hcju/isabelleps huggingface dataset:
tfr\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('fun, ('fun, 'atom) Term.term \<times> nat) stateful_strand_step list" shows "tfr\<^sub>s\<^sub>s\<^sub>t S \<equiv> tfr\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>s\<^sub>t S \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t S) \<and> list_all tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p S"
f_analytic: shows "f analytic_on {s. 1 \<le> Re s}"
multiplicity_normalize_left: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.multiplicity times one (normalize p) x = factorial_semiring.multiplicity times one p x"
deg_less1: fixes n :: "nat" assumes "0 < n" shows "degree (hensel_fpxs1 $ n) < degree g"
cases: fixes x :: "(String.literal \<times> String.literal) set \<times> ('a \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" obtains sub :: "(String.literal \<times> String.literal) set" and tcs :: "'a \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option" where "x = (sub, tcs)"
q2_def: shows "q2 = p - monom_mult (lookup p (t2 \<oplus> lt f2) / lc f2) t2 f2"
fst_lens_def: shows "fst\<^sub>L = \<lparr>lens_get = fst, lens_put = \<lambda>(\<sigma>, \<rho>) u. (u, \<rho>)\<rparr>"
\<alpha>nl_simps_2: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and x :: "('c, 'f) label list" shows "\<alpha>nl fg (LOC x) = \<alpha>n fg x"
c_def: shows "c = (\<lambda>n. ln (real n) - (\<Sum>r<n. inverse (x + real r)))"
wto: shows "wset_thread_ok (wset S) (thr S)"
list_ex_cong: fixes xs :: "'c list"   and ys :: "'c list"   and f :: "'c \<Rightarrow> bool"   and g :: "'c \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> f x = g x" shows "list_ex f xs = list_ex g ys"
lim_def: shows "local.lim = extend_measure (space (Pi\<^sub>M I M)) generator (\<lambda>x. x) mu_G"
direct_decomp_def: fixes A :: "'b set"   and ss :: "'b set list" shows "direct_decomp A ss = bij_betw sum_list (listset ss) A"
l1_defs_1: shows "l1 \<equiv> \<lparr>init = l1_init, trans = l1_trans, obs = id\<rparr>"
P2_proj_P1: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and i :: "nat" assumes "\<psi> \<in> \<P>\<^sup>2" shows "\<psi> i \<in> \<P>"
dim_basis: fixes A :: "'c set" assumes "finite A"   and "ker.basis A" shows "ker.dim = card A"
Transset_TC: fixes a :: "V" shows "Transset (TC a)"
1: shows "map_indets (the_inv_into X f) (map_indets f p) = p"
commute_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "commute R S = square R S S R"
ell2_norm_triangle_2: fixes x :: "'c \<Rightarrow> complex"   and y :: "'c \<Rightarrow> complex" assumes "has_ell2_norm x"   and "has_ell2_norm y" shows "ell2_norm (\<lambda>i. x i + y i) \<le> ell2_norm x + ell2_norm y"
3: shows "stake (length u) v \<preceq>\<^sub>F w (l (length u))"
diff_face_not_in_1: shows "a \<notin> G.face_cycle_set a'"
fold1_conv_fold: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and rbt :: "'b set_rbt" shows "fold1 f rbt = fold f (tl (RBT_Set2.keys rbt)) (hd (RBT_Set2.keys rbt))"
properties_lambda_gt_0: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b"   and lambda :: "nat"   and mu :: "nat" assumes "properties f x0 lambda mu" shows "0 < lambda"
sym_is_binary_functor: shows "binary_functor (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>B) F.sym"
K'_def: fixes s :: "'s" shows "K' s \<equiv> case map_of (map (\<lambda>(s, \<mu>). (s, pmf_of_alist \<mu>)) K) s of None \<Rightarrow> return_pmf (hd S) | Some s \<Rightarrow> s"
index_mat_swaprows_3: fixes k :: "nat"   and l :: "nat"   and A :: "'a mat" shows "dim_col (swaprows k l A) = dim_col A"
