hcju/isabelleps huggingface dataset:
homogeneous_ideal: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and n :: "nat" assumes "\<And>f. f \<in> F \<Longrightarrow> homogeneous f"   and "p \<in> ideal F" shows "hom_component p n \<in> ideal F"
rel_FGco_def: fixes L1 :: "'n \<Rightarrow> 'aa \<Rightarrow> bool"   and Co1 :: "'o \<Rightarrow> 'ab \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'ad \<Rightarrow> bool"   and Co3 :: "'r \<Rightarrow> 'ae \<Rightarrow> bool"   and Co4 :: "'s \<Rightarrow> 'af \<Rightarrow> bool"   and Co5 :: "'p \<Rightarrow> 'ac \<Rightarrow> bool"   and Co6 :: "'w \<Rightarrow> 'ai \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'ag \<Rightarrow> bool"   and Contra2 :: "'u \<Rightarrow> 'ah \<Rightarrow> bool"   and Contra3 :: "'x \<Rightarrow> 'aj \<Rightarrow> bool"   and Contra4 :: "'y \<Rightarrow> 'ak \<Rightarrow> bool" shows "rel_FGco L1 Co1 Co2 Co3 Co4 Co5 Co6 Contra1 Contra2 Contra3 Contra4 = rel_F L1 Co1 Co5 (rel_G Co1 Co2 Co3 Co4 Contra1 Contra2) Co3 Co6 Contra1 Contra3 Contra4"
cod_cod: fixes f :: "'s" assumes "S.arr f" shows "S.cod (S.cod f) = S.cod f"
simps: fixes xs :: "(nat \<times> 'b) list"   and m :: "nat option list"   and i :: "nat"   and j :: "nat" shows "idx_pqueue_swap_fun (xs, m) i j = (list_swap xs i j, m[fst (xs ! i) := Some j, fst (xs ! j) := Some i])"
mult_right_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq zero c" shows "less_eq (times a c) (times b c)"
assms_3: fixes i :: "nat" shows "coeff p i * coeff p (i + 2) \<le> (coeff p (i + 1))\<^sup>2"
root_condE: fixes plr :: "int poly \<times> rat \<times> rat"   and x :: "real" assumes "root_cond plr x" obtains "real_of_rat (rai_lb plr) \<le> x" and "x \<le> real_of_rat (rai_ub plr)" and "poly (real_of_int_poly (poly_real_alg_1 plr)) x = 0"
intros_4: fixes s :: "dB"   and s' :: "dB"   and t :: "dB"   and t' :: "dB" assumes "s => s'"   and "t => t'" shows "Abs s \<degree> t => s'[t'/0]"
vfsequence_vdomain: shows "\<D>\<^sub>\<circ> xs\<^sub>1 = vcard xs\<^sub>1"
Cons_1: fixes L' :: "atomUni list" assumes "convert_atom_list var L (xs' @ x # xs) = Some L'"   and "l' \<in> set L'" shows "\<exists>l\<in>set L. convert_atom var l (xs' @ x # xs) = Some l'"
invariant_def: fixes A :: "('a, 'b) automaton"   and I :: "'b \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" shows "(A \<TTurnstile> (I \<rightarrow>) P) = (\<forall>s\<in>reachable A I. P s)"
inter_sctn_specs_inter_sctn_spec: fixes d :: "nat" assumes "d = DIM('c)" shows "(inter_sctn_specs d, inter_sctn_spec) \<in> \<langle>lv_rel\<rangle>set_rel \<rightarrow> \<langle>lv_rel\<rangle>sctn_rel \<rightarrow> \<langle>\<langle>lv_rel\<rangle>set_rel\<rangle>nres_rel"
lcm_neg1: fixes a :: "'a"   and b :: "'a" shows "lcm (- a) b = lcm a b"
m_ivl_def: fixes ivl :: "ivl" shows "m_ivl ivl = (case ivl of I l h \<Rightarrow> (case l of None \<Rightarrow> 0 | Some x \<Rightarrow> 1) + (case h of None \<Rightarrow> 0 | Some x \<Rightarrow> 1))"
greaterThanAtMost_empty_iff2: fixes k :: "'c multiset"   and l :: "'c multiset" shows "({} = subset_mset.greaterThanAtMost k l) = (\<not> k \<subset># l)"
simps: fixes a :: "'b"   and r :: "('c \<times> 'b) set" shows "(a \<in> Range r) = (\<exists>aa b. a = b \<and> (aa, b) \<in> r)"
trans: fixes r :: "'e"   and s :: "'e"   and t :: "'e" assumes "r = s"   and "s = t" shows "r = t"
multiplicity_eqI: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and n :: "nat"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times (power.power one times p n) x"   and "\<not> dvd.dvd times (power.power one times p (Suc n)) x" shows "factorial_semiring.multiplicity times one p x = n"
