hcju/isabelleps huggingface dataset:
CallRedsNone: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and vs :: "val list"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool"   and C :: "char list"   and fs :: "Objects.fields"   and M :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>addr a,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>es,s\<^sub>1,b\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs,s\<^sub>2,b\<^sub>2\<rangle>"   and "hp s\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "\<nexists>b Ts T m D. P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> \<langle>e\<bullet>M(es),s,b\<rangle> \<rightarrow>* \<langle>THROW NoSuchMethodError,s\<^sub>2,b\<^sub>2\<rangle>"
\<psi>_def: fixes x :: "'c"   and h :: "'d" shows "\<psi> x h = \<epsilon>.map x \<cdot>\<^sub>C F h"
universal: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "C.commutative_square f g h k" shows "\<exists>!l. \<p>\<^sub>1[f, g] \<cdot> l = h \<and> \<p>\<^sub>0[f, g] \<cdot> l = k"
select_convs_2: fixes vertex :: "'b Point"   and slope :: "'b"   and more :: "'c" shows "slope \<lparr>vertex = vertex, slope = slope, \<dots> = more\<rparr> = slope"
wall_crossings_are_walls: fixes X :: "'b set set"   and H :: "'b set set set"   and Cs :: "'b set list" assumes "ThinChamberComplex X"   and "H \<in> set (ThinChamberComplex.wall_crossings X Cs)"   and "H \<noteq> {}" shows "H \<in> (\<Union>(f, g)\<in>ThinChamberComplex.foldpairs X. {{f \<turnstile> ChamberComplex.chamber_system X, g \<turnstile> ChamberComplex.chamber_system X}})"
vfsequence_vdomain: shows "\<D>\<^sub>\<circ> (xs #\<^sub>\<circ> x) = vcard (xs #\<^sub>\<circ> x)"
induct: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x1 :: "('c, 'd) rose_tree\<^sub>m"   and x2 :: "('c, 'd) rose_tree\<^sub>m"   and P :: "('c, 'd) rose_tree\<^sub>m \<Rightarrow> ('c, 'd) rose_tree\<^sub>m \<Rightarrow> bool" assumes "blinding_of_tree h bo x1 x2"   and "\<And>t1 t2. blinding_of_blindable (map_prod h (map (hash_tree h))) (rel_prod bo (list_all2 (\<lambda>x1 x2. blinding_of_tree h bo x1 x2 \<and> P x1 x2))) t1 t2 \<Longrightarrow> P (Tree\<^sub>m t1) (Tree\<^sub>m t2)" shows "P x1 x2"
substitute_induct: fixes P :: "('b \<Rightarrow> 'b ttree) \<Rightarrow> 'b set \<Rightarrow> 'b ttree \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> 'b ttree"   and a1 :: "'b set"   and a2 :: "'b ttree"   and a3 :: "'b list" assumes "\<And>f T t. P f T t []"   and "\<And>f T t x xs. P (f_nxt f T x) T (nxt t x \<otimes>\<otimes> f x) xs \<Longrightarrow> P f T t (x # xs)" shows "P a0 a1 a2 a3"
in_homE: fixes f :: "'j"   and a :: "'j"   and b :: "'j" assumes "J.in_hom f a b" obtains "J.arr f" and "J.dom f = a" and "J.cod f = b"
real_cond_exp_intg_fpos_1: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "integrable M (\<lambda>x. f x * g x)"   and "\<And>x. 0 \<le> f x"   and "f \<in> borel_measurable F"   and "g \<in> borel_measurable M" shows "integrable M (\<lambda>x. f x * real_cond_exp M F g x)"
that: fixes ys :: "'a list"   and y :: "'a" assumes "l = ys @ [y]" shows "thesis"
