hcju/isabelleps huggingface dataset:
Sup: fixes M :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "Complete_Partial_Order.chain (\<le>) M"   and "\<forall>x\<in>M. x \<in> ccpo_class.iterates f" shows "Sup M \<in> ccpo_class.iterates f"
Class_commutes_with_composition: fixes a :: "'a"   and b :: "'a" assumes "a \<in> G"   and "b \<in> G" shows "contained.Class a [\<cdot>] contained.Class b = contained.Class (a \<cdot> b)"
lta_distincts: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C LtA D E F" shows "A \<noteq> B \<and> C \<noteq> B \<and> D \<noteq> E \<and> F \<noteq> E \<and> D \<noteq> F"
finite_sc: fixes s :: "'s" assumes "s \<in> S" shows "finite (set_pmf (sc s))"
ide_cod: fixes f :: "'a" assumes "L.arr f" shows "L.ide (L.cod f)"
isomorphic_implies_hpar_3: fixes f :: "'a"   and f' :: "'a" assumes "S.isomorphic f f'" shows "S.src f = S.src f'"
assms_2: shows "cs \<equiv> init_fin_lift_state Running (start_state JVM_local_start (J2JVM P) C M vs)"
equivp_symp: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "equivp R"   and "R x y" shows "R y x"
prems_1: fixes f :: "'a match_expr \<Rightarrow> 'a match_expr option"   and m :: "'a match_expr"   and m' :: "'a match_expr" assumes "f \<in> set (f__ # fs)"   and "normalized_nnf_match m"   and "f m = Some m'" shows "matches \<gamma> m' a p = matches \<gamma> m a p"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) Adj.FG.map D.map Adj.\<epsilon>"
simps_18: fixes x7 :: "'b ptrm" shows "PSnd x7 \<noteq> PUnit"
vint_le_def: shows "vint_le = ZFC_in_HOL.set {[a, b]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> \<int>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat> \<and> int_of_vint a \<le> int_of_vint b}"
all_dghms_in_Vset: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "all_dghms \<alpha> \<in>\<^sub>\<circ> Vset \<beta>"
simps_81: fixes f1 :: "'d \<Rightarrow> 'a"   and f2 :: "'e \<Rightarrow> 'b"   and f3 :: "'f \<Rightarrow> 'c"   and x1 :: "('d, 'e, 'f) val" shows "map_expr f1 f2 f3 (VE x1) = VE (map_val f1 f2 f3 x1)"
hfvss: shows "hfvss \<equiv> negHorbit_list hs induced_vector vs"
sum_of_aezfun_is_aezfun: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> aezfun_set"   and "g \<in> aezfun_set" shows "f + g \<in> aezfun_set"
get_owner_document_ptr_in_heap: fixes get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   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 get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent known_ptr type_wf get_disconnected_nodes get_disconnected_nodes_locs get_root_node get_owner_document"   and "h \<turnstile> ok get_owner_document ptr" shows "ptr |\<in>| object_ptr_kinds h"
pbval_bval_part: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int" shows "pbval b (part s) = bval b s"
elims: fixes x :: "char"   and xa :: "trm option"   and y :: "game option" assumes "ODEo x xa = y" obtains xb :: "char" and \<theta> :: "trm" where "x = xb__" and "xa = Aterm \<theta>" and "y = Agame (ODE xb__ \<theta>)"   | xb :: "char" where "x = xb" and "xa = undeft" and "y = undefg"
aff_dim_cball: fixes e :: "real"   and a :: "'c" assumes "0 < e" shows "aff_dim (cball a e) = int DIM('c)"
fundadjset_ex1_eq_S_image: fixes D :: "'a set" assumes "D \<in> adjacentset C0 - {C0}" shows "\<exists>!s. s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) \<and> D = s `\<rightarrow> C0"
self_append_conv: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs = append xs ys) = (ys = [])"
fdts'_1: shows "P \<turnstile> D has_fields FDTs'"
\<phi>'\<psi>': shows "internal_equivalence f g' \<phi>' \<psi>'"
jozsa_transform_coeff_3: fixes i :: "nat"   and j :: "nat" assumes "i < dim_row U\<^sub>f \<and> j < dim_col U\<^sub>f" shows "1 \<le> j \<and> i = j - 1 \<and> even i \<longrightarrow> U\<^sub>f $$ (i, j) = of_nat (f (i div 2))"
