hcju/isabelleps huggingface dataset:
nonneg_power_le: fixes a :: "'a"   and b :: "'a"   and n :: "nat" assumes "(0::'a) \<le> a"   and "(0::'a) \<le> b" shows "(a ^ n \<le> b ^ n) = (n = 0 \<or> a \<le> b)"
step1_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "step1 r = (\<lambda>ys xs. \<exists>us z z' vs. xs = us @ z # vs \<and> r z' z \<and> ys = us @ z' # vs)"
function_uminus_eval'': fixes f :: "'c \<Rightarrow> 'a"   and a :: "'c" assumes "f \<in> carrier F"   and "a \<in> S" shows "(\<ominus>\<^bsub>F\<^esub> f) a = \<ominus> f a"
card_cosets_limit: fixes p :: "nat"   and x :: "nat" assumes "euler_witness_context p"   and "euler_witness (int x) p"   and "x \<in> carrier (euler_witness_context.G p)" shows "2 \<le> card (rcosets\<^bsub>euler_witness_context.G p\<^esub> euler_witness_context.H p)"
ring_ops_poly: shows "ring_ops (poly_ops ops) poly_rel"
plus: fixes x :: "V"   and y :: "V" shows "x + y = x \<union>\<^sub>\<circ> set ((+) x ` elts y)"
cond_hoare_r: fixes b :: "'a upred"   and p :: "'a upred"   and S :: "'a hrel"   and q :: "'a upred"   and T :: "'a hrel" assumes "\<lbrace>b \<and> p\<rbrace> S \<lbrace>q\<rbrace>\<^sub>u"   and "\<lbrace>\<not> b \<and> p\<rbrace> T \<lbrace>q\<rbrace>\<^sub>u" shows "\<lbrace>p\<rbrace> S \<triangleleft> b \<triangleright>\<^sub>r T \<lbrace>q\<rbrace>\<^sub>u"
wf_inv_image: fixes r :: "('g \<times> 'g) set"   and f :: "'h \<Rightarrow> 'g" assumes "wf r" shows "wf (inv_image r f)"
rep_eq: fixes x :: "\<Pi>\<^sub>1 \<Rightarrow> \<o>" shows "eval\<o> (forall\<^sub>1 x) = (\<lambda>s w. \<forall>xa. eval\<o> (x xa) s w)"
quotient_is_ring: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "ideal I R" shows "ring (R Quot I)"
tan_monotone: fixes y :: "real"   and x :: "real" assumes "- (pi / 2) < y"   and "y < x"   and "x < pi / 2" shows "tan y < tan x"
setc1: shows "(\<Sum>s\<in>set (map (consistent_sign_vec qs) (characterize_root_list_p p)). z_R I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}))) = (\<Sum>s\<in>set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = 1} \<union> set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = - 1}. z_R I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}))) + (\<Sum>s\<in>set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = 0}. z_R I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s})))"
set_of': fixes t :: "'a tree" shows "set (set_of' t) = set_of t"
irreducible_uminus: fixes a :: "'a" shows "irreducible (- a) = irreducible a"
traversed_states_def: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "traversed_states s as \<equiv> set (state_list s as)"
assms_2: fixes x :: "'a"   and y :: "'a" assumes "Q x y" shows "P x y"
Abs_ffilter: fixes f :: "'a \<Rightarrow> bool"   and s :: "'a fset"   and s' :: "'a fset" shows "(ffilter f s = s') = ({e \<in> fset s. f e} = fset s')"
norm_cblinfun_bound: fixes b :: "real"   and f :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "0 \<le> b"   and "\<And>x. norm (f *\<^sub>V x) \<le> b * norm x" shows "norm f \<le> b"
lunit'_simps_1: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.ide f" shows "S.arr (S.lunit' f)"
elem_of_symcl_1: fixes a :: "'c"   and b :: "'c"   and Rel :: "('c \<times> 'c) set" assumes "(a, b) \<in> Rel" shows "(a, b) \<in> symcl Rel"
s3: fixes y :: "real" assumes "\<And>x. x \<in> (\<lambda>t. r * \<parallel>f *\<^sub>v t\<parallel>) ` ball (0::'a) 1 \<Longrightarrow> x \<le> y" shows "r * (SUP t\<in>ball (0::'a) 1. \<parallel>f *\<^sub>v t\<parallel>) \<le> y"
