hcju/isabelleps huggingface dataset:
ele_n_prodTr1: fixes n :: "nat"   and J :: "nat \<Rightarrow> 'a set"   and a :: "'a" shows "(\<forall>k\<le>n. ideal R (J k)) \<and> a \<in> i\<Pi>\<^bsub>R,n\<^esub> J \<longrightarrow> (\<forall>k\<le>n. a \<in> J k)"
assms_1: shows "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> Sec"
subst_introduces_no_lids: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and e :: "('a, 'b, 'c) expr"   and x :: "'c"   and e' :: "('a, 'b, 'c) expr" assumes "substitution subst" shows "LID\<^sub>E (subst e x e') \<subseteq> LID\<^sub>E e \<union> LID\<^sub>E e'"
b: shows "(\<lambda>n. dist (rep_metric_completion x n) (rep_metric_completion z n) + dist (rep_metric_completion y n) (rep_metric_completion z n)) \<longlonglongrightarrow> dist x z + dist y z"
borel_adapt_stoch_proc_borel_measurable: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure"   and X :: "'d \<Rightarrow> 'c \<Rightarrow> 'e" assumes "filtrated_prob_space M F"   and "borel_adapt_stoch_proc F X" shows "\<forall>n. X n \<in> borel_measurable M"
ps_def: shows "ps \<equiv> smods_ext p (pderiv p)"
vp_t: shows "v' \<in> T (shift (snd \<Gamma> - k) 0 (fst (fst \<Gamma> ! n))) \<eta>"
inv_solve_right: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<in> carrier M"   and "b \<in> carrier M"   and "c \<in> carrier M" shows "(a = b \<oplus>\<^bsub>M\<^esub> \<ominus>\<^bsub>M\<^esub> c) = (b = a \<oplus>\<^bsub>M\<^esub> c)"
collapse: fixes rule :: "'c rule" shows "Rule (get_match rule) (get_action rule) = rule"
sample_env_def: fixes sample :: "'i pmf \<Rightarrow> ('i \<Rightarrow> 'h) \<Rightarrow> 'h"   and p :: "'i pmf"   and f :: "'i \<Rightarrow> ('g, 'h) envT" shows "sample_env sample p f = EnvT (\<lambda>r. sample p (\<lambda>x. run_env (f x) r))"
in_mat_kernel_berlekamp_resulting_mat: fixes x :: "'a mod_ring vec" assumes "(berlekamp_mat u)\<^sup>T *\<^sub>v x = x"   and "x \<in> carrier_vec (degree u)" shows "x \<in> mat_kernel (berlekamp_resulting_mat u)"
lebesgue_on_mono: fixes P :: "'b \<Rightarrow> bool"   and S :: "'b set"   and Q :: "'b \<Rightarrow> bool" assumes "almost_everywhere (lebesgue_on S) P"   and "\<And>x. \<lbrakk>P x; x \<in> S\<rbrakk> \<Longrightarrow> Q x" shows "almost_everywhere (lebesgue_on S) Q"
zcf_eq_zero_unique: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'c"   and h :: "nat \<Rightarrow> 'c" assumes "UP_cring R"   and "f \<in> carrier (UP R)"   and "g \<in> carrier (UP R) \<and> f = X_poly R \<otimes>\<^bsub>UP R\<^esub> g"   and "h \<in> carrier (UP R) \<and> f = X_poly R \<otimes>\<^bsub>UP R\<^esub> h" shows "h = g"
diff_numeral_special_2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and m :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "minus (numeral.numeral one plus m) one = neg_numeral.sub minus plus one m num.One"
set_plus_intro2: fixes b :: "'a"   and C :: "'a set"   and a :: "'a" assumes "b \<in> C" shows "a + b \<in> a +o C"
abs_eq: fixes xa :: "'b"   and x :: "('b \<times> real) nonzero" shows "chart_basis xa (Proj x) = chart_basis_nonzero xa x"
pointbased_def: fixes f :: "'b set \<Rightarrow> 'c set" shows "pointbased f = (\<exists>F. f = pointbase F)"
