hcju/isabelleps huggingface dataset:
maxsimp_vertex_conv_special_coset: fixes w :: "'w"   and X :: "'w set" assumes "w \<in> W"   and "X \<in> dual_order.poset_simplex_map \<P> {w}" shows "\<exists>s\<in>S. X = w +o \<langle>S - {s}\<rangle>"
primepow: fixes p :: "nat"   and k :: "nat" assumes "prime p"   and "0 < k" shows "\<exists>g. coprime (p ^ k) g \<and> ord (p ^ k) g = Carmichael (p ^ k)"
Infinitesimal_mult: fixes x :: "'b star"   and y :: "'b star" assumes "x \<in> Infinitesimal"   and "y \<in> Infinitesimal" shows "x * y \<in> Infinitesimal"
goodAbs_def: shows "goodAbs \<equiv> qGoodAbs /// alphaAbsGood"
fconst_on_iff: fixes a :: "V"   and c :: "V"   and A :: "V" shows "([a, c]\<^sub>\<circ> \<in>\<^sub>\<circ> fconst_on A c) = (a \<in>\<^sub>\<circ> A)"
codomains_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C" shows "partial_magma.codomains C f = (if partial_magma.arr C f then {partial_magma.cod C f} else {})"
obj_fields_marked_invL_def: shows "gc.obj_fields_marked_invL = (\<lambda>s. (atS gc gc.obj_fields_marked_locs s \<longrightarrow> gc.obj_fields_marked s\<down> \<and> gc_tmp_ref s\<down> \<in> gc_W s\<down>) \<and> (atS gc gc.obj_fields_marked_good_ref_locs s \<longrightarrow> obj_at_field_on_heap (\<lambda>r. gc_ref s\<down> = Some r \<or> marked r s\<down>) (gc_tmp_ref s\<down>) (gc_field s\<down>) s\<down>) \<and> (atS gc gc.mark_loop_mo_locs s \<longrightarrow> (\<forall>x. gc_ref s\<down> \<noteq> None \<and> (gc_the_ref s\<down> reaches x) s\<down> \<longrightarrow> obj_at (\<lambda>s. True) x s\<down>)) \<and> (at gc gc.mark_loop_fields s \<longrightarrow> gc_tmp_ref s\<down> \<in> gc_W s\<down>) \<and> (at gc gc.mark_loop_mark_field_done s \<longrightarrow> gc_ref s\<down> \<noteq> None \<longrightarrow> marked (gc_the_ref s\<down>) s\<down>) \<and> (at gc gc.mark_loop_blacken s \<longrightarrow> gc_field_set s\<down> = {}) \<and> (atS gc gc.ghost_honorary_grey_empty_locs s \<longrightarrow> gc_ghost_honorary_grey s\<down> = {}))"
square_part_nonzero: fixes n :: "'b" assumes "n \<noteq> (0::'b)" shows "square_part n = normalize (\<Prod>p\<in>prime_factors n. p ^ (multiplicity p n div 2))"
inj_map_strong: fixes x :: "'b multiset"   and xa :: "'b multiset"   and f :: "'b \<Rightarrow> 'c"   and fa :: "'b \<Rightarrow> 'c" assumes "\<And>z za. \<lbrakk>z \<in># x; za \<in># xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "image_mset f x = image_mset fa xa" shows "x = xa"
gso_oc_projection: fixes i :: "nat" assumes "i < length fs" shows "gso i = oc_projection (gso ` {0..<i}) (fs ! i)"
simps_2: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'f"   and x1 :: "'g"   and x2 :: "'h" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
good_qop_pickInp_inp: shows "qGood (qOp delta (pickInp inp) (pickBinp binp))"
pareto_separable_onD: fixes A :: "'x set"   and B :: "'x set"   and C :: "'x set"   and a :: "'x"   and b :: "'x" assumes "pareto_separable_on A"   and "B \<subseteq> A"   and "C \<subseteq> A"   and "a \<in> A"   and "b \<in> A"   and "a \<noteq> b"   and "Xd a = Xd b"   and "Xh a = Xh b"   and "a \<in> Ch (Xh b) (insert a (insert b B))" shows "b \<notin> Ch (Xh b) (insert a (insert b C))"
mono_times_nat: fixes n :: "nat" assumes "0 < n" shows "mono ((*) n)"
