hcju/isabelleps huggingface dataset:
mono_Mndet: fixes A :: "'a set" shows "monofun (mndet A)"
sifum_compositionality_cont: fixes cmds :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) list" assumes "list_all com_sifum_secure cmds"   and "\<forall>mem. True \<longrightarrow> sound_mode_use (cmds, mem)" shows "prog_sifum_secure_cont cmds"
assms_3: shows "min_maxsimpchain (x # ys @ [y])"
antisym_llexord: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "antisymp r"   and "\<And>x. \<not> r x x" shows "antisymp (llexord r)"
borel_measurable_simple_function: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "simple_function M f" shows "f \<in> borel_measurable M"
boolean_implies_below: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> y \<le> x \<leadsto> y"
simps: fixes p :: "'node"   and R :: "'node set"   and a :: "'node" shows "not_blocked p R a = ((\<exists>Sl q. a = q \<and> Sl \<in> slices p \<and> (\<forall>q\<in>Sl \<inter> W. q \<notin> R \<and> \<not> blocking_min R q) \<and> q \<in> Sl) \<or> (\<exists>p' Sl q. a = q \<and> not_blocked p R p' \<and> p' \<in> W \<and> Sl \<in> slices p' \<and> (\<forall>q\<in>Sl \<inter> W. q \<notin> R \<and> \<not> blocking_min R q) \<and> q \<in> Sl))"
updates_simps_2: fixes ks :: "'c list"   and ps :: "('c \<times> 'd) list" shows "updates ks [] ps = ps"
weak_action_modality_eqvt: fixes p :: "perm"   and \<alpha> :: "'act"   and x :: "('idx, 'pred, 'act) formula" shows "p \<bullet> \<langle>\<langle>\<alpha>\<rangle>\<rangle>x = \<langle>\<langle>p \<bullet> \<alpha>\<rangle>\<rangle>p \<bullet> x"
length_last_def: fixes xs :: "'a list" shows "length_last xs = (length xs, last xs)"
work_alt_def: fixes xs :: "int list" shows "work xs = (if 0 \<in> set xs then fail else return (prod_list xs))"
simps_3: fixes \<eta> :: "nat" shows "valid_mac_query \<eta> Forward = True"
h_is_min: shows "h = fMin (fset_of_list (h # t))"
q'_def: shows "q' \<equiv> map_poly inv_embed cq"
inj_on_Inner: fixes \<Delta> :: "('a, 'b) network_scheme"   and A :: "'c set" assumes "flow_attainability \<Delta>" shows "inj_on Inner A"
pred_set: fixes P :: "'c \<Rightarrow> bool" shows "pred_fun (\<lambda>_. True) P = (\<lambda>x. Ball (range x) P)"
bot_finite_def: shows "\<bottom> = \<Sqinter> UNIV"
unpadr_padr: fixes m :: "nat"   and v :: "'b vec" shows "unpadr m (v @\<^sub>v 0\<^sub>v m) = v"
zero_le_dist: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.metric_space dist uniformity open" shows "0 \<le> dist x y"
inl_ncoll: fixes y :: "real \<times> real"   and X :: "'a \<times> (real \<times> real) pdevs"   and z :: "real \<times> real" assumes "y \<in> set (inl (snd X))"   and "z \<in> set (inl (snd X))"   and "y \<noteq> z" shows "det3 0 y z \<noteq> 0"
is_max: fixes \<aa> :: "'a set" assumes "lideal \<aa> B addB multB zeroB oneB"   and "\<aa> \<noteq> B"   and "I \<subseteq> \<aa>" shows "I = \<aa>"
least: fixes f :: "'f" shows "least f = (w (f, 0) = w0 \<and> (\<forall>g. w (g, 0) = w0 \<longrightarrow> pr_weak' (g, 0) (f, 0)))"
s_2: shows "(w \<circ> r \<circ> s) \<longlonglongrightarrow> liminf (w \<circ> r)"
mult_right_cancel: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_no_zero_divisors_cancel plus zero times"   and "c \<noteq> zero" shows "(times a c = times b c) = (a = b)"
p: shows "p \<in> \<PP> \<and> pvalid p (finished_item \<alpha>)"
that: fixes \<Gamma>' :: "(Vars.var \<times> exp) list"   and S' :: "stack_elem list" assumes "c = (\<Gamma>', e, S')"   and "\<Gamma> = restrictA (- set r) \<Gamma>'"   and "S = restr_stack (- set r) S' @ map Dummy (rev r)" shows "thesis"
