hcju/isabelleps huggingface dataset:
get_dom_component_subset: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_dom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"   and "ptr' \<in> set c" shows "h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c"
injective_def: fixes f :: "'a \<Rightarrow> 'b" shows "injective f \<equiv> \<forall>x y. f x = f y \<longrightarrow> x = y"
assms: fixes U :: "'w set" assumes "RModule FG (\<star>) U \<and> U \<subseteq> W" shows "U = 0 \<or> U = W"
Sigma_msetD2: fixes a :: "'b"   and b :: "'c"   and A :: "'b multiset"   and B :: "'b \<Rightarrow> 'c multiset" assumes "(a, b) \<in># Sigma_mset A B" shows "b \<in># B a"
rel_inject_2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "'a \<times> nat"   and y2 :: "'b \<times> nat" shows "rel_SDG_node R (Formal_in x2) (Formal_in y2) = rel_prod R (=) x2 y2"
sphere_params_on_sphere: fixes x :: "real"   and \<alpha> :: "real"   and \<beta> :: "real"   and y :: "real"   and z :: "real" assumes "x = cos \<alpha> * cos \<beta>"   and "y = cos \<alpha> * sin \<beta>"   and "z = sin \<alpha>" shows "x * x + y * y + z * z = 1"
diff_eq_diff_less_eq: fixes plus :: "'y \<Rightarrow> 'y \<Rightarrow> 'y"   and zero :: "'y"   and minus :: "'y \<Rightarrow> 'y \<Rightarrow> 'y"   and uminus :: "'y \<Rightarrow> 'y"   and less_eq :: "'y \<Rightarrow> 'y \<Rightarrow> bool"   and less :: "'y \<Rightarrow> 'y \<Rightarrow> bool"   and a :: "'y"   and b :: "'y"   and c :: "'y"   and d :: "'y" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "minus a b = minus c d" shows "less_eq a b = less_eq c d"
take_initial_segment: fixes xs :: "'a list"   and ys :: "'a list"   and k :: "nat" assumes "initial_segment xs ys"   and "k \<le> length xs" shows "take k xs = take k ys"
list_all2_in_setE: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and x :: "'a" assumes "list_all2 P xs ys"   and "x \<in> set xs" obtains y :: "'b" where "y \<in> set ys" and "P x y"
hom_induced_restrict: fixes p :: "int"   and X :: "'a topology"   and S :: "'a set"   and Y :: "'b topology"   and T :: "'b set" shows "hom_induced p X (topspace X \<inter> S) Y (topspace Y \<inter> T) = hom_induced p X S Y T"
rightUniqueFunctionAfterInverse: fixes f :: "('c \<times> 'd) set"   and Y :: "'d set" assumes "runiq f" shows "f `` f\<inverse> `` Y \<subseteq> Y"
iso_incl_refl: fixes A :: "'c ring" shows "A \<lesssim> A"
Nil_in_shuffles: fixes xs :: "'b list"   and ys :: "'b list" shows "(\<epsilon> \<in> shuffles xs ys) = (xs = \<epsilon> \<and> ys = \<epsilon>)"
Min_le_iff: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "A \<noteq> {}" shows "(Min A \<le> x) = (\<exists>a\<in>A. a \<le> x)"
PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m_1_dir2: fixes ps :: "nat list" assumes "0 < length ps" shows "LPHIL0\<^sub>n\<^sub>o\<^sub>r\<^sub>m (ps ! 0) \<sqsubseteq> PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m 1 ps"
Nonce_Nonce: fixes X :: "nonce_t"   and X' :: "nonce_t" shows "(Nonce X = Nonce X') = (X = X')"
strict_sorted_list_of_set: fixes A :: "'a word set" shows "signed.strict_sorted (signed.sorted_list_of_set A)"
