hcju/isabelleps huggingface dataset:
nonsingular_def: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ell_field divide inverse times one plus zero minus uminus" shows "ell_field.nonsingular times one plus zero a b = (plus (times (numeral.numeral one plus (num.Bit0 (num.Bit0 num.One))) (power.power one times a 3)) (times (numeral.numeral one plus (num.Bit1 (num.Bit1 (num.Bit0 (num.Bit1 num.One))))) (power.power one times b 2)) \<noteq> zero)"
singletonD: fixes b :: "'f"   and a :: "'f" assumes "b \<in> {a}" shows "b = a"
simps_2: fixes i :: "int"   and j :: "int"   and l :: "int tree"   and k :: "int"   and r :: "int tree" shows "wpl i j \<langle>l, k, r\<rangle> = wpl i (k - 1) l + wpl (k + 1) j r + w i j"
mult_norm_sub_in_sub: fixes N :: "'a set"   and K :: "'a set"   and H :: "'a set" assumes "N \<lhd> G\<lparr>carrier := K\<rparr>"   and "subgroup H (G\<lparr>carrier := K\<rparr>)"   and "subgroup K G" shows "subgroup (N <#> H) (G\<lparr>carrier := K\<rparr>)"
InitMsgs: shows "{msg. msg \<in># msgs init} = {msg. msg \<in># msgs init \<and> (\<exists>p v. <p, v> = msg)} \<union> {msg. msg \<in># msgs init \<and> (\<exists>v. <\<bottom>, outM v> = msg)} \<union> {msg. msg \<in># msgs init \<and> (\<exists>p v. <p, inM v> = msg)}"
coeff_less_poly_subdegree: fixes k :: "nat"   and p :: "'b poly" assumes "k < poly_subdegree p" shows "coeff p k = (0::'b)"
add_get: fixes xa :: "char list"   and e :: "'b environment"   and U :: "'b" assumes "xa \<notin> env_dom e"   and "ok e" shows "the e\<lparr>xa:U\<rparr>!xa = U"
gf_G_nonneg: fixes z :: "real"   and x :: "'s"   and y :: "'s" assumes "0 \<le> z"   and "z < 1" shows "0 \<le> gf_G x y z"
assms_2: shows "m1 f|` S' = m2 f|` S'"
prv_eqv_prv: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "bprv \<phi>"   and "bprv (eqv \<phi> \<chi>)" shows "bprv \<chi>"
delete_block_design: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and bl :: "'a set" assumes "design point_set block_collection" shows "design point_set (incidence_system.del_block block_collection bl)"
simps_1: fixes atm :: "atom" shows "Example_PL.SC_discr (Atm atm) = Example_PL.presAtm atm"
m_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<otimes> y \<otimes> z = x \<otimes> (y \<otimes> z)"
distinct_1: fixes x1 :: "'c" shows "Action x1 \<noteq> NoAction"
U_idem_s_prod: shows "Multirelations.U \<cdot> Multirelations.U = Multirelations.U"
wellformed1_sorted_auxQ: fixes p :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "wellformed_policy1 p" shows "wellformed_policy1 (qsort p l)"
maxsimpD_simplex: fixes x :: "'a set" assumes "folding_g.chamber x" shows "x \<in> X"
step_1: fixes p :: "'a \<Rightarrow>\<^sub>0 'b" assumes "n = card (keys p)" shows "P p"
scale_cancel_left: fixes x :: "'a \<Rightarrow> real"   and y :: "'a \<Rightarrow> real"   and a :: "real" assumes "x \<in> sub.diff_fun_space"   and "y \<in> sub.diff_fun_space" shows "(a *\<^sub>R x = a *\<^sub>R y) = (x = y \<or> a = 0)"
mk_symmetric_def: fixes G :: "('c, 'd) pre_digraph" shows "mk_symmetric G \<equiv> \<lparr>pverts = verts G, parcs = \<Union>e\<in>arcs G. {(tail G e, head G e), (head G e, tail G e)}\<rparr>"