intro: fixes \<alpha> :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set" assumes "\<alpha> \<in> S \<rightarrow>\<^sub>E T" shows "map \<alpha> S T"
insert_not_elem: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<notin> A"   and "A \<noteq> {}" shows "Max (insert x A) = max x (Max A)"
assms: shows "\<forall>ops\<in>set \<pi>. \<forall>op\<in>set ops. op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"
singular_chain_subtopology: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "singular_chain p (subtopology X S) c = (singular_chain p X c \<and> (\<forall>f\<in>Poly_Mapping.keys c. f ` standard_simplex p \<subseteq> S))"
one_cinner_a_scaleC_one: fixes a :: "'a" shows "((1::'a) \<bullet>\<^sub>C a) *\<^sub>C (1::'a) = a"
bangPres: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim> Q" shows "!P \<sim> !Q"
\<phi>_4: fixes x :: "'a" assumes "x \<in> A" shows "\<phi> x = 1"
assms_3: shows "integrable (Pi\<^sub>M (insert i I) M) f"
equivclp_trans: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "equivclp r a b"   and "equivclp r b c" shows "equivclp r a c"
equiv_subst_left: fixes z :: "'a"   and x :: "'a"   and y :: "'a"   and comp :: "comp" assumes "cmp z x = Equiv" shows "(cmp z y = comp) = (cmp x y = comp)"
timpl_closure_Fun_inv: fixes s :: "(('e, 'f, 'g) prot_fun, 'h) Term.term"   and f :: "('e, 'f, 'g) prot_fun"   and T :: "(('e, 'f, 'g) prot_fun, 'h) Term.term list"   and TI :: "('g set \<times> 'g set) set" assumes "s \<in> timpl_closure (Fun f T) TI" shows "\<exists>g S. s = Fun g S"
CatchMatch: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and s' :: "'e"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> =n\<Rightarrow> Abrupt s'"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal s'\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Catch c\<^sub>1 c\<^sub>2,Normal s\<rangle> =n\<Rightarrow> t"
inj_img_insertE: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'b"   and B :: "'b set" assumes "inj_on f A"   and "x \<notin> B"   and "insert x B = f ` A" obtains x' :: "'a" and A' :: "'a set" where "x' \<notin> A'" and "A = insert x' A'" and "x = f x'" and "B = f ` A'"
linear_subspace_image: fixes f :: "'d \<Rightarrow> 'e"   and S :: "'d set" assumes "clinear f"   and "csubspace S" shows "csubspace (f ` S)"
DF_unfold: fixes A :: "'a set" shows "DF A = (\<sqinter>z\<in>A \<rightarrow>  DF A)"
sel_rews_6: fixes a :: "'a"   and slist :: "[:'a:]" assumes "a \<noteq> \<bottom>" shows "stail\<cdot>(a :# slist) = slist"
finite_list_subset: fixes A :: "'a set"   and n :: "nat" assumes "finite A"   and "n \<le> card A" obtains xs :: "'a list" where "set xs \<subseteq> A" and "distinct xs" and "length xs = n"
rexp_DA_axioms: shows "rexp_DA \<sigma> wf_atom project lookup init delta final wf_state id L n"
outputPres: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name" assumes "(P, Q) \<in> Rel" shows "a{b}.P \<leadsto><Rel> a{b}.Q"
crel_vs_elim: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and v :: "'f"   and v\<^sub>T :: "('b, 'g) state"   and M :: "'b" assumes "crel_vs R v v\<^sub>T"   and "cmem M"   and "P M" obtains v' :: "'g" and M' :: "'b" where "run_state v\<^sub>T M = (v', M')" and "R v v'" and "cmem M'" and "P M'"
