hcju/isabelleps huggingface dataset:
Gcd_greatest: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and a :: "'b" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "\<And>b. b \<in> A \<Longrightarrow> dvd.dvd times a b" shows "dvd.dvd times a (Gcd A)"
IH: fixes out :: "'c"   and c :: "('a, 'c, 'e) rpv"   and input :: "'e"   and gpv' :: "('b, 'd, 'f) gpv" assumes "IO out c \<in> set_spmf p"   and "input \<in> responses_\<I> \<I> out"   and "rel_gpv'' A C R (c input) gpv'" shows "lossless_gpv \<I>' gpv'"
l2_inv4_def: shows "l2_inv4 \<equiv> {s. \<forall>Rb. test_ended s \<longrightarrow> guessed_frame test xsk = Some (NonceF (Rb $ sk)) \<longrightarrow> progress s Rb = Some {xpkE, xsk}}"
propa_ss1': shows "\<forall>(q, \<tau>)\<in>set (step (hd b) (a ! hd b)). ss ! q = hd b # a ! hd b \<squnion>\<^bsub>f\<^esub> a ! q"
rewriting_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "rewriting R \<equiv> (\<forall>t t' u. R t t' \<longrightarrow> R (app t u) (app t' u)) \<and> (\<forall>u u' t. R u u' \<longrightarrow> R (app t u) (app t u'))"
dvd_diff: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "dvd.dvd times x y"   and "dvd.dvd times x z" shows "dvd.dvd times x (minus y z)"
locally_faithful_def: shows "locally_faithful \<equiv> \<forall>f g \<mu> \<mu>'. \<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C g\<guillemotright> \<and> \<guillemotleft>\<mu>' : f \<Rightarrow>\<^sub>C g\<guillemotright> \<and> F \<mu> = F \<mu>' \<longrightarrow> \<mu> = \<mu>'"
finite_list: fixes A :: "'k set" assumes "finite A" shows "\<exists>xs. set xs = A"
zabs_def: fixes i :: "int" shows "\<bar>i\<bar> = (if i < 0 then - i else i)"
sub_monom_1: fixes g :: "nat \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "g \<in> carrier (UP R)"   and "a \<in> carrier R" shows "Cring_Poly.compose R (monom (UP R) a n) g = to_polynomial R a \<otimes>\<^bsub>UP R\<^esub> g [^]\<^bsub>UP R\<^esub> n"
min_dist_z_iff: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and v :: "nat"   and v' :: "nat" assumes "Graph.connected c v v'" shows "(Graph.min_dist c v v' = 0) = (v' = v)"
intros: fixes uu :: "nat" shows "Regions.isConst (Regions.intv.Const uu)"
leoa: fixes w :: "'b word"   and x :: "'b word"   and y :: "'b word" assumes "w = x OR y" shows "y = w AND y"
submap_imp_state_succ_submap: fixes a :: "('c, bool) fmap \<times> ('c, bool) fmap"   and s1 :: "('c, bool) fmap"   and s2 :: "('c, bool) fmap" assumes "fst a \<subseteq>\<^sub>f s1"   and "s1 \<subseteq>\<^sub>f s2" shows "state_succ s1 a \<subseteq>\<^sub>f state_succ s2 a"
is_tdghm_op: shows "op_tdghm \<NN> : op_dghm \<GG> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M op_dghm \<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> op_dg \<BB>"
eval_simps'_3: fixes f :: "'a E.term" assumes "E.Arr f" shows "trg\<^sub>B (EVAL f) = EVAL (E.Trg f)"
bigo_add_commute_imp: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> g +o O(h)" shows "g \<in> f +o O(h)"
proper_map_alt: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "proper_map X Y f = (closed_map X Y f \<and> (\<forall>K. compactin Y K \<longrightarrow> compactin X {x \<in> topspace X. f x \<in> K}))"
ideal_subset1: fixes I :: "'a set" assumes "ideal R I" shows "I \<subseteq> carrier R"
assms: fixes n :: "nat"   and e :: "'a" shows "P (LnChain n, e)"
simps_3: fixes x :: "'a"   and y :: "'a" shows "Some x \<triangle> Some y = Some (x \<triangle> y)"
SN_SN_part_UNIV_conv: fixes r :: "('b \<times> 'b) set" shows "SN r = (SN_part r = UNIV)"
