hcju/isabelleps huggingface dataset:
get_M_pure: fixes ptr :: "(_) element_ptr"   and getter :: "('z, 'x, 'y, 'aa, 'ad option) RElement_scheme \<Rightarrow> 'ag"   and h :: "(_) heap" shows "pure (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h"
dterms_psimps_10: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and pn :: "'a" assumes "wellformed \<Gamma>" shows "dterms \<Gamma> (call(pn)) = dterms \<Gamma> (\<Gamma> pn)"
\<sigma>_set_def: shows "\<sigma>_set = lset (path_strategies P)"
affine_Int: fixes s :: "'a set"   and t :: "'a set" assumes "affine s"   and "affine t" shows "affine (s \<inter> t)"
paths_mono_iff: fixes t :: "'b ttree"   and t' :: "'b ttree" shows "(paths t \<sqsubseteq> paths t') = (t \<sqsubseteq> t')"
generate_one_switched_eqI: fixes A :: "'a set"   and a :: "'a"   and B :: "'a set"   and b :: "'a" assumes "A \<subseteq> carrier G"   and "a \<in> A"   and "B = A - {a} \<union> {b}"   and "b \<in> generate G A"   and "a \<in> generate G B" shows "generate G A = generate G B"
val_exp: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and z :: "int" assumes "valuation K v"   and "x \<in> carrier K"   and "x \<noteq> \<zero>" shows "z *\<^sub>a v x = v (x\<^bsub>K\<^esub>\<^bsup>z\<^esup>)"
regular_def: fixes R :: "(ZF \<times> ZF) set" shows "regular R \<equiv> \<forall>A. A \<noteq> Empty \<longrightarrow> (\<exists>x. Elem x A \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> \<not> Elem y A))"
fix_least_below: fixes F :: "'d \<rightarrow> 'd"   and x :: "'d" assumes "F\<cdot>x \<sqsubseteq> x" shows "fix\<cdot>F \<sqsubseteq> x"
restrict_upd_same: fixes m :: "'b \<Rightarrow> 'c option"   and x :: "'b"   and y :: "'c" shows "m(x \<mapsto> y) |` (- {x}) = m |` (- {x})"
lt_in_keys: fixes p :: "'t \<Rightarrow>\<^sub>0 'd" assumes "p \<noteq> 0" shows "lt p \<in> keys p"
path_conforms_with_strategy_irrelevant: fixes p :: "Player"   and P :: "'a llist"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v :: "'a"   and w :: "'a" assumes "path_conforms_with_strategy p P \<sigma>"   and "v \<notin> lset P" shows "path_conforms_with_strategy p P (\<sigma>(v := w))"
check_ipresIGAbs: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "ipresIGAbs check check (errMOD MOD) MOD"
concurrent_append_store_independent_technical: fixes i :: "'a"   and mo :: "'a"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and e :: "'b"   and r :: "'a" assumes "i = mo"   and "xs prefix of j"   and "(i, Append i e) \<in> set (node_deliver_messages xs)"   and "(r, Store e mo r) \<in> set (node_deliver_messages xs)" shows "hb (i, Append i e) (r, Store e mo r)"
matrix_to_iarray_fst_Gauss_Jordan_column_k_PA: fixes i :: "nat"   and A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and P :: "(('b, 'd) vec, 'd) vec" assumes "i \<le> nrows A"   and "k < ncols A" shows "matrix_to_iarray (fst (Gauss_Jordan_column_k_PA (P, i, A) k)) = fst (Gauss_Jordan_column_k_iarrays_PA (matrix_to_iarray P, i, matrix_to_iarray A) k)"
snoclr: fixes xs :: "'a list"   and ys :: "'b list"   and x :: "'a"   and y :: "'b" assumes "P xs ys" shows "P (xs @ [x]) (ys @ [y])"
rotate1_append: fixes xs :: "'a list"   and ys :: "'a list" shows "rotate1 (xs @ ys) = (if xs = [] then rotate1 ys else tl xs @ ys @ [hd xs])"
thetaIfTT_RetrT: shows "thetaIfTT \<subseteq> RetrT (thetaIfTT \<union> BisT)"
set_9: fixes x2 :: "('e, 'd) Term.term" shows "vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (receive\<langle>x2\<rangle>) = fv x2"
C: fixes x :: "hf"   and y :: "hf"   and z :: "hf" assumes "C x y z"   and "P y"   and "P z" shows "P x"
