hcju/isabelleps huggingface dataset:
a_remove_child_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr" shows "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_child get_child_nodes set_child_nodes get_owner_document get_disconnected_nodes set_disconnected_nodes ptr child = Heap_Error_Monad.bind (get_child_nodes ptr) (\<lambda>children. if child \<notin> set children then error NotFoundError else Heap_Error_Monad.bind (get_owner_document (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 child)) (\<lambda>owner_document. Heap_Error_Monad.bind (get_disconnected_nodes owner_document) (\<lambda>disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes owner_document (child # disc_nodes)) (\<lambda>_. set_child_nodes ptr (remove1 child children)))))"
lt: shows "\<lhd>\<^sub>l\<^sub>b (lt dir) (\<langle>\<V> s'\<rangle> x\<^sub>i) (LB dir s' x\<^sub>i)"
wp1While: fixes b :: "bexp"   and c :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" shows "wp\<^sub>1 (WHILE b DO c) Q l s = (if bval b s then wp\<^sub>1 c (wp\<^sub>1 (WHILE b DO c) Q) l s else Q l s)"
bounded_Si: shows "\<exists>B. \<forall>T. \<bar>Si T\<bar> \<le> B"
M_2: shows "\<forall>l. M l \<subseteq> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>))"
sorted_list_of_set_1: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'g set" assumes "class.linorder less_eq less"   and "finite A" shows "set (linorder.sorted_list_of_set less_eq A) = A"
sorted_keys: fixes t :: "('b, 'c) RBT.rbt" shows "sorted (RBT.keys t)"
assoc'_naturality: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" assumes "B.arr \<mu>"   and "B.arr \<nu>"   and "B.arr \<tau>"   and "src\<^sub>B \<mu> = trg\<^sub>B \<nu>"   and "src\<^sub>B \<nu> = trg\<^sub>B \<tau>" shows "B.\<a>' (B.cod \<mu>) (B.cod \<nu>) (B.cod \<tau>) \<cdot>\<^sub>B (\<mu> \<star>\<^sub>B \<nu> \<star>\<^sub>B \<tau>) = ((\<mu> \<star>\<^sub>B \<nu>) \<star>\<^sub>B \<tau>) \<cdot>\<^sub>B B.\<a>' (B.dom \<mu>) (B.dom \<nu>) (B.dom \<tau>)"
assms_1: shows "fst args0 \<in> keys (rep_list (snd args0))"
quant_eqI: fixes x :: "'c['d, 'e]"   and y :: "'c['d, 'e]" assumes "\<lbrakk>x\<rbrakk>\<^sub>Q = \<lbrakk>y\<rbrakk>\<^sub>Q" shows "x = y"
rep_eq: fixes x :: "'c set"   and xa :: "'c ell2" shows "Rep_ell2 (trunc_ell2 x xa) = (\<lambda>i. if i \<in> x then Rep_ell2 xa i else 0)"
sim_cfunc_adv: fixes p :: "'s_core spmf"   and q :: "'s_core' spmf"   and ia :: "'iadv_core"   and s1 :: "'s_core"   and s2 :: "'s_core'"   and oa :: "'oadv_core"   and s1' :: "'s_core"   and s2' :: "'s_core'" assumes "S p q"   and "ia \<in> IA"   and "s1 \<in> set_spmf p"   and "s2 \<in> set_spmf q"   and "(oa, s1') \<in> set_spmf (cfunc_adv core1 s1 ia)"   and "(oa, s2') \<in> set_spmf (cfunc_adv core2 s2 ia)" shows "trace_eq_simcl S (cond_spmf_fst (p \<bind> (\<lambda>s1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (q \<bind> (\<lambda>s2. cfunc_adv core2 s2 ia)) oa)"
hom_add_eq_zero: fixes x :: "'a poly"   and y :: "'a poly"   and a :: "'a" assumes "x + y = 0" shows "poly x a + poly y a = (0::'a)"
pow_zero: shows "\<forall>i. B ^ i \<noteq> 0"
one_side_reflexivity: fixes A :: "'p"   and P :: "'p"   and Q :: "'p" assumes "\<not> Col A P Q" shows "P Q OS A A"
inf1D1: fixes A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "inf A B x" shows "A x"
map_def: fixes f :: "'a \<times> 'a" shows "B.VV.map f = (if B.VV.arr f then f else B.VV.null)"