appKM_def: shows "appKM \<equiv> \<Lambda> fK xK. bindK\<cdot>fK\<cdot>(\<Lambda> (ValKF\<cdot>f). f\<cdot>xK)"
max_reg_list_def: fixes g :: "vname gexp list" shows "max_reg_list g = fold max (map GExp.max_reg g) None"
circline_symmetric_inv_homo_disc: fixes a :: "complex_homo" shows "circline_symmetric a (inversion a) unit_circle"
apd2: fixes y :: "'a"   and x :: "'a" shows "ar (y \<cdot> x) \<le> ar (ar ar y \<cdot> x)"
that: fixes q :: "'state" assumes "en a q"   and "int q \<noteq> int (ex a q)" shows "thesis"
rep_eq: fixes x :: "nat"   and xa :: "uint" shows "Rep_uint (drop_bit x xa) = drop_bit x (Rep_uint xa)"
edge_succ_permutes: shows "edge_succ HM permutes arcs H"
one_zero: shows "\<one>\<^bsub>S\<^esub> \<noteq> \<zero>\<^bsub>S\<^esub>"
proj3_inv: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dests :: "nat \<Rightarrow> nat option" assumes "dip \<in> kD rt" shows "\<pi>\<^sub>3 (the (invalidate rt dests dip)) = \<pi>\<^sub>3 (the (rt dip))"
map_cong: fixes x :: "'b \<Rightarrow> 'c"   and ya :: "'b \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "x = ya"   and "\<And>z. z \<in> range ya \<Longrightarrow> f z = g z" shows "f \<circ> x = g \<circ> ya"
DBM_zone_repr_def: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'e DBMEntry"   and v :: "'d \<Rightarrow> nat"   and n :: "nat" shows "[M]\<^bsub>v,n\<^esub> = {u. u \<turnstile>\<^bsub>v,n\<^esub> M}"
out_in_indep: fixes x :: "'a \<Longrightarrow> 'c"   and y :: "'d \<Longrightarrow> 'b" shows "$x\<acute> \<bowtie> $y"
fls_lr_inverse_shift_nonzero_2: fixes f :: "'a fls"   and m :: "int"   and x :: "'a" assumes "f \<noteq> 0" shows "fls_right_inverse (fls_shift m f) x = fls_shift (- m) (fls_right_inverse f x)"
impossible_Cons: fixes xs :: "'b list"   and ys :: "'b list"   and x :: "'b" assumes "order.greater_eq (length ys) (length xs)" shows "(xs = x # ys) = False"
assms_2: shows "g ` A \<subseteq> eball (0::'b) (fps_conv_radius f)"
61420: fixes x :: "'a"   and y :: "'a" shows "- (- - x \<sqinter> y) = - (- - y \<sqinter> x)"
ZAssign: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and x :: "char list"   and a :: "aexp" shows "\<turnstile>\<^sub>Z {\<lambda>s. eSuc (P (s[a/x]))} x ::= a {P}"
infdist_union_min: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" assumes "A \<noteq> {}"   and "B \<noteq> {}" shows "infdist x (A \<union> B) = min (infdist x A) (infdist x B)"
that_2: assumes "\<exists>n\<in>I - S. \<forall>Sl\<in>slices n. Sl \<inter> S \<inter> I \<noteq> {}" shows "thesis"
arr_char: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "arr f = (f \<noteq> Null \<and> Dom f \<in> {A. |A| <o \<AA>} \<and> Cod f \<in> {A. |A| <o \<AA>} \<and> Map f \<in> extensional (Dom f) \<inter> (Dom f \<rightarrow> Cod f))"
abs_eq: fixes xa :: "perm"   and x :: "'b \<times> 'c" shows "xa \<bullet> abs_residual x = abs_residual (xa \<bullet> x)"
small_big_append: fixes \<pi> :: "char list \<Rightarrow> com option"   and cs\<^sub>1 :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)"   and cs\<^sub>2 :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)"   and s\<^sub>3 :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "small_step \<pi> cs\<^sub>1 = Some cs\<^sub>2"   and "\<pi>: cs\<^sub>2 \<Rightarrow> s\<^sub>3" shows "\<pi>: cs\<^sub>1 \<Rightarrow> s\<^sub>3"
coprime_divisors: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a c"   and "dvd.dvd times b d"   and "algebraic_semidom.coprime times one c d" shows "algebraic_semidom.coprime times one a b"
strong_fermat_witness_def: shows "local.strong_fermat_witness = (\<not> local.strong_fermat_liar)"
