hcju/isabelleps huggingface dataset:
eval_agg_def: fixes n :: "nat"   and g0 :: "bool"   and y :: "nat"   and \<omega> :: "Formula.agg_type \<times> event_data"   and b :: "nat"   and f :: "Formula.trm"   and rel :: "event_data option list set" shows "eval_agg n g0 y \<omega> b f rel = (if g0 \<and> rel = empty_table then singleton_table n y (eval_agg_op \<omega> {}) else (\<lambda>k. let group = Set.filter (\<lambda>x. drop b x = k) rel; M = (\<lambda>y. (y, ecard (Set.filter (\<lambda>x. meval_trm f x = y) group))) ` meval_trm f ` group in k[y := Some (eval_agg_op \<omega> M)]) ` drop b ` rel)"
measurable_count_space_eq_countable: fixes A :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and M :: "'c measure" assumes "countable A" shows "(f \<in> M \<rightarrow>\<^sub>M count_space A) = (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
sum_in_spanI: fixes q :: "('b \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'c"   and B :: "('b \<Rightarrow>\<^sub>0 'c) set" shows "(\<Sum>b\<in>B. q b \<cdot> b) \<in> phull B"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "natural_isomorphism (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) trg\<^sub>B B.map \<phi>" shows "equivalence_functor (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) trg\<^sub>B"
take_rbl_plus: fixes xs :: "bool list"   and ys :: "bool list" shows "\<forall>n b. take n (rbl_plus b xs ys) = rbl_plus b (take n xs) (take n ys)"
comp_arr_inv': fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "category.iso C f" shows "C f (category.inv C f) = partial_magma.cod C f"
sorted_distinct_set_unique: fixes xs :: "'a list"   and ys :: "'a list" assumes "sorted xs"   and "distinct xs"   and "sorted ys"   and "distinct ys"   and "set xs = set ys" shows "xs = ys"
dg_Obj_vsubset_Vset: shows "\<BB>\<lparr>Obj\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
pre_F: shows "pre_splitFace g' v a f' [countVertices g'..<countVertices g' + n]"
generate_valid_topology_nodes: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph" shows "nodes (generate_valid_topology M G) = nodes G"
tiny_digraph_op: shows "tiny_digraph \<alpha> (op_dg \<AA>)"
homeomorphism_apply1: fixes S :: "'b set"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b"   and x :: "'b" assumes "homeomorphism S T f g"   and "x \<in> S" shows "g (f x) = x"
FWbisimulation_flip: shows "FWbisimulation final2 r2 final1 r1 (\<lambda>t. flip (bisim t))"
assms: fixes y :: "xml" assumes "y \<in> set (Xml.children x)" shows "monotone sum_bot.le_fun sum_bot_ord (p1 y)"
le_supI1: fixes x :: "'c \<times> 'd"   and a :: "'c \<times> 'd"   and b :: "'c \<times> 'd" assumes "less_eq_prod x a" shows "less_eq_prod x (sup_prod a b)"
pwd_spy_iv_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "case pwd_spy_iv n of (s, S, A, B, C, D) \<Rightarrow> pwd_spy_iv_state n (S, A, B, C, D) \<subseteq> s \<and> {Key (Auth_ShaKey n)} \<in> key_sets (PriKey S) (crypts (Log -` spied s))"
list_app_typeE: fixes e :: "nat \<Rightarrow> LambdaType.type"   and t :: "dB"   and ts :: "dB list"   and T :: "LambdaType.type" assumes "e \<turnstile> t \<degree>\<degree> ts : T" obtains Ts :: "LambdaType.type list" where "e \<turnstile> t : Ts \<Rrightarrow> T" and "e \<tturnstile> ts : Ts"
continuous_map_add: fixes X :: "'a topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_map X euclidean f"   and "continuous_map X euclidean g" shows "continuous_map X euclidean (\<lambda>x. f x + g x)"
