hcju/isabelleps huggingface dataset:
valid_subst: fixes \<Delta> :: "binding list"   and X :: "tyvrs"   and Q :: "ty"   and \<Gamma> :: "binding list"   and P :: "ty" assumes "\<turnstile> (\<Delta> @ TVarB X Q # \<Gamma>) ok"   and "P closed_in \<Gamma>" shows "\<turnstile> (\<Delta>[X \<mapsto> P]\<^sub>e @ \<Gamma>) ok"
closed_subspace_closed_sum: fixes A :: "'d set"   and B :: "'d set" assumes "csubspace A"   and "csubspace B" shows "closed_csubspace (A +\<^sub>M B)"
sorted_snd_alist_of: fixes q :: "('d, 'c) pq" shows "sorted (map snd (alist_of q))"
upred_eq_false: fixes p :: "'d upred" shows "(p =\<^sub>u false) = (\<not> p)"
path_component_of_subset_topspace: fixes X :: "'b topology"   and x :: "'b" shows "path_component_of_set X x \<subseteq> topspace X"
cases: fixes x :: "'a gState_scheme" obtains v :: "(String.literal, variable) assoc_list" and ch :: "channel list" and t :: "bool" and p :: "pState list" and m :: "'a" where "x = \<lparr>gState.vars = v, channels = ch, timeout = t, procs = p, \<dots> = m\<rparr>"
spoly_same: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "punit.spoly p p = 0"
pickE_def: fixes rho :: "'d \<Rightarrow> 'e \<Rightarrow> 'f set option" shows "pickE rho \<equiv> \<lambda>xs. lift pick (rho xs)"
string_binop_type_det: fixes op :: "string_binop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'b OCL_Types.type"   and \<rho>\<^sub>1 :: "'c OCL_Types.type"   and \<rho>\<^sub>2 :: "'c OCL_Types.type" assumes "string_binop_type op \<tau> \<sigma> \<rho>\<^sub>1"   and "string_binop_type op \<tau> \<sigma> \<rho>\<^sub>2" shows "\<rho>\<^sub>1 = \<rho>\<^sub>2"
prob_with_noop_def: fixes \<Pi> :: "('a, 'b, 'c) sas_plus_problem_scheme" shows "prob_with_noop \<Pi> \<equiv> \<lparr>sas_plus_problem.variables_of = \<Pi>\<^sub>\<V>\<^sub>+, operators_of = empty_sasp_action # \<Pi>\<^sub>\<O>\<^sub>+, initial_of = \<Pi>\<^sub>I\<^sub>+, goal_of = \<Pi>\<^sub>G\<^sub>+, range_of = range_of \<Pi>\<rparr>"
f_join_aux_nth: fixes i :: "nat"   and I :: "nat set"   and n :: "nat"   and xs :: "'b list" assumes "i < card (I \<inter> {n..<n + length xs})" shows "f_join_aux xs n I ! i = xs ! (((I \<inter> {n..<n + length xs}) \<rightarrow> i) - n)"
INF_eq_bot_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "(Inf (f ` A) = bot) = (\<forall>x. less bot x \<longrightarrow> (\<exists>i\<in>A. less (f i) x))"
Domain_unfold: fixes r :: "('c \<times> 'd) set" shows "Domain r = {x. \<exists>y. (x, y) \<in> r}"
finite_measure_Union: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> sets M"   and "B \<in> sets M"   and "A \<inter> B = {}" shows "Sigma_Algebra.measure M (A \<union> B) = Sigma_Algebra.measure M A + Sigma_Algebra.measure M B"
convex_closure: fixes S :: "'a set" assumes "convex S" shows "convex (closure S)"
Controllable_Interruptible_Separation_Kernel_axioms: shows "Controllable_Interruptible_Separation_Kernel kstep output_f s0 current cswitch kinvolved ifp vpeq AS_set invariant AS_precondition aborting waiting set_error_code"
close_app_neq: fixes A :: "'a list"   and x :: "'a"   and a :: "'a"   and B :: "'a list" assumes "List.member A x"   and "x \<noteq> a" shows "close (A @ B) a \<noteq> B"
invmfun_bijec: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N"   and "bijec\<^bsub>M,N\<^esub> f" shows "bijec\<^bsub>N,M\<^esub> invmfun R M N f"
infnorm_set_image: fixes x :: "'b" shows "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} = (\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
red_Nml: fixes a :: "'a local.term" assumes "Nml a" shows "a\<^bold>\<down> = a"
