hcju/isabelleps huggingface dataset:
remap_lang: fixes TA :: "('a, 'b) tree_automaton_rec"   and f :: "'a \<Rightarrow> 'c" assumes "tree_automaton TA"   and "inj_on f (ta_rstates TA)" shows "ta_lang (ta_remap f TA) = ta_lang TA"
in_pmdl_srtc: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "p \<in> pmdl F" shows "relation.srtc (red F) p 0"
holomorphic_higher_deriv: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and n :: "nat" assumes "f holomorphic_on S"   and "open S" shows "(deriv ^^ n) f holomorphic_on S"
f13: shows "\<not> (\<exists>(a', b', c')\<in>set c. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f \<noteq> 0))"
one_divide_eq_0_iff: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and a :: "'e" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(divide one a = zero) = (a = zero)"
first_edge_in_edges: fixes xs :: "'a list" assumes "tl xs \<noteq> []" shows "first_edge_of_walk xs \<in> edges_of_walk xs"
lt_trans: fixes eq :: "'r \<rightarrow> 'r \<rightarrow> tr"   and compare :: "'r \<rightarrow> 'r \<rightarrow> Ordering"   and x :: "'r"   and y :: "'r"   and z :: "'r" assumes "class.Ord_linear eq compare"   and "Ord.lt compare\<cdot>x\<cdot>y = TT"   and "Ord.lt compare\<cdot>y\<cdot>z = TT" shows "Ord.lt compare\<cdot>x\<cdot>z = TT"
FAcc: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs' :: "char list list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and D :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and Ds :: "char list list"   and Cs :: "char list list"   and fs :: "char list \<Rightarrow> val option"   and F :: "char list"   and v :: "val" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref (a, Cs'),(h, l)\<rangle>"   and "h a = \<lfloor>(D, S)\<rfloor>"   and "Ds = Cs' @\<^sub>p Cs"   and "(Ds, fs) \<in> S"   and "fs F = \<lfloor>v\<rfloor>" shows "P,E \<turnstile> \<langle>e\<bullet>F{Cs},s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,(h, l)\<rangle>"
verit_sum_simplify: fixes a :: "'e" shows "a + (0::'e) = a"
LSM_sub_submodule: fixes M :: "('e, 'a, 'f) Module_scheme"   and T :: "'e set"   and N :: "'e set" assumes "R module M"   and "T \<subseteq> carrier M"   and "submodule R M N"   and "T \<subseteq> N" shows "LSM\<^bsub>R\<^esub> M T \<subseteq> N"
nat_pow_distrib: fixes x :: "'a mod_ring poly"   and y :: "'a mod_ring poly"   and n :: "nat" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "(x \<otimes>\<^bsub>R\<^esub> y) [^]\<^bsub>R\<^esub> n = x [^]\<^bsub>R\<^esub> n \<otimes>\<^bsub>R\<^esub> y [^]\<^bsub>R\<^esub> n"
zhmset_of_enat_lt_iff_lt: fixes m :: "enat"   and n :: "enat" shows "(zhmset_of_enat m < zhmset_of_enat n) = (m < n)"
dual_disjunctive: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<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 dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> disjunctive.disjunctive sup sup times" shows "dual x \<in> conjunctive.conjunctive inf inf times"
SigmaD1: fixes a :: "'g"   and b :: "'h"   and A :: "'g set"   and B :: "'g \<Rightarrow> 'h set" assumes "(a, b) \<in> Sigma A B" shows "a \<in> A"
uncountable_infinite: fixes A :: "'b set" assumes "uncountable A" shows "infinite A"
asym_greater: fixes a :: "'a"   and b :: "'a" assumes "cmp a b = Greater" shows "cmp b a \<noteq> Greater"
monotone_convergence_increasing: fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and S :: "'a set"   and g :: "'a \<Rightarrow> real" assumes "\<And>k. f k integrable_on S"   and "\<And>k x. x \<in> S \<Longrightarrow> f k x \<le> f (Suc k) x"   and "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x"   and "bounded (range (\<lambda>k. integral S (f k)))" shows "g integrable_on S \<and> (\<lambda>k. integral S (f k)) \<longlonglongrightarrow> integral S g"