cbox_def: fixes a :: "'c"   and b :: "'c" shows "cbox a b = {x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i}"
d_reachable_reachable: fixes M :: "('e, 'f, 'd) FSM"   and p :: "'d" shows "d_reachable M p \<subseteq> reachable M p"
continuous_on_hurwitz_zeta: fixes a :: "real"   and A :: "complex set" assumes "0 < a"   and "1 \<notin> A" shows "continuous_on A (hurwitz_zeta a)"
bind_split: fixes P :: "'c option \<Rightarrow> bool"   and m :: "'d option"   and f :: "'d \<Rightarrow> 'c option" shows "P (m \<bind> f) = ((m = None \<longrightarrow> P None) \<and> (\<forall>v. m = Some v \<longrightarrow> P (f v)))"
mem_existence_ivl_subset: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl t0 x0" shows "t \<in> T"
upper_triangularD: fixes A :: "'a mat"   and j :: "nat"   and i :: "nat" assumes "upper_triangular A"   and "j < i"   and "i < dim_row A" shows "A $$ (i, j) = (0::'a)"
subprob_space_bind: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b measure"   and N :: "'b measure" assumes "subprob_space M"   and "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N" shows "subprob_space (M \<bind> f)"
preorder_on_axioms: shows "Order_Predicates.preorder_on (inv f -` carrier) (map_relation (inv f) le)"
link_tree_invar: fixes t1 :: "('c, 'd) SkewBinomialTree"   and t2 :: "('c, 'd) SkewBinomialTree" assumes "tree_invar t1"   and "tree_invar t2"   and "rank t1 = rank t2" shows "tree_invar (link t1 t2)"
inv_equality: fixes G :: "('f, 'g) monoid_scheme"   and y :: "'f"   and x :: "'f" assumes "Group.group G"   and "y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "inv\<^bsub>G\<^esub> x = y"
coeffs_uminus: fixes p :: "'b poly" shows "coeffs (- p) = map uminus_class.uminus (coeffs p)"
simps: fixes x :: "udom"   and k :: "udom \<rightarrow> udom\<cdot>lifted" shows "bindU\<cdot>(Lifted\<cdot>x)\<cdot>k = k\<cdot>x"
"1_1": shows "fst a \<subseteq>\<^sub>f s"
LPDs_MSkip: fixes n :: "nat" shows "LPDs (MSkip n) \<subseteq> MSkip ` {i. i \<le> n}"
tl_zero: fixes P :: "'b \<Rightarrow> bool"   and ys :: "'b list"   and j :: "nat" assumes "P (ys ! Suc j)"   and "Suc j < length ys"   and "ys \<noteq> []" shows "P (tl ys ! j)"
s_mul_ext_bottom_strict: fixes A :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" shows "({#}, A) \<notin> s_mul_ext ns s"
remove_all_distinct: fixes xs :: "'d list"   and x :: "'d" assumes "distinct xs" shows "distinct (x # remove_all x xs)"
Infinitesimal_monad_zero_iff: fixes x :: "'a star" shows "(x \<in> Infinitesimal) = (x \<in> monad 0)"
establish_invarI_ND: fixes I :: "('v, 'es) state_scheme \<Rightarrow> bool" assumes "\<And>u v s. on_discover param u v s = on_new_root param v s"   and "on_init param \<le>\<^sub>n SPEC (\<lambda>x. I (empty_state x))"   and "\<And>s s' v. \<lbrakk>DFS_invar G param s; I s; cond s; \<not> is_break param s; v \<notin> dom (discovered s); discovered s' = discovered s(v \<mapsto> counter s); finished s' = finished s; counter s' = Suc (counter s); stack s' = v # stack s; back_edges s' = back_edges s; cross_edges s' = cross_edges s; tree_edges s \<subseteq> tree_edges s'; state.more s' = state.more s\<rbrakk> \<Longrightarrow> on_new_root param v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u. \<lbrakk>DFS_invar G param s; I s; cond s; \<not> is_break param s; stack s \<noteq> []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s\<rbrakk> \<Longrightarrow> on_finish param u s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u v. \<lbrakk>DFS_invar G