hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "char list \<Rightarrow> secLevel option"   and a2 :: "expr"   and a3 :: "secLevel" assumes "a1 \<turnstile> a2 : a3" obtains \<Gamma> :: "char list \<Rightarrow> secLevel option" and V :: "val" and lev :: "secLevel" where "a1 = \<Gamma>__" and "a2 = Val V" and "a3 = lev__"   | \<Gamma> :: "char list \<Rightarrow> secLevel option" and Vn :: "char list" and lev :: "secLevel" where "a1 = \<Gamma>'__" and "a2 = Var Vn" and "a3 = leva__" and "\<Gamma>'__ Vn = Some leva__"   | \<Gamma> :: "char list \<Rightarrow> secLevel option" and e1 :: "expr" and e2 :: "expr" and bop :: "bop" where "a1 = \<Gamma>''__" and "a2 = e1__ \<guillemotleft>bop__\<guillemotright> e2__" and "a3 = Low" and "\<Gamma>''__ \<turnstile> e1__ : Low" and "\<Gamma>''__ \<turnstile> e2__ : Low"   | \<Gamma> :: "char list \<Rightarrow> secLevel option" and e1 :: "expr" and e2 :: "expr" and lev :: "secLevel" and bop :: "bop" where "a1 = \<Gamma>'''__" and "a2 = e1a__ \<guillemotleft>bopa__\<guillemotright> e2a__" and "a3 = High" and "\<Gamma>'''__ \<turnstile> e1a__ : High" and "\<Gamma>'''__ \<turnstile> e2a__ : levb__"   | \<Gamma> :: "char list \<Rightarrow> secLevel option" and e1 :: "expr" and lev :: "secLevel" and e2 :: "expr" and bop :: "bop" where "a1 = \<Gamma>" and "a2 = e1 \<guillemotleft>bop\<guillemotright> e2" and "a3 = High" and "\<Gamma> \<turnstile> e1 : lev" and "\<Gamma> \<turnstile> e2 : High"
maps_over_def: fixes A :: "'c set"   and B :: "'d set" shows "maps_over A B = {m. dom m \<subseteq> A \<and> ran m \<subseteq> B}"
join_rtrancl_join: fixes a :: "'b"   and b :: "'b"   and A :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> A\<^sup>\<down>"   and "(c, b) \<in> A\<^sup>*" shows "(a, c) \<in> A\<^sup>\<down>"
min_mult_distrib_right: fixes x :: "'a"   and y :: "'a"   and p :: "'a" shows "min x y * p = (if (0::'a) \<le> p then min (x * p) (y * p) else max (x * p) (y * p))"
cases: fixes a1 :: "'b com option list"   and a2 :: "'b"   and a3 :: "'b com option list"   and a4 :: "'b" assumes "(a1, a2) -pe\<rightarrow> (a3, a4)" obtains Ps :: "'b com option list" and s :: "'b" and t :: "'b" where "a1 = Ps" and "a2 = s" and "a3 = Ps" and "a4 = t"
hyps: fixes j :: "nat" shows "(ord_of i = ord_of j) = (i = j)"
return_valid: fixes p :: "'p" shows "proc_of fg (return fg p) = p"
oldest_token_equal: fixes q :: "'b"   and n :: "nat"   and i :: "nat"   and q' :: "'b" assumes "oldest_token q n = Some i"   and "oldest_token q' n = Some i" shows "q = q'"
A: shows "f = rec_nat s (\<lambda>i t. let e = SOME e. e \<in> ev ` A \<and> t @ [e] \<in> T P in t @ [e])"
assms_2: shows "[M]\<^bsub>v,n\<^esub> = {}"
set_zip_rightD: fixes x :: "'b"   and y :: "'c"   and xs :: "'b list"   and ys :: "'c list" assumes "(x, y) \<in> set (zip xs ys)" shows "y \<in> set ys"
unifiers_Pairs: fixes AAA :: "('a, 'b) Term.term set set" assumes "finite AAA"   and "\<forall>AA\<in>AAA. finite AA" shows "unifiers (set (Pairs AAA)) = {\<sigma>. is_unifiers \<sigma> AAA}"
declassified\<^sub>l\<^sub>s\<^sub>t_def: fixes \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and \<I> :: "'var \<Rightarrow> ('fun, 'a) Term.term" shows "declassified\<^sub>l\<^sub>s\<^sub>t \<A> \<I> \<equiv> {t. (\<star>, receive\<langle>t\<rangle>\<^sub>s\<^sub>t) \<in> set \<A>} \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>"