1: shows "stfx.one_stalk V \<in> stfx.carrier_stalk"
integrable_cong_AE: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. f x = g x" shows "integrable M f = integrable M g"
bij_betw_inv_into_right: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and A' :: "'e set"   and a' :: "'e" assumes "bij_betw f A A'"   and "a' \<in> A'" shows "f (inv_into A f a') = a'"
uw\<theta>: shows "ide w \<and> \<guillemotleft>w.the_\<theta> : s\<^sub>0 \<star> w \<Rightarrow> r\<^sub>0\<guillemotright>"
map_comp: fixes g1 :: "'d \<Rightarrow> 'a"   and g2 :: "'e \<Rightarrow> 'b"   and f1 :: "'f \<Rightarrow> 'd"   and f2 :: "'g \<Rightarrow> 'e"   and v :: "('f, 'g, 'c) var0" shows "map_var0 g1 g2 (map_var0 f1 f2 v) = map_var0 (g1 \<circ> f1) (g2 \<circ> f2) v"
norm_char: fixes a :: "'a" shows "cmod (\<chi> a) = (if a \<in> carrier G then 1 else 0)"
measure_preserving_preserves_integral_1: fixes T :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "T \<in> measure_preserving M N"   and "integrable N f" shows "integrable M (\<lambda>x. f (T x))"
mem_appL: fixes A :: "'a list"   and x :: "'a"   and B :: "'a list" assumes "List.member A x" shows "List.member (A @ B) x"
parts_crypt_pubkey_start: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and SK :: "nat option \<times> nat set \<times> nat set"   and C :: "nat" assumes "s \<turnstile> s'"   and "Crypt (SesK SK) (PubKey C) \<in> parts (used s')"   and "Crypt (SesK SK) (PubKey C) \<notin> parts (used s)" shows "C \<in> snd (snd SK) \<and> ((\<exists>n. (Owner n, SesKey SK) \<in> s') \<or> (\<exists>n B. (Asset n, Token n (Auth_PriK n) B C SK) \<in> s')) \<or> SesKey SK \<in> spied s'"
flush_step_preserves_read_tmps: fixes m :: "nat \<Rightarrow> nat"   and sb :: "'a memref list"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option) \<times> (nat \<Rightarrow> bool option)"   and m' :: "nat \<Rightarrow> nat"   and sb' :: "'a memref list"   and \<O>' :: "nat set \<times> (nat \<Rightarrow> bool option) \<times> (nat \<Rightarrow> bool option)"   and n :: "nat" assumes "(m, sb, \<O>) \<rightarrow>\<^sub>f (m', sb', \<O>')"   and "\<forall>i\<in>read_tmps sb. i < n" shows "\<forall>i\<in>read_tmps sb'. i < n"
ideD_2: fixes a :: "'c" assumes "Cop.ide a" shows "Cop.dom a = a"
flip: fixes prod :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "bounded_bilinear prod" shows "bounded_bilinear (\<lambda>x y. prod y x)"
coeff_iff_polynomial_cond: fixes K :: "'a set"   and p1 :: "'a list"   and p2 :: "'a list" assumes "polynomial K p1"   and "polynomial K p2" shows "(p1 = p2) = (local.coeff p1 = local.coeff p2)"
M: fixes T :: "'b \<Rightarrow> 'b"   and M :: "'b measure"   and Z :: "real measure"   and f :: "'b \<Rightarrow> real"   and B :: "nat \<Rightarrow> real" assumes "pmpt_limit T M Z f B" shows "f \<in> borel_measurable M"
incid_C_AB: fixes A :: "Points"   and B :: "Points"   and l :: "Lines"   and C :: "Points" assumes "A \<noteq> B"   and "incid A l"   and "incid B l"   and "incid C l" shows "incid C (line A B)"
eventually_defs: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and w :: "nat \<Rightarrow> 'a" shows "(\<diamond>F) w = (\<exists>n. F (w |\<^sub>s n))"