simps_1: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "tMap f (tInit x) = tInit (f x)"
cong_diff_iff_cong_0_nat': fixes a :: "nat"   and b :: "nat"   and m :: "nat" shows "[nat \<bar>int a - int b\<bar> = 0] (mod m) = [a = b] (mod m)"
ce_rel_to_set_srj: shows "surj ce_rel_to_set"
alphaAbs_all_qFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" shows "alphaAbs_all_qFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<forall>y. qFresh xs y X \<and> qFresh xs y X' \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
restrict_map_def: fixes m :: "'e \<Rightarrow> 'f option"   and A :: "'e set" shows "m |` A = (\<lambda>x. if x \<in> A then m x else None)"
P2: shows "Col A B X \<and> A B Perp C X"
INF_le_SUP: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<noteq> {}" shows "less_eq (Inf (f ` A)) (Sup (f ` A))"
logD_def: fixes fs :: "int vec list" shows "logD fs = (if \<alpha> = 4 / 3 then D fs else nat \<lfloor>log (1 / real_of_rat reduction) (real (D fs))\<rfloor>)"
d1_2: fixes K :: "'a set" assumes "K \<in> p1" shows "K \<subseteq> s1"
assms_2: shows "fun_eq_on f g (\<Union> X)"
closed_add_rng_of_frac: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" assumes "(r, s) \<in> carrier rel"   and "(r', s') \<in> carrier rel" shows "(r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s') \<in> set_class_of\<^bsub>rel\<^esub>"
minus_mat_def: fixes A :: "'a mat"   and B :: "'a mat" shows "A - B \<equiv> mat (dim_row B) (dim_col B) (\<lambda>ij. A $$ ij - B $$ ij)"
setVar'_gState_inv: fixes prog :: "program"   and g :: "'a gState_scheme"   and gl :: "bool"   and v :: "String.literal"   and idx :: "integer option"   and val :: "integer"   and p :: "pState" assumes "gState_inv prog g" shows "gState_inv prog (fst (setVar' gl v idx val g p))"
spies_partsEs_1: fixes A :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Says A B X \<in> set evs" obtains "X \<in> parts (knows Spy evs)"
equivalence_functor_axioms: shows "equivalence_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) (\<lambda>f. f \<otimes>\<^sub>C C.cod \<iota>\<^sub>C)"
map2_map_map: fixes h :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and f :: "'g \<Rightarrow> 'e"   and xs :: "'g list"   and g :: "'g \<Rightarrow> 'f" shows "map2 h (map f xs) (map g xs) = map (\<lambda>x. h (f x) (g x)) xs"
axioms_1: fixes L :: "('a, 'b) gorder_scheme" assumes "weak_partial_order L" shows "equivalence L"
bvars\<^sub>s\<^sub>s\<^sub>t_Cons: fixes a :: "('b, 'a) stateful_strand_step"   and A :: "('b, 'a) stateful_strand_step list" shows "bvars\<^sub>s\<^sub>s\<^sub>t (a # A) = set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p a) \<union> bvars\<^sub>s\<^sub>s\<^sub>t A"
R0_def: shows "R0 = {(i1, i2). \<exists>\<iota>' \<iota>'' b' b'' c1' c1'' c2' c2''. i1 = [if\<^bsub>\<iota>'\<^esub> b' then c1' else c2' fi] \<and> i2 = [if\<^bsub>\<iota>''\<^esub> b'' then c1'' else c2'' fi] \<and> \<iota>' \<notin> set (PPc c1) \<union> set (PPc c2) \<and> \<iota>'' \<notin> set (PPc c1) \<union> set (PPc c2) \<and> ([c1'], [c1'']) \<in> R1 \<and> ([c2'], [c2'']) \<in> R2 \<and> b' \<equiv>\<^bsub>d\<^esub> b''}"
x1_ge: fixes b :: "real" assumes "b \<in> set bs" shows "2 * x\<^sub>0 * inverse b \<le> x\<^sub>1"
tAct_3: shows "\<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2)"
bags_union: shows "\<Union> {bag t |t. t \<in> V\<^bsub>T\<^esub>} = V"
spec_ops_ref_less: fixes xs :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set"   and oid :: "'c"   and oper :: "'d"   and r :: "'c" assumes "spec_ops xs deps"   and "(oid, oper) \<in> set xs"   and "r \<in> deps oper" shows "r < oid"