iimg_def: fixes B :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and a :: "'a" shows "iimg B f a \<equiv> if a \<in> f ` B then a else f a"
ocondition_NF_wp: fixes L :: "'c \<Rightarrow> bool"   and l :: "'c \<Rightarrow> 'd option"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and R :: "'c \<Rightarrow> bool"   and r :: "'c \<Rightarrow> 'd option"   and C :: "'c \<Rightarrow> bool" assumes "ovalidNF L l Q"   and "ovalidNF R r Q" shows "ovalidNF (\<lambda>s. if C s then L s else R s) (ocondition C l r) Q"
sup_matrix_def: fixes f :: "'c \<times> 'c \<Rightarrow> 'd"   and g :: "'c \<times> 'c \<Rightarrow> 'd" shows "f \<oplus> g = (\<lambda>e. f e \<squnion> g e)"
in_listsD: fixes xs :: "'b list"   and A :: "'b set" assumes "xs \<in> lists A" shows "\<forall>x\<in>set xs. x \<in> A"
coeff_in_ring: fixes a :: "'f \<Rightarrow> 'a"   and A :: "'f set"   and x :: "'f" assumes "a \<in> A \<rightarrow> carrier K"   and "x \<in> A" shows "a x \<in> carrier K"
intro: fixes \<Gamma> :: "'b Inference_System.inference set"   and I_of :: "'b literal multiset set \<Rightarrow> 'b set" assumes "Inference_System.counterex_reducing_inference_system \<Gamma> I_of" shows "standard_redundancy_criterion_counterex_reducing \<Gamma> I_of"
map_graph_def: fixes f :: "('e \<times> 'd) set"   and G :: "('c, 'e) labeled_graph" shows "map_graph f G = LG (on_triple f `` edges G) (f `` vertices G)"
simps_12: fixes var :: "nat"   and i :: "nat"   and ux :: "atom fm" shows "grab_atoms var (AllN i ux) = None"
simps_3: fixes l :: "'b \<Rightarrow> 'b"   and r :: "'b \<Rightarrow> 'b"   and s :: "'b" shows "right (tree_iterate l r s) = tree_iterate l r (r s)"
prime_sum_upto_altdef2: fixes f :: "nat \<Rightarrow> 'a"   and x :: "real" shows "prime_sum_upto f x = (\<Sum>p | prime p \<and> p \<le> nat \<lfloor>x\<rfloor>. f p)"
only_blue_eyes_leave: fixes n :: "nat"   and p :: "'person"   and w :: "'person \<Rightarrow> 'color" assumes "local.leaves n p w"   and "local.valid w" shows "w p = blue"
induct: fixes P1 :: "'b exp \<Rightarrow> bool"   and P2 :: "'b exp list \<Rightarrow> bool"   and exp_list :: "'b exp list" assumes "\<And>x. P1 (new x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (Cast x1 x2)"   and "\<And>x. P1 (Val x)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x3\<rbrakk> \<Longrightarrow> P1 (x1 \<guillemotleft>x2\<guillemotright> x3)"   and "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (x1:=x2)"   and "\<And>x1 x2 x3. P1 x1 \<Longrightarrow> P1 (x1\<bullet>x2{x3})"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2{x3} := x4)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P2 x3\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2(x3))"   and "\<And>x1 x2 x3. P1 x3 \<Longrightarrow> P1 {x1:x2; x3}"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P1 (if (x1) x2 else x3)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (while (x1) x2)"   and "\<And>x. P1 x \<Longrightarrow> P1 (throw x)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (try x1 catch(x2 x3) x4)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P2 exp_list"
scale_scale: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'d"   and b :: "'d"   and x :: "'e" assumes "module scale" shows "scale a (scale b x) = scale (a * b) x"
mu_le: shows "\<mu> c s v \<le> \<mu> c s u + ereal (c e)"