override_on_cont1: fixes m :: "'c \<Rightarrow> 'd"   and S :: "'c set" shows "cont (\<lambda>x. x ++\<^bsub>S\<^esub> m)"
cont_spmf: fixes x :: "'c" shows "cont lub_spmf (ord_spmf (=)) Sup (\<le>) (\<lambda>p. spmf p x)"
monoE: fixes f :: "'a \<Rightarrow> 'c"   and x :: "'a"   and y :: "'a" assumes "mono f"   and "x \<le> y" obtains "f x \<le> f y"
simps: fixes xs :: "'c list" shows "IArray.list_of (IArray xs) = xs"
hyps: fixes x :: "'a" assumes "\<lbrakk>rtrancl_path r x (as @ [y]) y; rtrancl_path r y ys z\<rbrakk> \<Longrightarrow> thesis"   and "rtrancl_path r x (as @ y # ys) z" shows "thesis"
n_kat_21_opp: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "y \<cdot> !z \<le> !x \<cdot> y" shows "!x \<cdot> y \<cdot> !z = y \<cdot> !z"
norm_mult_numeral2: fixes a :: "'a"   and w :: "num" shows "norm (a * numeral w) = norm a * numeral w"
pseudoinverse': shows "\<forall>\<omega>\<in>{a<..<b}. \<forall>x. (\<omega> \<le> f x) = (I \<omega> \<le> x)"
rel_cset_csingle_iff: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'f"   and y :: "'g" shows "rel_cset R (csingle x) (csingle y) = R x y"
IH1: fixes s :: "'a net_state"   and s' :: "'a net_state" assumes "wf_net_tree p1"   and "s \<in> reachable (pnet np p1) TT"   and "(s, \<tau>, s') \<in> automaton.trans (pnet np p1)" shows "\<exists>i\<in>net_ips s. (\<forall>j. j \<noteq> i \<longrightarrow> netmap s' j = netmap s j) \<and> net_ip_action np \<tau> i p1 s s'"
gt_na_poss: fixes a :: "ant"   and m :: "nat" assumes "na a < m" shows "0 < m"
part_not_discrete_card: fixes X :: "'b set" assumes "part X"   and "\<not> discrete X" shows "card {f. CK_nf_pos f} = 3"
big_iff_small_WT: fixes P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "wwf_J_prog P"   and "P,E \<turnstile> e :: T"   and "P,shp s \<turnstile>\<^sub>b (e,b) \<surd>" shows "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle> = (P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',False\<rangle> \<and> final e')"
min_maxsimpchainD_maxsimpchain: fixes xs :: "'a set list" assumes "min_maxsimpchain xs" shows "maxsimpchain xs"
Anfang: fixes \<DD> :: "'b formula" shows "[\<DD>] \<Rightarrow> [\<DD>]"
normal_in_normal_set_mult: fixes N :: "'a set"   and H :: "'a set" assumes "N \<lhd> G"   and "subgroup H G" shows "N \<lhd> G\<lparr>carrier := N <#> H\<rparr>"
trms\<^sub>s\<^sub>s\<^sub>t_unlabel_subset_if_no_label: fixes l :: "'a"   and A :: "('a strand_label \<times> ('b, 'c) stateful_strand_step) list"   and l' :: "'a" assumes "\<not> list_ex (is_LabelN l) A" shows "trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj l A) \<subseteq> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj l' A)"
swaprows_mat: fixes A :: "'b mat"   and n :: "nat"   and nc :: "nat"   and k :: "nat"   and l :: "nat" assumes "A \<in> carrier_mat n nc"   and "k < n"   and "l < n" shows "swaprows k l A = swaprows_mat n k l * A"
unique_session_keys: fixes A :: "agent"   and nb :: "msg"   and B :: "agent"   and K :: "nat"   and na :: "msg"   and X :: "msg"   and evs :: "event list"   and A' :: "agent"   and nb' :: "msg"   and B' :: "agent"   and na' :: "msg"   and X' :: "msg" assumes "Says Server A \<lbrace>nb, Crypt (shrK A) \<lbrace>Agent B, Key K, na\<rbrace>, X\<rbrace> \<in> set evs"   and "Says Server A' \<lbrace>nb', Crypt (shrK A') \<lbrace>Agent B', Key K, na'\<rbrace>, X'\<rbrace> \<in> set evs"   and "evs \<in> yahalom" shows "A = A' \<and> B = B' \<and> na = na' \<and> nb = nb'"
