hcju/isabelleps huggingface dataset:
elims: fixes x :: "'k list"   and xa :: "'k list"   and y :: "bool option" assumes "subset_eq_mset_impl x xa = y" obtains ys :: "'k list" where "xa__ = []" and "xa = ys__" and "y = Some (ys__ \<noteq> [])"   | x :: "'k" and xs :: "'k list" and ys :: "'k list" where "xa__ = x # xs" and "xa = ys" and "y = (case List.extract ((=) x) ys of None \<Rightarrow> None | Some (ys1, x, ys2) \<Rightarrow> subset_eq_mset_impl xs (ys1 @ ys2))"
assms: shows "Fn x T A \<rightarrow>\<beta> M'"
inject_2: fixes x21 :: "secprop set"   and x22 :: "key"   and x23 :: "payload"   and y21 :: "secprop set"   and y22 :: "key"   and y23 :: "payload" shows "(DynCh x21 x22 x23 = DynCh y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
divide_self: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> '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" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero" shows "divide a a = one"
bv_or_set_of_bv: fixes A :: "bool list bdd list \<times> bool list"   and p :: "bool list"   and q :: "bool list" assumes "nfa_is_node A p"   and "nfa_is_node A q" shows "set_of_bv (bv_or p q) = set_of_bv p \<union> set_of_bv q"
not_Some_eq: fixes x :: "'g option" shows "(\<forall>y. x \<noteq> \<lfloor>y\<rfloor>) = (x = None)"
preserves_iso: fixes f :: "'b" assumes "C.iso f" shows "D.iso (G f)"
hconverse_def: fixes r :: "hf" shows "hconverse r = \<lbrace>z . w \<^bold>\<in> r, \<exists>x y. w = \<langle>x, y\<rangle> \<and> z = \<langle>y, x\<rangle>\<rbrace>"
image_filter_correct_aux_1: fixes s :: "'s1"   and f :: "'a \<Rightarrow> 'b option" assumes "invar1 s" shows "\<alpha>2 (image_filter f s) = {b. \<exists>a\<in>\<alpha>1 s. f a = Some b}"
simps: fixes a1 :: "'f"   and a2 :: "'g list"   and a3 :: "'f"   and t :: "('f \<times> 'g \<times> 'f) set" shows "((a1, a2, a3) \<in> trcl t) = ((\<exists>c. a1 = c \<and> a2 = [] \<and> a3 = c) \<or> (\<exists>c a c' w c''. a1 = c \<and> a2 = a # w \<and> a3 = c'' \<and> (c, a, c') \<in> t \<and> (c', w, c'') \<in> trcl t))"
equal_rat_def: fixes a :: "rat"   and b :: "rat" shows "equal_class.equal a b = (quotient_of a = quotient_of b)"
nested_prop_atoms\<^sub>\<mu>_def: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" shows "nested_prop_atoms\<^sub>\<mu> \<phi> X = {\<psi>[X]\<^sub>\<mu> |\<psi>. \<psi> \<in> nested_prop_atoms \<phi>}"
ipresIGSwapAbs_termMOD: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'varSort, 'k, 'l, 'var, 'm, 'h, 'n) model_scheme" shows "ipresIGSwapAbs hA termMOD MOD = ipresSwapAbs hA MOD"
eval: shows "P \<turnstile> \<langle>body,(h, [this # pns [\<mapsto>] vs])\<rangle> \<Rightarrow> \<langle>e',(h', l')\<rangle>"
is_pappus1_def: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and Q :: "Points"   and R :: "Points" shows "is_pappus1 A B C A' B' C' P Q R \<equiv> distinct6 A B C A' B' C' \<longrightarrow> col A B C \<longrightarrow> col A' B' C' \<longrightarrow> is_a_proper_intersec P A B' A' B \<longrightarrow> is_a_proper_intersec Q B C' B' C \<longrightarrow> is_a_proper_intersec R A C' A' C \<longrightarrow> col P Q R"
distr_id: fixes N :: "'a measure" shows "distr N N (\<lambda>x. x) = N"
divide_cong: shows "\<forall>\<^sub>F x in at_top. poly q x / poly p x = qq x / pp x * dd x"
freshChainAppend: fixes xvec :: "name list"   and yvec :: "name list"   and C :: "'d" shows "(xvec @ yvec) \<sharp>* C = (xvec \<sharp>* C \<and> yvec \<sharp>* C)"
uminus_rel_frac: fixes r :: "'a"   and s :: "'a" assumes "(r, s) \<in> R \<times> (R\<setminus>I)" shows "cq.uminus_rel (cq.frac r s) = cq.frac (comm.additive.inverse r) s"
