hcju/isabelleps huggingface dataset:
invsL_def: shows "gc.invsL = (\<lambda>s. gc.fM_fA_invL s \<and> gc_mark.mark_object_invL s \<and> gc.gc_W_empty_invL s \<and> gc.handshake_invL s \<and> gc.obj_fields_marked_invL s \<and> gc.phase_invL s \<and> gc.sweep_loop_invL s \<and> gc.tso_lock_invL s)"
perfect_hiding: fixes \<A> :: "('a \<Rightarrow> ((nat \<times> nat) \<times> 'c) spmf) \<times> ('a \<Rightarrow> 'c \<Rightarrow> bool spmf)" shows "spmf (ped_commit.hiding_game_ind_cpa \<A>) True - 1 / 2 = 0"
mu_exchange_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b \<Rightarrow> 'b"   and u :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'b" assumes "class.order less_eq less"   and "ord.galois less_eq l u"   and "ord.isotone less_eq g"   and "ord.isotone less_eq h"   and "order.has_least_prefixpoint less_eq (l \<circ> h)"   and "order.has_least_prefixpoint less_eq (h \<circ> l)"   and "order.has_least_prefixpoint less_eq (h \<circ> g)"   and "order.has_least_fixpoint less_eq (g \<circ> h)"   and "order.has_least_fixpoint less_eq (h \<circ> g)"   and "ord.lifted_less_eq less_eq (l \<circ> h \<circ> g) (g \<circ> h \<circ> l)" shows "less_eq (order.the_least_fixpoint less_eq (h \<circ> l)) (order.the_least_fixpoint less_eq (h \<circ> g))"
algebraic_int_diff: fixes x :: "'a"   and y :: "'a" assumes "algebraic_int x"   and "algebraic_int y" shows "algebraic_int (x - y)"
test_dist_var2: fixes p :: "'a"   and q :: "'a"   and r :: "'a" assumes "test p"   and "test q"   and "test r" shows "!(p \<cdot> q + !p \<cdot> r) = p \<cdot> !q + !p \<cdot> !r"
prime_harmonic_lower: fixes n :: "nat" assumes "2 \<le> n" shows "ln (harm n) - ln (5 / 3) \<le> (\<Sum>p\<leftarrow>primes_upto n. 1 / real p)"
tOp_assocTr42: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G"   and "d \<in> carrier G" shows "a \<cdot> b \<cdot> c \<cdot> d = a \<cdot> (b \<cdot> c) \<cdot> d"
pderiv_nonzero: shows "pderiv (prod_list fss) \<noteq> 0"
process_mru_is_Some: fixes v_f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b option"   and a :: "'c"   and rv :: "'a \<times> 'b" shows "(process_mru v_f a = Some rv) = (vote_set v_f {a} \<noteq> {} \<and> rv = Max_by fst (vote_set v_f {a}))"
"5_2": shows "a \<notin> set v\<^sub>1"
measurable_cong_simp: fixes M :: "'d measure"   and N :: "'d measure"   and M' :: "'e measure"   and N' :: "'e measure"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "M = N"   and "M' = N'"   and "\<And>w. w \<in> space M \<Longrightarrow> f w = g w" shows "(f \<in> M \<rightarrow>\<^sub>M M') = (g \<in> N \<rightarrow>\<^sub>M N')"
assms_2: shows "A \<subseteq>\<^sub>p B"
segment_le_mono: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<preceq> b = (segment D a \<subseteq> segment D b)"
s': shows "s' |\<in>| (fst \<circ> tree.root) |`| cont t"
catch_def: fixes m :: "'c + 'b"   and f :: "'c \<Rightarrow> 'a + 'b" shows "(try m catch f) = (case m of Inl e \<Rightarrow> f e | Inr x \<Rightarrow> Inr x)"
similarity_decomposition: fixes a :: "complex"   and b :: "complex" assumes "a \<noteq> 0" shows "moebius_similarity a b = moebius_translation b + moebius_rotation (arg a) + moebius_dilatation (cmod a)"
rel_conversep: fixes R :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "fmrel R\<inverse>\<inverse> = (fmrel R)\<inverse>\<inverse>"
IN_j: fixes x :: "'v option" shows "d_IN j x \<le> \<alpha>"
take_incr_non_empty: fixes xs :: "int list" assumes "xs \<noteq> []" shows "take_incr xs \<noteq> []"
strategy_attractsE: fixes A :: "'a set"   and W :: "'a set" assumes "strategy_attracts p \<sigma> A W"   and "v \<in> A" shows "visits_via P A W"
