hcju/isabelleps huggingface dataset:
hyps_1: shows "x \<Rightarrow>\<^sub>d\<^sup>* y"
in_next_plane0I: fixes g' :: "graph"   and n :: "nat"   and v :: "nat"   and f :: "face"   and g :: "graph"   and p :: "nat" assumes "g' \<in> set (generatePolygon n v f g)"   and "f \<in> set (nonFinals g)"   and "v \<in> \<V> f"   and "3 \<le> n"   and "n < 4 + p" shows "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"
at_set_avoiding3: fixes xs :: "atom set"   and c :: "'e"   and x :: "'f" assumes "finite xs"   and "finite (supp c)"   and "finite (supp x)"   and "xs \<sharp>* x" shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p \<and> supp p = xs \<union> p \<bullet> xs"
omega_pres_conj: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<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 dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra_fusion one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> conjunctive.conjunctive inf inf times" shows "omega x \<in> conjunctive.conjunctive inf inf times"
ring_hom_one: fixes h :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme" assumes "h \<in> ring_hom R S" shows "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>"
vconn_implies_hpar_2: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>C g\<guillemotright>" shows "trg\<^sub>C \<mu> = trg\<^sub>C f"
count_filter_mset: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c multiset"   and a :: "'c" shows "count (filter_mset P M) a = (if P a then count M a else 0)"
lift_def: fixes f :: "'b \<Rightarrow> bool"   and w :: "(bool \<times> 'b) list" shows "C2.lift f w = C2.m_concat (map (C2.lift_gi f) w)"
strict_iff_order: fixes a :: "'g multiset"   and b :: "'g multiset" shows "(a \<subset># b) = (a \<subseteq># b \<and> a \<noteq> b)"
maximal_ideal_proper: fixes mx :: "'a set" assumes "maximal_ideal R mx" shows "1\<^sub>r \<notin> mx"
chine_composite_1: shows "HHH\<mu>\<nu>\<pi>\<rho>.chine = \<pi>.leg0 \<cdot> \<mu>\<nu>_\<pi>.prj\<^sub>0 \<down>\<down> \<rho>.leg1"
filterlim_at_top: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "filterlim f at_top F = (\<forall>Z. \<forall>\<^sub>F x in F. Z \<le> f x)"
right_total_alt_def: fixes R :: "'i \<Rightarrow> 'j \<Rightarrow> bool" shows "right_total R = ((=) \<le> R\<inverse>\<inverse> OO R)"
poly_0_coeff_0: fixes p :: "'a poly" shows "poly p (0::'a) = constant_term p"
Max_insert2: fixes A :: "'a DBMEntry set"   and a :: "'a DBMEntry" assumes "finite A"   and "\<And>b. b \<in> A \<Longrightarrow> b \<preceq> a" shows "linordered_monoid.Max (insert a A) = a"
Some: fixes sa :: "'s"   and s'' :: "'s'" assumes "(sa, None) \<in> terminates_on local.flatten_raw" shows "(sa, Some s'') \<in> terminates_on local.flatten_raw"
subspaceI: fixes S :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "0 \<in> S"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x + y \<in> S"   and "\<And>c x. x \<in> S \<Longrightarrow> c \<cdot> x \<in> S" shows "phull.subspace S"
Cons: fixes xs :: "char list exp list"   and ys :: "val list"   and y :: "val" assumes "map_val xs ys" shows "map_val (Val y # xs) (y # ys)"
Gauss_Jordan_iarrays_def: fixes A :: "'b iarray iarray" shows "Gauss_Jordan_iarrays A = Gauss_Jordan_upt_k_iarrays A (ncols_iarray A - 1)"
vconverse_atI: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"   and "\<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> = b" shows "\<langle>b, a\<rangle> \<in>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>)\<inverse>\<^sub>\<circ>"
list_all2_list_all: fixes f :: "'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list" shows "list_all2 (\<lambda>x. f) xs ys = (length xs = length ys \<and> list_all f ys)"
