hcju/isabelleps huggingface dataset:
xysp: shows "xysp = map (\<lambda>(ax, ay). (case ax of Num_arc_delta x \<Rightarrow> x, case ay of Num_arc_delta y \<Rightarrow> y)) (filter (\<lambda>(ax, ay). ax \<noteq> MinInfty_delta \<and> ay \<noteq> MinInfty_delta) xys)"
bs: fixes b :: "'a poly"   and i :: "nat" assumes "(b, i) \<in> set bs" shows "degree b \<noteq> 0"
to_bl_bin: fixes w :: "'b word" shows "bl_to_bin (to_bl w) = uint w"
snd_swap: fixes x :: "'a \<times> 'b" shows "snd (prod.swap x) = fst x"
simps_2: fixes C :: "ty"   and X :: "ctrm set" shows "NEGn (NOT C) X = AXIOMSn (NOT C) \<union> BINDINGn (NOT C) X \<union> NOTLEFT (NOT C) (\<parallel><C>\<parallel>)"
ctr_1: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g21 :: "'a \<Rightarrow> 'b"   and g22 :: "'a \<Rightarrow> 'a" assumes "p a" shows "unfold_llist p g21 g22 a = LNil"
1: fixes acc0 :: "(bool \<times> 'b \<times> 'c) list" shows "set acc0 \<subseteq> set (fold (\<lambda>(ic, q_in_bs, p, q) ps'. if \<not> ic \<and> c q_in_bs ps' p q then ps' else (ic, p, q) # ps') ps acc0)"
the_greatest_fixpoint_def: fixes f :: "'a \<Rightarrow> 'a" shows "\<nu> f = (THE x. is_greatest_fixpoint f x)"
IH: shows "C M2 M1 \<Omega> V m (Suc k) \<subseteq> append_sets V (lists_of_length (inputs M2) k)"
P4: shows "Bet B' A' A1 \<and> Cong A' A1 B A"
not_nonzero_memE: fixes a :: "nat \<Rightarrow> int" assumes "a \<notin> nonzero Zp"   and "a \<in> carrier Zp" shows "a = \<zero>"
inv_project_eq: fixes h :: "'c \<Rightarrow> 'd" assumes "bij h" shows "inv (project (\<lambda>(x, u). h x) UNIV) = extend (\<lambda>(x, u). h x)"
prob_space_P: fixes J :: "'i set" assumes "finite J"   and "J \<subseteq> I" shows "prob_space (P J)"
lt_lin: fixes p :: "tm" assumes "tmbound0 p" shows "islin (lt p)"
proper_it'D: fixes it :: "'f \<Rightarrow> ('g \<Rightarrow> bool) \<Rightarrow> ('h \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and it' :: "'f \<Rightarrow> ('i \<Rightarrow> bool) \<Rightarrow> ('h \<Rightarrow> 'i \<Rightarrow> 'i) \<Rightarrow> 'i \<Rightarrow> 'i"   and s :: "'f" assumes "proper_it' it it'" shows "proper_it (it s) (it' s)"
continuous_on_diff: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. f x - g x)"
fds_shift_add: fixes c :: "'b"   and f :: "'b fds"   and g :: "'b fds" shows "fds_shift c (f + g) = fds_shift c f + fds_shift c g"
wn_q_eq_wn_p: fixes z :: "complex" assumes "z \<in> inside (path_image p)" shows "winding_number q z = winding_number p z"
lemma_2_5_9_b: fixes a :: "'a"   and b :: "'a" shows "a \<le> b l\<rightarrow> a"
a: shows "swaps ss init ! s \<in> set (swaps ss init)"
valid_empty_edges_iff_exists_offending_flows: fixes G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" assumes "sinvar_mono"   and "wf_graph G"   and "\<not> sinvar G nP" shows "sinvar \<lparr>nodes = nodes G, edges = {}\<rparr> nP = (set_offending_flows G nP \<noteq> {})"
C: shows "set_mset M \<subseteq> r \<down>m {#\<alpha>#}"
word_of_nat_le: fixes n :: "nat"   and x :: "'b word" assumes "n \<le> unat x" shows "word_of_nat n \<le> x"
ordertype_VWF_mono: fixes X :: "V set"   and Y :: "V set" assumes "X \<subseteq> Y"   and "small Y" shows "ordertype X VWF \<le> ordertype Y VWF"
Ord_not_hpair: fixes x :: "hf" assumes "Ord x" shows "\<not> is_hpair x"
in_set_member: fixes x :: "'a"   and xs :: "'a list" shows "x mem xs = List.member xs x"
