hcju/isabelleps huggingface dataset:
intro: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "known_ptr = ShadowRootClass.known_ptr"   and "type_wf = ShadowRootClass.type_wf"   and "set_child_nodes = Shadow_DOM.set_child_nodes"   and "set_child_nodes_locs = Shadow_DOM.set_child_nodes_locs" shows "l_set_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf known_ptr set_child_nodes set_child_nodes_locs"
Lambert_W'_nz: fixes x :: "real" shows "Lambert_W' x \<noteq> 0"
simps_9: fixes v :: "'c match_expr"   and rs :: "'c rule list" shows "process_ret (Rule v Empty # rs) = Rule v Empty # process_ret rs"
a1': shows "s |\<^sub>s n \<approx> t |\<^sub>s m"
RF_f: shows "x \<notin> RF\<^sup>\<circ> (TER f)"
distinct_67: fixes x51 :: "'a fm"   and x52 :: "'a fm"   and x91 :: "nat"   and x92 :: "'a fm" shows "Or x51 x52 \<noteq> ExN x91 x92"
reach\<^sub>t_def: fixes \<Sigma> :: "'e set"   and \<delta> :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and q\<^sub>0 :: "'d" shows "reach\<^sub>t \<Sigma> \<delta> q\<^sub>0 = {run\<^sub>t \<delta> q\<^sub>0 w n |w n. range w \<subseteq> \<Sigma>}"
le_supE: fixes a :: "'a"   and b :: "'a"   and x :: "'a" assumes "a + b \<le> x" obtains "a \<le> x" and "b \<le> x"
eq_combination_ax_def: shows "eq_combination_ax \<equiv> mk_eq Core.f Core.g \<longmapsto> mk_eq Core.x Core.y \<longmapsto> mk_eq (Core.f $ Core.x) (Core.g $ Core.y)"
function_uminus_add_r: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'a" assumes "a \<in> S"   and "f \<in> carrier F" shows "f \<oplus>\<^bsub>F\<^esub> function_uminus S R f = \<zero>\<^bsub>F\<^esub>"
intros_4: fixes u :: "'c \<Rightarrow> 'b"   and c :: "'c"   and d :: "'b" assumes "- u c < d" shows "dbm_entry_val u None (Some c) (Lt d)"
implicit_span_with: fixes U\<^sub>M :: "'d set"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" assumes "module_on U\<^sub>M scale" shows "span.with (0::'d) (+) scale = module_on.span scale"
euler_genus_def: shows "G.euler_genus \<equiv> (int (2 * card G.sccs) - int (card G.isolated_verts) - G.euler_char) div 2"
img_in_bij: fixes G :: "('c, 'd) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and M :: "'e set"   and g :: "'c" assumes "group_action G \<phi> M"   and "g \<in> carrier G" shows "\<phi> g \<in> Bij M"
ImageI: fixes a :: "'a"   and b :: "'b"   and r :: "('a \<times> 'b) set"   and A :: "'a set" assumes "(a, b) \<in> r"   and "a \<in> A" shows "b \<in> r `` A"
tOp_mHom_closed: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g"   and g :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N"   and "g \<in> mHom R M N" shows "tOp_mHom R M N f g \<in> mHom R M N"
preserves_dom: fixes f :: "'a" assumes "S.arr f" shows "S.dom (L f) = L (S.dom f)"
const_fun_def: fixes c :: "'f" shows "const_fun c = (\<lambda>x. c)"
length_remdups_card_conv: fixes xs :: "'c list" shows "length (remdups xs) = card (set xs)"
shift1_inv_some_snd_4: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "shift1_inv (qs, B)"   and "1 < card B"   and "(h, U) = (SOME b. b \<in> B \<and> card (snd b) = Suc m)" shows "card U = Suc m"
2: shows "\<guillemotleft>rep r : tab\<^sub>1 r \<star> (tab\<^sub>0 r)\<^sup>* \<Rightarrow> r\<guillemotright> \<and> local.iso (rep r)"