alist_split_1: shows "val = fst \<circ> alist"
VCG: fixes sys :: "('b, 'c, 'a, 'd, 'e, 'f) pre_system_scheme"   and I :: "('b, 'c, 'a, 'd, 'e) system_state \<Rightarrow> bool" assumes "\<forall>s. initial_state sys s \<longrightarrow> I \<lparr>GST = s, HST = []\<rparr>"   and "\<And>p. \<forall>(c, aft)\<in>vcg_fragments (PGMs sys p). PGMs sys, p, aft \<turnstile> \<lbrace>I\<rbrace> c" shows "sys \<Turnstile>\<^bsub>pre\<^esub> I"
quasi_inverses_symmetric: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.quasi_inverses f g" shows "Span.quasi_inverses g f"
sorted_WP2: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and l :: "(int \<times> int \<times> protocol) set set set list" assumes "FWNormalisationCore.sorted p l"   and "all_in_list p l"   and "distinct p"   and "allNetsDistinct p"   and "singleCombinators p" shows "wellformed_policy2Pr p"
ListUpdate_E: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "('a \<times> 'b) list" shows "ListUpdateE f xs = ListUpdate f xs (\<lambda>e x. e)"
wf_eq_check_append: fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and t :: "('d, 'c) Term.term"   and t' :: "('d, 'c) Term.term"   and S' :: "('d, 'c) strand_step list" assumes "wf\<^sub>s\<^sub>t V (S @ \<langle>check: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S')" shows "wf\<^sub>s\<^sub>t V (S @ S')"
P1_p2D: fixes x :: "'a \<times> 'b \<Rightarrow>\<^sub>L 'a" assumes "x \<in> P1" shows "fst x \<notin> p2"
unpadr_def: fixes m :: "nat"   and v :: "'c vec" shows "unpadr m v \<equiv> vec (dim_vec v - m) (($) v)"
rw: shows "snd (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k]) = fst (snd (foldl echelon_form_of_column_k_iarrays (matrix_to_iarray A, 0, bezout) [0..<Suc k]))"
simE_1: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and a :: "subject"   and x :: "name"   and Q' :: "pi" assumes "P \<leadsto>\<^sub>l[Rel] Q"   and "Q \<longmapsto>\<^sub>l a\<guillemotleft>x\<guillemotright> \<prec> Q'"   and "x \<sharp> P" shows "\<exists>P'. P \<longmapsto>\<^sub>l a\<guillemotleft>x\<guillemotright> \<prec> P' \<and> derivative P' Q' a x Rel"
compress_irule_set_eq: fixes rs :: "(name \<times> (Term_Class.term list \<times> pterm) fset) fset" shows "compress_irule_set rs = rs"
sub_polyfun: fixes a :: "nat \<Rightarrow> 'b"   and x :: "'b"   and n :: "nat"   and y :: "'b" shows "(\<Sum>i\<le>n. a i * x ^ i) - (\<Sum>i\<le>n. a i * y ^ i) = (x - y) * (\<Sum>j<n. \<Sum>k = Suc j..n. a k * y ^ (k - Suc j) * x ^ j)"
residue_field_cd: fixes R :: "('c, 'd) Ring_scheme"   and mx :: "'c set" assumes "Ring R"   and "maximal_ideal R mx" shows "Corps (R /\<^sub>r mx)"
RETURN_def: fixes x :: "'b" shows "RETURN x \<equiv> RES {x}"
type_less_x_Integer: fixes \<tau> :: "'a basic_type" assumes "\<tau> < Integer" obtains "\<tau> = OclVoid"   | "\<tau> = UnlimitedNatural"
MGT_implies_complete: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and P :: "'d set"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>Z. \<Gamma>,{}\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>c \<down> Normal s} c {t. \<Gamma>\<turnstile> \<langle>c,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>c,Normal Z\<rangle> \<Rightarrow> Abrupt t}"   and "\<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,{}\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
select_convs_1: fixes carrier :: "'c set"   and more :: "'d" shows "carrier \<lparr>carrier = carrier, \<dots> = more\<rparr> = carrier"
llist_all2_conv_lzip: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist" shows "llist_all2 P xs ys = (llength xs = llength ys \<and> (\<forall>(x, y)\<in>lset (lzip xs ys). P x y))"
inj_on_minus_set: fixes A :: "'b set" shows "inj_on ((-) A) (Pow A)"