psi_elem: fixes x :: "'b sort_key extended \<times> 'b sort_key extended"   and i :: "'b" assumes "is_interval x" shows "elem \<lbrakk>\<Psi> x i\<rbrakk> x"
S_satisfies_def: fixes P :: "'state"   and \<phi> :: "unit" shows "P \<turnstile>\<^sub>S \<phi> = False"
assms_1: shows "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> i"
assms_1: shows "(f \<longlongrightarrow> l) net"
skl1_obs_secrecy: shows "oreach skl1 \<subseteq> l1_secrecy"
FORWEAK_invarI: fixes X :: "'e set"   and d :: "'f nres"   and P :: "'f \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f nres"   and I :: "'f \<Rightarrow> 'e set \<Rightarrow> bool"   and c :: "'f \<Rightarrow> 'f \<Rightarrow> 'f nres" assumes "X = {} \<Longrightarrow> d \<le> SPEC P"   and "\<And>a. a \<in> X \<Longrightarrow> f a \<le> SPEC (\<lambda>x. I x (X - {a}))"   and "\<And>a. a \<in> X \<Longrightarrow> f a \<le> SPEC (\<lambda>x. I x X)"   and "\<And>a aa b ba. \<lbrakk>I aa b; a \<in> b\<rbrakk> \<Longrightarrow> f a \<le> SPEC (\<lambda>xb. c aa xb \<le> SPEC (\<lambda>r. I r (b - {a})))"   and "\<And>a aa b ba. \<lbrakk>I aa b; a \<in> b\<rbrakk> \<Longrightarrow> f a \<le> SPEC (\<lambda>xb. c aa xb \<le> SPEC (\<lambda>r. I r b))"   and "\<And>aa. I aa {} \<Longrightarrow> P aa" shows "FORWEAK X d f c \<le> SPEC P"
p\<^sub>1: shows "on_curve a b p\<^sub>1"
solves_ode_congI: fixes x :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "real set"   and X :: "'a set"   and y :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "real set"   and Y :: "'a set" assumes "(x solves_ode f) T X"   and "\<And>t. t \<in> T \<Longrightarrow> x t = y t"   and "\<And>t. t \<in> T \<Longrightarrow> f t (x t) = g t (x t)"   and "T = S"   and "X = Y" shows "(y solves_ode g) S Y"
Existence_def: fixes x :: "e" shows "E! x \<equiv> \<lambda>w. \<exists>xa. xa actualizedAt w \<and> xa = x"
succ_eqpoll_cong: fixes a :: "V"   and b :: "V" assumes "elts a \<approx> elts b" shows "elts (ZFC_in_HOL.succ a) \<approx> elts (ZFC_in_HOL.succ b)"
sanity_wf_ruleset_wf_chain: fixes \<Gamma> :: "(char list \<times> 'c rule list) list"   and rs :: "'c rule list" assumes "sanity_wf_ruleset \<Gamma>"   and "rs \<in> ran (map_of \<Gamma>)" shows "wf_chain (map_of \<Gamma>) rs"
elims_2: fixes x :: "('c, 'd) trie" assumes "invar_trie x" obtains vo :: "'d option" and kts :: "('c \<times> ('c, 'd) trie) list" where "x = Trie vo kts" and "distinct (map fst kts) \<and> (\<forall>(k, t)\<in>set kts. \<not> is_empty_trie t \<and> invar_trie t)"
hom_deg_set_alt_homogeneous_set: fixes A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and z :: "nat" assumes "homogeneous_set A" shows "hom_deg_set z A = {p \<in> A. homogeneous p \<and> (p = 0 \<or> poly_deg p = z)}"
expand: fixes prod :: "'d \<times> 'e"   and prod' :: "'d \<times> 'e" assumes "fst prod = fst prod' \<and> snd prod = snd prod'" shows "prod = prod'"
sum_upto_cong': fixes x :: "real"   and f :: "nat \<Rightarrow> 'a"   and f' :: "nat \<Rightarrow> 'a"   and x' :: "real" assumes "\<And>n. \<lbrakk>0 < n; real n \<le> x\<rbrakk> \<Longrightarrow> f n = f' n"   and "x = x'" shows "sum_upto f x = sum_upto f' x'"