meet_def': fixes x :: "'c"   and y :: "'c" shows "x \<sqinter> y = \<Sqinter>{x, y}"
map_le_iff_map_add_commute: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" shows "(f \<subseteq>\<^sub>m f ++ g) = (f ++ g = g ++ f)"
vertical_composite_inverse_iso: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "natural_isomorphism A B F G \<tau>" shows "vertical_composite.map A B (inverse_transformation.map A B G \<tau>) \<tau> = G"
streams_mono2: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T" shows "streams S \<subseteq> streams T"
prob_eq_1: assumes "{x} \<in> measure_pmf.events \<mu>" shows "(measure_pmf.prob \<mu> {x} = 1) = (AE xa in measure_pmf \<mu>. xa \<in> {x})"
wf_cdecl_mdecl: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and m :: "mdecl" assumes "wf_cdecl G (C, c)"   and "m \<in> set (methods c)" shows "wf_mdecl G C m"
Mp_smult_2: fixes m :: "int"   and a :: "int"   and f :: "int poly" shows "poly_mod.Mp m (smult a (poly_mod.Mp m f)) = poly_mod.Mp m (smult a f)"
y_split_1: shows "y = g \<diamondop> Opaque y'"
subdiv_distinct_arcs: shows "distinct [uv, rev_G uv, uw, rev_H uw, vw, rev_H vw]"
f2: shows "AA (R \<cdot> 1\<^sub>\<pi>) A a \<subseteq> A \<and> (a, AA (R \<cdot> 1\<^sub>\<pi>) A a) \<in> R \<cdot> 1\<^sub>\<pi>"
properfactor_fmset_ne: fixes a :: "'a"   and b :: "'a"   and as :: "'a list"   and bs :: "'a list" assumes "properfactor G a b"   and "wfactors G as a"   and "wfactors G bs b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "fmset G as \<noteq> fmset G bs"
simps: fixes f :: "'op"   and ts :: "'op rose_tree list" shows "local.wf (T f ts) = (length ts = arity f \<and> Ball (set ts) local.wf)"
distr_def: fixes M :: "'c measure"   and N :: "'b measure"   and f :: "'c \<Rightarrow> 'b" shows "distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f -` A \<inter> space M))"
fun\<pi>: shows "\<pi> \<in> elts \<alpha> \<rightarrow> elts (vcard \<alpha>)"
order_refl: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f" assumes "class.preorder less_eq less" shows "less_eq x x"
attract: fixes y :: "'c"   and x :: "'c"   and z :: "'c" assumes "y \<le> x"   and "x \<le> y"   and "z \<le> y" shows "z \<le> x"
double_gauss_sum: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "times (numeral.numeral one plus (num.Bit0 num.One)) (comm_monoid_add.sum plus zero (semiring_1.of_nat one plus zero) {0..n}) = times (semiring_1.of_nat one plus zero n) (plus (semiring_1.of_nat one plus zero n) one)"
simps_4: fixes test :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'e"   and r :: "'d regex"   and s :: "'d regex" shows "rpd test i (Times r s) = TimesL r (rpd test i s) \<union> (if eps test i s then rpd test i r else {})"
cwiseext_trans_strong: fixes zs :: "'b list"   and ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>z\<in>set zs. \<forall>y\<in>set ys. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "cwiseext gt zs ys"   and "cwiseext gt ys xs" shows "cwiseext gt zs xs"
Fun_def: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and f :: "'b" assumes "set_category_given_img S img \<AA>" shows "set_category_given_img.Fun S img f = restrict (img \<circ> S f \<circ> inv_into {fa. partial_magma.in_hom S fa (set_category_data.unity S) (partial_magma.dom S f)} img) (set_category_data.set S img (partial_magma.dom S f))"
intros_6: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<sqsubset> \<sigma>" shows "Set \<tau> \<sqsubset> Set \<sigma>"