legendre_identity': fixes x :: "real" shows "fact (nat \<lfloor>x\<rfloor>) = (\<Prod>p\<in>{p. prime p \<and> real p \<le> x}. p ^ legendre_aux x p)"
List_distinct: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref"   and ps :: "ref list" assumes "List p h ps" shows "distinct ps"
static_def: fixes s :: "'b list"   and rs :: "'b list" shows "static s rs = (set rs \<subseteq> set s)"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "\<And>x y. hom (x * y) = hom x * hom y" shows "times_hom hom"
prod_list_neutral: fixes xs :: "'a list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x = (1::'a)" shows "prod_list xs = (1::'a)"
hyps: fixes xss :: "'a llist llist" shows "lconcat (ltake (enat n) xss) = ltake (\<Sum>i<n. llength (lnth xss i)) (lconcat xss)"
removeNones_vol_not_f12: fixes x :: "nat" assumes "x \<in> set (removeNones vol)" shows "x \<notin> \<V> f12"
cbelow_on_mono: fixes A :: "'b set"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b"   and rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and B :: "'b set" assumes "cbelow_on A ord z rel a b"   and "A \<subseteq> B" shows "cbelow_on B ord z rel a b"
id_embed: fixes r :: "('c \<times> 'c) set" shows "embed r r id"
simps_2: fixes mds :: "Mode \<Rightarrow> 'Var set"   and a :: "'Var ModeUpd"   and as :: "'Var ModeUpd list" shows "mds \<oplus> (a # as) = update_modes a mds \<oplus> as"
stake_snth_charact: fixes n :: "nat"   and w :: "'b stream"   and x :: "'b stream" assumes "stake n w = stake n x" shows "\<forall>i<n. w !! i = x !! i"
Iff_def: fixes A :: "fmla"   and B :: "fmla" shows "A IFF B = (A IMP B) AND (B IMP A)"
dimension_is_inj: fixes n :: "nat"   and E :: "'a set"   and m :: "nat" assumes "dimension n K E"   and "dimension m K E" shows "n = m"
in_Above_under: fixes a :: "'c"   and r :: "('c \<times> 'c) set" assumes "a \<in> Field r" shows "a \<in> Above r (under r a)"
trans_ParCR: fixes c2 :: "('test, 'atom) com"   and s :: "'state"   and c2' :: "('test, 'atom) com"   and s' :: "'state"   and c1 :: "('test, 'atom) com" assumes "(c2, s) \<rightarrow>c (c2', s')" shows "(c1 | c2, s) \<rightarrow>c (c1 | c2', s')"
and_mask_eq_iff_le_mask: fixes w :: "'b word"   and n :: "nat" shows "(w && mask n = w) = (w \<le> mask n)"
assms_8: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) d = Dv"
iso_assoc\<^sub>A\<^sub>W\<^sub>C: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "f \<star> g \<noteq> null"   and "g \<star> h \<noteq> null" shows "local.iso \<a>[f, g, h]"
IH: fixes t :: "('a, 'b) tm" assumes "\<not> list_all (\<lambda>x. x = dir.Left) p" shows "head (emb_step_at p d t) = head t"
sentence_iff: fixes G :: "('d, 'c) labeled_graph"   and e\<^sub>1 :: "'d allegorical_term"   and e\<^sub>2 :: "'d allegorical_term" shows "G \<Turnstile> e\<^sub>1 \<sqsubseteq> e\<^sub>2 = (:G:\<lbrakk>e\<^sub>1\<rbrakk> \<subseteq> :G:\<lbrakk>e\<^sub>2\<rbrakk>)"
closest_supchamberD_2: fixes F :: "'a set"   and D :: "'a set" assumes "F \<in> X"   and "domain.chamber D" shows "order.greater_eq (domain.closest_supchamber F D) F"
Beta_pole1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> \<int>\<^sub>\<le>\<^sub>0" shows "Beta x y = (0::'a)"
card_mnm': fixes m :: "nat"   and n :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "m < n" shows "card {i. i < n \<and> P i} = card {i. i < m \<and> P i} + card {i. m \<le> i \<and> i < n \<and> P i}"
