hcju/isabelleps huggingface dataset:
bind_returns_heap_E: fixes h :: "'ap"   and f :: "('ap, 'aq, 'as) prog"   and g :: "'as \<Rightarrow> ('ap, 'aq, 'ar) prog"   and h'' :: "'ap" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>h h''" obtains x :: "'as" and h' :: "'ap" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> g x \<rightarrow>\<^sub>h h''"
ordIso_or_ordLess: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Well_order r"   and "Well_order r'" shows "r =o r' \<or> r <o r' \<or> r' <o r"
g: shows "finprod G g (insert x F) = g x \<otimes> finprod G g F"
continuous_within_sequentially: fixes a :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous (at a within s) f = (\<forall>x. (\<forall>n. x n \<in> s) \<and> x \<longlonglongrightarrow> a \<longrightarrow> (f \<circ> x) \<longlonglongrightarrow> f a)"
Mapping_lookup_filter_keys: fixes k :: "'c"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('c, 'd) mapping" assumes "k \<in> Mapping.keys (Mapping.filter f m)" shows "Mapping.lookup (Mapping.filter f m) k = Mapping.lookup m k"
cases: fixes a :: "'d"   and r :: "('d \<times> 'e) set" assumes "a \<in> Domain r" obtains a :: "'d" and b :: "'e" where "aa__ = a" and "(a, b) \<in> r"
save_rs: fixes far :: "nat"   and gs :: "recf list"   and ffp :: "nat"   and xs :: "nat list"   and f :: "recf"   and anything :: "nat list" assumes "far = length gs"   and "ffp \<le> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))"   and "far < ffp" shows "{\<lambda>nl. nl = map (\<lambda>i. rec_exec i xs) gs @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @ xs @ anything} mv_box far (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))) {\<lambda>nl. nl = map (\<lambda>i. rec_exec i xs) gs @ 0 \<up> (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) - length gs) @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything}"
finite_int_iff_bounded_le: fixes S :: "int set" shows "finite S = (\<exists>k. abs ` S \<subseteq> {..k})"
span_clauses_2: fixes S :: "'a set" shows "(0::'a) \<in> rational_vector.span S"
ide_src: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "span_vertical_category C"   and "partial_magma.arr (span_vertical_category.vcomp C) \<mu>" shows "partial_magma.ide (span_vertical_category.vcomp C) (span_vertical_category.src C \<mu>)"
dual_gfp: shows "complete_lattice.gfp Inf (\<lambda>x y. y \<le> x) = lfp"
CARD_valid_states: fixes PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set" assumes "finite PROB" shows "card (valid_states PROB) = 2 ^ card (prob_dom PROB)"
less_imp_inverse_less: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a b"   and "less zero a" shows "less (inverse b) (inverse a)"
representation_scale: fixes basis :: "'b set"   and v :: "'b"   and r :: "'a" assumes "independent basis"   and "v \<in> span basis" shows "representation basis (r *s v) = (\<lambda>b. r * representation basis v b)"
cos_le_zero: assumes "- x \<in> {pi / 2..3 * pi / 2}" shows "cos (- x) \<le> 0"
id_nec3_2: fixes v :: "i"   and x :: "\<nu>"   and y :: "\<nu>" shows "[\<^bold>\<diamond>x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P \<^bold>\<equiv> x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P in v]"
