hcju/isabelleps huggingface dataset:
signed_take_bit_eq: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and n :: "nat"   and w :: "'c word" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_1.signed one plus zero minus (take_bit n w) = (if n < LENGTH('c) then take_bit n (ring_1.signed one plus zero minus w) else ring_1.signed one plus zero minus w)"
1: shows "\<nu>.leg0 \<cdot> Prj\<^sub>0\<^sub>1 = \<pi>.leg1 \<cdot> Prj\<^sub>0"
\<sigma>_finite_img: shows "finite (subst_range \<sigma>)"
not_subseq_length: fixes ys :: "'b list"   and xs :: "'b list" assumes "length ys < length xs" shows "\<not> subseq xs ys"
da_elim_cases_51: fixes Env :: "env"   and B :: "lname set"   and vn :: "lname"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>LVar vn\<rangle>\<guillemotright> A" obtains "A = \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
h_eq: fixes z :: "complex" shows "h z = (h (z / 2) + h ((z + 1) / 2)) / 2"
DomainI: fixes a :: "'c"   and b :: "'d"   and r :: "('c \<times> 'd) set" assumes "(a, b) \<in> r" shows "a \<in> Domain r"
1: shows "\<forall>x\<in>s. (f x \<in> f ` (s - B)) = (x \<in> s - B)"
assms_3: shows "\<not> S2 x \<le> S1 x"
n: shows "S n = (ARG_MIN Rep_Arity x. x \<in> range S)"
elims_2: fixes x :: "'a \<times> 'b"   and xa :: "'a \<times> 'b" assumes "alpha_residual x xa" obtains \<alpha>1 :: "'a" and P1 :: "'b" and \<alpha>2 :: "'a" and P2 :: "'b" where "x = (\<alpha>1, P1)" and "xa = (\<alpha>2, P2)" and "[bn \<alpha>1]set. (\<alpha>1, P1) = [bn \<alpha>2]set. (\<alpha>2, P2)"
suffix_subseq_join: fixes i :: "nat"   and j :: "nat"   and v :: "nat \<Rightarrow> 'a" assumes "i \<le> j" shows "v [i \<rightarrow> j] \<frown> suffix j v = suffix i v"
fls_times_nth_1: fixes f :: "'a fls"   and g :: "'a fls"   and n :: "int" shows "fls_nth (f * g) n = (\<Sum>i = fls_subdegree f + fls_subdegree g..n. fls_nth f (i - fls_subdegree g) * fls_nth g (fls_subdegree g + n - i))"
exhaust: fixes y :: "'a simplex_checkpoint" obtains x :: "(nat, 'a \<times> QDelta) mapping \<times> (nat, 'a \<times> QDelta) mapping" where "y = Simplex_Checkpoint x"
NotActualForallD: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" assumes "\<not> [\<^bold>\<A>(\<^bold>\<forall>x. \<phi> x) in v]" shows "\<not> [\<^bold>\<forall>x. \<^bold>\<A>\<phi> x in v]"
prod_cases3: fixes y :: "'c \<times> 'd \<times> 'e" obtains a :: "'c" and b :: "'d" and c :: "'e" where "y = (a, b, c)"
lubmap_below_cong: fixes m :: "'e \<Rightarrow> 'f option"   and f1 :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and f2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'g" assumes "\<And>k v. m k = Some v \<Longrightarrow> f1 k v \<sqsubseteq> f2 k v" shows "lub (mapCollect f1 m) \<sqsubseteq> lub (mapCollect f2 m)"
unify_sound: fixes E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and cs :: "('c \<times> ('d, 'c) Term.term) list" assumes "unify E [] = Some cs" shows "is_imgu (subst_of cs) (set E)"
induct: fixes P :: "'d list \<Rightarrow> 'd list \<Rightarrow> bool"   and a0 :: "'d list"   and a1 :: "'d list" assumes "\<And>b. P [] b"   and "\<And>a. P a []"   and "\<And>a l b r. \<lbrakk>P l (b # r); P (a # l) r\<rbrakk> \<Longrightarrow> P (a # l) (b # r)" shows "P a0 a1"
dd_def: shows "dd = (\<lambda>x. if degree q < degree p then 1 / x ^ (degree p - degree q) else x ^ (degree q - degree p))"