zerocomb: shows "lincomb c (W \<union> X) = 0\<^sub>v n"
i: shows "mset_set (g ` (E - {x} \<union> {x})) = mset_set (g ` (E - {x}) \<union> {g x})"
total_on_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "total_on A r = (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> x \<succeq>[r] y \<or> y \<succeq>[r] x)"
transitive_def: fixes P :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "transitive P = (\<forall>x y z. P x y \<longrightarrow> P y z \<longrightarrow> P x z)"
waveD_OUT: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "wave \<Gamma> f"   and "x \<notin> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)" shows "d_OUT f x = 0"
emeasure_count_space: fixes X :: "'c set"   and A :: "'c set" assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then of_nat (card X) else \<infinity>)"
widen_Array_Array: fixes G :: "prog"   and T :: "ty"   and T' :: "ty" assumes "G\<turnstile>T.[]\<preceq>T'.[]" shows "G\<turnstile>T\<preceq>T'"
subst_img_comp_subset': fixes t :: "('a, 'b) Term.term"   and \<theta>1 :: "'b \<Rightarrow> ('a, 'b) Term.term"   and \<theta>2 :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "t \<in> subst_range (\<theta>1 \<circ>\<^sub>s \<theta>2)" shows "t \<in> subst_range \<theta>2 \<or> (\<exists>t'\<in>subst_range \<theta>1. t = t' \<cdot> \<theta>2)"
3: shows "\<guillemotleft>chine_hcomp \<nu> \<mu> : \<nu>.dom.leg0 \<down>\<down> \<mu>.dom.leg1 \<rightarrow>\<^sub>C \<nu>.cod.leg0 \<down>\<down> \<mu>.cod.leg1\<guillemotright>"
If: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and b :: "bexp"   and c\<^sub>1 :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c\<^sub>2 :: "com" assumes "\<turnstile>\<^sub>3\<^sub>a {\<lambda>(s, n). P (s, n) \<and> lmaps_to_axpr b True s} c\<^sub>1 { Q}"   and "\<turnstile>\<^sub>3\<^sub>a {\<lambda>(s, n). P (s, n) \<and> lmaps_to_axpr b False s} c\<^sub>2 { Q}" shows "\<turnstile>\<^sub>3\<^sub>a {(\<lambda>(s, n). P (s, n) \<and> vars b \<subseteq> dom s) \<and>* $ 1} IF b THEN c\<^sub>1 ELSE c\<^sub>2 { Q}"
composable_arrsE: fixes gf :: "V"   and \<CC> :: "V" assumes "gf \<in>\<^sub>\<circ> composable_arrs \<CC>" obtains g :: "V" and f :: "V" and a :: "V" and b :: "V" and c :: "V" where "gf = [g, f]\<^sub>\<circ>" and "g : b \<mapsto>\<^bsub>\<CC>\<^esub> c" and "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b"
SMP_wt_instances_subset: fixes M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and N :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "\<forall>t\<in>M. \<exists>s\<in>N. \<exists>\<delta>. t = s \<cdot> \<delta> \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"   and "t \<in> SMP M" shows "t \<in> SMP N"
subsumes_update_equality: fixes t1 :: "transition"   and c :: "nat \<Rightarrow>f value option"   and t2 :: "transition" assumes "subsumes t1 c t2" shows "\<forall>i. can_take_transition t2 i c \<longrightarrow> (\<forall>r'. evaluate_updates t1 i c $ r' = evaluate_updates t2 i c $ r' \<or> evaluate_updates t2 i c $ r' = None)"
fVectorSpace_indspace: shows "VectorSpace (\<currency>\<currency>) indV"
incr_in_r: fixes j :: "nat" assumes "t (Suc j) \<le> t1" shows "dist (x (t j)) (grid_function (discrete_evolution incr) (x (t 0)) t j) \<le> \<bar>r\<bar>"
semantics_in_vertices_2: fixes A :: "('d, 'e) labeled_graph"   and a :: "'e"   and b :: "'e"   and e :: "'d allegorical_term" assumes "graph A"   and "(a, b) \<in> :A:\<lbrakk>e\<rbrakk>" shows "b \<in> vertices A"
x1_x2_def: shows "R_Chop(ext v,x1,x2)"
valid_ownership_def: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "valid_ownership \<S> ts \<equiv> (outstanding_non_volatile_refs_owned_or_read_only \<S> ts \<and> outstanding_volatile_writes_unowned_by_others ts) \<and> read_only_reads_unowned ts \<and> ownership_distinct ts"