Cauchy_has_contour_integral_higher_derivative_circlepath: fixes z :: "complex"   and r :: "real"   and f :: "complex \<Rightarrow> complex"   and w :: "complex"   and k :: "nat" assumes "continuous_on (cball z r) f"   and "f holomorphic_on ball z r"   and "w \<in> ball z r" shows "((\<lambda>u. f u / (u - w) ^ Suc k) has_contour_integral complex_of_real (2 * pi) * \<i> / fact k * (deriv ^^ k) f w) (circlepath z r)"
topspace_subtopology: fixes U :: "'b topology"   and V :: "'b set" shows "topspace (subtopology U V) = topspace U \<inter> V"
SN_on_O_push: fixes X :: "'a set" assumes "SN_on R X" shows "SN_on (R O S\<^sup>*) X"
below_val_Zp_zero: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier Zp"   and "x (Suc n) \<noteq> 0" shows "val_Zp x \<le> eint (int n)"
cos_angle_opposite2: fixes H :: "ocircline"   and H' :: "ocircline" shows "cos_angle H (opposite_ocircline H') = - cos_angle H H'"
simps_4: fixes a :: "'a"   and \<nu> :: "'a set" shows "af_G_letter (np(a)) \<nu> = (if a \<notin> \<nu> then true else false)"
cinner_ket_right: fixes \<psi> :: "'c ell2"   and i :: "'c" shows "\<psi> \<bullet>\<^sub>C ket i = cnj (Rep_ell2 \<psi> i)"
1: fixes a :: "'a \<times> 'b"   and b :: "'a \<times> 'b" shows "set xs \<subseteq> insert a (insert b (set xs))"
lincomb: shows "lincomb_list c (factorization_lattice u (n - d) k) = g"
rev_finite_subset: fixes B :: "'g set"   and A :: "'g set" assumes "finite B"   and "A \<subseteq> B" shows "finite A"
rw2: shows "map (\<lambda>x. IArray.of_fun (\<lambda>i. A $ mod_type_class.from_nat x $ mod_type_class.from_nat i) CARD('c)) [0..<CARD('b)] ! mod_type_class.to_nat b !! mod_type_class.to_nat j = A $ b $ j"
floor_less_one: fixes x :: "'b" shows "(\<lfloor>x\<rfloor> < 1) = (x < (1::'b))"
parts_connection_ssh_def: shows "parts_connection_ssh \<equiv> \<lparr>pc_iiface = ''1'', pc_oiface = ''1'', pc_proto = TCP, pc_sport = 10000, pc_dport = 22\<rparr>"
L: shows "K \<subseteq> \<Union> (Z ` L)"
space_restrict_space: fixes M :: "'c measure"   and \<Omega> :: "'c set" shows "space (restrict_space M \<Omega>) = \<Omega> \<inter> space M"
image_is_empty: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(f ` A = {}) = (A = {})"
carr: shows "{fs ! i, M.sumlist (map (\<lambda>j. - \<mu> i j \<cdot>\<^sub>v gso j) [0..<l]), - \<mu> i l \<cdot>\<^sub>v gso l} \<subseteq> carrier_vec n"
Sup_finite_empty: shows "\<Squnion> {} = \<Sqinter> UNIV"
transcendental_imp_trivial_ker: fixes K :: "'a set"   and x :: "'a" assumes "(transcendental over K) x" shows "a_kernel (K [X]) R (\<lambda>p. local.eval p x) = {[]}"
le_antisym: fixes m :: "nat"   and n :: "nat" assumes "order.greater_eq n m"   and "order.greater_eq m n" shows "m = n"
simps_1: fixes s :: "'b sparc_state" shows "delayed_write_all [] s = s"
assms: shows "\<not> R\<^sup>+\<^sup>+ x y"
sgn_less_eq_1_rat: fixes a :: "rat"   and b :: "rat" assumes "sgn a = 1"   and "a \<le> b" shows "sgn b = 1"
eq_\<O>\<^sub>X: fixes U :: "'b set" assumes "U \<subseteq> X" shows "local.ind_sheaf U = \<O>\<^sub>X U"
smaps_over_un: fixes m1 :: "'c \<Rightarrow> 'd set"   and A :: "'c set"   and B :: "'d set"   and m2 :: "'c \<Rightarrow> 'd set" assumes "m1 \<in> smaps_over A B"   and "m2 \<in> smaps_over A B" shows "m1 \<union>. m2 \<in> smaps_over A B"
assms_2: shows "t \<noteq> \<langle>\<rangle>"
diag_block_mat_square: fixes As :: "'a mat list" assumes "Ball (set As) square_mat" shows "square_mat (diag_block_mat As)"
bin_cat_assoc: fixes n :: "nat"   and z :: "int"   and m :: "nat"   and y :: "int"   and x :: "int" shows "concat_bit n z (concat_bit m y x) = concat_bit (m + n) (concat_bit n z y) x"