M_def: fixes l :: "nat" shows "M l = mat K K (\<lambda>(i, j). if i = j \<and> l \<le> i then 1 else 0)"
pell_valuation_cnj_solution: fixes z :: "nat \<times> nat" assumes "solution z" shows "pell_valuation (pell_cnj (case z of (x, y) \<Rightarrow> (int x, int y))) = 1 / pell_valuation (case z of (x, y) \<Rightarrow> (int x, int y))"
foldl_create_initial_object_False: fixes h :: "'heap"   and ads :: "'addr list"   and Cs :: "String.literal list" shows "foldl create_initial_object (h, ads, False) Cs = (h, ads, False)"
degree_pdevs_of_list_le_length: fixes xs :: "'b list" shows "degree (pdevs_of_list xs) \<le> length xs"
finite_Lam: fixes rhs :: "'b trm set" assumes "finite rhs" shows "finite {r. Lam r \<in> rhs}"
preserves_dom: fixes f :: "'a" assumes "C.arr f" shows "D.dom (F f) = F (C.dom f)"
ex2: fixes X :: "'b program set" assumes "\<forall>GG. finite GG \<and> GG \<inter> X \<noteq> {} \<longrightarrow> OK GG (\<lambda>G. G) \<longrightarrow> (\<Squnion>G\<in>GG. G) \<in> X" shows "ex_prop X"
arg_cong2: fixes a :: "'a"   and b :: "'a"   and c :: "'b"   and d :: "'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'd" assumes "a = b"   and "c = d" shows "f a c = f b d"
alg_min_alg: fixes s1 :: "('f, 'd, 'e) F1 \<Rightarrow> 'd"   and s2 :: "('f, 'd, 'e) F2 \<Rightarrow> 'e" shows "alg (min_alg1 s1 s2) (min_alg2 s1 s2) s1 s2"
LTL_prop_entailment_monotonI: fixes S :: "'b ltl set"   and \<phi> :: "'b ltl"   and S' :: "'b ltl set" assumes "S \<Turnstile>\<^sub>P \<phi>"   and "S \<subseteq> S'" shows "S' \<Turnstile>\<^sub>P \<phi>"
twotox: fixes bs :: "nat list"   and y :: "nat"   and init :: "nat list"   and x :: "nat"   and h :: "nat list"   and as :: "nat list" assumes "count_list bs y \<le> 1"   and "distinct init"   and "x \<in> set init"   and "y \<in> set init"   and "x \<notin> set bs"   and "x \<noteq> y" shows "x < y in s_TS init h (as @ [x] @ bs @ [x]) (length (as @ [x] @ bs @ [x]))"
beh_in_reach: fixes b :: "'a list"   and T :: "('a, 'b) TS_scheme"   and s :: "'a" assumes "b \<in> beh T"   and "s \<in> set b" shows "s \<in> reach T"
conjugate_neg: fixes conjugate :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.conjugatable_ring conjugate plus zero minus uminus times" shows "conjugate (uminus a) = uminus (conjugate a)"
plossless_inline_invariant: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and \<I>' :: "('d, 'e) \<I>"   and s :: "('a, 'b, 'd, 'e) converter" assumes "plossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s x. \<lbrakk>x \<in> outs_\<I> \<I>; plossless_converter \<I> \<I>' s \<and> \<I>, \<I>' \<turnstile>\<^sub>C s \<surd>\<rbrakk> \<Longrightarrow> plossless_gpv \<I>' (run_converter s x)"   and "plossless_converter \<I> \<I>' s \<and> \<I>, \<I>' \<turnstile>\<^sub>C s \<surd>" shows "plossless_gpv \<I>' (inline run_converter gpv s)"
g_2: shows "linear_on (tangent_space p) (sub.tangent_space p) (*\<^sub>R) (*\<^sub>R) g"
qty_mult_comp_def: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and qty :: "nat \<Rightarrow> 'd \<Rightarrow> real" shows "qty_mult_comp pf1 qty = (\<lambda>x n w. pf1 x n w * qty n w)"
euclid_5__original_euclid: assumes "Euclid5" shows "EuclidSParallelPostulate"
fg_norm: shows "norm2 f = (cmod (lead_coeff f))\<^sup>2 * norm2 (\<Prod>a\<leftarrow>complex_roots_complex f. if 1 < cmod a then [:- 1, cnj a:] else [:- a, 1:])"
set_borel_measurable_sets: fixes M :: "'a measure"   and X :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'b set" assumes "set_borel_measurable M X f"   and "B \<in> sets borel"   and "X \<in> sets M" shows "f -` B \<inter> X \<in> sets M"