coeff_monom: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a" shows "MPoly_Type.coeff (MPoly_Type.monom m a) m = a"
node3_def: fixes nd1 :: "('c, 'd) Node"   and nd2 :: "('c, 'd) Node"   and nd3 :: "('c, 'd) Node" shows "node3 nd1 nd2 nd3 = Node3 (gmn nd1 + gmn nd2 + gmn nd3) nd1 nd2 nd3"
h2: shows "\<forall>x y. f y \<leftarrow> f x = f y \<leftarrow> x"
entails_def: fixes N1 :: "'f set"   and N2 :: "'f set" shows "(N1 \<Turnstile>\<inter>\<G> N2) = (\<forall>q\<in>Q. entails_q q (no_labels.\<G>_Fset_q q N1) (no_labels.\<G>_Fset_q q N2))"
finvimage_iff: fixes a :: "V"   and r :: "V"   and A :: "V" shows "(a \<in>\<^sub>\<circ> r -`\<^sub>\<bullet> A) = (\<exists>b\<in>elts A. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r)"
isomorphic_reflexive: fixes f :: "'b" assumes "B.ide f" shows "B.isomorphic f f"
MakeCat_def: fixes C :: "('d, 'e, 'f) Category_scheme" shows "MakeCat C \<equiv> \<lparr>Obj = obj\<^bsub>C\<^esub>, Mor = mor\<^bsub>C\<^esub>, Dom = restrict (Dom C) (mor\<^bsub>C\<^esub>), Cod = restrict (Cod C) (mor\<^bsub>C\<^esub>), Id = restrict (Category.Id C) (obj\<^bsub>C\<^esub>), Comp = \<lambda>f g. (\<lambda>(x, y)\<in>{(f, g) |f g. f \<approx>>\<^bsub>C\<^esub> g}. x ;;\<^bsub>C\<^esub> y) (f, g), \<dots> = Category.more C\<rparr>"
left_adjoint_extends_to_adjoint_pair: fixes f :: "'b" assumes "D.is_left_adjoint f" shows "D.adjoint_pair f f\<^sup>*\<^sup>D"
part_brn_disj2: fixes cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and n1 :: "nat"   and n2 :: "nat" assumes "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "n1 < length cl"   and "n2 < length cl"   and "n1 \<noteq> n2" shows "local.shift cl n1 ` P n1 \<inter> local.shift cl n2 ` P n2 = {}"
single_zero: fixes k :: "'a" shows "monomial (0::'b) k = 0"
select_convs_4: fixes Pol :: "('c \<times> 'c) set"   and Map :: "'b \<Rightarrow> 'c"   and Doms :: "'c set"   and List :: "'b list"   and ListOp :: "'b list option"   and Set :: "'b set"   and more :: "'d" shows "List \<lparr>Pol = Pol, Map = Map, Doms = Doms, List = List, ListOp = ListOp, Set = Set, \<dots> = more\<rparr> = List"
foot_def: shows "foot = (1::'a) / (3::'a) *\<^sub>Q yard"
simps_10: fixes p :: "fm"   and q :: "fm" shows "bound0 (Imp p q) = (bound0 p \<and> bound0 q)"
simps_10: fixes v :: "'a rexp"   and va :: "'a rexp" shows "local.le_rexp One (Times v va) = True"
arr_iff_in_hom: fixes f :: "'a \<times> 'a" shows "B.VV.arr f = B.VV.in_hom f (B.VV.dom f) (B.VV.cod f)"
set_ConsD: fixes y :: "'l"   and x :: "'l"   and xs :: "'l list" assumes "y \<in> set (x # xs)" shows "y = x \<or> y \<in> set xs"
totient_formula2: fixes n :: "nat" shows "real (totient n) = real n * (\<Prod>p\<in>prime_factors n. 1 - 1 / real p)"
g_secure_counterexample: shows "g_secure (c_process g_step g_out g_s\<^sub>0) (c_dom g_D)"
types_agree_imp_e_typing: fixes t :: "t"   and v :: "v"   and \<S> :: "s_context"   and \<C> :: "t_context" assumes "types_agree t v" shows "\<S>\<bullet>\<C> \<turnstile> [$C v] : [] _> [t]"
dom_cod: fixes f :: "'b" assumes "J0.arr f" shows "J0.dom (J0.cod f) = J0.cod f"
is_extensional: fixes f :: "'a" assumes "\<not> arr f" shows "trg f = null"