bdd_aboveI: fixes A :: "'a set"   and M :: "'a" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<le> M" shows "bdd_above A"
swap_sum_parallel_converter2: fixes conv1 :: "('h, 'j, 'm, 'o) converter"   and conv2 :: "('i, 'k, 'l, 'n) converter" shows "map_converter id id swap_sum swap_sum (conv1 |\<^sub>= conv2) = map_converter swap_sum swap_sum id id (conv2 |\<^sub>= conv1)"
ts''_def_2: shows "\<C> \<turnstile> [Br_if i] : ts'' _> ts'"
content_def: fixes p :: "'b poly" shows "content p = gcd_list (coeffs p)"
Suc: shows "T ^^ n \<in> tail_algebra (m + n) \<rightarrow>\<^sub>M tail_algebra m"
WT_converter_parallel_converter2: fixes \<I>1 :: "('m, 'n) \<I>"   and \<I>2 :: "('o, 'p) \<I>"   and conv1 :: "('m, 'n, 'o, 'p) converter"   and \<I>1' :: "('q, 'r) \<I>"   and \<I>2' :: "('s, 't) \<I>"   and conv2 :: "('q, 'r, 's, 't) converter" assumes "\<I>1, \<I>2 \<turnstile>\<^sub>C conv1 \<surd>"   and "\<I>1', \<I>2' \<turnstile>\<^sub>C conv2 \<surd>" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>1', \<I>2 \<oplus>\<^sub>\<I> \<I>2' \<turnstile>\<^sub>C conv1 |\<^sub>= conv2 \<surd>"
preserves_arr: fixes f :: "'a" assumes "A.arr f" shows "A.arr (A.map f)"
is_group: shows "Group.group (H\<lparr>carrier := generate H {b}\<rparr>)"
equiv_rcong: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "subgroup H G"   and "Group.group G" shows "equiv (carrier G) rcong\<^bsub>G\<^esub> H"
closed_env_def: fixes \<rho> :: "(nat \<times> exp) list" shows "closed_env \<rho> \<equiv> \<forall>x v. x \<in> sdom \<rho> \<longrightarrow> lookup \<rho> x = Some v \<longrightarrow> FV v = {}"
frombd_F12_tobd_F12: fixes b :: "'d"   and x :: "('e, 'd, 'f) F1"   and r2 :: "(bd_type_F2 \<times> bd_type_F2) set" assumes "b \<in> F1set2 x" shows "fromCard (F1set2 x) (dir_image (bd_F1 +c r2) Abs_bd_type_F) (toCard (F1set2 x) (dir_image (bd_F1 +c r2) Abs_bd_type_F) b) = b"
simps_12: fixes x1 :: "'a"   and x31 :: "'a local.term"   and x32 :: "'a local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> x31 \<^bold>\<otimes> x32"
mono: fixes A :: "'a set"   and x :: "'a"   and B :: "'a set" assumes "MaxR_opt A = Some x" shows "\<exists>y. MaxR_opt (A \<union> B) = Some y \<and> (x, y) \<in> r"
not_None_eq: fixes x :: "'a option" shows "(x \<noteq> None) = (\<exists>y. x = Some y)"
pos_le_divide_eq: 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 less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less_eq a (divide b c) = less_eq (times a c) b"
de_Morgan_conj: fixes P :: "bool"   and Q :: "bool" shows "(\<not> (P \<and> Q)) = (\<not> P \<or> \<not> Q)"
prepend_cong1_prepend: fixes X :: "'b llist \<Rightarrow> bool"   and ys :: "'b llist"   and xs :: "'b list" assumes "prepend_cong1 X ys" shows "prepend_cong1 X (prepend xs ys)"
Domain_def: shows "r2s \<equiv> \<lambda>r. {x. Domainp (\<lambda>x xa. (x, xa) \<in> r) x}"
prot_atom_UNIV: shows "UNIV = range prot_atom.Atom \<union> {Value, SetType, AttackType, Bottom, OccursSecType}"
word_log2_def: fixes w :: "'b word" shows "word_log2 w \<equiv> size w - 1 - word_clz w"
conj_left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b AND a AND c = a AND b AND c"
insertion_single: fixes f :: "nat \<Rightarrow> 'b"   and v :: "nat"   and n :: "nat"   and a :: "'b" shows "insertion f (monom (Poly_Mapping.single v n) a) = a * f v ^ n"