strict_trans2: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<prec> a"   and "c \<preceq> b" shows "c \<prec> a"
Broadcast_Deliver_prefix_closed: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and r :: "'a"   and ix :: "'a set"   and e :: "'b"   and j :: "nat"   and i :: "'a" assumes "xs @ [Broadcast (r, Rem ix e)] prefix of j"   and "i \<in> ix" shows "Deliver (i, Add i e) \<in> set xs"
F_inf_def: fixes f :: "'s \<Rightarrow> ennreal" shows "F_inf f = (\<lambda>s\<in>S. if s \<in> S2 then 1 else if s \<in> S1 then \<Sqinter>D\<in>K s. integral\<^sup>N (measure_pmf D) f else 0)"
that: fixes a :: "V"   and b :: "V" assumes "x = \<langle>a, b\<rangle>"   and "a \<in>\<^sub>\<circ> A"   and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "thesis"
less_eq_multiset_def: fixes M' :: "'k multiset"   and M :: "'k multiset" shows "(M' \<le> M) = (M' < M \<or> M' = M)"
main: fixes pa :: "complex poly"   and i :: "nat" assumes "(pa, i) \<in> set pis" shows "set (roots_of_complex_main pa) = {x. poly pa x = 0}"
f_g: fixes y :: "'b" assumes "y \<in> B" shows "f (g y) = y"
map_of_set_simps_2: fixes m :: "'a \<Rightarrow> unit option" shows "map_of_set (dom m) = m"
that: fixes L :: "real" assumes "X \<longlonglongrightarrow> L" shows "thesis"
assms_2: fixes i :: "Integer" assumes "defined i"   and "\<lbrakk>i\<rbrakk> < 0" shows "P i"
that: fixes c :: "'a" assumes "homotopic_with_canon (\<lambda>x. True) U T f (\<lambda>x. c)" shows "thesis"
distinct_lroft_s3: fixes amr :: "('b \<times> 32 simple_match \<times> char list \<times> simple_action) list"   and ifs :: "char list list" assumes "distinct (map fst amr)"   and "distinct ifs" shows "distinct (lr_of_tran_s3 ifs amr)"
IH_1: fixes ds :: "bool list" assumes "internal ds T\<^sub>1" shows "path ds T\<^sub>1"
ereal_open_atLeast: fixes x :: "ereal" shows "open {x..} = (x = - \<infinity>)"
gcs_plus_left: fixes u :: "'a"   and s :: "'a"   and t :: "'a" shows "gcs (u + s) (u + t) = u + gcs s t"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "\<And>A zs ys xs gt. \<lbrakk>finite A; zs \<in> lists A; ys \<in> lists A; xs \<in> lists A; \<forall>x\<in>A. \<not> gt x x; \<forall>z\<in>A. \<forall>y\<in>A. \<forall>x\<in>A. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x; ext gt zs ys; ext gt ys xs\<rbrakk> \<Longrightarrow> ext gt zs xs" shows "ext_irrefl_before_trans_axioms ext"
Kas_never_says_current_timestamp: fixes evs :: "event list"   and T :: "nat"   and X :: "msg" assumes "CT evs = T"   and "Number T \<in> parts {X}"   and "evs \<in> kerbV" shows "\<forall>A. Says Kas A X \<notin> set evs"
asian_option_def: fixes p :: "real"   and M :: "bool stream measure"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real"   and Mkt :: "(bool stream, 'b) discrete_market"   and r :: "real"   and risk_free_asset :: "'b"   and stk :: "'b"   and G :: "nat \<Rightarrow> bool stream measure"   and T :: "nat" assumes "CRR_market p M geom_proc u d init Mkt r risk_free_asset stk G" shows "CRR_market.asian_option Mkt stk T = (\<lambda>w. (\<Sum>i = 1..T. prices Mkt stk i w) / real T)"