ticks_image: fixes f :: "nat \<Rightarrow> nat"   and r :: "'b run"   and n :: "nat"   and c :: "clock" assumes "dilating_fun f r"   and "hamlet (Rep_run r n c)" shows "\<exists>n\<^sub>0. f n\<^sub>0 = n"
HT_to_mod: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" shows "HT \<pi> P c Q = HT_mods \<pi> (ANALYZE (lhsv \<pi> c)) P c Q"
inverse_functors_axioms: shows "inverse_functors (\<cdot>) (\<cdot>\<acute>) local.inv \<Phi>"
rep_eq: fixes x :: "'a ccsubspace" shows "space_as_set (- x) = orthogonal_complement (space_as_set x)"
beta_upright_0: fixes vb :: "'c zmultiset" shows "beta_upright {#}\<^sub>z vb"
approx_empty: shows "approx Map.empty = (\<lambda>_. True)"
ffunion_mono: fixes A :: "'a fset fset"   and B :: "'a fset fset" assumes "A |\<subseteq>| B" shows "ffUnion A |\<subseteq>| ffUnion B"
pt_list_inst: assumes "pt TYPE('a) TYPE('b)" shows "pt TYPE('a list) TYPE('b)"
path_defs_1: fixes g :: "real \<Rightarrow> 'b" shows "path g = continuous_on {0..1} g"
degrees_leI: fixes X :: "'b pdevs list"   and K :: "nat" assumes "\<And>x. x \<in> set X \<Longrightarrow> degree x \<le> K" shows "degrees X \<le> K"
decr: fixes js :: "'ord stream" assumes "alw (\<lambda>js. (shd (stl js), shd js) \<in> Ord) js"   and "alw (ev (\<lambda>js. shd (stl js) \<noteq> shd js)) js" shows "alw (\<lambda>js. (shd (stl js), shd js) \<in> Ord \<and> shd (stl js) \<noteq> shd js) (decr (shd js) js)"
d1: shows "subst_domain \<delta>' \<subseteq> fv s \<union> fv t - subst_domain \<theta>"
Standard_of_hypreal: fixes r :: "real star" assumes "r \<in> Standard" shows "of_hypreal r \<in> Standard"
rref: shows "reduced_row_echelon_form_upt_k (Gauss_Jordan A) (ncols (Gauss_Jordan A))"
mem_hequiv_rel: fixes a :: "'b"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and A :: "'b set"   and b :: "'b" assumes "a \<in> proj (hequiv_rel A H) b" shows "a \<in> A"
elim: fixes c :: "'a" assumes "b = a \<otimes> c"   and "c \<in> carrier G" shows "P"
meas_merge: shows "(\<lambda>x. merge (shift_var_set V) (Suc ` V') (x, \<rho>)) \<in> state_measure (shift_var_set V) (case_nat t \<Gamma>) \<rightarrow>\<^sub>M state_measure (shift_var_set (V \<union> V')) (case_nat t \<Gamma>)"
BC_wrt_phi_eq_BC: shows "BC_wrt \<phi> = BC"
parAssoc: fixes P :: "ccs"   and Q :: "ccs"   and R :: "ccs" shows "P \<parallel> Q \<parallel> R \<sim> P \<parallel> (Q \<parallel> R)"
circline_set_y_axis: shows "circline_set y_axis = of_complex ` {x. is_imag x} \<union> {\<infinity>\<^sub>h}"
Scons_inject2: fixes M :: "('b, 'c) Old_Datatype.node set"   and N :: "('b, 'c) Old_Datatype.node set"   and M' :: "('b, 'c) Old_Datatype.node set"   and N' :: "('b, 'c) Old_Datatype.node set" assumes "Scons M N = Scons M' N'" shows "N = N'"
atom_is_singleton: fixes p :: "'a" assumes "p \<noteq> 0"   and "\<forall>x\<le>p. x = 0 \<or> x = p" shows "singleton p"
IH_1: assumes "\<not> complete l" shows "2 ^ min_height l < size1 l"
plus_list_ub1: fixes xs :: "'a list"   and ys :: "'a list" assumes "set xs \<subseteq> A"   and "set ys \<subseteq> A"   and "length xs = length ys" shows "xs [\<sqsubseteq>\<^bsub>r\<^esub>] xs [\<squnion>\<^bsub>f\<^esub>] ys"
\<R>\<^sub>1_elim: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" obtains \<Gamma>' :: "'Var \<Rightarrow> Sec option" where "c\<^sub>2 = c\<^sub>1" and "\<turnstile> \<Gamma>' {c\<^sub>1} \<Gamma>" and "mds_consistent mds \<Gamma>'" and "mem\<^sub>1 =\<^bsub>\<Gamma>'\<^esub> mem\<^sub>2"