mono_ctble_discont: fixes f :: "real \<Rightarrow> real" assumes "mono f" shows "countable {a. \<not> isCont f a}"
aux: shows "wf_until_aux \<sigma> n R pos \<phi>'' I \<psi>'' aux (Monitor.progress \<sigma> (formula.Until \<phi>''' I \<psi>'') j)"
pair_size_def: fixes p :: "'c \<times> 'd" shows "pair_size p = size (fst p) + size (snd p) + 1"
all_total_def: fixes F :: "'b program" shows "all_total F \<equiv> \<forall>act\<in>Acts F. Domain act = UNIV"
OutOfMemory_def: shows "OutOfMemory \<equiv> ''OutOfMemory''"
intro: fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> int set" assumes "Gamma_semilattice num' plus' \<gamma>" shows "Abs_Int num' plus' \<gamma>"
simps_10: fixes x3 :: "msg"   and x1 :: "agr_key" shows "Hash x3 \<noteq> AgrKey x1"
conv_1: shows "g \<turnstile> a-as\<rightarrow>n"
zero_lowner_le_positiveI: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "positive A" shows "0\<^sub>m n n \<le>\<^sub>L A"
smallest: shows "\<forall>k<n. \<nexists>S u. finite S \<and> card S = k \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y"
FW_cyc_free_preservation: fixes n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len m i i xs" shows "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len (FW m n) i i xs"
irr: fixes b :: "'a" assumes "b dvd a"   and "\<not> a dvd b" shows "b dvd (1::'a)"
dvd_mult2: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times a b" shows "dvd.dvd times a (times b c)"
that: fixes M :: "nat list set" assumes "M \<in> [WW]\<^bsup>m\<^esup>"   and "\<And>U. U \<in> [M]\<^bsup>2\<^esup> \<Longrightarrow> Form l U \<and> list.set (inter_scheme l U) \<subseteq> N" shows "thesis"
src_vcomp: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" assumes "seq \<nu> \<mu>" shows "src (\<nu> \<bullet> \<mu>) = src \<nu> \<bullet> src \<mu>"
loopfree_digraphI_app_iso: fixes G :: "('c, 'd) pre_digraph"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "loopfree_digraph G"   and "pre_digraph.digraph_isomorphism G hom" shows "loopfree_digraph (app_iso hom G)"
map_poly_comm_monoid_add_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "map_poly_comm_monoid_add_hom hom \<equiv> comm_monoid_add_hom hom"
project_ncoord_aform_def: fixes x :: "'a \<times> 'a pdevs"   and b :: "nat"   and n :: "real" shows "project_ncoord_aform x b n = project_coord_aform (Sctn (Basis_list ! b) n) x"
squarefree_part_def: fixes n :: "nat" shows "squarefree_part n = {p. p \<in># prime_factorization n \<and> odd (multiplicity p n)}"
comp2_reachable: fixes C1 :: "('e set \<times> 'e set \<times> 'e set) \<times> 'f set \<times> ('f \<times> 'e \<times> 'f) set"   and C2 :: "('e set \<times> 'e set \<times> 'e set) \<times> 'g set \<times> ('g \<times> 'e \<times> 'g) set"   and s :: "'f \<times> 'g" assumes "reachable (C1 || C2) s" shows "reachable C2 (snd s)"
0: fixes f :: "'fsym" assumes "f \<in> fsymsPB \<Phi>" shows "intF1 f = intF2 f"
last_P_new_2: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and v :: "'c" assumes "DisjointPathsPlusOne G v0 v1 paths P_new"   and "v \<in> set (butlast P_new)" shows "\<not> DisjointPaths.hitting_paths v0 v1 paths v"