assms_3: fixes a :: "'a"   and b :: "'e" assumes "R' a b" shows "(R1 OO R2) a b"
unwrap_def: shows "unwrap \<equiv> \<Lambda> f x xs. f\<cdot>(x :@ xs)"
select_convs_2: fixes ok :: "bool"   and more :: "'b" shows "more \<lparr>ok = ok, \<dots> = more\<rparr> = more"
EQ: shows "P\<^sub>1 \<union> wrap B\<^sub>1 \<union> wrap {} \<union> (P\<^sub>2 \<union> wrap B\<^sub>2) \<union> wrap {u} \<union> {{v} |v. v \<in> V - {u}} = P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2 \<union> {{v} |v. v \<in> V}"
inverts_2: fixes discr :: "bool discr"   and discr' :: "bool discr" shows "(CB\<cdot>discr \<sqsubseteq> CB\<cdot>discr') = (discr \<sqsubseteq> discr')"
gr_zeroI: fixes n :: "'d multiset" assumes "n = {#} \<Longrightarrow> False" shows "{#} \<subset># n"
start_heap_classes: fixes P :: "'a prog"   and a :: "nat"   and C :: "char list"   and fs :: "Objects.fields" assumes "start_heap P a = \<lfloor>(C, fs)\<rfloor>" shows "C \<in> sys_xcpts"
simps_5: shows "\<stileturn> \<noteq> \<turnstile>"
make_ground_subst: fixes C :: "'a literal multiset"   and \<sigma> :: "'s" assumes "is_ground_cls (C \<cdot> \<sigma>)" shows "\<exists>\<tau>. is_ground_subst \<tau> \<and> C \<cdot> \<tau> = C \<cdot> \<sigma>"
ergodic_qmpt_def: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" shows "ergodic_qmpt M T \<equiv> qmpt M T \<and> ergodic_qmpt_axioms M T"
bounded_MPLS_contains_supremum: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" assumes "finite PROB"   and "\<exists>k. \<forall>x\<in>MPLS PROB. x < k" shows "Sup (MPLS PROB) \<in> MPLS PROB"
fsyms_satL_cong: fixes intT :: "'tp \<Rightarrow> 'b \<Rightarrow> bool"   and intF1 :: "'fsym \<Rightarrow> 'b list \<Rightarrow> 'b"   and intP :: "'psym \<Rightarrow> 'b list \<Rightarrow> bool"   and intF2 :: "'fsym \<Rightarrow> 'b list \<Rightarrow> 'b"   and l :: "('fsym, 'psym) lit"   and \<xi> :: "Preliminaries.var \<Rightarrow> 'b" assumes "Struct wtFsym wtPsym arOf resOf intT intF1 intP"   and "Struct wtFsym wtPsym arOf resOf intT intF2 intP"   and "\<And>f. f \<in> fsymsL l \<Longrightarrow> intF1 f = intF2 f" shows "Struct.satL intF1 intP \<xi> l = Struct.satL intF2 intP \<xi> l"
loc_srj_1: shows "nat_to_sch (c_pair 2 0) = Base_fst"
intro: fixes \<alpha> :: "'d \<Rightarrow> 'c"   and S :: "'d set"   and T :: "'c set" assumes "\<alpha> ` S = T" shows "surjective_map_axioms \<alpha> S T"
hoare_demonic: fixes P :: "'a set"   and R :: "'a \<Rightarrow> 'b set"   and Q :: "'b set" shows "\<Turnstile> P{| [: R :] |}Q = (\<forall>s. s \<in> P \<longrightarrow> R s \<subseteq> Q)"
tendsto_Liminf: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "F \<noteq> bot"   and "Limsup F f = Liminf F f" shows "(f \<longlongrightarrow> Liminf F f) F"
min_rvar_incdec_eq_Some_incdec: fixes dir :: "('i, 'a) Direction"   and s :: "('i, 'a) state"   and eq :: "nat \<times> linear_poly"   and x\<^sub>j :: "nat" assumes "min_rvar_incdec_eq dir s eq = Inr x\<^sub>j" shows "0 < coeff (rhs eq) x\<^sub>j \<and> \<lhd>\<^sub>u\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> x\<^sub>j) (UB dir s x\<^sub>j) \<or> coeff (rhs eq) x\<^sub>j < 0 \<and> \<rhd>\<^sub>l\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> x\<^sub>j) (LB dir s x\<^sub>j)"
comparables_iff: fixes T :: "nat set"   and S :: "nat set"   and M :: "nat set" shows "(T \<in> comparables S M) = (finite T \<and> (init_segment T S \<or> init_segment S T \<and> T \<subseteq> S \<union> M))"