ord_option_simps_1: fixes ord :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d option" shows "ord_option ord None x = True"
rev_cases: fixes xs :: "'g list" obtains "xs = []"   | ys :: "'g list" and y :: "'g" where "xs = ys @ [y]"
zin: shows "z \<in> inside (path_image (subpath t 0 q +++ linepath (a - complex_of_real d) (a + complex_of_real e)))"
qequiv_sym: fixes a :: "'a['b, 'c]"   and b :: "'a['d, 'c]" assumes "a \<cong>\<^sub>Q b" shows "b \<cong>\<^sub>Q a"
rev_steps_app: fixes ok :: "nat"   and f :: "'b list"   and f' :: "'b list"   and r :: "'b list"   and r' :: "'b list" assumes "st_inv (Rev ok f f' r r')" shows "steps (length f + 1) (Rev ok f f' r r') = App (length f + ok) (rev f @ f') (rev r @ r')"
l_get_dom_component_get_scdom_component\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component_get_scdom_component\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM_SC_DOM_Components.get_scdom_component Shadow_DOM_SC_DOM_Components.is_strongly_scdom_component_safe Shadow_DOM_SC_DOM_Components.is_weakly_scdom_component_safe Shadow_DOM.get_owner_document get_disconnected_nodes get_disconnected_nodes_locs Shadow_DOM.to_tree_order Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs Shadow_DOM_DOM_Components.get_dom_component Shadow_DOM_DOM_Components.is_strongly_dom_component_safe Shadow_DOM_DOM_Components.is_weakly_dom_component_safe Shadow_DOM.get_root_node Shadow_DOM.get_root_node_locs Shadow_DOM.get_ancestors Shadow_DOM.get_ancestors_locs Shadow_DOM.get_element_by_id Shadow_DOM.get_elements_by_tag_name"
zero_not_one: shows "\<zero>\<^bsub>UP R\<^esub> \<noteq> \<one>\<^bsub>UP R\<^esub>"
pmc_vacuity: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "\<not> K \<turnstile> \<phi>" shows "K \<div>\<^bsub>\<gamma>\<^sub>T\<^sub>R\<^esub> \<phi> = K"
cInf_le_finite: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "finite X"   and "x \<in> X" shows "less_eq (Inf X) x"
ereal_abs_leI: fixes x :: "ereal"   and y :: "ereal" assumes "x \<le> y"   and "- x \<le> y" shows "\<bar>x\<bar> \<le> y"
set_plus_mono4_b: fixes a :: "'a"   and C :: "'a set"   and x :: "'a"   and D :: "'a set" assumes "a \<in> C"   and "x \<in> a +o D" shows "x \<in> D + C"
bis_raw_coind2: fixes Retr :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   and theta :: "('a \<times> 'a) set" assumes "Bisim.mono Retr"   and "theta \<subseteq> Retr theta"   and "theta\<inverse> \<subseteq> Retr (theta\<inverse>)" shows "theta \<subseteq> bis Retr"
intro: fixes Q :: "'a set"   and Bot_F :: "'b set"   and Inf_F :: "'b inference set"   and Bot_G :: "'c set"   and entails_q :: "'a \<Rightarrow> 'c set \<Rightarrow> 'c set \<Rightarrow> bool"   and Inf_G_q :: "'a \<Rightarrow> 'c inference set"   and Red_I_q :: "'a \<Rightarrow> 'c set \<Rightarrow> 'c inference set"   and Red_F_q :: "'a \<Rightarrow> 'c set \<Rightarrow> 'c set"   and \<G>_F_q :: "'a \<Rightarrow> 'b \<Rightarrow> 'c set"   and \<G>_I_q :: "'a \<Rightarrow> 'b inference \<Rightarrow> 'c inference set option"   and Prec_F_g :: "'c \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>q\<in>Q. tiebreaker_lifting Bot_F Inf_F Bot_G (entails_q q) (Inf_G_q q) (Red_I_q q) (Red_F_q q) (\<G>_F_q q) (\<G>_I_q q) Prec_F_g" shows "lifting_intersection_axioms Inf_F Bot_G Q Inf_G_q entails_q Red_I_q Red_F_q Bot_F \<G>_F_q \<G>_I_q Prec_F_g"
that: fixes r :: "'a set set" assumes "p \<subseteq> r"   and "r division_of T" shows "thesis"
case_prodE: fixes c :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and p :: "'e \<times> 'f" assumes "case p of (a, b) \<Rightarrow> c a b" obtains x :: "'e" and y :: "'f" where "p = (x, y)" and "c x y"
