hcju/isabelleps huggingface dataset:
kruskal_vc_2: fixes minarc :: "'b \<Rightarrow> 'b"   and sum :: "'b \<Rightarrow> 'b"   and plus :: "'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 bot :: "'b"   and top :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and conv :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "class.m_kleene_algebra minarc sum plus inf less_eq less sup bot top uminus one times conv star"   and "m_kleene_algebra.kruskal_spanning_invariant inf less_eq uminus one times conv star f g h"   and "h \<noteq> bot" shows "(less_eq (minarc h) (uminus (times (star (conv f)) (star f))) \<longrightarrow> m_kleene_algebra.kruskal_spanning_invariant inf less_eq uminus one times conv star (sup (sup (inf f (uminus (times (times top (minarc h)) (star (conv f))))) (conv (inf f (times (times top (minarc h)) (star (conv f)))))) (minarc h)) g (inf (inf h (uminus (minarc h))) (uminus (conv (minarc h)))) \<and> card {x. x = uminus (uminus x) \<and> less_eq x (uminus (uminus h)) \<and> less_eq x (uminus (minarc h)) \<and> less_eq x (uminus (conv (minarc h)))} < card {x. x = uminus (uminus x) \<and> less_eq x (uminus (uminus h))}) \<and> (\<not> less_eq (minarc h) (uminus (times (star (conv f)) (star f))) \<longrightarrow> m_kleene_algebra.kruskal_spanning_invariant inf less_eq uminus one times conv star f g (inf (inf h (uminus (minarc h))) (uminus (conv (minarc h)))) \<and> card {x. x = uminus (uminus x) \<and> less_eq x (uminus (uminus h)) \<and> less_eq x (uminus (minarc h)) \<and> less_eq x (uminus (conv (minarc h)))} < card {x. x = uminus (uminus x) \<and> less_eq x (uminus (uminus h))})"
singular_relboundary: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "singular_relboundary p X S c = (\<exists>d e. singular_chain (Suc p) X d \<and> singular_chain p (subtopology X S) e \<and> chain_boundary (Suc p) d + e = c)"
isState_purgeIdle: fixes sll :: "('St, 'U, 'C) state list" shows "(\<exists>sl. purgeIdle sl = sll) = list_all isState sll"
inf_eq_top_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> y = \<top>) = (x = \<top> \<and> y = \<top>)"
rsp: shows "(1, 0) \<approx> (1, 0)"
wf_RESTR: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula" assumes "local.wf idx \<phi>" shows "local.wf idx (RESTR \<phi>)"
is_generalised_output_of_subsumes: fixes t' :: "transition"   and t :: "transition"   and r :: "nat"   and p :: "nat"   and v :: "value"   and c :: "nat \<Rightarrow>f value option" assumes "is_generalised_output_of t' t r p"   and "Outputs t ! p = L v"   and "c $ r = Some v" shows "subsumes t' c t"
dg_is_arrD_1: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
preserves_inverse_arrows: fixes f :: "'d"   and g :: "'d" assumes "D.inverse_arrows f g" shows "C.inverse_arrows (F f) (F g)"
state_eq: fixes \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<nu>' :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "\<And>i. fst \<nu> $ i = fst \<nu>' $ i"   and "\<And>i. snd \<nu> $ i = snd \<nu>' $ i" shows "\<nu> = \<nu>'"
Eq\<^sub>0I: fixes x :: "\<o>"   and y :: "\<o>"   and v :: "i" assumes "x = y" shows "[x \<^bold>=\<^sub>0 y in v]"
mult: fixes a :: "nat"   and b :: "nat" shows "liouville_lambda (a * b) = liouville_lambda a * liouville_lambda b"
src_cod: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "src\<^sub>C (C.cod \<mu>) = src\<^sub>C \<mu>"