shows_between_def: fixes l :: "char list \<Rightarrow> char list"   and p :: "char list \<Rightarrow> char list"   and r :: "char list \<Rightarrow> char list" shows "shows_between l p r = l \<circ> (p \<circ> r)"
substT_type: fixes \<Delta> :: "binding list"   and Q :: "POPLmark.type"   and \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmark.type"   and P :: "POPLmark.type" assumes "\<Delta> @ TVarB Q \<Colon> \<Gamma> \<turnstile> t : T"   and "\<Gamma> \<turnstile> P <: Q" shows "\<Delta>[0 \<mapsto>\<^sub>\<tau> P]\<^sub>e @ \<Gamma> \<turnstile> t[\<parallel>\<Delta>\<parallel> \<mapsto>\<^sub>\<tau> P] : T[\<parallel>\<Delta>\<parallel> \<mapsto>\<^sub>\<tau> P]\<^sub>\<tau>"
intros: fixes t :: "dag"   and nodes :: "ref set"   and p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref" assumes "set_of t \<subseteq> nodes"   and "Dag p low high t"   and "t \<noteq> Tip" shows "Dagsp nodes low high t"
hyps_2: shows "[a] @- v\<^sub>2 =\<^sub>I l @- u"
val_simps_3: fixes x :: "'b" shows "Time_Monad.val (tick x) = x"
stepsI: fixes x :: "'b"   and xs :: "'b list" assumes "B x (hd xs)"   and "B.steps xs" shows "B.steps (x # xs)"
resr_eq: fixes x :: "'a" shows "residual ((\<cdot>) x) = inv_resr x"
def_X': fixes x :: "'b" assumes "x \<in> \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X \<tau>)\<rceil>\<rceil>" shows "x \<noteq> null"
lunit_coherence2: fixes a :: "'c" assumes "C.ide a" shows "F (C'.lunit a) \<cdot>\<^sub>D \<phi> (snd (C'.unity, a), fst (C'.unity, a)) = monoidal_category.lunit (\<cdot>\<^sub>D) D'.T D'.\<alpha> F'.\<iota>\<^sub>1 (F a)"
rep_eq: shows "Rep_lifted_pair top = (\<lambda>f. (top, Filters.filter.Abs_filter {top}))"
ffb_loopI: fixes P :: "'a set"   and I :: "'a \<Rightarrow> bool"   and Q :: "'a set"   and F :: "'a \<Rightarrow> 'a set" assumes "P \<subseteq> {s. I s}"   and "{s. I s} \<subseteq> Q"   and "{s. I s} \<subseteq> fb\<^sub>\<F> F {s. I s}" shows "P \<subseteq> fb\<^sub>\<F> (LOOP F INV I ) Q"
prems_4: shows "\<forall>e\<in>pred c1. finite (Bexp.vars e)"
measurable_pair_iff: fixes f :: "'c \<Rightarrow> 'd \<times> 'e"   and M :: "'c measure"   and M1 :: "'d measure"   and M2 :: "'e measure" shows "(f \<in> M \<rightarrow>\<^sub>M M1 \<Otimes>\<^sub>M M2) = (fst \<circ> f \<in> M \<rightarrow>\<^sub>M M1 \<and> snd \<circ> f \<in> M \<rightarrow>\<^sub>M M2)"
pseudo_proj_True_stake_image: fixes n :: "nat"   and w :: "bool stream"   and x :: "bool stream" assumes "stake n w = stake n x" shows "pseudo_proj_True n w = pseudo_proj_True n x"
Ri_indep: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set"   and N' :: "'b literal multiset set"   and N :: "'b literal multiset set" assumes "redundancy_criterion \<Gamma> Rf Ri"   and "N' \<subseteq> Rf N" shows "Ri N \<subseteq> Ri (N - N')"
combine_rbt_greater: fixes k :: "'a"   and l :: "('a, 'c) rbt"   and r :: "('a, 'c) rbt" assumes "k \<guillemotleft>| l"   and "k \<guillemotleft>| r" shows "k \<guillemotleft>| combine l r"