l3_inv10_trans: shows "{l3_inv10} TS.trans l3 {> l3_inv10}"
rep_eq: shows "blinfun_apply fst_blinfun = fst"
tuple_simps_2: fixes f :: "'a"   and g :: "'a" assumes "span f g" shows "local.dom \<langle>f, g\<rangle> = local.dom f"
Collect_case_prodD: fixes x :: "'b \<times> 'd"   and A :: "'b \<Rightarrow> 'd \<Rightarrow> bool" assumes "x \<in> {(x, y). A x y}" shows "A (fst x) (snd x)"
partition_onI: fixes P :: "'c set set"   and A :: "'c set" assumes "\<Union> P = A"   and "\<And>p q. \<lbrakk>p \<in> P; q \<in> P; p \<noteq> q\<rbrakk> \<Longrightarrow> disjnt p q"   and "{} \<notin> P" shows "partition_on A P"
take_all_iff: fixes n :: "nat"   and xs :: "'d list" shows "(take n xs = xs) = (length xs \<le> n)"
sqrt_sqrt_mult: fixes x :: "real"   and y :: "real" assumes "0 \<le> x" shows "sqrt x * (sqrt x * y) = x * y"
pickQFresh: fixes V :: "'var set"   and XS :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort" assumes "finite V \<or> |V| <o |UNIV|"   and "finite XS \<or> |XS| <o |UNIV|"   and "Ball XS qGood" shows "pickQFresh xs V XS \<notin> V \<and> (\<forall>X\<in>XS. qAFresh xs (pickQFresh xs V XS) X \<and> qFresh xs (pickQFresh xs V XS) X)"
one_div_numeral: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'b \<times> 'b"   and divmod_step :: "num \<Rightarrow> 'b \<times> 'b \<Rightarrow> 'b \<times> 'b"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "divide one (numeral.numeral one plus n) = fst (divmod num.One n)"
model_for_cnf_is_model_of_all_subsets: fixes \<nu> :: "'b \<Rightarrow> bool"   and \<F> :: "'b literal set set"   and \<F>' :: "'b literal set set" assumes "cnf_semantics \<nu> \<F>"   and "\<F>' \<subseteq> \<F>" shows "cnf_semantics \<nu> \<F>'"
min_gallery_betw_CCons_reduce: fixes w :: "'a set"   and x :: "'a set"   and ys :: "'a set list"   and z :: "'a set" assumes "min_gallery (w # x # ys @ [z])" shows "min_gallery (x # ys @ [z])"
method_exit_unique: 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 n :: "'b"   and n' :: "'b" assumes "Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"   and "CFGExit.method_exit sourcenode kind valid_edge Exit n"   and "CFGExit.method_exit sourcenode kind valid_edge Exit n'"   and "get_proc n = get_proc n'" shows "n = n'"
vector_smult_lzero: fixes x :: "('a, 'b) vec" shows "(0::'a) *s x = 0"
simps_1: fixes r :: "'b list"   and d :: "'b list"   and n :: "nat" shows "mod_poly_one_main_list r d (Suc n) = (let a = hd r; rr = tl (if a = (0::'b) then r else minus_poly_rev_list r (map ((*) a) d)) in mod_poly_one_main_list rr d n)"
perm_8: fixes pi :: "(name \<times> name) list"   and x2 :: "pi"   and x1 :: "pi" shows "pi \<bullet> x2 \<parallel> x1 = (pi \<bullet> x2) \<parallel> (pi \<bullet> x1)"
healthy_feasibleD: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "healthy t" shows "feasible t"
base: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" assumes "P \<omega>" shows "ev_at P 0 \<omega>"
double_dual_iso_apply: fixes \<chi> :: "('a \<Rightarrow> complex) \<Rightarrow> complex"   and x :: "'a \<Rightarrow> complex" assumes "character (Characters G) \<chi>" shows "G'.double_dual_iso x \<chi> = \<chi> x"
inv_in_hom: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "CC.iso f"   and "CC.in_hom f a b" shows "CC.in_hom (CC.inv f) b a"
exists_MST: fixes g :: "'a ugraph"   and w :: "'a set \<Rightarrow> nat" assumes "Undirected_Graph.connected g" shows "\<exists>t. is_MST w g t"
zero_less_norm_iff: fixes x :: "'a" shows "(0 < norm x) = (x \<noteq> (0::'a))"