Ninv_def: shows "Ninv = (\<lambda>(a, b). if a < 0 then (- b, \<bar>a\<bar>) else (b, a))"
assms_1: shows "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> x"
that: fixes d :: "'a" assumes "fst x = fst (normalize_quot x) * d"   and "snd x = snd (normalize_quot x) * d"   and "d dvd fst x"   and "d dvd snd x"   and "d \<noteq> (0::'a)" shows "thesis"
assms_2: fixes n :: "nat" assumes "n < length cl"   and "n < length dl" shows "cl ! n \<approx>01 dl ! n"
inf: shows "(\<forall>(a, b, c)\<in>set eq. a = 0 \<and> b = 0 \<and> c = 0) = (\<forall>x\<in>set eq. case x of (a, b, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c = 0)"
Z: shows "e2ennreal (esssup M (\<lambda>x. ereal 0)) = 0"
A: shows "countable A \<and> A \<subseteq> sets (distr M N f) \<and> \<Union> A = space (distr M N f) \<and> (\<forall>a\<in>A. emeasure (distr M N f) a \<noteq> \<infinity>)"
tendsto_add: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and b :: "'b" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> a + b) F"
mconn_pointwise: fixes E :: "('b \<times> 'b) set" shows "mconn E = {(u, v). (u, v) \<in> E\<^sup>* \<and> (v, u) \<in> E\<^sup>*}"
map_ident: fixes t :: "'b rexp" shows "map_rexp (\<lambda>x. x) t = t"
asIMOD_igVar: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and ys :: "'varSort"   and y :: "'var" shows "igVar (asIMOD SEM) ys y = (\<lambda>val. if sWlsVal SEM val then val ys y else undefined)"
digits_def: fixes base :: "nat" shows "digits base \<equiv> 0 < base"
good_item_simps_33: fixes rhoa :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xsa :: "'varSort"   and x1 :: "'var"   and xa :: "'var" assumes "goodEnv rhoa" shows "goodEnv (rhoa &[x1 // xa]_xsa)"
monad_simps_4: fixes v :: "'c option" shows "assert_opt v \<equiv> case v of None \<Rightarrow> fail | Some v \<Rightarrow> return v"
mult_assoc: fixes a :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> bool" shows "((a \<and>* b) \<and>* c) = (a \<and>* b \<and>* c)"
zr_rewrite: fixes z :: "nat"   and x :: "nat"   and c :: "nat"   and r :: "nat" assumes "z = (x * c + r) mod order \<G>"   and "r < order \<G>" shows "(z + order \<G> * x * c - x * c) mod order \<G> = r"
mult_pos_pos: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) < a"   and "(0::'a) < b" shows "(0::'a) < a * b"
start_value_main: fixes x :: "int"   and p :: "nat" assumes "0 \<le> x"   and "0 < p" shows "x \<le> start_value x p ^ p \<and> 0 \<le> start_value x p"
finite_insert: fixes a :: "'c"   and A :: "'c set" shows "finite (a \<triangleright> A) = finite A"
is_element_ptr_def: fixes ptr :: "(_) element_ptr" shows "is_element_ptr\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = (case ptr of element_ptr.Ref x \<Rightarrow> True | element_ptr.Ext element_ptr \<Rightarrow> False)"
IH_2: fixes V :: "nat set"   and \<sigma> :: "nat \<Rightarrow> val" assumes "randomfree e2"   and "free_vars e2 \<subseteq> V"   and "\<sigma> \<in> space (state_measure V (\<lambda>a. case a of 0 \<Rightarrow> t1 | Suc x \<Rightarrow> \<Gamma> x))" shows "return_val (expr_sem_rf \<sigma> e2) = expr_sem \<sigma> e2"
