hcju/isabelleps huggingface dataset:
has_as_product_def: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'b \<Rightarrow> 'a"   and a :: "'a" shows "has_as_product J D a \<equiv> \<exists>\<pi>. product_cone J (\<cdot>) D a \<pi>"
PRJ\<^sub>1_def: shows "Maps.PRJ\<^sub>1 \<equiv> \<lambda>K H. if Maps.cospan K H then \<lbrakk>\<lbrakk>prj\<^sub>1 (Maps.REP H) (Maps.REP K)\<rbrakk>\<rbrakk> else Maps.null"
sup_continuousD: fixes F :: "'b \<Rightarrow> 'c"   and M :: "nat \<Rightarrow> 'b" assumes "sup_continuous F"   and "incseq M" shows "F (Sup (range M)) = (SUP i. F (M i))"
si_neq: fixes x :: "'a event"   and s :: "'a event list"   and X :: "'a event set"   and y :: "'a event"   and t :: "'a event list" shows "setinterleaving (x # s, X, y # t) = (if x \<in> X then if y \<in> X then if x = y then {x # u |u. u setinterleaves ((s, t), X)} else {} else {y # u |u. u setinterleaves ((x # s, t), X)} else if y \<notin> X then {x # u |u. u setinterleaves ((s, y # t), X)} \<union> {y # u |u. u setinterleaves ((x # s, t), X)} else {x # u |u. u setinterleaves ((s, y # t), X)})"
le_add_iff1: fixes a :: "'b"   and e :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" shows "(a * e + c \<le> b * e + d) = ((a - b) * e + c \<le> d)"
classD1: shows "class P D = \<lfloor>(Bs', fs', ms')\<rfloor>"
cnj_of_nat: fixes n :: "nat" shows "Quaternions.cnj (of_nat n) = of_nat n"
surjective: fixes r :: "('d, 'e, 'f) category_scheme" shows "r = \<lparr>ob = Ob\<^bsub>r\<^esub>, ar = Ar\<^bsub>r\<^esub>, dom = category.dom r, cod = cod r, id = category.id r, comp = (\<bullet>\<^bsub>r\<^esub>), \<dots> = category.more r\<rparr>"
a_add_idem: fixes x :: "'a" shows "ad x \<oplus> ad x = ad x"
closed_sub: fixes k :: "nat"   and m :: "nat"   and p :: "('a, 'b) form"   and c :: "'a" assumes "k \<le> m" shows "closed (Suc m) p = closed m (p[App c []/k])"
no_zeros: fixes r :: "nat" assumes "r \<in> set rs" shows "0 < r"
has_integral_negligible_cbox: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a" assumes "negligible S"   and "\<And>x. x \<notin> S \<Longrightarrow> f x = (0::'b)" shows "(f has_integral (0::'b)) (cbox a b)"
VectorSpace_Im: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" assumes "VectorSpaceHom smult V smult' T" shows "VectorSpace smult' (T ` V)"
LeastI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and k :: "'b" assumes "class.wellorder less_eq less"   and "P k" shows "P (ord.Least less_eq P)"
fds_deriv_def: fixes f :: "'b fds" shows "fds_deriv f = fds (\<lambda>n. - ln (real n) *\<^sub>R fds_nth f n)"
linorder_neq_iff: fixes x :: "'d"   and y :: "'d" shows "(x \<noteq> y) = (x < y \<or> y < x)"
abs_pos: fixes a :: "'a" assumes "(0::'a) \<le> a" shows "\<bar>a\<bar> = a"
join_ac_1: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "bounded_lattice L"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<squnion>\<^bsub>L\<^esub> y \<squnion>\<^bsub>L\<^esub> z = x \<squnion>\<^bsub>L\<^esub> (y \<squnion>\<^bsub>L\<^esub> z)"
balance_left_inv2_with_inv1_2: fixes lt :: "('a, 'b) rbt"   and rt :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b" assumes "inv2 lt"   and "inv2 rt"   and "bheight lt + 1 = bheight rt"   and "inv1 rt" shows "inv2 (balance_left lt k v rt)"