not_infinity_eq: fixes x :: "eint" shows "(x \<noteq> \<infinity>) = (\<exists>i. x = eint i)"
weak_seq_inf_distrib: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice"   and d\<^sub>0 :: "'b::refinement_lattice"   and d\<^sub>1 :: "'b::refinement_lattice" assumes "seq_distrib_left (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "(seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c::'b::refinement_lattice) ((d\<^sub>0::'b::refinement_lattice) \<sqinter> (d\<^sub>1::'b::refinement_lattice)) \<sqsubseteq> seq c d\<^sub>0 \<sqinter> seq c d\<^sub>1"
lin_independent_ConsD1: fixes m :: "'v"   and ms :: "'v list" assumes "R_lin_independent (m # ms)" shows "R_lin_independent ms"
simps: fixes r :: "'b \<Rightarrow> nat option"   and \<nu> :: "'a"   and q :: "'b" shows "pre_ranks r \<nu> q = {i. \<exists>q'. r q' = Some i \<and> q = \<delta> q' \<nu>} \<union> (if q = q\<^sub>0 then {max_rank} else {})"
deriv_j_ladder_shift_j: fixes index :: "nat"   and L :: "(nat \<times> nat \<times> nat) list"   and d :: "nat" assumes "index < length L" shows "deriv_j (ladder_shift_j d L ! index) = deriv_j (L ! index) - d"
prems_1: shows "f \<in> insert a A \<rightarrow> carrier G"
I_subst\<^sub>21: fixes f :: "atom fm"   and xs :: "'a list"   and l :: "nat"   and u :: "nat"   and x :: "'a" assumes "nqfree f"   and "xs ! l < xs ! u"   and "DLO.I (subst\<^sub>2 l u f) xs"   and "xs ! l < x"   and "x < xs ! u" shows "DLO.I f (x # xs)"
sig_crit'_sym: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_crit' bs (Inl (p, q))" shows "sig_crit' bs (Inl (q, p))"
preserves_section_retraction: fixes e :: "'a \<times> 'a"   and m :: "'a \<times> 'a" assumes "C.VV.ide (C.VV.comp e m)" shows "C.ide ((fst e \<star>\<^sub>C snd e) \<cdot>\<^sub>C (fst m \<star>\<^sub>C snd m))"
malt_SE_def: fixes S :: "('b \<Rightarrow> ('c \<times> 'b) option) list" shows "\<Sqinter>\<^sub>S\<^sub>E S = foldr (\<sqinter>\<^sub>S\<^sub>E) S fail\<^sub>S\<^sub>E"
le_sum_encode_Inl: fixes x :: "nat"   and y :: "nat" assumes "x \<le> y" shows "x \<le> sum_encode (Inl y)"
A': shows "A = A \<inter># M + (A - M) \<inter># N"
subformulas\<^sub>\<mu>\<^sub>\<nu>_subfrmlsn: fixes \<phi> :: "'b ltln" shows "subformulas\<^sub>\<mu> \<phi> \<union> subformulas\<^sub>\<nu> \<phi> \<subseteq> subfrmlsn \<phi>"
basic_just_sp_pred_axioms_def: fixes G :: "('c, 'd) pre_digraph"   and dist :: "'c \<Rightarrow> ereal"   and c :: "'d \<Rightarrow> real"   and s :: "'c"   and num :: "'c \<Rightarrow> enat"   and pred :: "'c \<Rightarrow> 'd option" shows "basic_just_sp_pred_axioms G dist c s num pred \<equiv> \<forall>v. v \<in> verts G \<longrightarrow> v \<noteq> s \<longrightarrow> num v \<noteq> \<infinity> \<longrightarrow> (\<exists>e\<in>arcs G. e = the (pred v) \<and> v = head G e \<and> dist v = dist (tail G e) + ereal (c e) \<and> num v = num (tail G e) + enat 1)"
simps_1: fixes s :: "'b" shows "tLength (tInit s) = 0"
supp_abs_residual_pair: fixes \<alpha> :: "'a"   and P :: "'b" shows "supp \<langle>\<alpha>,P\<rangle> = supp (\<alpha>, P) - bn \<alpha>"
edka3_refine: shows "edka3 \<le> \<Down> Id edka2"
add_image_real_of_ereal_image_greaterThanLessThan: fixes c :: "real"   and l :: "ereal"   and u :: "ereal" shows "(+) c ` real_of_ereal ` {l<..<u} = real_of_ereal ` {ereal c + l<..<ereal c + u}"
lemma_1: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R"   and "is_pappus" shows "col M N P \<or> incid A (line B' C') \<or> incid C' (line A B)"
select_convs_3: fixes mag :: "'b"   and dim :: "(int, 'c) dimvec"   and more :: "'a" shows "Quantity.more \<lparr>mag = mag, dim = dim, \<dots> = more\<rparr> = more"