locally_finite_onE: fixes X :: "'c set"   and I :: "'d set"   and U :: "'d \<Rightarrow> 'c set"   and p :: "'c" assumes "locally_finite_on X I U"   and "p \<in> X" obtains N :: "'c set" where "p \<in> N" and "open N" and "finite {i \<in> I. U i \<inter> N \<noteq> {}}"
symrun_interp_expansion: fixes \<Gamma>\<^sub>1 :: "'b constr list"   and \<Gamma>\<^sub>2 :: "'b constr list" shows "\<lbrakk>\<lbrakk> \<Gamma>\<^sub>1 @ \<Gamma>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk>\<lbrakk> \<Gamma>\<^sub>1 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk>\<lbrakk> \<Gamma>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
simps_1: fixes x :: "'b" shows "is_a_constant (Var x) = False"
distinct_3: fixes x1 :: "b_e"   and x3 :: "cl" shows "$x1 \<noteq> Callcl x3"
writerT_left_unit_invar_lemma: fixes m :: "'e\<cdot>('f, 'g) writerT" assumes "invar m" shows "runWriterT\<cdot>m \<bind> (\<Lambda> (Writer\<cdot>w\<cdot>x). return\<cdot>(Writer\<cdot>w\<cdot>x)) = runWriterT\<cdot>m"
Pi_I': fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'b \<Rightarrow> 'c set" assumes "\<And>x. x \<in> A \<longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
sup_idem: fixes x :: "'e multiset" shows "x \<union># x = x"
Gromov_converging_at_boundary_imp_not_constant: fixes x :: "'a" shows "\<not> Gromov_converging_at_boundary (\<lambda>n. x)"
Rep_CoCall_inject: fixes x :: "CoCalls"   and y :: "CoCalls" shows "(Rep_CoCall x = Rep_CoCall y) = (x = y)"
refl: fixes x :: "'b" assumes "x \<in> carrier" shows "R i x x"
avl_delete_1: fixes t :: "(('c \<times> 'd) \<times> nat) tree"   and x :: "'c" assumes "avl t" shows "avl (AVL_Map.delete x t)"
closedin_Inter: fixes K :: "'c set set"   and U :: "'c topology" assumes "K \<noteq> {}"   and "\<And>S. S \<in> K \<Longrightarrow> closedin U S" shows "closedin U (\<Inter> K)"
m_comm: fixes x :: "nat"   and y :: "nat" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"
Domain_Rel_cases: fixes x0 :: "'pub0"   and x1 :: "'pub1" assumes "(x0, x1) \<in> Domain Rel_OR" shows "(\<exists>w0. (x0, w0) \<in> Rel0 \<and> x1 \<in> valid_pub1) \<or> (\<exists>w1. (x1, w1) \<in> Rel1 \<and> x0 \<in> valid_pub0)"
rel_sel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b tree"   and b :: "'c tree" shows "rel_tree R a b = (R (root a) (root b) \<and> rel_tree R (left a) (left b) \<and> rel_tree R (right a) (right b))"
hyps: fixes xa :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "xa \<in> fmran' \<pi>" shows "xa \<squnion> \<sigma> = \<sigma> \<squnion> xa"
seqp_broadcastTE: fixes \<xi> :: "'e"   and l :: "'h"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<xi>' :: "'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "((\<xi>, {l}broadcast(s\<^sub>m\<^sub>s\<^sub>g) . p), a, \<xi>', q) \<in> seqp_sos \<Gamma>" obtains "a = broadcast (s\<^sub>m\<^sub>s\<^sub>g \<xi>)" and "\<xi>' = \<xi>" and "q = p"
msg_fresh_1: fixes \<sigma> :: "nat \<Rightarrow> state"   and hops :: "nat"   and rreqid :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool" shows "msg_fresh \<sigma> (Rreq hops rreqid dip dsn dsk oip osn sip handled) = (1 \<le> osn \<and> (sip \<noteq> oip \<longrightarrow> oip \<in> kD (rt (\<sigma> sip)) \<and> osn \<le> nsqn (rt (\<sigma> sip)) oip \<and> (nsqn (rt (\<sigma> sip)) oip = osn \<longrightarrow> the (dhops (rt (\<sigma> sip)) oip) \<le> hops \<or> the (flag (rt (\<sigma> sip)) oip) = Aodv_Basic.inv)))"