elements_four_block_mat: fixes A :: "'a mat"   and nr1 :: "nat"   and nc1 :: "nat"   and B :: "'a mat"   and nc2 :: "nat"   and C :: "'a mat"   and nr2 :: "nat"   and D :: "'a mat" assumes "A \<in> carrier_mat nr1 nc1"   and "B \<in> carrier_mat nr1 nc2"   and "C \<in> carrier_mat nr2 nc1"   and "D \<in> carrier_mat nr2 nc2" shows "elements_mat (four_block_mat A B C D) \<subseteq> elements_mat A \<union> elements_mat B \<union> elements_mat C \<union> elements_mat D"
distinct_13: fixes x21 :: "nat"   and x22 :: "nat"   and x31 :: "nat set"   and x32 :: "nat set"   and x33 :: "'a" shows "x21:deliver(x22) \<noteq> x31\<not>x32:arrive(x33)"
circ_plus_one: fixes x :: "'a" shows "x\<^sup>\<star> = (1::'a) \<squnion> x\<^sup>\<star>"
ta_3: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>a"
sep_add_commute: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and sep_disj :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.pre_sep_algebra plus zero sep_disj"   and "sep_disj x y" shows "plus x y = plus y x"
starP_star_n: fixes P :: "'b \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'b" shows "(*p* P) (star_n X) = (\<forall>\<^sub>F n in \<U>. P (X n))"
rel_generatI: fixes x :: "('g, 'h, 'i) generat"   and y :: "('j, 'k, 'l) generat"   and A :: "'g \<Rightarrow> 'j \<Rightarrow> bool"   and Out :: "'h \<Rightarrow> 'k \<Rightarrow> bool"   and R :: "'i \<Rightarrow> 'l \<Rightarrow> bool" assumes "is_Pure x = is_Pure y"   and "\<lbrakk>is_Pure x; is_Pure y\<rbrakk> \<Longrightarrow> A (generat.result x) (generat.result y)"   and "\<lbrakk>\<not> is_Pure x; \<not> is_Pure y\<rbrakk> \<Longrightarrow> Out (generat.output x) (generat.output y) \<and> R (continuation x) (continuation y)" shows "rel_generat A Out R x y"
satPB_e: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and parOf :: "'b \<Rightarrow> 'c list"   and D :: "univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi> :: "('a, 'b) lit list set" assumes "UM_Struct wtFsym wtPsym arOf parOf D intF intP" shows "U.Struct.satPB intF intP \<xi> \<Phi> = M.Struct.satPB intF intP \<xi> \<Phi>"
qe_inv: shows "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm bbs bs (qe p) = Ifm bbs bs (E p)"
rpr_less_not: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "rpr A r"   and "hasw [x, y] A"   and "\<not> x \<^bsub>r\<^esub>\<prec> y" shows "y \<^bsub>r\<^esub>\<preceq> x"
prob_space_def: fixes M :: "'c measure" shows "prob_space M \<equiv> finite_measure M \<and> prob_space_axioms M"
arith_extra_simps_18: fixes a :: "'a" shows "(0::'a) * a = (0::'a)"
combine_nrows_max: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and A :: "'c matrix"   and B :: "'d matrix" assumes "f (0::'c) (0::'d) = (0::'b)" shows "nrows (combine_matrix f A B) \<le> max (nrows A) (nrows B)"
simps_2: fixes cfg1 :: "('p, 'v, 's) configuration"   and p :: "'p"   and v :: "'v"   and cfg2 :: "('p, 'v, 's) configuration" shows "cfg1 \<turnstile> <p, v> \<mapsto> cfg2 = ((\<forall>s. (s = p \<longrightarrow> states cfg2 p = trans p (states cfg1 p) (Value v)) \<and> (s \<noteq> p \<longrightarrow> states cfg2 s = states cfg1 s)) \<and> enabled cfg1 <p, v> \<and> msgs cfg2 = sends p (states cfg1 p) (Value v) \<union># msgs cfg1 -# <p, v>)"
swap: fixes y :: "'c"   and x :: "'c"   and l :: "'c list" shows "y # x # l <~~> x # y # l"
set_mode_get_disconnected_nodes: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) document_ptr" assumes "l_set_mode_get_disconnected_nodes type_wf set_mode set_mode_locs get_disconnected_nodes get_disconnected_nodes_locs" shows "\<forall>w\<in>set_mode_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
