarget r p \<in> gnodes A w V"
valuate_opposite_val: fixes p :: "linear_poly"   and v :: "nat \<Rightarrow> 'a" shows "p \<lbrace> \<lambda>x. - v x \<rbrace> = - p \<lbrace> v \<rbrace>"
nonpos_Reals_numeral_I: fixes w :: "num" shows "numeral w \<notin> \<real>\<^sub>\<le>\<^sub>0"
is_K2_isometry_z_non_zero: fixes p :: "proj2"   and J :: "cltn2" assumes "p \<in> hyp2 \<union> S"   and "is_K2_isometry J" shows "z_non_zero (apply_cltn2 p J)"
K: shows "(\<Union>xa\<in>V - R `` {x} - R `` {y}. {per_union R x y `` {xa}}) = (\<Union>x\<in>V - R `` {x} - R `` {y}. {R `` {x}})"
subgroupE_3: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set"   and a :: "'e" assumes "Group.group G"   and "subgroup H G"   and "a \<in> H" shows "inv\<^bsub>G\<^esub> a \<in> H"
i_set_Mult_closed: fixes I :: "nat set"   and k :: "nat" assumes "I \<in> i_set" shows "I \<otimes> k \<in> i_set"
dvd_add_triv_right_iff: fixes 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 a :: "'a"   and b :: "'a" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "dvd.dvd times a (plus b a) = dvd.dvd times a b"
absorb_iff2: fixes b :: "'b"   and a :: "'b" shows "order.greater_eq a b = (inf a b = b)"
PHILs_is_PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m: shows "PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m N (replicate N 0) = fold (\<lambda>i P. P|||RPHIL i) [1..<N] LPHIL0"
collinear_closed_segment: fixes a :: "'b"   and b :: "'b" shows "collinear (closed_segment a b)"
N0_3: shows "n0 = 0 \<or> \<not> (\<forall>n'\<ge>n0 - 1. n' < length trace \<longrightarrow> execMsg n' \<noteq> msg)"
simps_3: fixes v :: "char list" shows "fromC (B v) = None"
broadcast_before_delivery: fixes m :: "'a"   and i :: "nat" assumes "Deliver m \<in> set (history i)" shows "\<exists>j. Broadcast m \<sqsubset>\<^sup>j Deliver m"
RSpan_contains_spanset: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and ms :: "'b list" assumes "RModule R smult M"   and "set ms \<subseteq> M" shows "set ms \<subseteq> R_scalar_mult.RSpan R smult ms"
dominates_def: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" shows "dominates F f g = (\<forall>p. (\<lambda>x. g x powr p) \<in> o[F](f))"
independent_if_scalars_zero: fixes A :: "('a, 'b) vec set" assumes "finite A"   and "\<And>f x. \<lbrakk>(\<Sum>x\<in>A. f x *s x) = 0; x \<in> A\<rbrakk> \<Longrightarrow> f x = (0::'a)" shows "vec.independent A"
fetch_def: shows "fetch = State_Monad.get \<bind> (\<lambda>s. State_Monad.set (stl s) \<bind> (\<lambda>_. pure (shd s)))"
prems: shows "t \<in> trms\<^sub>s\<^sub>s\<^sub>t (s # S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
zero_power2: shows "(0::'a)\<^sup>2 = (0::'a)"
integral_nonneg: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> real" assumes "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x" shows "0 \<le> integral\<^sup>L M f"
vsv_vcard_vdomain: shows "vcard (\<D>\<^sub>\<circ> xs) = vcard xs"
cases: fixes \<I>_adv :: "('a, 'b) \<I>"   and \<I>_usr :: "('c, 'd) \<I>"   and I :: "'e \<Rightarrow> bool"   and core :: "('e, 'f, 'a, 'c, 'b, 'd) core" assumes "WT_core \<I>_adv \<I>_usr I core" obtains "\<And>s e s'. \<lbrakk>s' \<in> set_spmf (cpoke core s e); I s\<rbrakk> \<Longrightarrow> I s'" and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (cfunc_adv core s x); x \<in> outs_\<I> \<I>_adv; I s\<rbrakk> \<Longrightarrow> y \<in> responses_\<I> \<I>_adv x \<and> I s'" and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (cfunc_usr core s x); x \<in> outs_\<I> \<I>_usr; I s\<rbrakk> \<Longrightarrow> y \<in> responses_\<I> \<I>_usr x \<and> I s'"