CFG_SSA_wf_code_def: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> ('b, 'd set) mapping"   and phis :: "'a \<Rightarrow> ('b \<times> 'd, 'd list) mapping" shows "CFG_SSA_wf_code \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis \<equiv> CFG_SSA_code \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis \<and> CFG_SSA_wf \<alpha>e \<alpha>n invar inEdges' Entry defs (usesOf \<circ> uses) (\<lambda>g. Mapping.lookup (phis g))"
pwd_spy_i_def: fixes n :: "nat" shows "pwd_spy_i n \<equiv> (insert (Spy, Crypt (Auth_ShaKey n) (Auth_PriKey n)) s\<^sub>0, Auth_PriK n, 0, 0, 0, 0)"
invar_trie_update: fixes t :: "('c, 'd) Trie.trie"   and ks :: "'c list"   and v :: "'d" assumes "invar_trie t" shows "invar_trie (update_trie ks v t)"
\<nu>: shows "\<guillemotleft>\<nu> : r \<Rightarrow> g \<star> f\<^sub>a\<guillemotright> \<and> \<guillemotleft>\<nu> : trg f \<rightarrow> trg g\<guillemotright> \<and> arr \<nu> \<and> src \<nu> = trg f \<and> trg \<nu> = trg g \<and> local.dom \<nu> = r \<and> cod \<nu> = g \<star> f\<^sub>a"
pair_list_dgrad_sig_set: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" assumes "rb_aux_inv (bs, ss, ps)"   and "p \<in> set ps" shows "poly_of_pair p \<in> dgrad_sig_set' (length fs) dgrad"
him_cnj: fixes q :: "quat" shows "HIm (Quaternions.cnj q) = - HIm q"
assms_1: shows "f \<in> carrier L \<rightarrow> carrier L"
poly_cond2D_2: fixes p :: "int poly" assumes "poly_cond2 p" shows "1 < degree p"
disc_2: fixes x21 :: "'e"   and x22 :: "('e, 'f) Term.term list" shows "is_Fun (Fun x21 x22)"
reds_hext_incr: fixes E :: "char list \<Rightarrow> ty option"   and es :: "expr 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 es' :: "expr 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 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" assumes "P,E \<turnstile> \<langle>es,(h, l)\<rangle> [\<rightarrow>] \<langle>es',(h', l')\<rangle>" shows "h \<unlhd> h'"
simps: fixes v :: "'a"   and n :: "nat" shows "time (??.Time_Monad.tm.TM v n) = n"
subcomplex_chamber: fixes Y :: "'a set set"   and C :: "'a set" assumes "ChamberSubcomplex Y"   and "SimplicialComplex.maxsimp Y C" shows "chamber C"
MajoritySet_def: shows "MajoritySet = {D. IsMajority D}"
D_p_3: shows "\<forall>E\<in>{E \<in> N_of_state (Sup_state Sts) \<union> P_of_state (Sup_state Sts) \<union> Q_of_state (Sup_state Sts). subsumes E C}. \<not> strictly_subsumes E D"
IH\<alpha>rec: fixes \<nu> :: "variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real"   and X :: "(variable \<Rightarrow> real) set" assumes "Uvariation \<nu> \<omega> (fst (usubstappp \<sigma> U \<alpha>))"   and "snd (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<alpha>) \<noteq> undefg" shows "(\<nu> \<in> game_sem I (the (snd (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<alpha>))) X) = (\<nu> \<in> game_sem (USubst.adjoint \<sigma> I \<omega>) \<alpha> X)"
pt_ex1_eqvt: fixes pi :: "('b \<times> 'b) list"   and P :: "'a \<Rightarrow> bool" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pi \<bullet> (\<exists>!x. P x) = (\<exists>!x. pi \<bullet> P (rev pi \<bullet> x))"
Goal: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<longmapsto>\<^sub>l a<\<nu>x> \<prec>\<^sub>l P'"   and "x \<sharp> P" shows "P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'"
intro: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c" assumes "category A"   and "category B"   and "constant_functor_axioms B b" shows "constant_functor A B b"
meet_def: fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y = (THE inf. is_inf x y inf)"