intros_2: fixes Pa :: "complex mat"   and U :: "complex mat" assumes "ps_P.is_quantum_predicate Pa" shows "ps_P.hoare_partial (adjoint U * Pa * U) (Utrans U) Pa"
owns_unshared_share_acquired: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list"   and a :: "nat" assumes "sharing_consistent \<S> \<O> sb"   and "a \<in> \<O>"   and "a \<notin> all_unshared sb" shows "a \<in> dom (share sb \<S>) \<union> acquired True sb \<O>"
value_gt_def: fixes a :: "value option"   and b :: "value option" shows "value_gt a b \<equiv> MaybeBoolInt (\<lambda>x y. y < x) a b"
\<beta>_numbound0: fixes p :: "fm" assumes "iszlfm p" shows "\<forall>b\<in>set (\<beta> p). numbound0 b"
FG: shows "(F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) \<circ> (G::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) = (\<lambda>x::'a::refinement_lattice. (c::'a::refinement_lattice)\<^sup>\<star> \<sqinter> c ; x)"
dags_in_lower_levels: fixes t1 :: "dag"   and t2 :: "dag" assumes "t1 \<in> Dags (fnct ` Nodes n levellista) low high"   and "t2 \<in> Dags (fnct ` Nodes n levellista) low high" shows "P t1 t2 var"
msubst_lit': fixes y :: "'c"   and v :: "('e, 'd) uexpr" shows "\<guillemotleft>y\<guillemotright>\<lbrakk>x\<rightarrow>v\<rbrakk> = \<guillemotleft>y\<guillemotright>"
orderE: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b" obtains "a = a \<sqinter> b"
rel_envTI: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and M :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j" assumes "(R ===> M) f g" shows "local.rel_envT R M (EnvT f) (EnvT g)"
fds_const_uminus: fixes b :: "'a" shows "fds_const (- b) = - fds_const b"
wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s'_def: fixes ar :: "'b \<Rightarrow> nat"   and M :: "('b, 'c) Term.term set" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s' ar M \<equiv> \<forall>t\<in>M. wf\<^sub>t\<^sub>r\<^sub>m' ar t"
times_ms_aux_singleton: fixes c :: "'a"   and e :: "real"   and xs :: "('a \<times> real) msllist" shows "times_ms_aux (MSLCons (c, e) MSLNil) xs = scale_shift_ms_aux (c, e) xs"
summands_valid: fixes A :: "'c set"   and a :: "'c \<Rightarrow> 'a" assumes "A \<subseteq> carrier M"   and "a \<in> A \<rightarrow> carrier R" shows "\<forall>v\<in>A. a v \<odot>\<^bsub>M\<^esub> v \<in> carrier M"
owner_ii_used: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and A :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "(Owner n, \<lbrace>Num 1, PubKey A\<rbrace>) \<in> s \<longrightarrow> PriKey A \<in> used s"
trace_refine_def: fixes P :: "'a process"   and Q :: "'a process" shows "P \<sqsubseteq>\<^sub>T Q \<equiv> T Q \<subseteq> T P"
newton_sqrt_aux_simps_2: fixes x :: "nat"   and n :: "nat" assumes "x \<le> (x + n div x) div 2" shows "newton_sqrt_aux x n = x"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_on_left_moving (as, lm) (s, l, r) ires = (inv_on_left_moving_norm (as, lm) (s, l, r) ires \<or> inv_on_left_moving_in_middle_B (as, lm) (s, l, r) ires)"
c_def: fixes xs :: "'a list"   and fs :: "'a list"   and i :: "nat" shows "??.Newton_Interpolation.c xs fs i = ??.Newton_Interpolation.xij_f xs fs 0 i"
conj_comp_iff: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'b" shows "(\<lambda>x. P x \<and> Q x) \<circ> g = (\<lambda>x. (P \<circ> g) x \<and> (Q \<circ> g) x)"
Class_equivalence: fixes S :: "'c set"   and E :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "equivalence S E"   and "a \<in> S"   and "b \<in> S" shows "(equivalence.Class S E a = equivalence.Class S E b) = ((a, b) \<in> E)"