ipurge_ref_aux_all: fixes U :: "'a set"   and D :: "'b \<Rightarrow> 'a"   and X :: "'b set"   and xs :: "'b list"   and I :: "('a \<times> 'a) set" assumes "\<And>u. u \<in> U \<Longrightarrow> \<not> (\<exists>v\<in>D ` (X \<union> set xs). (u, v) \<in> I)" shows "ipurge_ref_aux I D U xs X = X"
degree_rhs_card: shows "degree (\<Prod>x\<in>UNIV. [:0, 1:] - [:x:]) = CARD('a)"
less_prod_def: fixes p :: "'a \<times> 'b"   and q :: "'a \<times> 'b" shows "(p < q) = strict (\<le>) p q"
left_inverse_linear: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec"   and g :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec" assumes "linear (*s) (*s) f"   and "g \<circ> f = id" shows "linear (*s) (*s) g"
connected_iff_interval: fixes U :: "'c set" shows "connected U = (\<forall>x\<in>U. \<forall>y\<in>U. \<forall>z\<ge>x. z \<le> y \<longrightarrow> z \<in> U)"
wlsFSwSb_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "wlsFSwSb MOD \<equiv> wlsFSw MOD \<and> gSubstAllPresGWlsAll MOD \<and> gSubstCls MOD"
defs_uses_disjoint': fixes n :: "'node"   and g :: "'g"   and v :: "'var \<times> 'node \<times> Def" assumes "n \<in> set (\<alpha>n g)"   and "v \<in> defs' g n"   and "v \<in> uses' g n" shows "False"
hyps: fixes ys :: "'a list"   and thesis :: "bool" assumes "\<And>ys1 ys2. \<lbrakk>ys1 \<in> listset xs1; ys2 \<in> listset xs2; ys = ys1 @ ys2\<rbrakk> \<Longrightarrow> thesis"   and "ys \<in> listset (xs1 @ xs2)" shows "thesis"
hseq_char: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.hseq \<nu> \<mu> = (Span.arr \<mu> \<and> Span.arr \<nu> \<and> Span.src \<nu> = Span.trg \<mu>)"
assms_2: fixes C :: "('l, 'x) labeled_graph" assumes "consequence_graph Rs C" shows "maintained (S, G) C"
simps: fixes a1 :: "('a, 'b, 'c) psi"   and a2 :: "('a, 'b, 'c) psi" shows "bangPred a1 a2 = ((\<exists>P. a1 = P \<and> a2 = !P) \<or> (\<exists>P. a1 = P \<and> a2 = P \<parallel> !P))"
mapl_FGl_def: fixes l1 :: "'p \<Rightarrow> 'ac"   and l2 :: "'q \<Rightarrow> 'ad"   and l3 :: "'w \<Rightarrow> 'ae" shows "mapl_FGl l1 l2 l3 = map_FGl l1 l2 l3 id id id id id id id id"
empty_natural: fixes f :: "'n \<Rightarrow> 'p"   and g :: "'q \<Rightarrow> 'o" shows "(\<lambda>_. {}) \<circ> f = (`) g \<circ> (\<lambda>_. {})"
trans: fixes P :: "('a, 'b) Proc"   and Q :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b"   and SRel :: "('a \<times> 'a) set"   and TRel :: "('b \<times> 'b) set"   and R :: "('a, 'b) Proc" assumes "(P, Q) \<in> encoding.indRelSTEQ Enc SRel TRel"   and "(Q, R) \<in> encoding.indRelSTEQ Enc SRel TRel" shows "(P, R) \<in> encoding.indRelSTEQ Enc SRel TRel"
elims_3: fixes x :: "'a"   and xa :: "'a interval" assumes "\<not> x \<in>\<^sub>i xa" obtains y :: "'a" and lx :: "'a" and ux :: "'a" where "x = y" and "xa = Interval lx ux" and "\<not> (lx \<le> y \<and> y \<le> ux)"
pelims: fixes x :: "complex \<times> complex"   and xa :: "complex \<times> complex"   and y :: "complex" assumes "x *\<^sub>v\<^sub>v xa = y"   and "mult_vv_dom (x, xa)" obtains x :: "complex" and y :: "complex" and a :: "complex" and b :: "complex" where "xa__ = (x, y)" and "xa = (a, b)" and "ya__ = x * a + y * b" and "mult_vv_dom ((x, y), a, b)"
simps_9: fixes v :: "nat" shows "\<top>\<^sub>2 +\<^sub>2 [v]\<^sub>2 = \<top>\<^sub>2"
complete_of_proper: fixes S :: "'a set" assumes "proper S" shows "complete S"
induces_unit_counit_adjunction: shows "unit_counit_adjunction (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G \<eta> \<epsilon>"