DmdImpl2: fixes F :: "'a \<Rightarrow> bool"   and sigma :: "behavior"   and G :: "'a \<Rightarrow> bool" assumes "sigma \<Turnstile> \<diamond>F"   and "sigma \<Turnstile> \<box>(F \<longrightarrow> G)" shows "sigma \<Turnstile> \<diamond>G"
cobounded2: fixes a :: "'a word"   and b :: "'a word" shows "signed.min a b \<le>s b"
measure_induct: fixes f :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
linear: shows "char_poly A = (\<Prod>a\<leftarrow>as. [:- a, 1::'a:])"
one_element_vec_access: fixes i :: "nat"   and n :: "nat"   and e :: "'a" assumes "i < n" shows "one_element_vec n e $ i = e"
point_add: fixes xA :: "real"   and yA :: "real"   and xB :: "real"   and yB :: "real" shows "Point xA yA + Point xB yB = Point (xA + xB) (yA + yB)"
CHAOS_DF_refine_FD: fixes A :: "'d set" shows "CHAOS A \<sqsubseteq>\<^sub>F\<^sub>D DF A"
norm_memb_eq_rep: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f"   and "t \<in> f" shows "free_monoidal_category.norm C t = free_monoidal_category.rep C f"
frequently_cofinite: fixes P :: "'a \<Rightarrow> bool" shows "Inf_many P = infinite {x. P x}"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_Cons_5: fixes l :: "'d strand_label"   and t :: "('e, 'f) Term.term"   and s :: "('e, 'f) Term.term"   and A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, delete\<langle>t,s\<rangle>) # A) = (l, delete\<langle>t,s\<rangle>) # dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
trdsp_def: fixes bs :: "('t \<Rightarrow>\<^sub>0 'd) list"   and p :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e" shows "trdsp bs p \<equiv> trd bs (spoly (fst (fst p)) (fst (snd p)))"
pw_leq_step: fixes x :: "nat \<times> nat"   and y :: "nat \<times> nat"   and X :: "(nat \<times> nat) multiset"   and Y :: "(nat \<times> nat) multiset" assumes "(x, y) \<in> pair_leq"   and "pw_leq X Y" shows "pw_leq ({#x#} + X) ({#y#} + Y)"
unat_mod: fixes x :: "'a word"   and y :: "'a word" shows "unat (x mod y) = unat x mod unat y"
sel_2: fixes A :: "'c llist set" assumes "\<not> (\<forall>x\<in>A. lnull x)" shows "ltl (lSup A) = lSup (ltl ` (A \<inter> {xs. \<not> lnull xs}))"
collapse: fixes tree :: "'b tree" shows "Node (root tree) (left tree) (right tree) = tree"
extend: shows "\<forall>X\<in>CC. \<forall>Y\<in>CC. X \<sqsubseteq>\<^sup>s Y \<longrightarrow> X \<union> Y \<in> CC"
simps_2: fixes n :: "nat"   and w :: "'a vec"   and u :: "'a vec"   and us :: "'a vec list" shows "adjuster n w (u # us) = - (w \<bullet>c u) / (u \<bullet>c u) \<cdot>\<^sub>v u + adjuster n w us"
diamond_fin_word_step: fixes a :: "'action"   and v :: "'action list"   and p :: "'state" assumes "Ind {a} (set v)"   and "en a p"   and "path v p" shows "path v (ex a p)"
SN_on_subset2: fixes A :: "'a set"   and B :: "'a set"   and r :: "('a \<times> 'a) set" assumes "A \<subseteq> B"   and "SN_on r B" shows "SN_on r A"
1: shows "(x\<^sup>+ \<cdot> 1') ; 1 \<le> (x\<^sup>\<star> \<cdot> x\<^sup>T) ; 1"
dvd_times_right_cancel_iff: 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 a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "a \<noteq> zero" shows "dvd.dvd times (times b a) (times c a) = dvd.dvd times b c"