tjs_disc_dw_tw: fixes x :: "'v"   and u :: "'v"   and y :: "'v" assumes "x \<in> set (dropWhile ((\<noteq>) u) (tj_stack s))"   and "y \<in> set (takeWhile ((\<noteq>) u) (tj_stack s))" shows "\<delta> s x < \<delta> s y"
closureE: fixes x :: "'b"   and T :: "'b set set"   and a :: "'b set"   and c :: "'b set" assumes "x \<in> closure\<^bsub>T\<^esub> a" obtains "\<not> c closed\<^bsub>T\<^esub>"   | "\<not> a \<subseteq> c"   | "x \<in> c"
Maclaurin2: fixes h :: "real"   and diff :: "nat \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and n :: "nat" assumes "0 < h"   and "diff 0 = f"   and "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> (diff m has_real_derivative diff (Suc m) t) (at t)" shows "\<exists>t>0. t \<le> h \<and> f h = (\<Sum>m<n. diff m 0 / fact m * h ^ m) + diff n t / fact n * h ^ n"
matcher_def: fixes r :: "'a rexp"   and s :: "'a list" shows "matcher r s = nullable (derivs s r)"
assms_2: shows "\<D>\<^sub>\<circ> hgf = 3\<^sub>\<nat>"
label_incr_start_Node_smaller_rev: fixes n :: "label"   and i :: "nat"   and l :: "nat" assumes "n \<oplus> i = Label l" shows "n = Label (l - i)"
phiDefs_def: fixes phis :: "'b \<Rightarrow> 'c \<times> 'a \<Rightarrow> 'a list option"   and g :: "'b"   and n :: "'c" shows "CFG_SSA_base.phiDefs phis g n \<equiv> {v. (n, v) \<in> dom (phis g)}"
le_antisym: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D"   and "a \<preceq> b"   and "b \<preceq> a" shows "a = b"
derives1_implies_derives: fixes a :: "symbol list"   and b :: "symbol list" assumes "derives1 a b" shows "derives a b"
strict_coboundedI2: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "b < c" shows "min a b < c"
hyps_4: fixes k :: "('fun, 'var) Term.term" assumes "k \<in> set K" shows "ik\<^sub>e\<^sub>s\<^sub>t A \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c k \<cdot> \<I>"
groupWIs_same_fw_not2: fixes A :: "'a wordinterval list"   and c :: "parts_connection"   and rs :: "'a simple_rule list"   and B :: "'a wordinterval list" assumes "A \<in> set (groupWIs c rs)"   and "B \<in> set (groupWIs c rs)"   and "A \<noteq> B" shows "\<forall>ip1\<in>wordinterval_list_to_set A. \<forall>ip2\<in>wordinterval_list_to_set B. \<not> same_fw_behaviour_one ip1 ip2 c rs"
preferred_alts_lift_pref_profile: fixes i :: "'a"   and x :: "'b" assumes "i \<in> agents'"   and "x \<in> alts'" shows "preferred_alts (R' i) x = (if i \<in> agents \<and> x \<in> alts then preferred_alts (R i) x else alts')"
kill_short_\<alpha>: fixes G :: "nat set \<times> nat set set"   and k :: "nat" assumes "finite (uverts G)" shows "\<alpha> (kill_short G k) \<le> \<alpha> G"
z_pos: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec"   and i :: "'a" assumes "perron_frobenius A" shows "0 < perron_frobenius.z A $h i"
simps_1: fixes e :: "'b" shows "prefixToElement e [] = []"
runit_in_hom: fixes a :: "'d" assumes "ide a" shows "\<guillemotleft>\<r>[a] : a \<otimes> \<I> \<rightarrow> a\<guillemotright>"
countable_basis_at_decseq: fixes "open" :: "'c set \<Rightarrow> bool"   and x :: "'c" assumes "class.first_countable_topology open" obtains A :: "nat \<Rightarrow> 'c set" where "\<And>i. open (A i)" and "\<And>i. x \<in> A i" and "\<And>S. \<lbrakk>open S; x \<in> S\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F i in sequentially. A i \<subseteq> S"