principal_dchar_def: fixes n :: "nat" shows "principal_dchar n = (\<lambda>k. if coprime k n then 1 else 0)"
postdominate_refl: fixes n :: "'node" assumes "valid_node n"   and "n \<noteq> (_Exit_)" shows "n postdominates n"
ord_of_nat_le_omega: fixes n :: "nat" shows "ord_of_nat n \<le> \<omega>"
simpnum_ci: fixes bs :: "int list"   and t :: "Cooper.num" shows "Inum bs (simpnum t) = Inum bs t"
with_proj_simps_4: fixes G :: "'c pair_pre_digraph" shows "tail (with_proj G) = fst"
simps_5: fixes t :: "'c local.term" shows "Dom \<^bold>\<l>\<^bold>[t\<^bold>] = \<^bold>\<I> \<^bold>\<otimes> Dom t"
inject_7: fixes x71 :: "key"   and x72 :: "msg"   and y71 :: "key"   and y72 :: "msg" shows "(Crypt x71 x72 = Crypt y71 y72) = (x71 = y71 \<and> x72 = y72)"
arr_char: fixes f :: "'a" shows "Right.arr f = right (src \<mu>) f"
finite_acyclic_wf_converse: fixes r :: "('x \<times> 'x) set" assumes "finite r"   and "acyclic r" shows "wf (r\<inverse>)"
dg_1_is_arrI: fixes a :: "V"   and \<aa> :: "V"   and b :: "V"   and f :: "V"   and \<ff> :: "V" assumes "a = \<aa>"   and "b = \<aa>"   and "f = \<ff>" shows "f : a \<mapsto>\<^bsub>dg_1 \<aa> \<ff>\<^esub> b"
assms: shows "{min a b..max a b} \<subseteq> A"
meld_correct_1: fixes q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list"   and q' :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" assumes "SkewBinomialHeapStruc.invar (map BsSkewBinomialHeapStruc.bsmapt q)"   and "SkewBinomialHeapStruc.invar (map BsSkewBinomialHeapStruc.bsmapt q')" shows "SkewBinomialHeapStruc.invar (map BsSkewBinomialHeapStruc.bsmapt (BsSkewBinomialHeapStruc.meld q q'))"
union'_def: fixes A :: "'a set"   and B :: "'a set" shows "union' A B = A \<union> B"
accessible_def: shows "Accessible.accessible \<equiv> {q. Accessible.left_lang q \<noteq> {}}"
post_meas1_def: fixes n :: "nat"   and v :: "complex mat"   and i :: "nat" shows "post_meas1 n v i \<equiv> complex_of_real (1 / sqrt (prob1 n v i)) \<cdot>\<^sub>m |vec (2 ^ n) (\<lambda>j. if select_index n i j then v $$ (j, 0) else 0)\<rangle>"
subcringI: fixes H :: "'a set" assumes "subring H R"   and "\<And>h1 h2. \<lbrakk>h1 \<in> H; h2 \<in> H\<rbrakk> \<Longrightarrow> h1 \<otimes> h2 = h2 \<otimes> h1" shows "subcring H R"
simps_1: fixes uu :: "'b ext0" shows "(ext0.Bot \<le> uu) = True"
vimage_set_def: fixes A :: "V" shows "r `\<^sub>\<circ> A = set {b. \<exists>a\<in>elts (A \<inter>\<^sub>\<circ> \<D>\<^sub>\<circ> r). r\<lparr>a\<rparr> = b}"
foldl_Nil: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and a :: "'b" shows "foldl f a [] = a"
m_cf_len: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c" shows "fst (m_cf S c) = fst c"
many: shows "many ((\<noteq>) CHR ''<'') tss = Inr (x1, x2)"
simps_14: fixes v :: "'a rexp"   and va :: "'a rexp" shows "local.le_rexp (Times v va) One = False"
column_add_row_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and n :: "'e"   and m :: "'e"   and q :: "'d"   and i :: "'f" shows "column_add_row A n m q i = (\<chi>j. if j = n then A $ i $ n + A $ i $ m * q else A $ i $ j)"
index_eq_some: fixes l :: "'a list"   and n :: "nat"   and x :: "'a" shows "(index l n = Some x) = (n < length l \<and> x = l ! n)"
comp_char: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" shows "VVV.comp g f = (if VVV.arr f \<and> VVV.arr g \<and> CCC.seq g f then CCC.comp g f else CCC.null)"
