hcju/isabelleps huggingface dataset:
fst_gb_schema_dummy_isGB_init: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> 'd"   and rc :: "nat" assumes "struct_spec sel ap ab compl"   and "compl_conn compl"   and "is_Groebner_basis (fst ` set gs)"   and "unique_idx (gs @ ab gs [] bs data) data"   and "rem_comps_spec (gs @ ab gs [] bs data) (rc, data)" shows "is_Groebner_basis (fst ` set (fst (gb_schema_dummy (rc, data) (set bs \<times> (set gs \<union> set bs) -\<^sub>p set (ap gs [] [] bs data)) (ab gs [] bs data) (ap gs [] [] bs data))))"
which_node_intra_standard_control_dependence_source: fixes nx :: "'node"   and as :: "'edge list"   and a :: "'edge"   and as' :: "'edge list"   and n :: "'node"   and n' :: "'node"   and a' :: "'edge" assumes "nx -as @ a # as'\<rightarrow>\<^sub>\<iota>* n"   and "sourcenode a = n'"   and "sourcenode a' = n'"   and "n \<notin> set (sourcenodes (a # as'))"   and "valid_edge a'"   and "intra_kind (kind a')"   and "inner_node n"   and "\<not> method_exit n"   and "\<not> n postdominates targetnode a'"   and "\<forall>ax ax'. ax \<in> set as' \<and> sourcenode ax = sourcenode ax' \<and> valid_edge ax' \<and> intra_kind (kind ax') \<longrightarrow> n postdominates targetnode ax'" shows "n' controls n"
PO_hoare_defs_1: fixes pre :: "'a set"   and R :: "('a \<times> 'a) set"   and post :: "'a set" shows "{pre} R {> post} \<equiv> R `` pre \<subseteq> post"
sum_prodE: fixes p :: "'c \<times> 'd + 'e" obtains a :: "'c" and b :: "'d" where "p = Inl (a, b)"   | j :: "'e" where "p = Inr j"
last_message_hold_init_length: fixes xs :: "'b message_af list"   and m :: "'b message_af" shows "length (last_message_hold_init xs m) = length xs"
projection_on_empty_trace: fixes X :: "'a set" shows "[] \<upharpoonleft> X = []"
compile_cn_gs_correct': fixes n :: "nat"   and gs :: "recf list"   and xs :: "nat list"   and ft :: "nat"   and ffp :: "nat"   and anything :: "nat list" assumes "\<forall>g\<in>set (take n gs). terminate g xs \<and> (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_exec g xs # 0 \<up> (xb - Suc xa) @ xc}))"   and "ft = max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))" shows "{\<lambda>nl. nl = xs @ 0 # 0 \<up> (ft + length gs) @ anything} cn_merge_gs (map rec_ci (take n gs)) ft {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything}"
setum_permutations_compose_left: fixes q :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and f :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b" assumes "q permutes S" shows "sum f {p. p permutes S} = (\<Sum>p | p permutes S. f (q \<circ> p))"
mat0_row: fixes i :: "nat"   and nr :: "nat"   and ze :: "'c"   and nc :: "nat" assumes "i < nr" shows "row (mat0I ze nr nc) i = vec0I ze nc"
incl_HK: fixes J :: "'a set" assumes "subfield J R"   and "I \<subseteq> J" shows "generate_field (R\<lparr>carrier := J\<rparr>) I \<subseteq> J"
init_fin_descend_thr_eq_None: fixes ts :: "'e \<Rightarrow> (('f \<times> 'c) \<times> 'd \<Rightarrow>f nat) option"   and t :: "'e" shows "(init_fin_descend_thr ts t = None) = (ts t = None)"
sample_env_def: fixes p :: "'p pmf"   and f :: "'p \<Rightarrow> ('c, 'm) envT" shows "local.sample_env p f = EnvT (\<lambda>r. sample p (\<lambda>x. run_env (f x) r))"
obj_class_CInst: fixes C :: "qtname"   and vs :: "vn \<Rightarrow> val option" shows "obj_class \<lparr>tag = CInst C, values = vs\<rparr> = C"
coclop_Fix_range: fixes f :: "'b \<Rightarrow> 'b" assumes "coclop f" shows "Fix f = range f"
snd_image_mset_times_mset: fixes A :: "'b multiset"   and B :: "'a multiset" shows "image_mset snd (A \<times># B) = (if A = {#} then {#} else repeat_mset (size A) B)"