param s; I s; cond s; \<not> is_break param s; stack s \<noteq> []; (u, v) \<in> pending s; u = hd (stack s); v \<in> dom (discovered s); v \<in> dom (finished s); s' = cross_edge u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)\<rbrakk> \<Longrightarrow> on_cross_edge param u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u v. \<lbrakk>DFS_invar G param s; I s; cond s; \<not> is_break param s; stack s \<noteq> []; (u, v) \<in> pending s; u = hd (stack s); v \<in> dom (discovered s); v \<notin> dom (finished s); s' = back_edge u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)\<rbrakk> \<Longrightarrow> on_back_edge param u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))" shows "is_invar I"
gflimuls_eq: shows "(\<Union>Cl\<in>Ql. \<G>_F (fst Cl)) = G"
surjective: fixes r :: "('c, 'd) monoid_scheme" shows "r = \<lparr>carrier = carrier r, mult = (\<otimes>\<^bsub>r\<^esub>), one = \<one>\<^bsub>r\<^esub>, \<dots> = monoid.more r\<rparr>"
lfinite_eq_range_llist_of: fixes xs :: "'c llist" shows "lfinite xs = (xs \<in> range llist_of)"
fcd_le: fixes x :: "complex" assumes "x \<in> interior S \<and> x \<in> interior {x. d \<bullet> x \<le> k}" shows "f field_differentiable at x"
axioms_1: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "group_hom G H h" shows "Group.group G"
FHModule_indspace: shows "FGModule H (\<currency>) indV"
independentD: fixes s :: "'b set"   and t :: "'b set"   and u :: "'b \<Rightarrow> 'a"   and v :: "'b" assumes "s \<subseteq> U\<^sub>V\<^sub>S"   and "\<not> local.dependent s"   and "finite t"   and "t \<subseteq> s"   and "(\<Sum>v\<in>t. u v *s v) = (0::'b)"   and "v \<in> t" shows "u v = (0::'a)"
in_net_ips_fst_init_missing: fixes i :: "nat"   and s :: "'s net_state" assumes "i \<in> net_ips s" shows "fst (initmissing (netgmap sr s)) i = the (fst (netgmap sr s) i)"
times_trilean_assoc: fixes a :: "trilean"   and b :: "trilean"   and c :: "trilean" shows "a \<and>? b \<and>? c = a \<and>? (b \<and>? c)"
assms_1: shows "interaction_bounded_by' (\<lambda>_. True) gpv n"
local_property_def: fixes P :: "(nat \<Rightarrow> 'f \<Rightarrow> 'g) \<Rightarrow> bool" shows "local_property P \<equiv> \<forall>\<sigma> \<tau>. locally_similar \<sigma> \<tau> \<longrightarrow> P \<sigma> \<longrightarrow> P \<tau>"
wf_cdecl_wt_init: fixes G :: "prog"   and C :: "qtname"   and c :: "class" assumes "wf_cdecl G (C, c)" shows "\<lparr>prg = G, cls = C, lcl = Map.empty\<rparr>\<turnstile>init c\<Colon>\<surd>"
fd_eq_fdia: fixes x :: "'a"   and y :: "'a" shows "domain_semiringl.fd (\<cdot>) d x y \<equiv> antidomain_semiringl.fdia a (\<cdot>) x y"
subst_cls_list_empty: fixes \<sigma> :: "'s" shows "[] \<cdot>cl \<sigma> = []"
snapshot_stable_ver_3: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and p :: "'d"   and j :: "nat" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "\<not> ps (distributed_system.s channel trans send recv init t i) p \<noteq> None"   and "j \<le> i" shows "\<not> ps (distributed_system.s channel trans send recv init t j) p \<noteq> None"
less: shows "((length xs + 4) div 5 - 1) div 2 < nat \<lceil>real (length xs) / 5\<rceil>"
weakSimWeakEqSim: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" assumes "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q" shows "P \<leadsto><Rel> Q"
space_uniform_count_measure: fixes A :: "'c set" shows "space (uniform_count_measure A) = A"
inject_2: fixes x3 :: "'a"   and y3 :: "'a" shows "(Goes_wrong x3 = Goes_wrong y3) = (x3 = y3)"