reduction_trc_eval: fixes t :: "db"   and t' :: "db"   and v :: "db" assumes "t \<rightarrow>\<^sub>v\<^sup>* t'"   and "t' \<Down> v" shows "t \<Down> v"
preserves_cod: fixes f :: "'c" assumes "Cop.arr f" shows "S.cod (Y a f) = Y a (Cop.cod f)"
extremum_uniqueI: fixes a :: "'d upred" assumes "a \<sqsubseteq> true" shows "a = true"
field_def: fixes P :: "'a prog"   and C :: "char list"   and F :: "char list" shows "field P C F \<equiv> THE (D, b, T). P \<turnstile> C sees F,b:T in D"
cmem_def: fixes M :: "'b" shows "cmem M \<equiv> \<forall>param\<in>dom (local.map_of M). local.map_of M param = Some (dp param)"
prv_imp_refl: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "bprv (imp \<phi> \<phi>)"
simps_2: fixes x :: "'d list"   and xs :: "'d list list" shows "concat (x # xs) = x @ concat xs"
isShortV_def: fixes v :: "Value" shows "isShortV v = (case v of shortV s \<Rightarrow> True | _ \<Rightarrow> False)"
inj_smap_graph: shows "inj (\<lambda>f. {(x, y). y = f x \<and> y \<noteq> {}})"
sumpar: fixes \<rho> :: "complex mat" assumes "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>" shows "\<forall>n. partial_density_operator (matrix_sum d (\<lambda>n. denote_while_n (M 0) (M 1) (denote S) n \<rho>) n)"
spectrum_def: shows "Spec \<equiv> {I. pr_ideal R I (+) (\<cdot>) \<zero> \<one>}"
INT_I: fixes A :: "'b set"   and b :: "'c"   and B :: "'b \<Rightarrow> 'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> b \<in> B x" shows "b \<in> \<Inter> (B ` A)"
null_uni_uniq_4: fixes v :: "i" shows "[\<^bold>\<exists>y. y\<^sup>P \<^bold>= (\<^bold>\<iota>x. TAO_8_Definitions.Universal (x\<^sup>P)) in v]"
intros_3: fixes P :: "'a eclause set"   and S :: "'a eclause set"   and C :: "'a eclause"   and S' :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list"   and C' :: "'a literal set" assumes "\<forall>x. x \<in> P \<longrightarrow> derivable_ecl x S"   and "derivable C P S' \<sigma> FirstOrder C'" shows "derivable_ecl C S"
no_jmp: shows "fst s0 \<noteq> Some (Jump j)"
straighten_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "straighten w1 w2" shows "kauff_mat w1 = kauff_mat w2"
abst_dbtm_graph_aux_def: shows "abst_dbtm_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>abst_dbtm_aux name i. x1 = (name, i, DBZero) \<and> x2 = DBZero) \<or> (\<exists>abst_dbtm_aux name i name'. x1 = (name, i, DBVar name') \<and> x2 = (if name = name' then DBInd i else DBVar name')) \<or> (\<exists>abst_dbtm_aux name i j. x1 = (name, i, DBInd j) \<and> x2 = DBInd j) \<or> (\<exists>abst_dbtm_aux name i t1 t2. x1 = (name, i, DBEats t1 t2) \<and> x2 = DBEats (((\<lambda>uu_. uu_) \<circ> abst_dbtm_aux) (name, i, t1)) (((\<lambda>uu_. uu_) \<circ> abst_dbtm_aux) (name, i, t2)) \<and> p (name, i, t1) (((\<lambda>uu_. uu_) \<circ> abst_dbtm_aux) (name, i, t1)) \<and> p (name, i, t2) (((\<lambda>uu_. uu_) \<circ> abst_dbtm_aux) (name, i, t2))))"
multrow_transfer: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "((R ===> R ===> R) ===> (=) ===> R ===> mat_rel R ===> mat_rel R) mat_multrow_gen mat_multrow_gen"