leg_simps_2: shows "C.dom \<nu>\<mu>.cod.leg0 = \<nu>\<mu>.cod.apex"
his_le: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and x :: "nat \<times> 'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'a" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "x \<in> Blockchain.his active cmp bc mining t n nid" shows "fst x \<le> n"
abs_eq: fixes xa :: "'c \<Rightarrow> 'd"   and x :: "'c \<Rightarrow> 'd" assumes "eq_onp (\<lambda>x. x \<in> bcontfun) xa xa"   and "eq_onp (\<lambda>x. x \<in> bcontfun) x x" shows "Bcontfun xa + Bcontfun x = Bcontfun (\<lambda>xb. xa xb + x xb)"
space_pre_sigma: fixes T :: "'a \<Rightarrow> 't" shows "space (pre_sigma T) = \<Omega>"
simps_2: fixes n :: "nat" shows "f (Suc n) = (SOME x. At x (Suc n) \<and> Follows x (f n))"
HAEvents_SAEvents_SAs: fixes HA :: "('e, 'd, 'f) hierauto" shows "\<Union> (SAEvents ` SAs HA) \<subseteq> HAEvents HA"
distinct_2: fixes x21 :: "'c"   and x22 :: "'c llist" shows "LCons x21 x22 \<noteq> LNil"
cases: fixes x :: "game option" obtains \<alpha> :: "game" where "x = Agame \<alpha>"   | "x = undefg"
sf: shows "vs2.span T \<subseteq> f ` vs1.span S"
inclusion_L: fixes Cn :: "'b set \<Rightarrow> 'b set"   and A :: "'b set" assumes "Tarskian_logic Cn" shows "A \<subseteq> Cn A"
product_two_squares_int: fixes m :: "int"   and n :: "int" assumes "is_sum2sq_int m"   and "is_sum2sq_int n" shows "is_sum2sq_int (m * n)"
la_def: shows "la = snd (n_as ! fst (j_to_jb_index n_as j))"
flow_initial_time_if: fixes x0a :: "'a"   and t0 :: "real"   and x0 :: "'a \<Rightarrow>\<^sub>L 'a" shows "Y.mvar.flow x0a t0 x0 t0 = (if t0 \<in> Y.existence_ivl0 x0a \<and> x0 \<in> UNIV then x0 else 0)"
2: shows "{i \<in> liset A (lselect s w). enat i < esize s} = liset A (lselect s w)"
B_def: fixes n :: "nat" shows "B n = snd ((bisect ^^ n) (a, b))"
dg_SemiCAT_def: fixes \<alpha> :: "V" shows "dg_SemiCAT \<alpha> = [ZFC_in_HOL.set {\<CC>. semicategory \<alpha> \<CC>}, all_smcfs \<alpha>, \<lambda>\<FF>\<in>\<^sub>\<circ>all_smcfs \<alpha>. \<FF>\<lparr>HomDom\<rparr>, \<lambda>\<FF>\<in>\<^sub>\<circ>all_smcfs \<alpha>. \<FF>\<lparr>HomCod\<rparr>]\<^sub>\<circ>"
fps_shift_times_X_power'': fixes n :: "nat"   and f :: "'b fps" shows "fps_shift n (fps_X ^ n * f) = f"
binop_type_2: fixes wf_md :: "'addr J_prog \<Rightarrow> String.literal \<Rightarrow> (String.literal list \<times> 'addr expr) mdecl \<Rightarrow> bool"   and h :: "'heap"   and v1 :: "'addr val"   and T1 :: "ty"   and v2 :: "'addr val"   and T2 :: "ty"   and bop :: "bop"   and T :: "ty"   and a :: "'addr" assumes "wf_prog wf_md P"   and "preallocated h"   and "typeof\<^bsub>h\<^esub> v1 = \<lfloor>T1\<rfloor>"   and "typeof\<^bsub>h\<^esub> v2 = \<lfloor>T2\<rfloor>"   and "P \<turnstile> T1\<guillemotleft>bop\<guillemotright>T2 : T"   and "binop bop v1 v2 = \<lfloor>Inr a\<rfloor>" shows "P,h \<turnstile> Addr a :\<le> Class Throwable"
rational_preference_axioms_def: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set" shows "rational_preference_axioms carrier relation \<equiv> total_on carrier relation"
replicate_count_mset_eq_filter_eq: fixes xs :: "'c list"   and k :: "'c" shows "replicate (count (mset xs) k) k = filter ((=) k) xs"