produces_imp_Pos_in_lits: fixes N :: "'b literal multiset set"   and C :: "'b literal multiset"   and A :: "'b" assumes "production N C = {A}" shows "Pos A \<in># C"
map_nth_eq_take: fixes n :: "nat"   and xs :: "'a list" assumes "n < length xs" shows "map ((!) xs) [0..<n] = take n xs"
finI1: fixes K :: "'c \<Rightarrow> 'c DBMEntry"   and g :: "'a \<times> 'c \<Rightarrow> 'd"   and f :: "(nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry) \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "theLe \<circ> K = id" shows "finite (g ` {(c', d) |c' d. 0 < v c' \<and> v c' \<le> n \<and> c \<noteq> c' \<and> f M (v c') = K d})"
carrier_polynomial_shell: fixes K :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) set"   and p :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list" assumes "subring K (L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>)"   and "p \<in> carrier (K [X]\<^bsub>L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>\<^esub>)" shows "p \<in> carrier (poly_ring (L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>))"
simps: fixes p :: "('b, 'c) Combinators list" shows "net_list p = remdups (net_list_aux p)"
flush_all_until_volatile_nth_update_unused: fixes i :: "nat"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and p' :: "'b"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and \<D>' :: "'c"   and \<O>' :: "'d"   and \<R>' :: "'e"   and m :: "nat \<Rightarrow> nat" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ts[i := (p', is', \<theta>', sb, \<D>', \<O>', \<R>')]) m = flush_all_until_volatile_write ts m"
isomorphic_implies_hpar_4: fixes f :: "'a"   and f' :: "'a" assumes "S.isomorphic f f'" shows "S.trg f = S.trg f'"
inverse_transformation_axioms: shows "inverse_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<Phi>'.Y a) F \<Phi>"
Ugp_def: fixes G :: "('c, 'd) Group_scheme" shows "Ugp G \<equiv> Group G \<and> carrier G = {\<one>\<^bsub>G\<^esub>}"
insert: fixes A :: "'g set"   and x :: "'g"   and g :: "'g \<Rightarrow> 'h" assumes "finite A"   and "x \<notin> A" shows "prod g (insert x A) = g x * prod g A"
carrier_vs_is_self: fixes W :: "'c set" shows "carrier (vs W) = W"
add_cancel_right_left: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and minus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and a :: "'f"   and b :: "'f" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(a = plus b a) = (b = zero)"
sub_closed: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "Cring_Poly.compose R q p \<in> carrier P"
lt_of_comp_def: fixes acomp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" shows "lt_of_comp acomp x y = (case acomp x y of Lt \<Rightarrow> True | _ \<Rightarrow> False)"
elims_3: fixes x :: "'a extT"   and xa :: "'a extT" assumes "\<not> x \<le> xa" obtains x :: "'a" and y :: "'a" where "xa__ = extT.Val x" and "xa = extT.Val y" and "\<not> x \<le> y"   | uu_ :: "'a" where "xa__ = extT.Top" and "xa = extT.Val uu_"
funion_absorb2: fixes B :: "'a fset"   and A :: "'a fset" assumes "B |\<subseteq>| A" shows "A |\<union>| B = A"
assms_5: fixes xs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "xs \<noteq> []"   and "P x y xs" shows "P (- x) (- y) (map uminus xs)"
range_ex1_eq: fixes f :: "'b \<Rightarrow> 'c"   and b :: "'c" assumes "inj f" shows "(b \<in> Set.range f) = (\<exists>!x. b = f x)"
of_nat_less_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m < n)"
of_int_power_le_of_int_cancel_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "int"   and b :: "int"   and w :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (ring_1.of_int one plus zero minus x) (power.power one times (ring_1.of_int one plus zero minus b) w) = (x \<le> b ^ w)"
fold_insert_idem: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'a set"   and z :: "'b"   and x :: "'a" assumes "comp_fun_idem f"   and "finite A" shows "Finite_Set.fold f z (insert x A) = f x (Finite_Set.fold f z A)"