unique_connecting_path_properties_1: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "w \<in> V\<^bsub>G\<^esub>" shows "Graph.walk G (v \<leadsto>\<^bsub>G\<^esub> w) \<and> distinct (v \<leadsto>\<^bsub>G\<^esub> w)"
ident: shows "\<forall>x\<in>{a, b, c, d}. r x = x"
arev_neq: fixes a :: "'a \<times> 'a" assumes "a \<in> arcs (with_proj G)" shows "swap_in (parcs G) a \<noteq> a"
l22: shows "sqrt (b\<^sup>2 - 4 * a * c) \<in> radical_sqrt"
Max_wrt_prefs_finite: fixes R :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "total_preorder_on alts R" shows "finite (Max_wrt R)"
tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_cons: fixes G :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and d :: "('fun, 'var) Term.term \<times> ('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term" assumes "G \<in> set (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F D)"   and "d \<in> set D" shows "(pair (s, t), pair d) # G \<in> set (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s ((s, t) # F) D)"
IH: shows "sum_list (map f xs) = (\<Sum>x\<in>set xs. count_list xs x * f x)"
assms: shows "if P x then \<nexists>y. P y else \<forall>y. \<not> P y"
Lt_le_LtI: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b" shows "Lt a \<le> Lt b"
focus_flatten: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 ('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and X :: "'a set" assumes "p \<in> P[X]"   and "lookup p ` keys p \<subseteq> P[- X]" shows "focus X (flatten p) = p"
NSconvergent_NSLIMSEQ_iff: fixes X :: "nat \<Rightarrow> 'a" shows "NSconvergent X = X \<longlonglongrightarrow>\<^sub>N\<^sub>S nslim X"
is_cdi_def: fixes i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" shows "(i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k) = (is_path \<pi> \<and> k < i \<and> \<pi> i \<noteq> local.return \<and> (\<forall>j\<in>{k..i}. \<pi> j \<noteq> ipd (\<pi> k)))"
isomorphic_symmetric: fixes f :: "'c \<times> 'c"   and g :: "'c \<times> 'c" assumes "C.VV.isomorphic f g" shows "C.VV.isomorphic g f"
hyps_4: fixes b :: "binding list" assumes "\<turnstile> b ok"   and "T\<^sub>2 closed_in b" shows "b\<turnstile>T\<^sub>2<:T\<^sub>2"
wqo_onI: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "transp_on P A"   and "almost_full_on P A" shows "wqo_on P A"
mult_onel: fixes x :: "'a" shows "1\<^sub>\<pi> \<parallel> x = x"
iso_f_def: shows "iso_f = \<lparr>iso_verts = id, iso_arcs = \<lambda>a. (tail G a, f a, head G a), iso_head = snd \<circ> snd, iso_tail = fst\<rparr>"
absorb1: fixes b :: "'k"   and a :: "'k" assumes "b \<le> a" shows "max a b = a"
bisimScopeExtChain: fixes xvec :: "name list"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P" shows "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>P \<parallel> Q \<sim> P \<parallel> (\<lparr>\<nu>*xvec\<rparr>Q)"
res_lim_Zp_cauchy: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "is_Zp_cauchy s" obtains N :: "nat" where "\<And>m. N < m \<longrightarrow> s m k = res_lim s k"
NI_indirect_sources_def: shows "NI_indirect_sources \<equiv> \<forall>execs a n. run n (Some s0) execs \<rightharpoonup> (\<lambda>s_f. B2 (run n (Some s0) (ipurge_l execs (current s_f))) (run n (Some s0) (ipurge_r execs (current s_f))) (\<lambda>s_l s_r. output_f s_l a = output_f s_r a))"
is_digit_expand: fixes n :: "nat" shows "is_digit n = (n = 0 \<or> n = 1 \<or> n = 2 \<or> n = 3 \<or> n = 4 \<or> n = 5 \<or> n = 6 \<or> n = 7 \<or> n = 8 \<or> n = 9)"
Fun_mkArr: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" assumes "S.arr (S.mkArr A B F)" shows "S.Fun (S.mkArr A B F) = restrict F A"