assms_1: shows "(A, B) \<in> ns_mul_ext ns s"
RHS3: shows "Cring_Poly.compose R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) q) r = to_polynomial R a \<otimes>\<^bsub>P\<^esub> Cring_Poly.compose R (q [^]\<^bsub>P\<^esub> n) r"
simps_1: fixes p :: "'c node ref option" shows "os_list [] p = \<up> (p = None)"
rep_not_zero: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and design_rep_number :: "int" assumes "constant_rep_design point_set block_collection design_rep_number" shows "design_rep_number \<noteq> 0"
select_convs_1: fixes worlds :: "'b set"   and relations :: "'c \<Rightarrow> ('b \<times> 'b) set"   and valuation :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and more :: "'e" shows "worlds \<lparr>worlds = worlds, relations = relations, valuation = valuation, \<dots> = more\<rparr> = worlds"
eq_scons: fixes w :: "'b stream"   and a :: "'b"   and v :: "'b stream" shows "(w = a ## v) = (a = shd w \<and> v = stl w)"
basis_row_space_subset_row_space: fixes A :: "(('a, 'b) vec, 'c) vec" shows "basis_row_space A \<subseteq> row_space A"
arr_cod_iff_arr: fixes f :: "'a \<times> 'a" shows "CC.arr (CC.cod f) = CC.arr f"
condCatch_def: fixes c\<^sub>1 :: "('d, 'e, 'f) com"   and b :: "'d set"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "condCatch c\<^sub>1 b c\<^sub>2 = Catch c\<^sub>1 (Cond b c\<^sub>2 Throw)"
z_def: shows "z \<equiv> getFr [xx, yy, zz] [t1, t2] []"
transI: fixes r :: "('a \<times> 'a) set" assumes "\<And>x y z. (x, y) \<in> r \<and> (y, z) \<in> r \<Longrightarrow> (x, z) \<in> r" shows "trans r"
fa_closed: shows "f \<bullet> a \<in> carrier Zp"
and_P2_security: fixes n :: "nat"   and m1 :: "bool \<times> bool"   and m2 :: "bool \<times> bool" shows "and_secret_sharing.perfect_sec_P2 n m1 m2"
ActualForallE: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" assumes "[\<^bold>\<A>(\<^bold>\<forall>x. \<phi> x) in v]" shows "[\<^bold>\<forall>x. \<^bold>\<A>\<phi> x in v]"
cyclic_order_is_ord: fixes g :: "'a" assumes "g \<in> carrier G" shows "ord g = order (subgroup_generated G {g})"
right_neutral: fixes a :: "('a \<times> 'a set) set" shows "a \<parallel> 1\<^sub>\<pi> = a"
IH: fixes x :: "'a" assumes "x \<in> X"   and "X - {x} \<subseteq> carrier"   and "\<not> indep (X - {x})" shows "\<exists>C. circuit C \<and> C \<subseteq> X - {x}"
InvariantCllCharacterizationAfterApplyLearn: fixes state :: "State" assumes "InvariantUniq (getM state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantUniqC (getC state)"   and "getConflictFlag state" shows "let state' = applyLearn state in InvariantCllCharacterization (getCl state') (getCll state') (getC state') (getM state')"
assms_1: shows "(\<box>(\<lambda>s. enabled' s \<longrightarrow> enabled s)) \<sigma>"
assms_1: shows "F \<in> A \<longmapsto> B"
cp\<^sub>P\<^sub>R\<^sub>E: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "PRE self a1 \<tau> = PRE (\<lambda>_. self \<tau>) (\<lambda>_. a1 \<tau>) \<tau>"
setroots: shows "set (characterize_root_list_p p) = {x. poly p x = 0}"
cong: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "L F f = L F g"
vec_to_iarray_greatest_not_zero: fixes A :: "('e, 'd) vec" assumes "\<exists>b. A $ b \<noteq> (0::'e)" shows "greatest_not_zero (vec_to_iarray A) = mod_type_class.to_nat (GREATEST b. A $ b \<noteq> (0::'e))"
unwinding_theorem_BSI: fixes ur :: "('s \<times> 's) set" assumes "modified_view.lrb ur"   and "modified_view.osc ur" shows "BSI \<V>' Tr\<^bsub>(induceES SES)\<^esub>"