elem_setmonotone: fixes f :: "'b set \<Rightarrow> 'b set"   and x :: "'b"   and X :: "'b set" assumes "setmonotone f"   and "x \<in> X" shows "x \<in> f X"
Iod_le: fixes T :: "'a set"   and a :: "'a"   and b :: "'a" assumes "T \<subseteq> carrier D"   and "a \<in> T"   and "b \<in> T" shows "a \<preceq>\<^bsub>Iod D T\<^esub> b = a \<preceq> b"
adds_gcs: fixes u :: "'a"   and s :: "'a"   and t :: "'a" assumes "u adds s"   and "u adds t" shows "u adds gcs s t"
card_b_finite: fixes k :: "nat"   and u :: "nat \<times> nat" shows "finite (b k u)"
conjunctive_def: fixes S :: "'d \<Rightarrow> 'e" shows "conjunctive S = (\<forall>Q. S (Inf Q) = Inf (S ` Q))"
memoized_success: fixes dp\<^sub>T :: "'a \<Rightarrow> 'b Heap"   and x :: "'a" assumes "consistentDP dp\<^sub>T" shows "success (dp\<^sub>T x) empty"
S: fixes x :: "real" assumes "x \<in> {0..1} - S" shows "g differentiable at x"
cbi_esdovovisiffidi: shows "b\<inverse> O (e \<union> s \<union> d \<union> ov \<union> ov\<inverse> \<union> s\<inverse> \<union> f \<union> f\<inverse> \<union> d\<inverse>) \<subseteq> b\<inverse> \<union> m\<inverse> \<union> ov\<inverse> \<union> f \<union> d"
types_agree_def: fixes t :: "t"   and v :: "v" shows "types_agree t v = (typeof v = t)"
of_bool_eq_1: fixes one :: "'d"   and zero :: "'d" assumes "class.zero_neq_one one zero" shows "zero_neq_one.of_bool one zero False = zero"
cut_less_Un: fixes A :: "'b set"   and B :: "'b set"   and t :: "'b" shows "(A \<union> B) \<down>< t = A \<down>< t \<union> B \<down>< t"
upd_nested_insert: fixes d :: "'a"   and f :: "'a \<Rightarrow> 'a"   and m :: "('b, ('c, 'a) mapping) mapping"   and x :: "'b \<times> 'c"   and A :: "('b \<times> 'c) set" assumes "d = f d"   and "\<And>x. f (f x) = f x" shows "upd_nested m d f (insert x A) = upd_nested_step d f x (upd_nested m d f A)"
case_prodI2: fixes p :: "'x \<times> 'y"   and c :: "'x \<Rightarrow> 'y \<Rightarrow> bool" assumes "\<And>a b. p = (a, b) \<Longrightarrow> c a b" shows "case p of (a, b) \<Rightarrow> c a b"
run_reachable: fixes i :: "nat" shows "reachable (s i)"
gen_UNION_def: fixes tol :: "'j \<Rightarrow> 'k list"   and emp :: "'i"   and un :: "'l \<Rightarrow> 'i \<Rightarrow> 'i"   and X :: "'j"   and f :: "'k \<Rightarrow> 'l" shows "gen_UNION tol emp un X f \<equiv> fold (un \<circ> f) (tol X) emp"
l_inv_of_ZX: shows "ZX\<^sup>\<dagger> * ZX = 1\<^sub>m 2"
val_of_spec: fixes e :: "'a exp"   and v :: "val" assumes "val_of e = \<lfloor>v\<rfloor>" shows "e = Val v"
CatExpDom: fixes f :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and A :: "('g, 'i, 'k) Category_scheme"   and B :: "('h, 'j, 'l) Category_scheme" assumes "f \<in> mor\<^bsub>CatExp A B\<^esub>" shows "dom\<^bsub>CatExp A B\<^esub> f = NTDom f"
a: shows "a \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
simps_1: fixes k :: "'a" shows "rbt_lookup Empty k = None"
last_nth: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "last xs = xs ! (length xs - Suc 0)"