card_fset_mono: fixes xs :: "'b fset"   and ys :: "'b fset" assumes "xs |\<subseteq>| ys" shows "card_fset xs \<le> card_fset ys"
CollectD: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "a \<in> {x. P x}" shows "P a"
pre_oldest_configuration_finite: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "finite (semi_mojmir.pre_oldest_tokens \<delta> q\<^sub>0 w q n)"
steph: fixes amount :: "nat"   and var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" assumes "amount \<le> var + 1" shows "(\<exists>xs. length xs = var + 1 \<and> eval (list_conj (map fm.Atom L @ F)) (xs @ \<Gamma>)) = (\<exists>xs. length xs = var + 1 \<and> eval (step amount var L F) (xs @ \<Gamma>))"
e_1: shows "xs @ rs = aform_vals e (XS @ map pdevs_of_real rs)"
Min_Di'_le: fixes i :: "nat"   and a :: "nat list"   and e :: "nat"   and b :: "nat list"   and y :: "nat list" assumes "i < length a"   and "e \<in> hlde_ops.Di' a b i y"   and "length y \<le> length b" shows "Min (hlde_ops.Di' a b i y) \<le> Max (set b)"
path_Entry_Cons: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge Entry as n'"   and "n' \<noteq> Entry" obtains n :: "'b" and a :: "'a" where "sourcenode a = Entry" and "targetnode a = n" and "CFG.path sourcenode targetnode valid_edge n (tl as) n'" and "valid_edge a" and "a = hd as"
liveVal_inv: fixes v :: "'val" assumes "liveVal g v"   and "v \<noteq> chooseNext g" shows "step.liveVal g v"
Class_self: fixes a :: "'a set \<times> 'b" assumes "a \<in> Sigma I \<FF>1" shows "a \<in> e1.Class a"
inverse_function_theorem: fixes U :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and f' :: "'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a"   and x0 :: "'a"   and invf :: "'a \<Rightarrow>\<^sub>L 'a" assumes "open U"   and "\<And>x. x \<in> U \<Longrightarrow> (f has_derivative blinfun_apply (f' x)) (at x)"   and "continuous_on U f'"   and "x0 \<in> U"   and "invf o\<^sub>L f' x0 = id_blinfun" obtains U' :: "'a set" and V :: "'a set" and g :: "'a \<Rightarrow> 'a" and g' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where "open U'" and "U' \<subseteq> U" and "x0 \<in> U'" and "open V" and "f x0 \<in> V" and "homeomorphism U' V f g" and "\<And>y. y \<in> V \<Longrightarrow> (g has_derivative g' y) (at y)" and "\<And>y. y \<in> V \<Longrightarrow> g' y = inv (blinfun_apply (f' (g y)))" and "\<And>y. y \<in> V \<Longrightarrow> bij (blinfun_apply (f' (g y)))"
AR_imp_retract: fixes S :: "'a set"   and U :: "'a set" assumes "AR S \<and> closedin (top_of_set U) S" shows "S retract_of U"
typeAmic_def: fixes n :: "nat"   and m :: "nat" shows "typeAmic n m = [card {i. \<exists>N. n = N * gcd n m \<and> prime i \<and> i dvd N \<and> \<not> i dvd gcd n m}, card {j. \<exists>M. m = M * gcd n m \<and> prime j \<and> j dvd M \<and> \<not> j dvd gcd n m}]"
a_prop: shows "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>a\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F) in v]"
set_def: fixes a :: "'d array"   and x :: "'d list" shows "Array.set a x = arrays_update (\<lambda>h. h(TYPEREP('d) := (h TYPEREP('d))(addr_of_array a := map to_nat x)))"
f: fixes v :: "'a" assumes "v' = v" shows "f v = f' v"
infdist_zero: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "infdist a A = 0"
mon_c_empty: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme" shows "mon_c fg {#} = {}"
simps_2: fixes F :: "'b formula"   and G :: "'b formula" shows "is_nnf (F \<^bold>\<and> G) = (is_nnf F \<and> is_nnf G)"