simps_10: fixes vi :: "16 word"   and vj :: "16 word"   and m :: "of_match_field set" shows "prerequisites (L4Src vi vj) m = (\<exists>proto\<in>{TCP, UDP, L4_Protocol.SCTP}. let v = IPv4Proto proto in v \<in> m \<and> prerequisites v m)"
abs_eq: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "eq_onp semilattice_set x x" shows "set_fold1 (Abs_semilattice_set x) = semilattice_set.F x"
blockmat_make_vert: fixes n :: "nat" shows "blockmat (make_vert_block n) = make_vert_equiv n"
leg1_simps_3: shows "src (g \<star> n) = src m"
b_least_less_impl_eq: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat" assumes "b_least f x < x" shows "b_least f x = (LEAST y. f x y \<noteq> 0)"
assms_2: fixes x :: "'a" assumes "a < x"   and "x < b" shows "isCont f x"
topology_inverse': fixes U :: "'d set \<Rightarrow> bool" assumes "istopology U" shows "openin (topology U) = U"
nths_vars1': shows "nths ps2_P.dims0 ps2_P.vars1' = ps2_P.dims1"
psubstBinp_preserves_wls: fixes delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "wlsBinp delta binp"   and "wlsEnv rho" shows "wlsBinp delta binp %%[rho]"
basis_reduction_add_rows_loop_bound: fixes i :: "nat"   and fs :: "int vec list"   and j :: "nat"   and fs' :: "int vec list" assumes "LLL_bound_invariant False True i fs"   and "\<mu>_small_row i fs j"   and "\<mu>_bound_row_inner fs i j"   and "basis_reduction_add_rows_loop i fs j = fs'"   and "i < m"   and "j \<le> i" shows "LLL_bound_invariant True False i fs'"
coeff_in_ring2: fixes x :: "'e"   and A :: "'e set"   and a :: "'e \<Rightarrow> 'a" assumes "x \<in> A"   and "a \<in> A \<rightarrow> carrier R" shows "a x \<in> carrier R"
isRecvMarker_def: fixes event :: "('a, 'b, 'c) event" shows "isRecvMarker event = (\<exists>x51 x52 x53. event = RecvMarker x51 x52 x53)"
brc_step_def: fixes rqrm :: "('a, ('a, 'b) ta_rule dlist) HashMap.hashmap" shows "brc_step rqrm \<equiv> \<lambda>(Q, W, rcm). ls.iterate (rqrm_lookup rqrm (hd W)) brc_inner_step (Q, tl W, rcm)"
le_supI1: fixes x :: "'e multiset"   and a :: "'e multiset"   and b :: "'e multiset" assumes "x \<subseteq># a" shows "x \<subseteq># a \<union># b"
add_nonpos_neg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq a zero"   and "less b zero" shows "less (plus a b) zero"
enabled_qsend_indis: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and \<omega> :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream"   and n :: "nat"   and m :: "nat"   and t :: "'state" assumes "enabled cf \<omega>"   and "qsend (cf ## \<omega>) \<le> enat n"   and "qsend (cf ## \<omega>) \<le> enat m" shows "(snd ((cf ## \<omega>) !! n) \<approx> t) = (snd ((cf ## \<omega>) !! m) \<approx> t)"
def_body': fixes x :: "Set('d)"   and y :: "'e state \<times> 'e state \<Rightarrow> 'd"   and \<tau> :: "'e state \<times> 'e state" assumes "x \<noteq> \<bottom>"   and "x \<noteq> null"   and "(\<upsilon> y) \<tau> = true \<tau>" shows "\<lfloor>\<lfloor>y \<tau> \<in> \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e x\<rceil>\<rceil>\<rfloor>\<rfloor> \<noteq> \<bottom> \<and> \<lfloor>\<lfloor>y \<tau> \<in> \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e x\<rceil>\<rceil>\<rfloor>\<rfloor> \<noteq> null"
nucleus_alt_def_cor1: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "nucleus f" shows "f (x \<rightarrow> y) \<le> x \<rightarrow> f y"
assert_def: fixes p :: "'d"   and q :: "'d" shows "{. p .} q \<equiv> p \<sqinter> q"
executable_dRETURN: fixes prog :: "(integer \<times> edge list) iarray iarray"   and g :: "gState\<^sub>I" shows "nres_of (dRETURN (executable_impl prog g)) \<le> executable prog g"
