hcju/isabelleps huggingface dataset:
iMODb_append_union: fixes r :: "nat"   and m :: "nat"   and c :: "nat"   and c' :: "nat" shows "[ r, mod m, c ] \<union> [ r + m * c, mod m, c' ] = [ r, mod m, c + c' ]"
arctan_lbound: fixes y :: "real" shows "- (pi / 2) < arctan y"
up_filter_dist_inf: fixes x :: "'c"   and y :: "'c" shows "up_filter (x \<sqinter> y) = up_filter x \<squnion> up_filter y"
T_insert3_amor: fixes n :: "nat"   and t :: "'a tree"   and a :: "'a"   and t' :: "'a tree"   and n' :: "nat" assumes "n = size t"   and "bal_i (size t) (height t)"   and "insert3 a (t, n) = (t', n')" shows "real (T_insert3 a (t, n)) + \<Phi> t' - \<Phi> t \<le> (6 * e + 2) * real (height t + 1) + 1"
zero_not_in_Basis: shows "(0::'b) \<notin> {1::'b}"
simps_11: fixes v :: "'b ltln" shows "Unf (X\<^sub>n v) = X\<^sub>n v"
bisimTransitive: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim> Q"   and "\<Psi> \<rhd> Q \<sim> R" shows "\<Psi> \<rhd> P \<sim> R"
bullet_app_red: fixes t :: "CL"   and u :: "CL" shows "(` t u, bullet_app t u) \<in> red\<^sup>="
id3: shows "of_int_poly (pCons b c2 - smult (coeff (pCons b c2) (degree q) div lead_coeff q) q) = pCons (rat_of_int b) (of_int_poly c2) - smult (coeff (pCons (rat_of_int b) (of_int_poly c2)) (degree q) / coeff (of_int_poly q) (degree q)) (of_int_poly q)"
perm_simps_1: fixes p :: "perm"   and tm1 :: "tm"   and tm2 :: "tm" shows "p \<bullet> tm1 IN tm2 = (p \<bullet> tm1) IN (p \<bullet> tm2)"
isometry_def: fixes f :: "'c \<Rightarrow> 'd" shows "isometry f \<equiv> isometry_on UNIV f \<and> surj f"
path_weightp_distinct_def: fixes l1 :: "'loc"   and l2 :: "'loc"   and s :: "'sum" shows "path_weightp_distinct l1 l2 s \<equiv> \<exists>xs. distinct xs \<and> path l1 l2 xs \<and> s = sum_weights (map (\<lambda>(s, l, t). l) xs)"
S5Basic_9: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<diamond>\<^bold>\<diamond>\<phi> \<^bold>\<equiv> \<^bold>\<diamond>\<phi> in v]"
not_in_kD_case: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and en :: "'a"   and es :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set \<Rightarrow> 'a" assumes "dip \<notin> kD rt" shows "(case rt dip of None \<Rightarrow> en | Some r \<Rightarrow> es r) = en"
o_x_axis_pos_oriented: shows "pos_oriented o_x_axis"
hnr_RETURN_pass: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and x :: "'d"   and p :: "'c" shows "hn_refine (hn_ctxt R x p) (return p) (hn_invalid R x p) R (RETURN x)"
PTrans: shows "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'"
Ext_ZF_hull_def: fixes R :: "(ZF \<times> ZF) set"   and s :: "ZF" shows "Ext_ZF_hull R s \<equiv> SeqSum (Ext_ZF_n R s)"
assms: shows "mcont lub ord lSup (\<sqsubseteq>) f"
ide_in_hom: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "partial_magma C" shows "partial_magma.ide C a = partial_magma.in_hom C a a a"
simps_17: fixes V :: "'g"   and o' :: "('f, 'g, 'h) exp"   and e :: "('f, 'g, 'h) exp"   and A :: "'f set option" shows "\<D> (sync\<^bsub>V\<^esub> (o') e) A = (\<D> o' A \<and> \<D> e (A \<squnion> \<A> o'))"
hyp: fixes L :: "'a language"   and K :: "'a language" assumes "R L K" shows "\<oo> L = \<oo> K \<and> rel_fun (=) (regular_cong R) (\<dd> L) (\<dd> K)"
erf_coeffs_def: fixes n :: "nat" shows "erf_coeffs n = (if odd n then 2 / sqrt pi * (- 1) ^ (n div 2) / (real n * fact (n div 2)) else 0)"
int_neg_cyc: fixes v :: "'a" assumes "v \<in> verts G"   and "dist v = - \<infinity>" shows "fst ` C \<inter> pwalk_verts v \<noteq> {}"
map_le_trans: fixes m1 :: "'f \<Rightarrow> 'g option"   and m2 :: "'f \<Rightarrow> 'g option"   and m3 :: "'f \<Rightarrow> 'g option" assumes "m1 \<subseteq>\<^sub>m m2"   and "m2 \<subseteq>\<^sub>m m3" shows "m1 \<subseteq>\<^sub>m m3"