token_stays_in_sink: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and x :: "nat"   and n :: "nat"   and m :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 q"   and "semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n = q" shows "semi_mojmir_def.token_run \<delta> q\<^sub>0 w x (n + m) = q"
0: fixes ps :: "(int \<times> int) list" assumes "ps \<in> {ps. distinct ps}" shows "real (time (closest_pair_tm ps)) \<le> (closest_pair_time \<circ> length) ps"
Suc_1: fixes p :: "nat"   and k :: "nat" assumes "Suc p \<le> Suc n" shows "smap_stoch_proc (Suc n) f k w ! p = f (k + p) w"
new_character_data_def: shows "new_character_data = Heap_Error_Monad.bind get_heap (\<lambda>h. Heap_Error_Monad.bind (return (new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a h)) (\<lambda>(new_ptr, h'). Heap_Error_Monad.bind (return_heap h') (\<lambda>_. return new_ptr)))"
find_adds_NoneE: fixes fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and u :: "'t"   and f :: "'t \<Rightarrow>\<^sub>0 'c" assumes "find_adds fs u = None"   and "f \<in> set fs" obtains "f = 0"   | "f \<noteq> 0" and "\<not> lt f adds\<^sub>t u"
assms_4: shows "d \<le> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>"
dsjAssoc: fixes A :: "fmla"   and B :: "fmla"   and C :: "fmla" shows "A OR B OR C IMP (A OR B) OR C \<in> boolean_axioms"
assms_2: shows "g \<turnstile> n-ns\<rightarrow>m"
cases: fixes a1 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a2 :: "nat \<Rightarrow> nat"   and a3 :: "pr_operation'"   and a4 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a5 :: "nat \<Rightarrow> nat" assumes "((a1, a2), a3, a4, a5) \<in> pr_algo_lts'" obtains f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and e :: "nat \<times> nat" where "a1 = fb__" and "a2 = lb__" and "a3 = NONSAT_PUSH'" and "a4 = push_effect fb__ e__" and "a5 = lb__" and "Height_Bounded_Labeling c s t fb__ lb__" and "nonsat_push_precond fb__ lb__ e__"   | f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and e :: "nat \<times> nat" where "a1 = fc__" and "a2 = lc__" and "a3 = SAT_PUSH' e" and "a4 = push_effect fc__ e" and "a5 = lc__" and "Height_Bounded_Labeling c s t fc__ lc__" and "sat_push_precond fc__ lc__ e"   | f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and u :: "nat" where "a1 = f" and "a2 = l" and "a3 = RELABEL'" and "a4 = f" and "a5 = relabel_effect f l u" and "Height_Bounded_Labeling c s t f l" and "relabel_precond f l u"
absolutely_integrable_component_lbound: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f absolutely_integrable_on A"   and "g integrable_on A"   and "\<And>x b. \<lbrakk>x \<in> A; b \<in> Basis\<rbrakk> \<Longrightarrow> f x \<bullet> b \<le> g x \<bullet> b" shows "g absolutely_integrable_on A"
wt: fixes I :: "nat set" assumes "I \<in> P - {I0}"   and "W cf1 I0 < 1"   and "W cf2 (F I0) < 1" shows "W cf1 I / (1 - W cf1 I0) = W cf2 (F I) / (1 - W cf2 (F I0))"
iMODb_subset_conv': fixes c :: "nat"   and c' :: "nat"   and r' :: "nat"   and m' :: "nat"   and r :: "nat"   and m :: "nat" assumes "0 < c"   and "0 < c'" shows "([ r', mod m', c' ] \<subseteq> [ r, mod m, c ]) = (r' mod m = r mod m \<and> r \<le> r' \<and> m' mod m = 0 \<and> r' + m' * c' \<le> r + m * c)"
Class_equivalence: fixes a :: "'a"   and b :: "'a" assumes "a \<in> S"   and "b \<in> S" shows "(Class a = Class b) = ((a, b) \<in> E)"
Disjoint_commute_D: fixes t :: "tm"   and u :: "tm"   and H :: "fm set"   and A :: "fm" assumes "insert (Disjoint t u) H \<turnstile> A" shows "insert (Disjoint u t) H \<turnstile> A"