L_infinity_zero_space: fixes M :: "'b measure" shows "zero_space\<^sub>N (\<LL> \<infinity> M) = {f \<in> borel_measurable M. AE x in M. f x = 0}"
refl_onD: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and a :: "'a" assumes "refl_on A r"   and "a \<in> A" shows "a \<^bsub>r\<^esub>\<preceq> a"
supto_bottom_iff: fixes i :: "Integer"   and j :: "Integer" shows "(supto\<cdot>i\<cdot>j = \<bottom>) = (i = \<bottom> \<or> j = \<bottom>)"
fds_deriv_fds_integral: fixes f :: "'b fds"   and c :: "'b" assumes "fds_nth f (Suc 0) = (0::'b)" shows "fds_deriv (fds_integral c f) = f"
mconn_sym: fixes E :: "('b \<times> 'b) set" shows "mconn E = (mconn E)\<inverse>"
cut_cut': fixes S :: "'v vertex set" assumes "MFMC_Network.cut \<Delta>'' S" shows "MFMC_Network.cut \<Delta> (Vertex -` S)"
subset: shows "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set (reduction_signs_R p qs signs subsets (M_mat_R signs subsets))"
actHn: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "Blockchain active cmp pin pout bc mining honest cb" shows "\<exists>nid. honest nid \<and> active nid (t n) \<and> active nid (t (Suc n))"
lcons_correct: fixes a :: "'a \<times> 'b"   and t :: "('a, 'b) FingerTree" shows "FingerTree.toList (FingerTree.lcons a t) = a # FingerTree.toList t"
assms_2: assumes "ide a"   and "src a = a"   and "trg a = a" shows "T"
domD: fixes a :: "'e"   and m :: "'e \<Rightarrow> 'f option" assumes "a \<in> dom m" shows "\<exists>b. m a = Some b"
fmrestrict_fset_dom: fixes m :: "'e \<rightharpoonup>\<^sub>f 'f" shows "fmrestrict_fset (fmdom m) m = m"
continuous_on_flow0: fixes x0 :: "'a" shows "continuous_on (existence_ivl0 x0) (flow0 x0)"
real_image_ereal_ivl: fixes a :: "ereal"   and b :: "ereal" shows "real_of_ereal ` {a<..<b} = (if a < b then if a = - \<infinity> then if b = \<infinity> then UNIV else {..<real_of_ereal b} else if b = \<infinity> then {real_of_ereal a<..} else {real_of_ereal a<..<real_of_ereal b} else {})"
im_sheaf_axioms: shows "Comm_Ring.im_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S' is_open' f"
Cons: shows "monotone option.le_fun option_ord (\<lambda>f. mapM (\<lambda>y. C y f) B)"
tensor_vec_dim: fixes dims :: "nat list"   and vars :: "nat set"   and v1 :: "'a vec"   and v2 :: "'a vec" shows "dim_vec (partial_state.tensor_vec dims vars v1 v2) = state_sig.d dims"
iSince_def: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "iSince I P Q \<equiv> \<diamond> t I. Q t \<and> (\<box> t' I \<down>> t. P t')"
hyps_4: shows "roomk s r1 \<in> {k, k1}"
sa_iso_left_invertible: fixes x :: "'c" shows "sa_iso_inv (sa_iso x) = x"
assms: shows "z \<in> set (zippers_view (p, t))"
qSwapInp_preserves_qGoodInp: fixes qinp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm option"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "qGoodInp qinp" shows "qGoodInp (qSwapInp xs x1 x2 qinp)"
D_diag_nonnegI: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat"   and i :: "nat" assumes "cycle_free M n"   and "i \<le> n" shows "(0::'b) \<le> D M i i n"
mono_Span_append_1: fixes Us :: "'a list"   and Vs :: "'a list" assumes "set Us \<subseteq> carrier R"   and "set Vs \<subseteq> carrier R" shows "Span K Us \<subseteq> Span K (Us @ Vs)"