tendsto_inverse_0_at_top: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter" assumes "filterlim f at_top F" shows "((\<lambda>x. inverse (f x)) \<longlongrightarrow> 0) F"
cfs_zero: fixes n :: "nat" shows "\<zero>\<^bsub>P\<^esub> n = \<zero>"
extend_ensures_slice: fixes F :: "'a program"   and A :: "'c set"   and B :: "'c set"   and y :: "'b" assumes "extend h F \<in> A ensures B" shows "F \<in> slice A y ensures project_set h B"
assoc: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c"   and c :: "'b \<times> 'c" shows "inf_prod (inf_prod a b) c = inf_prod a (inf_prod b c)"
inres_SPEC: fixes M :: "'a nres"   and x :: "'a"   and \<Phi> :: "'a \<Rightarrow> bool" assumes "inres M x"   and "M \<le> SPEC \<Phi>" shows "\<Phi> x"
tendsto_nonzero_sgnx: fixes f :: "real \<Rightarrow> real"   and c :: "real"   and F :: "real filter" assumes "(f \<longlongrightarrow> c) F"   and "F \<noteq> bot"   and "c \<noteq> 0" shows "sgnx f F = sgn c"
elims_1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a list"   and y :: "bool" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq x = y" obtains "x = []" and "y"   | xa :: "'a" and ys :: "'a list" where "x = xa # ys" and "y = (Ball (set ys) (less_eq xa) \<and> linorder.sorted less_eq ys)"
aff_dim_halfspace_ge: fixes r :: "real"   and a :: "'b" shows "aff_dim {x. r \<le> a \<bullet> x} = (if a = (0::'b) \<and> 0 < r then - 1 else int DIM('b))"
Some: fixes ord :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" assumes "ord x y" shows "ord_option ord (Some x) (Some y)"
dpd1: fixes x :: "'a" shows "x \<le> x \<cdot> ar ar x"
encode_Suc_exp: fixes e :: "nat"   and n :: "nat" shows "local.encode (Suc e) n = local.encode e (base * n)"
foldli_append: fixes xs1 :: "'b list"   and xs2 :: "'b list"   and c :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and \<sigma> :: "'a" shows "foldli (xs1 @ xs2) c f \<sigma> = foldli xs2 c f (foldli xs1 c f \<sigma>)"
proj_onto_sum_rw: fixes v :: "'b"   and A :: "'b set" shows "(\<Sum>x\<in>A. (x \<bullet> v / (x \<bullet> x)) *\<^sub>R x) = (\<Sum>x\<in>A. (v \<bullet> x / (x \<bullet> x)) *\<^sub>R x)"
is_aligned_andI1: fixes x :: "'b word"   and n :: "nat"   and y :: "'b word" assumes "is_aligned x n" shows "is_aligned (x AND y) n"
insert_iff: fixes a :: "'c"   and b :: "'c"   and A :: "'c set" shows "(a \<in> b \<triangleright> A) = (a = b \<or> a \<in> A)"
bj_2: shows "(a, i) \<noteq> (b, j)"
applicableDecide_def: fixes state :: "State"   and decisionVars :: "nat set" shows "applicableDecide state decisionVars \<equiv> \<exists>state'. appliedDecide state state' decisionVars"
valid_prefix_implies_ltl: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set"   and i :: "nat" assumes "af \<phi> (prefix i w) \<sim> true\<^sub>n" shows "w \<Turnstile>\<^sub>n \<phi>"
ipt_iprange_compress_src_matching: fixes \<alpha> :: "action \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> bool"   and ml :: "'c ipt_iprange negation_type list"   and a :: "action"   and p :: "('c, 'd) tagged_packet_scheme" shows "match_list (common_matcher, \<alpha>) (map (Match \<circ> Src) (ipt_iprange_compress ml)) a p = matches (common_matcher, \<alpha>) (alist_and (NegPos_map Src ml)) a p"
sg1: shows "a (Suc (t + 3 + k + d)) = (if i (t + 3 + k + d) = [] then [] else [sc_ack])"