smod_int_compares_7: fixes a :: "int"   and b :: "int" assumes "a \<le> 0"   and "b < 0" shows "a smod b \<le> 0"
F_def: shows "F \<equiv> \<lambda>X. {x. X \<in> XS \<and> \<not> qAFresh xs x X}"
higher_deriv_pre_zeta_1_1: fixes k :: "nat" shows "(deriv ^^ k) (pre_zeta 1) 1 = (- 1) ^ k * stieltjes_gamma k"
join_3: shows "s r' = Some (\<sigma>', \<tau>', VE v)"
simps_10: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "page_t \<Rightarrow> 'a"   and x3 :: "page_t" shows "(case BUF x3 of PREP \<Rightarrow> f1 | WAIT \<Rightarrow> f2 | BUF x \<Rightarrow> f3 x) = f3 x3"
formulaTrueRemoveAll: fixes formula :: "Literal list list"   and valuation :: "Literal list"   and clause :: "Literal list" assumes "formulaTrue formula valuation" shows "formulaTrue (removeAll clause formula) valuation"
g: fixes x :: "nat" shows "(if x = length vs then Some j else (vs @ [0] \<odot> 0\<^sup>\<infinity>) x) = (vs @ [j] \<odot> 0\<^sup>\<infinity>) x"
splitFace_holds_minGraphProps: fixes g' :: "graph"   and v :: "nat"   and a :: "nat"   and f' :: "face"   and n :: "nat" assumes "pre_splitFace g' v a f' [countVertices g'..<countVertices g' + n]"   and "minGraphProps g'" shows "minGraphProps (snd (snd (splitFace g' v a f' [countVertices g'..<countVertices g' + n])))"
bit_exp_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and m :: "nat"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m) n = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m \<noteq> zero \<and> m = n)"
usubst_upd_pr_var_id: fixes x :: "'b \<Longrightarrow> 'c" assumes "vwb_lens x" shows "[x \<mapsto>\<^sub>s &x] = id"
transfer_pnet_action: fixes s :: "'s net_state"   and n :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'g"   and \<zeta> :: "'l net_state"   and a :: "'m node_action"   and s' :: "'s net_state" assumes "s \<in> reachable (pnet np n) TT"   and "netgmap sr s = netmask (net_tree_ips n) (\<sigma>, \<zeta>)"   and "wf_net_tree n"   and "(s, a, s') \<in> automaton.trans (pnet np n)" obtains \<sigma>'' :: "nat \<Rightarrow> 'g" and \<zeta>'' :: "'l net_state" where "((\<sigma>, \<zeta>), a, \<sigma>'', \<zeta>'') \<in> automaton.trans (opnet onp n)" and "\<forall>j. j \<notin> net_ips \<zeta> \<longrightarrow> \<sigma>'' j = \<sigma> j" and "netgmap sr s' = netmask (net_tree_ips n) (\<sigma>'', \<zeta>'')"
endswith_eq_sdrop: fixes pat :: "[:'r:]"   and xs :: "[:'r:]" shows "endswith\<cdot>pat\<cdot>xs = eq\<cdot>pat\<cdot>(sdrop\<cdot>(slength\<cdot>xs - slength\<cdot>pat)\<cdot>xs)"
Pbij_Dom_def: fixes \<beta> :: "(Location \<times> Location) set" shows "Pbij_Dom \<beta> = {l. \<exists>ll. (l, ll) \<in> \<beta>}"
rv_edge_slice_kinds: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'a"   and S :: "'b SDG_node set"   and s :: "(('c \<Rightarrow> 'd option) \<times> 'e) list"   and s' :: "(('c \<Rightarrow> 'd option) \<times> 'e) list"   and as :: "'a list"   and asx :: "'a list" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "valid_edge a"   and "intra_kind (kind a)"   and "\<forall>V\<in>SDG.relevant_vars sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S (CFG_node (sourcenode a)). fst (hd s) V = fst (hd s') V"   and "CFG.preds procs (SDG.slice_kinds sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S (a # as)) s"   and "CFG.preds procs (SDG.slice_kinds sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S (a # asx)) s'" shows "\<forall>V\<in>SDG.relevant_vars sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S (CFG_node (targetnode a)). fst (hd (CFG.transfer procs (SDG.slice_kind sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S a) s)) V = fst (hd (CFG.transfer procs (SDG.slice_kind sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S a) s')) V"