l_inv: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
norm_diff_triangle_le: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and e1 :: "real"   and z :: "'a"   and e2 :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm (minus x y) \<le> e1"   and "norm (minus y z) \<le> e2" shows "norm (minus x z) \<le> e1 + e2"
sub_distribI: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>P s. sound P \<Longrightarrow> t P s \<ominus> 1 \<le> t (\<lambda>s'. P s' \<ominus> 1) s" shows "sub_distrib t"
iso_compose: fixes f :: "'d \<Rightarrow> 'e"   and A :: "('d, 'f) gorder_scheme"   and B :: "('e, 'g) gorder_scheme"   and g :: "'e \<Rightarrow> 'h"   and C :: "('h, 'i) gorder_scheme" assumes "f \<in> carrier A \<rightarrow> carrier B"   and "isotone A B f"   and "g \<in> carrier B \<rightarrow> carrier C"   and "isotone B C g" shows "isotone A C (g \<circ> f)"
exhaust: fixes y :: "inv_mode" obtains "y = Static"   | "y = SuperM"   | "y = IntVir"
continuous_onI_mono: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "open (f ` A)"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<le> y\<rbrakk> \<Longrightarrow> f x \<le> f y" shows "continuous_on A f"
insert: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c" assumes "ListMem x xs" shows "ListMem x (y # xs)"
left_idem: fixes a :: "'t"   and b :: "'t" shows "ord_term_lin.max a (ord_term_lin.max a b) = ord_term_lin.max a b"
eval_LAM: fixes x :: "name"   and t :: "lam" shows "LAM [x].t \<Down> LAM [x].t"
s_def: fixes t :: "real" shows "ego.s t = (if t \<le> 0 then s\<^sub>e else if t \<le> ego.t_stop then ego.p t else ego.p_max)"
minor_eq_col_row: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and j :: "'d" shows "minor M i j = upd_col (upd_row M i (axis j (1::'c))) j (axis i (1::'c))"
IntI: fixes c :: "'x"   and A :: "'x set"   and B :: "'x set" assumes "c \<in> A"   and "c \<in> B" shows "c \<in> A \<inter> B"
append_eq_append_conv_if: fixes xs\<^sub>1 :: "'e list"   and xs\<^sub>2 :: "'e list"   and ys\<^sub>1 :: "'e list"   and ys\<^sub>2 :: "'e list" shows "(xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>1 @ ys\<^sub>2) = (if length xs\<^sub>1 \<le> length ys\<^sub>1 then xs\<^sub>1 = take (length xs\<^sub>1) ys\<^sub>1 \<and> xs\<^sub>2 = drop (length xs\<^sub>1) ys\<^sub>1 @ ys\<^sub>2 else take (length ys\<^sub>1) xs\<^sub>1 = ys\<^sub>1 \<and> drop (length ys\<^sub>1) xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>2)"
the_\<theta>_simps_4: shows "local.dom \<chi>.the_\<theta> = u\<^sub>0 \<star> \<chi>.chine"
Pow_not_empty: fixes A :: "'a set" shows "Pow A \<noteq> {}"
select_convs_1: fixes para :: "char list list"   and body :: "expr"   and more :: "'a" shows "para \<lparr>para = para, body = body, \<dots> = more\<rparr> = para"
subjects_reduction: 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 es :: "expr list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "expr list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and Ts :: "ty list" assumes "wf_C_prog P"   and "P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>] \<langle>es',s'\<rangle>"   and "P,E \<turnstile> s \<surd>"   and "P,E,hp s \<turnstile> es [:] Ts" shows "P,E,hp s' \<turnstile> es' [:]\<^bsub>NT\<^esub> Ts"
intro: fixes A :: "ordinal set" assumes "\<And>g. \<forall>n. g n \<in> A \<Longrightarrow> oLimit g \<in> A"   and "\<And>x. \<exists>y\<in>A. x < y" shows "normal_set A"
tensor_vec_from_lookup_Nil: fixes e :: "nat list \<Rightarrow> 'b" shows "tensor_vec_from_lookup [] e = [e []]"