polynomial_split_on_var: fixes p :: "'a mpoly"   and x' :: "nat" obtains q :: "'a mpoly" and r :: "'a mpoly" where "p = MPoly_Type.monom (monomial (Suc 0) x') (1::'a) * q + r" and "x' \<notin> vars r"
distinct_4: fixes x3 :: "'b formula"   and x1 :: "'b" shows "\<^bold>\<not> x3 \<noteq> Atom x1"
TensorDiag_assoc: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" assumes "Diag t"   and "Diag u"   and "Diag v" shows "(t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> v = t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> v"
that: fixes U :: "'x set"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "U \<subseteq> X"   and "(monomial (1::'a) t, U) \<in> set (snd (split 0 X S))"   and "poly_deg g = Suc (deg_pm t)" shows "thesis"
the_lubI_ex: fixes A :: "'b set" assumes "\<exists>x. lub A x" shows "lub A (\<Squnion>A)"
lmap_transfer: fixes A :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'h \<Rightarrow> bool" shows "((A ===> B) ===> llist_all2 A ===> llist_all2 B) lmap lmap"
ex_forward: fixes P' :: "'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "\<exists>x. P' x"   and "\<And>x. P' x \<Longrightarrow> P x" shows "\<exists>x. P x"
top_in_filter: fixes F :: "'a set" assumes "ord_class.filter F" shows "top \<in> F"
flush_cache_all_def: fixes state :: "'b sparc_state" shows "flush_cache_all state \<equiv> state\<lparr>cache := cache state\<lparr>icache := empty_cache, dcache := empty_cache\<rparr>\<rparr>"
congs_pres_nodes: fixes F\<^sub>1 :: "'a list"   and F\<^sub>2 :: "'a list" assumes "F\<^sub>1 \<cong> F\<^sub>2" shows "set F\<^sub>1 = set F\<^sub>2"
IH_3: fixes as :: "accent list"   and bs :: "accent list" assumes "d # ds = as @ ([Acute] @ [Grave]) @ bs" shows "thesis"
induct: fixes x1 :: "nat \<Rightarrow> classDef option"   and x2 :: "nat"   and x3 :: "nat"   and P :: "(nat \<Rightarrow> classDef option) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" assumes "x1 \<turnstile> x2 <: x3"   and "\<And>CT C. P CT C C"   and "\<And>CT C D E. \<lbrakk>CT \<turnstile> C <: D; P CT C D; CT \<turnstile> D <: E; P CT D E\<rbrakk> \<Longrightarrow> P CT C E"   and "\<And>CT C CDef D. \<lbrakk>CT C = Some CDef; cSuper CDef = D\<rbrakk> \<Longrightarrow> P CT C D" shows "P x1 x2 x3"
data_dependency_consistent_mono: fixes T :: "nat set"   and "is" :: "instr list"   and T' :: "nat set" assumes "data_dependency_consistent_instrs T is"   and "T \<subseteq> T'" shows "data_dependency_consistent_instrs T' is"
N_P: shows "\<forall>x\<ge>N. P (2 * x) \<and> P (2 * x + 1)"
assms_2: assumes "\<Phi>" shows "S \<le> \<Down> R S'"
eqI: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "bijection f"   and "f a = f b" shows "a = b"
leg0_simps_4: shows "local.dom r\<^sub>0 = r\<^sub>0"
iso_lunit: fixes f :: "'a" assumes "B.ide f" shows "B.iso (B.lunit f)"
finite_Di: fixes i :: "nat"   and y :: "nat list" shows "finite (Di i y)"
bounded_linear_bounded_clinear: fixes A :: "'a \<Rightarrow> 'b" assumes "bounded_linear A"   and "\<forall>c x. A (c *\<^sub>C x) = c *\<^sub>C A x" shows "bounded_clinear A"
reflect_poly_smult: fixes c :: "'a"   and p :: "'a poly" shows "reflect_poly (smult c p) = smult c (reflect_poly p)"
inj_on_inv_vreal_of_real_impl: shows "inj_on (inv vreal_of_real_impl) (range vreal_of_real_impl)"
drop_bit_Suc: fixes n :: "nat"   and a :: "'a" shows "drop_bit (Suc n) a = drop_bit n (a div (2::'a))"
not_less: fixes x :: "'t"   and y :: "'t" shows "(\<not> x \<prec>\<^sub>t y) = (y \<preceq>\<^sub>t x)"
desargues_config_not_col_6: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R" shows "\<not> col C C' A'"