simps_1: fixes x :: "'a" shows "selem\<cdot>x\<cdot>[::] = FF"
continuous_on_tendsto_compose: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a"   and l :: "'a"   and F :: "'c filter" assumes "continuous_on s f"   and "(g \<longlongrightarrow> l) F"   and "l \<in> s"   and "\<forall>\<^sub>F x in F. g x \<in> s" shows "((\<lambda>x. f (g x)) \<longlongrightarrow> f l) F"
prod_dvd_prod_subset: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and B :: "'c set"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.comm_monoid_mult times one"   and "finite B"   and "A \<subseteq> B" shows "dvd.dvd times (comm_monoid_mult.prod times one f A) (comm_monoid_mult.prod times one f B)"
SPLIT_ACCORDING_def: fixes m :: "'d list"   and l :: "'e list" shows "SPLIT_ACCORDING m l \<equiv> length l = length m"
not_mod_2_eq_0_eq_1: fixes a :: "'a" shows "(a mod (2::'a) \<noteq> (0::'a)) = (a mod (2::'a) = (1::'a))"
ipresIGVar_def: fixes h :: "'h \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'l, 'm, 'n, 'o, 'h, 'p, 'q) model_scheme"   and MOD' :: "('r, 's, 'l, 't, 'u, 'o, 'i, 'v, 'w) model_scheme" shows "ipresIGVar h MOD MOD' \<equiv> \<forall>xs x. h (igVar MOD xs x) = igVar MOD' xs x"
sum_list_replicate: fixes n :: "nat"   and c :: "'b" shows "sum_list (replicate n c) = of_nat n * c"
mahler_graeffe: fixes f :: "complex poly"   and c :: "complex"   and as :: "complex list"   and m :: "nat" assumes "f = Polynomial.smult c (\<Prod>a\<leftarrow>as. [:- a, 1:])" shows "mahler_measure_poly (graeffe_poly c as m) = mahler_measure_poly f ^ 2 ^ m"
ge_refl: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.non_strict_order less_eq" shows "less_eq x x"
acomplete_Node_if_wbal1: fixes l :: "'b tree"   and r :: "'b tree"   and x :: "'b" assumes "acomplete l"   and "acomplete r"   and "size l = size r + 1" shows "acomplete \<langle>l, x, r\<rangle>"
ch2ch_monofun: fixes f :: "'c \<Rightarrow> 'd"   and Y :: "nat \<Rightarrow> 'c" assumes "monofun f"   and "chain Y" shows "chain (\<lambda>i. f (Y i))"
four_elem_comm: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R"   and "d \<in> carrier R" shows "a \<ominus> c \<oplus> b \<ominus> d = a \<oplus> b \<ominus> c \<ominus> d"
is_extensional: fixes f :: "'a" assumes "\<not> arr f" shows "\<d>[f] = null"
I_def: fixes u :: "complex" shows "I u = exp u * (\<Sum>j\<le>degree f. poly ((pderiv ^^ j) f) 0) - (\<Sum>j\<le>degree f. poly ((pderiv ^^ j) f) u)"
dg_GRPH_is_arrD: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<^bsub>dg_GRPH \<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
UN_I: fixes a :: "'e"   and A :: "'e set"   and b :: "'f"   and B :: "'e \<Rightarrow> 'f set" assumes "a \<in> A"   and "b \<in> B a" shows "b \<in> \<Union> (B ` A)"
use_fps: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "weak_partial_order L"   and "x \<in> fps L f" shows "f x .=\<^bsub>L\<^esub> x"
simps_2: fixes f :: "'l \<Rightarrow> 'm \<Rightarrow> 'k"   and x1 :: "'l"   and x2 :: "'m" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