Opsem_def: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and v :: "Val" shows "Opsem M l s h v = (\<exists>n. (M, l, s, n, h, v) \<in> Exec)"
amult_pos_mono_l: fixes w :: "int"   and x :: "ant"   and y :: "ant" assumes "0 < w" shows "(ant w * x \<le> ant w * y) = (x \<le> y)"
2: shows "fmrestrict_set vs (fst a) \<subseteq>\<^sub>f s"
t_heap_of_A_log_bound: fixes xs :: "'b list" shows "t_heap_of_A xs \<le> length xs * (nat \<lceil>log 2 (real (length xs + 1))\<rceil> + 1)"
mem_Collect_eq: fixes a :: "'d"   and P :: "'d \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
simps_5: fixes v :: "variable"   and va :: "typ"   and uw :: "nat" shows "loose_bvar (Fv v va) uw = False"
h3: shows "lose t = [False] \<longrightarrow> ack t = [y] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = y"
assms_2: shows "f ` (P .\<^bold>\<le> x) = Pow A"
zero_cvec_def: shows "0\<^sub>v = (0, 1)"
weak_unit_self_right: fixes a :: "'a" assumes "weak_unit a" shows "right a a"
Abs_ty_inverse: fixes y :: "ty_Rep" assumes "y \<in> Collect ty_Rep_set" shows "Rep_ty (Abs_ty y) = y"
X: shows "X = prod_emb UNIV (\<lambda>_. M) J (Pi\<^sub>E J F)"
g: fixes x :: "complex" assumes "x \<in> A" shows "(g has_field_derivative deriv f x / f x) (at x)"
S_def: shows "S \<equiv> \<lambda>n. x ^ n /\<^sub>R fact n"
cong_scalar_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a"   and d :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a" shows "unique_euclidean_semiring.cong modulo (times d b) (times d c) a"
distinct_19: fixes x51 :: "pdf_type"   and x52 :: "pdf_type" shows "REAL \<noteq> PRODUCT x51 x52"
card_lists_distinct_length_eq: fixes A :: "'c set"   and k :: "nat" assumes "finite A"   and "k \<le> card A" shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1..card A}"
compact_imp_fip: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and F :: "'b set set" assumes "class.topological_space open"   and "topological_space.compact open S"   and "\<And>T. T \<in> F \<Longrightarrow> topological_space.closed open T"   and "\<And>F'. \<lbrakk>finite F'; F' \<subseteq> F\<rbrakk> \<Longrightarrow> S \<inter> \<Inter> F' \<noteq> {}" shows "S \<inter> \<Inter> F \<noteq> {}"
simps: fixes x :: "'a" shows "the_res (dRETURN x) = x"
pmdl_full_pmdl: fixes K :: "'k set" shows "pmdl (full_pmdl K) = full_pmdl K"
SA_trans_TVar: fixes X :: "tyvrs"   and S :: "ty"   and \<Gamma> :: "binding list"   and T :: "ty" assumes "TVarB X S \<in> set \<Gamma>"   and "\<Gamma>\<turnstile>S<:T" shows "\<Gamma>\<turnstile>Tvar X<:T"
simplex_extremal_le: fixes S :: "'a set" assumes "finite S"   and "S \<noteq> {}" shows "\<exists>u\<in>S. \<exists>v\<in>S. \<forall>x\<in>convex hull S. \<forall>y\<in>convex hull S. norm (x - y) \<le> norm (u - v)"
labels_def: shows "labels = fold_tree (\<lambda>x. [x]) (@)"
ex_lderivs_Pr: fixes ass :: "'a list"   and r :: "'b rexp" shows "\<exists>s. lderivs ass (Pr r) = Pr s"
relAPP_def: fixes f :: "('e \<times> 'f) set \<Rightarrow> 'd"   and x :: "('e \<times> 'f) set" shows "\<langle>x\<rangle>f \<equiv> f x"
rsum_right_distrib: fixes c :: "real"   and D :: "(real \<times> real \<times> real) list"   and f :: "real \<Rightarrow> real" shows "c * rsum D f = rsum D (\<lambda>x. c * f x)"