finite_measure_subalgebra_axioms_def: fixes M :: "'c measure"   and F :: "'c measure" shows "finite_measure_subalgebra_axioms M F \<equiv> subalgebra M F"
intro: assumes "\<And>y x. PP y x \<Longrightarrow> \<exists>z. PP z x \<and> \<not> O z y" shows "MM1_axioms"
converse_converse: fixes r :: "('e \<times> 'f) set" shows "(r\<inverse>)\<inverse> = r"
siftDown_in_tree_set: fixes v :: "'c"   and t :: "'c Tree" shows "in_tree v t = in_tree v (siftDown t)"
b_e_cases: obtains "b_e = Binop_i t iop"   | "b_e = Binop_f t fop"   | "b_e = Relop_i t irop"   | "b_e = Relop_f t frop"
Dag_RefD: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag" assumes "Dag p l r t"   and "p \<noteq> Null" shows "\<exists>lt rt. t = Node lt p rt \<and> Dag (l p) l r lt \<and> Dag (r p) l r rt"
Cd_lad: fixes X :: "'a set"   and d :: "'b" shows "lad_on X (Cd d)"
BisT_MtransC: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "c \<approx>T d"   and "mustT c s"   and "mustT d t"   and "s \<approx> t"   and "(c, s) \<rightarrow>*c (c', s')" obtains d' :: "('test, 'atom) com" and t' :: "'state" where "(d, t) \<rightarrow>*c (d', t')" and "s' \<approx> t'" and "c' \<approx>T d'"
indRelSTEQ_impl_SRel_and_TRel_reflect_barbs_1: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_reflects_binary_pred (indRelSTEQ SRel TRel) (\<lambda>b. hasBarb b (STCalWB SWB TWB))" shows "rel_reflects_binary_pred SRel (\<lambda>b. hasBarb b SWB)"
preserves_trg: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<mu> :: "'c" assumes "weak_arrow_of_homs C src\<^sub>C trg\<^sub>C D src\<^sub>D trg\<^sub>D F"   and "partial_magma.arr C \<mu>" shows "trg\<^sub>D (F \<mu>) = weak_arrow_of_homs.map\<^sub>0 src\<^sub>D F (trg\<^sub>C \<mu>)"
\<I>simple_def: shows "\<I>simple \<equiv> {(\<langle>c, mds, mem\<rangle>\<^sub>C, \<langle>c', mds', mem'\<rangle>\<^sub>C) |c mds mem c' mds' mem'. c = c'}"
real_norm_def: fixes r :: "real" shows "\<parallel>r\<parallel> = \<bar>r\<bar>"
main: fixes i :: "nat" shows "in_interval ((local.tighten_bound_root ^^ i) (il, ir)) (root n (real_of_1 x)) \<and> sub_interval ((local.tighten_bound_root ^^ i) (il, ir)) (il, ir) \<and> (i \<noteq> 0 \<longrightarrow> sub_interval ((local.tighten_bound_root ^^ i) (il, ir)) ((local.tighten_bound_root ^^ (i - 1)) (il, ir))) \<and> snd ((local.tighten_bound_root ^^ i) (il, ir)) - fst ((local.tighten_bound_root ^^ i) (il, ir)) \<le> (ir - il) / 2 ^ i"
cfg_approx_def: fixes prog :: "'a program_scheme" shows "cfg_approx prog \<equiv> \<Union> {approx_reachable (body p) |p. p \<in> set (program.processes prog)}"
lappend_code_1: fixes ys :: "'b llist" shows "<> $ ys = ys"
Collect_subset: fixes A :: "'e set"   and P :: "'e \<Rightarrow> bool" shows "{x \<in> A. P x} \<subseteq> A"
eq: shows "sig_of_pair p = term_of_pair (0::'a, j)"
polyhedral_cone_def: fixes A :: "'a mat" shows "next_dim.polyhedral_cone A = {x \<in> carrier_vec (n + 1). A *\<^sub>v x \<le> 0\<^sub>v (dim_row A)}"
foundation18': fixes \<tau> :: "'d state \<times> 'd state"   and X :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "(\<tau> \<Turnstile> \<upsilon> X) = (X \<tau> \<noteq> \<bottom>)"