transition_idx_less: fixes i :: "nat"   and j :: "nat"   and s :: "nat \<Rightarrow> 'b"   and T :: "nat \<Rightarrow> ('b \<times> 'b) set"   and k :: "nat"   and n :: "nat" assumes "i < j"   and "(s j, s i) \<in> T k"   and "k < n" shows "transition_idx s T {i, j} < n"
assms: shows "Sstd.load p\<^sub>1 s\<^sub>1"
norm_infsetsum_bound: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "norm (infsetsum f A) \<le> (\<Sum>\<^sub>ax\<in>A. norm (f x))"
ridmap_def: fixes R :: "('c, 'd) Ring_scheme" shows "ridmap R = (\<lambda>x\<in>carrier R. x)"
rtrancl_is_final: fixes a :: "'a"   and b :: "'a" assumes "a \<rightarrow>\<^sup>* b"   and "is_final a" shows "a = b"
compatibility_def: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat"   and q :: "(symbol \<times> character list) list"   and q' :: "(symbol \<times> character list) list"   and t :: "symbol \<times> character list" assumes "p \<in> \<P> k u"   and "q \<in> \<P> k u"   and "charslength p = k"   and "q = q' @ [t]"   and "charslength q' = k"   and "admissible (p @ [t])" shows "p @ [t] \<in> \<P> k u"
arr_dom: fixes f :: "'a" assumes "arr f" shows "arr (local.dom f)"
that: assumes "map_iterator_genord iti (map_of lc) (\<lambda>(k, v) (k', v'). R (k, \<alpha> v) (k', \<alpha> v'))"   and "(map_option \<alpha> \<circ>\<circ> map_of) lc = m"   and "\<And>k v. map_of lc k = Some v \<Longrightarrow> invar v" shows "thesis"
list_assn_aux_ineq_len: fixes l :: "'b list"   and li :: "'c list"   and A :: "'b \<Rightarrow> 'c \<Rightarrow> assn" assumes "length l \<noteq> length li" shows "list_assn A l li = false"
respond_certificate: fixes A :: "agent"   and B :: "msg"   and NA :: "msg"   and P :: "msg"   and RA :: "msg"   and K :: "nat"   and evs :: "event list" assumes "(Hash[Key (shrK A)] \<lbrace>Agent A, B, NA, P\<rbrace>, RA, K) \<in> respond evs" shows "Crypt (shrK A) \<lbrace>Key K, B, NA\<rbrace> \<in> parts {RA}"
assms_2: shows "inv_imps_work_sum (prop_config c1 p)"
total_pseudo_ordered_set_axioms: shows "total_pseudo_ordered_set P (\<sqsubseteq>)"
a_coset_add_assoc: fixes M :: "'a set"   and g :: "'a"   and h :: "'a" assumes "M \<subseteq> carrier R"   and "g \<in> carrier R"   and "h \<in> carrier R" shows "M +> g +> h = M +> g \<oplus> h"
dilated_strict_prefix: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and n :: "nat"   and c :: "clock" assumes "dilating f sub r" shows "{i. i < f n \<and> hamlet (Rep_run r i c)} = f ` {i. i < n \<and> hamlet (Rep_run sub i c)}"
return\<^sub>C_def: fixes upd :: "('e list \<Rightarrow> 'e list) \<Rightarrow> 'd control_state_scheme \<Rightarrow> 'd control_state_scheme"   and rhs :: "'d control_state_scheme \<Rightarrow> 'e" shows "return\<^sub>C upd rhs = (\<lambda>\<sigma>. if exec_stop \<sigma> then Some ((), \<sigma>) else (assign_local upd rhs ;- set_return_status) \<sigma>)"
word_sle_eq: fixes a :: "'b word"   and b :: "'b word" shows "(a \<le>s b) = (sint a \<le> sint b)"
hyps: shows "[:b, 1::'a:] ^ length (filter ((=) b) as) dvd (\<Prod>a\<leftarrow>as. [:a, 1::'a:])"
length_prefixes: fixes xs :: "'b list" shows "length (prefixes xs) = length xs + 1"
l_type_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_axioms: shows "l_type_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t type_wf"
u_simps_3: shows "trg u = src (k\<^sup>* \<star> h)"