known_ptr_not_shadow_root_ptr: fixes ptr :: "(_) object_ptr" assumes "local.a_known_ptr ptr"   and "\<not> (case cast ptr of None \<Rightarrow> False | Some shadow_root_ptr \<Rightarrow> is_shadow_root_ptr shadow_root_ptr)" shows "DocumentClass.known_ptr ptr"
lexordp_eq_antisym: fixes xs :: "'a list"   and ys :: "'a list" assumes "lexordp_eq xs ys"   and "lexordp_eq ys xs" shows "xs = ys"
l_set_mode_get_disconnected_nodes_axioms_def: fixes set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_mode_get_disconnected_nodes_axioms set_mode_locs get_disconnected_nodes_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_mode_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
hcomplex_Infinitesimal_iff: fixes x :: "complex star" shows "(x \<in> Infinitesimal) = (hRe x \<in> Infinitesimal \<and> hIm x \<in> Infinitesimal)"
Basis_euclidean_space_def: shows "Basis \<equiv> range euclidean_space_basis_vector"
3: shows "list_of w \<preceq>\<^sub>F\<^sub>I v"
more_rejective_def: fixes p :: "'d \<Rightarrow> 'e decision option"   and q :: "'d \<Rightarrow> 'e decision option" shows "p \<sqsubseteq>\<^sub>D q = (\<forall>x. case q x of \<bottom> \<Rightarrow> True | \<lfloor>allow y\<rfloor> \<Rightarrow> True | \<lfloor>deny y\<rfloor> \<Rightarrow> \<exists>z. p x = \<lfloor>deny z\<rfloor>)"
simps: fixes a :: "'c list"   and A :: "'c set"   and n :: "nat" shows "(a \<in> nlists A n) = (\<exists>xs. a = xs \<and> set xs \<subseteq> A \<and> length xs = n)"
id_FSwImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'g, 'e, 'f, 'h) model_scheme" shows "FSwImorph id id MOD MOD"
chambercomplex_image: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexMorphism X Y f" shows "ChamberComplex (f \<turnstile> X)"
0: assumes "Ide u" shows "Can (u\<^bold>\<down>) \<and> u\<^bold>\<down> \<in> VHom u \<^bold>\<lfloor>u\<^bold>\<rfloor>"
\<alpha>ah_tl: fixes m :: "'b"   and e :: "'b set \<times> 'b set"   and x :: "'b"   and w :: "('b set \<times> 'b set) list" assumes "m \<notin> fst e"   and "x \<in> \<alpha>ah w m" shows "x \<in> \<alpha>ah (e # w) m"
extendAssoc: fixes S :: "'b sequent"   and R :: "'b sequent"   and c :: "'b sequent" shows "extend S (extend R c) = extend (extend S R) c"
ortho_1: fixes w :: "(real, 3) vec" assumes "w \<in> {proj2_rep p, proj2_rep q}" shows "orthogonal (proj2_line_rep (proj2_line_through p q)) w"
hyps_1: shows "P \<turnstile> \<langle>e,(h, l(V := None), sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>"
angle_refl_mid: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b" shows "angle a b a = 0"
Suc_times_numeral_mod_eq: fixes k :: "num"   and n :: "nat" assumes "numeral k \<noteq> 1" shows "Suc (numeral k * n) mod numeral k = 1"
multiplicative_function_axioms: fixes n :: "nat" shows "multiplicative_function (ind (is_nth_power n))"
in_set_nthsD: fixes x :: "'a"   and xs :: "'a list"   and I :: "nat set" assumes "x \<in> set (nths xs I)" shows "x \<in> set xs"
reachable_n: fixes A :: "('e, 'f) ioa"   and s :: "'e"   and a :: "'f"   and t :: "'e" assumes "reachable A s"   and "s \<midarrow>a\<midarrow>A\<longrightarrow> t" shows "reachable A t"
signed_take_bit_eq_if_positive: fixes a :: "'a"   and n :: "nat" assumes "\<not> bit a n" shows "signed_take_bit n a = take_bit n a"
ipassmt_disjoint_matcheq_iifce_dstip: fixes ipassmt :: "iface \<Rightarrow> ('c word \<times> nat) list option"   and ifce :: "iface"   and i_ips :: "('c word \<times> nat) list"   and p :: "('c, 'd) simple_packet_scheme"   and p_ips :: "('c word \<times> nat) list" assumes "ipassmt_sanity_nowildcards ipassmt"   and "ipassmt_sanity_disjoint ipassmt"   and "ipassmt ifce = Some i_ips"   and "ipassmt (Iface (p_oiface p)) = Some p_ips \<and> p_dst p \<in> ipcidr_union_set (set p_ips)" shows "match_iface ifce (p_oiface p) = (p_dst p \<in> ipcidr_union_set (set i_ips))"