next_action_def: fixes s :: "'state_t"   and execs :: "'dom_t \<Rightarrow> 'action_t list list" shows "next_action s execs = fst (control s (current s) (execs (current s)))"
\<sigma>_wt: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma>"
ennreal_less_numeral_iff: fixes n :: "real"   and w :: "num" shows "(ennreal n < numeral w) = (n < numeral w)"
simps_1: fixes uu :: "'b \<times> 'b"   and uv :: "'b" shows "co_path uu uv [] = []"
prog_ext_def: fixes para :: "char list list"   and body :: "expr"   and more :: "'a" shows "\<lparr>para = para, body = body, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons prog_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple para body) more"
Am_ub: fixes t :: "'b tree" assumes "bst t"   and "t \<noteq> \<langle>\<rangle>" shows "Am t \<le> log \<alpha> (real (size1 t) / 2) + 1"
CONS_wrt_def: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" shows "CONS_wrt \<psi> \<equiv> {U. \<exists>s. learn_cons \<psi> U s}"
inf_divide_inf: shows "\<infinity>\<^sub>h :\<^sub>h \<infinity>\<^sub>h = 1\<^sub>h"
transitive: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and R :: "('a, 'b, 'c) psi"   and Rel'' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<Psi> \<rhd> P \<leadsto>[Rel] Q"   and "\<Psi> \<rhd> Q \<leadsto>[Rel'] R"   and "eqvt Rel''"   and "{(\<Psi>, P, R) |\<Psi> P R. \<exists>Q. (\<Psi>, P, Q) \<in> Rel \<and> (\<Psi>, Q, R) \<in> Rel'} \<subseteq> Rel''" shows "\<Psi> \<rhd> P \<leadsto>[Rel''] R"
isomorphic_transitive: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "D.isomorphic f g"   and "D.isomorphic g h" shows "D.isomorphic f h"
assms_3: shows "P \<Turnstile> \<langle>\<langle>\<tau>\<rangle>\<rangle>Conj (binsert (Pred \<phi>) (bsingleton x))"
cases: fixes x :: "'b heap \<times> 'b heap" obtains h :: "'b heap" where "x = (h, heap.Empty)"   | v :: "'b" and va :: "'b heap list" where "x = (heap.Empty, Hp v va)"   | xa :: "'b" and hsx :: "'b heap list" and y :: "'b" and hsy :: "'b heap list" where "x = (Hp xa hsx, Hp y hsy)"
distinct_selsort_inl: fixes X :: "'a \<times> (real \<times> real) pdevs" shows "distinct (ccw.selsort 0 (inl (snd X)))"
vimage2pI: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'd"   and x :: "'f"   and g :: "'g \<Rightarrow> 'e"   and y :: "'g" assumes "R (f x) (g y)" shows "BNF_Def.vimage2p f g R x y"
F1: shows "\<forall>x\<^sub>1. (1::'a) * x\<^sub>1 = x\<^sub>1"
exec: shows "(ta', xcp', h', (stk', loc', C, M, pc') # frs) \<in> exec_instr (ins ! pc) P t h stk loc C M pc frs"
chine_composite: shows "H\<mu>H\<nu>\<pi>.chine = \<mu>.leg0 \<down>\<down> \<nu>\<pi>.leg1"
preserves_seq: fixes g :: "'a"   and f :: "'a" assumes "seq g f" shows "seq (R g) (R f)"
Ext_foo_dyn_accessible: fixes S :: "qtname" shows "tprg \<turnstile> (Ext, mdecl Ext_foo) in Ext dyn_accessible_from S"
Re_complex_of_real: fixes z :: "real" shows "Re (cor z) = z"
elems: fixes X :: "'a set" assumes "X \<in># Cs" shows "\<exists>x. (x \<in> carrier G \<and> irreducible G x) \<and> X = assocs G x"
app_vlrestriction_def: fixes r :: "V"   and A :: "V" shows "r \<restriction>\<^sup>l\<^sub>\<circ> A = set {\<langle>a, b\<rangle> |a b. a \<in>\<^sub>\<circ> A \<and> \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r}"
maxsimpD_maximal: fixes x :: "'b set"   and z :: "'b set" assumes "codomain.chamber x"   and "z \<in> Y"   and "order.greater_eq z x" shows "z = x"
map_simp: fixes f :: "'d" assumes "D.arr f" shows "D.map f = f"