abelian_groupE_3: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "abelian_group R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<oplus>\<^bsub>R\<^esub> z = x \<oplus>\<^bsub>R\<^esub> (y \<oplus>\<^bsub>R\<^esub> z)"
simps: fixes idx :: "nat"   and "is" :: "int list"   and uu :: "int"   and uv :: "int" shows "wf0 idx (presb.Eq is uu uv) = (length is = idx)"
U'_def: shows "U' \<equiv> {y \<in> topspace X'. g y \<in> U''}"
qe: shows "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm bs (qe p) = Ifm bs (E p)"
line_integral_distrib_1: fixes f :: "'a \<Rightarrow> 'a"   and basis :: "'a set"   and g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "line_integral_exists f basis g1"   and "line_integral_exists f basis g2"   and "valid_path g1"   and "valid_path g2" shows "line_integral f basis (g1 +++ g2) = line_integral f basis g1 + line_integral f basis g2"
nb\<^sub>1: shows "\<forall>op\<in>set (\<Psi>\<^sub>\<O>\<^sub>+). \<exists>op'\<in>set ((\<phi> \<Psi> )\<^sub>\<O>). op' = \<phi>\<^sub>O \<Psi> op"
assms_3: shows "r \<le> x\<^sup>\<star> ; q"
inv_imps_work_sum_zcount: fixes c :: "('loc, 't) configuration" shows "inv_imps_work_sum c = inv_imps_work_sum_zcount c"
linorder_cases: fixes x :: "'t"   and y :: "'t" obtains "x \<prec>\<^sub>t y"   | "x = y"   | "y \<prec>\<^sub>t x"
red_mult_scalar_lt: fixes f :: "'a \<Rightarrow>\<^sub>0 'c"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and c :: "'c" assumes "f \<noteq> 0" shows "(punit.red {f})\<^sup>*\<^sup>* (local.punit.mult_scalar p (monomial c (punit.lt f))) (local.punit.monom_mult (- c / punit.lc f) (0::'a) (local.punit.mult_scalar p (punit.tail f)))"
intro: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "\<Omega> \<in> M" shows "algebra_axioms \<Omega> M"
removeKeyList_eq: fixes as :: "'a list"   and ps :: "('a \<times> 'b) list" shows "removeKeyList as ps = filter (\<lambda>p. \<forall>a\<in>set as. a \<noteq> fst p) ps"
encR: fixes S :: "'procS" shows "SourceTerm S \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R TargetTerm (\<lbrakk>S\<rbrakk>)"
simps_10: fixes v :: "nat \<Rightarrow> 'a"   and l1 :: "linear_poly"   and l2 :: "linear_poly" shows "v \<Turnstile>\<^sub>c EQPP l1 l2 = (l1 \<lbrace> v \<rbrace> = l2 \<lbrace> v \<rbrace>)"
bisim1Throw2: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list" shows "P,throw e,h \<turnstile> (Throw a, xs) \<leftrightarrow> ([Addr a], xs, length (compE2 e), \<lfloor>a\<rfloor>)"
a_star_ImpL: fixes M :: "trm"   and M' :: "trm"   and N :: "trm"   and N' :: "trm"   and a :: "coname"   and y :: "name"   and z :: "name" assumes "M \<longrightarrow>\<^sub>a* M'"   and "N \<longrightarrow>\<^sub>a* N'" shows "ImpL <a>.M y.N z \<longrightarrow>\<^sub>a* ImpL <a>.M' y.N' z"
H: shows "(Set{\<two>} \<doteq> null) = false"
msubset_of_functions_of: fixes M :: "'c multiset"   and B :: "'c set"   and A :: "'d set" assumes "set_mset M \<subseteq> B"   and "size M = card A"   and "finite A" shows "msubset_of A (functions_of A M) = M"
Abs_dlist_inject: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> {xs. distinct xs}"   and "y \<in> {xs. distinct xs}" shows "(Abs_dlist x = Abs_dlist y) = (x = y)"
exhaust: fixes y :: "('e, 'f, 'g, 'h) prot_transaction" obtains x1 :: "((('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) list" and x2 :: "('h strand_label \<times> (('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step) list" and x3 :: "('h strand_label \<times> (('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step) list" and x4 :: "('h strand_label \<times> (('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step) list" and x5 :: "('h strand_label \<times> (('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step) list" and x6 :: "('h strand_label \<times> (('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step) list" where "y = Transaction x1 x2 x3 x4 x5 x6"