less_multiset_doubletons: fixes y :: "'a"   and t :: "'a"   and s :: "'a"   and x :: "'a" assumes "y < t \<or> y < s"   and "x < t \<or> x < s" shows "{#y, x#} < {#t, s#}"
const_res_subseq: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "is_subseq_of s (const_res_subseq k s)"
extremum: fixes a :: "'a upred" shows "true \<sqsubseteq> a"
closedin_product_topology_singleton: fixes f :: "'c \<Rightarrow> 'd"   and I :: "'c set"   and X :: "'c \<Rightarrow> 'd topology" assumes "f \<in> extensional I" shows "closedin (product_topology X I) {f} = (\<forall>i\<in>I. closedin (X i) {f i})"
test_kat_2_opp: fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "test p"   and "test q" shows "(x \<cdot> q = p \<cdot> x \<cdot> q) = (x \<cdot> q \<le> p \<cdot> x)"
simps_2: fixes n :: "nat"   and x :: "'g" shows "replicate (Suc n) x = x # replicate n x"
\<Sigma>_prot0: shows "\<Sigma>0.\<Sigma>_protocol"
set: fixes x :: "'c list" shows "set_iarray (IArray x) = set x"
card_face_cycle_sets: shows "card GM.face_cycle_sets = card H.face_cycle_sets"
eq: shows "int (nat (fst (bezw t (order \<G>)) mod int (order \<G>))) = fst (bezw t (order \<G>)) mod int (order \<G>)"
Word_compare: fixes E :: "('c, 'd) Order_scheme" assumes "Worder E" shows "(\<exists>a\<in>carrier D. ord_equiv (Iod D (segment D a)) E) \<or> ord_equiv D E \<or> (\<exists>b\<in>carrier E. ord_equiv D (Iod E (segment E b)))"
assms_4: fixes m :: "'a match_expr" shows "matches \<gamma> (optimize m) p = matches \<gamma> m p"
subst_4: fixes x :: "name"   and s :: "trm"   and y :: "name"   and v :: "trm"   and t :: "trm" assumes "x \<sharp> (s, y, v)" shows "(s to x in t)[y::=v] = s[y::=v] to x in t[y::=v]"
is_prefix_cancel: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" shows "is_prefix (a @ b) (a @ c) = is_prefix b c"
card_older_seniors_monotonic: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and m :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "x \<le> n" shows "card (semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x (n + m)) \<le> card (semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n)"
fmap_ext: fixes m :: "('a, 'b) fmap"   and n :: "('a, 'b) fmap" assumes "\<And>x. fmlookup m x = fmlookup n x" shows "m = n"
exhaust: fixes y :: "'d option" obtains "y = \<bottom>"   | x2 :: "'d" where "y = \<lfloor>x2\<rfloor>"
not_outside_1: fixes x :: "'b"   and y :: "'b" assumes "R i x y" shows "x \<in> carrier"
nn_integral_eq_simple_integral: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "simple_function M f" shows "integral\<^sup>N M f = integral\<^sup>S M f"
count_Bag: fixes xs :: "('a, nat) alist" shows "count (Bag xs) = count_of (alist.impl_of xs)"
simps_1: fixes vs :: "real list" shows "interpret_floatariths [] vs = []"
vlepoll_vsubset: fixes A :: "V"   and B :: "V" assumes "A \<subseteq>\<^sub>\<circ> B" shows "A \<lesssim>\<^sub>\<circ> B"
neval_ZeroE: fixes x :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and y :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "x \<leadsto>\<^bsup>0\<^esup> y" obtains "y = x"