induct: fixes P :: "'d decision \<Rightarrow> bool"   and decision :: "'d decision" assumes "\<And>x. P (allow x)"   and "\<And>x. P (deny x)" shows "P decision"
assms_1: shows "(f \<longlongrightarrow> 0) F"
comp_of_ord_eq_comp_of_ords: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "antisymp le" shows "comp_of_ord le = comp_of_ords le (\<lambda>x y. le x y \<and> \<not> le y x)"
can_occur_Recv: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and i :: "nat"   and p :: "'a"   and q :: "'a"   and s :: "'b"   and s' :: "'b"   and m :: "'c"   and c :: "('a, 'b, 'c) configuration" assumes "distributed_system channel"   and "distributed_system.can_occur channel trans send recv (Recv i p q s s' m) c" shows "states c p = s \<and> channel i = Some (q, p) \<and> (\<exists>xs. msgs c i = Msg m # xs) \<and> recv i p q s s' m"
distinct_9: fixes x1 :: "'c" shows "Suspend x1 \<noteq> WokenUp"
existsB: fixes y :: "nat" assumes "y \<in> B" shows "\<exists>x. P x y"
less_num_inner_nodes_label: fixes l :: "nat"   and c :: "cmd" assumes "l < #:c" shows "\<exists>c'. labels c l c'"
subfieldI: fixes K :: "'a set" assumes "subcring K R"   and "Units (R\<lparr>carrier := K\<rparr>) = K - {\<zero>}" shows "subfield K R"
that: fixes V :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "T \<subseteq> V"   and "openin (top_of_set U) V"   and "continuous_on V g"   and "g ` V \<subseteq> S"   and "\<And>x. x \<in> T \<Longrightarrow> g x = f x" shows "thesis"
LPDs_def: fixes r :: "mregex" shows "LPDs r = (\<Union>i. LPDi i r)"
case_2: fixes f1 :: "vname \<Rightarrow> 'a"   and f2 :: "sig \<Rightarrow> 'a"   and x2 :: "sig" shows "(case mid x2 of fid x \<Rightarrow> f1 x | mid x \<Rightarrow> f2 x) = f2 x2"
only_horizontal_division: shows "only_horizontal_division (boundary C) two_chain_typeII"
Basis_imp: fixes k :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "real" assumes "k \<in> Basis" shows "g (cbox a b) = g (cbox a b \<inter> {x. x \<bullet> k \<le> c}) \<^bold>* g (cbox a b \<inter> {x. c \<le> x \<bullet> k})"
elims_2: fixes x :: "'a val" assumes "is_Addr x" obtains a :: "'a" where "x = Addr a"
power_diff_conv_inverse: fixes x :: "'a"   and m :: "nat"   and n :: "nat" assumes "x \<noteq> (0::'a)"   and "m \<le> n" shows "x ^ (n - m) = x ^ n * inverse x ^ m"
i1: shows "PROP P \<Longrightarrow> PROP Q"
insert_list_def: fixes x :: "'a"   and xs :: "'a list" shows "insert_list x xs = (if x \<in> set xs then xs else x # xs)"
star_sim1: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<cdot> z \<le> z \<cdot> y" shows "x\<^sup>\<star> \<cdot> z \<le> z \<cdot> y\<^sup>\<star>"
tpOfV_pred_tpOfV: shows "tpOfV_pred tpOfV"
spr_jview_tObsC_trans: fixes a :: "'a"   and t :: "('a, 'es, 'as) BEState Trace"   and t' :: "('a, 'es, 'as) BEState Trace"   and a' :: "'a"   and t'' :: "('a, 'es, 'as) BEState Trace" assumes "spr_jview a t = spr_jview a t'"   and "spr_jview a' t' = spr_jview a' t''" shows "tObsC t = tObsC t''"
S4: shows "f x (b_least2 f x y2) \<noteq> 0"