f_def: shows "f = (\<lambda>x. ln (real x) / (ln (ln (real x)))\<^sup>2)"
a: shows "\<Q> k \<subseteq> \<P> (Suc k) 0"
has_derivative_zero_unique_strong_connected: fixes S :: "'a set"   and K :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and c :: "'a"   and y :: "'b"   and x :: "'a" assumes "connected S"   and "open S"   and "finite K"   and "continuous_on S f"   and "c \<in> S"   and "f c = y"   and "\<And>x. x \<in> S - K \<Longrightarrow> (f has_derivative (\<lambda>h. 0::'b)) (at x within S)"   and "x \<in> S" shows "f x = y"
sat_Ands: fixes \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and l :: "formula list" shows "sat \<sigma> V v i (Ands l) = (\<forall>\<phi>\<in>set l. sat \<sigma> V v i \<phi>)"
RhsEqVal_axioms: shows "RhsEqVal rhs_eq_val"
wf_bs_coefficients: fixes bs :: "'b list"   and p :: "'b poly" assumes "wf_bs bs p" shows "\<forall>c\<in>set (coefficients p). wf_bs bs c"
DirProd_def: fixes G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme" shows "G \<times>\<times> H = \<lparr>carrier = carrier G \<times> carrier H, mult = \<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h'), one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>"
norm_triangle_ineq3: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "\<bar>norm a - norm b\<bar> \<le> norm (minus a b)"
prime_imp_power_coprime: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and a :: "'a"   and m :: "nat" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "\<not> dvd.dvd times p a" shows "algebraic_semidom.coprime times one a (power.power one times p m)"
subst_supportD: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "\<theta> supports \<delta>" shows "\<theta> \<preceq>\<^sub>\<circ> \<delta>"
INF_lower: fixes i :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "i \<in> A" shows "Inf (f ` A) \<le> f i"
less_llist_def: fixes xs :: "'a llist"   and ys :: "'a llist" shows "(xs < ys) = lstrict_prefix xs ys"
scale_eq_0_iff: fixes x :: "'a"   and a :: "real" assumes "x \<in> S" shows "(a *\<^sub>R x = (0::'a)) = (a = 0 \<or> x = (0::'a))"
interval_bij_bij: fixes a :: "'a"   and b :: "'a"   and u :: "'a"   and v :: "'a"   and x :: "'a" assumes "\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i \<and> u \<bullet> i < v \<bullet> i" shows "interval_bij (a, b) (u, v) (interval_bij (u, v) (a, b) x) = x"
is_arrow_of_tabulations_in_maps: shows "arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg r r.tab (tab\<^sub>0 r) (tab\<^sub>1 r) r r.tab (tab\<^sub>0 r) (tab\<^sub>1 r) r"
countable_trees: fixes A :: "'b set" assumes "countable A" shows "countable (trees A)"
ide_hcomp: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "B.ide \<nu>"   and "B.ide \<mu>"   and "src\<^sub>B \<nu> = trg\<^sub>B \<mu>" shows "B.ide (\<nu> \<star>\<^sub>B \<mu>)"
inject: fixes x2 :: "'e"   and y2 :: "'e" shows "(Some x2 = Some y2) = (x2 = y2)"
lemma_tan_add1: fixes x :: "'a"   and y :: "'a" assumes "cos x \<noteq> (0::'a)"   and "cos y \<noteq> (0::'a)" shows "(1::'a) - tan x * tan y = cos (x + y) / (cos x * cos y)"