absorb2: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a" shows "a \<sqinter> b = b"
length_quicksort_accp: fixes ac :: "'a list"   and xs :: "'a list" assumes "quicksort_acc_quicksort_part_dom (Inl (ac, xs))" shows "length (quicksort_acc ac xs) = length ac + length xs"
assms_2: shows "\<forall>F x z. x \<notin> dom F \<longrightarrow> P F \<longrightarrow> P (F(x \<mapsto> z))"
that: fixes ns :: "'b" assumes "s = NodeS i ns R"   and "(\<sigma>, ns) \<in> init (onp i)" shows "thesis"
star_star_plus: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<star> + y\<^sup>\<star> = x\<^sup>+ + y\<^sup>\<star>"
vlrestriction_vsubset_vrange: fixes r :: "V"   and A :: "V" assumes "vbrelation r"   and "\<D>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> A" shows "r \<restriction>\<^sup>l\<^sub>\<circ> A = r"
E_wf_2: fixes G :: "'b graph" assumes "wf_graph G" shows "snd ` edges G \<subseteq> nodes G"
meets_irrefl: fixes p :: "'a" assumes "\<I> p" shows "\<not> p \<parallel> p"
ide_leg0: shows "ide (tab\<^sub>0 (r \<star> s))"
boffa_var: fixes x :: "'a" assumes "transitive x" shows "x\<^sup>\<star> = 1' + x"
infinite_chain_relation_measure: fixes P :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat \<Rightarrow> bool"   and R :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat \<Rightarrow> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat \<Rightarrow> bool"   and m :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat \<Rightarrow> 'b"   and mR :: "('b \<times> 'b) set"   and k :: "nat" assumes "\<And>St St'. \<lbrakk>P St; R St St'\<rbrakk> \<Longrightarrow> (m St', m St) \<in> mR"   and "chain R (ldrop (enat k) Sts)"   and "llength Sts = \<infinity>"   and "\<And>i. P (lnth (ldrop (enat k) Sts) i)" shows "chain (\<lambda>x y. (x, y) \<in> mR)\<inverse>\<inverse> (lmap m (ldrop (enat k) Sts))"
whileAnnoG_def: fixes gs :: "('f \<times> 'd set) list"   and b :: "'d set"   and I :: "'d set"   and V :: "('d \<times> 'd) set"   and c :: "('d, 'e, 'f) com" shows "whileAnnoG gs b I V c = while gs b c"
simps: fixes \<sigma> :: "nat \<Rightarrow> linear_poly"   and rel :: "le_rel"   and p :: "linear_poly"   and c :: "'a" shows "subst_poly_lec \<sigma> (Le_Constraint rel p c) = Le_Constraint rel (subst_poly \<sigma> p) c"
simps_4: fixes uu :: "'a" shows "Val uu + Top = Top"
simply_connected_eq_homeomorphic_to_disc: fixes S :: "complex set" assumes "open S" shows "simply_connected S = (S = {} \<or> S homeomorphic ball 0 1)"
omega_sum_less: fixes ms :: "nat list" shows "omega_sum ms < \<omega> \<up> ord_of_nat (length ms)"
ab_semigroup_mult_ow: shows "class.ab_semigroup_mult = ab_semigroup_mult_ow UNIV"
sum_multiset_singleton: fixes A :: "'c set" shows "(\<Sum>n\<in>A. {#n#}) = mset_set A"
fmset_listassoc_cong: fixes as :: "'a list"   and bs :: "'a list" assumes "as [\<sim>] bs"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "fmset G as = fmset G bs"
eval_fds_altdef: fixes f :: "'b fds"   and s :: "'b" assumes "fds_abs_converges f s" shows "eval_fds f s = (\<Sum>\<^sub>an. fds_nth f n / nat_power n s)"
collapse: fixes inference :: "'b Inference_System.inference" shows "old_Infer (old_side_prems_of inference) (old_main_prem_of inference) (old_concl_of inference) = inference"
simps_1: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.list_comb app f [] = f"
personal_quorums_axioms: shows "personal_quorums quorum_of"
is_top_sorted_isolated_constraint: fixes R' :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set"   and x :: "'a"   and X :: "'a set"   and l :: "'a list" assumes "R' \<subseteq> R \<union> {x} \<times> X"   and "R' \<inter> UNIV \<times> {x} = {}"   and "x \<notin> set l"   and "is_top_sorted R l" shows "is_top_sorted R' l"