h2: shows "well_def_signs (length (qs1 @ qs2)) (get_signs (snd (combine_systems p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2))))"
as_bs_4: shows "v = bs \<bullet>\<cdot> vs"
nat_of_integer_of_nat: fixes n :: "nat" shows "nat_of_integer (of_nat n) = n"
r_univ_def: shows "r_univ \<equiv> Cn 2 r_dec [Cn 2 r_result [Mn 2 (Cn 3 r_not [r_result]), recf.Id 2 0, recf.Id 2 1]]"
Collect_mono_iff: fixes P :: "'x \<Rightarrow> bool"   and Q :: "'x \<Rightarrow> bool" shows "(Collect P \<subseteq> Collect Q) = (\<forall>x. P x \<longrightarrow> Q x)"
Un_iff: fixes c :: "'g"   and A :: "'g set"   and B :: "'g set" shows "(c \<in> A \<union> B) = (c \<in> A \<or> c \<in> B)"
ground_subset: fixes M :: "('c, 'd) Term.term set"   and N :: "('c, 'd) Term.term set" assumes "M \<subseteq> N"   and "ground N" shows "ground M"
spmf_spmf_of_pmf: fixes p :: "'a pmf"   and x :: "'a" shows "spmf (spmf_of_pmf p) x = pmf p x"
cases: fixes a :: "exp" assumes "val(a)" obtains vs :: "exp list" and C :: "nat" where "a = New C vs" and "vals(vs)"
permutes_implies_image_mset_eq: fixes p :: "'c \<Rightarrow> 'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd" assumes "p permutes A"   and "\<And>x. x \<in> A \<Longrightarrow> f x = f' (p x)" shows "image_mset f' (mset_set A) = image_mset f (mset_set A)"
inverse_ereal_ge0I: fixes x :: "ereal" assumes "0 \<le> x" shows "0 \<le> inverse x"
fdia_mult: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows " |x \<cdot> y\<rangle> z =  |x\<rangle>  |y\<rangle> z"
type_update_select_type_length3: fixes tn :: "t list"   and tm :: "t list" assumes "type_update_select (Type tn) = Type tm" shows "\<exists>t ts'. tn = ts' @ [t, t, T_i32]"
lsb_int_def: fixes i :: "int" shows "lsb i = i !! 0"
OrL_inv': fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "F \<^bold>\<or> G, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "F, \<Gamma> \<Rightarrow> \<Delta> \<down> n \<and> G, \<Gamma> \<Rightarrow> \<Delta> \<down> n"
RedT: shows "s -\<triangleright>ttas\<rightarrow>* s'"
QgrpUnit_3: fixes G :: "('e, 'f) Group_scheme"   and E :: "('g, 'h) Group_scheme"   and H :: "'e set"   and H1 :: "'e set" assumes "Group G"   and "Ugp E"   and "G \<guillemotright> H "   and "G \<guillemotright> H1 "   and "\<natural>\<^bsub>G\<^esub>H \<triangleright> H1"   and "(\<natural>\<^bsub>G\<^esub>H / H1) \<cong> E" shows "H = H1"
Step_def: fixes A :: "('state \<Rightarrow> 'b) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> 'answer \<times> 'b)"   and s :: "'state \<times> 'b"   and r :: "'request" shows "Step A s r = (let (a, is') = snd A s r in (step (fst s) r a, is'))"
Supraclassical_logic_def: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL \<equiv> Tarskian_logic Cn \<and> Supraclassical_logic_axioms Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"
complex_roots_int_1: fixes p :: "int poly" shows "Polynomial.smult (complex_of_int (lead_coeff p)) (\<Prod>a\<leftarrow>complex_roots_int p. [:- a, 1:]) = of_int_poly p"
length_dominant_term: fixes F :: "'b" shows "length (snd (dominant_term F)) = expansion_level TYPE('b)"
to_fract_sum: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "to_fract (sum f A) = (\<Sum>x\<in>A. to_fract (f x))"