intros_2: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and effect_apply :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and L :: "'c \<times> 'd fs_set \<times> 'd \<Rightarrow> 'd fs_set"   and F :: "'d fs_set"   and P :: "'a"   and Q :: "'a"   and f :: "'d" assumes "effect_nominal_ts satisfies transition effect_apply L"   and "effect_nominal_ts.FL_bisimilar satisfies transition effect_apply L F P Q"   and "f \<in>\<^sub>f\<^sub>s F" shows "effect_nominal_ts.L_bisimilar satisfies transition effect_apply L (AC (f, F, effect_apply f P)) (AC (f, F, effect_apply f Q))"
y_1: shows "y \<in> \<omega>_limit_set x"
\<phi>_simps_5: shows "cod r\<^sub>0s\<^sub>1.\<phi> = s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0"
rel_state_state_of: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e"   and b :: "'f Heap" assumes "crel_vs1 R a b" shows "local.rel_state (=) (state_of b) b"
nres_relD: fixes c :: "'a nres"   and a :: "'b nres"   and R :: "('a \<times> 'b) set" assumes "(c, a) \<in> \<langle>R\<rangle>nres_rel" shows "c \<le> \<Down> R a"
distinct_mset_inter_remdups_mset: fixes A :: "'c multiset"   and B :: "'c multiset" assumes "distinct_mset A" shows "A \<inter># remdups_mset B = A \<inter># B"
leq_add_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.add_linorder_min plus minus zero less_eq less" shows "less_eq x (plus x y)"
length_weave: fixes A :: "nat set"   and xs :: "'a list"   and ys :: "'a list" shows "length (weave A xs ys) = length xs + length ys"
cert_target: fixes pc' :: "nat"   and s' :: "'a"   and pc :: "nat" assumes "(pc', s') \<in> set (step pc (\<tau>s ! pc))"   and "pc' \<noteq> pc + 1"   and "pc < length \<tau>s"   and "pc' < length \<tau>s" shows "c ! pc' = \<tau>s ! pc'"
VCG_step: fixes sys :: "('h, 'i, 'g, 'j, 'k, 'l) pre_system_scheme"   and pre :: "('h, 'i, 'g, 'j, 'k) system_state \<Rightarrow> bool"   and post :: "('h, 'i, 'g, 'j, 'k) system_state \<Rightarrow> bool"   and p :: "'g set"   and sh' :: "('h, 'i, 'g, 'j, 'k) system_state"   and sh :: "('h, 'i, 'g, 'j, 'k) system_state" assumes "\<And>p. \<forall>(c, aft)\<in>vcg_fragments (PGMs sys p). PGMs sys, p, aft \<turnstile> \<lbrace>pre\<rbrace> c \<lbrace>post\<rbrace>"   and "system_step p sh' sh"   and "reachable_state sys sh"   and "pre sh" shows "post sh'"
cont2contlubE: fixes f :: "'e \<Rightarrow> 'f"   and Y :: "nat \<Rightarrow> 'e" assumes "cont f"   and "chain Y" shows "f (\<Squnion>i. Y i) = (\<Squnion>i. f (Y i))"
inj_vimage_image_eq: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj f" shows "f -` f ` A = A"
r2s_surj: shows "\<forall>A. \<exists>R\<subseteq>Id. A = r2s R"
c2: assumes "sqn rt ip < \<pi>\<^sub>2 r" shows "P (rt(ip \<mapsto> r))"
conv_le_abs_conv_abscissa: fixes f :: "'a fds" shows "conv_abscissa f \<le> abs_conv_abscissa f"
rank_le_nc: fixes A :: "'a mat"   and n :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat n nc" shows "vec_space.rank n A \<le> nc"
is_RB_uptD2: fixes d :: "'a \<Rightarrow> nat"   and canon :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and v :: "'t" assumes "is_RB_upt d canon G u"   and "v \<prec>\<^sub>t u"   and "d (pp_of_term v) \<le> dgrad_max d"   and "component_of_term v < length fs" shows "is_RB_in d canon G v"