count_le_replicate_mset_subset_eq: fixes n :: "nat"   and M :: "'a multiset"   and x :: "'a" shows "(n \<le> count M x) = (replicate_mset n x \<subseteq># M)"
set_2: fixes x2 :: "'b" shows "set_option (Some x2) = {x2}"
simps_1: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('b, 'a, 'c) Stmt" assumes "sifum_modes aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B" shows "sifum_modes.annotate c [] = c"
simps_14: fixes x41 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool"   and x42 :: "'d"   and x43 :: "('a \<Rightarrow> 'b option) \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'a \<Rightarrow> 'b option"   and x2 :: "('a \<Rightarrow> 'b option) \<Rightarrow> bool" shows "x41\<hookleftarrow>\<^bsub>x42\<^esub>x43 \<noteq> (x2)\<^sub>\<surd>"
VDMIff: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c1 :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and c2 :: "IMP"   and B :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr" assumes "G \<rhd> c1 : A"   and "G \<rhd> c2 : B" shows "G \<rhd> Iff b c1 c2 : (\<lambda>s t. (evalB b s \<longrightarrow> A s t) \<and> (\<not> evalB b s \<longrightarrow> B s t))"
simple_function_Pair: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "simple_function M f"   and "simple_function M g" shows "simple_function M (\<lambda>x. (f x, g x))"
word_upto_alt: fixes a :: "'b word"   and b :: "'b word" assumes "a \<le> b" shows "word_upto a b = map word_of_nat [unat a..<Suc (unat b)]"
max_step1_axioms: shows "max_step1 t (t0 + e2') B' (euler_C TYPE('a)) (r - b)"
simps_3: fixes x2 :: "'a" shows "\<lfloor>x2\<rfloor> \<noteq> \<bottom>"
gen_lower_bound_decreasing: fixes x :: "real"   and a :: "real"   and fl :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "x \<le> a"   and "\<And>y. \<lbrakk>x \<le> y; y \<le> a\<rbrakk> \<Longrightarrow> ((\<lambda>x. fl x - f x) has_real_derivative g y) (at y)"   and "\<And>y. \<lbrakk>x \<le> y; y \<le> a\<rbrakk> \<Longrightarrow> 0 \<le> g y"   and "fl a = f a" shows "fl x \<le> f x"
zero_l_neutral_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" shows "zero_l_neutral f \<equiv> \<forall>a. f (0::'b) a = a"
differentiable_within_subset: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and s :: "'c set"   and t :: "'c set" assumes "f differentiable at x within s"   and "t \<subseteq> s" shows "f differentiable at x within t"
Inj: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> H" shows "X \<in> parts H"
bounded_plus: fixes S :: "'a set"   and T :: "'a set" assumes "bounded S"   and "bounded T" shows "bounded ((\<lambda>(x, y). x + y) ` (S \<times> T))"
pderiv_coeffs_def: fixes xs :: "'b list" shows "pderiv_coeffs xs = pderiv_coeffs_code (1::'b) (tl xs)"
iMax_def: fixes i :: "nat set" shows "iMax i \<equiv> if finite i then enat (Max i) else \<infinity>"
bot_acom_def: shows "\<bottom>\<^sub>c = anno None"
fold_A_facet_distance_map: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set"   and F :: "'b set" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z"   and "SimplicialComplex.maxsimp X F" shows "ChamberComplex.face_distance X z ((ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A D \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B C) ` F) = ChamberComplex.face_distance X z F"
eq_divide_imp: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "c \<noteq> zero"   and "times a c = b" shows "a = divide b c"