const2_def: fixes x :: "'f" shows "const2 x \<equiv> \<lambda>_ _. x"
map_scale_single: fixes k :: "'d"   and x :: "'c"   and l :: "'d" shows "k \<cdot> monomial l x = monomial (k * l) x"
restrict_shared_def: fixes \<S> :: "nat \<Rightarrow> bool option"   and A :: "nat set"   and L :: "nat set" shows "\<S> \<ominus>\<^bsub>A\<^esub> L \<equiv> \<lambda>a. if a \<in> L then None else case \<S> a of None \<Rightarrow> None | Some writeable \<Rightarrow> Some (a \<in> A \<or> writeable)"
connected_sphere_eq: fixes a :: "'a"   and r :: "real" shows "connected (sphere a r) = (2 \<le> DIM('a) \<or> r \<le> 0)"
Mul_Collector: fixes n :: "nat" shows "\<turnstile> Mul_Collector n  \<lbrace>False\<rbrace>"
pre: fixes F :: "'b \<Rightarrow> bool" assumes "\<turnstile> F" shows "|~ F"
Carmichael_not_primepow: fixes n :: "nat" assumes "Carmichael_number n" shows "\<not> primepow n"
t_design_axioms_def: fixes u_block_size :: "int"   and grouping :: "int" shows "t_design_axioms u_block_size grouping \<equiv> grouping \<le> u_block_size"
monad_state_altc_optionT: fixes return :: "'a option \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a option \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and altc :: "'d cset \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_state_altc return bind get put altc" shows "monad_state_altc (return_option return) (bind_option return bind) (get_option get) (put_option put) (altc_option altc)"
qtableI: fixes n :: "nat"   and A :: "nat set"   and X :: "'a option list set"   and P :: "'a option list \<Rightarrow> bool"   and Q :: "'a option list \<Rightarrow> bool" assumes "table n A X"   and "\<And>x. \<lbrakk>x \<in> X; wf_tuple n A x; P x\<rbrakk> \<Longrightarrow> Q x"   and "\<And>x. \<lbrakk>wf_tuple n A x; P x; Q x\<rbrakk> \<Longrightarrow> x \<in> X" shows "qtable n A P Q X"
list_all2_dropI: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list"   and n :: "nat" assumes "list_all2 P xs ys" shows "list_all2 P (drop n xs) (drop n ys)"
assms_2: shows "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"
pos_divideR_less_eq: fixes c :: "real"   and b :: "'a"   and a :: "'a" assumes "0 < c" shows "(b /\<^sub>R c < a) = (b < c *\<^sub>R a)"
constrains_weaken_L: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set"   and B :: "'b set" assumes "F \<in> A co A'"   and "B \<subseteq> A" shows "F \<in> B co A'"
ht_distinct_def: fixes l :: "('a \<times> 'b) list list" shows "ht_distinct l \<equiv> \<forall>i<length l. distinct (map fst (l ! i))"
assms_1: shows "prv (imp \<phi>1 (imp \<phi>2 (imp \<phi>3 \<phi>4)))"
b_assn_invalid_merge2: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" shows "hn_invalid (b_assn A P) x y \<or>\<^sub>A hn_invalid A x y \<Longrightarrow>\<^sub>t hn_invalid A x y"
absI: fixes implem :: "chan \<Rightarrow> msg"   and M :: "msg"   and x :: "tag"   and A :: "agent"   and B :: "agent"   and H :: "msg set" assumes "semivalid_implem implem"   and "M \<in> payload"   and "implem (Chan x A B M) \<in> H" shows "Chan x A B M \<in> semivalid_implem.abs implem H"
66082: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<squnion> (y \<squnion> - z)) = - - z \<sqinter> - (x \<squnion> y)"