sheaf_spec_is_presheaf: shows "presheaf_of_rings cring0.spectrum cring0.is_zariski_open cring0.sheaf_spec cring0.sheaf_spec_morphisms cring0.\<O>b cring0.add_sheaf_spec cring0.mult_sheaf_spec cring0.zero_sheaf_spec cring0.one_sheaf_spec"
interleavings_length: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "interleavings xs ys zs" shows "length xs + length ys = length zs"
asc: fixes as :: "'a list"   and a :: "'a"   and bs :: "'a list"   and b :: "'a" assumes "wfactors G as a"   and "wfactors G bs b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "a \<sim> b = essentially_equal G as bs"
matrix_to_iarray_mat2matofpoly: fixes A :: "(('c, 'd) vec, 'e) vec" shows "matrix_to_iarray (mat2matofpoly A) = mat2matofpoly_iarrays (matrix_to_iarray A)"
fresh_star_Un: fixes as :: "atom set"   and bs :: "atom set"   and x :: "'d" shows "(as \<union> bs) \<sharp>* x = (as \<sharp>* x \<and> bs \<sharp>* x)"
push_precond_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat" shows "push_precond f l \<equiv> \<lambda>(u, v). (0::'capacity) < excess f u \<and> (u, v) \<in> cfE_of f \<and> l u = l v + 1"
not_Cons_self2: fixes x :: "'g"   and xs :: "'g list" shows "x # xs \<noteq> xs"
trancl_subseteq: fixes R :: "('d \<times> 'd) set"   and Q :: "('d \<times> 'd) set"   and S :: "'d \<times> 'd" assumes "R \<subseteq> Q"   and "S \<in> R\<^sup>*" shows "S \<in> Q\<^sup>*"
set2_F_map: fixes l1 :: "'n \<Rightarrow> 'y"   and l2 :: "'o \<Rightarrow> 'x"   and l3 :: "'p \<Rightarrow> 'z" shows "set2_F \<circ> mapl_F l1 l2 l3 = (`) l2 \<circ> set2_F"
emeasure_lim_emb: fixes J :: "'i set"   and X :: "('i \<Rightarrow> 'a) set" assumes "finite J"   and "J \<subseteq> I"   and "X \<in> sets (Pi\<^sub>M J M)" shows "emeasure local.lim (emb I J X) = emeasure (Pi\<^sub>M J M) X"
less_4: fixes xs :: "'a list" assumes "ls @ xs = ls'"   and "xs \<noteq> []" shows "thesis"
vector_space_on_axioms: shows "vector_space_on U\<^sub>M\<^sub>_\<^sub>2 (*s\<^sub>2)"
relcomp_distrib: fixes R :: "('d \<times> 'f) set"   and S :: "('f \<times> 'e) set"   and T :: "('f \<times> 'e) set" shows "R O (S \<union> T) = R O S \<union> R O T"
list_swap_def: fixes xs :: "'b list"   and i :: "nat"   and j :: "nat" shows "list_swap xs i j = xs[i := xs ! j, j := xs ! i]"
assms_1: shows "a \<in> carrier (DirProds G I)"
bet_eq2: shows "between (vertices f) v u = between (verticesFrom f v') v u"
extremum_unique: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and a :: "'b" assumes "ordering_top less_eq less top" shows "less_eq top a = (a = top)"
corrKey_Dom_keySetup: fixes K :: "key" assumes "K \<in> corrKey" shows "K \<in> Domain keySetup"
sorted_sublist_wrt_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and lo :: "nat"   and hi :: "nat" shows "sorted_sublist_wrt R xs lo hi = sorted_wrt R (WB_Sort.sublist xs lo hi)"
set_of_appr_project: fixes xrs :: "real list"   and xs :: "'b list"   and "is" :: "nat list" assumes "xrs \<in> set_of_appr xs"   and "\<And>i. i \<in> set is \<Longrightarrow> i < length xs" shows "map ((!) xrs) is \<in> set_of_appr (map ((!) xs) is)"
coeff_0_reflect_poly_0_iff: fixes p :: "'c poly" shows "(coeff (reflect_poly p) 0 = (0::'c)) = (p = 0)"
fract_poly_eq_iff: fixes p :: "'b poly"   and q :: "'b poly" shows "(map_poly to_fract p = map_poly to_fract q) = (p = q)"