rawpsubstT_Var_not: fixes x :: "'var"   and txs :: "('trm \<times> 'var) list" assumes "x \<in> var"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "x \<notin> snd ` set txs" shows "rawpsubstT (Var x) txs = Var x"
pred_sp: shows "\<forall>l\<in>dom f. \<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. the (f l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> the (f' l) = \<sigma>[s,p] t'')"
Tau_def: fixes x1 :: "pi" shows "\<tau>.x1 \<equiv> Abs_pi (Tau_Rep (Rep_pi x1))"
preserves_arr: fixes f :: "'a" assumes "Right_a.arr f" shows "Right_a.arr (H\<^sub>R a f)"
mono_invE: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less"   and "order.mono less_eq f"   and "f x < f y" obtains "less_eq x y"
schroeder_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x ; y \<cdot> z = 0) = (y \<cdot> x\<^sup>\<smile> ; z = 0)"
delay_bisimulation_final_base_flip: fixes t :: "'t" shows "delay_bisimulation_final_base (r2 t) (r1 t) (flip (bisim t)) \<tau>move2 \<tau>move1 (\<lambda>(x2, m). final2 x2) (\<lambda>(x1, m). final1 x1)"
unitary11_comp: fixes k1 :: "complex"   and M1 :: "complex \<times> complex \<times> complex \<times> complex"   and a1 :: "complex"   and b1 :: "complex"   and k2 :: "complex"   and M2 :: "complex \<times> complex \<times> complex \<times> complex"   and a2 :: "complex"   and b2 :: "complex" assumes "k1 \<noteq> 0"   and "M1 = k1 *\<^sub>s\<^sub>m (a1, b1, cnj b1, cnj a1)"   and "mat_det (a1, b1, cnj b1, cnj a1) \<noteq> 0"   and "k2 \<noteq> 0"   and "M2 = k2 *\<^sub>s\<^sub>m (a2, b2, cnj b2, cnj a2)"   and "mat_det (a2, b2, cnj b2, cnj a2) \<noteq> 0" shows "\<exists>k a b. k \<noteq> 0 \<and> M1 *\<^sub>m\<^sub>m M2 = k *\<^sub>s\<^sub>m (a, b, cnj b, cnj a) \<and> mat_det (a, b, cnj b, cnj a) \<noteq> 0"
\<iota>_inf: shows "\<iota> \<in> Inf"
finite_UN: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and M :: "'d measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> fmeasurable M" shows "\<Union> (A ` I) \<in> fmeasurable M"
simps_5: fixes a :: "MIR.num" shows "zsplit0 (Neg a) = (let (i', a') = zsplit0 a in (- i', Neg a'))"
contractible_imp_connected: fixes S :: "'a set" assumes "contractible S" shows "connected S"
interpretLow: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and s1 :: "char list \<Rightarrow> val option"   and s2 :: "char list \<Rightarrow> val option"   and e :: "expr" assumes "\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2"   and "\<forall>V\<in>fv e. \<Gamma> V = Some Low" shows "\<lbrakk>e\<rbrakk>s1 = \<lbrakk>e\<rbrakk>s2"
ideal_generated_singleton: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "ring.ideal_generated plus zero uminus times {a} = {times k a |k. k \<in> UNIV}"
L_def: shows "L \<equiv> Dirichlet_L n \<chi> 1"
allE: fixes P :: "'e \<Rightarrow> bool"   and x :: "'e" assumes "\<forall>x. P x" obtains "P x"
lalternate_finite: fixes xs :: "'b llist" shows "lfinite (lalternate xs) = lfinite xs"
CHECK_refine: fixes \<Phi>i :: "bool"   and \<Phi> :: "bool"   and msgi :: "'c"   and msg :: "'d"   and E :: "('c \<times> 'd) set" assumes "\<Phi>i = \<Phi>"   and "\<not> \<Phi> \<Longrightarrow> (msgi, msg) \<in> E" shows "CHECK \<Phi>i msgi \<le> \<Down>\<^sub>E E unit_rel (CHECK \<Phi> msg)"
expansion_factor_1: fixes p :: "proj2"   and J :: "cltn2" assumes "p \<in> hyp2 \<union> S"   and "is_K2_isometry J" shows "expansion_factor p J \<noteq> 0"