sup_commute: fixes x :: "'e multiset"   and y :: "'e multiset" shows "x \<union># y = y \<union># x"
ws_cdecl_all: fixes G :: "prog" assumes "G = tprg" shows "\<forall>(C, c)\<in>set Classes. ws_cdecl G C (super c)"
to_nat_on_finite: fixes S :: "'b set" assumes "finite S" shows "bij_betw (to_nat_on S) S {..<card S}"
continuous_on_translation_eq: fixes A :: "'a set"   and a :: "'b"   and g :: "'a \<Rightarrow> 'b" shows "continuous_on A ((+) a \<circ> g) = continuous_on A g"
nth_normalize_less_zero_iff: fixes p :: "(real, 2) vec"   and i :: "2" shows "(normalize p $ i < 0) = (p $ i < 0)"
nonzero_mult_divide_mult_cancel_left: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and 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 c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times c a) (times c b) = divide a b"
\<gamma>_belowI: fixes i :: "int"   and iv :: "ivl"   and j :: "int" assumes "i \<in> \<gamma>_ivl iv"   and "j \<le> i" shows "j \<in> \<gamma>_ivl (below iv)"
singleton_inject: fixes a :: "'c"   and b :: "'c" assumes "{a} = {b}" shows "a = b"
get_parent_wf_is_l_get_parent_wf: shows "l_get_parent_wf ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel Shadow_DOM.get_child_nodes Shadow_DOM.get_parent"
perm_xs: shows "perm_restrict f (set xs) permutes set xs"
keys_of_def: fixes A :: "agent" shows "keys_of A \<equiv> insert (priK A) {shrK B C |B C. B = A \<or> C = A}"
LIMSEQ_subseq_LIMSEQ: fixes X :: "nat \<Rightarrow> 'b"   and L :: "'b"   and f :: "nat \<Rightarrow> nat" assumes "X \<longlonglongrightarrow> L"   and "strict_mono f" shows "(X \<circ> f) \<longlonglongrightarrow> L"
wf_tree: fixes v :: "'vertex"   and p :: "('form, 'var) antecedent"   and t :: "'vertex"   and pth :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) list" assumes "valid_in_port (v, p)"   and "terminal_path v t pth" shows "local.wf (tree v p pth)"
ordertype_VWF_mono: fixes X :: "V set"   and Y :: "V set" assumes "X \<subseteq> Y"   and "small Y" shows "tp X \<le> tp Y"
Sum_any_right_distrib: fixes g :: "'b \<Rightarrow> 'c"   and r :: "'c" assumes "finite {a. g a \<noteq> (0::'c)}" shows "r * Sum_any g = (\<Sum>n. r * g n)"
simps: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and V :: "nat set"   and s :: "(nat set \<times> 'b option list set) set" assumes "getIJ getIJ" shows "New_max.projectQuery V s = New_max.projectTable V ` s"
embedS_def: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and f :: "'d \<Rightarrow> 'e" shows "embedS r r' f \<equiv> embed r r' f \<and> \<not> bij_betw f (Field r) (Field r')"
ln_at_0: shows "filterlim ln at_bot (at_right 0)"
comp_cod_arr: fixes f :: "'a"   and b :: "'a" assumes "B.arr f"   and "B.cod f = b" shows "b \<cdot>\<^sub>B f = f"
lab_def_2: shows "\<S>\<bullet>\<C> \<turnstile> [Label n es' lfilledk] : ts' _> ts''"
Up: fixes Y :: "char list"   and Xs :: "char list set" assumes "Y \<in> Xs"   and "finite Xs" shows "fresh_string Xs Y = fresh_string (Xs - {Y}) (upChar Y)"
ANR_def: fixes S :: "'a set" shows "ANR S \<equiv> \<forall>U S'. S homeomorphic S' \<and> closedin (top_of_set U) S' \<longrightarrow> (\<exists>T. openin (top_of_set U) T \<and> S' retract_of T)"
Ostrowski_baseTr1: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and l :: "nat" assumes "vals_nonequiv K (Suc n) vv"   and "l \<le> Suc n" shows "0 < vv l (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l)"
assms_2: fixes \<sigma> :: "nat \<Rightarrow> 'g" assumes "P \<sigma>"   and "fst (initmissing (netgmap sr s)) = \<sigma>" shows "Q \<sigma>"