IH: fixes accu :: "('a \<times> 'a) list" assumes "valid_reqs (get_IFS M)"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "set accu \<subseteq> E"   and "set Es \<subseteq> E"   and "\<forall>e\<in>E - (set accu \<union> set Es). \<not> all_security_requirements_fulfilled (get_IFS M) (stateful_policy_to_network_graph \<lparr>hosts = V, flows_fix = E, flows_state = {e} \<union> set accu\<rparr>)" shows "let stateful = set (filter_IFS_no_violations_accu \<lparr>nodes = V, edges = E\<rparr> M accu Es) in \<forall>e\<in>E - stateful. \<not> all_security_requirements_fulfilled (get_IFS M) (stateful_policy_to_network_graph \<lparr>hosts = V, flows_fix = E, flows_state = {e} \<union> stateful\<rparr>)"
match_term_list_Some_matchrel: fixes P :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) list"   and \<sigma> :: "'d \<Rightarrow> ('e, 'f) Term.term option"   and \<tau> :: "'d \<Rightarrow> ('e, 'f) Term.term option" assumes "match_term_list P \<sigma> = Some \<tau>" shows "((mset P, \<sigma>), {#}, \<tau>) \<in> matchrel\<^sup>*"
insert_inorder: fixes t :: "'a btree"   and k :: "nat"   and x :: "'a" assumes "sorted_less (inorder t)" shows "inorder (local.insert k x t) = ins_list x (inorder t)"
nset_m_m: fixes m :: "nat" shows "nset m m = {m}"
simps: fixes \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'd"   and q :: "'c"   and n :: "nat" shows "semi_mojmir_def.oldest_token \<delta> q\<^sub>0 w q n = (if semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n \<noteq> {} then Some (Min (semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n)) else None)"
vsubst_subst_compose1: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "good X"   and "good Y" shows "X #[Y / y]_ys #[y1 // y]_ys = X #[Y #[y1 // y]_ys / y]_ys"
simps_5: fixes SK :: "nat" shows "InvK (ShaK SK) = ShaK SK"
focus_empty: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "focus {} p = monomial p 0"
sets_Collect_imp: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "{x \<in> \<Omega>. P x} \<in> M"   and "{x \<in> \<Omega>. Q x} \<in> M" shows "{x \<in> \<Omega>. Q x \<longrightarrow> P x} \<in> M"
first_summand_in_sum: fixes x :: "i"   and y :: "i" shows "P x (x \<oplus> y)"
simps_3: fixes x :: "Vars.var" shows "ESem (Var x) = (\<Lambda> \<rho>. C_case\<cdot>(\<rho> x))"
dgrad_p_set_closed_plus: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "p \<in> punit.dgrad_p_set d m"   and "q \<in> punit.dgrad_p_set d m" shows "p + q \<in> punit.dgrad_p_set d m"
new_element_get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and getter :: "('z, 'x, 'y, 'ag option) RShadowRoot_scheme \<Rightarrow> 'aj" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter) h h'"
Q_defined: fixes u :: "'v"   and v :: "'v" assumes "(u, v) \<in> edges rg"   and "u \<notin> S (A (Q' Q \<pi> u) (\<pi>' Q \<pi> u))"   and "v \<in> S (A (Q' Q \<pi> u) (\<pi>' Q \<pi> u))" shows "Q' Q \<pi> u u \<noteq> \<infinity>"
foldl_def: shows "foldl \<equiv> \<lambda>uu uua uub. rec_list (\<lambda>f a. a) (\<lambda>x xs xsa f a. xsa f (f a x)) uub uu uua"
simps_6: shows "- \<infinity> + - \<infinity> = - \<infinity>"