lemma_2_6_20_a: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a l\<rightarrow> b \<sqinter> c = (a l\<rightarrow> b) \<sqinter> (a l\<rightarrow> c)"
rell_G_mono_strong: fixes L1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and L2 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and x :: "('n, 'p, 'r, 's, 't, 'u, 'v) G"   and y :: "('o, 'q, 'r, 's, 't, 'u, 'v) G"   and L1' :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and L2' :: "'p \<Rightarrow> 'q \<Rightarrow> bool" assumes "rell_G L1 L2 x y"   and "\<And>a b. \<lbrakk>a \<in> set1_G x; b \<in> set1_G y; L1 a b\<rbrakk> \<Longrightarrow> L1' a b"   and "\<And>a b. \<lbrakk>a \<in> set2_G x; b \<in> set2_G y; L2 a b\<rbrakk> \<Longrightarrow> L2' a b" shows "rell_G L1' L2' x y"
cindex_polyE_inverse_add_cross: fixes a :: "real"   and b :: "real"   and p :: "real poly"   and q :: "real poly" assumes "a < b"   and "coprime p q" shows "cindex_polyE a b q p + cindex_polyE a b p q = real_of_int (cross_alt p q a b) / 2"
axioms_2: fixes L :: "('c, 'd) gorder_scheme" assumes "lower_semilattice L" shows "lower_semilattice_axioms L"
subst_set_offending_flows: fixes nP :: "'v \<Rightarrow> 'a" shows "SecurityInvariant_withOffendingFlows.set_offending_flows (\<lambda>G nP. c_sinvar m G) G nP = c_offending_flows m G"
spoly_def: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "i.punit.spoly p q = (let v1 = i.lpp p; v2 = i.lpp q in if punit.component_of_term v1 = punit.component_of_term v2 then let t1 = punit.pp_of_term v1; t2 = punit.pp_of_term v2; l = lcs t1 t2 in punit.monom_mult ((1::'b) / lookup p v1) (l - t1) p - punit.monom_mult ((1::'b) / lookup q v2) (l - t2) q else 0)"
tr_sem_equiv': fixes D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and A :: "('fun, 'var) stateful_strand_step list"   and M :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<forall>(t, t')\<in>set D. (fv t \<union> fv t') \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}"   and "fv\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}"   and "ground M"   and "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)" shows "\<lbrakk>M; set D \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I>; A\<rbrakk>\<^sub>s \<I> = (\<exists>A'\<in>set (tr A D). \<lbrakk>M; A'\<rbrakk>\<^sub>d \<I>)"
evnotp2: shows "\<forall>\<^sub>F t in at_right 0. flow0 x0 t \<notin> p2"
pathin_subtopology: fixes X :: "'a topology"   and S :: "'a set"   and g :: "real \<Rightarrow> 'a" shows "pathin (subtopology X S) g = (pathin X g \<and> (\<forall>x\<in>{0..1}. g x \<in> S))"
nat_filtration_borel_measurable_singleton: fixes f :: "bool stream \<Rightarrow> 'b"   and n :: "nat"   and x :: "bool stream" assumes "f \<in> borel_measurable (nat_filtration n)" shows "f -` {f x} \<in> sets (nat_filtration n)"
phiDefs_in_allDefs: fixes v :: "'var \<times> 'node \<times> Def"   and g :: "'g"   and n :: "'node" assumes "v \<in> braun_ssa.phiDefs g n" shows "v \<in> braun_ssa.allDefs g n"
iso_in_homE: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "VoV.iso_in_hom f a b" obtains "VoV.in_hom f a b" and "VoV.iso f"
cons_eqvt: fixes pi :: "('b \<times> 'b) list"   and x :: "'a"   and xs :: "'a list" shows "pi \<bullet> (x # xs) = pi \<bullet> x # pi \<bullet> xs"