frac_distr_2: fixes d :: "real"   and a :: "real" assumes "0 < d"   and "d < 1 - frac a" shows "frac a + d = frac (a + d)"
newQuery_equiv_def: fixes V :: "nat set"   and Q :: "(nat set \<times> 'a option list set) set"   and st :: "nat set"   and t :: "'a option list" shows "newQuery V Q (st, t) = projectQuery V ((\<lambda>tab. semiJoin tab (st, t)) ` Q)"
distinct_3: fixes x1 :: "bool"   and x31 :: "'b aexp"   and x32 :: "'b aexp" shows "Bc x1 \<noteq> Gt x31 x32"
simps: fixes v :: "nat"   and L :: "atom list"   and F :: "atom fm list" shows "luckyFind v L F = (case find_lucky_eq v L of None \<Rightarrow> None | Some p \<Rightarrow> Some (elimVar v L F (Eq p)))"
zp: shows "\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
assms_4: shows "mod_type_class.to_nat b + 1 < k"
rtranclp_induct2: fixes r :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"   and ax :: "'a"   and ay :: "'b"   and bx :: "'a"   and "by" :: "'b"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* (ax, ay) (bx, by)"   and "P ax ay"   and "\<And>a b aa ba. \<lbrakk>r\<^sup>*\<^sup>* (ax, ay) (a, b); r (a, b) (aa, ba); P a b\<rbrakk> \<Longrightarrow> P aa ba" shows "P bx by"
red_single_lt_distinct_lt: fixes f :: "'a \<Rightarrow>\<^sub>0 'c"   and g :: "'a \<Rightarrow>\<^sub>0 'c"   and h :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "punit.red_single f g h t"   and "g \<noteq> 0"   and "punit.lt g \<noteq> punit.lt f" shows "punit.lt f = local.punit.splus t (punit.lt h)"
rotate1_rotate_swap: fixes n :: "nat"   and xs :: "'g list" shows "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
Methd: fixes G :: "prog"   and s0 :: "st"   and D :: "qtname"   and sig :: "sig"   and v :: "val"   and s1 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>body G D sig-\<succ>v\<rightarrow> s1" shows "G\<turnstile>Norm s0 \<midarrow>Methd D sig-\<succ>v\<rightarrow> s1"
parallel_converter2_eq_\<I>_cong: fixes \<I>1 :: "('a, 'b) \<I>"   and \<I>1' :: "('c, 'd) \<I>"   and conv1 :: "('a, 'b, 'c, 'd) converter"   and conv1' :: "('a, 'b, 'c, 'd) converter"   and \<I>2 :: "('e, 'f) \<I>"   and \<I>2' :: "('g, 'h) \<I>"   and conv2 :: "('e, 'f, 'g, 'h) converter"   and conv2' :: "('e, 'f, 'g, 'h) converter" assumes "\<I>1,\<I>1' \<turnstile>\<^sub>C conv1 \<sim> conv1'"   and "\<I>2,\<I>2' \<turnstile>\<^sub>C conv2 \<sim> conv2'" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>2,\<I>1' \<oplus>\<^sub>\<I> \<I>2' \<turnstile>\<^sub>C conv1 |\<^sub>= conv2 \<sim> conv1' |\<^sub>= conv2'"
step_1_main_inv: fixes i :: "nat"   and j :: "nat"   and A :: "'b mat"   and n :: "nat" assumes "i \<le> j"   and "A \<in> carrier_mat n n"   and "??.Jordan_Normal_Form_Existence.inv_all n ??.Jordan_Normal_Form_Existence.uppert A"   and "inv_part n ??.Jordan_Normal_Form_Existence.diff_ev A i j" shows "??.Jordan_Normal_Form_Existence.inv_all n ??.Jordan_Normal_Form_Existence.uppert (??.Jordan_Normal_Form_Existence.step_1_main n i j A) \<and> ??.Jordan_Normal_Form_Existence.inv_all n ??.Jordan_Normal_Form_Existence.diff_ev (??.Jordan_Normal_Form_Existence.step_1_main n i j A)"
relators_3: fixes rs :: "'g list" assumes "rs \<in> R" shows "proper_signed_list (map (charpair S) rs)"
under_arg: fixes i :: "nat" assumes "head t = head s"   and "num_args t = num_args s"   and "args t ! i \<rightarrow>\<^sub>e\<^sub>m\<^sub>b args s ! i"   and "\<And>j. \<lbrakk>j < num_args t; i \<noteq> j\<rbrakk> \<Longrightarrow> args t ! j = args s ! j" shows "P"
refl: fixes A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and l :: "'e"   and u :: "'f \<Rightarrow> 'g" shows "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>* \<langle>l, u\<rangle>"