ev_block_def: fixes n :: "nat"   and A :: "'a mat" shows "local.ev_block n A = (\<forall>i j. i < n \<longrightarrow> j < n \<longrightarrow> A $$ (i, i) = A $$ (j, j))"
of_nat_max: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "nat"   and y :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "semiring_1.of_nat one plus zero (max x y) = ord.max less_eq (semiring_1.of_nat one plus zero x) (semiring_1.of_nat one plus zero y)"
in_cong: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and h :: "'b \<Rightarrow> 'c" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "(f \<in> \<Omega>[F](h)) = (g \<in> \<Omega>[F](h))"
incidB_lAB: fixes B :: "Points"   and A :: "Points" shows "incid B (line A B)"
DPR: fixes ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "ring_ops ops R" shows "Domainp R = arith_ops_record.DP ops"
imp_ex: fixes P :: "'a \<Rightarrow> bool"   and Q :: "bool" shows "((\<exists>x. P x) \<longrightarrow> Q) = (\<forall>x. P x \<longrightarrow> Q)"
t_wcode_prepare_def: shows "t_wcode_prepare \<equiv> [(W1, 2), (L, 1), (L, 3), (R, 2), (R, 4), (W0, 3), (R, 4), (R, 5), (R, 6), (R, 5), (R, 7), (R, 5), (W1, 7), (L, 0)]"
sup_bot_left: fixes x :: "'a" shows "bot \<squnion> x = x"
join_def: fixes x :: "'b"   and y :: "'b" shows "x \<squnion> y = (if \<exists>z. {x, y} <<| z then lub {x, y} else x)"
z'_def: shows "z' \<equiv> getFr [xx, yy, z] [t1, t2] []"
wp_loop_step_bounded: fixes Q :: "'a \<Rightarrow> real"   and b :: "real"   and t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "nneg Q"   and "bounded_by b Q"   and "healthy t"   and "healthy (wp body)" shows "bounded_by b (wp (body ;; Embed t \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) Q)"
comp_red_basis_aux_irred: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and ys :: "('t \<Rightarrow>\<^sub>0 'c) list"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_minimal_basis (set (xs @ ys))"   and "distinct (xs @ ys)"   and "\<And>y. y \<in> set ys \<Longrightarrow> \<not> is_red (set (xs @ ys) - {y}) y"   and "p \<in> set (comp_red_basis_aux xs ys)" shows "\<not> is_red (set (comp_red_basis_aux xs ys) - {p}) p"
quarter_discrim_invariant: fixes t :: "(real, 3) vec"   and i :: "real"   and a :: "(real, 3) vec"   and r :: "(real, 3) vec" assumes "t = i *\<^sub>R a + r" shows "quarter_discrim a t = quarter_discrim a r"
a: shows "P s s' \<and> (x1 ::= x2, s) \<Rightarrow> t \<and> (c', s') \<Rightarrow> t'"
get_scdom_component_impl: shows "get_scdom_component = local.a_get_scdom_component"
const_list_cons_last_2: fixes es :: "e list"   and e :: "e" assumes "const_list (es @ [e])" shows "is_const e"
P1_ideal_model_def: fixes M :: "'a \<times> 'a"   and \<sigma> :: "bool"   and z :: "'f"   and \<A> :: "('a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'f \<Rightarrow> (('a \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> ((('a \<times> 'a) \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('g \<Rightarrow> 'e spmf)" shows "P1_ideal_model M \<sigma> z \<A> = (let (\<A>1, \<A>2, \<A>3) = \<A> in sample_uniform (order \<G>) \<bind> (\<lambda>r. sample_uniform (order \<G>) \<bind> (\<lambda>\<alpha>0. sample_uniform (order \<G>) \<bind> (\<lambda>\<alpha>1. let h0 = \<^bold>g [^] \<alpha>0; h1 = \<^bold>g [^] \<alpha>1; a = \<^bold>g [^] r; b0 = h0 [^] r; b1 = h1 [^] r \<otimes> \<^bold>g in \<A>1 M h0 h1 a b0 b1 z \<bind> (\<lambda>((in1, in2, in3), s). let (h, a, b) = (h0 \<otimes> inv h1, a, b0 \<otimes> inv b1) in assert_spmf ((in1, in2, in3) = (h, a, b)) \<bind> (\<lambda>_. \<A>2 h0 h1 a b0 b1 M s \<bind> (\<lambda>(((w0, z0), w1, z1), s'). let x0' = z0 \<otimes> inv w0 [^] \<alpha>0; x1' = z1 \<otimes> inv w1 [^] \<alpha>1 in mal_def.trusted_party (x0', x1') \<sigma> \<bind> (\<lambda>(uu_, f_out2). \<A>3 s' \<bind> (\<lambda>adv_out. return_spmf (adv_out, f_out2))))))))))"