cases: fixes s :: "'b + 'c"   and a :: "'b" assumes "setlp s a" obtains x :: "'b" where "a = x" and "s = Inl x"
int_pow_pow: fixes x :: "'a"   and n :: "int"   and m :: "int" assumes "x \<in> carrier \<G>" shows "(x [^] n) [^] m = x [^] (n * m)"
sq_mtx_inv_idempotent: fixes A :: "'a sq_mtx" assumes "mtx_invertible A" shows "A\<^sup>-\<^sup>1\<^sup>-\<^sup>1 = A"
is_nextElem_between_empty': fixes vs :: "'a list"   and a :: "'a"   and b :: "'a" assumes "between vs a b = []"   and "distinct vs"   and "a \<in> set vs"   and "b \<in> set vs"   and "a \<noteq> b" shows "is_nextElem vs a b"
set_2: fixes x21 :: "'x"   and x22 :: "'x list" shows "set (x21 # x22) = insert x21 (set x22)"
disjoint_mono: fixes a :: "'a set"   and a' :: "'a set"   and b :: "'a set"   and b' :: "'a set" assumes "a \<subseteq> a'"   and "b \<subseteq> b'"   and "a' \<inter> b' = {}" shows "a \<inter> b = {}"
sum_vector_def: fixes \<alpha> :: "(nat \<times> int) list \<Rightarrow> real"   and \<beta> :: "(nat \<times> int) list \<Rightarrow> real" shows "sum_vector \<alpha> \<beta> \<equiv> \<lambda>p. \<alpha> p + \<beta> p"
stake_run: fixes execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'a stream"   and p :: "'b" assumes "\<And>k. transition_system.path execute enabled (stake k r) p" shows "transition_system.run execute enabled r p"
simps_1: fixes x :: "('b, 'c) Combinators"   and y :: "('b, 'c) Combinators" shows "first_destNet (x \<oplus> y) = first_destNet x"
E27: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<diamond>\<box>\<diamond>F = \<box>\<diamond>F"
is_Prim0_Trg: fixes t :: "'a local.term" shows "is_Prim\<^sub>0 (Trg t)"
l_remove_child_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog" shows "l_remove_child type_wf known_ptr known_ptrs remove_child remove_child_locs get_owner_document get_child_nodes get_disconnected_nodes \<equiv> l_known_ptrs known_ptr known_ptrs \<and> l_remove_child_axioms type_wf known_ptrs remove_child remove_child_locs get_owner_document get_child_nodes get_disconnected_nodes"
stack_xlift_compxE2: fixes n :: "nat"   and e :: "'a expr1"   and pc :: "nat"   and d :: "nat" shows "stack_xlift n (compxE2 e pc d) = compxE2 e pc (n + d)"
all_zero: shows "\<forall>m\<ge>(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows). A $ m $ mod_type_class.from_nat k = (0::'a)"
add_right_cancel: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.cancel_semigroup_add plus" shows "(plus b a = plus c a) = (b = c)"
simps_4: fixes h1 :: "'b heap"   and h2 :: "'b heap" shows "U Merge [h1, h2] = log 2 (real (size_hp h1 + size_hp h2 + 1)) + 2"
remove_comp_trading_strat: fixes pf1 :: "'d \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'d" assumes "trading_strategy pf1" shows "trading_strategy (qty_rem_comp pf1 x)"
Abs_rename_lst: fixes p :: "perm"   and bs :: "atom list"   and x :: "'a" assumes "(p \<bullet> set bs) \<sharp>* x" shows "\<exists>q. [bs]lst. x = [p \<bullet> bs]lst. q \<bullet> x \<and> q \<bullet> bs = p \<bullet> bs"
tla_defs_2: fixes f :: "'a \<Rightarrow> 'b" shows "$f \<equiv> %s. f (first s)"
cod_closed: fixes f :: "'a" assumes "left b f" shows "left b (cod f)"