bi_unique_alt_def: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "bi_unique A = (left_unique A \<and> right_unique A)"
inj_map_bij: fixes f :: "'c \<Rightarrow> 'd option" assumes "inj_on f (dom f)" shows "bij_betw f (dom f) (Some ` ran f)"
prems: shows "\<langle>c\<^sub>1;; c\<^sub>2',s\<rangle> \<rightarrow> \<langle>c\<^sub>2,s\<^sub>2\<rangle>"
InvariantEquivalentZLAfterApplyBackjump: fixes state :: "State"   and F0 :: "Literal list list" assumes "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "getConflictFlag state"   and "InvariantUniqC (getC state)"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantCEntailed (getConflictFlag state) F0 (getC state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "InvariantEquivalentZL (getF state) (getM state) F0"   and "isUIP (opposite (getCl state)) (getC state) (getM state)"   and "0 < currentLevel (getM state)" shows "let state' = applyBackjump state in InvariantEquivalentZL (getF state') (getM state') F0"
deriv_inv: fixes x :: "real" assumes "x \<in> {a..b}" shows "(inverse has_real_derivative - inverse (x * x)) (at x within {a..b})"
witness_2: shows "t = results_in t' (sum_weights (map (\<lambda>(s, l, t). l) xs))"
get_map_elem: fixes M :: "('a \<times> 'b) set"   and k :: "'a"   and v :: "'b" assumes "is_map M"   and "(k, v) \<in> M" shows "get_map M k = (k, v)"
sel_3: fixes x31 :: "poscheckvariant"   and x32 :: "('e, 'f) Term.term"   and x33 :: "('e, 'f) Term.term" shows "the_check \<langle>x31: x32 \<doteq> x33\<rangle> = x31"
lang_lderiv: fixes \<Sigma> :: "nat \<Rightarrow> 'a set"   and project :: "'a \<Rightarrow> 'a"   and embed :: "'a \<Rightarrow> 'a list"   and wf_atom :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and r :: "'b rexp"   and w :: "'a"   and lookup :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "Pi_Derivatives.embed \<Sigma> project embed"   and "alphabet.wf wf_atom n r"   and "w \<in> \<Sigma> n" shows "project.lang \<Sigma> project lookup n (embed.lderiv lookup embed w r) = lQuot w (project.lang \<Sigma> project lookup n r)"
simps_7: fixes cl :: "('a, 'b, 'c) cmd list" shows "noWhile (ParT cl) = Ball (set cl) noWhile"
sort_key_simps_2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and xs :: "'d list" assumes "class.linorder less_eq less" shows "linorder.sort_key less_eq f (x # xs) = linorder.insort_key less_eq f x (linorder.sort_key less_eq f xs)"
assms_6: shows "v\<^sup>T = r\<^sup>T * t\<^sup>\<star>"
sqrt_aux_2: fixes n :: "nat" shows "{m. m\<^sup>2 \<le> n} \<noteq> {}"
new_vars_private_RelS_Eg1Eg2: shows "new_vars_private RelS_Eg1Eg2"
I_out_cb_def: fixes s :: "'b list" shows "I_out_cb s \<equiv> \<lambda>(\<ff>s, i, j). length s + 1 = length \<ff>s \<and> (\<forall>jj<j. \<ff>s ! jj = \<ff> s jj) \<and> \<ff>s ! (j - 1) = i \<and> 0 < j"
from_nat_into_def: fixes S :: "'b set"   and n :: "nat" shows "from_nat_into S n = (if n \<in> to_nat_on S ` S then inv_into S (to_nat_on S) n else SOME s. s \<in> S)"
inv\<^sub>3E_1: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set" assumes "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V" shows "inv\<^sub>2 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"
int_pow_int: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and n :: "nat" shows "x [^]\<^bsub>G\<^esub> int n = x [^]\<^bsub>G\<^esub> n"
pure_B_conv: fixes g :: "('c \<Rightarrow> 'a) list"   and f :: "('b \<Rightarrow> 'c) list"   and x :: "'b list" shows "[\<lambda>g f x. g (f x)] \<diamondop> g \<diamondop> f \<diamondop> x = g \<diamondop> (f \<diamondop> x)"