strict_sorted_equal: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "class.linorder less_eq less"   and "linorder.strict_sorted less xs"   and "linorder.strict_sorted less ys"   and "set ys = set xs" shows "ys = xs"
rel_fun_def: fixes A :: "'n \<Rightarrow> 'p \<Rightarrow> bool"   and B :: "'o \<Rightarrow> 'q \<Rightarrow> bool" shows "A ===> B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
assms_3: shows "x \<in> state_set (state_list s as)"
assms: shows "(if \<sigma> then b0 else b1) = False"
add: shows "\<Prod>\<^sub># F ddvd \<Prod>\<^sub># {#f \<in># F. \<not> f dvd (1::'a)#}"
is_red_singletonD: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "punit.is_red {p} q"   and "p \<in> F" shows "punit.is_red F q"
seqE: fixes g :: "'a"   and f :: "'a" assumes "B.seq g f" obtains "B.arr f" and "B.arr g" and "B.dom g = B.cod f"
continuous_imp_integrable: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on (cbox a b) f" shows "integrable (lebesgue_on (cbox a b)) f"
assms_3: fixes s :: "complex Matrix.vec" shows "U * (|v\<rangle> \<Otimes> |s\<rangle>) = |v\<rangle> \<Otimes> |v\<rangle>"
representation_zero: fixes basis :: "'b set" shows "crepresentation basis (0::'b) = (\<lambda>b. 0)"
1: shows "local.inv f = local.inv (g \<cdot> f) \<cdot> g"
card_le_Suc0_iff_eq: fixes A :: "'b set" assumes "finite A" shows "(card A \<le> Suc 0) = (\<forall>a1\<in>A. \<forall>a2\<in>A. a1 = a2)"
hcomp_vcomp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data"   and \<mu>' :: "'b arrow_of_spans_data"   and \<nu>' :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "partial_magma.arr (span_vertical_category.vcomp C) \<mu>"   and "partial_magma.arr (span_vertical_category.vcomp C) \<nu>"   and "span_vertical_category.src C \<nu> = span_vertical_category.trg C \<mu>"   and "partial_magma.arr (span_vertical_category.vcomp C) \<mu>'"   and "partial_magma.arr (span_vertical_category.vcomp C) \<nu>'"   and "span_vertical_category.src C \<nu>' = span_vertical_category.trg C \<mu>'"   and "partial_magma.arr (span_vertical_category.vcomp C) (span_vertical_category.vcomp C \<mu>' \<mu>)"   and "partial_magma.arr (span_vertical_category.vcomp C) (span_vertical_category.vcomp C \<nu>' \<nu>)" shows "span_bicategory.hcomp C prj0 prj1 (span_vertical_category.vcomp C \<nu>' \<nu>) (span_vertical_category.vcomp C \<mu>' \<mu>) = span_vertical_category.vcomp C (span_bicategory.hcomp C prj0 prj1 \<nu>' \<mu>') (span_bicategory.hcomp C prj0 prj1 \<nu> \<mu>)"
R_def: shows "R \<equiv> Min {r, fps_conv_radius f, fps_conv_radius g}"
1: fixes xs :: "nat list"   and x :: "nat" assumes "xs \<noteq> []"   and "x = hd xs" shows "x # tl xs = xs"
Bernstein_Poly_def: fixes j :: "nat"   and p :: "nat"   and c :: "real"   and d :: "real" shows "Bernstein_Poly j p c d = smult (real (p choose j) / (d - c) ^ p) (monom 1 j \<circ>\<^sub>p [:- c, 1:] * monom 1 (p - j) \<circ>\<^sub>p [:d, - 1:])"
tree_def: fixes g :: "'b ugraph" shows "tree g \<equiv> Undirected_Graph.connected g \<and> cycle_free g"