lookup_alt_def: fixes return :: "int \<Rightarrow> 'b"   and fail :: "'b"   and get :: "(('c \<Rightarrow> int option) \<Rightarrow> 'b) \<Rightarrow> 'b"   and x :: "'c" shows "memo_exp_base.lookup return fail get x = get (\<lambda>s. case apply s x of None \<Rightarrow> fail | Some y \<Rightarrow> return y)"
seq_Sup_distl: fixes P :: "('a, 'c) urel"   and A :: "('c, 'b) urel set" shows "P ;; \<Or>A = \<Or>((;;) P ` A)"
int1: shows "integrable (restr_to_subalg M F) (\<lambda>x. g x * q (real_cond_exp M F X x))"
sym_is_binary_functor: shows "binary_functor (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) F'.sym"
min_states_def: fixes M :: "'b dfa" assumes "dfa M" shows "dfa.min_states M \<equiv> card (UNIV // eq_app_right (dfa.language M))"
mult_zero_left: fixes a :: "'b" shows "(0::'b) * a = (0::'b)"
fsubset_funion_eq: fixes A :: "'b fset"   and B :: "'b fset" shows "(A |\<subseteq>| B) = (A |\<union>| B = B)"
class_of_def: fixes U :: "'a set"   and s :: "'b" shows "\<lfloor>U, s\<rfloor> \<equiv> rel.Class (U, s)"
LeastMethodDef_unique_def: 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 C :: "char list"   and M :: "char list"   and Cs :: "char list list" shows "LeastMethodDef_unique P C M Cs = (\<forall>Cs' mthd'. MethodDefs' P C M Cs' mthd' \<longrightarrow> (leq_path1p P C)\<^sup>*\<^sup>* Cs Cs')"
K_time_split: fixes t :: "real"   and j :: "real"   and f :: "real \<times> 'a \<Rightarrow> ennreal"   and s :: "'a" assumes "t \<le> j"   and "f \<in> borel_measurable S" shows "\<integral>\<^sup>+ x. f x * indicator {j<..} (fst x) \<partial>K (t, s) = integral\<^sup>N (K (j, s)) f * emeasure (exponential (escape_rate s)) {j - t<..}"
gram_schmidt_fs_int_axioms: shows "gram_schmidt_fs_int n (map of_int_hom.vec_hom fs)"
rel_eq: shows "Resumption.resumption.rel_resumption (=) (=) = (=)"
monic_irreducible_factorization: fixes p :: "'a poly" assumes "monic p" shows "\<exists>as f. finite as \<and> p = (\<Prod>a\<in>as. a ^ Suc (f a)) \<and> as \<subseteq> {q. irreducible q \<and> monic q}"
add: fixes x :: "'a ell2"   and y :: "'a ell2" shows "C1 (x + y) = C1 x + C1 y"
subspace_dim_equal: fixes S :: "'b set"   and T :: "'b set" assumes "subspace S"   and "subspace T"   and "S \<subseteq> T"   and "dim T \<le> dim S" shows "S = T"
strong_del_block_wf: fixes b :: "'a set" shows "incidence_system (\<V> - b) (str_del_block b)"
lnull_imp_lfinite: fixes xs :: "'b llist" assumes "lnull xs" shows "lfinite xs"
cwiseext_hd_or_tl: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and ys :: "'b list"   and x :: "'b"   and xs :: "'b list" assumes "cwiseext gt (y # ys) (x # xs)" shows "gt y x \<or> cwiseext gt ys xs"
underS_def: fixes r :: "('c \<times> 'c) set"   and a :: "'c" shows "underS r a \<equiv> {b. b \<noteq> a \<and> (b, a) \<in> r}"
sort_oalist_def: fixes xs :: "('a \<times> 'c) list" shows "sort_oalist xs = foldr update_by_pair xs []"
direct_decomp_ideal_insert: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite X"   and "F \<subseteq> P[X]"   and "f \<in> P[X]" shows "direct_decomp (ideal (insert f F) \<inter> P[X]) [ideal F \<inter> P[X], (*) f ` normal_form ((ideal F \<div> f) \<inter> P[X]) ` P[X]]"
value_flow_cleanup: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and \<Delta> :: "('a, 'b) network_scheme" assumes "\<And>x. f (x, source \<Delta>) = 0" shows "value_flow \<Delta> (cleanup f) = value_flow \<Delta> f"
Derives1_skip_prefix: fixes a :: "symbol list"   and i :: "nat"   and b :: "symbol list"   and r :: "symbol \<times> symbol list"   and c :: "symbol list" assumes "length a \<le> i"   and "Derives1 (a @ b) i r (a @ c)" shows "Derives1 b (i - length a) r c"