order_trans: fixes x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" assumes "x \<subseteq># y"   and "y \<subseteq># z" shows "x \<subseteq># z"
step_push_ubx2: fixes F :: "'fenv_ubx"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and pc :: "nat"   and b :: "'ubx2"   and H :: "'henv"   and \<Sigma> :: "('dyn, 'ubx1, 'ubx2) unboxed list"   and st :: "('fun, ('dyn, 'ubx1, 'ubx2) unboxed) frame list" assumes "Fubx_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = IPushUbx2 b" shows "Subx.step (State F H (Frame f pc \<Sigma> # st)) (State F H (Frame f (Suc pc) (OpUbx2 b # \<Sigma>) # st))"
dim_ty_sem_DimInv_def: fixes x :: "('a\<^sup>-\<^sup>1) itself" shows "dim_ty_sem x = inverse QD('a)"
ZObis_matchT_ZO: fixes s :: "('test, 'atom) com"   and t :: "('test, 'atom) com" assumes "s \<approx>01 t" shows "matchT_ZO s t"
simps: fixes uu :: "name"   and rhs :: "sterm" shows "srule (uu, rhs) = (pre_strong_term_class.wellformed rhs \<and> closed rhs \<and> is_abs rhs)"
functor_axioms: shows "functor A2xA1.comp (\<cdot>\<^sub>B) F.sym"
simps_4: fixes \<Sigma> :: "nat \<Rightarrow> 'c set"   and project :: "'c \<Rightarrow> 'c"   and embed :: "'c \<Rightarrow> 'c list"   and lookup :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'d" assumes "Pi_Derivatives.embed \<Sigma> project embed" shows "embed.lderiv lookup embed a (Atom b) = (if lookup b a then One else Zero)"
Id_def: shows "Id = {p. \<exists>x. p = (x, x)}"
rank_take_cols: fixes inds :: "nat list"   and n :: "nat" assumes "distinct inds" shows "vec_space.rank n (take_cols (1\<^sub>m n) inds) = length (filter (\<lambda>y. y < n) inds)"
fun_type': shows "\<forall>f\<in>\<Union> (range \<Sigma>n) - \<C>. \<forall>X. \<Gamma> (Fun f X) = Fun f (map \<Gamma> X)"
disc_2: fixes x21 :: "'b"   and x22 :: "'b llist" shows "\<not> lnull (x21 ## x22)"
set_concat: fixes xs :: "'b list list" shows "set (concat xs) = \<Union> (sset xs)"
UNIF_empty_imp_is_mgu_compose: fixes ss :: "('c \<Rightarrow> ('d, 'c) Term.term) list"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "UNIF ss E {#}" shows "is_mgu (compose ss) (set_mset E)"
AA32: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<turnstile> \<box>\<diamond>\<box>[A]_v = \<diamond>\<box>[A]_v"
inter_inter_aform_plane_ortho: fixes p :: "nat"   and Z :: "'a \<times> 'a pdevs"   and n :: "'a"   and g :: "real"   and X :: "'a \<times> 'a pdevs" assumes "inter_aform_plane_ortho p Z n g = Some X" shows "Affine Z \<inter> plane n g \<subseteq> Affine X"
count_mset_set_3: fixes x :: "'b"   and A :: "'b set" assumes "x \<notin> A" shows "count (mset_set A) x = 0"
Sup_insert: fixes a :: "'b"   and A :: "'b set" shows "\<Squnion> insert a A = a \<squnion> \<Squnion> A"
fin: fixes i :: "nat" assumes "i \<in> I" shows "emeasure M (F i) \<noteq> \<infinity>"
invertible_mult_column: fixes q :: "'a"   and k :: "'a"   and a :: "'b" assumes "q * k = (1::'a)"   and "k * q = (1::'a)" shows "invertible (mult_column (mat (1::'a)) a q)"
a: shows "(as, x) \<approx>res (=) supp p (as', x')"
Can_implies_Arr: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Can C t" shows "monoidal_language.Arr C t"
\<psi>_\<phi>: fixes f :: "'d"   and b :: "'d"   and a :: "'d" assumes "\<guillemotleft>f : b \<rightarrow>\<^sub>D a\<guillemotright>" shows "HomD.\<psi> (b, a) (inD f) = f"