isEmpty_def: fixes t :: "('a, 'b) FingerTree" shows "FingerTree.isEmpty t \<equiv> FingerTreeStruc.isEmpty (Rep_FingerTree t)"
The_on_equality: fixes a :: "'c"   and U :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "a \<in> U"   and "P a"   and "\<And>x. \<lbrakk>x \<in> U; P x\<rbrakk> \<Longrightarrow> x = a" shows "the (THE x on U. P x) = a"
cod_mkarr: fixes t :: "'c local.term" assumes "Arr t" shows "\<F>C.cod (\<F>C.mkarr t) = \<F>C.mkarr (Cod t)"
scaleR_sum_left: fixes f :: "'b \<Rightarrow> real"   and A :: "'b set"   and x :: "'a" shows "sum f A *\<^sub>R x = (\<Sum>a\<in>A. f a *\<^sub>R x)"
surjective: fixes r :: "('c, 'd) ParityGame_scheme" shows "r = \<lparr>verts = V\<^bsub>r\<^esub>, arcs = E\<^bsub>r\<^esub>, player0 = V0\<^bsub>r\<^esub>, priority = \<omega>\<^bsub>r\<^esub>, \<dots> = ParityGame.ParityGame.more r\<rparr>"
is_path_simps_1: fixes G :: "('a, 'b) graph"   and v :: "'a" assumes "valid_graph G" shows "valid_graph.is_path G v [] v = (v \<in> nodes G)"
span_eq: fixes Sa :: "(('c \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'c"   and T :: "(('c \<Rightarrow> real) \<Rightarrow> real) set" assumes "Sa \<subseteq> manifold_eucl.dest.tangent_space p"   and "T \<subseteq> manifold_eucl.dest.tangent_space p" shows "(span Sa = span T) = (Sa \<subseteq> span T \<and> T \<subseteq> span Sa)"
sqsubset_is_subset: fixes A :: "'a set"   and B :: "'a set" shows "(A \<sqsubseteq> B) = (A \<subseteq> B)"
L_eq_D: shows "L.T y = distr (K.T x) (stream_space (count_space UNIV)) (smap f)"
methd_declared_in_declclass: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd" assumes "methd G C sig = Some m"   and "ws_prog G"   and "is_class G C" shows "G\<turnstile>Methd sig  m declared_in declclass m"
eq_on_cyclic_on_iff1: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and x :: "'c" assumes "cyclic_on f S"   and "x \<in> S" obtains "f x \<in> S" and "(f x = x) = (card S = 1)"
l_child_lower_bound: shows "(p < l_child l p) = (l \<le> p)"
cases: fixes x :: "(('d \<times> 'e) \<times> 'f \<times> 'd \<times> 'e) tree" obtains "x = \<langle>\<rangle>"   | l :: "(('d \<times> 'e) \<times> 'f \<times> 'd \<times> 'e) tree" and xa :: "'d \<times> 'e" and uu_ :: "'f" and mkp :: "'d \<times> 'e" and r :: "(('d \<times> 'e) \<times> 'f \<times> 'd \<times> 'e) tree" where "x = \<langle>l, (xa, uu_, mkp), r\<rangle>"
cases: fixes x :: "'v \<times> ('v \<times> 'w \<times> 'v) list \<times> 'v" obtains v :: "'v" and v' :: "'v" where "x = (v__, [], v'__)"   | v :: "'v" and v1 :: "'v" and w :: "'w" and v2 :: "'v" and p :: "('v \<times> 'w \<times> 'v) list" and v' :: "'v" where "x = (v, (v1, w, v2) # p, v')"
simps: fixes a1 :: "cld"   and a2 :: "meth_def list" shows "class_methods a1 a2 = (\<exists>dcl cl fds meth_defs. a1 = cld_def dcl cl fds meth_defs \<and> a2 = meth_defs)"
par_com_validity_def: fixes Ps :: "'b com option list"   and pre :: "'b set"   and rely :: "('b \<times> 'b) set"   and guar :: "('b \<times> 'b) set"   and post :: "'b set" shows "\<Turnstile> Ps SAT [pre, rely, guar, post] \<equiv> \<forall>s. par_cp Ps s \<inter> par_assum (pre, rely) \<subseteq> par_comm (guar, post)"
subdiv_ends_H_rev_1: shows "tail H (rev_H uw) = w"
left_commute: fixes f :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and b :: "'b::type"   and a :: "'b::type"   and c :: "'b::type" assumes "abel_semigroup (f::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(f::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (b::'b::type) (f (a::'b::type) (c::'b::type)) = f a (f b c)"