Cons_3: shows "(a, b) = chophelper (t # L) L'"
differing_finite: fixes mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" shows "finite (differing_vars mem\<^sub>1 mem\<^sub>2)"
grey_protects_white_dequeue_Mark: fixes fl :: "bool"   and s :: "'f process_name \<Rightarrow> ('e, 'f, 'g, 'h) Model.local_state"   and r :: "'h"   and p :: "'f process_name"   and w :: "'h"   and ws :: "('e, 'g, 'h) mem_store_action list" assumes "fl = sys_fM s"   and "r \<in> ghost_honorary_grey (s p)" shows "(\<exists>g. (g grey_protects_white w) (s(sys := s sys\<lparr>heap := (sys_heap s)(r := map_option (obj_mark_update (\<lambda>_. fl)) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)\<rparr>))) = (\<exists>g. (g grey_protects_white w) s)"
succ_vars: fixes s :: "('i, 'a) state"   and s' :: "('i, 'a) state" assumes "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s" obtains x\<^sub>i :: "nat" and x\<^sub>j :: "nat" where "x\<^sub>i \<in> lvars (\<T> s)" and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" and "x\<^sub>j \<in> rvars (\<T> s)" and "lvars (\<T> s') = lvars (\<T> s) - {x\<^sub>i} \<union> {x\<^sub>j}" and "rvars (\<T> s') = rvars (\<T> s) - {x\<^sub>j} \<union> {x\<^sub>i}"
sup_state_opt_any_Some: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and s :: "ty list \<times> ty err list"   and X :: "(ty list \<times> ty err list) option" shows "P \<turnstile> \<lfloor>s\<rfloor> \<le>' X = (\<exists>s'. X = \<lfloor>s'\<rfloor> \<and> P \<turnstile> s \<le>\<^sub>i s')"
EnsuresLeadsto: fixes A :: "state \<times> state \<Rightarrow> bool"   and P :: "state \<Rightarrow> bool"   and Q :: "state \<Rightarrow> bool" assumes "\<turnstile> A \<and> $P \<longrightarrow> Q$" shows "\<turnstile> \<box>A \<longrightarrow> (P \<leadsto> Q)"
IH2: fixes s :: "'a net_state" assumes "s \<in> init (pnet np p2)"   and "netmap s i = Some v" shows "v \<in> init (np i)"
f3: shows "rk {b, c, \<alpha>} = 2"
vec_of_basis_enum_cspan: fixes X :: "'b set" assumes "length canonical_basis = n" shows "vec_of_basis_enum ` cspan X = local.span (vec_of_basis_enum ` X)"
strip_guards_simps_1: fixes F :: "'f set" shows "strip_guards F Skip = Skip"
Quotient_trm: shows "Quotient (\<approx>) abs_trm rep_trm cr_trm"
in_hom_def: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright> \<equiv> a \<in> domains f \<and> b \<in> codomains f"
CHAOS_unfold: fixes A :: "'a set" shows "CHAOS A = (STOP |-| Mprefix A (\<lambda>z. CHAOS A))"
nth_upt_zero: fixes i :: "nat"   and n :: "nat" assumes "i < length [0..<n]" shows "[0..<n] ! i = i"
left_mult_zero_mat': fixes A :: "'b mat"   and n :: "nat"   and nr :: "nat" assumes "dim_row A = n" shows "0\<^sub>m nr n * A = 0\<^sub>m nr (dim_col A)"
is_empty_def: fixes m :: "('c, 'd) mapping" shows "Mapping.is_empty m = (Mapping.keys m = {})"
rtrancl_last_visit': fixes q :: "'a"   and q' :: "'a"   and R :: "('a \<times> 'a) set"   and S :: "'a set" assumes "(q, q') \<in> R\<^sup>*" obtains "(q, q') \<in> (R - UNIV \<times> S)\<^sup>*"   | qt :: "'a" where "qt \<in> S" and "(q, qt) \<in> R\<^sup>*" and "(qt, q') \<in> (R - UNIV \<times> S)\<^sup>*"
card_of_Field_ordIso: fixes r :: "('c \<times> 'c) set" assumes "Card_order r" shows "|Field r| =o r"
rotate2_eq: fixes x :: "(real, 2) vec"   and y :: "(real, 2) vec" shows "(rotate2 x = rotate2 y) = (x = y)"
simps_1: fixes U :: "'f set" shows "atU_s U [] = False"
lesspoll_def: fixes A :: "'a set"   and B :: "'b set" shows "A \<prec> B \<equiv> A \<lesssim> B \<and> \<not> A \<approx> B"
prime_power_inj'_2: 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 q :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q"   and "power.power one times p m = power.power one times q n"   and "0 < m"   and "0 < n" shows "m = n"