a: shows "(\<lambda>n. dist (rep_metric_completion x n) (rep_metric_completion y n)) \<longlonglongrightarrow> dist x y"
Rep_triangle: fixes x :: "triangle" shows "Rep_triangle x \<in> triangle"
sum_list_subst_cls_list_subst_cls: fixes Cs :: "'a literal multiset list"   and \<eta> :: "'s" shows "sum_list (Cs \<cdot>cl \<eta>) = sum_list Cs \<cdot> \<eta>"
prems_1: fixes x :: "Vars.var"   and ea :: "exp"   and aa :: "'a" assumes "(x, ea) \<in> set as" shows "eqvt_at transform_sumC (aa, ea)"
x_y_def: shows "x_y \<equiv> [:[:0::'b, 1::'b:], - 1:]"
dual\<^sub>s\<^sub>t_trms_eq: fixes S :: "('a, 'b) strand_step list" shows "trms\<^sub>s\<^sub>t (dual\<^sub>s\<^sub>t S) = trms\<^sub>s\<^sub>t S"
msb_bin_sc: fixes n :: "nat"   and b :: "bool"   and x :: "int" shows "msb (bin_sc n b x) = msb x"
assms_4: shows "limsup (\<lambda>n. \<integral>\<^sup>+ x. ennreal (norm (F n x)) \<partial>M) \<le> \<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M"
bicategory_axioms: shows "bicategory (\<cdot>) (\<star>\<^sub>B) \<a>\<^sub>B \<i> trg src"
remove_cycles_ends_cycle: fixes xs :: "'b list"   and x :: "'b"   and ys :: "'b list" assumes "remove_cycles xs x ys \<noteq> rev ys @ xs" shows "x \<in> set xs"
simps_11: fixes jmps :: "jump set"   and C :: "qtname" shows "jumpNestingOkS jmps (Init C) = True"
order_power_n_n: fixes a :: "'a"   and n :: "nat" shows "Polynomial.order a ([:- a, 1::'a:] ^ n) = n"
iwlsFSw_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme" shows "iwlsFSw MOD \<equiv> igWlsAllDisj MOD \<and> igWlsAbsIsInBar MOD \<and> igConsIPresIGWls MOD \<and> igSwapAllIPresIGWlsAll MOD \<and> igFreshCls MOD \<and> igSwapCls MOD \<and> igAbsCongS MOD"
rng_to_rng_of_frac_def: fixes r :: "'a" shows "rng_to_rng_of_frac r \<equiv> r |\<^bsub>rel\<^esub> \<one>"
invertible_commutes_with_inverse: fixes \<eta> :: "'c \<Rightarrow> 'd"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and M' :: "'d set"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d"   and a :: "'c" assumes "monoid_homomorphism \<eta> M composition unit M' composition' unit'"   and "monoid.invertible M composition unit a"   and "a \<in> M" shows "\<eta> (monoid.inverse M composition unit a) = monoid.inverse M' composition' unit' (\<eta> a)"
reducedly_dynamically_complete: fixes B :: "'a"   and Ns :: "'a set llist" assumes "B \<in> Bot"   and "chain (\<rhd>) Ns"   and "reduc_fair Ns"   and "lhd Ns \<Turnstile> {B}" shows "\<exists>i\<in>{i. enat i < llength Ns}. \<exists>B'\<in>Bot. B' \<in> lnth Ns i"
wf_matchF_aux_def: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex"   and aux :: "(nat \<times> event_data option list set list \<times> event_data option list set) list"   and ne :: "nat"   and k :: "nat" shows "wf_matchF_aux \<sigma> V n R I r aux ne k = (case to_mregex r of (mr, \<phi>s) \<Rightarrow> list_all2 (\<lambda>x i. case x of (t, rels, rel) \<Rightarrow> t = \<tau> \<sigma> i \<and> list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) \<phi>s rels \<and> qtable n (fv_regex r) (mem_restr R) (\<lambda>v. \<exists>j\<ge>i. j < ne + length aux + k \<and> mem (\<tau> \<sigma> j - \<tau> \<sigma> i) I \<and> Regex.match (Formula.sat \<sigma> V (map the v)) r i j) rel) aux [ne..<ne + length aux])"
sel_2: fixes z :: "quat" shows "Im1 (Quaternions.cnj z) = - Im1 z"
weak_equality_spec_def: fixes p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset" shows "weak_equality_spec p q = SPEC (\<lambda>r. r \<longrightarrow> p = q)"