WriteS4: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds"   and l :: "Locs" shows "\<turnstile> $ImpInv rmhist p \<and> Write rmCh mm ires p l \<longrightarrow> $S4 rmhist p"
has_witness_two: fixes n :: "nat"   and S :: "'b set" assumes "has n S"   and "2 \<le> n" shows "\<exists>x y. hasw [x, y] S"
simps: fixes n :: "nat"   and p :: "'person"   and w :: "'person \<Rightarrow> 'color"   and w' :: "'person \<Rightarrow> 'color" shows "local.possible n p w w' = (local.valid w \<and> local.valid w' \<and> (\<forall>p'. p' \<noteq> p \<longrightarrow> w p' = w' p') \<and> (\<forall>n'<n. \<forall>p'. local.leaves n' p' w = local.leaves n' p' w'))"
intro: fixes S :: "'c set" assumes "subspace S" shows "real_vector_space_on S"
ker_of_idmap: fixes A :: "('c, 'd) Ring_scheme" assumes "Ring A" shows "ker\<^bsub>A,A\<^esub> ridmap A = {\<zero>\<^bsub>A\<^esub>}"
select_in_set: fixes k :: "nat"   and xs :: "'b list" assumes "k < length xs" shows "select k xs \<in> set xs"
ta_bisim_\<epsilon>_2: fixes b :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and ta :: "('f, 'a, 'b, 'c, 'g, 'h) thread_action" shows "ta_bisim b ta \<lbrace>\<rbrace> = (ta = \<lbrace>\<rbrace>)"
invertible_set_def: fixes R :: "('a sequent list \<times> 'a sequent) set" shows "invertible_set R \<equiv> \<forall>(ps, c)\<in>R. invertible (ps, c) R"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b stream"   and b :: "'c stream" assumes "stream_all2 R a b" obtains x1a :: "'b" and x2a :: "'b stream" and y1a :: "'c" and y2a :: "'c stream" where "a = x1a ## x2a" and "b = y1a ## y2a" and "R x1a y1a" and "stream_all2 R x2a y2a"
totatives_prime_power_Suc: fixes p :: "nat"   and n :: "nat" assumes "prime p" shows "totatives (p ^ Suc n) = {0<..p ^ Suc n} - (*) p ` {0<..p ^ n}"
simps_1: fixes x :: "('b, 'c) Combinators"   and y :: "('b, 'c) Combinators" shows "first_bothNet (x \<oplus> y) = first_bothNet x"
measurable_Pair2': fixes y :: "'a"   and M2 :: "'a measure"   and M1 :: "'b measure" assumes "y \<in> space M2" shows "(\<lambda>x. (x, y)) \<in> M1 \<rightarrow>\<^sub>M M1 \<Otimes>\<^sub>M M2"
birkhoff_sum_mono: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real"   and n :: "nat"   and x :: "'b" assumes "qmpt M T"   and "\<And>x. f x \<le> g x" shows "qmpt.birkhoff_sum T f n x \<le> qmpt.birkhoff_sum T g n x"
simply_connected_eq_winding_number_zero: fixes S :: "complex set" assumes "open S" shows "simply_connected S = (connected S \<and> (\<forall>g z. path g \<and> path_image g \<subseteq> S \<and> pathfinish g = pathstart g \<and> z \<notin> S \<longrightarrow> winding_number g z = 0))"
pt_set_inst: assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pt TYPE('a set) TYPE('b)"
mult_poly_spec_def: fixes p :: "int mpoly"   and q :: "int mpoly" shows "mult_poly_spec p q = SPEC (\<lambda>r. p * q - r \<in> More_Modules.ideal polynomial_bool)"
closed_sum_def: fixes A :: "'b set"   and B :: "'b set" shows "A +\<^sub>M B = closure (A + B)"
proj_arcs_H_simps_6: fixes a :: "'b" assumes "a \<in> arcs G" shows "proj_arcs_H a = a"
set_l_is: shows "set (List.product [0..<t] (\<Pi>\<^sub>\<V>)) = {0..<t} \<times> set (\<Pi>\<^sub>\<V>)"
prems_2: shows "length m2 = nc \<and> (\<forall>a\<in>set m2. vec nr a)"
grounding_of_cls_def: fixes C :: "'a literal multiset" shows "grounding_of_cls C = {C \<cdot> \<sigma> |\<sigma>. is_ground_subst \<sigma>}"
partial_equiv_funI: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x y. \<lbrakk>x \<in> domain; y \<in> domain; x \<sim> y\<rbrakk> \<Longrightarrow> f x \<sim> g y" shows "f \<sim> g"