remove1Prio_remove1: fixes bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.remove1Prio (SkewBinomialHeapStruc.prio (SkewBinomialHeapStruc.getMinTree bq)) bq = remove1 (SkewBinomialHeapStruc.getMinTree bq) bq"
power_odd_inj: fixes k :: "nat"   and a :: "'a"   and b :: "'a" assumes "odd k"   and "a ^ k = b ^ k" shows "a = b"
naturality: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "\<Phi>\<^sub>H (C.VV.cod f) \<cdot>\<^sub>D H.H\<^sub>DoFF.map f = H.FoH\<^sub>C.map f \<cdot>\<^sub>D \<Phi>\<^sub>H (C.VV.dom f)"
Entry_in_graph: fixes g :: "'g" shows "Entry g \<in> set (\<alpha>n g)"
close_preserved: fixes \<Gamma> :: "(W.var \<times> tyS) list"   and T :: "ty"   and \<theta> :: "(W.tvar \<times> ty) list" assumes "(ftv T - ftv \<Gamma>) \<sharp>* \<theta>" shows "\<theta><close \<Gamma> T> = close (\<theta><\<Gamma>>) (\<theta><T>)"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'d tree23 \<Rightarrow> 'c"   and f2 :: "'d tree23 \<Rightarrow> 'd \<Rightarrow> 'd tree23 \<Rightarrow> 'c"   and upI :: "'d upI" shows "P (case upI of TI x \<Rightarrow> f1 x | OF x xa xb \<Rightarrow> f2 x xa xb) = ((\<forall>x1. upI = TI x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22 x23. upI = OF x21 x22 x23 \<longrightarrow> P (f2 x21 x22 x23)))"
prems_5: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "j < length (tx # txs)" shows "snd ((tx # txs) ! j) \<notin> FvarsT (fst ((tx # txs) ! i))"
circline_set_moebius_circline_I: fixes z :: "complex_homo"   and H :: "circline"   and M :: "moebius" assumes "z \<in> circline_set H" shows "moebius_pt M z \<in> circline_set (moebius_circline M H)"
carrier: shows "vec_of_basis_enum ` X \<subseteq> carrier_vec n"
product_sigma_finite_interval: shows "product_sigma_finite (\<lambda>b. interval_measure (\<lambda>x. x))"
unique_eigenvector_real: fixes A :: "real mat"   and n :: "nat"   and lam :: "real"   and v :: "real Matrix.vec"   and w :: "real Matrix.vec" assumes "A \<in> carrier_mat n n"   and "order lam (char_poly A) = 1"   and "eigenvector A v lam"   and "eigenvector A w lam" shows "\<exists>a. v = a \<cdot>\<^sub>v w"
finter_iff: fixes c :: "'b"   and A :: "'b fset"   and B :: "'b fset" shows "(c |\<in>| A |\<inter>| B) = (c |\<in>| A \<and> c |\<in>| B)"
approx_minus_iff2: fixes x :: "'a star"   and y :: "'a star" shows "(x \<approx> y) = (- y + x \<approx> 0)"
cases: fixes x :: "('b \<times> 'b) \<times> 'b \<times> ('b \<times> 'b) list" obtains uu_ :: "'b \<times> 'b" and uv_ :: "'b" where "x = (uu_, uv_, [])"   | u :: "'b" and v :: "'b" and w :: "'b" and e :: "'b \<times> 'b" and es :: "('b \<times> 'b) list" where "x = ((u, v), w, e # es)"
finite_fv\<^sub>s\<^sub>n\<^sub>d: fixes x :: "('d, 'c) strand_step" shows "finite (fv\<^sub>s\<^sub>n\<^sub>d x)"
PiE_iff: fixes f :: "'a \<Rightarrow> 'b"   and I :: "'a set"   and X :: "'a \<Rightarrow> 'b set" shows "(f \<in> Pi\<^sub>E I X) = ((\<forall>i\<in>I. f i \<in> X i) \<and> f \<in> extensional I)"
lcons_def: fixes a :: "'c \<times> 'd"   and t :: "('c, 'd) FingerTree" shows "lcons a t \<equiv> Abs_FingerTree (FingerTreeStruc.lcons a (Rep_FingerTree t))"
simps_5: fixes I :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and b :: "bexp"   and C :: "acom"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "pre ({I} WHILE b DO C) Q = (I \<and>* $ 1)"