lookup_add_monom_2: fixes lp :: "(nat, rat) fmap"   and v :: "nat"   and c :: "rat" assumes "get_var_coeff lp v + c = 0" shows "fmlookup (add_monom c v lp) v = None"
WHILEI_body_trimono: fixes I :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'm" shows "trimono (WHILEI_body bind return I b f)"
quot_Disj: fixes A :: "fm"   and B :: "fm" shows "\<guillemotleft>A OR B\<guillemotright> = Q_Disj \<guillemotleft>A\<guillemotright> \<guillemotleft>B\<guillemotright>"
simps_2: fixes f :: "'a \<Rightarrow> 'a"   and n :: "nat"   and x :: "'a" shows "funpower f (Suc n) x = f (funpower f n x)"
jkbpC_tLength_inv: fixes t :: "('a, 'es, 'as) BEState Trace"   and n :: "nat" assumes "t \<in> SPR.jkbpC"   and "tLength t = n" shows "t \<in> SPR.jkbpCn n"
set_delete_aux: fixes xs :: "('a \<times> 'b) list"   and k :: "'a" assumes "distinct (map fst xs)" shows "list.set (AList.delete_aux k xs) = list.set xs - {k} \<times> UNIV"
sumZeroLeft: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "Id \<subseteq> Rel" shows "P \<oplus> \<zero> \<leadsto>[Rel] P"
that: fixes P :: "'a mat" assumes "P \<in> Units (ring_mat TYPE('a) (dim_row A) b)"   and "row_echelon A = P * A" shows "thesis"
c1: shows "\<forall>l s. wp\<^sub>t (WHILE b DO c) Q l s \<longrightarrow> (\<exists>n. wpw b c n Q l s)"
att_and_fst_lem: fixes x1 :: "'a attree"   and x2a :: "'a attree list"   and x :: "'a set \<times> 'a set"   and xa :: "'a" assumes "\<turnstile>x1 # x2a \<oplus>\<^sub>\<and>\<^bsup>x\<^esup>"   and "xa \<in> fst (attack (x1 # x2a \<oplus>\<^sub>\<and>\<^bsup>x\<^esup>))" shows "xa \<in> fst (attack x1)"
open_map_fst: fixes X :: "'d topology"   and Y :: "'e topology" shows "open_map (prod_topology X Y) X fst"
N_def: shows "N = (LEAST n. 0 < n \<and> fds_nth h n \<noteq> (0::'a))"
indep_vars_subset: fixes M' :: "'c \<Rightarrow> 'd measure"   and X :: "'c \<Rightarrow> 'a \<Rightarrow> 'd"   and I :: "'c set"   and J :: "'c set" assumes "indep_vars M' X I"   and "J \<subseteq> I" shows "indep_vars M' X J"
step_def: shows "step \<equiv> {(c, c'). \<exists>a. astep c a c'}"
zero_le_divide_1_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq zero (divide one a) = less_eq zero a"
surj_one: fixes x :: "'a" assumes "is_sur x" shows "1 ; x = 1"
induct: fixes P :: "('a \<times> 'b) list \<Rightarrow> bool"   and a0 :: "('a \<times> 'b) list" assumes "P []"   and "\<And>uu_ x xs. P xs \<Longrightarrow> P ((uu_, x) # xs)" shows "P a0"
add_right_cancel: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and b :: "'k"   and a :: "'k"   and c :: "'k" assumes "class.cancel_semigroup_add plus" shows "(plus b a = plus c a) = (b = c)"
representation_ne_zero: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and basis :: "'d set"   and v :: "'d"   and b :: "'d" assumes "module scale"   and "module.representation scale basis v b \<noteq> (0::'c)" shows "b \<in> basis"
coinduct: fixes R :: "('d, 'e, 'f) gpv \<Rightarrow> ('d, 'e, 'f) gpv \<Rightarrow> bool"   and gpv :: "('d, 'e, 'f) gpv"   and gpv' :: "('d, 'e, 'f) gpv" assumes "R gpv gpv'"   and "\<And>gpv gpv'. R gpv gpv' \<Longrightarrow> rel_spmf (rel_generat (=) (=) (rel_fun (=) R)) (the_gpv gpv) (the_gpv gpv')" shows "gpv = gpv'"