conservativeI2: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "qmpt M T"   and "\<And>A. \<lbrakk>A \<in> sets M; 0 < emeasure M A\<rbrakk> \<Longrightarrow> \<exists>n>0. (T ^^ n) -` A \<inter> A \<noteq> {}" shows "conservative M T"
simps_7: shows "subcomp sA7 = {sA71, sA72}"
Int_insert_right_if0: fixes a :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "a \<notin> A" shows "A \<inter> Set.insert a B = A \<inter> B"
assms: shows "\<turnstile> p \<rhd> u \<Rightarrow> \<theta>"
vsv_ex1_app2: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> I" shows "(f\<lparr>a\<rparr> = b) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> f)"
nxt_subset_greaterThan: fixes m :: "nat"   and n :: "nat"   and N :: "nat set" assumes "m \<le> n" shows "nxt N n \<subseteq> {m<..}"
assms_1: fixes a :: "'a" assumes "a \<in> A" shows "a .\<in> B"
mset_poly_rel_minus: fixes a :: "char list multiset"   and b :: "int"   and v' :: "int mpoly"   and yc :: "(char list multiset \<times> int) list"   and r' :: "int mpoly"   and r :: "(char list list \<times> int) list" assumes "({#(a, b)#}, v') \<in> mset_poly_rel"   and "(mset yc, r') \<in> mset_poly_rel"   and "(r, yc) \<in> \<langle>unsorted_term_poly_list_rel \<times>\<^sub>r int_rel\<rangle>list_rel" shows "(add_mset (a, b) (mset yc), v' + r') \<in> mset_poly_rel"
subfield_gen_incl: fixes H :: "'a set"   and K :: "'a set"   and I :: "'a set" assumes "subfield H R"   and "subfield K R"   and "I \<subseteq> H"   and "I \<subseteq> K" shows "generate_field (R\<lparr>carrier := K\<rparr>) I \<subseteq> generate_field (R\<lparr>carrier := H\<rparr>) I"
ordinal_utility_left: fixes x :: "'a pmf"   and y :: "'a pmf" assumes "x \<succeq> y" shows "measure_pmf.expectation y u \<le> measure_pmf.expectation x u"
proots_ball_def: fixes p :: "complex poly"   and z0 :: "complex"   and r :: "real" shows "proots_ball p z0 r = proots_count p (ball z0 r)"
monom_closed: fixes a :: "nat \<Rightarrow> int"   and n :: "nat" assumes "a \<in> carrier Zp" shows "monom (UP Zp) a n \<in> carrier (UP Zp)"
changes_rev: fixes xs :: "'a list" shows "changes (rev xs) = changes xs"
tensor_mat_zero1: fixes m1 :: "'b mat" shows "ps_P.tensor_mat (0\<^sub>m ps_P.d1 ps_P.d1) m1 = 0\<^sub>m ps_P.d ps_P.d"
HMA_V_def: shows "HMA_Connect.HMA_V = (\<lambda>v w. v = HMA_Connect.from_hma\<^sub>v w)"
eqs_def: fixes h :: "'b \<Rightarrow> int"   and x :: "'b" shows "eqs h x \<equiv> {y. h y = h x}"
set_pmf_wt: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state" shows "set_pmf (wt_pmf cf) = (if proper (fst cf) then {i. i < brn (fst cf) \<and> 0 < wt (fst cf) (snd cf) i} else {0})"
digraph_axioms: shows "digraph \<alpha> (dg_Rel \<alpha>)"
mult_less_cancel_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times a c) (times b c) = ((less_eq zero c \<longrightarrow> less a b) \<and> (less_eq c zero \<longrightarrow> less b a))"
U_def: shows "U \<equiv> \<lambda>x. {xa \<in> S. \<forall>y. y \<in> S \<and> norm (y - xa) < 1 / real x \<longrightarrow> norm (y - xa) \<le> real x * norm (f y - f xa)}"
that: fixes as :: "'edge list" assumes "parent_node n -as\<rightarrow>\<^bsub>sl\<^esub>* parent_node n'" shows "thesis"
rev_exhaust: fixes xs :: "'c list" obtains "xs = []"   | ys :: "'c list" and y :: "'c" where "xs = ys @ [y]"
order_refl: fixes x :: "('e, 'f) Term.term" shows "x \<sqsubseteq> x"