vrange_atD: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vreal_mult" shows "\<exists>a\<in>elts (\<D>\<^sub>\<circ> vreal_mult). vreal_mult\<lparr>a\<rparr> = b"
simps: fixes s1 :: "('a, 'e, 'f, 'g) SLin_state_scheme"   and s2 :: "('a, 'h, 'i, 'j) SLin_state_scheme" shows "P21 (s1, s2) = Ball (abortVals s2) ((\<preceq>) \<Sqinter> abortVals s1)"
Q_hit_edges_disjoint: shows "\<Union> (H.edges_of_walk ` (Q - {Q_hit})) \<inter> H.edges_of_walk Q_hit = {}"
of_int_eq_iff: fixes w :: "int"   and z :: "int" shows "(of_int w = of_int z) = (w = z)"
RSadj_snd: fixes x :: "'sz"   and y :: "'sz"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" shows "RSadj x y (snd \<nu>) = snd (Radj x y \<nu>)"
vrange_atD: fixes b :: "V"   and A :: "V"   and f :: "V \<Rightarrow> V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (VLambda A f)" shows "\<exists>a\<in>elts (\<D>\<^sub>\<circ> (VLambda A f)). VLambda A f\<lparr>a\<rparr> = b"
update_convs_8: fixes dests' :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> nat option"   and ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and more :: "'a" shows "dests_update dests' \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests' dests, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr>"
floor_divide_upper: fixes q :: "'a"   and p :: "'a" assumes "(0::'a) < q" shows "p < (of_int \<lfloor>p / q\<rfloor> + (1::'a)) * q"
ipresIGFresh_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'varSort, 'k, 'l, 'var, 'h, 'm, 'n) model_scheme" shows "ipresIGFresh h termMOD MOD = ipresFresh h MOD"
invertible_commutes_with_inverse: fixes a :: "'a" assumes "maxI.additive.invertible a"   and "a \<in> A" shows "f (maxI.additive.inverse a) = fiso.target.additive.inverse (f a)"
subst_bop: fixes \<sigma> :: "'b \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'a"   and u :: "('d, 'c) uexpr"   and v :: "('e, 'c) uexpr" shows "\<sigma> \<dagger> bop f u v = bop f (\<sigma> \<dagger> u) (\<sigma> \<dagger> v)"
PLD_lemma: fixes F :: "'a program"   and G :: "'c program"   and C :: "'c set"   and A :: "'a set"   and B :: "'a set" assumes "extend h F \<squnion> G \<in> stable C"   and "F \<squnion> project h C G \<in> project_set h C \<inter> A \<longmapsto> B" shows "extend h F \<squnion> G \<in> C \<inter> extend_set h (project_set h C \<inter> A) \<longmapsto> extend_set h B"
rel_converter_simps: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'a \<Rightarrow> ('c \<times> ('a, 'c, 'e, 'g) converter, 'e, 'g) gpv"   and g :: "'b \<Rightarrow> ('d \<times> ('b, 'd, 'f, 'h) converter, 'f, 'h) gpv" shows "rel_converter A B C R (Converter f) (Converter g) = rel_fun A (rel_gpv'' (rel_prod B (rel_converter A B C R)) C R) f g"
regular_closed_sup: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_p_algebra.regular x"   and "matrix_p_algebra.regular y" shows "matrix_p_algebra.regular (x \<oplus> y)"
dim_vec_of_list_h: fixes h :: "'b poly"   and u :: "'c poly" assumes "degree h < degree u" shows "dim_vec (vec_of_list (coeffs h @ replicate (degree u - length (coeffs h)) (0::'b))) = degree u"
simps_2: fixes x :: "'c"   and a :: "'c"   and as :: "'c list" shows "mem_idx x (a # as) = (if x = a then Some 0 else map_option Suc (mem_idx x as))"
IH: fixes m :: "nat"   and cs :: "'a list" assumes "m < length cs"   and "n < m"   and "1 < length cs" shows "\<exists>xs ys zs. cs = xs @ cs ! n # ys @ cs ! m # zs"
rep_eq: fixes x :: "'b \<Rightarrow> bool"   and xa :: "'b multiset" shows "count (filter_mset x xa) = (\<lambda>xb. if x xb then count xa xb else 0)"