case_4: fixes f1 :: "bool \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and f2 :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b"   and f3 :: "'c \<Rightarrow> 'c \<Rightarrow> instr list \<Rightarrow> 'b"   and f4 :: "nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and x41 :: "nat set"   and x42 :: "nat set"   and x43 :: "nat set"   and x44 :: "nat set" shows "(case Ghost\<^sub>s\<^sub>b x41 x42 x43 x44 of Write\<^sub>s\<^sub>b x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Read\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f2 x xa xb xc | Prog\<^sub>s\<^sub>b x xa xb \<Rightarrow> f3 x xa xb | Ghost\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f4 x xa xb xc) = f4 x41 x42 x43 x44"
semiring_0_ow_axioms: shows "semiring_0_ow U (+\<^sub>o\<^sub>w) 0\<^sub>o\<^sub>w (*\<^sub>o\<^sub>w)"
divide_right_mono: 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"   and b :: "'c"   and c :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq a b"   and "less_eq zero c" shows "less_eq (divide a c) (divide b c)"
Ints_minus: fixes a :: "'a" assumes "a \<in> \<int>" shows "- a \<in> \<int>"
cases: fixes x :: "('a, 'b) tm" obtains uu_ :: "('a, 'b) hd" where "x = Hd uu_"   | s :: "('a, 'b) tm" and t :: "('a, 'b) tm" where "x = App s t"
insert_def: fixes x :: "'b"   and t :: "('b \<times> nat) tree" shows "insert x t = merge \<langle>\<langle>\<rangle>, (x, 1), \<langle>\<rangle>\<rangle> t"
WCFG_WhileBodyExit: fixes c' :: "cmd"   and n :: "w_node"   and et :: "(char list \<Rightarrow> val option) edge_kind"   and b :: "expr" assumes "c' \<turnstile> n -et\<rightarrow> (_Exit_)"   and "n \<noteq> (_Entry_)" shows "while (b) c' \<turnstile> n \<oplus> 2 -et\<rightarrow> (_ 0 _)"
drop_all_iff: fixes n :: "nat"   and xs :: "'b list" shows "(xs \<up> n = []) = (length xs \<le> n)"
o: shows "A *v x - b \<in> orthogonal_complement (col_space A)"
bot_apply: fixes x :: "'c" shows "\<bottom> x = \<bottom>"
sem_5bd2: fixes X :: "w \<Rightarrow> bool"   and Y :: "w \<Rightarrow> bool"   and Z :: "w \<Rightarrow> bool" shows "ob X Y \<and> X \<sqsubseteq> Z \<longrightarrow> ob Z ((Z \<sqinter> (\<sim>X)) \<squnion> Y)"
concat_map_foldr: fixes f :: "'e \<Rightarrow> 'd list"   and l :: "'e list" shows "concat (map f l) = foldr (\<lambda>x. (@) (f x)) l []"
Inf_eq_Sup: fixes A :: "'a set" shows "\<Sqinter>A = \<Squnion>{b. \<forall>a\<in>A. b \<le> a}"
g_nonneg: fixes i :: "nat"   and x :: "'a" assumes "i \<in> N"   and "x \<in> carrier" shows "0 \<le> g i x"
holomorphic_on_exp: fixes s :: "complex set" shows "exp holomorphic_on s"
R'_eq_R: shows "H.R = (\<lambda>f. f \<otimes> \<I>)"
disc_1: fixes x :: "('f, 'g, 'h) Resumption.resumption"   and f :: "'f \<Rightarrow> ('i, 'g, 'h) Resumption.resumption" assumes "Resumption.resumption.is_Done x"   and "Resumption.resumption.result x \<noteq> None \<longrightarrow> Resumption.resumption.is_Done (f (the (Resumption.resumption.result x)))" shows "Resumption.resumption.is_Done (x \<bind> f)"
true: assumes "do_if v e1 e2 = Some e1" shows "thesis"
Entry_empty: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and Def :: "'h \<Rightarrow> 'i set"   and Use :: "'h \<Rightarrow> 'i set"   and ParamDefs :: "'h \<Rightarrow> 'i list"   and ParamUses :: "'h \<Rightarrow> 'i set list" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Def Use ParamDefs ParamUses" shows "Def Entry = {} \<and> Use Entry = {}"
hyps: shows "length (blockmat ls) = 2 ^ nat (codomain_block ls)"
transfer: shows "rel_fun (=) (pcr_ell2 (=)) (\<lambda>x y. if x = y then 1 else 0) ket"
da_NewC: fixes A :: "assigned"   and B :: "lname set"   and Env :: "env"   and C :: "qtname" assumes "A = \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>" shows "Env\<turnstile> B \<guillemotright>\<langle>NewC C\<rangle>\<guillemotright> A"