brec_invar_initial: fixes \<delta> :: "('a, 'b) ta_rule dlist"   and Qi :: "('a, unit) HashMap.hashmap" assumes "ls.invar \<delta>"   and "hs.invar Qi" shows "brec_initial Qi \<delta> \<in> brec_invar_add (hs.\<alpha> Qi)"
add_less_imp_less_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less"   and "less (plus c a) (plus c b)" shows "less a b"
inject: fixes x1 :: "face list"   and x2 :: "nat"   and x3 :: "face list list"   and x4 :: "nat list"   and y1 :: "face list"   and y2 :: "nat"   and y3 :: "face list list"   and y4 :: "nat list" shows "(Graph x1 x2 x3 x4 = Graph y1 y2 y3 y4) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4)"
valid_subgraph: fixes G :: "('a, 'b) graph"   and H :: "('a, 'b) graph" assumes "valid_graph G"   and "subgraph H G" shows "valid_graph H"
A_trusts_YM3: fixes A :: "agent"   and B :: "agent"   and K :: "nat"   and na :: "msg"   and nb :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace> \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> yahalom" shows "Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace> \<in> set evs"
simps: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and ta :: "float"   and p :: "float poly"   and e :: "float interval"   and t :: "taylor_model" shows "tm_comp prec ord I a ta (TaylorModel p e) t = (let t_sub_ta = tm_sub t (tm_const ta) in Let (eval_poly_at_tm prec ord I a p t_sub_ta) (tm_inc_err e))"
transpos_hom: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i \<noteq> j" shows "transpos i j \<in> {i. i \<le> n} \<rightarrow> {i. i \<le> n}"
is_sig_GB_is_Groebner_basis: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set" assumes "dickson_grading d"   and "hom_grading d"   and "G \<subseteq> dgrad_max_set d"   and "\<And>u. is_sig_GB_in d G u" shows "punit.is_Groebner_basis (rep_list ` G)"
compare_double_refl: fixes x :: "double" shows "compare_double x x = 0"
unit_closed: shows "stfx.class_of V (one_str\<^sub>Y V) \<in> stfx.carrier_direct_lim"
A_to_prodag_mem: fixes A :: "('c, 'd) aGroup_scheme"   and I :: "'e set"   and B :: "'e \<Rightarrow> ('f, 'g) aGroup_scheme"   and S :: "'e \<Rightarrow> 'c \<Rightarrow> 'f"   and x :: "'c" assumes "aGroup A"   and "\<forall>k\<in>I. aGroup (B k)"   and "\<forall>k\<in>I. S k \<in> aHom A (B k)"   and "x \<in> carrier A" shows "A_to_prodag A I S B x \<in> carr_prodag I B"
map_default_in_finfun: fixes f :: "'a \<Rightarrow> 'b option"   and b :: "'b" assumes "finite (dom f)" shows "map_default b f \<in> finfun"
low_proc_non0_def: fixes p :: "prog" shows "low_proc_non0 p \<equiv> \<forall>\<sigma>\<in>states p. \<forall>\<sigma>'. the (\<sigma> ''lowIn'') \<noteq> 0 \<and> body p \<rhd> \<sigma> \<leadsto> Some \<sigma>' \<longrightarrow> the (\<sigma>' ''lowOut'') = the (\<sigma> ''lowIn'') + 1"
assms_1: shows "equivalent_on \<sigma> \<eta> (vars_of_lit l) I"
taylor_def: fixes R :: "('c, 'd) ring_scheme" assumes "UP_cring R" shows "UP_cring.taylor R \<equiv> taylor_expansion R"
idx_sequence_def: fixes idx :: "nat \<Rightarrow> nat" shows "idx_sequence idx \<equiv> idx 0 = 0 \<and> (\<forall>n. idx n < idx (Suc n))"
transfer_rule_numeral_uint: shows "rel_fun (=) cr_uint16 numeral numeral"
cases: fixes a1 :: "'a sequent list \<times> 'a sequent"   and a2 :: "'a form" assumes "rightPrincipal a1 a2" obtains C :: "'a sequent" and F :: "'a" and Fs :: "'a form list" and Ps :: "'a sequent list" where "a1 = (Ps, C)" and "a2 = Compound F Fs" and "C = ( \<Empt> \<Rightarrow>* \<LM> Compound F Fs  \<RM>)"
inj_unsigned: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.semiring_char_0 one times plus zero" shows "inj (semiring_1.unsigned one plus zero)"