dedekind: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x ; y \<cdot> z \<le> (x \<cdot> z ; y\<^sup>\<smile>) ; (y \<cdot> x\<^sup>\<smile> ; z)"
le_Cons: fixes x :: "nat"   and xs :: "nat list"   and y :: "nat"   and ys :: "nat list" shows "(x # xs \<le>\<^sub>v y # ys) = (x \<le> y \<and> xs \<le>\<^sub>v ys)"
d\<mu>_def: fixes i :: "nat"   and j :: "nat" shows "fs_init.d\<mu> i j = int_of_rat (rat_of_int (fs_init.d fs_init (Suc j)) * gs.\<mu> i j)"
assms_2: shows "Suc i = j \<and> Suc j = k \<and> (finite X \<longrightarrow> k < card X)"
hyps_3: shows "hyps (nodeOf v\<^sub>1) p\<^sub>1 = None"
rev_def: shows "rev \<equiv> rec_list [] (\<lambda>x xs xsa. xsa @ [x])"
DERIV_isCont: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a" assumes "(f has_field_derivative D) (at x)" shows "isCont f x"
c_converg: assumes "n \<noteq> 0" shows "eval c [f \<triangleright> n] \<down>"
H: fixes p :: "'b pl"   and H :: "'b pl set" assumes "p \<in> H" shows "H |- p"
unitary11_gen_iff: fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "unitary11_gen M = (\<exists>k a b. k \<noteq> 0 \<and> mat_det (a, b, cnj b, cnj a) \<noteq> 0 \<and> M = k *\<^sub>s\<^sub>m (a, b, cnj b, cnj a))"
DBM_reset_empty'': fixes n :: "nat"   and v :: "'c \<Rightarrow> nat"   and c :: "'c"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and d :: "'d"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "v c \<le> n"   and "DBM_reset M n (v c) d M'" shows "([M]\<^bsub>v,n\<^esub> = {}) = ([M']\<^bsub>v,n\<^esub> = {})"
SeqCTerm_def: fixes vf :: "bool"   and s :: "hf"   and k :: "hf"   and t :: "hf" shows "SeqCTerm vf s k t \<equiv> BuildSeq (\<lambda>u. u = 0 \<or> vf \<and> is_Var u) (\<lambda>u v w. u = q_Eats v w) s k t"
taylor_eq_1: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier P"   and "a \<in> carrier R" shows "T\<^bsub>a\<^esub> f \<ominus>\<^bsub>P\<^esub> trms_of_deg_leq 1 (T\<^bsub>a\<^esub> f) = shift 2 (T\<^bsub>a\<^esub> f) \<otimes>\<^bsub>P\<^esub> X_poly R [^]\<^bsub>P\<^esub> 2"
measurable_convex: fixes S :: "'a set" assumes "convex S"   and "bounded S" shows "S \<in> lmeasurable"
split_asm: fixes P :: "'k \<Rightarrow> bool"   and f :: "'l \<Rightarrow> 'm \<Rightarrow> 'k"   and prod :: "'l \<times> 'm" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<nexists>x1 x2. prod = (x1, x2) \<and> \<not> P (f x1 x2))"
assms: shows "low_equal s1 s2 \<and> get_S (cpu_reg_val PSR s1) = 0 \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> t1 = snd (fst (write_state_reg_instr instr s1)) \<and> t2 = snd (fst (write_state_reg_instr instr s2))"
lunit_simps_3: fixes f :: "'a" assumes "S.ide f" shows "S.trg (S.lunit f) = S.trg f"
Star_1: fixes s :: "'a regex" assumes "\<forall>x\<in>atms r. safe Strict x \<longrightarrow> safe Lax x"   and "local.safe_regex Futu Strict r"   and "s \<in> lpd test i r" shows "local.safe_regex Futu Strict s"
map_upd_in_expansion_map_swap: fixes tab :: "'b \<Rightarrow> 'a option"   and x :: "'b"   and y :: "'a"   and vn :: "'b"   and z :: "'a"   and tab' :: "'b \<Rightarrow> 'a option" assumes "(tab(x \<mapsto> y)) vn = Some z"   and "tab vn \<noteq> Some z" shows "(tab'(x \<mapsto> y)) vn = Some z"
rep_eq: fixes x :: "'a fset" shows "fBex x = Bex (fset x)"
divide_le_eq_numeral1_2: fixes b :: "'a"   and w :: "num"   and a :: "'a" shows "(b / - numeral w \<le> a) = (a * - numeral w \<le> b)"
v11_vid_on: fixes A :: "V" shows "v11 (vid_on A)"