cntCE: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set set"   and T :: "'c set set"   and m :: "'c set" assumes "f \<in> cnt S T" obtains "\<not> m open\<^bsub>T\<^esub>" and "f \<in> carrier\<^bsub>S\<^esub> \<rightarrow> carrier\<^bsub>T\<^esub>"   | "carrier\<^bsub>S\<^esub> \<inter> f -` m open\<^bsub>S\<^esub>" and "f \<in> carrier\<^bsub>S\<^esub> \<rightarrow> carrier\<^bsub>T\<^esub>"
indep_def: fixes A :: "'a set"   and B :: "'a set" shows "indep A B = (prob (A \<inter> B) = prob A * prob B)"
f2f2': shows "1 / (f2 - f2') \<le> l0"
ennreal_approx_SUP: fixes A :: "'b set"   and f :: "'b \<Rightarrow> ennreal"   and x :: "ennreal" assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<le> x"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>i\<in>A. x \<le> f i + ennreal e" shows "x = Sup (f ` A)"
parts_dec: fixes s' :: "(agent \<times> msg) set"   and X :: "msg"   and s :: "(agent \<times> msg) set"   and K :: "key"   and Y :: "msg" assumes "s' = insert (Spy, X) s \<and> (Spy, Crypt K X) \<in> s \<and> (Spy, Key (InvK K)) \<in> s"   and "Y \<in> parts_msg X" shows "Y \<in> parts (used s)"
rel_pmf_restrict_relpI: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c pmf"   and y :: "'d pmf"   and P :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "rel_pmf R x y"   and "pred_pmf P x"   and "pred_pmf Q y" shows "rel_pmf (R \<upharpoonleft> P \<otimes> Q) x y"
s_has_vector_derivative: fixes t :: "real" assumes "0 \<le> t"   and "v / a \<le> 0"   and "a \<noteq> 0" shows "(s has_vector_derivative s' t) (at t within {0..})"
\<psi>: shows "\<guillemotleft>\<psi> : local.map (trg\<^sub>C \<mu>) \<rightarrow>\<^sub>D trg\<^sub>D (local.map \<mu>)\<guillemotright> \<and> D.iso \<psi>"
wm_to_state_works_for_wf_wm_closed: fixes M :: "object atom formula set"   and fmla :: "object atom formula" assumes "wf_world_model M"   and "fmla \<in> close_world M" shows "close_eq (wm_to_state M) \<Turnstile> fmla"
emeasure_eq_measure: fixes A :: "'s set" shows "emeasure (stat C) A = ennreal (stat.prob A)"
Inf_spec: fixes X :: "'b" assumes "X \<in> XX" shows "Inf_spec X \<le> SPEC (Q X)"
arith_simps_52: fixes m :: "num"   and n :: "num" shows "numeral m - numeral n = neg_numeral_class.sub m n"
integrable_on_insert_iff: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and X :: "'a set" shows "(f integrable_on insert x X) = (f integrable_on X)"
mult_mm_non_zero_r: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" assumes "A \<noteq> mat_zero"   and "mat_det B \<noteq> 0" shows "A *\<^sub>m\<^sub>m B \<noteq> mat_zero"
wf_called_by_chain: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "finite (calls_chain \<Gamma>)"   and "wf (calls_chain \<Gamma>)" shows "wf (called_by_chain \<Gamma>)"
balanced_pos_and_neg_terms_cancel_out2: assumes "is_balanced" shows "(\<Sum>k<2 ^ n. (- int 1) ^ (f k + 1)) = 0"
invarR: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and lv :: "nat"   and r :: "('b \<times> nat) tree" assumes "ASSUMPTION (invar \<langle>l, (a, lv), r\<rangle>)" shows "invar r"
is_Yield_def: fixes step :: "('c, 'd) step" shows "is_Yield step = (\<exists>x31 x32. step = Yield x31 x32)"
oeq: shows "ordertype (elts \<beta>) VWF = \<beta>"
Span_base_incl: fixes Us :: "'a list" assumes "set Us \<subseteq> carrier R" shows "set Us \<subseteq> Span K Us"
reindex_nontrivial: fixes A :: "'c set"   and h :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'e" assumes "finite A"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y; h x = h y\<rbrakk> \<Longrightarrow> g (h x) = (0::'e)" shows "sum g (h ` A) = sum (g \<circ> h) A"