normalize_poly_def: fixes p :: "'a poly" shows "normalize p = p div [:unit_factor (Polynomial.lead_coeff p):]"
sel_5: fixes x1 :: "real"   and x2 :: "real"   and x3 :: "real"   and x4 :: "real"   and x5 :: "real"   and x6 :: "real"   and x7 :: "real"   and x8 :: "real" shows "Im4 (Octo x1 x2 x3 x4 x5 x6 x7 x8) = x5"
inf_commute: fixes inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.semilattice_inf inf less_eq less" shows "inf x y = inf y x"
all_paths_def: fixes G :: "('c, 'd) Graph_scheme" assumes "Digraph G" shows "Digraph.all_paths G \<equiv> {xs |xs. Digraph.walk G xs \<and> distinct xs}"
pgen_lossless_gpv_def: fixes fail :: "ennreal"   and \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv" shows "pgen_lossless_gpv fail \<I> gpv = (expectation_gpv fail \<I> (\<lambda>_. 1) gpv = 1)"
gap_Suc_1: fixes b :: "nat"   and i :: "nat"   and t :: "nat"   and k :: "nat" assumes "b = block i t"   and "k = gap i t"   and "change_on_0 b k" shows "gap i (Suc t) = e_length b"
zero_le_even_power': fixes 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 sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (power.power one times a (2 * n))"
Suc_innermost_id2': fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and j' :: "nat"   and i' :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "j' < j"   and "i' \<le> i" shows "fw m n (Suc k) i' j' i j = fw m n k n n i j"
comp_res: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>T ; - (x ; y) \<le> - y"
invariant_restrict_inD: fixes A :: "('a, 'b) automaton"   and P :: "'a \<Rightarrow> bool"   and QI :: "'b \<Rightarrow> bool" assumes "A \<TTurnstile> P" shows "A \<TTurnstile> (QI \<rightarrow>) P"
characteristic_weak_formula_is_characteristic: fixes Q :: "'state"   and P :: "'state" shows "Q \<Turnstile> characteristic_weak_formula P = P \<approx>\<cdot> Q"
standard_decompE: fixes k :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set"   and d :: "nat" assumes "standard_decomp k ps"   and "(h, U) \<in> set (ps\<^sub>+)"   and "k \<le> d"   and "d \<le> poly_deg h" obtains h' :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" and U' :: "'x set" where "(h', U') \<in> set ps" and "poly_deg h' = d" and "card U \<le> card U'"
rel_pmf_return_pmf2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and M :: "'c pmf"   and x :: "'d" shows "rel_pmf R M (return_pmf x) = (\<forall>a\<in>set_pmf M. R a x)"
max_dist_def: fixes src :: "nat" shows "max_dist src \<equiv> Max (min_dist src ` V)"
cring_simprules_12: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "cring R"   and "x \<in> carrier R" shows "\<one>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = x"
upper_closure: shows "\<pi>\<^sub>* \<in> carrier \<Y> \<rightarrow> carrier \<X>"
open_INT: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'a set" assumes "finite A"   and "\<forall>x\<in>A. open (B x)" shows "open (\<Inter> (B ` A))"
coupling_def: fixes Q :: "'b \<Rightarrow> nat option"   and V :: "'b \<Rightarrow> nat option"   and D :: "'b \<Rightarrow> enat"   and S :: "'b set" shows "coupling Q V D S \<equiv> D = Dijkstra_Abstract.enat_of_option \<circ> V ++ Q \<and> S = dom V \<and> dom V \<inter> dom Q = {}"
path_split2: fixes \<beta> :: "'e list"   and \<alpha> :: "'e list"   and s :: "'s" assumes "s0\<^bsub>SES\<^esub> (\<beta> @ \<alpha>)\<Longrightarrow>\<^bsub>SES\<^esub> s" shows "\<exists>s'\<in>S\<^bsub>SES\<^esub>. s0\<^bsub>SES\<^esub> \<beta>\<Longrightarrow>\<^bsub>SES\<^esub> s' \<and> s' \<alpha>\<Longrightarrow>\<^bsub>SES\<^esub> s \<and> reachable SES s'"