g_def: shows "g = (\<lambda>z. 1 / (z ^ n * (exp z - 1)))"
cball_diff_eq_sphere: fixes a :: "'c"   and r :: "real" shows "cball a r - ball a r = sphere a r"
nopathinE2: shows "(u, v) \<notin> connected E2"
cong_0_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b zero a = dvd.dvd times a b"
size_mset_tree: fixes t :: "'b tree" assumes "invar_tree t" shows "size (mset_tree t) = 2 ^ rank t"
igSubstAbsIPresIGWlsAbs_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igSubstAbsIPresIGWlsAbs MOD \<equiv> \<forall>ys Y y us s A. isInBar (us, s) \<and> igWls MOD (asSort ys) Y \<and> igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD (us, s) (igSubstAbs MOD ys Y y A)"
bitset_2n1: fixes n :: "nat" shows "bitset (2 * n + 1) = insert 0 (Suc ` bitset n)"
simps_42: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'b"   and f3 :: "('c, 'd) fm \<Rightarrow> 'b"   and f4 :: "('c, 'd) fm \<Rightarrow> ('c, 'd) fm \<Rightarrow> 'b"   and f5 :: "('c, 'd) fm \<Rightarrow> 'b"   and f6 :: "'d \<Rightarrow> ('c, 'd) fm \<Rightarrow> 'b"   and x61 :: "'d"   and x62 :: "('c, 'd) fm" shows "(case \<^bold>@ x61 x62 of Pro x \<Rightarrow> f1 x | Nom x \<Rightarrow> f2 x | \<^bold>\<not> x \<Rightarrow> f3 x | x \<^bold>\<or> xa \<Rightarrow> f4 x xa | \<^bold>\<diamond> x \<Rightarrow> f5 x | \<^bold>@ x xa \<Rightarrow> f6 x xa) = f6 x61 x62"
tarjan_back_def: fixes u :: "'v"   and v :: "'v"   and s :: "'v tarjan_state" shows "tarjan_back u v s = (if \<delta> s v < \<delta> s u \<and> v \<in> set (tj_stack s) then let ul' = min (\<zeta> s u) (\<delta> s v) in NOOP (s\<lparr>lowlink := lowlink s(u \<mapsto> ul')\<rparr>) else NOOP s)"
foo: shows "mult r \<subseteq> mult (r\<^sup>+)"
ideal_Union: fixes I :: "nat \<Rightarrow> 'a set" assumes "\<forall>A\<in>{I n |n. n \<in> UNIV}. ideal A"   and "\<forall>n. I n \<subseteq> I (n + 1)" shows "ideal (\<Union> {I n |n. n \<in> UNIV})"
AtParallelExprs: fixes i :: "nat"   and cs :: "('d, 'e, 'f) com list"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and as :: "(('d, 'e, 'f) ann \<times> 'd set \<times> 'd set) list"   and a :: "'d set \<times> ('d, 'e, 'f) com" assumes "i < length cs"   and "atomicsR \<Gamma> \<Theta> (fst (as ! i)) (cs ! i) a" shows "atomicsR \<Gamma> \<Theta> (AnnPar as) (Parallel cs) a"
f_ax: fixes n :: "nat" shows "f (f n) < f (Suc n)"
hom_mult: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and y :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"   and "y \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "hom_boundary (int n) (nsphere n) (equator n) (x \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> y) = hom_boundary (int n) (nsphere n) (equator n) x \<otimes>\<^bsub>homology_group (int n - 1) (subtopology (nsphere n) (equator n))\<^esub> hom_boundary (int n) (nsphere n) (equator n) y"
mset_sorted_list_of_multiset: fixes M :: "'a multiset" shows "mset (sorted_list_of_multiset M) = M"
closure_bounded_linear_image_subset_eq: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "bounded_linear f" shows "closure (f ` closure S) = closure (f ` S)"
is_lasso_def: fixes ma :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'c"   and m :: "'b"   and L :: "('Q \<times> nat) lasso" shows "gb_graph.is_lasso (b_graph.to_gbg_ext (degeneralize_ext ma) m) L \<equiv> graph.is_lasso_pre (b_graph.to_gbg_ext (degeneralize_ext ma) m) L \<and> (\<forall>A\<in>gbg_F (b_graph.to_gbg_ext (degeneralize_ext ma) m). set (lasso_cycle L) \<inter> A \<noteq> {})"