pred_mono_strong: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c pmf"   and Pa :: "'c \<Rightarrow> bool" assumes "pred_pmf P x"   and "\<And>z. \<lbrakk>z \<in> set_pmf x; P z\<rbrakk> \<Longrightarrow> Pa z" shows "pred_pmf Pa x"
cases: fixes a1 :: "sequent list"   and a2 :: "sequent" assumes "provRulesp a1 a2" obtains c :: "sequent" and F :: "char list" and x :: "NominalSequents.var" and A :: "form" and ps :: "sequent list" where "a1 = ps" and "a2 = c" and "NominalSequents.mset c = \<LM> F \<nabla> [x].A  \<RM>" and "ps \<noteq> []" and "x \<sharp> set_of_prem ( ps - A )"
carrier_eq: shows "\<Union> (set As) = {x. of_weak_ranking As x x}"
servK_authentic_ter: fixes A :: "agent"   and authK :: "nat"   and Ta :: "msg"   and authTicket :: "msg"   and evs :: "event list"   and servK :: "nat"   and B :: "agent"   and Ts :: "msg" assumes "Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Ta\<rbrace>, authTicket\<rbrace> \<in> set evs"   and "Crypt authK \<lbrace>Key servK, Agent B, Ts\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "evs \<in> kerbV" shows "Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Ts\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Ts\<rbrace>\<rbrace> \<in> set evs"
bot_boolean_subset_def: shows "bot \<equiv> Abs_boolean_subset bot"
dlist_eq_def: fixes xs :: "'b list"   and ys :: "'b list" shows "dlist_eq xs ys = (remdups xs = remdups ys)"
IH: fixes gpv :: "('a, 'b, 'c) gpv" assumes "\<I>' \<turnstile>g gpv \<surd>" shows "expectation_gpv' gpv \<le> rhs (mk_lossless_gpv (responses_\<I> \<I>') x gpv)"
dilating_def: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run" shows "dilating f sub r \<equiv> dilating_fun f r \<and> (\<forall>n c. time (Rep_run sub n c) = time (Rep_run r (f n) c)) \<and> (\<forall>n c. hamlet (Rep_run sub n c) = hamlet (Rep_run r (f n) c))"
bernpoly_1': fixes n :: "nat" shows "bernpoly n (1::'a) = of_real (bernoulli' n)"
Carmichael_number_iff: fixes n :: "nat" shows "Carmichael_number n = (n \<noteq> 1 \<and> \<not> prime n \<and> squarefree n \<and> (\<forall>p\<in>#prime_factorization n. p - 1 dvd n - 1))"
integral_mono_AE: fixes M :: "'e measure"   and f :: "'e \<Rightarrow> real"   and g :: "'e \<Rightarrow> real" assumes "integrable M f"   and "integrable M g"   and "AE x in M. f x \<le> g x" shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
analz_UN_analz_lemma: fixes X :: "msg"   and H :: "'a \<Rightarrow> msg set"   and A :: "'a set" assumes "X \<in> analz (\<Union>i\<in>A. analz (H i))" shows "X \<in> analz (\<Union> (H ` A))"
union_mset_add_mset_right: fixes A :: "'b multiset"   and a :: "'b"   and B :: "'b multiset" shows "A + (a, B) = a, A + B"
Max_ge: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "x \<in> A" shows "less_eq x (linorder.Max less_eq A)"
group_closure_mult_right: fixes t :: "int"   and S :: "int set"   and s :: "int" assumes "t \<in> group_closure S" shows "s * t \<in> group_closure S"
up_set_up_set: fixes X :: "'a set" shows "is_up_set (\<Up>X)"
adjacentchain_preimage: fixes Cs :: "'a set list" assumes "order.greater_eq (folding.\<C> - f \<turnstile> folding.\<C>) (set Cs)"   and "adjacentchain (f \<Turnstile> Cs)" shows "adjacentchain Cs"
det_nfa_reach: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat"   and bd :: "nat option bdd"   and ls :: "bool list list"   and q1 :: "bool list"   and j :: "nat"   and bss :: "bool list list" assumes "subset_DFS A n"   and "subset_dfs A (nfa_startnode A) = (bd, ls)"   and "q1 = ls ! j"   and "j < length ls"   and "list_all (is_alph n) bss" shows "nfa_reach A (nfa_startnode A) bss q1 = dfa_reach (det_nfa A) 0 bss j"