sets_sup_measure': fixes B :: "'c measure"   and A :: "'c measure" assumes "sets B = sets A" shows "sets (sup_measure' A B) = sets A"
parts_con: fixes s' :: "(agent \<times> msg) set"   and X :: "msg"   and Y :: "msg"   and s :: "(agent \<times> msg) set"   and Z :: "msg" assumes "s' = insert (Spy, \<lbrace>X, Y\<rbrace>) s \<and> (Spy, X) \<in> s \<and> (Spy, Y) \<in> s"   and "Z \<in> parts_msg X \<or> Z \<in> parts_msg Y" shows "Z \<in> parts (used s)"
sub_design_def: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" shows "sub_design \<U> \<A> \<V> \<B> \<equiv> sub_incidence_system \<U> \<A> \<V> \<B> \<and> design \<V> \<B>"
reflection: shows "rel_reflects_binary_pred Rel (\<lambda>b. hasBarb b (STCalWB SWB TWB))"
rho_char: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c"   and y :: "'d" assumes "triple phi" shows "up_filter (triple.rho phi x y) = phi x \<sqinter> up_filter y"
monad_fmap: fixes f :: "'c \<rightarrow> 'a"   and xs :: "'c\<cdot>'b" shows "fmap\<cdot>f\<cdot>xs = xs \<bind> (\<Lambda> x. return\<cdot>(f\<cdot>x))"
keys_of_monomial: fixes c :: "'b"   and t :: "'c" assumes "c \<noteq> (0::'b)" shows "keys (monomial c t) = {t}"
basis_partition_1: shows "basis1 \<union> basis2 = basis"
0: shows "X_poly R \<otimes>\<^bsub>P\<^esub> g = X_poly R \<otimes>\<^bsub>P\<^esub> h"
R43_ad_1: shows "pmf (sds R43) a = 1 / 2"
separated_by_in_other: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b"   and y :: "'b" assumes "separated_by {A, B} x y"   and "x \<notin> A" shows "x \<in> B \<and> y \<in> A"
process_charn: fixes P :: "'a process" shows "([], {}) \<in> F P \<and> (\<forall>s X. (s, X) \<in> F P \<longrightarrow> front_tickFree s) \<and> (\<forall>s t. (s @ t, {}) \<notin> F P \<or> (s, {}) \<in> F P) \<and> (\<forall>s X Y. (s, Y) \<notin> F P \<or> \<not> X \<subseteq> Y \<or> (s, X) \<in> F P) \<and> (\<forall>s X Y. (s, X) \<in> F P \<and> (\<forall>c. c \<in> Y \<longrightarrow> (s @ [c], {}) \<notin> F P) \<longrightarrow> (s, X \<union> Y) \<in> F P) \<and> (\<forall>s X. (s @ [tick], {}) \<in> F P \<longrightarrow> (s, X - {tick}) \<in> F P) \<and> (\<forall>s t. s \<notin> D P \<or> \<not> tickFree s \<or> \<not> front_tickFree t \<or> s @ t \<in> D P) \<and> (\<forall>s X. s \<notin> D P \<or> (s, X) \<in> F P) \<and> (\<forall>s. s @ [tick] \<notin> D P \<or> s \<in> D P)"
neg'_mult_resl: fixes y :: "'a"   and x :: "'a" shows "inv_resr (- x) y = inv_resl x (-' y)"
2: shows "gcs (pp_of_term u) s = (0::'a)"
fst_hyp_edge_at: fixes c :: "'form"   and "is" :: "nat list"   and n :: "nat"   and s :: "'subst" shows "fst (hyp_edge_at c is n s) = hyp_edge_from c is n s"
right_unique_alt_def: fixes R :: "'i \<Rightarrow> 'j \<Rightarrow> bool" shows "right_unique R = (R\<inverse>\<inverse> OO R \<le> (=))"
finite_total_preorder_on_axioms: shows "finite_total_preorder_on {} (\<lambda>_ _. False)"
Diff_subset: fixes A :: "'b set"   and B :: "'b set" shows "order.greater_eq A (A - B)"
ge_trans: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x"   and "z \<le> y" shows "z \<le> x"