Cons_prefix_Cons: fixes x :: "'e"   and xs :: "'e list"   and y :: "'e"   and ys :: "'e list" shows "prefix (x # xs) (y # ys) = (x = y \<and> prefix xs ys)"
axis: fixes x :: "complex" assumes "is_real x"   and "0 < Re x"   and "Re x < 1" shows "P 0\<^sub>h (of_complex x)"
P_def: shows "P = (\<exists>b\<subseteq>U\<^sub>V\<^sub>S. \<not> local.dependent b \<and> local.span b = local.span V)"
that: fixes a :: "'a"   and b :: "'a" assumes "p = [:a, b:]"   and "b \<noteq> (0::'a)" shows "thesis"
simps_8: fixes t :: "'c local.term" shows "Can \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[t\<^bold>] = Can t"
acyclicP_converse: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "acyclicP r\<inverse>\<inverse> = acyclicP r"
radical_def: fixes x :: "'a" shows "radical x = (if x = (0::'a) then 0::'a else \<Prod>(prime_factors x))"
inject_5: fixes x1 :: "name"   and x2 :: "('a, 'b, 'c) psi"   and y1 :: "name"   and y2 :: "('a, 'b, 'c) psi" shows "(\<lparr>\<nu>x1\<rparr>x2 = \<lparr>\<nu>y1\<rparr>y2) = ([x1].x2 = [y1].y2)"
string_ternop_type_det: fixes op :: "string_ternop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'b OCL_Types.type"   and \<rho> :: "'c OCL_Types.type"   and \<upsilon>\<^sub>1 :: "'d OCL_Types.type"   and \<upsilon>\<^sub>2 :: "'d OCL_Types.type" assumes "string_ternop_type op \<tau> \<sigma> \<rho> \<upsilon>\<^sub>1"   and "string_ternop_type op \<tau> \<sigma> \<rho> \<upsilon>\<^sub>2" shows "\<upsilon>\<^sub>1 = \<upsilon>\<^sub>2"
span_finite: fixes S :: "'b set" assumes "finite S" shows "span S = range (\<lambda>u. \<Sum>v\<in>S. u v *s v)"
authKeys_are_not_AKcryptSK: fixes K :: "nat"   and evs :: "event list" assumes "K \<in> authKeys evs \<union> range shrK"   and "evs \<in> kerbIV_gets" shows "\<forall>SK. \<not> AKcryptSK SK K evs \<and> K \<in> symKeys"
assms_2: shows "\<forall>xs\<in>set xss. finite xs"
sim_fI: fixes M :: "('b, 'd, 'c) KripkeStructure"   and f :: "'c \<Rightarrow> 'e"   and M' :: "('b, 'd, 'e) KripkeStructure" assumes "\<And>a u v. (u, v) \<in> relations M a \<Longrightarrow> (f u, f v) \<in> relations M' a" shows "sim_f M M' f"
cut_ge_Min_all: fixes t :: "'a"   and I :: "'a set" assumes "t \<le> iMin I" shows "I \<down>\<ge> t = I"
preserves_dom: fixes f :: "'a" assumes "\<phi>.A.arr f" shows "\<phi>.B.dom (\<phi> f) = F (\<phi>.A.dom f)"
sup_absorb2: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.semilattice_sup sup less_eq less"   and "less_eq x y" shows "sup x y = y"
simps_1: fixes l :: "'b set list" shows "qsort [] l = []"
encode_problem_forall_step_decoded_plan_is_serializable_i: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<Pi> t" shows "\<Pi>\<^sub>G \<subseteq>\<^sub>m execute_serial_plan (\<Pi>\<^sub>I) (concat (\<Phi>\<inverse> \<Pi> \<A> t))"
lproject_prefix: fixes u :: "'a llist"   and v :: "'a llist"   and A :: "'a set" assumes "u \<le> v" shows "lproject A u \<le> lproject A v"
semilattice_set_apply': fixes f :: "'b semilattice_set" shows "semilattice_set (semilattice_set_apply f)"
set_alist_product: fixes xs :: "('a \<times> 'b) list"   and ys :: "('c \<times> 'd) list" shows "set (local.alist_product xs ys) = (\<lambda>((a, b), c, d). ((a, c), f a b c d)) ` (set xs \<times> set ys)"
indep_sets2_eq: fixes A :: "'a set set"   and B :: "'a set set" shows "indep_set A B = (A \<subseteq> events \<and> B \<subseteq> events \<and> (\<forall>a\<in>A. \<forall>b\<in>B. prob (a \<inter> b) = prob a * prob b))"