prefix_Par_Int_skip2: fixes a :: "'a"   and A :: "'a set"   and P :: "'a process" assumes "a \<in> A" shows "(a \<rightarrow> P \<lbrakk>A\<rbrakk> SKIP) = STOP"
intro: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> carrier; y \<in> carrier\<rbrakk> \<Longrightarrow> le x y \<or> le y x" shows "total_preorder_on_axioms carrier le"
le_less_linear: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.linorder less_eq less" shows "less_eq x y \<or> less y x"
is_weakly_scdom_component_safe_def: shows "is_weakly_scdom_component_safe \<equiv> l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_weakly_scdom_component_safe get_owner_document get_disconnected_nodes to_tree_order"
iso_tuple_update_accessor_eq_assist_idI: fixes v' :: "'a"   and f :: "'a \<Rightarrow> 'a"   and v :: "'a" assumes "v' = f v" shows "Record.iso_tuple_update_accessor_eq_assist Fun.id Fun.id v f v' v"
M_def: shows "M = C * (\<Sum>a\<in>basis. \<parallel>f a\<parallel>)"
wellformed: fixes b :: "('a \<times> 'b) set" assumes "b \<in># f bl" shows "b \<subseteq> bl \<times> I"
memoization_correct_1: fixes lookup :: "'a \<Rightarrow> ('b, 'c option) state"   and update :: "'a \<Rightarrow> 'c \<Rightarrow> ('b, unit) state"   and P :: "'b \<Rightarrow> bool"   and empty :: "'b"   and dp :: "'a \<Rightarrow> 'c"   and dp\<^sub>T :: "'a \<Rightarrow> ('b, 'c) state"   and x :: "'a"   and v :: "'c"   and m :: "'b" assumes "dp_consistency_empty lookup update P empty"   and "dp_consistency.consistentDP lookup P dp dp\<^sub>T"   and "run_state (dp\<^sub>T x) empty = (v, m)" shows "dp x = v"
main: fixes pa :: "'a fract poly"   and qa :: "'a fract poly" assumes "content_ff pa =dff 1"   and "content_ff qa =dff 1" shows "content_ff (pa * qa) =dff 1"
sq_norm_poly_def: fixes p :: "'d poly" shows "\<parallel>p\<parallel>\<^sup>2 \<equiv> sum_list (map sq_norm (coeffs p))"
MP_Rel_def: fixes m :: "int"   and f :: "int poly"   and f' :: "'a mod_ring poly" assumes "poly_mod_type TYPE('a) m" shows "poly_mod_type.MP_Rel m f f' \<equiv> poly_mod.Mp m f = to_int_poly f'"
INJ: shows "inj_on g B \<and> g ` B \<subseteq> A"
rcis_zero_arg: fixes r :: "real" shows "rcis r 0 = cor r"
equalityI: fixes A :: "'h set"   and B :: "'h set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
elims_1: fixes x :: "('b \<times> nat) option"   and xa :: "'b"   and xb :: "lock_action"   and y :: "bool" assumes "lock_action_ok x xa xb = y" obtains l :: "('b \<times> nat) option" and t :: "'b" where "x = la__" and "xa = tb__" and "xb = Lock" and "y = may_lock la__ tb__"   | l :: "('b \<times> nat) option" and t :: "'b" where "x = lb__" and "xa = tc__" and "xb = Unlock" and "y = has_lock lb__ tc__"   | l :: "('b \<times> nat) option" and t :: "'b" where "x = lc__" and "xa = td__" and "xb = UnlockFail" and "y = (\<not> has_lock lc__ td__)"   | l :: "('b \<times> nat) option" and t :: "'b" where "x = l" and "xa = t" and "xb = ReleaseAcquire" and "y"
mem_Collect_eq: fixes a :: "'l"   and P :: "'l \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
closure_insert: fixes x :: "'a"   and s :: "'a set" shows "closure (insert x s) = insert x (closure s)"
gH_rel_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "ps_prod.gH_rel x y = (ps_prod.gL_rel x y \<and> ps_prod.gR_rel x y)"