pred_set: fixes P :: "'c \<Rightarrow> bool" shows "pred_state P = (\<lambda>x. Ball (set_state x) P)"
boundedI: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "A \<noteq> {}"   and "\<And>a. a \<in> A \<Longrightarrow> x \<le> a" shows "x \<le> Min A"
assms_2: fixes x :: "'b"   and y :: "'b" assumes "x \<in> keys t"   and "y \<in> keys u" shows "x < y"
sup_V_0_right: fixes a :: "V" shows "a \<union>\<^sub>\<circ> 0 = a"
int_le_induct: fixes i :: "int"   and k :: "int"   and P :: "int \<Rightarrow> bool" assumes "i \<le> k"   and "P k"   and "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P (i - 1)" shows "P i"
prems_4: shows "list.set ns \<subseteq> {0<..}"
n_add_comp: fixes x :: "'a" shows "!x + t x = (1::'a)"
theFT_def: fixes cl :: "('d, 'e, 'f) cmd list" shows "theFT cl \<equiv> {n. n < length cl \<and> finished (cl ! n)}"
iptables_bigstep_induct: fixes \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b"   and rs :: "'a rule list"   and s :: "state"   and t :: "state"   and P :: "'a rule list \<Rightarrow> state \<Rightarrow> state \<Rightarrow> bool" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"   and "\<And>t. P [] t t"   and "\<And>m a. \<lbrakk>matches \<gamma> m p; a = Accept\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided (Decision FinalAllow)"   and "\<And>m a. \<lbrakk>matches \<gamma> m p; a = Drop \<or> a = Reject\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided (Decision FinalDeny)"   and "\<And>m a. \<lbrakk>matches \<gamma> m p; a = Log \<or> a = Empty\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided Undecided"   and "\<And>m a. \<not> matches \<gamma> m p \<Longrightarrow> P [Rule m a] Undecided Undecided"   and "\<And>rs X. P rs (Decision X) (Decision X)"   and "\<And>rs rs\<^sub>1 rs\<^sub>2 t t'. \<lbrakk>rs = rs\<^sub>1 @ rs\<^sub>2; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> t; P rs\<^sub>1 Undecided t; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t'; P rs\<^sub>2 t t'\<rbrakk> \<Longrightarrow> P rs Undecided t'"   and "\<And>m a chain rs\<^sub>1 m' rs\<^sub>2. \<lbrakk>matches \<gamma> m p; a = Call chain; \<Gamma> chain = Some (rs\<^sub>1 @ [Rule m' Return] @ rs\<^sub>2); matches \<gamma> m' p; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> Undecided; P rs\<^sub>1 Undecided Undecided\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided Undecided"   and "\<And>m a chain rs t. \<lbrakk>matches \<gamma> m p; a = Call chain; \<Gamma> chain = Some rs; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> t; P rs Undecided t\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided t" shows "P rs s t"
simps_2: fixes w :: "nat"   and t\<^sub>1 :: "'a tree"   and t\<^sub>2 :: "'a tree" shows "height (Node w t\<^sub>1 t\<^sub>2) = max (height t\<^sub>1) (height t\<^sub>2) + 1"
slp_of_fa_bin_slp_of_fa_rev_bin: fixes Binop :: "floatarith \<Rightarrow> floatarith \<Rightarrow> floatarith"   and a :: "floatarith"   and b :: "floatarith"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list"   and M2 :: "(floatarith, nat) mapping"   and slp2 :: "floatarith list" shows "slp_of_fa_rev_bin Binop a b M slp (length slp) M2 slp2 (length slp2) = (let (M, slp') = slp_of_fa_bin Binop a b M (rev slp) M2 (rev slp2) in (M, rev slp', length slp'))"
f2: shows "\<forall>x0 x1. (\<exists>v2 v3 v4. (v3 \<in> game_sem v2 x1 v4) \<noteq> (v3 \<in> game_sem v2 x1 (selectlike v4 v3 x0))) = ((rr x0 x1 \<in> game_sem (ii x0 x1) x1 (FF x0 x1)) \<noteq> (rr x0 x1 \<in> game_sem (ii x0 x1) x1 (selectlike (FF x0 x1) (rr x0 x1) x0)))"