prems: shows "(\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<N>\<lbrace>\<Gamma>\<rbrace>\<^esub>)\<cdot>(C\<cdot>r) \<noteq> \<bottom>"
borel_measurable_power: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and n :: "nat" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. f x ^ n) \<in> borel_measurable M"
bind_eq_Some_conv: fixes f :: "'n option"   and g :: "'n \<Rightarrow> 'm option"   and x :: "'m" shows "(Option.bind f g = Some x) = (\<exists>y. f = Some y \<and> g y = Some x)"
msgs_unchanged_for_other_is: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and i :: "nat"   and i' :: "nat" assumes "c \<turnstile> ev \<mapsto> c'"   and "regular_event ev"   and "getId ev = i"   and "i' \<noteq> i" shows "msgs c i' = msgs c' i'"
mono: fixes C1 :: "'a acom"   and C2 :: "'a acom" assumes "P C1"   and "P C2"   and "C1 \<le> C2" shows "f C1 \<le> f C2"
image_mono: fixes A :: "'f set"   and B :: "'f set"   and f :: "'f \<Rightarrow> 'g" assumes "A \<subseteq> B" shows "f ` A \<subseteq> f ` B"
deleted_ids_def: fixes ops :: "('c \<times> ('c, 'd) list_op) list" shows "deleted_ids ops \<equiv> List.map_filter (\<lambda>oper. case oper of (oid, Insert ref x) \<Rightarrow> Map.empty x | (oid, Delete ref) \<Rightarrow> Some ref) ops"
z: shows "z = real_alg_2 (cf_pos_poly (poly_mult_rat r1 p2), l, r)"
D_def: shows "D = (1 + CB * (lambda + C + norm (f (0::'a))) ^ DIM('b)) / Ca"
getMinTree_min_prio: fixes bq :: "('c, 'd) SkewBinomialTree list"   and y :: "'c \<times> 'd" assumes "queue_invar bq"   and "y \<in># queue_to_multiset bq" shows "prio (getMinTree bq) \<le> snd y"
dns1: fixes x :: "('b \<times> 'b) set" shows "rel_antidomain_kleene_algebra.ads_d x O x = x"
atMost_eq_iff: fixes x :: "'e"   and y :: "'e" shows "({..x} = {..y}) = (x = y)"
update_PSR_icc_1: fixes w' :: "word32"   and w :: "word32" assumes "w' = w AND 4279238655 \<and> get_S w = 0" shows "get_S w' = 0"
fresh_ssubst_fm: fixes i :: "'a"   and V :: "name set"   and A :: "fm" assumes "atom i \<sharp> p \<bullet> V"   and "V \<subseteq> Vs" shows "atom i \<sharp> ssubst \<lfloor>A\<rfloor>V V F"
c_def: shows "c = (\<lambda>x. if x \<le> k then c\<^sub>1 x else c\<^sub>2 (m + x - k))"
run_propag_on_create_graph: fixes \<sigma> :: "nat \<Rightarrow> 'a node"   and k :: "nat" assumes "ipath gba.E \<sigma>" shows "\<sigma> k \<in> qs \<and> name (\<sigma> k) \<in> incoming (\<sigma> (k + 1))"
DisjointPaths_reduce: fixes paths' :: "'a list set" assumes "paths' \<subseteq> Q" shows "DisjointPaths H_x v0 v1 paths'"
q64_array_def: shows "q64_array = IArray [True, True, False, False, True, False, False, False, False, True, False, False, False, False, False, False, True, True, False, False, False, False, False, False, False, True, False, False, False, False, False, False, False, True, False, False, True, False, False, False, False, True, False, False, False, False, False, False, False, True, False, False, False, False, False, False, False, True, False, False, False, False, False, False, False]"
Well_order_extend: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and B :: "'c set" assumes "well_order_on A r"   and "A \<subseteq> B" shows "\<exists>r'. well_order_on B r' \<and> r \<subseteq> r'"
rb_aux_term1_def: shows "rb_aux_term1 \<equiv> {(x, y). \<exists>z. x = z # y}"