append_strict1: fixes ys :: "['b]" shows "\<bottom> ++ ys = \<bottom>"
n_dist_var2': fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "!(!x \<cdot> !y + t x \<cdot> !z) = !x \<cdot> t y + t x \<cdot> t z"
a: shows " t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t2"
weakCongSum1: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs"   and Q :: "ccs" assumes "P \<Longrightarrow>\<alpha> \<prec> P'" shows "P \<oplus> Q \<Longrightarrow>\<alpha> \<prec> P'"
intros: fixes P :: "'a prog"   and C :: "char list"   and D :: "char list"   and rest :: "fdecl list \<times> 'a mdecl list" assumes "class P C = \<lfloor>(D, rest)\<rfloor>"   and "C \<noteq> Object" shows "P \<turnstile> C \<prec>\<^sup>1 D"
in_set_product_lists_length: fixes xs :: "'g list"   and xss :: "'g list list" assumes "xs \<in> set (product_lists xss)" shows "length xs = length xss"
roots1: fixes c :: "complex poly"   and l :: "nat" assumes "(c, l) \<in> set cs" shows "\<forall>x\<in>ball 0 (1 / R). poly c x \<noteq> 0"
da_elim_cases_7: fixes Env :: "env"   and B :: "lname set"   and c1 :: "stmt"   and c2 :: "stmt"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>c1;; c2\<rangle>\<guillemotright> A" obtains C1 :: "assigned" and C2 :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1" and "Env\<turnstile> nrm C1 \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2" and "nrm A = nrm C2" and "brk A = brk C1 \<Rightarrow>\<inter>  brk C2"
comm_groupE_3: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "comm_group G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"
distinct_list_of_dlist: fixes dxs :: "'a dlist" shows "distinct (list_of_dlist dxs)"
simps: fixes P1 :: "'b \<Rightarrow> bool"   and P2 :: "'c \<Rightarrow> bool"   and a :: "'b \<times> 'c" shows "pred_prod P1 P2 a = (\<exists>a b. a = (a, b) \<and> P1 a \<and> P2 b)"
omit_all_is: fixes p :: "('b, 'c) fm"   and ps :: "('b, 'c) fm list"   and xs :: "nat set"   and q :: "('b, 'c) fm" assumes "all_is p ps xs"   and "q \<in>. ps"   and "q \<notin> set (omit xs ps)" shows "q = p"
add: shows "\<Prod>\<^sub># (image_mset Mp b) =m \<Prod>\<^sub># b"
Kleene_iter_gpfp: fixes f :: "'c \<Rightarrow> 'c"   and p :: "'c"   and k :: "nat" assumes "mono f"   and "p \<le> f p" shows "p \<le> (f ^^ k) top"
l: fixes i :: "'a" assumes "i \<in> I"   and "a < i"   and "i \<le> b" shows "(f \<longlongrightarrow> l i) (at_left i)"
closure_imp_adh: fixes A :: "'a llist set"   and x :: "'a llist" assumes "A \<subseteq> carrier"   and "x \<in> closure A" shows "x adh A"
PO_def: fixes cl :: "'b potype" shows "PO cl \<equiv> cl \<in> PartialOrder"
arg_cong2: fixes a :: "'k"   and b :: "'k"   and c :: "'l"   and d :: "'l"   and f :: "'k \<Rightarrow> 'l \<Rightarrow> 'm" assumes "a = b"   and "c = d" shows "f a c = f b d"
emb_carrier: fixes f :: "'e \<Rightarrow> 'a"   and D :: "'e set"   and R :: "'a set" assumes "f \<in> D \<rightarrow> R" shows "emb f D \<in> D \<rightarrow> R"
H_def: shows "H = {x \<in> insert_messages M. a pm < a \<lbrakk>I x\<rbrakk> \<and> a \<lbrakk>I x\<rbrakk> < a sm}"
tautology_monotonous: fixes atom_ordering :: "('a \<times> 'a) set"   and C :: "'b Literal set"   and D :: "'b Literal set" assumes "propositional_atoms atom_ordering"   and "tautology C"   and "C \<subseteq> D" shows "tautology D"
snoc_2: shows "\<exists>a\<in>set (xs @ [x]). P a"
plus_enat_simps_2: fixes q :: "enat" shows "\<infinity> + q = \<infinity>"