runit_def: fixes f :: "'a" shows "Right.runit f \<equiv> THE \<mu>. Right.in_hom \<mu> (Right.R f) f \<and> Right.R \<mu> = Right.comp (f \<star> \<i>[src f]) \<a>[f, src f, src f]"
filterlim_sup1: fixes f :: "'a \<Rightarrow> 'b"   and G1 :: "'b filter"   and F :: "'a filter"   and G2 :: "'b filter" assumes "filterlim f G1 F" shows "filterlim f (sup G1 G2) F"
connected_complement_bounded_convex: fixes S :: "'a set" assumes "bounded S"   and "convex S"   and "2 \<le> DIM('a)" shows "connected (- S)"
I_out_cb_def: fixes s :: "'a list" shows "I_out_cb s \<equiv> \<lambda>(\<ff>s, i, j). length s + 1 = length \<ff>s \<and> (\<forall>jj<j. \<ff>s ! jj = \<ff> s jj) \<and> \<ff>s ! (j - 1) = i \<and> 0 < j"
terminal_char1: fixes t :: "'s" shows "terminal t = (ide t \<and> (\<exists>!x. x \<in> local.set t))"
flatten_set_group: fixes G :: "('c set, 'd) monoid_scheme"   and rep :: "'c set \<Rightarrow> 'e" assumes "Group.group G"   and "inj_on rep (carrier G)" shows "Group.group (flatten G rep)"
finprod_minus: fixes a :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "a \<in> A"   and "f \<in> A \<rightarrow> carrier G"   and "finite A" shows "finprod G f A = f a \<otimes> finprod G f (A - {a})"
value_less_eq1: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and y :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "y \<in> carrier K"   and "v x < v y" shows "v x = v (y \<plusminus> x)"
zmset_of_inject: fixes M :: "'b multiset"   and N :: "'b multiset" shows "(zmset_of M = zmset_of N) = (M = N)"
bdd_finite_1: fixes A :: "'d multiset set" assumes "finite A" shows "subset_mset.bdd_above A"
inject_9: fixes x111 :: "('a, 'b, 'c) com"   and x112 :: "('a, 'b, 'c) com"   and y111 :: "('a, 'b, 'c) com"   and y112 :: "('a, 'b, 'c) com" shows "(Catch x111 x112 = Catch y111 y112) = (x111 = y111 \<and> x112 = y112)"
pairwise_empty: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "pairwise P {}"
lead_coeff_list_def: fixes p :: "'a poly" shows "lead_coeff p = (if coeffs p = [] then 0::'a else last (coeffs p))"
shadow_root_eq_h: fixes character_data_ptr :: "(_) element_ptr"   and shadow_root_opt :: "(_) shadow_root_ptr option" shows "h \<turnstile> get_shadow_root character_data_ptr \<rightarrow>\<^sub>r shadow_root_opt = h' \<turnstile> get_shadow_root character_data_ptr \<rightarrow>\<^sub>r shadow_root_opt"
divide_nonneg_nonpos: fixes x :: "'d"   and y :: "'d" assumes "(0::'d) \<le> x"   and "y \<le> (0::'d)" shows "x / y \<le> (0::'d)"
RSpan_single: fixes m :: "'m" shows "RSpan [m] = {r \<cdot> m |r. r \<in> R}"
lift_Def_setp_lift_Def_set_eq: fixes Def :: "'f \<Rightarrow> 'g set"   and E :: "'f"   and X :: "'f"   and H :: "'g set"   and L :: "'g set" shows "lift_Def_setp Def E X (\<lambda>x. x \<in> H) L = (\<lambda>x xa. (x, xa) \<in> lift_Def_set Def E X H L)"
R1_def: fixes x :: "nat"   and y :: "nat" shows "R1 x y = TI \<bind> (\<lambda>((c1, d1), c2, d2). let e2 = (x + c1) mod q; e1 = (y + (q - c2)) mod q; s1 = (x * e1 + (q - d1)) mod q; s2 = (e2 * c2 + (q - d2)) mod q in return_spmf ((x, c1, d1, e1), s1, s2))"
lessThan_def: fixes u :: "'a" shows "{..<u} \<equiv> {x. x < u}"
f: shows "f \<in> [T]\<^bsup>k\<^esup> \<rightarrow> D"
simps_10: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and f3 :: "'c \<Rightarrow> 'a"   and x1 :: "'b" shows "(case Oldf x1 of Oldf x \<Rightarrow> f1 x | Tag x \<Rightarrow> f2 x | Wit x \<Rightarrow> f3 x) = f1 x1"
assms: shows "f \<in> eq_upto (BAD P) g i"
lle: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "l j \<le> l i"