cod_eqI: fixes b :: "'b"   and f :: "'b" assumes "J0.ide b"   and "J0.seq b f" shows "J0.cod f = b"
ccpo: fixes leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lub :: "'c set \<Rightarrow> 'c" assumes "partial_function_definitions leq lub" shows "class.ccpo (fun_lub lub) (fun_ord leq) (mk_less (fun_ord leq))"
F_mono: fixes x :: "real"   and y :: "real" assumes "a \<le> x"   and "x \<le> y" shows "F x \<le> F y"
fmdrop_set_null: fixes m :: "('c, 'd) fmap" shows "fmdrop_set {} m = m"
map_prod_simp: fixes f :: "'f \<Rightarrow> 'd"   and g :: "'g \<Rightarrow> 'e"   and a :: "'f"   and b :: "'g" shows "map_prod f g (a, b) = (f a, g b)"
lexordp_eq_simps_1: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list" shows "ord.lexordp_eq less [] ys = True"
inf_commute: fixes x :: "'b"   and y :: "'b" shows "x \<squnion> y = y \<squnion> x"
assms_2: shows "\<forall>op\<in>set ops. op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"
intTF_def: shows "M.M.intTF \<equiv> InfModel.intTF"
concurrent_set_def: fixes x :: "'a"   and xs :: "'a list" shows "concurrent_set x xs \<equiv> \<forall>y\<in>set xs. x \<parallel> y"
apply_guards_false_condense: fixes G :: "vname gexp list"   and s :: "vname \<Rightarrow> value option" shows "\<exists>g. (\<not> apply_guards G s) = (gval g s = false)"
map_sel_2: fixes a :: "'c llist"   and f :: "'c \<Rightarrow> 'd" assumes "a \<noteq> LNil" shows "un_LCons2 (lmap f a) = lmap f (un_LCons2 a)"
that: fixes \<delta> :: "fm" assumes "{} \<turnstile> \<delta> IFF \<alpha>(i::=\<guillemotleft>\<delta>\<guillemotright>)"   and "supp \<delta> = supp \<alpha> - {atom i}" shows "thesis"
x_def: shows "x = Abs_freelist (rev (alternating_list m s t) @ alternating_list m s t)"
valuate_update_x: fixes lp :: "linear_poly"   and x :: "nat"   and v1 :: "nat \<Rightarrow> 'a"   and v2 :: "nat \<Rightarrow> 'a" assumes "\<forall>y\<in>vars lp. y \<noteq> x \<longrightarrow> v1 y = v2 y" shows "lp \<lbrace> v1 \<rbrace> + coeff lp x *R (v2 x - v1 x) = lp \<lbrace> v2 \<rbrace>"
separable_imp_square_free: fixes f :: "'c poly" assumes "separable f" shows "square_free f"
simps: fixes Ax :: "('a, 'b) Axioms"   and a :: "('a, 'b) Language" assumes "Axioms Ax" shows "(a \<in> Axioms.Theory Ax) = ((\<exists>A. a = A \<and> A \<in> aAxioms Ax) \<or> (\<exists>A e B. a = Vx : A \<turnstile> e\<equiv>e : B \<and> Sig aS\<^bsub>Ax\<^esub> \<triangleright> Vx : A \<turnstile> e : B) \<or> (\<exists>A e1 e2 B. a = Vx : A \<turnstile> e2\<equiv>e1 : B \<and> Vx : A \<turnstile> e1\<equiv>e2 : B \<in> Axioms.Theory Ax) \<or> (\<exists>A e1 e2 B e3. a = Vx : A \<turnstile> e1\<equiv>e3 : B \<and> Vx : A \<turnstile> e1\<equiv>e2 : B \<in> Axioms.Theory Ax \<and> Vx : A \<turnstile> e2\<equiv>e3 : B \<in> Axioms.Theory Ax) \<or> (\<exists>A e1 e2 B f C. a = Vx : A \<turnstile> f E@ e1\<equiv>f E@ e2 : C \<and> Vx : A \<turnstile> e1\<equiv>e2 : B \<in> Axioms.Theory Ax \<and> f \<in> Sig aS\<^bsub>Ax\<^esub> : B \<rightarrow> C) \<or> (\<exists>A e1 B e2 e3 C. a = Vx : A \<turnstile> sub e1 in e2\<equiv>sub e1 in e3 : C \<and> Sig aS\<^bsub>Ax\<^esub> \<triangleright> Vx : A \<turnstile> e1 : B \<and> Vx : B \<turnstile> e2\<equiv>e3 : C \<in> Axioms.Theory Ax))"
axioms_3: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "isomorphic_categories A B" shows "isomorphic_categories_axioms A B"