measurable_distr_eq1: fixes Mf :: "'c measure"   and Nf :: "'a measure"   and f :: "'c \<Rightarrow> 'a"   and Mf' :: "'b measure" shows "distr Mf Nf f \<rightarrow>\<^sub>M Mf' = Nf \<rightarrow>\<^sub>M Mf'"
is_faithful: fixes f :: "'c"   and f' :: "'c" assumes "C.par f f'"   and "local.map f = local.map f'" shows "f = f'"
nb: shows "evalDdb e\<cdot>env_empty_db \<noteq> \<bottom>"
wp_comp: fixes x :: "'b"   and y :: "'b"   and p :: "'b Assertion" shows "wp (x * y) p = wp x (wp y p)"
primitive_abs_int_poly: fixes p :: "int poly" shows "primitive (abs_int_poly p) = primitive p"
is_bicategory: shows "bicategory (\<cdot>) (\<star>) \<a> \<i> src trg"
ccw'_not_coll_3: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "ccw' a b c" shows "det3 b a c \<noteq> 0"
lunionM_return_empty2: fixes x :: "'m" shows "lunionM x (return []) = x"
og_mat_size: shows "M_mat signs subsets \<in> carrier_mat (length signs) (length signs)"
\<phi>: shows "\<guillemotleft>\<l>\<^sub>D[g] \<cdot>\<^sub>D (\<sigma>.counit a' \<star>\<^sub>D g) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a', \<sigma>.map\<^sub>0 a', g] \<cdot>\<^sub>D (\<tau>\<^sub>0 a' \<star>\<^sub>D (\<sigma>.map\<^sub>0 a' \<star>\<^sub>D \<r>\<^sub>D[g]) \<cdot>\<^sub>D (\<sigma>.map\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<sigma>.counit a) \<cdot>\<^sub>D (\<sigma>.map\<^sub>0 a' \<star>\<^sub>D \<a>\<^sub>D[g, \<tau>\<^sub>0 a, \<sigma>.map\<^sub>0 a]) \<cdot>\<^sub>D \<a>\<^sub>D[\<sigma>.map\<^sub>0 a', g \<star>\<^sub>D \<tau>\<^sub>0 a, \<sigma>.map\<^sub>0 a]) : \<tau>\<^sub>0 a' \<star>\<^sub>D (\<sigma>.map\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<tau>\<^sub>0 a) \<star>\<^sub>D \<sigma>.map\<^sub>0 a \<Rightarrow>\<^sub>D g\<guillemotright>"
prod_2_list_def: fixes x :: "'f \<Rightarrow> 'b decision option"   and y :: "('g \<Rightarrow> 'c decision option) list" shows "x \<Otimes>\<^sub>2\<^sub>L y = (x \<Otimes>\<^sub>L y) (\<Otimes>\<^sub>2)"
cnf_semantics: fixes F :: "'a formula"   and \<A> :: "'a \<Rightarrow> bool" assumes "is_nnf F" shows "cnf_semantics \<A> (cnf F) = \<A> \<Turnstile> F"
val_Zp_pow': fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> nonzero Zp" shows "val_Zp (x [^] n) = eint (int n) * val_Zp x"
domain: shows "Domainp pcr_float = (\<lambda>x. \<exists>y. x = y \<and> y \<in> float)"
a_ensure_pre_insertion_validity_def: fixes node :: "(_) node_ptr"   and parent :: "(_) object_ptr"   and child_opt :: "(_) node_ptr option" shows "local.a_ensure_pre_insertion_validity node parent child_opt = Heap_Error_Monad.bind (if is_character_data_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r parent then error HierarchyRequestError else return ()) (\<lambda>_. Heap_Error_Monad.bind (get_ancestors_si parent) (\<lambda>ancestors. Heap_Error_Monad.bind (if cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node \<in> set ancestors then error HierarchyRequestError else return ()) (\<lambda>_. Heap_Error_Monad.bind (case child_opt of None \<Rightarrow> return () | Some child \<Rightarrow> Heap_Error_Monad.bind (get_parent child) (\<lambda>child_parent. if child_parent \<noteq> Some parent then error NotFoundError else return ())) (\<lambda>_. Heap_Error_Monad.bind (get_child_nodes parent) (\<lambda>children. Heap_Error_Monad.bind (if children \<noteq> [] \<and> is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r parent then error HierarchyRequestError else return ()) (\<lambda>_. if is_character_data_ptr\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node \<and> is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r parent then error HierarchyRequestError else return ()))))))"