in_homE: fixes f :: "'s"   and a :: "'s"   and b :: "'s" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>S b\<guillemotright>" obtains "S.arr f" and "S.dom f = a" and "S.cod f = b"
rtos_hom3: fixes r :: "nat"   and s :: "nat"   and i :: "nat" assumes "0 < r"   and "0 < s"   and "i \<le> r * s - Suc 0" shows "rtos r s i div r = i mod s"
lt_plus_distinct_eq_max: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.lt p \<noteq> punit.lt q" shows "punit.lt (p + q) = ordered_powerprod_lin.max (punit.lt p) (punit.lt q)"
tan_diff: shows "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)"
A: fixes x :: "'a list"   and y :: "'a list" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
Lambda: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp"   and L :: "Vars.var list" shows "\<Gamma> : Lam [x]. e \<Down>\<^bsub>L\<^esub> \<Gamma> : Lam [x]. e"
integral_cnj: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> complex" shows "CLINT x|M. cnj (f x) = cnj (integral\<^sup>C M f)"
center_def: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "center x y z = (SOME t. t \<in> {x--y} \<inter> {x--z} \<inter> {y--z})"
HFail_Inv2a_dblock_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HFail s s' p"   and "Inv2a_innermost s q (dblock s q)" shows "Inv2a_innermost s' q (dblock s' q)"
axioms_2: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" assumes "monoid f z" shows "monoid_axioms f z"
mset_lit_subst: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and L :: "'b literal"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.mset_lit (subst_lit L \<sigma>) = {#x \<lhd> \<sigma>. x \<in># basic_superposition.mset_lit L#}"
hi: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (reduced_homology_group (int p) (nsphere p))" shows "hom_induced (int p) (nsphere p) {} (nsphere p) {} f x = x [^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> d"
assms: shows "finite (A \<times> B) \<and> card A * card B \<le> 1"
bounded_def: fixes P :: "'b \<Rightarrow> real" shows "Expectations.bounded P \<equiv> \<exists>b. bounded_by b P"
P: shows "bij_betw p {0..<card {x \<in> A. P x}} {x \<in> A. P x}"
higher_deriv_Polygamma_approx: fixes j :: "nat"   and i :: "nat"   and m :: "nat" shows "(deriv ^^ j) (Polygamma_approx i m) = Polygamma_approx (j + i) m"
iso_fgraph_refl: fixes g :: "'a list list" shows "g \<simeq> g"
sum_constant: fixes one :: "'q"   and times :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and y :: "'q"   and A :: "'r set" assumes "class.semiring_1 one times plus zero" shows "comm_monoid_add.sum plus zero (\<lambda>x. y) A = times (semiring_1.of_nat one plus zero (card A)) y"
sub: shows "set (solve_depressed_cubic_complex (cor e) (cor f)) \<subseteq> \<real>"
rep_eq: fixes x :: "'a regular"   and xa :: "'a regular" shows "Rep_regular (x \<sqinter> xa) = Rep_regular x \<sqinter> Rep_regular xa"
R23sD: fixes s :: "l2_state"   and t :: "l3_state" assumes "(s, t) \<in> R23s" shows "ik s = ik t \<inter> payload \<and> secret s = secret t \<and> progress s = progress t \<and> signalsInit s = signalsInit t \<and> signalsResp s = signalsResp t \<and> signalsInit2 s = signalsInit2 t \<and> signalsResp2 s = signalsResp2 t \<and> chan s = local.abs (ik t) \<and> l2_state.bad s = l3_state.bad t"
assms_1: shows "(\<lambda>x. f x (h1 x)) \<sim>[F] (\<lambda>x. g x (h1 x))"
irreducible_imp_primitive: fixes f :: "'b poly" assumes "irreducible f"   and "Polynomial.degree f \<noteq> 0" shows "Missing_Polynomial_Factorial.primitive f"
promela_language_ltl_def: fixes promg :: "program \<times> expr iarray \<times> gState" shows "promela_language_ltl promg \<equiv> let (prog, APs, g) = promg in {promela_props_ltl APs \<circ> r |r. promela_is_run_ltl promg r}"