bet3__bet: fixes A :: "'p"   and B :: "'p"   and E :: "'p"   and D :: "'p"   and C :: "'p" assumes "Bet A B E"   and "Bet A D E"   and "Bet B C D" shows "Bet A C E"
pos_2: fixes k :: "nat" assumes "k < m" shows "0 \<le> d fs' k"
fg: fixes x :: "'n" assumes "x \<in> S" shows "norm (f x) \<le> g x \<bullet> k"
ideal_prod_assoc: fixes I :: "'a set"   and J :: "'a set"   and K :: "'a set" assumes "ideal I R"   and "ideal J R"   and "ideal K R" shows "I \<cdot> J \<cdot> K = I \<cdot> (J \<cdot> K)"
AA_E: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "fixed_point_proof A f"   and "X \<in> fixed_point_proof.AA A less_eq f" obtains "X \<subseteq> A" and "f ` X \<subseteq> X" and "\<And>Y s. \<lbrakk>Y \<subseteq> X; extreme_bound A less_eq Y s\<rbrakk> \<Longrightarrow> s \<in> X"
eq: fixes x :: "V" assumes "x \<in> U" shows "ordermap (insert \<alpha> U) VWF x = ordermap U VWF x"
del_left_braun_size: fixes t :: "'b tree"   and x :: "'b"   and t' :: "'b tree" assumes "del_left t = (x, t')"   and "braun t"   and "t \<noteq> \<langle>\<rangle>" shows "braun t' \<and> size t = size t' + 1"
Nil: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list" assumes "[] \<in> \<S>" shows "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet>\<^sub>c (update\<^sub>s\<^sub>t \<S> [], \<A>)"
concat_conv_foldr: fixes xss :: "'c list list" shows "concat xss = foldr (@) xss []"
not_g_image: fixes r :: "real" shows "0 \<notin> path_image (g r)"
assms_5: shows "InvariantCEntailed (getConflictFlag state) F0 (getC state)"
assms_1: shows "lambda C -quasi_isometry_between UNIV UNIV f"
mult_aform_def: fixes x :: "(real \<times> real pdevs) \<times> real"   and y :: "(real \<times> real pdevs) \<times> real" shows "mult_aform x y = ((fst (fst x) * fst (fst y), add_pdevs (scaleR_pdevs (fst (fst y)) (snd (fst x))) (scaleR_pdevs (fst (fst x)) (snd (fst y)))), Radius (fst x) * Radius (fst y) + \<bar>snd x\<bar> * (\<bar>fst (fst y)\<bar> + Radius (fst y)) + \<bar>snd y\<bar> * (\<bar>fst (fst x)\<bar> + Radius (fst x)) + \<bar>snd x\<bar> * \<bar>snd y\<bar>)"
same_diag_ev_block: fixes n :: "nat"   and A :: "'b mat"   and B :: "'b mat" assumes "??.Jordan_Normal_Form_Existence.same_diag n A B"   and "??.Jordan_Normal_Form_Existence.ev_block n A" shows "??.Jordan_Normal_Form_Existence.ev_block n B"
rq_reassoc: fixes u :: "'b list"   and v :: "'b list"   and w :: "'b list" assumes "u \<le>s v" shows "w \<cdot> v\<^sup><\<inverse>u  = (w \<cdot> v)\<^sup><\<inverse>u "
ce_rel_lm_27: fixes r :: "(nat \<times> nat) set" assumes "r \<in> ce_rels" shows "Range r \<in> ce_sets"
N_def: shows "N = (\<Sum>n. norm (f n))"
preserves_cod: fixes f :: "'c" assumes "Cop.arr f" shows "S.cod (\<Psi>.map f) = Y a (Cop.cod f)"
A_1: fixes p :: "real poly"   and x :: "real" assumes "p \<in> ps"   and "p \<noteq> 0"   and "poly p x = 0" shows "l < x"
div_mult_div_if_dvd: 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 b :: "'b"   and a :: "'b"   and d :: "'b"   and c :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b a"   and "dvd.dvd times d c" shows "times (divide a b) (divide c d) = divide (times a c) (times b d)"
poly_fun_uminus_1: fixes f :: "int \<Rightarrow> int" assumes "poly_fun f" shows "poly_fun (\<lambda>x. - f x)"