phl_cons2: fixes w :: "'a"   and x :: "'a"   and z :: "'a"   and y :: "'a" assumes "w \<le> x"   and "\<lparr>z\<rparr>y\<lparr>y \<cdot> w\<rparr>" shows "\<lparr>z\<rparr>y\<lparr>y \<cdot> x\<rparr>"
length_splice: fixes xs :: "'b list"   and ys :: "'b list" shows "\<^bold>|splice xs ys\<^bold>| = \<^bold>|xs\<^bold>| + \<^bold>|ys\<^bold>|"
inclusion_is_functor: shows "functor C (\<cdot>) inclusion_of_generators"
committed_w: shows "w \<in> committed (J (Suc n))"
assms: shows "h \<turnstile> ok create_character_data document_ptr text"
injects: fixes a :: "'b"   and a' :: "'b" shows "(Just\<cdot>a = Just\<cdot>a') = (a = a')"
y_def: shows "y = (t1 * int u2 - t2 * int u1) div k"
F_def: shows "drop n D = derivation_shift F 0 (Suc j)"
finfun_apply_inverse: fixes x :: "'d \<Rightarrow>f 'e" shows "Abs_finfun (finfun_apply x) = x"
SAT_def: fixes \<Phi> :: "('fsym, 'psym) lit list set" shows "SAT \<Phi> \<equiv> \<forall>\<xi>. wtE \<xi> \<longrightarrow> satPB \<xi> \<Phi>"
sin_cos_squared_add2: fixes x :: "'d" shows "(cos x)\<^sup>2 + (sin x)\<^sup>2 = (1::'d)"
singleton_inject: fixes a :: "'b"   and b :: "'b" assumes "{a} = {b}" shows "a = b"
unitary_range: fixes U :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "unitary U" shows "U *\<^sub>S \<top> = \<top>"
exesorts_def: fixes exesignature :: "exesignature" shows "exesorts exesignature = (case exesignature of ExeSignature x1 x2 x3 \<Rightarrow> x3)"
snd_conv: fixes x1 :: "'l"   and x2 :: "'k" shows "snd (x1, x2) = x2"
skip_joiner: fixes \<pi> :: "(char list \<times> 'a list) prefix" assumes "safe_formula \<phi>"   and "gen_unique \<phi>" shows "joiner (\<lambda>k. M (splitter \<pi> k)) = skip_joiner (\<lambda>k. M (splitter \<pi> k))"
simps_2: fixes k :: "'d"   and ks :: "'d list"   and f :: "'e option \<Rightarrow> 'e"   and v :: "'e option"   and ps :: "('d \<times> ('d, 'e) trie) list" shows "update_with_trie (k # ks) f (Trie v ps) = Trie v (AList.update_with_aux empty_trie k (update_with_trie ks f) ps)"
is_moebius_comp: fixes m1 :: "complex_homo \<Rightarrow> complex_homo"   and m2 :: "complex_homo \<Rightarrow> complex_homo" assumes "is_moebius m1"   and "is_moebius m2" shows "is_moebius (m1 \<circ> m2)"
simps_4: fixes a :: "('a, 'b) Combinators"   and b :: "('a, 'b) Combinators" shows "rotatePolicy (a \<oplus> b) = rotatePolicy b \<oplus> rotatePolicy a"
ide_def: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "ide a \<equiv> a \<cdot> a \<noteq> null \<and> (\<forall>f. (f \<cdot> a \<noteq> null \<longrightarrow> f \<cdot> a = f) \<and> (a \<cdot> f \<noteq> null \<longrightarrow> a \<cdot> f = f))"
numeral_eq_neg_one_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "Num.num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(uminus (numeral.numeral one plus n) = uminus one) = (n = num.One)"
fds_nth_of_real: fixes f :: "real fds"   and n :: "nat" shows "fds_nth (fds_of_real f) n = of_real (fds_nth f n)"
maxsimpchain_Cons_reduce: fixes x :: "'a set"   and xs :: "'a set list" assumes "gallery (x # xs)" shows "gallery xs"
Par_ZObis_discrL2: fixes indis :: "('a \<times> 'a) set"   and c2 :: "('b, 'c) com"   and d :: "('b, 'c) com"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c1 :: "('b, 'c) com" assumes "PL_Indis indis"   and "(c2, d) \<in> PL_Indis.ZObis tval aval indis"   and "PL_Indis.discr tval aval indis c1" shows "(c1 | c2, d) \<in> PL_Indis.ZObis tval aval indis"
fds_nth_Suc_0_exp: fixes f :: "'a fds" shows "fds_nth (fds_exp f) (Suc 0) = exp (fds_nth f (Suc 0))"