F: shows "(map (inv q) (filter (\<lambda>x. x \<in> range q) xs), inv q ` Y) \<in> failures Q"
subst_apply_eq_Var: fixes s :: "('c, 'e) Term.term"   and \<sigma> :: "'e \<Rightarrow> ('c, 'd) Term.term"   and x :: "'d" assumes "s \<cdot> \<sigma> = Var x" obtains y :: "'e" where "s = Var y" and "\<sigma> y = Var x"
univ_poly_a_inv_def': fixes K :: "'a set"   and p :: "'a list" assumes "subring K R"   and "p \<in> carrier (K [X])" shows "\<ominus>\<^bsub>K [X]\<^esub> p = map (a_inv R) p"
empty_map_add: fixes m :: "'c \<Rightarrow> 'd option" shows "m \<Oplus> \<emptyset> = m"
dghm_ArrMap_in_Vset: fixes \<beta> :: "V" assumes "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<FF>\<lparr>ArrMap\<rparr> \<in>\<^sub>\<circ> Vset \<beta>"
unordered_pair_equality: fixes p :: "'a"   and q :: "'a" shows "{p, q} = {q, p}"
basis_reduction_main: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list"   and fs' :: "int vec list" assumes "LLL_invariant upw i fs"   and "local.basis_reduction_main (upw, i, fs) = fs'" shows "LLL_invariant True m fs'"
take_eq_Nil: fixes n :: "nat"   and xs :: "'b list" shows "(take n xs = \<epsilon>) = (n = 0 \<or> xs = \<epsilon>)"
transD: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "trans r"   and "(x, y) \<in> r"   and "(y, z) \<in> r" shows "(x, z) \<in> r"
canon_rewriter_unique: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and u :: "'t"   and a :: "'t \<Rightarrow>\<^sub>0 'b"   and b :: "'t \<Rightarrow>\<^sub>0 'b" assumes "rb_aux_inv1 bs"   and "is_canon_rewriter rword (set bs) u a"   and "is_canon_rewriter rword (set bs) u b" shows "a = b"
atMost_shift: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "sum g {..n} = g 0 + (\<Sum>i<n. g (Suc i))"
ccFromList_append: fixes xs :: "Vars.var list"   and ys :: "Vars.var list" shows "ccFromList (xs @ ys) = ccFromList xs \<squnion> ccFromList ys \<squnion> set xs G\<times> set ys"
bcontfun\<^sub>N_2: fixes f :: "'c \<Rightarrow> 'd" shows "Norm bcontfun\<^sub>N f = (if f \<in> bcontfun then norm (Bcontfun f) else 0)"
normal_inv_iff: fixes N :: "'a set" shows "N \<lhd> add_monoid G = (subgroup N (add_monoid G) \<and> (\<forall>x\<in>carrier G. \<forall>h\<in>N. x \<oplus> h \<oplus> \<ominus> x \<in> N))"
gcd_ff_list_exists: fixes X :: "'b fract list" shows "\<exists>g. gcd_ff_list X g"
con: shows "continuous_on UNIV (Functions I i \<circ> (\<lambda>x. \<chi>i. sterm_sem I (args i) x))"
subst_eqvt': fixes pi :: "(vrs \<times> vrs) list"   and T :: "ty"   and X :: "tyvrs"   and T' :: "ty" shows "pi \<bullet> T[X \<mapsto> T']\<^sub>\<tau> = (pi \<bullet> T)[pi \<bullet> X \<mapsto> pi \<bullet> T']\<^sub>\<tau>"
le_cases: fixes x :: "'d word"   and y :: "'d word" obtains "x \<le>s y"   | "y \<le>s x"
get_elements_by_class_name_result_in_tree_order: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and name :: "char list"   and results :: "(_) element_ptr list"   and to :: "(_) object_ptr list"   and result :: "(_) element_ptr" assumes "h \<turnstile> get_elements_by_class_name ptr name \<rightarrow>\<^sub>r results"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to"   and "result \<in> set results" shows "cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r result \<in> set to"