MGF_Loop: fixes A :: "triple set"   and c :: "stmt"   and Z :: "state"   and x :: "vname" assumes "\<forall>Z. A \<turnstile> {(=) Z} c {\<lambda>t. \<exists>n. Z -c-n\<rightarrow> t}" shows "A \<turnstile> {(=) Z} While (x) c {\<lambda>t. \<exists>n. Z -While (x) c-n\<rightarrow> t}"
maximal_conditional_maximal: fixes C :: "'a set"   and m :: "'a set" assumes "C \<in> carrier fTo D"   and "maximal\<^bsub>Iod fTo D {S \<in> carrier fTo D. C \<subseteq> S}\<^esub> m" shows "maximal\<^bsub>fTo D\<^esub> m"
distinct_7: fixes x2 :: "('a \<Rightarrow> 'b option) \<Rightarrow> bool"   and x31 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool"   and x32 :: "'c"   and x33 :: "'d"   and x34 :: "(('a \<Rightarrow> 'b option) \<Rightarrow> 'b option) list" shows "(x2)\<^sub>\<surd> \<noteq> x31:x32\<hookrightarrow>\<^bsub>x33\<^esub>x34"
has_domain_iff_has_codomain: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C" shows "(partial_magma.domains C f \<noteq> {}) = (partial_magma.codomains C f \<noteq> {})"
prod_topology_def: fixes X :: "'c topology"   and Y :: "'d topology" shows "prod_topology X Y \<equiv> topology (arbitrary union_of (\<lambda>U. U \<in> {S \<times> T |S T. openin X S \<and> openin Y T}))"
assms: shows "fst args0 \<in> keys (rep_list (snd args0))"
2: shows "eval (Mn 1 r_search_enum) [b] \<down>"
add_strict_left_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a < b" shows "c + a < c + b"
blinfun_apply_blinfun_compose: fixes a :: "'d \<Rightarrow>\<^sub>L 'b"   and b :: "'c \<Rightarrow>\<^sub>L 'd"   and c :: "'c" shows "blinfun_apply (a o\<^sub>L b) c = blinfun_apply a (blinfun_apply b c)"
abs_eq: shows "heterogenous_cblinfun_id = CBlinfun (if bounded_clinear heterogenous_identity then heterogenous_identity else (\<lambda>_. 0::'b))"
nf_inres_def: fixes m :: "'a nres"   and x :: "'a" shows "nf_inres m x \<equiv> nofail m \<and> inres m x"
writeCArray_def: fixes array :: "nat \<Rightarrow> 'a"   and index :: "nat"   and "value" :: "'a" shows "writeCArray array index value \<equiv> array(index := value)"
slim_paths_def: shows "S.slim_paths \<equiv> (\<lambda>e. (fst e, S.choose_iapath (fst e) (snd e), snd e)) ` parcs (gen_contr_graph (with_proj slim) (verts3 (with_proj slim)))"
size_mset_mono: fixes A :: "'k multiset"   and B :: "'k multiset" assumes "A \<subseteq># B" shows "size A \<le> size B"
axioms_1: fixes G :: "('b, 'c) monoid_scheme" assumes "Group.comm_monoid G" shows "Group.monoid G"
idx_pm_of_pm_plus: fixes xs :: "'d list"   and f :: "'d \<Rightarrow>\<^sub>0 'c"   and g :: "'d \<Rightarrow>\<^sub>0 'c" shows "idx_pm_of_pm xs (f + g) = idx_pm_of_pm xs f + idx_pm_of_pm xs g"
is_unit_gcd: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (gcd a b) one = algebraic_semidom.coprime times one a b"
vwb_lens_wb: fixes x :: "'d \<Longrightarrow> 'e" assumes "vwb_lens x" shows "wb_lens x"
copy_array_def: fixes a :: "'b array" shows "copy_array a = Array.freeze a \<bind> Array.of_list"
Spy_spies_bad_privateKey: fixes A :: "agent"   and b :: "keymode"   and evs :: "event list" assumes "A \<in> bad" shows "Key (privateKey b A) \<in> knows Spy evs"
exhaust_sel: fixes option :: "'b option" obtains "option = None"   | "option = Some (the option)"
eventually_in_cballs: fixes d :: "real"   and c :: "real"   and t0 :: "'b"   and x0 :: "'c" assumes "0 < d"   and "0 < c" shows "\<forall>\<^sub>F e in at_right 0. cball t0 (c * e) \<times> cball x0 e \<subseteq> cball (t0, x0) d"
sup_bot_left: fixes x :: "'e multiset" shows "{#} \<union># x = x"