cases: fixes a1 :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and a2 :: "'e set"   and a3 :: "'e \<Rightarrow> 'f" assumes "valid_abstraction a1 a2 a3" obtains A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)" and k :: "'e \<Rightarrow> 'f" and X :: "'e set" where "a1 = A" and "a2 = X" and "a3 = k" and "\<forall>(x, m)\<in>clkp_set A. m \<le> k x \<and> x \<in> X \<and> m \<in> \<nat>" and "collect_clkvt (trans_of A) \<subseteq> X" and "finite X"
INT_greatest: fixes A :: "'a set"   and C :: "'b set"   and B :: "'a \<Rightarrow> 'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x" shows "C \<subseteq> \<Inter> (B ` A)"
has_integral_empty: fixes f :: "'b \<Rightarrow> 'c" shows "(f has_integral (0::'c)) {}"
right_assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a + b + c = a + (b + c)"
mix_pmf_comp_left_div: fixes \<alpha> :: "real"   and \<beta> :: "real"   and p :: "'b pmf"   and q :: "'b pmf"   and e :: "'b" assumes "\<alpha> \<in> {0..1}"   and "\<beta> \<in> {0..1}"   and "\<beta> < \<alpha>" shows "pmf (mix_pmf (\<beta> / \<alpha>) (mix_pmf \<alpha> p q) q) e = \<beta> * pmf p e + pmf q e - \<beta> * pmf q e"
simps: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and a :: "('a, 'b, 'c) state list" shows "GM_sec_model.wffp do a = ((\<exists>s. a = [s]) \<or> (\<exists>s' s sl. a = s # s' # sl \<and> s' \<in> GM_sec_model.\<delta> do s \<and> GM_sec_model.wffp do (s' # sl)))"
elims_3: fixes x :: "('d, 'e) fmap"   and xa :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list" assumes "\<not> sat_precond_as x xa" obtains s :: "('d, 'e) fmap" and a :: "('d, 'e) fmap \<times> ('d, 'e) fmap" and as :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list" where "x = s" and "xa = a # as" and "\<not> (fst a \<subseteq>\<^sub>f s \<and> sat_precond_as (state_succ s a) as)"
mult_left_mono_neg: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<le> a"   and "c \<le> (0::'a)" shows "c * a \<le> c * b"
simps_1: fixes C :: "char list" shows "getbase (Repeats C) = C"
s_x_zero: fixes x :: "'a" shows "1\<^sub>\<sigma> \<sqinter> x \<cdot> (0::'a) = (0::'a)"
obtain_v1_v2_1: shows "v1 \<in> vs"
ae_zero: shows "AE x1 in lborel. emeasure (lborel_f n) {x \<in> space (lborel_f n). x(n := x1) \<in> A} = 0"
h'_def: shows "h' \<equiv> \<lambda>z. if snd z \<in> S then h z else (f \<circ> snd) z"
intro: fixes M :: "('h, 'i, 'j) Module_scheme"   and R :: "('i, 'k) Ring_scheme" assumes "aGroup M"   and "Module_axioms M R" shows "R module M"
replace_primitive_matchexpr_def: fixes disc_sel :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and replace_f :: "'c negation_type \<Rightarrow> 'b match_expr"   and m :: "'b match_expr" shows "replace_primitive_matchexpr disc_sel replace_f m \<equiv> let (as, rst) = primitive_extractor disc_sel m in if as = [] then m else MatchAnd (andfold_MatchExp (map replace_f as)) rst"
seq_mult_closed: fixes s :: "nat \<Rightarrow> 'a"   and s' :: "nat \<Rightarrow> 'a" assumes "s \<in> carrier R\<^bsup>\<omega>\<^esup>"   and "s' \<in> carrier R\<^bsup>\<omega>\<^esup>" shows "s \<otimes>\<^bsub>R\<^bsup>\<omega>\<^esup>\<^esub> s' \<in> carrier R\<^bsup>\<omega>\<^esup>"
square_free_iff_separable_GFp: fixes f :: "'a mod_ring poly" assumes "degree f < CARD('a)" shows "square_free f = separable f"