assms_1: shows "(\<^bold>\<not> (\<^bold>\<diamond> p)) at a in (ps, a) # branch"
tail_punit_def: fixes cmp_term :: "'a nat_term_order" shows "tail_punit cmp_term \<equiv> ordered_term.tail fst (le_of_nat_term_order cmp_term)"
per2__sams: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "B \<noteq> C"   and "D \<noteq> E"   and "E \<noteq> F"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "Tarski_neutral_dimensionless.Per Bet Cong D E F" shows "Tarski_neutral_dimensionless.SAMS Bet Cong A B C D E F"
G1_G: shows "gs2.gso i = gs1.gso i"
indicator_mult_ennreal: fixes A :: "'a set"   and x :: "'a"   and r :: "real" shows "indicator A x * ennreal r = ennreal (indicat_real A x * r)"
e: shows "\<forall>x\<in>s. e * norm x \<le> norm (f x)"
summable: fixes a :: "nat \<Rightarrow> real" assumes "a \<longlonglongrightarrow> 0"   and "\<And>n. 0 \<le> a n"   and "\<And>n. a (Suc n) \<le> a n" shows "summable (\<lambda>n. (- 1) ^ n * a n)"
H_cond_iff: fixes p :: "'a"   and r :: "'a"   and x :: "'a"   and y :: "'a"   and q :: "'a" shows "H p (if r then x else y) q = (H (\<tt>\<tt> p \<cdot> \<tt>\<tt> r) x q \<and> H (\<tt>\<tt> p \<cdot> !r) y q)"
lve_edge: fixes valid_edge :: "'a \<Rightarrow> bool"   and a :: "'a"   and src :: "'a \<Rightarrow> 'b"   and E :: "'b"   and trg :: "'a \<Rightarrow> 'b"   and X :: "'b"   and e :: "'b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node"   and knd :: "'a \<Rightarrow> 'c edge_kind" assumes "valid_edge a"   and "src a \<noteq> E \<or> trg a \<noteq> X"   and "e = (LDCFG_node.Node (src a), knd a, LDCFG_node.Node (trg a))" shows "lift_valid_edge valid_edge src trg knd E X e"
unch_def: fixes v :: "'a \<Rightarrow> 'b" shows "Unchanged v \<equiv> v$ = $v"
index_update_vec1: fixes i :: "nat"   and v :: "'b vec"   and a :: "'b" assumes "i < dim_vec v" shows "(v |\<^sub>v i \<mapsto> a) $ i = a"
simps: fixes a :: "trm" shows "val a = (\<exists>x e. a = Lam [x].e)"
order_less: assumes "\<not> prime n" shows "order G < n - 1"
ms_aux_hd_exp_MSLCons: fixes x :: "'a \<times> real"   and xs :: "('a \<times> real) msllist" shows "ms_aux_hd_exp (MSLCons x xs) = Some (snd x)"
le_rem_simps_2: fixes a :: "'a" shows "le_rem_s (ENV a) = a"
bspec: fixes A :: "'f set"   and P :: "'f \<Rightarrow> bool"   and x :: "'f" assumes "\<forall>x\<in>A. P x"   and "x \<in> A" shows "P x"
leg1_composite_2: shows "HH\<nu>\<pi>\<rho>.leg1 = \<nu>.leg1 \<cdot> \<nu>\<pi>\<rho>.Prj\<^sub>1\<^sub>1"
one_QDelta_def: shows "1 = QDelta 1 0"
resolution_example2: shows "resolution_deriv {{Nb, Na}, {Pax}, {Pa}, {Na, Pb, Naa}} {{Nb, Na}, {Pax}, {Pa}, {Na, Pb, Naa}, {Na, Pb}, {Na}, {}}"
productive_imp_produces_Max_atom: fixes N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "production N C \<noteq> {}" shows "production N C = {Max (atms_of C)}"
II: fixes it :: "'d set"   and \<sigma> :: "'c" assumes "it \<subseteq> dom m"   and "it \<noteq> dom m"   and "\<not> c \<sigma>"   and "I it \<sigma>" shows "P \<sigma>"