is_cfailed_def: fixes code_status :: "'a code_status" shows "is_cfailed code_status = (\<exists>x1. code_status = CFAILED x1)"
simps_8: fixes c :: "int"   and e :: "MIR.num" shows "\<beta> (Ge (CN 0 c e)) = [Sub (C (- 1)) e]"
nn_integral_distr: fixes T :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and M' :: "'d measure"   and f :: "'d \<Rightarrow> ennreal" assumes "T \<in> M \<rightarrow>\<^sub>M M'"   and "f \<in> borel_measurable (distr M M' T)" shows "integral\<^sup>N (distr M M' T) f = \<integral>\<^sup>+ x. f (T x) \<partial>M"
snoc_1: fixes c' :: "('p, 's, 'm) configuration" assumes "trace c t c'"   and "snd (cs c cid) \<noteq> NotStarted"   and "channel cid = Some (p, q)" shows "snd (cs c' cid) \<noteq> NotStarted"
trans_restrictPI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "transp r" shows "transp (r |` A)"
assms: shows "\<not> halts tcontra [code_tcontra]"
first_isIn: fixes x :: "('b, 'c) Combinators" assumes "\<not> member DenyAll x" shows "(first_srcNet x, first_destNet x) \<in> sdnets x"
Y_cts_cnv: fixes \<omega> :: "real" assumes "\<omega> \<in> {0<..<1}"   and "isCont M.I \<omega>" shows "(\<lambda>n. \<mu>.I n \<omega>) \<longlonglongrightarrow> M.I \<omega>"
strong_reduction_simulation_impl_SOCom: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel"   and "\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel\<^sup>*"   and "strong_reduction_simulation Rel (STCal Source Target)" shows "\<forall>S S'. S \<longmapsto>Source S' \<longrightarrow> (\<exists>T. \<lbrakk>S\<rbrakk> \<longmapsto>Target T \<and> (\<lbrakk>S'\<rbrakk>, T) \<in> TRel\<^sup>*)"
IH: fixes P :: "'a simple_packet set" assumes "p \<notin> P" shows "simple_fw (rmshadow' rs P) p = simple_fw rs p"
sets_null_measure: fixes M :: "'c measure" shows "sets (null_measure M) = sets M"
HInfinite_def: shows "HInfinite = {x. \<forall>r\<in>\<real>. r < hnorm x}"
Inf_atLeast: fixes x :: "'a" shows "\<Sqinter> {x..} = x"
aadd_two_pos: fixes x :: "ant"   and y :: "ant" assumes "0 \<le> x"   and "0 \<le> y" shows "0 \<le> x + y"
Lxy_mono: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" assumes "{x, y} \<subseteq> set xs"   and "distinct xs" shows "x < y in xs = x < y in Lxy xs {x, y}"
a: shows "(sgn (ipoly p y) = sgn (ipoly p r)) \<noteq> (real_of_1 (p, l, r) < real_of_rat y)"
rel_flip: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c set['e]"   and b :: "'d set['e]" shows "rel_bset R\<inverse>\<inverse> a b = rel_bset R b a"
simps_1: fixes I :: "'c set"   and R :: "('c, 'd) Ring_scheme" shows "I \<^bsup>\<diamondsuit>R 0\<^esup> = carrier R"
foldl_conv_fold: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'g"   and s :: "'g"   and xs :: "'h list" shows "foldl f s xs = fold (\<lambda>x s. f s x) xs s"
hom_uminus: fixes x :: "'b" shows "[:- x:] = - [:x:]"