endo_inj_surj: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'c" assumes "finite A"   and "f ` A \<subseteq> A"   and "inj_on f A" shows "f ` A = A"
conf_hext: fixes h :: "'heap"   and h' :: "'heap"   and v :: "'addr val"   and T :: "ty" assumes "h \<unlhd> h'"   and "P,h \<turnstile> v :\<le> T" shows "P,h' \<turnstile> v :\<le> T"
Span_hom: fixes K :: "'a set"   and Us :: "'a list" assumes "K \<subseteq> carrier R"   and "set Us \<subseteq> carrier R" shows "S.Span (h ` K) (map h Us) = h ` R.Span K Us"
ivl_subset: fixes i :: "'a"   and j :: "'a"   and m :: "'a"   and n :: "'a" shows "({i..<j} \<subseteq> {m..<n}) = (j \<le> i \<or> m \<le> i \<and> j \<le> n)"
addN_def: fixes u :: "'g \<Rightarrow> ('node, 'val set) mapping"   and g :: "'g"   and n :: "'node" shows "uninst_code.ssa.addN u g n \<equiv> fold (\<lambda>v. Mapping.map_default v {} (insert n)) (sorted_list_of_set ((usesOf \<circ> u) g n))"
that: assumes "rtrancl_path r x (xs @ [y]) y"   and "rtrancl_path r y ys z" shows "thesis"
count_a: assumes "count Zs a \<le> count (mset xs) a" shows "\<exists>z. gt z a \<and> count (mset xs) z < count Zs z"
i_Exec_Stream_Acc_Output_eq_last_message_hold: fixes k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e" assumes "0 < k" shows "i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c = (output_fun \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) c) \<longmapsto> k \<div>\<^sub>l k"
Cignorabimus_init: fixes \<alpha> :: "game"   and V :: "variable set"   and x :: "variable" assumes "FVG \<alpha> \<subseteq> V"   and "x \<notin> V" shows "{x} \<in> Cignorabimus \<alpha> V"
subspace_span: fixes S :: "'d set" shows "csubspace (cspan S)"
COD_in_cod: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f" shows "monoidal_language.Cod C (free_monoidal_category.rep C f) \<in> partial_magma.cod (free_monoidal_category.comp C) f"
put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_put_ptrs: fixes document_ptr :: "(_) document_ptr"   and document :: "('o, 'm, 'n, 'v option) RDocument_scheme"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "put document_ptr document h = h'" shows "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|cast document_ptr|}"
minit_safe_def: fixes \<phi> :: "'b MFOTL.formula" shows "minit_safe \<phi> = (if mmonitorable_exec \<phi> then minit \<phi> else undefined)"
Tw_ord_inj: fixes E :: "('e, 'f) Order_scheme" assumes "Worder E"   and "\<forall>a\<in>carrier D. \<exists>b\<in>carrier E. ord_equiv (Iod D (segment D a)) (Iod E (segment E b))" shows "ord_inj D E (Tw\<^bsub>D,E\<^esub>)"
eq_norm: fixes x :: "int" shows "uint (word_of_int x) = take_bit LENGTH('b) x"
Prefixes_def: fixes A :: "'b list set" shows "Prefixes A = {w. \<exists>q. w @ q \<in> A}"
set_sort: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "set (linorder.sort_key less_eq f xs) = set xs"
spec_def: fixes x :: "'a"   and p :: "'a"   and q :: "'a" shows "(x \<le> R p q) = H p x q"
part_circlepath_def: fixes z :: "complex"   and r :: "real"   and s :: "real"   and t :: "real" shows "part_circlepath z r s t \<equiv> \<lambda>x. z + complex_of_real r * exp (\<i> * complex_of_real (linepath s t x))"
rel_refl: fixes Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a list" assumes "\<And>x. Ra x x" shows "list_all2 Ra x x"
ind_add_str_def: fixes V :: "'b set" shows "ind_add_str V \<equiv> add_str (X \<inter> V)"
comp_arr_inv: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "category C"   and "category.inverse_arrows C f g" shows "C f g = partial_magma.dom C g"
f0: shows "int (\<Sum>i<n. bin_rep n m ! i * 2 ^ (n - 1 - i)) = (\<Sum>i<n. int (m mod 2 ^ (n - i)) - int (m mod 2 ^ (n - 1 - i)))"