unit_char_4: fixes a :: "'a" assumes "B.obj a" shows "\<exists>!\<psi>. \<guillemotleft>\<psi> : UP.map\<^sub>0 a \<Rightarrow> UP a\<guillemotright> \<and> local.iso \<psi> \<and> \<psi> \<cdot> \<i> (UP.map\<^sub>0 a) = (UP \<i>\<^sub>B[a] \<cdot> cmp\<^sub>U\<^sub>P (a, a)) \<cdot> (\<psi> \<star> \<psi>)"
fg_genTr: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and i :: "'d \<Rightarrow> 'd"   and s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and z :: "'d"   and n :: "nat" assumes "fgmodule_condition R f i s A z" shows "\<forall>x. x \<in> add_set f (aug_pm_set z i (s_set R s A)) n \<longrightarrow> x \<in> linear_span R (fgmodule R A z i f s) (carrier R) A"
l_inv_ex: fixes x :: "'c" assumes "x \<in> carrier S" shows "\<exists>y\<in>carrier S. y \<oplus>\<^bsub>S\<^esub> x = \<zero>\<^bsub>S\<^esub>"
sym_def: fixes a1 :: "'a"   and a0 :: "'a" shows "local.sym a1 a0 \<equiv> if ide a0 \<and> ide a1 then tuple (pr0 a1 a0) (pr1 a1 a0) else null"
Rep_inject: fixes x :: "'b word"   and y :: "'b word" shows "(sint x = sint y) = (x = y)"
it_to_sorted_list_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a set" shows "it_to_sorted_list R s \<equiv> SPEC (\<lambda>l. distinct l \<and> s = set l \<and> sorted_wrt R l)"
flatten_plus: fixes p :: "'b \<Rightarrow>\<^sub>0 'b \<Rightarrow>\<^sub>0 'c"   and q :: "'b \<Rightarrow>\<^sub>0 'b \<Rightarrow>\<^sub>0 'c" shows "flatten (p + q) = flatten p + flatten q"
assms_1: fixes \<phi> :: "'a fm" assumes "nqfree \<phi>" shows "qfree (qe \<phi>)"
one_dim_inverse: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<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 dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and canonical_basis :: "'a list"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "complex" assumes "class.one_dim scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner canonical_basis divide inverse one times" shows "inverse (scaleC a one) = scaleC (inverse a) one"
span_neg: fixes U :: "'a vec set"   and v :: "'a vec" assumes "U \<subseteq> carrier_vec n"   and "v \<in> local.span U" shows "\<ominus>\<^bsub>V\<^esub> v \<in> local.span U"
getlocs_def: fixes s :: "state" shows "loc s \<equiv> case s of st g l \<Rightarrow> l"
cond_distr: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and b :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool"   and c :: "'b \<times> 'b \<Rightarrow> bool"   and R :: "'b \<times> 'b \<Rightarrow> bool" shows "(P \<triangleleft> b \<triangleright>  (Q \<triangleleft> c \<triangleright>  R)) = (P \<triangleleft> b \<triangleright>  Q \<triangleleft> c \<triangleright>  (P \<triangleleft> b \<triangleright>  R))"
iff: shows "(u [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> Brouwer_degree2 n f = u [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> (a - b)) = (Brouwer_degree2 n f = a - b)"
tensor_prod_2: fixes x1 :: "complex"   and x2 :: "complex"   and x3 :: "complex"   and x4 :: "complex" shows "mult.vec_vec_Tensor (*) [x1, x2] [x3, x4] = [x1 * x3, x1 * x4, x2 * x3, x2 * x4]"
job_lower_bound_makespan: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat"   and x :: "nat" assumes "lb T A j"   and "x \<in> {1..j}" shows "t x \<le> makespan T"