strassen_too_small_def: fixes A :: "'b mat"   and B :: "'c mat" shows "strassen_too_small A B \<equiv> dim_row A < strassen_constant \<or> dim_col A < strassen_constant \<or> dim_col B < strassen_constant"
in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "arr f"   and "local.dom f = a"   and "cod f = b" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
Low_neq_Exit: assumes "L \<noteq> {}" shows "(_Low_) \<noteq> (_Exit_)"
elims: fixes x :: "('c, 'd) discrete_market"   and xa :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and xb :: "nat"   and y :: "'c \<Rightarrow> real" assumes "up_cl_proc x xa xb = y" obtains Mkt :: "('c, 'd) discrete_market" and p :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" where "x = Mkta__" and "xa = p__" and "xb = 0" and "y = val_process Mkta__ p__ 0"   | Mkt :: "('c, 'd) discrete_market" and p :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" and n :: "nat" where "x = Mkt" and "xa = p" and "xb = Suc n" and "y = (\<lambda>w. \<Sum>x\<in>support_set p. prices Mkt x (Suc n) w * p x (Suc n) w)"
le_sqrt_mult_imp_le: fixes a :: "real"   and b :: "real" assumes "0 \<le> a"   and "0 \<le> b"   and "a \<le> A"   and "b \<le> B" shows "a * b \<le> x"
resultant_def: fixes p :: "'c poly"   and q :: "'c poly" shows "resultant p q = det (sylvester_mat p q)"
dbrrr: shows "distinct (fst (TSdet init h (as @ x # bs @ [x]) (Suc (length as + length bs))))"
unit_simps_3: fixes a :: "'a" assumes "C.obj a" shows "trg\<^sub>C \<i>\<^sub>C[a] = a"
enc_atom_\<sigma>: fixes i :: "nat"   and w :: "'b list"   and I :: "(nat + nat set) list"   and n :: "nat"   and p :: "'a" assumes "i < length w"   and "length I = n"   and "p \<in> set \<Sigma>" shows "enc_atom I i p \<in> set (\<sigma> \<Sigma> n)"
HFail_HInv4b_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HFail s s' p"   and "p \<noteq> q"   and "HInv4b s q" shows "HInv4b s' q"
bidirected_digraph_def: fixes G :: "('c, 'd) pre_digraph"   and arev :: "'d \<Rightarrow> 'd" shows "bidirected_digraph G arev \<equiv> wf_digraph G \<and> bidirected_digraph_axioms G arev"
ennreal_less_top: fixes x :: "real" shows "ennreal x < \<top>"
assms_3: shows "\<lbrace>p \<cdot> b\<rbrace>x\<lbrace>p\<rbrace>"
subgroup_action: fixes H :: "'a set" assumes "subgroup H G" shows "group_action (G\<lparr>carrier := H\<rparr>) (conjugation_action (p ^ a)) (subgroups_of_size (p ^ a))"
class_leq_antisym: fixes cs :: "(String.literal \<times> String.literal) set"   and c1 :: "String.literal"   and c2 :: "String.literal" assumes "wf_subclass_loc cs"   and "class_leq cs c1 c2"   and "class_leq cs c2 c1" shows "c1 = c2"
ag_l_inv1: fixes A :: "('c, 'd) aGroup_scheme"   and x :: "'c" assumes "aGroup A"   and "x \<in> carrier A" shows "-\<^sub>a\<^bsub>A\<^esub> x \<plusminus>\<^bsub>A\<^esub> x = \<zero>\<^bsub>A\<^esub>"
fact3: shows "msb (~~ w + 1) = msb (~~ w)"
in_hhom_def: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a" shows "S.in_hhom \<mu> a b \<equiv> S.arr \<mu> \<and> S.src \<mu> = a \<and> S.trg \<mu> = b"
last_message_NoMsg_conv: fixes xs :: "'d message_af list" shows "(last_message xs = NoMsg) = (\<forall>i<length xs. xs ! i = NoMsg)"
cong_sym_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and b :: "'c"   and c :: "'c"   and a :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b c a = unique_euclidean_semiring.cong modulo c b a"
assms_3: shows "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = (pns, body) in D"
BigOr_is_not_cnf: fixes ls :: "'b formula list" shows "is_cnf \<^bold>\<Or>ls = (\<forall>l\<in>set ls. is_lit_plus l)"