word_cat_split_alt: fixes w :: "'a word"   and u :: "'b word"   and v :: "'c word" assumes "size w \<le> size u + size v"   and "word_split w = (u, v)" shows "word_cat u v = w"
strictify1: fixes f :: "'b \<rightarrow> 'a" shows "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>"
llength_lappend_start'E: fixes w :: "'b llist"   and n :: "nat" assumes "llength w = enat (Suc n)" obtains a :: "'b" and v :: "'b llist" where "w = <a> $ v" and "llength v = enat n"
update_convs_1: fixes rel' :: "('c \<times> 'c) set \<Rightarrow> ('c \<times> 'c) set"   and carrier :: "'c set"   and rel :: "('c \<times> 'c) set"   and more :: "'d" shows "rel_update rel' \<lparr>carrier = carrier, rel = rel, \<dots> = more\<rparr> = \<lparr>carrier = carrier, rel = rel' rel, \<dots> = more\<rparr>"
sin_expansion_lemma: fixes x :: "real"   and m :: "nat" shows "sin (x + real (Suc m) * pi / 2) = cos (x + real m * pi / 2)"
cmg: fixes i :: "nat" shows "continuous_map (Euclidean_space m) euclideanreal (g i)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs" shows "l_new_element_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"
intro: fixes f :: "'a \<Rightarrow> 'b" assumes "bij f" shows "rekey f"
SD_strict_singleton: fixes x :: "'alt"   and y :: "'alt" assumes "x \<in> alts"   and "y \<in> alts" shows "return_pmf x \<prec>[SD (Pareto R)] return_pmf y = x \<prec>[Pareto R] y"
integral_restrict_UNIV: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "S \<in> sets lebesgue" shows "LINT x|lebesgue. (if x \<in> S then f x else (0::'c)) = integral\<^sup>L (lebesgue_on S) f"
dvd_smult: fixes p :: "'c poly"   and q :: "'c poly"   and a :: "'c" assumes "p dvd q" shows "p dvd smult a q"
boundedI: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<preceq> b"   and "a \<preceq> c" shows "a \<preceq> b \<sqinter> c"
remdups_filter: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "remdups (filter P xs) = filter P (remdups xs)"
bibd_subset_occ: fixes x :: "'a set"   and bl :: "'a set" assumes "x \<subseteq> bl"   and "bl \<in># \<B>"   and "card x = 2" shows "int (size (filter_mset ((\<subseteq>) x) (remove1_mset bl \<B>))) = \<Lambda> - 1"
a_objects_exist_1: fixes v :: "i" shows "[\<^bold>\<box>(\<^bold>\<exists>x. \<lparr>A!,x\<^sup>P\<rparr>) in v]"
paste_outside_restrict: fixes P :: "('c \<times> 'd) set"   and X :: "'c set" shows "P = Outside' X P +< (P || X)"
weakBisimWeakUpto: fixes P :: "ccs"   and Q :: "ccs"   and X :: "(ccs \<times> ccs) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto>\<^sup>^<(weakBisimulation O X O bisim)> Q"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> (Q, P) \<in> X" shows "P \<approx> Q"
ds_subid_aux: fixes x :: "'a"   and y :: "'a" shows "d x \<cdot> y \<sqsubseteq> y"
hta_mem'_correct: fixes H :: "('a, 'b) hashedTa"   and t :: "'b tree" assumes "hashedTa H"   and "hta_has_idx_f H" shows "hta_mem' t H = (t \<in> ta_lang (hta_\<alpha> H))"
rewrite_MultiportPorts_def: fixes m :: "'b common_primitive match_expr" shows "rewrite_MultiportPorts m \<equiv> normalize_match (replace_primitive_matchexpr (is_MultiportPorts, multiportports_sel) rewrite_MultiportPorts_one m)"
union_with_outside: fixes S :: "'b set" shows "S \<union> outside S = - inside S"
wtprog: shows "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"
assms_1: shows "max_lideal I B addB multB zeroB oneB"