pure_conj_left: fixes P' :: "bool"   and Q' :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "((\<lambda>s. P' \<and> Q' s) \<and>* Q) = (\<lambda>s. P' \<and> (Q' \<and>* Q) s)"
is_unit_monom_trivial: fixes p :: "'b poly" assumes "is_unit p" shows "monom (lead_coeff p) 0 = p"
assms_2: shows "\<S>' = update\<^sub>s\<^sub>t \<S> (send\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"
alt_writer_def: fixes m :: "('w, 'a, 'm) writerT"   and m' :: "('w, 'a, 'm) writerT" shows "local.alt_writer m m' = WriterT (alt (run_writer m) (run_writer m'))"
sym_group_def: fixes n :: "nat" shows "sym_group n = \<lparr>carrier = {p. p permutes {1..n}}, monoid.mult = (\<circ>), one = id\<rparr>"
Op: fixes inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option"   and delta :: "'opSym" assumes "goodInp inp"   and "goodBinp binp"   and "liftAll phi inp"   and "liftAll phiAbs binp" shows "phi (Op delta inp binp)"
closed_no_lconsts: fixes f :: "'form" assumes "closed f" shows "lconsts f = {}"
order_trans1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.transitive less_eq less"   and "less x y"   and "less y z" shows "less x z"
that_1: fixes i :: "'a" shows "f i \<in> topspace (Y i) \<rightarrow> X"
divide_minus_right: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide a (uminus b) = uminus (divide a b)"
Un_absorb: fixes A :: "'b set" shows "(A::'b set) \<union> A = A"
lookup_tail: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "lookup (tail p) v = (if v \<prec>\<^sub>t lt p then lookup p v else (0::'c))"
onode_notucast: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d"   and d :: "nat"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd) \<times> 'e seq_action \<times> (nat \<Rightarrow> 'c) \<times> 'd) set"   and R :: "nat set"   and i :: "nat" assumes "((\<sigma>, s), \<not>unicast d, \<sigma>', s') \<in> S"   and "d \<notin> R"   and "\<forall>j. j \<noteq> i \<longrightarrow> \<sigma>' j = \<sigma> j" shows "((\<sigma>, NodeS i s R), \<tau>, \<sigma>', NodeS i s' R) \<in> onode_sos S"
dominates_induce_subgraphD: fixes u :: "'c"   and v :: "'c"   and G :: "('c, 'd) pre_digraph"   and S :: "'c set" assumes "u \<rightarrow>\<^bsub>G \<restriction> S\<^esub> v" shows "u \<rightarrow>\<^bsub>G\<^esub> v"
domintros: fixes data :: "nat \<times> nat \<times> 'd"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set" assumes "\<And>ad ae bb af bc ag ah bd xba ai be. \<lbrakk>data = (ad, ae, bb); ps \<noteq> []; (af, bc) = compl gs bs (ps -- sel gs bs ps (ae, bb)) (sel gs bs ps (ae, bb)) (ae, bb); \<not> ad \<le> count_const_lt_components (fst (compl gs bs (ps -- sel gs bs ps (ae, bb)) (sel gs bs ps (ae, bb)) (ae, bb))); (ag, ah, bd) = add_indices (compl gs bs (ps -- sel gs bs ps (ae, bb)) (sel gs bs ps (ae, bb)) (ae, bb)) (ae, bb); (xba, ai, be) = add_indices (compl gs bs (ps -- sel gs bs ps (ae, bb)) (sel gs bs ps (ae, bb)) (ae, bb)) (ae, bb)\<rbrakk> \<Longrightarrow> gb_schema_dummy_dom ((ad - count_const_lt_components (fst (compl gs bs (ps -- sel gs bs ps (ae, bb)) (sel gs bs ps (ae, bb)) (ae, bb))), ai, be), D \<union> (set xba \<times> (set gs \<union> set bs \<union> set xba) \<union> set (ps -- sel gs bs ps (ae, bb)) -\<^sub>p set (ap gs bs (ps -- sel gs bs ps (ae, bb)) xba (ai, be))), ab gs bs xba (ai, be), ap gs bs (ps -- sel gs bs ps (ae, bb)) xba (ai, be))" shows "gb_schema_dummy_dom (data, D, bs, ps)"