split_trace: 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 c :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and c' :: "('d, 'e, 'f) configuration"   and t' :: "('d, 'e, 'f) event list"   and t'' :: "('d, 'e, 'f) event list" assumes "distributed_system channel"   and "distributed_system.trace channel trans send recv c t c'"   and "t = t' @ t''" shows "\<exists>c''. distributed_system.trace channel trans send recv c t' c'' \<and> distributed_system.trace channel trans send recv c'' t'' c'"
row_space_eq: fixes A :: "(('a, 'b) vec, 'c) vec" shows "row_space A = {w. \<exists>y. Finite_Cartesian_Product.transpose A *v y = w}"
finite_measure_mono: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B"   and "B \<in> events" shows "prob A \<le> prob B"
dvd_Gcd_fin_iff: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "finite A" shows "dvd.dvd times b (semiring_gcd.Gcd_fin one zero gcd A) = (\<forall>a\<in>A. dvd.dvd times b a)"
iso_lunit: fixes a :: "'a" assumes "ide a" shows "local.iso \<l>[a]"
merge_in_space: fixes x :: "nat \<Rightarrow> val" assumes "x \<in> space (state_measure V \<Gamma>)" shows "merge V V' (x, \<rho>) \<in> space (dens_ctxt_measure \<Y> \<rho>)"
assms: shows "\<forall>p\<in>dom \<Gamma>. \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>Call p \<down> Normal s} Call p {t. \<Gamma>\<turnstile> \<langle>Call p,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>Call p,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
l_inv_ex: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp" shows "\<exists>y\<in>carrier Zp. y \<oplus> x = \<zero>"
zero_not_ind_mod: shows "\<not> principal.induced_modulus 0"
sound_modes_invariant: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val"   and k :: "nat"   and cms' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "sound_mode_use (cms, mem)"   and "(cms, mem) \<leadsto>\<^bsub>k\<^esub> (cms', mem')" shows "sound_mode_use (cms', mem')"
Acc: fixes G :: "prog"   and s0 :: "st"   and va :: "Term.var"   and n :: "nat"   and v :: "val"   and f :: "val \<Rightarrow> state \<Rightarrow> state"   and s1 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>va=\<succ>(v, f)\<midarrow>n\<rightarrow> s1" shows "G\<turnstile>Norm s0 \<midarrow>Acc va-\<succ>v\<midarrow>n\<rightarrow> s1"
simps_1: fixes x :: "Preliminaries.var" shows "Ik.wt (Var x) = True"
unit_vectors_is_lin_indpt: fixes n :: "nat" shows "\<not> module.lin_dep cpx_rng (module_cpx_vec n) (unit_vectors n)"
summable_nk: fixes x :: "'v"   and y :: "'v" shows "summable (\<lambda>i. \<bar>(k_i' i (x, y) - k_i' i (y, x)) * indicat_real \<^bold>E (x, y)\<bar>)"
negate_Integer_def: shows "negate = (\<Lambda> (MkI\<cdot>x). MkI\<cdot>(- x))"
less_oLimitI: fixes x :: "ordinal"   and f :: "nat \<Rightarrow> ordinal"   and n :: "nat" assumes "x < f n" shows "x < oLimit f"
int_simps_33: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and R :: "'b \<Rightarrow> bool" shows "\<turnstile> (P \<or> Q \<longrightarrow> R) = ((P \<longrightarrow> R) \<and> (Q \<longrightarrow> R))"
degree_coeff_poly_x_minus_y: fixes p :: "'a poly"   and i :: "nat" shows "degree (coeff (poly_x_minus_y p) i) = degree p - i"
vfsequence_pair_axioms: shows "vfsequence_pair xs\<^sub>1 xs\<^sub>2"
coeff_monom: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a"   and m' :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (MPoly_Type.monom m a) m' = (a when m' = m)"
lhd_llist_of_stream: fixes xs :: "'c stream" shows "lhd (llist_of_stream xs) = shd xs"