all_true_def: fixes s :: "bool set" shows "all_true s \<equiv> \<forall>x\<in>s. x"
r\<^sub>0s\<^sub>1_is_cospan: shows "cospan_of_maps_in_bicategory_of_spans (\<cdot>) (\<star>) \<a> \<i> src trg u\<^sub>1 t\<^sub>0"
apex_def: shows "trg_f.dom.apex \<equiv> C.dom trg_f.dom.leg0"
spec_is_affine_scheme: shows "affine_scheme R (+) (\<cdot>) \<zero> \<one> Spec is_zariski_open sheaf_spec sheaf_spec_morphisms \<O>b add_sheaf_spec mult_sheaf_spec zero_sheaf_spec one_sheaf_spec (identity Spec) (\<lambda>U. identity (\<O> U))"
meet_right: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a" assumes "weak_lower_semilattice L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqinter>\<^bsub>L\<^esub> y \<sqsubseteq>\<^bsub>L\<^esub> y"
valid_decomp_split_2: fixes U :: "'x set"   and S :: "('x \<Rightarrow>\<^sub>0 nat) set"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite X"   and "U \<subseteq> X"   and "finite S"   and "t \<in> .[X]" shows "valid_decomp X (snd (split t U S))"
mset_ns_mul_ext: fixes xs :: "'a list"   and ys :: "'a list"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool" shows "((mset xs, mset ys) \<in> ns_mul_ext {(x, y). snd (f x y)} {(x, y). fst (f x y)}) = snd (mul_ext f xs ys)"
sums_If_finite: fixes P :: "nat \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b" assumes "finite {r. P r}" shows "(\<lambda>r. if P r then f r else (0::'b)) sums (\<Sum>r | P r. f r)"
nat_filtration_not_borel_info: fixes f :: "bool stream \<Rightarrow> 'a"   and n :: "nat"   and N :: "'a measure" assumes "f \<in> nat_filtration n \<rightarrow>\<^sub>M N"   and "set_discriminating n f N" shows "f \<circ> pseudo_proj_True n = f"
less_csucc: fixes \<kappa> :: "V" assumes "Card \<kappa>" shows "\<kappa> < csucc \<kappa>"
mono: fixes c :: "'b \<times> 'c"   and a :: "'b \<times> 'c"   and d :: "'b \<times> 'c"   and b :: "'b \<times> 'c" assumes "less_eq_prod c a"   and "less_eq_prod d b" shows "less_eq_prod (sup_prod c d) (sup_prod a b)"
chain_then_restrict: fixes S :: "nat \<Rightarrow> ('e, 'f) labeled_graph"   and i :: "nat" assumes "chain S" shows "graph (S i)"
fresh_delete: fixes a :: "atom"   and \<Gamma> :: "('a \<times> 'b) list"   and v :: "'a" assumes "a \<sharp> \<Gamma>" shows "a \<sharp> delete v \<Gamma>"
lincomb_append: fixes rs :: "'f list"   and ms :: "'v list"   and ss :: "'f list"   and ns :: "'v list" assumes "length rs = length ms" shows "(rs @ ss) \<bullet>\<cdot> (ms @ ns) = rs \<bullet>\<cdot> ms + ss \<bullet>\<cdot> ns"
delay_bisimulation_diverge_compose: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim12 :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim12 :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and trsys3 :: "'k \<Rightarrow> 'l \<Rightarrow> 'k \<Rightarrow> bool"   and bisim23 :: "'i \<Rightarrow> 'k \<Rightarrow> bool"   and tlsim23 :: "'j \<Rightarrow> 'l \<Rightarrow> bool"   and \<tau>move3 :: "'k \<Rightarrow> 'l \<Rightarrow> 'k \<Rightarrow> bool" assumes "delay_bisimulation_diverge trsys1 trsys2 bisim12 tlsim12 \<tau>move1 \<tau>move2"   and "delay_bisimulation_diverge trsys2 trsys3 bisim23 tlsim23 \<tau>move2 \<tau>move3" shows "delay_bisimulation_diverge trsys1 trsys3 (bisim12 \<circ>\<^sub>B bisim23) (tlsim12 \<circ>\<^sub>B tlsim23) \<tau>move1 \<tau>move3"