pref_profile_from_tableI': fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and xss :: "('a \<times> 'b set list) list"   and agents :: "'a set"   and alts :: "'b set" assumes "R1 \<equiv> prefs_from_table xss"   and "prefs_from_table_wf agents alts xss" shows "pref_profile_wf agents alts R1"
inject: fixes x2 :: "'a set"   and y2 :: "'a set" shows "(RES x2 = RES y2) = (x2 = y2)"
finsertE: fixes a :: "'c"   and b :: "'c"   and A :: "'c fset" assumes "a |\<in>| finsert b A" obtains "a = b"   | "a |\<in>| A"
refines_def: fixes A :: "'c set set"   and B :: "'c set set" shows "(A refines B) = (\<forall>s\<in>A. \<exists>t. t \<in> B \<and> s \<subseteq> t)"
sync_commute: fixes P :: "'a process"   and A :: "'a set"   and Q :: "'a process" shows "(P \<lbrakk>A\<rbrakk> Q) = (Q \<lbrakk>A\<rbrakk> P)"
swap0_def: shows "swap0 \<equiv> map_fun id (map_fun id (map_fun lookup Abs_poly_mapping)) (\<lambda>b i p x. p (swap b i x))"
hyp_edge_at_def: fixes c :: "'form"   and "is" :: "nat list"   and n :: "nat"   and s :: "'subst" shows "hyp_edge_at c is n s = (hyp_edge_from c is n s, hyp_edge_to c is)"
assms_1: shows "q \<noteq> q\<^sub>0"
arcs_del_arc: fixes G :: "('d, 'c) pre_digraph"   and a :: "'c" shows "arcs (pre_digraph.del_arc G a) = arcs G - {a}"
snd_eqvt: fixes p :: "perm"   and x :: "'d \<times> 'c" shows "p \<bullet> snd x = snd (p \<bullet> x)"
arr_iff_in_hom: fixes f :: "'a1 \<times> 'a2" shows "A1xA2.arr f = \<guillemotleft>f : A1xA2.dom f \<rightarrow>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2 A1xA2.cod f\<guillemotright>"
eval_rest: shows "P \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
sup_continuous_def: fixes F :: "'a \<Rightarrow> 'b" shows "sup_continuous F = (\<forall>M. incseq M \<longrightarrow> F (Sup (range M)) = (SUP i. F (M i)))"
smcf_def: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> = [\<FF>\<lparr>ObjMap\<rparr>, \<FF>\<lparr>ArrMap\<rparr>, \<FF>\<lparr>HomDom\<rparr>, \<FF>\<lparr>HomCod\<rparr>]\<^sub>\<circ>"
maxindex_coefficients: fixes p :: "poly" shows "\<forall>c\<in>set (coefficients p). maxindex c \<le> maxindex p"
wadjust_on_left_moving_B_Bk_drop_one: fixes m :: "nat"   and rs :: "nat"   and c :: "cell list"   and list :: "cell list" assumes "wadjust_on_left_moving_B m rs (c, Bk # list)"   and "hd c = Bk" shows "wadjust_on_left_moving_B m rs (tl c, Bk # Bk # list)"
diff_union_swap2: fixes y :: "'a"   and M :: "'a multiset"   and x :: "'a" assumes "y \<in># M" shows "(x, M) - (y, {#}) = x, M - (y, {#})"
ker_in_carrier: shows "kernel G H h \<subseteq> carrier G"
out_var_def: fixes x :: "'d \<Longrightarrow> 'f" shows "$x\<acute> = x ;\<^sub>L snd\<^sub>L"
xivp_1: shows " D Y\<^sub>1 = (\<lambda>t. f t (Y\<^sub>1 t)) on U"
step_elim_cases_8: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and p :: "'a"   and s :: "('b, 'c) xstate"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (Call p, s) \<rightarrow> u" obtains bdy :: "('b, 'a, 'c) com" and sa :: "'b" where "s = Normal sa__" and "u = (bdy, Normal sa__)" and "\<Gamma> p = Some bdy"   | sa :: "'b" where "s = Normal sa" and "u = (Skip, Stuck)" and "\<Gamma> p = None"   | f :: "'c" where "s = Fault f__" and "u = (Skip, Fault f__)"   | "s = Stuck" and "u = (Skip, Stuck)"   | f :: "'b" where "s = Abrupt f" and "u = (Skip, Abrupt f)"
pred_cof_def: fixes L :: "char list set"   and t :: "sterm"   and t' :: "sterm" shows "pred_cof L t t' = (\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. t\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> t' = \<sigma>[s,p] t''))"
Setcompr_eq_image: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "{f x |x. x \<in> A} = f ` A"