prems_2: shows "P,shp s \<turnstile>\<^sub>b (RI (C,e) ; [] \<leftarrow> e',b) \<surd>"
rep_eq: fixes x :: "'d" shows "blinfun_apply (blinfun_mult_right x) = (*) x"
All_init_leD: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and t :: "Term.term"   and Q :: "'b assn" assumes "\<forall>n. G,A\<turnstile>{P \<and>. G\<turnstile>init\<le>n} t\<succ> {Q}" shows "G,A\<turnstile>{P} t\<succ> {Q}"
binrelchain_funcong_extra_condition_Cons_snoc: fixes Q :: "'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and zs :: "'b list"   and y :: "'b" assumes "\<And>x y. \<lbrakk>Q x; P x y\<rbrakk> \<Longrightarrow> Q y"   and "\<And>x y. \<lbrakk>Q x; P x y\<rbrakk> \<Longrightarrow> f y = f x"   and "Q x"   and "binrelchain P (x # zs @ [y])" shows "f y = f x"
e: shows "(2\<^sup>2 * (3 * 2 ^ (2 - 1) - 1) * (3 * 2\<^sup>2 - 1)) Amic (2\<^sup>2 * (9 * 2 ^ (2 * 2 - 1) - 1))"
crel_vs_execute_Some: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "['d]"   and heap :: "heap" assumes "crel_vs R a b"   and "P heap"   and "Q heap"   and "state_dp_consistency.cmem heap" obtains x :: "'d" and heap' :: "heap" where "execute b heap = Some (x, heap')" and "P heap'" and "Q heap'"
Collect_conv_if: fixes a :: "'e"   and P :: "'e \<Rightarrow> bool" shows "{x. x = a \<and> P x} = (if P a then {a} else {})"
wf_dia_inv': fixes V :: "node fset"   and \<Lambda> :: "node \<Rightarrow> assertion_gadget"   and E :: "(node fset \<times> command_gadget \<times> node fset) list" assumes "wf_dia (Graph V \<Lambda> E)" obtains "\<forall>e\<in>set E. wf_com (snd3 e)" and "\<forall>v\<in>fset V. wf_ass (\<Lambda> v)" and "acyclicity E" and "linearity E" and "\<forall>e\<in>set E. fst3 e |\<subseteq>| V \<and> thd3 e |\<subseteq>| V"
get_ancestors_si_locs_def: shows "get_ancestors_si_locs \<equiv> l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_si_locs Shadow_DOM.get_parent_locs get_host_locs"
minus_zero: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.group_add minus plus zero uminus" shows "uminus zero = zero"
trans: fixes x :: "real \<times> real"   and y :: "real \<times> real"   and z :: "real \<times> real" assumes "compare x y = Lt"   and "compare y z = Lt" shows "compare x z = Lt"
distinct_5: fixes x1 :: "'e"   and x4 :: "nat" shows "broadcast x1 \<noteq> \<not>unicast x4"
elims_3: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c list" assumes "class.linorder less_eq less"   and "\<not> linorder.sorted less_eq x" obtains x :: "'c" and ys :: "'c list" where "xa__ = x # ys" and "\<not> (Ball (set ys) (less_eq x) \<and> linorder.sorted less_eq ys)"
ab_prop: shows "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& \<lparr>A!,b\<^sup>P\<rparr> \<^bold>& a \<^bold>\<noteq> b \<^bold>& (\<^bold>\<lambda>z. \<lparr>\<^bold>\<lambda>\<^sup>2 (\<lambda>x y. \<^bold>\<forall>F. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>),z\<^sup>P,a\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>\<^bold>\<lambda>\<^sup>2 (\<lambda>x y. \<^bold>\<forall>F. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>),z\<^sup>P,b\<^sup>P\<rparr>) in v]"
intro: fixes Ch :: "'d \<Rightarrow> 'e set \<Rightarrow> 'e set" assumes "\<forall>h. lad (Ch h)" shows "ContractsWithLAD_axioms Ch"
assms_2: shows "z \<cdot> e' \<le> e"
seq_char: fixes g :: "'c local.term set \<times> 'c local.term set"   and f :: "'c local.term set \<times> 'c local.term set" shows "\<F>C.CC.seq g f = (\<F>C.seq (fst g) (fst f) \<and> \<F>C.seq (snd g) (snd f))"