\<beta>0: fixes i :: "nat" shows "\<beta> i \<noteq> (0::'a)"
WTrts_conv_list_all2: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and es :: "'addr expr list"   and Ts :: "ty list" shows "(P,E,h \<turnstile> es [:] Ts) = list_all2 (WTrt_syntax P E h) es Ts"
2: shows "\<forall>(t, t')\<in>set Di. fv t \<union> fv t' \<subseteq> X"
lb1_sgn: shows "\<forall>x\<le>lb1. \<forall>p\<in>set ps. sgn (poly p x) = sgn_neg_inf p"
unification: fixes L :: "fterm literal set"   and \<sigma> :: "char list \<Rightarrow> fterm" assumes "unification"   and "finite L"   and "unifier\<^sub>l\<^sub>s \<sigma> L" shows "\<exists>\<theta>. mgu\<^sub>l\<^sub>s \<theta> L"
diff_single_trivial: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<notin># M" shows "M - {#x#} = M"
meet_connection: fixes x :: "'a"   and y :: "'a" shows "(x \<sqsubseteq> y) = (x \<sqinter> y = x)"
Basis_le_norm: fixes b :: "'c"   and x :: "'c" assumes "b \<in> Basis" shows "\<bar>inner x b\<bar> \<le> norm x"
assms_3: assumes "r\<lparr>a\<rparr> = b" shows "P"
simps_2: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "find P (x # xs) = (if P x then Some x else find P xs)"
simps: fixes x :: "'b"   and t :: "('b \<times> real) tree"   and A :: "'b set" shows "insR x t A = distr (\<U> 0 1) (tree_sigma (count_space A \<Otimes>\<^sub>M lborel)) (\<lambda>p. ins x p t)"
cod_simp_4: shows "partial_magma.cod (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} {False, True} (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) parallel_pair.j1 = quiver.One"
MP: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla"   and H' :: "fmla set" assumes "H \<turnstile> A IMP B"   and "H' \<turnstile> A" shows "H \<union> H' \<turnstile> B"
bind_def: fixes M :: "'c nres"   and f :: "'c \<Rightarrow> 'b nres" shows "M \<bind> f \<equiv> case M of FAILi \<Rightarrow> FAIL | RES X \<Rightarrow> Sup (f ` X)"
extf_compat_snoc: fixes f :: "'s" shows "ext_compat_snoc (extf f)"
Keys_bad_def: fixes IK :: "msg set"   and Bad :: "agent set" shows "Keys_bad IK Bad \<equiv> IK \<inter> range LtK \<subseteq> range pubK \<union> \<Union> (keys_of ` Bad) \<and> range pubK \<subseteq> IK"
gcd_eq_1_imp_coprime: fixes a :: "'a"   and b :: "'a" assumes "gcd a b = (1::'a)" shows "coprime a b"
b_join_d: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b \<squnion> d_aux a b c = b \<squnion> c \<sqinter> a"
esqrt_nonneg: fixes x :: "ereal" shows "0 \<le> esqrt x"
catch_error: fixes e :: "'c"   and f :: "'c \<Rightarrow> 'a + 'b" shows "(try Inl e catch f) = f e"
permutes_inverses_2: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and x :: "'c" assumes "p permutes S" shows "inv p (p x) = x"
G_af\<^sub>G_simp: fixes \<phi> :: "'b ltl"   and w :: "'b set list" shows "\<^bold>G (af\<^sub>G \<phi> w) = \<^bold>G \<phi>"
dim_eq_size_basis: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and vs :: "'b list" assumes "VectorSpace smult V"   and "set vs \<subseteq> V \<and> V = R_scalar_mult.RSpan UNIV smult vs \<and> R_scalar_mult.R_lin_independent UNIV smult vs" shows "length vs = R_scalar_mult.dim_R UNIV smult V"
assms_2: shows "fv e \<subseteq> set (V # Vs)"
multiplicity_zero: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.multiplicity times one p zero = 0"