Vagree_statediff: fixes \<omega> :: "variable \<Rightarrow> real"   and \<omega>' :: "variable \<Rightarrow> real"   and S :: "variable set" assumes "Vagree \<omega> \<omega>' S" shows "statediff \<omega> \<omega>' \<subseteq> - S"
CodFunctor: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme"   and f :: "'f" assumes "Functor F"   and "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "cod\<^bsub>CatCod F\<^esub> (F ## f) = F @@ cod\<^bsub>CatDom F\<^esub> f"
int: shows "(\<lambda>x. f (t + x) - l) absolutely_integrable_on {- pi..pi}"
eventually_within_Un: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and s :: "'a set"   and t :: "'a set" shows "eventually P (at x within s \<union> t) = (eventually P (at x within s) \<and> eventually P (at x within t))"
elims: fixes x :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and xa :: "'a set"   and xb :: "('a, 'b) fmap"   and y :: "('a, 'b) fmap list" assumes "vars_change x xa xb = y" obtains vs :: "'a set" and s :: "('a, 'b) fmap" where "x = []" and "xa = vs__" and "xb = s__" and "y = []"   | a :: "('a, 'b) fmap \<times> ('a, 'b) fmap" and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" and vs :: "'a set" and s :: "('a, 'b) fmap" where "x = a # as" and "xa = vs" and "xb = s" and "y = (if fmrestrict_set vs (state_succ s a) \<noteq> fmrestrict_set vs s then state_succ s a # vars_change as vs (state_succ s a) else vars_change as vs (state_succ s a))"
directed_ub: fixes X :: "'a set" assumes "directed X" shows "\<forall>x\<in>X. \<forall>y\<in>X. \<exists>z\<in>X. x \<le> z \<and> y \<le> z"
has_density_space: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "has_density M N f" shows "space M = space N"
assms: shows "\<forall>n<length ws. P (ws ! n) (drop (Suc n) ws)"
insert_subsetI: fixes x :: "'a"   and A :: "'a set"   and X :: "'a set" assumes "x \<in> A"   and "X \<subseteq> A" shows "insert x X \<subseteq> A"
scc_root_disc_le: fixes v :: "'v"   and scc :: "'v set"   and x :: "'v" assumes "scc_root s v scc"   and "x \<in> scc"   and "x \<in> dom (discovered s)" shows "\<delta> s v \<le> \<delta> s x"
r_parallel''_4: fixes i :: "nat"   and x :: "nat"   and j :: "nat" assumes "eval r_phi [i, x] \<down> \<and> eval r_phi [j, x] \<down>" shows "eval r_parallel [i, j, x] \<down>= prod_encode (0, the (eval r_phi [i, x])) \<or> eval r_parallel [i, j, x] \<down>= prod_encode (1, the (eval r_phi [j, x]))"
assms_3: shows "card M = weight {i \<in> L ` V. 1 < i} (N V L)"
A: fixes x :: "name"   and y :: "name"   and n :: "trm"   and u :: "trm"   and v :: "trm"   and l :: "stack" assumes "x \<sharp> y"   and "x \<sharp> n"   and "t = u to x in v"   and "[y__]n__\<ggreater>L = [y]n\<ggreater>l"   and "r = (u to x in v to y in n) \<star> l" shows "P"
Goal: fixes R :: "pi"   and P :: "pi"   and Q :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "(R, expandSet P Q) \<in> sumComposeSet"   and "hnf P"   and "hnf Q"   and "a \<noteq> x" shows "P \<parallel> Q \<longmapsto> a<\<nu>x> \<prec> P' = R \<longmapsto> a<\<nu>x> \<prec> P'"
a'c: fixes z :: "complex" assumes "z \<in> closed_segment a' c" shows "k \<le> d \<bullet> z"
simps_1: fixes t :: "dag" shows "subdag Tip t = False"