pos_add: fixes X :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'b"   and Y :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "pos X"   and "pos Y" shows "pos (X + Y)"
fract_poly_eq_iff: fixes p :: "'a poly"   and q :: "'a poly" shows "(map_poly to_fract p = map_poly to_fract q) = (p = q)"
IH_1: assumes "invar l" shows "invar (update a b l)"
list_to_lpoly_dim_less: fixes cs :: "rat list" shows "dim_poly (list_to_lpoly cs) \<le> length cs"
eventually_True: fixes F :: "'c filter" shows "\<forall>\<^sub>F x in F. True"
ORRIGHT1_eqvt_name: fixes pi :: "(name \<times> name) list"   and A :: "ty"   and B :: "ty"   and X :: "ctrm set" shows "pi \<bullet> ORRIGHT1 (A OR B) X = ORRIGHT1 (A OR B) (pi \<bullet> X)"
linear_orderable_1_implies_2: fixes x :: "'a" assumes "linear_orderable_1 x" shows "linear_orderable_2 (x \<sqinter> - (1::'a))"
1: shows "hom_induced p (subtopology (nsphere n) {x. - 1 / 2 \<le> x k}) {x. - 1 / 2 \<le> x k \<and> x k \<le> 0} (nsphere n) {x. x k \<le> 0} id \<in> Group.iso (relative_homology_group p (subtopology (nsphere n) {x. - 1 / 2 \<le> x k}) {x. - 1 / 2 \<le> x k \<and> x k \<le> 0}) (relative_homology_group p (nsphere n) {x. x k \<le> 0})"
check_no_overlap_def: fixes \<gamma> :: "'d set \<Rightarrow> 'e \<Rightarrow> bool"   and ft :: "('d, 'f) flow_entry_match list" shows "check_no_overlap \<gamma> ft = (\<forall>a\<in>set ft. \<forall>b\<in>set ft. \<forall>p\<in>UNIV. ofe_prio a = ofe_prio b \<and> \<gamma> (ofe_fields a) p \<and> a \<noteq> b \<longrightarrow> \<not> \<gamma> (ofe_fields b) p)"
aux_lemma_lose_2: fixes d :: "nat"   and k :: "nat"   and lose :: "nat \<Rightarrow> bool list"   and t :: "nat" assumes "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]" shows "\<forall>x\<le>d + 1. lose (t + x) = [False]"
f_succ_finite: fixes \<delta> :: "('d, 'e) ta_rule set" assumes "finite \<delta>" shows "finite (f_succ \<delta>)"
map_def: shows "\<Psi>'.map = transformation_by_components.map (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) F (\<lambda>a. S.inv (\<Phi>' a))"
convex_affine_rel_frontier_Int: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "affine T"   and "interior S \<inter> T \<noteq> {}" shows "rel_frontier (S \<inter> T) = frontier S \<inter> T"
intro: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c" assumes "ParityGame G"   and "vm_path_axioms G P v0" shows "vm_path G P v0"
isomorphic_transitive: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "G.B.isomorphic f g"   and "G.B.isomorphic g h" shows "G.B.isomorphic f h"
blinfun_of_cblinfun_inj: fixes f :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and g :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "blinfun_of_cblinfun f = blinfun_of_cblinfun g" shows "f = g"
Fxyz: shows "finite_entropy (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Y x, Z x)) Pxyz"
assms: fixes i :: "'n" shows "norm (x $ i) \<le> norm (y $ i)"
rOfWax_def: fixes \<sigma> :: "'b" shows "rOfWax \<sigma> = Fn (Wit \<sigma>) []"
IH: fixes s :: "('a, 'b) fmap"   and vs :: "'a set" shows "exec_plan s (as_proj (rem_effectless_act as) vs) = exec_plan s (as_proj as vs)"