finite_measure_subalgebra_is_sigma_finite: fixes M :: "'c measure"   and F :: "'c measure" assumes "finite_measure_subalgebra M F" shows "sigma_finite_subalgebra M F"
path_connected_component: fixes S :: "'c set" shows "path_connected S = (\<forall>x\<in>S. \<forall>y\<in>S. path_component S x y)"
P5: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c"   and w :: "'b \<Rightarrow> 'd" shows "\<turnstile> \<box>[P]_v \<longrightarrow> \<box>[\<box>[P]_v]_w"
lt_def: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "ordered_term.lt term_of_pair ord_term p = (if p = 0 then ordered_term.min_term term_of_pair else linorder.Max ord_term (keys p))"
Pderiv_no_occurrence: fixes x :: "'b"   and R :: "'b rexp set" assumes "x \<notin> Atoms R" shows "Pderiv x R = {}"
ltln_strong_weak_2: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> M\<^sub>n \<psi> = w \<Turnstile>\<^sub>n F\<^sub>n \<phi> and\<^sub>n \<phi> R\<^sub>n \<psi>"
simps_5: fixes s_inst :: "t_context list"   and s_funcs :: "tf list"   and s_tab :: "nat list"   and s_mem :: "nat list"   and s_globs :: "tg list"   and more :: "'a" shows "s_globs \<lparr>s_inst = s_inst, s_funcs = s_funcs, s_tab = s_tab, s_mem = s_mem, s_globs = s_globs, \<dots> = more\<rparr> = s_globs"
gr_zeroI: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "'b" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "n = zero \<Longrightarrow> False" shows "less zero n"
dickson_less_pD3: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_less_p d m p q" shows "p \<prec>\<^sub>p q"
Quotient_commute: shows "Quotient (eq_onp (\<lambda>f. \<forall>x y. f x y = f y x)) Abs_commute apply_commute cr_commute"
ordertype_nat_imp_finite: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and m :: "nat" assumes "ordertype A r = ord_of_nat m"   and "small A"   and "wf r"   and "total_on A r" shows "finite A"
assms_8: fixes z :: "complex" assumes "z \<in> U" shows "f z = g z"
sup_ge2: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "class.semilattice_sup sup less_eq less" shows "less_eq y (sup x y)"
tri_def: fixes g :: "graph"   and v :: "nat" shows "tri g v \<equiv> |filter (\<lambda>f. final f \<and> triangle f) (facesAt g v)|"
closed_form_2: fixes z :: "complex" assumes "poly (reflect_poly q) z = 0" shows "degree (rs z) \<le> order z (reflect_poly q) - 1"
single_subset_wens_set: fixes act :: "('b \<times> 'b) set"   and B :: "'b set"   and init :: "'b set"   and allowed :: "('b \<times> 'b) set set" assumes "single_valued act" shows "insert (wens_single act B) (range (wens_single_finite act B)) \<subseteq> wens_set (mk_program (init, {act}, allowed)) B"
divisor_count_def: fixes n :: "nat" shows "divisor_count n = card {d. d dvd n}"
finite_imp_compact: fixes s :: "'a set" assumes "finite s" shows "compact s"