AVar: fixes G :: "prog"   and s0 :: "st"   and e1 :: "expr"   and a :: "val"   and s1 :: "state"   and e2 :: "expr"   and i :: "val"   and s2 :: "state"   and v :: "vvar"   and s2' :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e1-\<succ>a\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>e2-\<succ>i\<rightarrow> s2"   and "(v, s2') = avar G i a s2" shows "G\<turnstile>Norm s0 \<midarrow>e1.[e2]=\<succ>v\<rightarrow> s2'"
15: shows "irreflexive ((w \<sqinter> - prim_EP w v e)\<^sup>+ * prim_P w v e\<^sup>T\<^sup>\<star> * (w \<sqinter> - prim_EP w v e)\<^sup>\<star>)"
assms_3: shows "x # xs \<in> Joints (X # XS)"
ideD_1: fixes a :: "'a2" assumes "C2.ide a" shows "C2.arr a"
mpop_min_refines: fixes \<sigma> :: "('V \<Rightarrow> 'W infty option) \<times> ('V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option)"   and \<sigma>' :: "'V set \<times> ('V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option)" assumes "(\<sigma>, \<sigma>') \<in> br \<alpha>s dinvarm" shows "mpop_min \<sigma> \<le> \<Down> (br (\<lambda>(v, w, \<sigma>). (v, \<alpha>s \<sigma>)) (\<lambda>(v, w, \<sigma>). dinvarm \<sigma> \<and> w = mpath_weight' (snd \<sigma> v))) (pop_min \<sigma>')"
mkAutoSim_ec: fixes t :: "'s Trace"   and a :: "'a" assumes "t \<in> jkbpC" shows "simAbs (runJP mkAutoSim t a) = sim_equiv_class a t"
assms_3: fixes a :: "'c"   and b :: "'d" assumes "a \<in> set2_FGf x"   and "b \<in> set2_FGf y"   and "L2 a b" shows "L2' a b"
eventually_closed_segment: fixes X0 :: "'a set"   and x0 :: "'a"   and U :: "'a set" assumes "open X0"   and "x0 \<in> X0" shows "\<forall>\<^sub>F x in at x0 within U. closed_segment x0 x \<subseteq> X0"
assms: shows "knows P [kE m] \<or> knows Q [kE m]"
paths_tri2: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and ab :: "'b set"   and a :: "'b"   and b :: "'b"   and bc :: "'b set"   and c :: "'b"   and ca :: "'b set" assumes "MinkowskiPrimitive \<E> \<P>"   and "ab \<in> \<P> \<and> a \<in> ab \<and> b \<in> ab \<and> a \<noteq> b"   and "bc \<in> \<P> \<and> b \<in> bc \<and> c \<in> bc \<and> b \<noteq> c"   and "ca \<in> \<P> \<and> c \<in> ca \<and> a \<in> ca \<and> c \<noteq> a"   and "ab \<noteq> bc" shows "MinkowskiPrimitive.kinematic_triangle \<E> \<P> a b c"
from_nat_into_inj_infinite: fixes A :: "'d set"   and m :: "nat"   and n :: "nat" assumes "countable A"   and "infinite A" shows "(from_nat_into A m = from_nat_into A n) = (m = n)"
lnth_lappend2: fixes xs :: "'c llist"   and k :: "nat"   and n :: "nat"   and ys :: "'c llist" assumes "llength xs = enat k"   and "k \<le> n" shows "lnth (lappend xs ys) n = lnth ys (n - k)"
simps_3: fixes v :: "nat" shows "pnat_min pnat.PInfty (pnat v) = pnat v"
emeasure_pair_measure_Times: fixes A :: "'b set"   and N :: "'b measure"   and B :: "('i \<Rightarrow> 'a) set" assumes "A \<in> sets N"   and "B \<in> I.events" shows "emeasure (N \<Otimes>\<^sub>M Pi\<^sub>M I M) (A \<times> B) = emeasure N A * emeasure (Pi\<^sub>M I M) B"
log_exp: fixes n :: "nat" shows "local.log (2 ^ n) = n"
fls_X_inv_nonzero: shows "fls_X_inv \<noteq> 0"
execn_block: shows "\<Gamma>\<turnstile> \<langle>block init bdy return c,Normal s\<rangle> =n\<Rightarrow> t"
that: fixes r :: "real"   and \<alpha> :: "real" assumes "0 \<le> r"   and "\<alpha> \<in> {a..b}"   and "z = rcis r \<alpha>" shows "thesis"
star_sup_1: fixes star :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.left_kleene_algebra star one times sup less_eq less bot" shows "star (sup x y) = times (star x) (star (times y (star x)))"
path_component_refl: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> S" shows "path_component S x x"
b'_def: shows "b' = 0\<^sub>v nc @\<^sub>v b @\<^sub>v - b"