weak_respection_of_barbs_and_closures_5: fixes Rel :: "('b \<times> 'b) set"   and CWB :: "('b, 'c) calculusWithBarbs" assumes "rel_weakly_respects_barbs Rel CWB" shows "rel_weakly_respects_barbs (Rel\<^sup>*) CWB"
abs_eq: fixes xa :: "circline_mat"   and x :: "complex_homo_coords" shows "on_ocircline (abs_ocircline xa) (abs_complex_homo x) = on_circline_clmat_hcoords xa x"
congruenceWeakBisim: fixes P :: "pi"   and Q :: "pi" assumes "P \<simeq> Q" shows "P \<approx> Q"
simps_11: fixes v :: "'a acom"   and va :: "'a acom"   and vb :: "char list"   and vc :: "aexp"   and vd :: "'a" shows "(v;; va \<sqsubseteq> vb ::= vc {vd}) = False"
rep_eq: shows "fls_nth 1 = (\<lambda>k. if k = 0 then 1::'a else (0::'a))"
bisim1CASThrow3: fixes P :: "'b prog"   and h :: "'heap"   and e3 :: "'addr expr1"   and ad :: "'addr"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and e1 :: "'addr expr1"   and D :: "String.literal"   and F :: "String.literal"   and e2 :: "'addr expr1"   and v' :: "'addr val"   and v :: "'addr val" assumes "P,e3,h \<turnstile> (Throw ad, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>ad\<rfloor>)" shows "P,e1\<bullet>compareAndSwap(D\<bullet>F, e2, e3),h \<turnstile> (Throw ad, xs) \<leftrightarrow> (stk @ [v', v], loc, length (compE2 e1) + length (compE2 e2) + pc, \<lfloor>ad\<rfloor>)"
assume_seq: fixes b :: "'b upred"   and c :: "'b upred" shows "?[b] ;; ?[c] = ?[b \<and> c]"
TopoS_List_Impl_def: fixes default_node_properties :: "'a"   and sinvar_spec :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> bool"   and sinvar_impl :: "'b list_graph \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> bool"   and receiver_violation :: "bool"   and offending_flows_impl :: "'b list_graph \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> ('b \<times> 'b) list list"   and node_props_impl :: "('b, 'a) TopoS_Params \<Rightarrow> 'b \<Rightarrow> 'a"   and eval_impl :: "'b list_graph \<Rightarrow> ('b, 'a) TopoS_Params \<Rightarrow> bool" shows "TopoS_List_Impl default_node_properties sinvar_spec sinvar_impl receiver_violation offending_flows_impl node_props_impl eval_impl \<equiv> (SecurityInvariant sinvar_spec default_node_properties receiver_violation \<and> (\<forall>G nP. wf_list_graph G \<longrightarrow> sinvar_spec (list_graph_to_graph G) nP = sinvar_impl G nP)) \<and> (\<forall>G nP. wf_list_graph G \<longrightarrow> SecurityInvariant_withOffendingFlows.set_offending_flows sinvar_spec (list_graph_to_graph G) nP = set ` set (offending_flows_impl G nP)) \<and> (\<forall>P. SecurityInvariant.node_props_formaldef default_node_properties P = node_props_impl P) \<and> (\<forall>G P. (distinct (nodesL G) \<and> distinct (edgesL G) \<and> SecurityInvariant.eval sinvar_spec default_node_properties (list_graph_to_graph G) P) = eval_impl G P)"
simps_2: fixes F :: "'b formula"   and Fs :: "'b formula list" shows "\<^bold>\<And>F # Fs = F \<^bold>\<and> \<^bold>\<And>Fs"
cases: fixes x :: "('a, 'b) BinomialHeapStruc.BinomialTree list" obtains "x = []"   | t :: "('a, 'b) BinomialHeapStruc.BinomialTree" and q :: "('a, 'b) BinomialHeapStruc.BinomialTree list" where "x = t # q"
curry_case_prod: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "curry (\<lambda>(x, y). f x y) = f"