m_comm: fixes x :: "nat \<times> nat"   and y :: "nat \<times> nat" assumes "x \<in> carrier S"   and "y \<in> carrier S" shows "x \<otimes>\<^bsub>S\<^esub> y = y \<otimes>\<^bsub>S\<^esub> x"
Int_insert_left_if0: fixes a :: "'b"   and C :: "'b set"   and B :: "'b set" assumes "a \<notin> C" shows "insert a B \<inter> C = B \<inter> C"
set_to_map_insert: fixes kv :: "'g \<times> 'h"   and S :: "('g \<times> 'h) set" assumes "fst kv \<notin> fst ` S" shows "set_to_map (insert kv S) = set_to_map S(fst kv \<mapsto> snd kv)"
SemAssign: fixes t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and x :: "Var"   and e :: "Expr" assumes "t = (update (fst s) x (evalE e (fst s)), snd s)" shows " s , Assign x e \<rightarrow>\<^sub>1  t "
min_dist_le: fixes src :: "nat"   and v :: "nat"   and d' :: "nat" assumes "local.connected src v"   and "d' \<le> min_dist src v" shows "\<exists>v'. local.connected src v' \<and> min_dist src v' = d'"
b_least_def: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat" shows "b_least f x \<equiv> LEAST y. y = x \<or> y < x \<and> f x y \<noteq> 0"
dep_vars_Cons_cdep: fixes V :: "'var"   and a :: "'edge"   and as' :: "'edge list"   and n'' :: "'node"   and as'' :: "'edge list"   and n' :: "'node" assumes "V \<in> Use (sourcenode a)"   and "sourcenode a -a # as'\<rightarrow>\<^sub>c\<^sub>d n''"   and "n'' -as''\<rightarrow>\<^sub>d* n'" shows "(V, [], a # as' @ as'') \<in> dependent_live_vars n'"
image_affinity_atLeastAtMost_div_diff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and m :: "'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(\<lambda>x. minus (divide x m) c) ` ord.atLeastAtMost less_eq a b = (if ord.atLeastAtMost less_eq a b = {} then {} else if less_eq zero m then ord.atLeastAtMost less_eq (minus (divide a m) c) (minus (divide b m) c) else ord.atLeastAtMost less_eq (minus (divide b m) c) (minus (divide a m) c))"
simps_4: fixes A :: "agent"   and B :: "agent"   and M :: "msg" shows "asym.implSecure A B M = Sign (Aenc \<langle>SecureTag, Messages.Agent A, M\<rangle> (pubK B)) (priK A)"
card: shows "card {q. prime q \<and> q \<le> m} = n"
positively_invariant_eq_rev: fixes M :: "'a set" shows "positively_invariant M = rev.negatively_invariant M"
Domain_id_on: fixes R :: "('e \<times> 'g) set"   and S :: "('g \<times> 'f) set" shows "Domain (R O S) = Domain R \<inter> R\<inverse> `` Domain S"
word_cat_eq': fixes a :: "'d word"   and b :: "'e word" shows "word_cat a b = word_of_int (concat_bit LENGTH('e) (uint b) (uint a))"
uniform_limit_iff: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and S :: "'d set"   and l :: "'d \<Rightarrow> 'e"   and F :: "'c filter" shows "uniform_limit S f l F = (\<forall>e>0. \<forall>\<^sub>F n in F. \<forall>x\<in>S. dist (f n x) (l x) < e)"
cong_1_poly: fixes a :: "'b poly"   and b :: "'b poly" shows "[a = b] (mod 1)"
restrict1_image: fixes B :: "'b set"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'b" assumes "order.greater_eq A B" shows "restrict1 f A ` B = f ` B"
norm_diff_triangle_ineq: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (minus (plus a b) (plus c d)) \<le> norm (minus a c) + norm (minus b d)"
surfp_notin: fixes x :: "'a" assumes "norm x = 1"   and "x \<in> affine hull S" shows "surf (proj x) \<notin> rel_interior S"
simps_1: fixes x :: "'b equation" shows "positive_literal (Pos x) = True"
convol_expand_snd': fixes f :: "'d \<Rightarrow> 'e \<times> 'f"   and g :: "'d \<Rightarrow> 'e"   and h :: "'d \<Rightarrow> 'f" assumes "fst \<circ> f = g" shows "(h = snd \<circ> f) = (<g , h> = f)"
fermat_test_def: shows "fermat_test = primality_test (\<lambda>n a. fermat_liar a n)"
extremum: fixes a :: "'c multiset" shows "{#} \<subseteq># a"
primitive_part_mult: fixes p :: "'b poly"   and q :: "'b poly" shows "primitive_part (p * q) = primitive_part p * primitive_part q"