infsetsum_Sigma': fixes A :: "'d set"   and B :: "'d \<Rightarrow> 'e set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "countable A"   and "\<And>i. countable (B i)"   and "(\<lambda>(x, y). f x y) abs_summable_on Sigma A B" shows "(\<Sum>\<^sub>ax\<in>A. infsetsum (f x) (B x)) = (\<Sum>\<^sub>a(x, y)\<in>Sigma A B. f x y)"
mirror_pair_def: fixes p :: "'c \<times> 'b" shows "mirror_pair p \<equiv> (snd p, fst p)"
min_lvar_not_in_bounds_Some_min: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat" shows "min_lvar_not_in_bounds s = Some x\<^sub>i \<longrightarrow> (\<forall>x\<in>lvars (\<T> s). x < x\<^sub>i \<longrightarrow> in_bounds x \<langle>\<V> s\<rangle> (\<B> s))"
le_supI1: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<le> a" shows "x \<le> sup a b"
cases: fixes x :: "('a \<Rightarrow> (('b \<times> 'b) \<times> 'c, 'd, 'e) gpv) \<times> ('f \<Rightarrow> 'c \<Rightarrow> (bool, 'd, 'e) gpv)" obtains \<A>1 :: "'a \<Rightarrow> (('b \<times> 'b) \<times> 'c, 'd, 'e) gpv" and \<A>2 :: "'f \<Rightarrow> 'c \<Rightarrow> (bool, 'd, 'e) gpv" where "x = (\<A>1, \<A>2)"
le_imp_add: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "\<exists>c. b = a + c"
mult_le_mono2_hmset: fixes i :: "hmultiset"   and j :: "hmultiset"   and k :: "hmultiset" assumes "i \<le> j" shows "k * i \<le> k * j"
proj3_inv: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dests :: "nat \<Rightarrow> nat option" assumes "dip \<in> kD rt" shows "\<pi>\<^sub>3 (the (invalidate rt dests dip)) = \<pi>\<^sub>3 (the (rt dip))"
FGModuleHom_restrict0_GSubspace: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and U :: "'c set" assumes "FGModuleHom G smult V smult' T"   and "RModule (aezfun_setspan G) smult U \<and> U \<subseteq> V" shows "FGModuleHom G smult U smult' (T \<down> U)"
cd1_ent: shows "CCDD' \<Turnstile> EE"
path_valid_node_2: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as n'" shows "CFG.valid_node sourcenode targetnode valid_edge n'"
safe_l: shows "\<forall>x<l. safe_delayed (c' (n + x))"
inv_inv: fixes f :: "'a" assumes "local.iso f" shows "local.inv (local.inv f) = f"
l_sort_aux_set: fixes X :: "'b list \<times> 'b list \<times> 'b list" shows "l_sort_aux X \<in> l_sort_set X"
finite_in_arcs: fixes v :: "'a" shows "finite (in_arcs G v)"
remove_vertex_walk: fixes G :: "('a, 'b) Graph_scheme"   and xs :: "'a list"   and x :: "'a" assumes "Digraph G"   and "Digraph.walk G xs"   and "x \<notin> set xs" shows "Digraph.walk (Digraph.remove_vertex G x) xs"
wfp_on_mono: fixes A :: "'d set"   and B :: "'d set"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "A \<subseteq> B"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "wfp_on Q B" shows "wfp_on P A"
mset_map: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "mset (map f xs) = image_mset f (mset xs)"
comp_right_one: fixes x :: "'a" shows "x \<sqinter> top = x"
before_dist_fst: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "before vs ram1 ram2"   and "distinct vs" shows "fst (splitAt ram1 (fst (splitAt ram2 vs))) = fst (splitAt ram1 vs)"
rt_fresherI1: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<in> kD rt1"   and "dip \<in> kD rt2"   and "nsqn rt1 dip < nsqn rt2 dip" shows "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"
wf_operator_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.wf_operator problem \<equiv> \<lambda>(name, pres, effs, cost). ast_problem.wf_partial_state problem pres \<and> distinct (map (\<lambda>(uu_, v, uu_, uu_). v) effs) \<and> (\<forall>(epres, x, vp, v)\<in>set effs. ast_problem.wf_partial_state problem epres \<and> x < ast_problem.numVars problem \<and> v < ast_problem.numVals problem x \<and> (case vp of None \<Rightarrow> True | Some v \<Rightarrow> v < ast_problem.numVals problem x))"