less_pow2_len: fixes x :: "nat" shows "x < 2 ^ len x"
supp_convolution_subset_sum_supp: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "supp (convolution f g) \<subseteq> supp f + supp g"
stirling_weak: fixes x :: "real" assumes "1 \<le> x" shows "sum_upto (\<lambda>x. ln (real x)) x \<in> {x * ln x - x - ln x + 1..x * ln x}"
extensional_def: fixes A :: "'g set" shows "extensional A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = undefined}"
sorted_list_of_map_def: fixes m :: "'a \<Rightarrow> 'c option" shows "sorted_list_of_map m \<equiv> map (\<lambda>k. (k, the (m k))) (sorted_list_of_set (dom m))"
continuous_map_closedin_preimage_eq: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "continuous_map X Y f = (f ` topspace X \<subseteq> topspace Y \<and> (\<forall>U. closedin Y U \<longrightarrow> closedin X (topspace X \<inter> f -` U)))"
distinct_33: fixes x41 :: "'b"   and x42 :: "PreSimplyTyped.type"   and x43 :: "'b ptrm"   and x6 :: "'b ptrm" shows "PFn x41 x42 x43 \<noteq> PFst x6"
closed_except_simps_1: fixes x :: "name"   and S :: "name fset" shows "closed_except (Svar x) S = (x |\<in>| S)"
simps: fixes a1 :: "'b list"   and a2 :: "'c list"   and r :: "('b \<times> 'c) set" shows "((a1, a2) \<in> listrel r) = (a1 = [] \<and> a2 = [] \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> (x, y) \<in> r \<and> (xs, ys) \<in> listrel r))"
preserves_ide: fixes a :: "'c" assumes "C.ide a" shows "C.ide (\<I>\<^sub>C \<otimes>\<^sub>C a)"
east_west_tangents_distinct: shows "polar east \<noteq> polar west"
measure_empty: fixes M :: "'c measure" shows "Sigma_Algebra.measure M {} = 0"
dom_cod: fixes f :: "'a arrow_of_spans_data" assumes "V.arr f" shows "V.dom (V.cod f) = V.cod f"
keys_add: fixes f :: "'a \<Rightarrow>\<^sub>0 'b"   and g :: "'a \<Rightarrow>\<^sub>0 'b" shows "keys (f + g) \<subseteq> keys f \<union> keys g"
rtrancl_into_trancl2: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> r"   and "(b, c) \<in> r\<^sup>*" shows "(a, c) \<in> r\<^sup>+"
valid_edges: fixes nodes :: "'a stream"   and inPorts :: "'a \<Rightarrow> 'b fset"   and outPorts :: "'a \<Rightarrow> 'c fset"   and vertices :: "'d fset"   and nodeOf :: "'d \<Rightarrow> 'a"   and edges :: "(('d \<times> 'c) \<times> 'd \<times> 'b) set" assumes "Port_Graph nodes inPorts outPorts vertices nodeOf edges" shows "\<forall>(ps1, ps2)\<in>edges. Vertex_Graph.valid_out_port outPorts vertices nodeOf ps1 \<and> Vertex_Graph.valid_in_port inPorts vertices nodeOf ps2"
simps_2: fixes prec :: "nat"   and n :: "nat"   and a :: "floatarith"   and f :: "form"   and bs :: "float interval option list"   and ss :: "nat list" shows "approx_form prec (Assign (Var n) a f) bs ss = (case approx prec a bs of None \<Rightarrow> False | Some ivl \<Rightarrow> approx_form' prec f (ss ! n) n ivl bs ss)"
h_def: fixes "is" :: "int list" shows "h is = sum_list (concat (map (\<lambda>i. if 0 < i then [i] else []) is))"
integrable_pmf: fixes X :: "'a set"   and M :: "'a pmf" shows "integrable (count_space X) (pmf M)"
subset_oc_segment: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "({a<--<b} \<subseteq> {c--d}) = (a = b \<or> a \<in> {c--d} \<and> b \<in> {c--d})"
S_transform_weak_transition_iff: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and P :: "'a"   and \<alpha> :: "'c"   and P' :: "'a" assumes "weak_nominal_ts satisfies transition \<tau>" shows "weak_nominal_ts.weak_transition (nominal_ts.S_transition satisfies transition) (S_action.Act \<tau>) P (S_action.Act \<alpha>) P' = weak_nominal_ts.weak_transition transition \<tau> P \<alpha> P'"