enat_less_induct: fixes P :: "enat \<Rightarrow> bool"   and n :: "enat" assumes "\<And>n. \<forall>m<n. P m \<Longrightarrow> P n" shows "P n"
LIM_const_eq: fixes k :: "'c"   and L :: "'c"   and a :: "'b" assumes "(\<lambda>x. k) \<midarrow>a\<rightarrow> L" shows "k = L"
while_option_finite_subset_Some: fixes f :: "'b set \<Rightarrow> 'b set"   and C :: "'b set" assumes "mono f"   and "\<And>X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C"   and "finite C" shows "\<exists>P. while_option (\<lambda>A. f A \<noteq> A) f {} = Some P"
S_def: shows "S = {u \<in> U. w u \<le> real c / 2}"
det_nfa_reach: fixes bd :: "nat option bdd"   and ls :: "bool list list"   and q1 :: "bool list"   and j :: "nat"   and bss :: "bool list list" assumes "subset_dfs A (nfa_startnode A) = (bd, ls)"   and "q1 = ls ! j"   and "j < length ls"   and "list_all (is_alph n) bss" shows "nfa_reach A (nfa_startnode A) bss q1 = dfa_reach (det_nfa A) 0 bss j"
is_operator_applicable_in_def: fixes s :: "'c \<Rightarrow> 'd option"   and op :: "('c, 'd) sas_plus_operator" shows "SAS_Plus_Representation.is_operator_applicable_in s op \<equiv> map_of (sas_plus_operator.precondition_of op) \<subseteq>\<^sub>m s"
pw_abs_nofail: fixes R :: "('c \<times> 'b) set"   and M :: "'c nres" shows "nofail (\<Up> R M) = (nofail M \<and> (\<forall>x. inres M x \<longrightarrow> x \<in> Domain R))"
elims_2: fixes p :: "'b \<Rightarrow> bool"   and xa :: "'b list \<times> 'b set" assumes "While_Combinator.rtrancl_while_test p xa" obtains ws :: "'b list" and uu_ :: "'b set" where "xa = (ws, uu_)" and "ws \<noteq> [] \<and> p (hd ws)"
1: shows "closedin (top_of_set s) (s \<inter> h -` {..a})"
rel_predD: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'c set"   and B :: "'d set"   and x :: "'c"   and y :: "'d" assumes "rel_pred R A B"   and "R x y" shows "(x \<in> A) = (y \<in> B)"
PLemmas_15: fixes src_net :: "'b set set"   and dest_net :: "'b set set" shows "deny_all_from_to src_net dest_net = {pa. src pa \<sqsubset> src_net \<and> dest pa \<sqsubset> dest_net} \<triangleleft> D\<^sub>U"
sep_disj_commuteI: fixes x :: "'a"   and y :: "'a" assumes "x ## y" shows "y ## x"
X_sub: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "Cring_Poly.compose R p (X_poly R) = p"
halt_lemma2': fixes LE :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a"   and Q :: "'a \<Rightarrow> bool" assumes "wf LE"   and "\<forall>n. \<not> P (f n) \<and> Q (f n) \<longrightarrow> Q (f (Suc n)) \<and> (f (Suc n), f n) \<in> LE"   and "Q (f 0)" shows "\<exists>n. P (f n)"
prim_impl_def: shows "prim_impl = (let Qi = Q_update r 0 Q_empty; \<pi>i = M_empty; (Qi, \<pi>i) = outer_loop_impl Qi \<pi>i in \<pi>i)"
assms: shows "\<lless>\<pi>\<^sub>1, eP\<ggreater> P\<rightarrow> \<lless>\<pi>\<^sub>2, eP'\<ggreater>"
effect_freezeI: fixes h' :: "heap"   and h :: "heap"   and r :: "'b list"   and a :: "'b array" assumes "h' = h"   and "r = Array.get h a" shows "effect (Array.freeze a) h h' r"
signed_take_bit_def: fixes n :: "nat"   and a :: "'a" shows "signed_take_bit n a = take_bit n a OR of_bool (bit a n) * NOT (mask n)"
that: fixes u :: "int" assumes "u \<in> carrier (residue_ring (p ^ k)) \<and> u * p ^ m = x (m + k)" shows "thesis"
source_nondet_stimcomm: fixes B :: "'a"   and C :: "'a"   and A :: "'a" assumes "B \<rightarrow>\<^sub>\<S> C" shows "A + B \<rightarrow>\<^sub>\<S> C"
space_PiM: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" shows "space (Pi\<^sub>M I M) = (\<Pi>\<^sub>E i\<in>I. space (M i))"
Abs_G_induct: fixes P :: "'b G \<Rightarrow> bool"   and x :: "'b G" assumes "\<And>y. y \<in> G \<Longrightarrow> P (Abs_G y)" shows "P x"