2: shows "(\<lambda>n. snd (X n)) \<longlonglongrightarrow> lim (\<lambda>n. snd (X n))"
simps_4: fixes x2 :: "'d"   and x1 :: "'d" shows "Lt x2 \<noteq> Le x1"
P5: shows "A' = mult_row interchange_A i ((1::'a) / interchange_A $ i $ j)"
simps_1: fixes t :: "('e, 'f) Term.term"   and \<theta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "send\<langle>t\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta> = send\<langle>t \<cdot> \<theta>\<rangle>"
that: fixes a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "a \<in> P[U]"   and "p = a * h" shows "thesis"
modInequalityLemma: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "a \<noteq> b \<and> a mod c = b mod c" shows "a div c \<noteq> b div c"
distinct_3: fixes x3 :: "trm" shows "zer \<noteq> suc x3"
sub_mr_set_cos_def: fixes M :: "('e, 'f, 'g) Module_scheme"   and H :: "'e set"   and N :: "'e set" shows "N \<^sub>s/\<^sub>M H = {X. \<exists>n\<in>N. X = n \<uplus>\<^bsub>M\<^esub> H}"
get_owner_document_def: fixes ptr :: "(_) object_ptr" shows "get_owner_document ptr = invoke (local.CD.a_get_owner_document_tups @ local.a_get_owner_document_tups) ptr ()"
unrest_var_comp: fixes x :: "'a \<Longrightarrow> 'b"   and P :: "('c, 'b) uexpr"   and y :: "'d \<Longrightarrow> 'b" assumes "x \<sharp> P"   and "y \<sharp> P" shows "{x, y} \<sharp> P"
simps_1: fixes r :: "'a"   and h :: "'b" shows "|Inr (r, h)|\<^sub>r\<^sub>h = (r, h)"
nn_integral_0_1_ln_times_powr: fixes e :: "real" assumes "- 1 < e" shows "\<integral>\<^sup>+x\<in>{0<..<1}. ennreal (- ln x * x powr e)\<partial>lborel = ennreal (1 / (e + 1)\<^sup>2)"
fake_monotone_ik: fixes c :: "chan"   and bad :: "agent set"   and IK :: "msg set"   and chan :: "chan set"   and IK' :: "msg set" assumes "c \<in> fake bad IK chan"   and "IK \<subseteq> IK'" shows "c \<in> fake bad IK' chan"
weakFreshChainDerivative: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and yvec :: "name list" assumes "\<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'"   and "yvec \<sharp>* P"   and "yvec \<sharp>* \<alpha>"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "distinct (bn \<alpha>)" shows "yvec \<sharp>* P'"
simps_1: shows "BinomialHeapStruc.height_queue [] = 0"
black_heap_locs_def: shows "black_heap_locs = {sweep_idle, idle_noop_mfence, idle_noop_init_type}"
Abs_atoms_inverse: fixes y :: "'a set" assumes "y \<in> range atom_map" shows "Rep_atoms (Abs_atoms y) = y"
assms_3: fixes x :: "'b"   and y :: "'b" assumes "a = Inr x"   and "b = Inr y"   and "\<not> Q x y" shows "thesis"
units_of_carrier: fixes G :: "('d, 'e) monoid_scheme" shows "carrier (units_of G) = Units G"
sign_is_hom: fixes n :: "nat" shows "sign \<in> hom (sym_group n) sign_img"
nat_pow_0: fixes x :: "'a mod_ring poly" shows "[0] \<cdot>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
social_decision_scheme_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "social_decision_scheme agents alts sds \<equiv> election agents alts \<and> social_decision_scheme_axioms agents alts sds"
sep_add_disjI1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "sep_disj (plus x y) z"   and "sep_disj x y" shows "sep_disj (plus x z) y"
ZFfun_ext: fixes A :: "ZF"   and f :: "ZF \<Rightarrow> ZF"   and g :: "ZF \<Rightarrow> ZF"   and B :: "ZF" assumes "\<forall>x. x |\<in>| A \<longrightarrow> f x = g x" shows "ZFfun A B f = ZFfun A B g"
thetaIfZOT_sym: shows "sym thetaIfZOT"