bit_d: fixes x :: "'a"   and y :: "'a"   and x0 :: "'a"   and y0 :: "'a"   and s :: "('a list \<times> bool list \<times> 'a list) pmf"   and qs :: "'a list" assumes "x \<noteq> y"   and "{x, y} = {x0, y0}"   and "BIT_inv s x [x0, y0]"   and "set qs \<subseteq> {x, y}"   and "qs \<in> lang (seq [Atom x, Atom x])" shows "T\<^sub>p_on_rand' BIT s qs \<le> 175 / 10\<^sup>2 * real (T\<^sub>p [x, y] qs (OPT2 qs [x, y])) \<and> BIT_inv (Partial_Cost_Model.config'_rand BIT s qs) (last qs) [x0, y0] \<and> T\<^sub>p_on_rand' BIT s qs = 0"
rep_eq: fixes x :: "('d \<times> 'e) list" shows "Rep_slist (abs_slist_if x) = (if distinct (map fst x) \<and> rsorted (map fst x) then x else [])"
is_pot_ordD: fixes u :: "'t"   and v :: "'t" assumes "is_pot_ord"   and "component_of_term u < component_of_term v" shows "u \<prec>\<^sub>t v"
slice_kind_Pred_obs_nearer_SOME: fixes a :: "'edge"   and S :: "'node SDG_node set"   and Q :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool"   and m :: "'node"   and x :: "nat" assumes "sourcenode a \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "kind a = (Q)\<^sub>\<surd>"   and "m \<in> obs_intra (sourcenode a) \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "distance (targetnode a) m x"   and "distance (sourcenode a) m (x + 1)"   and "targetnode a = (SOME n'. \<exists>a'. sourcenode a = sourcenode a' \<and> distance (targetnode a') m x \<and> valid_edge a' \<and> intra_kind (kind a') \<and> targetnode a' = n')" shows "slice_kind S a = (\<lambda>s. True)\<^sub>\<surd>"
ln_fact_conv_sum_mangoldt: fixes x :: "real" shows "sum_upto (\<lambda>n. mangoldt n * real_of_int \<lfloor>x / real n\<rfloor>) x = ln (fact (nat \<lfloor>x\<rfloor>))"
interleave_valid_list: fixes xs :: "Vars.var list"   and ys :: "Vars.var list"   and zs :: "Vars.var list"   and S :: "Vars.var set"   and G :: "CoCalls"   and S' :: "Vars.var set"   and G' :: "CoCalls" assumes "xs \<in> ys \<otimes> zs"   and "ys \<in> valid_lists S G"   and "zs \<in> valid_lists S' G'" shows "xs \<in> valid_lists (S \<union> S') (G \<squnion> (G' \<squnion> S G\<times> S'))"
solvable_def: fixes G :: "('b, 'c) monoid_scheme" shows "solvable G = solvable_seq G (carrier G)"
simps_3: fixes sys :: "'a \<Rightarrow> ('b, 'c) process_term"   and a :: "'d"   and T :: "('d, 'a) process_term" shows "isACT sys (ACT a T) = True"
calculus_axioms: shows "calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F"
m2zz2mInv: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e" assumes "LSCategory C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "z2m\<^bsub>C\<^esub>m2z\<^bsub>C\<^esub>f = f"
elims_2: fixes m :: "nat"   and x :: "('b \<times> nat) tree" assumes "HBT m"   and "HBT.hbt m x" obtains "x = \<langle>\<rangle>"   | l :: "('b \<times> nat) tree" and a :: "'b" and n :: "nat" and r :: "('b \<times> nat) tree" where "x = \<langle>l, (a, n), r\<rangle>" and "\<bar>int (height l) - int (height r)\<bar> \<le> int m \<and> n = max (height l) (height r) + 1 \<and> HBT.hbt m l \<and> HBT.hbt m r"
e1': shows "fst (trunc_bound_eucl p (fst (trunc_bound_eucl p (alpha * fst X)) + zeta)) = fst (trunc_bound_eucl p (alpha * fst X)) + zeta + e1'"
fps_divide_def: fixes f :: "'b fps"   and g :: "'b fps" shows "f / g = fps_shift (subdegree g) (f * inverse (unit_factor g))"