length_Suc_rev_conv: fixes xs :: "'a list"   and n :: "nat" shows "(length xs = Suc n) = (\<exists>ys y. xs = ys @ [y] \<and> length ys = n)"
geqr: fixes x :: "'a" assumes "x \<in> W0 \<union> S" shows "g x = r x"
simps_11: fixes V :: "char list"   and T :: "ty"   and e :: "expr"   and A :: "char list set option" shows "\<D> {V:T; e} A = \<D> e (A \<ominus> V)"
im_neg: fixes g :: "'g" shows "T (- g) = - T g"
l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.known_ptr ShadowRootClass.type_wf Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs ShadowRootClass.known_ptrs Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel get_disconnected_nodes"
subdivision_gen_contr: fixes G :: "'b pair_pre_digraph"   and V :: "'b set" assumes "pair_pseudo_graph G"   and "pair_pre_digraph.is_slim G V" shows "subdivision_pair (gen_contr_graph (with_proj G) V) G"
vector_smult_rzero: fixes c :: "'a" shows "c *s 0 = 0"
cone_axioms: shows "D.cone (C.dom p0) (D.mkCone p0 p1)"
min_less_iff_conj: fixes z :: "'c word"   and x :: "'c word"   and y :: "'c word" shows "(z <s signed.min x y) = (z <s x \<and> z <s y)"
strict_iff_order: fixes b :: "'d list"   and a :: "'d list" shows "strict_prefix b a = (prefix b a \<and> a \<noteq> b)"
welltyped_open_Var_close_Var: fixes t :: "expr"   and T :: "Unary_PCF.type"   and xT :: "char list \<times> Unary_PCF.type" assumes "t ::: T" shows "open0_Var xT (close0_Var xT t) ::: T"
F2set1_incl_JF2set: fixes a :: "'d JF2" shows "F2set1 (dtor2 a) \<subseteq> JF2set a"
polyhedron_Int_affine: fixes S :: "'b set" shows "polyhedron S = (\<exists>F. finite F \<and> S = affine hull S \<inter> \<Inter> F \<and> (\<forall>h\<in>F. \<exists>a b. a \<noteq> (0::'b) \<and> h = {x. a \<bullet> x \<le> b}))"
G_def: shows "G \<equiv> \<lambda>x. \<Sum>n. moebius_mu (n + Suc (nat \<lfloor>x\<rfloor>)) / real ((n + Suc (nat \<lfloor>x\<rfloor>))\<^sup>2)"
ofilter_def: fixes A :: "('c \<Rightarrow> 'a) set" shows "rexpt.ofilter A \<equiv> A \<subseteq> Field rt.oexp \<and> (\<forall>a\<in>A. rexpt.under a \<subseteq> A)"
assms: shows "(P, Q) \<in> {(P, P') |P P'. P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'}\<^sup>*"
Inequality: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) extstrand_step list"   and X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> S"   and "ineq_model \<I> X F" shows "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> (S @ [Step \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])"
num: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream" assumes "ev (\<lambda>xs. P (shd xs)) xs" shows "(list_all (Not \<circ> P) (stake (num P xs) xs) \<and> P (xs !! num P xs)) \<and> (\<forall>m. list_all (Not \<circ> P) (stake m xs) \<and> P (xs !! m) \<longrightarrow> num P xs \<le> m)"
option_admissible: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "option.admissible (\<lambda>f. \<forall>x y. f x = Some y \<longrightarrow> P x y)"
split_zip_ex: fixes P :: "('d \<times> 'e) list \<Rightarrow> bool" shows "(\<exists>x. P x) = (\<exists>y z. length y = length z \<and> P (y || z))"
fds_truncate_deriv: fixes m :: "nat"   and f :: "'b fds" shows "fds_truncate m (fds_deriv f) = fds_deriv (fds_truncate m f)"