log_def: fixes a :: "real"   and x :: "real" shows "Transcendental.log a x = ln x / ln a"
hypsubst: fixes A :: "'k set"   and B :: "'k set"   and x :: "'k" assumes "A = B"   and "x \<in> B" obtains "x \<in> A"
atd_lem: fixes n :: "nat"   and xs :: "'b list"   and t :: "'b list"   and d :: "'b list" assumes "take n xs = t"   and "drop n xs = d" shows "xs = t @ d"
bdd_above_def: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set" assumes "class.preorder less_eq less" shows "preorder.bdd_above less_eq A = (\<exists>M. \<forall>x\<in>A. less_eq x M)"
proposition3_4_inv_step: fixes r :: "('a \<times> 'a) set"   and \<sigma> :: "'a list"   and \<beta> :: "'a"   and \<alpha> :: "'a" assumes "trans r"   and "irrefl r"   and "(r|\<sigma>| -s r \<down>l [\<beta>], {#\<alpha>#}) \<in> mul_eq r" shows "\<exists>\<sigma>1 \<sigma>2 \<sigma>3. \<sigma> = \<sigma>1 @ \<sigma>2 @ \<sigma>3 \<and> LD_1' r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3"
gauss_jordan_main_row_echelon: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'c mat"   and i :: "nat"   and j :: "nat"   and A' :: "'c mat"   and B' :: "'c mat"   and f :: "nat \<Rightarrow> nat" assumes "A \<in> carrier_mat nr nc"   and "gauss_jordan_main A B i j = (A', B')"   and "pivot_fun A f j"   and "\<And>i'. i' < i \<Longrightarrow> f i' < j"   and "\<And>i'. i \<le> i' \<Longrightarrow> f i' = j"   and "i \<le> nr"   and "j \<le> nc" shows "row_echelon_form A'"
ground\<^sub>l_vars\<^sub>l: fixes l :: "fterm literal" assumes "ground\<^sub>l l" shows "vars\<^sub>l l = {}"
xcpt_eff_def: fixes i :: "instr"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and pc :: "nat"   and \<tau> :: "ty list \<times> ty err list"   and et :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list" shows "xcpt_eff i P pc \<tau> et = (let (ST, LT) = \<tau> in map (\<lambda>(f, t, C, h, d). (h, \<lfloor>(Class C # drop (length ST - d) ST, LT)\<rfloor>)) (relevant_entries P i pc et))"
multiset_nonemptyE: fixes A :: "'b multiset" assumes "A \<noteq> \<Empt>" obtains x :: "'b" where "x \<in># A"
simps_2: fixes bs :: "real list" shows "Ifm bs F = False"
mdc_def: fixes S :: "char list \<times> char list list" shows "mdc S = fst S"
compact_sequentialE: fixes S :: "'b set"   and T :: "'b set" assumes "compact S"   and "infinite T"   and "T \<subseteq> S" obtains t :: "nat \<Rightarrow> 'b" and l :: "'b" where "\<And>n. t n \<in> T" and "\<And>n. t n \<noteq> l" and "t \<longlonglongrightarrow> l" and "l \<in> S"
r2f_rtrancl_hom_var: shows "\<F> \<circ> rtrancl = kstar \<circ> \<F>"
distinct_map_append: fixes ls :: "'a list list"   and xs :: "'a list" assumes "distinct ls" shows "distinct (map ((@) xs) ls)"
simps: fixes a1 :: "'a OCL_Types.type"   and a2 :: "'a OCL_Types.type" shows "to_nonunique_collection a1 a2 = ((\<exists>\<tau>. a1 = Collection \<tau> \<and> a2 = Bag \<tau>) \<or> (\<exists>\<tau>. a1 = Set \<tau> \<and> a2 = Bag \<tau>) \<or> (\<exists>\<tau>. a1 = OrderedSet \<tau> \<and> a2 = Sequence \<tau>) \<or> (\<exists>\<tau>. a1 = Bag \<tau> \<and> a2 = Bag \<tau>) \<or> (\<exists>\<tau>. a1 = Sequence \<tau> \<and> a2 = Sequence \<tau>))"