init: shows "nfa.init \<lparr>nfa.states = HF.Inl ` dfa.states MS \<union> HF.Inr ` dfa.states MT, init = {HF.Inl (dfa.init MS)}, final = HF.Inr ` dfa.final MT, nxt = \<lambda>q x. sum_case (\<lambda>qs. {HF.Inl (dfa.nxt MS qs x)}) (\<lambda>qt. {HF.Inr (dfa.nxt MT qt x)}) q, eps = (\<lambda>q. (HF.Inl q, HF.Inr (dfa.init MT))) ` dfa.final MS\<rparr> \<subseteq> nfa.states \<lparr>nfa.states = HF.Inl ` dfa.states MS \<union> HF.Inr ` dfa.states MT, init = {HF.Inl (dfa.init MS)}, final = HF.Inr ` dfa.final MT, nxt = \<lambda>q x. sum_case (\<lambda>qs. {HF.Inl (dfa.nxt MS qs x)}) (\<lambda>qt. {HF.Inr (dfa.nxt MT qt x)}) q, eps = (\<lambda>q. (HF.Inl q, HF.Inr (dfa.init MT))) ` dfa.final MS\<rparr>"
assms: shows "varset_action a (s - (vs \<union> vs'))"
fba_Diff: fixes S :: "'a set set"   and T :: "'a set set" assumes "S \<in> fba"   and "T \<in> fba" shows "S - T \<in> fba"
antisym_conv1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.order less_eq less"   and "\<not> less x y" shows "less_eq x y = (x = y)"
subset: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "subgroup H G" shows "H \<subseteq> carrier G"
circle_01_diam_def: shows "circle_01_diam = {x. cmod (x - of_nat 1 / of_nat 2) < real 1 / real 2}"
\<epsilon>_def: shows "\<epsilon> = 1 / real (2 * k)"
merge_tree: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option" assumes "merge_on UNIV h bo m" shows "merge_on UNIV (hash_tree h) (blinding_of_tree h bo) (merge_tree h m)"
Sum2: fixes Q :: "pi"   and Rs :: "residual"   and P :: "pi" assumes "Q \<longmapsto> Rs" shows "P \<oplus> Q \<longmapsto> Rs"
continuous_attains_inf: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "compact s"   and "s \<noteq> {}"   and "continuous_on s f" shows "\<exists>x\<in>s. \<forall>y\<in>s. f x \<le> f y"
AKcryptSK_Says: fixes authK :: "nat"   and servK :: "nat"   and S :: "agent"   and A :: "agent"   and X :: "msg"   and evs :: "event list" shows "AKcryptSK authK servK (Says S A X # evs) = (Tgs = S \<and> (\<exists>B Ts. X = Crypt authK \<lbrace>Key servK, Agent B, Number Ts, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace>\<rbrace>) \<or> AKcryptSK authK servK evs)"
top_unique: fixes a :: "'a" shows "(\<top> \<le> a) = (a = \<top>)"
filter_split': fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "filter P xs = ys @ zs" shows "\<exists>us vs. xs = us @ vs \<and> filter P us = ys \<and> filter P vs = zs"
self_finance_succ: fixes Mkt :: "('b, 'c) discrete_market"   and asset :: "'c"   and n :: "nat"   and w :: "'b"   and pf :: "'c \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real"   and v :: "real" assumes "prices Mkt asset (Suc n) w \<noteq> 0"   and "portfolio pf" shows "val_process Mkt (self_finance Mkt v pf asset) (Suc n) w = prices Mkt asset (Suc n) w * remaining_qty Mkt v pf asset (Suc n) w + cls_val_process Mkt pf (Suc n) w"
RDirProd_iso7: fixes R :: "('a, 'c) ring_scheme" shows "(\<lambda>a. (a, [])) \<in> ring_iso R (RDirProd R nil_ring)"
hyps: assumes "is_regular_sequence fs" shows "is_regular_sequence (removeAll 0 fs)"
simps: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a set"   and a2 :: "'a set" shows "max_extp R a1 a2 = (\<exists>X Y. a1 = X \<and> a2 = Y \<and> finite X \<and> finite Y \<and> Y \<noteq> {x. \<bottom> x} \<and> (\<forall>x. x \<in> X \<longrightarrow> Bex Y (R x)))"
fw_impl'_def: fixes M :: "nat \<times> nat \<Rightarrow> 'a" shows "local.fw_impl' M = fw' M n n"
add_closed: fixes g :: "'m"   and h :: "'m" assumes "g \<in> M"   and "h \<in> M" shows "g + h \<in> M"
x: shows "real_of_1 x = real_of_1 (p, l, r)"
smult_minus_1: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme"   and v :: "'j" assumes "module R M"   and "v \<in> carrier M" shows "\<ominus>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> v = \<ominus>\<^bsub>M\<^esub> v"
def_C': shows "C' = cl_ecl P1 - {L} \<union> (cl_ecl P2 - {M} \<union> {L'})"
unProg: fixes x :: "Prog" shows "unProg x \<in> {P. closed P}"