dpWeak: fixes \<Gamma> :: "('a, 'b) form multiset"   and \<Delta> :: "('a, 'b) form multiset"   and n :: "nat"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and R2 :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M :: "'b"   and N :: "'b"   and R1 :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and R3 :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and \<Gamma>' :: "('a, 'b) form multiset"   and \<Delta>' :: "('a, 'b) form multiset" assumes "( \<Gamma> \<Rightarrow>* \<Delta>, n) \<in> derivable (ext R R2 M N)"   and "R1 \<subseteq> upRules"   and "R2 \<subseteq> modRules2"   and "R3 \<subseteq> modRules2"   and "R = Ax \<union> R1 \<union> p_e R2 M N \<union> R3" shows "( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', n) \<in> derivable (ext R R2 M N)"
prod_mset_prime_factorization_nat: fixes n :: "nat" assumes "0 < n" shows "\<Prod>\<^sub># (prime_factorization n) = n"
map\<^sub>0_simps_1: fixes a :: "'a" assumes "C.obj a" shows "D.obj (F.map\<^sub>0 a)"
mm_equiv_intro: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and \<R> :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and lc\<^sub>1 :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)"   and lc\<^sub>2 :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)" assumes "sifum_security_init dma \<C>_vars eval"   and "sifum_security_init.strong_low_bisim_mm dma \<C>_vars eval \<R>"   and "(lc\<^sub>1, lc\<^sub>2) \<in> \<R>" shows "(lc\<^sub>1, lc\<^sub>2) \<in> sifum_security_init.mm_equiv dma \<C>_vars eval"
inj_iff_trivial_ker: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "group_hom G H h" shows "inj_on h (carrier G) = (kernel G H h = {\<one>\<^bsub>G\<^esub>})"
l12_18_a: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and P :: "'p" assumes "Cong A B C D"   and "Cong B C D A"   and "\<not> Col A B C"   and "B \<noteq> D"   and "Col A P C"   and "Col B P D" shows "A B Par C D"
that: fixes X :: "nat \<Rightarrow> real" assumes "incseq X"   and "\<And>i. a < ereal (X i)"   and "\<And>i. ereal (X i) < b"   and "(\<lambda>x. ereal (X x)) \<longlonglongrightarrow> b" shows "thesis"
set_pmf_of_set: fixes S :: "'b set" assumes "S \<noteq> {}"   and "finite S" shows "set_pmf (pmf_of_set S) = S"
inject_2: fixes x21 :: "'a"   and x22 :: "'a"   and y21 :: "'a"   and y22 :: "'a" shows "(Edge x21 x22 = Edge y21 y22) = (x21 = y21 \<and> x22 = y22)"
lass_val_of_eval: fixes e :: "char list exp"   and a :: "char list \<times> val"   and P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap" assumes "lass_val_of e = \<lfloor>a\<rfloor>"   and "P \<turnstile> \<langle>e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',(h', l', sh')\<rangle>" shows "e' = unit \<and> h' = h \<and> l' = l(fst a \<mapsto> snd a) \<and> sh' = sh"
measure_space_0': fixes \<Omega> :: "'c set" shows "measure_space \<Omega> {{}, \<Omega>} (\<lambda>x. 0)"
domain_axioms_def: fixes R :: "('a, 'b) ring_scheme" shows "domain_axioms R \<equiv> \<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub> \<and> (\<forall>a b. a \<otimes>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub> \<longrightarrow> a \<in> carrier R \<longrightarrow> b \<in> carrier R \<longrightarrow> a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>)"