insert_append: fixes ks :: "bool list"   and ks' :: "bool list"   and t :: "trie" shows "insert (ks @ ks') (prefix_trie ks t) = prefix_trie ks (insert ks' t)"
Notes_master_imp_Crypt_PMS: fixes C :: "agent"   and s :: "msg"   and A :: "agent"   and B :: "agent"   and PMS :: "nat"   and NA :: "nat"   and NB :: "nat"   and evs :: "event list" assumes "Notes C \<lbrace>s, Agent A, Agent B, Nonce (PRF (PMS, NA, NB))\<rbrace> \<in> set evs"   and "evs \<in> tls" shows "Crypt (pubK B) (Nonce PMS) \<in> parts (knows Spy evs)"
list_assn_pure_conv: fixes R :: "('e \<times> 'd) set" shows "list_assn (pure R) = pure (\<langle>R\<rangle>list_rel)"
def_hashmap_size_rat_def: shows "def_hashmap_size = (\<lambda>_. 10)"
step: shows "(cms\<^sub>1, mem\<^sub>1) \<leadsto> (cms\<^sub>1', mem\<^sub>1')"
all_in_listSubset: fixes b :: "('b, 'c) Combinators list"   and l :: "'b set list"   and a :: "('b, 'c) Combinators list" assumes "all_in_list b l"   and "singleCombinators a"   and "set a \<subseteq> set b" shows "all_in_list a l"
inf_complement_left_antitone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "- x \<le> - y" shows "- (- y \<sqinter> - z) \<le> - (- x \<sqinter> - z)"
fg_diff: fixes n :: "nat" shows "(\<Sum>i<2 * n. (- 1) ^ i * a i) - (\<Sum>i<2 * n + 1. (- 1) ^ i * a i) = - a (2 * n)"
i_take_the_conv: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "f \<Down> k = (THE xs. length xs = k \<and> (\<exists>g. xs \<frown> g = f))"
chine_is_induced_map: shows "\<chi>.is_induced_map \<chi>.chine"
1: shows "arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G y x g"
map_endpow_Suc: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "VectorSpaceEnd smult V T" shows "map (VectorSpaceEnd.endpow V T) [0..<Suc n] = id \<down> V # map ((\<circ>) T) (map (VectorSpaceEnd.endpow V T) [0..<n])"
carrier_Gp: fixes H :: "'a set" assumes "G \<guillemotright> H " shows "carrier (\<natural>H) = H"
length_ge_2_implies_tl_not_empty: fixes x :: "'a list" shows "2 \<le> length x \<longrightarrow> tl x \<noteq> []"
num_groups_lt_points: shows "int (card \<G>) \<le> \<v>"
poly_squarefree_decomp_order2: fixes p :: "'b poly"   and q :: "'b poly"   and d :: "'b poly"   and e :: "'b poly"   and r :: "'b poly"   and s :: "'b poly" assumes "pderiv p \<noteq> 0"   and "p = q * d"   and "pderiv p = e * d"   and "d = r * p + s * pderiv p" shows "\<forall>a. order a q = (if order a p = 0 then 0 else 1)"
Sink: fixes s :: "'state" assumes "s \<in> S" shows "\<exists>s'. s' \<in> \<delta> s \<inter> Sink"
map_1: fixes f1 :: "'d \<Rightarrow> 'a"   and f2 :: "'e \<Rightarrow> 'b"   and f3 :: "'f \<Rightarrow> 'c"   and x1 :: "'d" shows "map_generat f1 f2 f3 (Pure x1) = Pure (f1 x1)"
path_edges: fixes xs :: "'a list"   and v :: "'a"   and w :: "'a" assumes "path xs"   and "(v, w) \<in> edges_of_walk xs" shows "\<exists>xs_pre xs_post. xs = xs_pre @ v # w # xs_post \<and> (v, w) \<notin> edges_of_walk (xs_pre @ [v]) \<and> (v, w) \<notin> edges_of_walk (w # xs_post)"