uniform_discrete_imp_closed: fixes S :: "'c set" assumes "uniform_discrete S" shows "closed S"
IH: assumes "boundary_chain (set l)"   and "valid_chain_list l"   and "\<And>k \<gamma>. (k, \<gamma>) \<in> set l \<Longrightarrow> valid_path \<gamma>"   and "line_integral_exists F basis (rec_join l)"   and "finite basis" shows "\<forall>a\<in>set l. case a of (k, a) \<Rightarrow> line_integral_exists F basis a"
separatingD: fixes G :: "('k, 'l) graph_scheme"   and A :: "'k set"   and B :: "'k set"   and S :: "'k set"   and x :: "'k"   and p :: "'k list"   and y :: "'k" assumes "separating_gen G A B S"   and "path G x p y"   and "x \<in> A"   and "y \<in> B" shows "(\<exists>z\<in>set p. z \<in> S) \<or> x \<in> S"
assms: shows "f \<in> borel_measurable (lborel_f (Suc n))"
alphabet_swapFourSyms: fixes a :: "'a"   and t :: "'a tree"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<in> alphabet t"   and "b \<in> alphabet t"   and "c \<in> alphabet t"   and "d \<in> alphabet t" shows "alphabet (swapFourSyms t a b c d) = alphabet t"
simps_11: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x2 :: "'c"   and y2 :: "'d" shows "rel_option R (Some x2) (Some y2) = R x2 y2"
axioms_2: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog" assumes "l_get_host_wf heap_is_wellformed known_ptr known_ptrs type_wf get_host" shows "l_get_host_wf_axioms heap_is_wellformed known_ptrs type_wf get_host"
WeakI2: fixes Gamma :: "formula list"   and A :: "(formula list \<times> formula list set) set"   and Delta :: "formula list" assumes "Gamma \<in> deductions A"   and "Perms \<subseteq> A"   and "Weaks \<subseteq> A" shows "Gamma @ Delta \<in> deductions A"
zero_less_abs_iff: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less zero (abs a) = (a \<noteq> zero)"
not_eq_extremum: fixes a :: "'e multiset" shows "(a \<noteq> {#}) = ({#} \<subset># a)"
prems: fixes x :: "'a" assumes "x \<in> set (x__ # xs)" shows "r2 (f y) (f x) = r1 y x"
OclExcluding_valid_args_valid'': fixes X :: "'d state \<times> 'd state \<Rightarrow> Bag('e)"   and x :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "\<upsilon> X->excluding\<^sub>B\<^sub>a\<^sub>g(x) = (\<delta> X and \<upsilon> x)"
conjr2_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<rhd> z = (x \<rhd> z) \<squnion> (y \<rhd> z)"
PDG_scd: shows "PDG sourcenode targetnode kind valid_edge (_Entry_) Def Use state_val (_Exit_) standard_control_dependence"
Reals_0: shows "(0::'a) \<in> \<real>"
initialiI: fixes OI :: "(('a \<Rightarrow> 'b) \<times> 'c) set"   and i :: "'a"   and CI :: "('b \<times> 'c) set" assumes "\<And>\<sigma> p. (\<sigma>, p) \<in> OI \<Longrightarrow> (\<sigma> i, p) \<in> CI"   and "\<And>\<xi> p. (\<xi>, p) \<in> CI \<Longrightarrow> \<exists>\<sigma>. \<xi> = \<sigma> i \<and> (\<sigma>, p) \<in> OI" shows "initiali i OI CI"
neg_closed: fixes U :: "'c set"   and V :: "'c set"   and x :: "'c" assumes "Subspace.subspace U V"   and "vectorspace V"   and "x \<in> U" shows "- x \<in> U"
smap_union_mono: fixes ve1 :: "'a \<Rightarrow> 'b set"   and ve1' :: "'a \<Rightarrow> 'b set"   and ve2 :: "'a \<Rightarrow> 'b set"   and ve2' :: "'a \<Rightarrow> 'b set" assumes "ve1 \<subseteq>. ve1'"   and "ve2 \<subseteq>. ve2'" shows "ve1 \<union>. ve2 \<subseteq>. ve1' \<union>. ve2'"
exact_seq_imp_group_hom: fixes H :: "'b monoid"   and G :: "'b monoid"   and l :: "'b monoid list"   and q :: "('b \<Rightarrow> 'b) list"   and g :: "'b \<Rightarrow> 'b" assumes "exact_seq (G # l, q)  \<longlongrightarrow>\<^bsub>g\<^esub> H" shows "group_hom G H g"