compl_inf_bot: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and x :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf (uminus x) x = bot"
map_transform_Nil: fixes t :: "'a \<Rightarrow> exp \<Rightarrow> exp"   and ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>" shows "map_transform t ae [] = []"
OUTfromVCorrect2SYSTEM: shows "OUTfromVCorrect2SYSTEM"
irreducibleD: fixes C :: "'s option set"   and a :: "'s option"   and b :: "'s option" assumes "C \<in> UNIV // K2.communicating"   and "a \<in> C"   and "b \<in> C" shows "(a, b) \<in> K2.communicating"
loc_imps_fw_implication_in_M: fixes c :: "('loc, 't) configuration"   and loc1 :: "'loc"   and loc2 :: "'loc"   and M :: "'t zmultiset"   and xs :: "('loc \<times> 'sum \<times> 'loc) list"   and t :: "'t" assumes "inv_imps_work_sum c"   and "inv_implications_nonneg c"   and "loc_imps_fw c loc1 loc2 M xs"   and "0 < zcount (c_imp c loc1) t" shows "0 < zcount M (results_in t (sum_weights (map (\<lambda>(s, l, t). l) xs)))"
lookup_raw_map_val_raw: fixes xs :: "('a \<times> 'e) list \<times> 'o"   and f :: "'a \<Rightarrow> 'e \<Rightarrow> 'f"   and k :: "'a" assumes "oalist_inv xs"   and "f k (0::'e) = (0::'f)" shows "lookup_raw (map_val_raw f xs) k = f k (lookup_raw xs k)"
Inj: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> H" shows "X \<in> analz H"
Units_m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> Units G"   and "y \<in> Units G" shows "x \<otimes> y \<in> Units G"
bl_to_bin_aux_append: fixes bs :: "bool list"   and cs :: "bool list"   and w :: "int" shows "bl_to_bin_aux (bs @ cs) w = bl_to_bin_aux cs (bl_to_bin_aux bs w)"
assms_4: shows "g : a \<mapsto>\<^bsub>\<AA>\<^esub> c"
disjoint: shows "prime_factors x \<inter> prime_factors y = {}"
simps_6: fixes n :: "nat"   and r :: "'a atom rexp"   and s :: "'a atom rexp" shows "local.wf n (Times r s) = (local.wf n r \<and> local.wf n s)"
integral_discrete_difference: fixes X :: "'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "countable X"   and "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"   and "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"   and "\<And>x. \<lbrakk>x \<in> space M; x \<notin> X\<rbrakk> \<Longrightarrow> f x = g x" shows "integral\<^sup>L M f = integral\<^sup>L M g"
N_1: fixes n :: "nat"   and fs :: "'a vec list" assumes "gram_schmidt_fs_int n fs"   and "set fs \<noteq> {}" shows "(1::'a) \<le> gram_schmidt_fs_lin_indpt.N fs"
simps_2: fixes x :: "[:'a:]"   and xs :: "[:[:'a:]:]" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "sconcat\<cdot>(x :# xs) = x :@ sconcat\<cdot>xs"
path2_split_1: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and n' :: "'node"   and ns' :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns @ n' # ns'\<rightarrow>m" shows "g \<turnstile> n-ns @ [n']\<rightarrow>n'"
pruned_def: fixes g :: "'g" shows "other.pruned g = (\<forall>n\<in>set (\<alpha>n g). \<forall>val. val \<in> other.phiDefs g n \<longrightarrow> other.liveVal g val)"
covering_space_lift_inessential_function: fixes C :: "'a set"   and p :: "'a \<Rightarrow> 'b"   and S :: "'b set"   and U :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and a :: "'b" assumes "covering_space C p S"   and "homotopic_with_canon (\<lambda>x. True) U S f (\<lambda>x. a)" obtains g :: "'c \<Rightarrow> 'a" where "continuous_on U g" and "g ` U \<subseteq> C" and "\<And>y. y \<in> U \<Longrightarrow> p (g y) = f y"
powsum_01: fixes x :: "'a" shows "x\<^bsub>0\<^esub>\<^bsup>1\<^esup> = (1::'a) + x"
connectedin_Un: fixes X :: "'b topology"   and S :: "'b set"   and T :: "'b set" assumes "connectedin X S"   and "connectedin X T"   and "S \<inter> T \<noteq> {}" shows "connectedin X (S \<union> T)"