count_inI: fixes M :: "'e multiset"   and x :: "'e" assumes "count M x = 0 \<Longrightarrow> False" shows "x \<in># M"
opt_def: fixes A :: "'b set" shows "opt A = insert None {\<lfloor>y\<rfloor> |y. y \<in> A}"
a_ac_1: fixes x :: "'a vec"   and y :: "'a vec"   and z :: "'a vec" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n"   and "z \<in> carrier_vec n" shows "x + y + z = x + (y + z)"
bounded_segment_2: fixes u :: "'b"   and w :: "'b"   and z :: "'b" assumes "u \<in> {w<--<z}" shows "u \<in> {w--z}"
Rep_Abs_1: shows "Rep_bit0 (Abs_bit0 1) = 1"
elts_1: shows "elts 1 = {[]\<^sub>\<circ>}"
generated_ringI: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and C :: "'b set set"   and a :: "'b set" assumes "semiring_of_sets \<Omega> M"   and "finite C"   and "disjoint C"   and "C \<subseteq> M"   and "a = \<Union> C" shows "a \<in> semiring_of_sets.generated_ring M"
euclidean_eq_iff: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and x :: "'c"   and y :: "'c" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "(x = y) = (\<forall>b\<in>Basis. inner x b = inner y b)"
ground_arg: fixes s :: "('b, 'a) tm" assumes "ground s" shows "ground (arg s)"
finfun_const_eq_update: fixes b :: "'b"   and f :: "('a, 'b) finfun"   and a :: "'a"   and b' :: "'b" shows "(finfun_const b = finfun_update f a b') = (b = b' \<and> (\<forall>a'. a \<noteq> a' \<longrightarrow> finfun_apply f a' = b))"
not_irreducibleE: fixes x :: "'a" assumes "\<not> irreducible x" obtains "x = (0::'a)"   | "x dvd (1::'a)"   | a :: "'a" and b :: "'a" where "x = a * b" and "\<not> a dvd (1::'a)" and "\<not> b dvd (1::'a)"
find_base_vectors_i_def: fixes ops :: "'b arith_ops_record"   and A :: "'b mat" shows "find_base_vectors_i ops A \<equiv> find_base_vectors_gen (arith_ops_record.uminus ops) (arith_ops_record.zero ops) (arith_ops_record.one ops) A"
antipar_2: shows "src\<^sub>C g = trg\<^sub>C f"
elims: fixes x :: "init_call_status"   and y :: "char list list" assumes "ics_classes x = y" obtains C :: "char list" and Cs :: "char list list" where "x = Calling C Cs__" and "y = Cs__"   | Cs :: "char list list" and a :: "nat" where "x = Throwing Csa__ a" and "y = Csa__"   | Cs :: "char list list" where "x = Called Cs" and "y = Cs"   | "x = No_ics" and "y = []"
inj_on_disjoint_Un: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and B :: "'c set" assumes "inj_on f A"   and "inj_on g B"   and "f ` A \<inter> g ` B = {}" shows "inj_on (\<lambda>x. if x \<in> A then f x else g x) (A \<union> B)"
conv_abscissa_higher_deriv_le: fixes n :: "nat"   and f :: "'a fds" shows "conv_abscissa ((fds_deriv ^^ n) f) \<le> conv_abscissa f"
sum_subtractf: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "(\<Sum>x\<in>A. f x - g x) = sum f A - sum g A"
Skip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and Q :: "'e set"   and A :: "'e set" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (AnnExpr Q) Skip Q, A"
empty_Collect_eq: fixes P :: "'d \<Rightarrow> bool" shows "({} = Collect P) = (\<forall>x. \<not> P x)"
"3_3": assumes "x \<noteq> y"   and "\<not> rel x y"   and "\<And>y. \<lbrakk>y \<in> set ys; P y\<rbrakk> \<Longrightarrow> False" shows "filter P (merge_wrt rel (x # xs) ys) = filter P (x # xs)"
nprv_eql_transI: fixes F :: "'fmla set"   and t1 :: "'trm"   and t2 :: "'trm"   and t3 :: "'trm" assumes "nprv F (eql t1 t2)"   and "nprv F (eql t2 t3)"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "t3 \<in> trm"   and "F \<subseteq> fmla"   and "finite F" shows "nprv F (eql t1 t3)"