sublinear_subadd: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "sublinear t"   and "feasible t" shows "sub_add t"
ujk: shows "unat j * unat k < 2 ^ LENGTH('a)"
concurrent_ops_commute_appendD: fixes xs :: "'a list"   and ys :: "'a list" assumes "concurrent_ops_commute (xs @ ys)" shows "concurrent_ops_commute xs"
assms_1: shows "f \<in> nat_filtration n \<rightarrow>\<^sub>M N"
free_tv_o_subst: fixes S1 :: "nat \<Rightarrow> typ"   and S2 :: "nat \<Rightarrow> typ" shows "free_tv ($ S1 \<circ> S2) \<subseteq> free_tv S1 \<union> free_tv S2"
ring_simprules_8: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "ring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"
IH: shows "finprod (ideals_set R) I {..n} = \<Inter> (I ` {..n})"
heq_3: shows "map of_int_hom.vec_hom fs' ! (i - 1) = map of_int_hom.vec_hom fs ! i"
DF_AF: fixes k :: "nat"   and i :: "nat" shows "DF k i \<lless> AF k i"
casts_null: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list" shows "P \<turnstile> Class C casts Null to Null "
exhaust: fixes y :: "'b + 'c" obtains a :: "'b" where "y = Inl a"   | b :: "'c" where "y = Inr b"
wbt_delete: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "wbt t" shows "wbt (delete x t)"
bindWT_def: shows "bindWT = (\<Lambda> m k. WriterT\<cdot>(runWriterT\<cdot>m \<bind> (\<Lambda> (Writer\<cdot>w\<cdot>x). runWriterT\<cdot>(k\<cdot>x) \<bind> (\<Lambda> (Writer\<cdot>w'\<cdot>y). return\<cdot>(Writer\<cdot>(mappend\<cdot>w\<cdot>w')\<cdot>y)))))"
assms_1: shows "f \<in> {0..<n} \<rightarrow> {0..<m}"
truncate_up_rat_precision: fixes prec :: "nat"   and x :: "nat"   and y :: "nat" shows "truncate_up prec (real x / real y) = round_up (rat_precision prec (int x) (int y)) (real x / real y)"
Dg1_is_gamma3_pathimg: shows "path_image gamma3 = {p. \<exists>x. x \<in> cbox a b \<and> p = (x, g1 x)}"
vbrelation_vcomp_vid_on_right: fixes A :: "V" assumes "\<AA>\<lparr>Obj\<rparr> \<subseteq>\<^sub>\<circ> A" shows "\<FF>\<lparr>ObjMap\<rparr> \<circ>\<^sub>\<circ> vid_on A = \<FF>\<lparr>ObjMap\<rparr>"
sSplice_def: fixes a :: "'a"   and s :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState"   and s' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState" shows "sSplice a s s' \<equiv> s\<lparr>ps := (ps s)(a := ps s' a)\<rparr>"
add_minus_eq_minus: fixes x :: "'a"   and y :: "'a" assumes "x \<in> V"   and "y \<in> V"   and "x + y = (0::'a)" shows "x = - y"
swing_neg_def: fixes x :: "wall"   and y :: "wall" shows "swing_neg x y \<equiv> x = r_under_braid \<and> y = l_under_braid"
ev_alw_imp_alw_ev: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and xs :: "'b stream" assumes "ev (alw \<phi>) xs" shows "alw (ev \<phi>) xs"
mult_1s_1: fixes a :: "'d" shows "Numeral1 * a = a"
sin_integer_2pi: fixes n :: "real" assumes "n \<in> \<int>" shows "sin (2 * pi * n) = 0"
simps_8: fixes i :: "String.literal"   and p :: "fol fm" shows "free_in i (fm.Not p) = free_in i p"
is_rat_def: fixes x :: "real" shows "is_rat x = (x \<in> \<rat>)"
DropToShift: fixes l :: "nat"   and i :: "nat"   and list :: "'b list" assumes "l + i < length list" shows "drop l list ! i = list ! (l + i)"