\<mu>': fixes j :: "nat"   and i :: "nat" assumes "j \<le> i"   and "i < length (map of_int_hom.vec_hom fs)" shows "gs.\<mu>' i j = gs.d j * (map of_int_hom.vec_hom fs ! i \<bullet> map of_int_hom.vec_hom fs ! j) - gs.\<sigma> j i j"
simps: fixes a1 :: "('c \<Rightarrow> ('d, 'c) Term.term) list"   and a2 :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and a3 :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" shows "UNIF a1 a2 a3 = ((\<exists>E. a1 = [] \<and> a2 = E \<and> a3 = E) \<or> (\<exists>\<sigma> E E' ss E''. a1 = \<sigma> # ss \<and> a2 = E \<and> a3 = E'' \<and> UNIF1 \<sigma> E E' \<and> UNIF ss E' E''))"
comp_closed: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool"   and f :: "'b"   and g :: "'b" assumes "subcategory C Arr"   and "Arr f"   and "Arr g"   and "partial_magma.cod C f = partial_magma.dom C g" shows "Arr (C g f)"
normal_subgroup_set_mult_closed: fixes M :: "'a set"   and N :: "'a set" assumes "M \<lhd> G"   and "N \<lhd> G" shows "M <#>\<^bsub>G\<^esub> N \<lhd> G"
negateI: fixes P :: "'b \<Rightarrow> bool"   and s :: "'b" assumes "\<not> P s" shows "\<N> P s"
bounded_clinear_cinner_right: fixes a :: "'b" shows "bounded_clinear (cinner a)"
cons_match: fixes name :: "'a"   and rhs :: "'b"   and rest :: "('a \<times> 'b) list" shows "srewrite_step ((name, rhs) # rest) name rhs"
iT_Plus_iWeakUntil_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<W> t2 I \<oplus> k. Q t2) = (P (t1 + k). t1 \<W> t2 I. Q (t2 + k))"
standard_control_dependence_def_variant: fixes n :: "'node"   and n' :: "'node" shows "n controls\<^sub>s n' = (\<exists>as. n -as\<rightarrow>* n' \<and> n \<noteq> n' \<and> \<not> n' postdominates n \<and> n' \<notin> set (sourcenodes as) \<and> (\<forall>n''\<in>set (targetnodes as). n' postdominates n''))"
par_converg: assumes "n \<noteq> 0" shows "eval (Cn 1 r_parallel [at0, at1, c]) [f \<triangleright> n] \<down>"
assms_2: shows "t mod cycleLength (nC k) mem schedule (nC k)"
Rep_perm_add: fixes p1 :: "'b perm"   and p2 :: "'b perm" shows "Rep_perm (p1 + p2) = Rep_perm p1 \<circ> Rep_perm p2"
elims_3: fixes x :: "poly" assumes "\<not> isconstant x" obtains c :: "poly" and p :: "poly" where "x = CN c 0 p"
1: shows "is_least_prefixpoint (\<lambda>x. \<mu> f x) (\<mu> (\<lambda>x. \<mu> f x))"
Sum_any_def: fixes plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.Sum_any plus zero \<equiv> comm_monoid_fun.G plus zero"
is_arg_minI: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and m :: "'a \<Rightarrow> 'b" assumes "P x"   and "\<And>y. P y \<Longrightarrow> \<not> m y < m x" shows "is_arg_min m P x"
d_equal_def: fixes DA :: "'a \<Rightarrow> 'b"   and d :: "'b"   and m :: "'a \<Rightarrow> 'c"   and m' :: "'a \<Rightarrow> 'c" shows "WHATWHERE.d_equal DA d m m' \<equiv> \<forall>x. DA x \<le> d \<longrightarrow> m x = m' x"
total_less_nmultiset: fixes X :: "'a nmultiset"   and Y :: "'a nmultiset" assumes "\<not> X < Y"   and "Y \<noteq> X" shows "Y < X"
moebius_pt_def: shows "moebius_pt \<equiv> map_fun rep_moebius (map_fun rep_complex_homo abs_complex_homo) moebius_pt_mmat_hcoords"
AbGroup_subset_inner_dirsum: fixes Gs :: "'a set list"   and H :: "'a set" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "H \<in> set Gs" shows "H \<subseteq> (\<Oplus>G\<leftarrow>Gs. G)"
interaction_any_bounded_converter_converter_of_resource: fixes res :: "('a, 'b) resource"   and n :: "enat" shows "interaction_any_bounded_converter (converter_of_resource res) n"
ht_hash_ls_insls: fixes l :: "('a \<times> 'b) list list"   and ls :: "('a \<times> 'b) list" assumes "ht_hash l" shows "ht_hash (ls_insls ls l)"
pref_index: fixes u :: "'b list"   and w :: "'b list"   and i :: "nat" assumes "u \<le>p w"   and "i < \<^bold>|u\<^bold>|" shows "u ! i = w ! i"