deliverT: fixes \<xi> :: "'e"   and l :: "'h"   and s\<^sub>d\<^sub>a\<^sub>t\<^sub>a :: "'e \<Rightarrow> nat"   and p :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" shows "((\<xi>, {l}deliver(s\<^sub>d\<^sub>a\<^sub>t\<^sub>a) . p), deliver (s\<^sub>d\<^sub>a\<^sub>t\<^sub>a \<xi>), \<xi>, p) \<in> seqp_sos \<Gamma>"
bin_to_bl_or: fixes n :: "nat"   and a :: "int"   and b :: "int" shows "bin_to_bl n (a OR b) = map2 (\<or>) (bin_to_bl n a) (bin_to_bl n b)"
OclIncludes_execute\<^sub>S\<^sub>e\<^sub>t: fixes X :: "'a state \<times> 'a state \<Rightarrow> Set(Set('b))"   and x :: "'a state \<times> 'a state \<Rightarrow> Set('b)"   and y :: "'a state \<times> 'a state \<Rightarrow> Set('b)" shows "X->including\<^sub>S\<^sub>e\<^sub>t(x)->includes\<^sub>S\<^sub>e\<^sub>t(y) = (if \<delta> X then if x \<doteq> y then true else X->includes\<^sub>S\<^sub>e\<^sub>t(y) endif else invalid endif)"
ln_chain_dominates: fixes n :: "nat"   and m :: "nat" assumes "n < m" shows "dominates at_top (ln ^^ n) (ln ^^ m)"
assms_1: shows "N \<lhd> G\<lparr>carrier := K\<rparr>"
S_5: shows "(\<Sum>v\<in>S. u v *s v) = (0::'b)"
\<Sigma>_def: fixes f :: "'b \<Rightarrow> 'a"   and j :: "int"   and l :: "int" shows "\<Sigma> f j l = (if j < l then sum (f \<circ> of_int) {j..<l} else if l < j then - sum (f \<circ> of_int) {l..<j} else (0::'a))"
distinct_disc_12: fixes event :: "('d, 'e, 'f) event" assumes "isSnapshot event" shows "\<not> isSend event"
le_add_same_cancel2: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq a (plus b a) = less_eq zero b"
consistent_witness: fixes A :: "'b set"   and S :: "(('c, 'b) fm list \<times> 'b) set"   and block :: "('c, 'b) fm list \<times> 'b" assumes "infinite UNIV"   and "consistent A S"   and "finite (\<Union> (block_nominals ` S))"   and "block \<in> S"   and "finite A" shows "consistent A ({witness block (A \<union> \<Union> (block_nominals ` S))} \<union> S)"
refl: fixes a :: "'c DBMEntry" shows "a \<preceq> a"
plossless_gpv_map_gpv': fixes b :: "ennreal"   and \<I> :: "('a, 'b) \<I>"   and f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'a"   and h :: "'b \<Rightarrow> 'f"   and gpv :: "('d, 'e, 'f) gpv" shows "pgen_lossless_gpv b \<I> (map_gpv' f g h gpv) = pgen_lossless_gpv b (map_\<I> g h \<I>) gpv"
scaleR_mono: fixes 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 scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "real"   and b :: "real"   and x :: "'c"   and y :: "'c" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "a \<le> b"   and "less_eq x y"   and "0 \<le> b"   and "less_eq zero x" shows "less_eq (scaleR a x) (scaleR b y)"
order_trans: fixes x :: "'c Word.word"   and y :: "'c Word.word"   and z :: "'c Word.word" assumes "x \<le>s y"   and "y \<le>s z" shows "x \<le>s z"
sameDom_def: fixes inp1 :: "'a \<Rightarrow> 'b option"   and inp2 :: "'a \<Rightarrow> 'c option" shows "sameDom inp1 inp2 \<equiv> \<forall>i. (inp1 i = None) = (inp2 i = None)"
super_class: fixes X :: "'b set"   and elem :: "'b"   and P :: "'b set set" assumes "X \<in> partition_without elem P" obtains Z :: "'b set" where "Z \<in> P" and "X = Z - {elem}"
DBM_reset_diag_preservation: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and i :: "nat"   and d :: "'c"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "\<forall>k\<le>n. M k k \<le> \<one>"   and "DBM_reset M n i d M'" shows "\<forall>k\<le>n. M' k k \<le> \<one>"
fppvs_def: shows "fppvs = take (length vs - length us) ppvs"