eq: shows "p \<cdot> u \<cdot> q = t \<^sup>@ k"
singleton_insert_inj_eq': fixes a :: "'e"   and A :: "'e set"   and b :: "'e" shows "(insert a A = {b}) = (a = b \<and> A \<subseteq> {b})"
assms_6: shows "\<exists>z\<in>A. f z \<noteq> 0"
sup0: fixes a :: "'a" shows "a \<equiv> a \<squnion> bot"
assms_2: shows "n = IArray.length (matrix_to_iarray A)"
mod_0_imp_dvd: fixes a :: "'a"   and b :: "'a" assumes "a mod b = (0::'a)" shows "b dvd a"
nc_iff2: fixes x :: "'a" shows "(x \<le> nc) = (x \<cdot> (0::'a) = (0::'a))"
not_positive_Real: fixes X :: "nat \<Rightarrow> rat" assumes "Real.cauchy X" shows "(\<not> Real.positive (Real.Real X)) = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> r)"
incseq_def: fixes X :: "nat \<Rightarrow> 'c" shows "incseq X = (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n)"
tail_ord_p: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "tail p \<prec>\<^sub>p p"
FDOM_pre_subset_FDOM_valid_states_pair: fixes a :: "('a, 'b) fmap \<times> ('a, 'c) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set"   and s :: "('a, 'd) fmap" assumes "a \<in> PROB"   and "s \<in> valid_states PROB" shows "fmdom' (fst a) \<subseteq> fmdom' s"
inject_14: fixes x151 :: "floatarith"   and x152 :: "nat"   and y151 :: "floatarith"   and y152 :: "nat" shows "(x151 ^\<^sub>e x152 = y151 ^\<^sub>e y152) = (x151 = y151 \<and> x152 = y152)"
assms_2: fixes x :: "'a"   and h :: "'a" assumes "x \<in> carrier G"   and "h \<in> N" shows "x \<otimes> h \<otimes> inv x \<in> N"
0: fixes i :: "nat" assumes "i < m" shows "B4 $$ (i, m) = (0::'a)"
graph_union_semantics: fixes A :: "('b, 'a) labeled_graph"   and e :: "'b allegorical_term"   and B :: "('b, 'a) labeled_graph" shows "(:A:\<lbrakk>e\<rbrakk>) \<union> (:B:\<lbrakk>e\<rbrakk>) \<subseteq> :graph_union A B:\<lbrakk>e\<rbrakk>"
llist_all2_lconcatI: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xss :: "'c llist llist"   and yss :: "'d llist llist" assumes "llist_all2 (llist_all2 A) xss yss" shows "llist_all2 A (lconcat xss) (lconcat yss)"
C_eq_until_separatedQ: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and l :: "(int \<times> int \<times> protocol) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "Cp (list2FWpolicy (separate (qsort (removeShadowRules2 (remdups (rm_MT_rules Cp (insertDeny (removeShadowRules1 (policy2list p)))))) l))) = Cp p"
nofail_def: fixes S :: "'e nres" shows "nofail S \<equiv> S \<noteq> FAIL"
mult_rat_cancel: fixes c :: "int"   and a :: "int"   and b :: "int" assumes "c \<noteq> 0" shows "Fract (c * a) (c * b) = Fract a b"
eq_cnj_iff_real: fixes z :: "complex" shows "(cnj z = z) = is_real z"
Gauss_Jordan_upt_k_PA_def: fixes A :: "(('b, 'd) vec, 'c) vec"   and k :: "nat" shows "Gauss_Jordan_upt_k_PA A k = (let foldl = foldl Gauss_Jordan_column_k_PA (mat (1::'b), 0, A) [0..<Suc k] in (fst foldl, snd (snd foldl)))"
Suc_innermost_id1: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and j' :: "nat"   and i' :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "j' \<le> n"   and "i' < i" shows "fw m n (Suc k) i' j' i j = fw m n k i j i j"
is_extensional: fixes f :: "'a1 \<times> 'a2" assumes "\<not> A1xA2.arr f" shows "\<tau> f = B.null"
carrier_vecI: fixes v :: "'b vec"   and n :: "nat" assumes "dim_vec v = n" shows "v \<in> carrier_vec n"
simps_6: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlr"   and \<psi> :: "'b ltlr" shows "\<xi> \<Turnstile>\<^sub>r \<phi> or\<^sub>r \<psi> = (\<xi> \<Turnstile>\<^sub>r \<phi> \<or> \<xi> \<Turnstile>\<^sub>r \<psi>)"