rem_x: shows "odd (degree v' (rem_unPath (x # xs) G)) = odd (degree v' (del_unEdge x1 x2 x3 (rem_unPath xs G)))"
dd: fixes k' :: "nat"   and k :: "nat" assumes "k' < k" shows "d k' \<lless> d k"
exp_less_mono: fixes x :: "real"   and y :: "real" assumes "x < y" shows "exp x < exp y"
magQ_plus: fixes x :: "'c['d, 'e]"   and y :: "'c['d, 'e]" shows "\<lbrakk>x + y\<rbrakk>\<^sub>Q = \<lbrakk>x\<rbrakk>\<^sub>Q + \<lbrakk>y\<rbrakk>\<^sub>Q"
recurrent_iff_G_infinite: fixes x :: "'s" shows "recurrent x = (G x x = \<infinity>)"
equiv_domain_permutation: fixes A :: "'c set"   and B :: "'d set" shows "equiv (A \<rightarrow>\<^sub>E B) (domain_permutation A B)"
inverse_eq_iff_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> '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 a :: "'c"   and b :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(inverse a = inverse b) = (a = b)"
post_fun_def: fixes p :: "'c"   and q :: "'c" shows "post_fun p q = (if p \<le> q then \<top> else \<bottom>)"
some_geodesic_endpoints_3: fixes x :: "'c"   and S :: "'c set"   and y :: "'c" shows "{x--S--y} \<noteq> {}"
adjacent_half_chamber_system_image: fixes C :: "'a set"   and D :: "'a set" assumes "C \<in> f \<turnstile> folding.\<C>"   and "D \<in> folding.\<C> - f \<turnstile> folding.\<C>"   and "C \<sim> D" shows "f ` D = C"
inject_1: fixes x21 :: "char list"   and x22 :: "aexp"   and y21 :: "char list"   and y22 :: "aexp" shows "(x21 ::= x22 = y21 ::= y22) = (x21 = y21 \<and> x22 = y22)"
rule_sym: fixes u :: "'a"   and xs :: "'b list"   and ys :: "'b list"   and P :: "'b process"   and I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a" assumes "(u, xs, ys) \<in> rel_induct_aux P I D" shows "(u, ys, xs) \<in> rel_induct_aux P I D"
seq_comp_futures: fixes P :: "'c option process"   and Q :: "'c option process"   and xs :: "'c option list" assumes "weakly_sequential P" shows "futures (P ; Q) xs = {(ys, Y). (xs @ ys, Y) \<in> seq_comp_failures P Q}"
embedding_map_imp_homeomorphic_space: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" assumes "embedding_map X Y f" shows "X homeomorphic_space subtopology Y (f ` topspace X)"
inf_sup_ord_3: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "less_eq_prod x (sup_prod x y)"
structCongBisimSubst: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "P \<equiv>\<^sub>s Q" shows "P \<sim>\<^sub>s Q"
unstream_flatten_Some: fixes f :: "'h \<Rightarrow> 'g"   and g'' :: "('e, 'g) generator"   and g :: "('h, 'f) generator"   and s :: "'f"   and s' :: "'g" shows "unstream (flatten f g'' g) (s, Some s') = unstream g'' s' @ unstream (flatten f g'' g) (s, None)"
AllN_9: shows "\<forall>x. insertion (nth_default 0 ((drop z L)[var := x])) d = Dv"
assign_global_def: fixes upd :: "('d \<Rightarrow> 'd) \<Rightarrow> 'c control_state_scheme \<Rightarrow> 'c control_state_scheme"   and rhs :: "'c control_state_scheme \<Rightarrow> 'd" shows "assign_global upd rhs = assign (\<lambda>\<sigma>. upd (\<lambda>_. rhs \<sigma>) \<sigma>)"
zero_compare_simps_6: fixes a :: "'a"   and b :: "'a" shows "((0::'a) < a * b) = ((0::'a) < a \<and> (0::'a) < b \<or> a < (0::'a) \<and> b < (0::'a))"
assms_2: shows "normalize (\<Prod>\<^sub># P) = normalize n"