specify_left: fixes m :: "'c nres"   and \<Phi> :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd nres"   and M :: "'d nres" assumes "m \<le> SPEC \<Phi>"   and "\<And>x. \<Phi> x \<Longrightarrow> f x \<le> M" shows "m \<bind> f \<le> M"
convergent_prod_def: fixes f :: "nat \<Rightarrow> 'e" shows "convergent_prod f \<equiv> \<exists>M p. raw_has_prod f M p"
sel_1: fixes x11 :: "'a"   and x12 :: "nat"   and x13 :: "'a interval" shows "pos (INS x11 x12 x13) = x11"
nodes_connected_refl: fixes G :: "('a, 'b) graph"   and a :: "'a" assumes "valid_graph G"   and "a \<in> nodes G" shows "nodes_connected G a a"
lowner_lub_prop: shows "local.lowner_is_lub lowner_lub"
literal_of_asciis_Nil: shows "String.literal_of_asciis [] = STR ''''"
q: fixes j :: "'a \<Rightarrow> 'b" assumes "p j" shows "q (j \<circ> h)"
lfp_F_def: fixes ds :: "'b set" shows "lfp_F ds = map_prod id undual (lfp (F ds))"
a_rcos_const: fixes H :: "'e set"   and G :: "('e, 'f) ring_scheme"   and h :: "'e" assumes "abelian_subgroup H G"   and "h \<in> H" shows "H +>\<^bsub>G\<^esub> h = H"
get_parent_is_strongly_dom_component_safe_step: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_parent ptr' \<rightarrow>\<^sub>r Some parent" shows "(parent \<in> set c) = (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr' \<in> set c)"
factors_closed: fixes G :: "('c, 'd) monoid_scheme"   and fs :: "'c list"   and a :: "'c" assumes "Group.monoid G"   and "factors G fs a"   and "set fs \<subseteq> carrier G" shows "a \<in> carrier G"
hyps: shows "eval_poly (\<lambda>_. x) p \<le> poly_coeff_sum p * x ^ poly_degree p"
D_cyc_free_preservation: fixes n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and m' :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len m i i xs"   and "\<forall>i\<le>n. \<forall>j\<le>n. D m i j n = m' i j" shows "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len m' i i xs"
Pd'_d'_range: shows "Field (Pd' d') \<subseteq> {y. Xd y = d'}"
prems: fixes k :: "nat"   and z :: "'a" assumes "k \<in> {i..<j + 1}"   and "z \<in> atms (Star r)" shows "t k z = t' k z"
open_Int_closure_eq_empty: fixes S :: "'b set"   and T :: "'b set" assumes "open S" shows "(S \<inter> closure T = {}) = (S \<inter> T = {})"
queue_invar_simps_2: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.queue_invar (t # q) = (SkewBinomialHeapStruc.tree_invar t \<and> SkewBinomialHeapStruc.queue_invar q)"
set_iterateoi_I: fixes invar :: "'c \<Rightarrow> bool"   and itoi :: "'c \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and \<alpha> :: "'c \<Rightarrow> 'e set" assumes "\<And>s. invar s \<Longrightarrow> set_iterator_linord (itoi s) (\<alpha> s)" shows "set_iterateoi \<alpha> invar itoi"
zero_divides: fixes a :: "'a" shows "\<zero> divides a = (a = \<zero>)"
simps_2: fixes m :: "'b SecurityInvariant_configured"   and Ms :: "'b SecurityInvariant_configured list"   and G :: "'b graph" shows "generate_valid_topology (m # Ms) G = delete_edges (generate_valid_topology Ms G) (\<Union> (c_offending_flows m G))"
nn_integral_bind: fixes f :: "'a \<Rightarrow> ennreal"   and B :: "'a measure"   and N :: "'b \<Rightarrow> 'a measure"   and M :: "'b measure" assumes "f \<in> borel_measurable B"   and "N \<in> M \<rightarrow>\<^sub>M subprob_algebra B" shows "integral\<^sup>N (M \<bind> N) f = \<integral>\<^sup>+ x. integral\<^sup>N (N x) f \<partial>M"
T_a: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "ad p \<cdot> (p \<leadsto> x \<leadsto> q) = ad p"