assms_1: shows "\<forall>x\<in>S. x \<inter> P1 = {} \<and> x \<inter> P2 = {} \<and> (\<forall>y\<in>S. x \<noteq> y \<longrightarrow> x \<inter> y = {})"
store_local_label_empty_2: fixes i :: "nat"   and \<S> :: "s_context"   and s :: "s" assumes "i < length (s_inst \<S>)"   and "store_typing s \<S>" shows "local (s_inst \<S> ! i) = []"
sort_append_swap: fixes xs :: "'a list"   and ys :: "'a list" shows "sort (xs @ ys) = sort (ys @ xs)"
l2wi_append: fixes l1 :: "('b word \<times> 'b word) list"   and l2 :: "('b word \<times> 'b word) list" shows "wordinterval_to_set (l2wi (l1 @ l2)) = wordinterval_to_set (l2wi l1) \<union> wordinterval_to_set (l2wi l2)"
seq_rident: fixes x :: "'a" shows "x ; (1::'a) = x"
take_minus_one_conv_butlast: fixes n :: "nat"   and l :: "'c list" assumes "n \<le> length l" shows "take (n - Suc 0) l = butlast (take n l)"
mult_of_def: fixes R :: "('a, 'b) ring_scheme" shows "mult_of R \<equiv> \<lparr>carrier = carrier R - {\<zero>\<^bsub>R\<^esub>}, mult = (\<otimes>\<^bsub>R\<^esub>), one = \<one>\<^bsub>R\<^esub>\<rparr>"
vid_on_small: fixes A :: "V" shows "small {\<langle>a, a\<rangle> |a. a \<in>\<^sub>\<circ> A}"
tagged_fun_relD_both: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'c \<Rightarrow> 'd"   and A :: "('a \<times> 'c) set"   and B :: "('b \<times> 'd) set"   and x :: "'a"   and x' :: "'c" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(Let x f, Let x' f') \<in> B"
Suffixes_union: fixes A :: "'a list set"   and B :: "'a list set" shows "Suffixes (A \<union> B) = Suffixes A \<union> Suffixes B"
valid_gc_mmsaux_unfolded: fixes args :: "Monitor.args"   and cur :: "nat"   and nt :: "nat"   and gc :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and data_prev :: "(nat \<times> 'b option list set) queue"   and data_in :: "(nat \<times> 'b option list set) queue"   and tuple_in :: "('b option list, nat) mapping"   and tuple_since :: "('b option list, nat) mapping"   and ys :: "(nat \<times> 'b option list set) list" assumes "valid_mmsaux args cur (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) ys" shows "valid_mmsaux args cur (gc_mmsaux (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since)) ys"
continuous_within_topologicalE: fixes x :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'b set" assumes "continuous (at x within s) f"   and "open B"   and "f x \<in> B" obtains A :: "'a set" where "open A" and "x \<in> A" and "\<And>y. \<lbrakk>y \<in> s; y \<in> A\<rbrakk> \<Longrightarrow> f y \<in> B"
forest_mono: fixes X :: "'edge set"   and Y :: "'edge set" assumes "forest X"   and "Y \<subseteq> X" shows "forest Y"
subst_clss_id_subst: fixes CC :: "'a literal multiset set" shows "CC \<cdot>cs id_subst = CC"
Pi_cancel_merge_2: fixes J :: "'d set"   and I :: "'d set"   and x :: "'d \<Rightarrow> 'e"   and y :: "'d \<Rightarrow> 'e"   and B :: "'d \<Rightarrow> 'e set" assumes "J \<inter> I = {}" shows "(merge I J (x, y) \<in> Pi I B) = (x \<in> Pi I B)"
simps_3: fixes x :: "'c"   and y :: "'c" shows "weak_gt_arctic_delta (Num_arc_delta x) (Num_arc_delta y) = (y < x)"
iAll_singleton: fixes t :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<box> t' {t}. P t') = P t"
exhaust: fixes y :: "'b option" obtains "y = None"   | x2 :: "'b" where "y = Some x2"
item_\<alpha>_def: fixes x :: "item" shows "item_\<alpha> x = take (item_dot x) (item_rhs x)"
sup_isomorphism_ord_isomorphism: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "sup_homomorphism f"   and "bij f" shows "(x \<le> y) = (f x \<le> f y)"