map_closed: fixes a :: "'a set \<times> 'b" assumes "a \<in> Sigma I \<FF>" shows "Class a \<in> Partition"
bdd_below_def: fixes A :: "'t set" shows "ord_term_lin.bdd_below A = (\<exists>m. \<forall>x\<in>A. m \<preceq>\<^sub>t x)"
S_endomorphism: fixes s :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "ChamberComplexEndomorphism X ((\<rightarrow>) s)"
elims_3: fixes x :: "'e \<times> ('a, 'f, 'g, 'h) SLin_state_scheme" assumes "\<not> P12 x" obtains s1 :: "'e" and s2 :: "('a, 'f, 'g, 'h) SLin_state_scheme" where "x = (s1, s2)" and "\<not> (initialized s2 \<longrightarrow> \<Sqinter> initVals s2 \<preceq> dstate s2)"
Res: fixes S :: "pi"   and T :: "pi"   and x :: "name" assumes "(S, T) \<in> Rel'" shows "(<\<nu>x>S, <\<nu>x>T) \<in> Rel'"
extend_map_affine_to_sphere_cofinite_simple: fixes S :: "'a set"   and U :: "'b set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "compact S"   and "convex U"   and "bounded U"   and "aff_dim T \<le> aff_dim U"   and "S \<subseteq> T"   and "continuous_on S f"   and "f ` S \<subseteq> rel_frontier U" obtains K :: "'a set" and g :: "'a \<Rightarrow> 'b" where "finite K" and "K \<subseteq> T" and "disjnt K S" and "continuous_on (T - K) g" and "g ` (T - K) \<subseteq> rel_frontier U" and "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
bij_betw_iff_bijections: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" shows "bij_betw f A B = (\<exists>g. (\<forall>x\<in>A. f x \<in> B \<and> g (f x) = x) \<and> (\<forall>y\<in>B. g y \<in> A \<and> f (g y) = y))"
rel_S_neg_distr_cond'_stronger: fixes Co1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and Co1' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Co2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and Co2' :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Contra1 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra1' :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and Contra2 :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and Contra2' :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and tytok :: "('o \<times> 'p \<times> 'q \<times> 'r \<times> 's \<times> 't \<times> 'u) itself" assumes "rel_S_neg_distr_cond' Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok" shows "rel_G_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok"
foldl_conv_fold: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and s :: "'d"   and xs :: "'e list" shows "foldl f s xs = fold (\<lambda>x s. f s x) xs s"
vars_of_cl_lem: fixes L :: "'b literal"   and C :: "'b literal set" assumes "L \<in> C" shows "vars_of_lit L \<subseteq> vars_of_cl C"
exhaust: fixes y :: "ty_Rep" obtains "y = TBase_Rep"   | "y = TUnit_Rep"   | x31 :: "ty_Rep" and x32 :: "ty_Rep" where "y = Arrow_Rep x31 x32"
hyps_7: fixes \<Delta> :: "binding list"   and b :: "vrs"   and ba :: "trm" assumes "D @ VarB x U # \<Gamma> = \<Delta> @ VarB b U # \<Gamma>"   and "\<Gamma> \<turnstile> ba : U" shows "\<Delta> @ \<Gamma> \<turnstile> t1[b \<mapsto> ba] : (\<forall>X<:T11. T12)"
restrict_compose_right: fixes g :: "'d \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'd"   and S :: "'a set" shows "restrict (g \<circ> restrict f S) S = restrict (g \<circ> f) S"
lookup0_clearjunk0: fixes xs :: "('b, 'a) fmap"   and s :: "'b" shows "lookup0 xs s = lookup0 (clearjunk0 xs) s"
semi_inverse2: fixes x :: "'b" assumes "x \<in> carrier \<Y>" shows "\<pi>\<^sub>* x = \<pi>\<^sub>* (\<pi>\<^sup>* (\<pi>\<^sub>* x))"
power_over_fact_tendsto_0': fixes c :: "real"   and x :: "real" shows "(\<lambda>n. c * x ^ n / fact n) \<longlonglongrightarrow> 0"