open_ereal_def: fixes A :: "ereal set" shows "open A = (open (ereal -` A) \<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {ereal x<..} \<subseteq> A)) \<and> (- \<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A)))"
special_cosetsI: fixes T :: "'w set"   and w :: "'w" assumes "T \<in> Pow S"   and "w \<in> W" shows "w +o \<langle>T\<rangle> \<in> \<P>"
30469: fixes x :: "'a" shows "- x \<sqinter> (x \<squnion> - - x) = bot"
i_notin_suc_p: shows "to_F \<iota> \<notin> fst (lnth Ns (Suc p))"
dominant_term_ms_aux_bigo: fixes basis :: "(real \<Rightarrow> real) list"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" assumes "basis_wf basis"   and "is_expansion_aux xs f basis" shows "f \<in> O(eval_monom (1, snd (dominant_term_ms_aux xs)) basis)"
comparator_composition: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and f :: "'b \<Rightarrow> 'a" assumes "comparator cmp"   and "inj f" shows "comparator (\<lambda>x y. cmp (f x) (f y))"
enabled_def: fixes r :: "'rule"   and s :: "'state" shows "enabled r s \<equiv> \<exists>sl. effG eff' r s sl"
induct: fixes P :: "(('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat list \<times> 'c list \<Rightarrow> bool"   and a0 :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and a1 :: "'c \<Rightarrow> 'd"   and a2 :: "nat"   and a3 :: "nat"   and a4 :: "nat"   and a5 :: "nat \<times> nat list \<times> 'c list" assumes "\<And>index key p q r u xs. P index key p q r (u, [], xs)"   and "\<And>index key p q r u ns xs. P index key p q r (u, ns, xs) \<Longrightarrow> P index key p q r (u, 0 # ns, xs)"   and "\<And>index key p q r u ns xs. P index key p q r (u, ns, tl xs) \<Longrightarrow> P index key p q r (u, Suc 0 # ns, xs)"   and "\<And>index key p q r u n ns xs. (\<And>x xa xb y xaa xc ya xd yb. \<lbrakk>x = take (Suc (Suc n)) xs; xa = bn_comp n p q r; (xb, y) = xa; xaa = round_suc_suc index key x n xb u; (xc, ya) = xaa; (xd, yb) = ya\<rbrakk> \<Longrightarrow> P index key p q y (xc, ns, drop (Suc (Suc n)) xs)) \<Longrightarrow> P index key p q r (u, Suc (Suc n) # ns, xs)" shows "P a0 a1 a2 a3 a4 a5"
f: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "isCont f x"
cod_simp: fixes f :: "'a" assumes "hhom\<^sub>C.arr f" shows "hhom\<^sub>C.cod f = C.cod f"
vunion_in_VsetD_2: fixes a :: "V"   and b :: "V"   and \<alpha> :: "V" assumes "a \<union>\<^sub>\<circ> b \<in>\<^sub>\<circ> Vset \<alpha>" shows "b \<in>\<^sub>\<circ> Vset \<alpha>"
star_of_compare_numeral_6: fixes x :: "'a"   and k :: "num" shows "(star_of x = numeral k) = (x = numeral k)"
weight_del_edge: fixes x :: "'a"   and y :: "'a"   and T :: "'a ugraph"   and w :: "'a set \<Rightarrow> nat" assumes "(x, y) \<in> edges T" shows "weight w (restrict_edges T (- {(x, y), (y, x)})) = weight w T - w {x, y}"
stutter_equiv_element_left: fixes \<sigma> :: "nat \<Rightarrow> 'f"   and \<tau> :: "nat \<Rightarrow> 'f"   and n :: "nat" assumes "\<sigma> \<approx> \<tau>" obtains m :: "nat" where "\<sigma> m = \<tau> n" and "\<forall>j<m. \<exists>k<n. \<sigma> j = \<tau> k"
T_equiv: fixes x :: "dB"   and f :: "dB" shows "\<T> \<degree> x \<degree> f \<leftrightarrow> f \<degree> x"
basis_null_space_eq_basis_left_null_space_transpose: fixes A :: "(('a, 'b) vec, 'c) vec" shows "basis_null_space A = basis_left_null_space (Finite_Cartesian_Product.transpose A)"
R_trans: fixes b :: "real"   and a :: "real"   and c :: "real" assumes "b \<le> a"   and "c \<le> b" shows "c \<le> a"