returns_toE: fixes P :: "'a set"   and x :: "'a" assumes "returns_to P x" obtains t0 :: "real" and t1 :: "real" where "0 < t0" and "t0 \<le> t1" and "t1 \<in> existence_ivl0 x" and "flow0 x t1 \<in> P" and "\<And>t. \<lbrakk>0 < t; t < t0\<rbrakk> \<Longrightarrow> flow0 x t \<notin> P"
independent_mono: fixes A :: "'c set"   and B :: "'c set" assumes "vs2.independent A"   and "B \<subseteq> A" shows "vs2.independent B"
map_idI: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'a" assumes "\<And>x. x \<in> list.set xs \<Longrightarrow> f x = x" shows "map f xs = xs"
analz_hard: fixes A :: "agent"   and B :: "agent"   and M :: "nat"   and Na :: "nat"   and evs :: "event list"   and K :: "nat" assumes "Says A B \<lbrace>Nonce M, Agent A, Agent B, Crypt (shrK A) \<lbrace>Nonce Na, Nonce M, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs"   and "Crypt (shrK A) \<lbrace>Nonce Na, Key K\<rbrace> \<in> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> orb" shows "Says B A \<lbrace>Nonce M, Crypt (shrK A) \<lbrace>Nonce Na, Key K\<rbrace>\<rbrace> \<in> set evs"
bS: fixes x :: "real" assumes "x \<in> S" shows "x \<le> b"
dvd_associated2: fixes a :: "'b"   and b :: "'b" assumes "a dvd b"   and "b dvd a"   and "a \<noteq> (0::'b)" shows "\<exists>u. u dvd (1::'b) \<and> a = u * b"
inner_node_dyn_standard_control_dependence_predecessor: fixes n :: "'b" assumes "inner_node n" obtains n' :: "'b" and as :: "'a list" where "n' controls\<^sub>s n via as"
extreme_bound_singleton_refl: fixes x :: "'a" shows "local.extreme_bound {x} x = (x \<in> A \<and> x \<sqsubseteq> x)"
FreeUltrafilterNat_def: shows "\<U> = (SOME U. freeultrafilter U)"
type_less_eq_x_Bag_intro: fixes \<tau> :: "'a OCL_Types.type"   and \<rho> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> = Bag \<rho>"   and "\<rho> \<le> \<sigma>" shows "\<tau> \<le> Bag \<sigma>"
nextl_init_state: fixes xs :: "'a list" shows "ST.nextl (dfa.init \<lparr>dfa.states = dfa.states MS, init = dfa.init MS, final = dfa.states MS - dfa.final MS, nxt = dfa.nxt MS\<rparr>) xs \<in> dfa.states \<lparr>dfa.states = dfa.states MS, init = dfa.init MS, final = dfa.states MS - dfa.final MS, nxt = dfa.nxt MS\<rparr>"
equalityD2: fixes A :: "'e set"   and B :: "'e set" assumes "A = B" shows "B \<subseteq> A"
simps_6: fixes c :: "int"   and e :: "MIR.num" shows "iszlfm (Le (CN 0 c e)) = (\<lambda>bs. 0 < c \<and> numbound0 e \<and> isint e bs)"
upper_semilattice_axioms_def: fixes L :: "('c, 'd) gorder_scheme" shows "upper_semilattice_axioms L \<equiv> \<forall>x y. x \<in> carrier L \<longrightarrow> y \<in> carrier L \<longrightarrow> (\<exists>s. is_lub L s {x, y})"
NatTransComp_Comp1: fixes x :: "'g"   and f :: "('g, 'j, 'h, 'k, 'i, 'l) NatTrans"   and g :: "('g, 'j, 'h, 'k, 'i, 'l) NatTrans" assumes "x \<in> obj\<^bsub>NTCatDom f\<^esub>"   and "f \<approx>>\<bullet> g" shows "f \<bullet> g $$ x = f $$ x ;;\<^bsub>NTCatCod g\<^esub> (g $$ x)"
assms: shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<circ> A"
rcs_subset_elem: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set"   and a :: "'c"   and x :: "'c" assumes "Group G"   and "G \<guillemotright> H "   and "a \<in> carrier G"   and "x \<in> H \<bullet>\<^bsub>G\<^esub> a" shows "x \<in> carrier G"
sets_into_space: fixes x :: "'c set"   and M :: "'c measure" assumes "x \<in> sets M" shows "x \<subseteq> space M"
add1_zle_eq: fixes w :: "int"   and z :: "int" shows "(w + 1 \<le> z) = (w < z)"