create_claim_def: fixes ts :: "traffic"   and c :: "cars"   and n :: "nat"   and ts' :: "traffic" shows "ts \<^bold>\<midarrow>c( c, n ) \<^bold>\<rightarrow> ts' \<equiv> pos ts' = pos ts \<and> res ts' = res ts \<and> dyn ts' = dyn ts \<and> physical_size ts' = physical_size ts \<and> braking_distance ts' = braking_distance ts \<and> |clm ts c| = 0 \<and> |res ts c| = 1 \<and> (n + 1 \<^bold>\<in> res ts c \<or> n - 1 \<^bold>\<in> res ts c) \<and> clm ts' = (clm ts)(c := Abs_nat_int {n})"
digraphD_3: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "vsv (\<CC>\<lparr>Dom\<rparr>)"
hcmod_mult_i: fixes y :: "real star" shows "hcmod (iii * hcomplex_of_hypreal y) = \<bar>y\<bar>"
less_eq_sat_def: fixes x :: "'a sat"   and y :: "'a sat" shows "(x \<le> y) = (nat_of x \<le> nat_of y)"
minus_le_iff: fixes a :: "'a"   and b :: "'a" shows "(- a \<le> b) = (- b \<le> a)"
assms_3: fixes x :: "'a" assumes "x \<in> A" shows "{x} \<in> sets M"
L_flat_Prf1: fixes v :: "'b val"   and r :: "'b rexp" assumes "\<turnstile> v : r" shows "flat v \<in> lang r"
le_minus_one_simps_1: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (uminus one) zero"
bind_assoc: fixes m :: "'a + 'd"   and f :: "'d \<Rightarrow> 'a + 'c"   and g :: "'c \<Rightarrow> 'a + 'b" shows "m \<bind> f \<bind> g = m \<bind> (\<lambda>x. f x \<bind> g)"
pe_q': shows "set (vwalk_arcs (tl q)) \<subseteq> arcs_ends G"
iso_set_sym: fixes G :: "('c, 'd) monoid_scheme"   and h :: "'c \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme" assumes "Group.group G"   and "h \<in> Group.iso G H" shows "inv_into (carrier G) h \<in> Group.iso H G"
less_le_trans: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y < x"   and "z \<le> y" shows "z < x"
l1_step4_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and gnx :: "msg" shows "l1_step4 Rb A B gnx \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress s Rb = Some {xny, xgnx, xgny, xsk} \<and> guessed_frame Rb xgnx = Some gnx \<and> (can_signal s A B \<longrightarrow> (\<exists>Ra. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> in_progressS (progress s Ra) {xnx, xgnx, xgny, xsk, xEnd} \<and> guessed_frame Ra xgnx = Some gnx \<and> guessed_frame Ra xgny = Some (Exp Gen (NonceF (Rb $ ny))))) \<and> (Rb = test \<longrightarrow> Exp gnx (NonceF (Rb $ ny)) \<notin> synth (analz (ik s))) \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xny, xgnx, xgny, xsk, xEnd}), secret := {x. x = Exp gnx (NonceF (Rb $ ny)) \<and> Rb = test} \<union> secret s, signalsResp := if can_signal s A B then addSignal (signalsResp s) (Commit A B (Exp gnx (NonceF (Rb $ ny)))) else signalsResp s\<rparr>}"
gcd_mod_right: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)" shows "gcd a (b mod a) = gcd a b"
vintersection_vempty: fixes A :: "V" shows "0 \<subseteq>\<^sub>\<circ> A"
subcls_conv_subcls': fixes P :: "'a prog" shows "subcls P = subcls' P"
CNOT_def: shows "CNOT \<equiv> Matrix.mat 4 4 (\<lambda>(i, j). if i = 0 \<and> j = 0 then 1 else if i = 1 \<and> j = 1 then 1 else if i = 2 \<and> j = 3 then 1 else if i = 3 \<and> j = 2 then 1 else 0)"