map_poly_degree_eq: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b poly" assumes "f (lead_coeff p) \<noteq> (0::'a)" shows "degree (map_poly f p) = degree p"
cindex_combine: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and c :: "real"   and b :: "real" assumes "finite {x. jump f x \<noteq> 0 \<and> a < x \<and> x < c}"   and "a < b"   and "b < c" shows "cindex a c f = cindex a b f + jump f b + cindex b c f"
zero_not_nan: fixes x :: "('a, 'b) IEEE.float" assumes "IEEE.is_zero x" shows "\<not> IEEE.is_nan x"
triv_subalg_borel_eq: fixes F :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and c :: "'b" assumes "subalgebra M F"   and "sets F = {{}, space M}"   and "AEeq M f (\<lambda>x. c)"   and "f \<in> borel_measurable F" shows "\<forall>x\<in>space M. f x = c"
dfs_invariant: fixes S :: "'a set"   and xs :: "'a list" assumes "S \<subseteq> reach (set \<Sigma>) \<delta> q\<^sub>0"   and "list_all (\<lambda>q. q \<in> reach (set \<Sigma>) \<delta> q\<^sub>0) xs" shows "dfs S xs \<subseteq> reach (set \<Sigma>) \<delta> q\<^sub>0"
intro: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>s. system.reachable s\<^sub>0 lstep s \<Longrightarrow> \<exists>s' l. lstep l s s'" shows "df_system s\<^sub>0 lstep"
executionally_equivalent_acceptance_map: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and s1 :: "nat"   and r1 :: "nat \<Rightarrow>f value option"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and s2 :: "nat"   and r2 :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list \<times> value list) list" assumes "executionally_equivalent e1 s1 r1 e2 s2 r2 (map (\<lambda>(l, i, o). (l, i)) t)" shows "accepts_trace e2 s2 r2 t = accepts_trace e1 s1 r1 t"
space_as_set_inject: fixes x :: "'d ccsubspace"   and y :: "'d ccsubspace" shows "(space_as_set x = space_as_set y) = (x = y)"
Jeroslow_Diagonalization_axioms: shows "Jeroslow_Diagonalization var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv enc Ops tenc F encF N ssap"
cases: fixes x :: "ereal" obtains r :: "real" where "x = ereal r"   | "x = \<infinity>"   | "x = - \<infinity>"
PQ: fixes x :: "'a pmf"   and y :: "'b pmf"   and a :: "'a"   and b :: "'b" assumes "(x, y) \<in> set_pmf pq"   and "(a, b) \<in> set_pmf (PQ x y)" shows "P a b"
assms_1: shows "sKS s \<in> LocalSecrets PQ"
surj_directional_derivative: fixes k :: "enat"   and a :: "'b" assumes "k = \<infinity>" shows "range (directional_derivative k a) = manifold_eucl.tangent_space k a"
results_gpv_map_gpv_id: fixes \<I> :: "('e, 'f) \<I>"   and f :: "'g \<Rightarrow> 'd"   and gpv :: "('g, 'e, 'f) gpv" shows "results_gpv \<I> (map_gpv f id gpv) = f ` results_gpv \<I> gpv"
simps_3: fixes l :: "label" shows "(New \<sqsubset>l l) = False"
assms: shows "\<D>\<^sub>\<circ> r = set {a}"
left_add_twice: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_numeral one times plus" shows "plus a (plus a b) = plus (times (numeral.numeral one plus (num.Bit0 num.One)) a) b"
E: shows "j < offs (enum xs index key n mi ma) 0 ! i"
cms\<^sub>C'_def: shows "cms\<^sub>C' = cms\<^sub>C[n := cmn']"
vpairs_vunion: fixes A :: "V"   and B :: "V" shows "vpairs (A \<union>\<^sub>\<circ> B) = vpairs A \<union>\<^sub>\<circ> vpairs B"
increasing_measurable_info: fixes f :: "'a \<Rightarrow> 'b"   and s :: "nat"   and N :: "'b measure"   and t :: "nat" assumes "f \<in> F s \<rightarrow>\<^sub>M N"   and "s \<le> t" shows "f \<in> F t \<rightarrow>\<^sub>M N"
down_ucast_inj: fixes A :: "'b word set" assumes "is_down UCAST('a \<rightarrow> 'b)" shows "inj_on UCAST('b \<rightarrow> 'a) A"