hcju/isabelleps huggingface dataset:
rv_branching_edges_slice_kinds_False: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and backward_slice :: "'b set \<Rightarrow> 'b set"   and a :: "'a"   and ax :: "'a"   and n :: "'b"   and n'' :: "'b"   and S :: "'b set"   and as :: "'a list"   and s :: "'c"   and asx :: "'a list"   and s' :: "'c" assumes "BackwardSlice sourcenode targetnode kind valid_edge Entry Def Use state_val backward_slice"   and "valid_edge a"   and "valid_edge ax"   and "sourcenode a = n"   and "sourcenode ax = n"   and "targetnode a = n''"   and "targetnode ax \<noteq> n''"   and "preds (BackwardSlice.slice_kinds sourcenode targetnode kind valid_edge backward_slice S (a # as)) s"   and "preds (BackwardSlice.slice_kinds sourcenode targetnode kind valid_edge backward_slice S (ax # asx)) s'"   and "\<forall>V\<in>BackwardSlice.relevant_vars sourcenode targetnode valid_edge Def Use backward_slice S n. state_val s V = state_val s' V" shows "False"
gf_G_def: fixes K :: "'b \<Rightarrow> 'b pmf"   and s :: "'b"   and t :: "'b"   and z :: "'a" shows "MC_syntax.gf_G K s t z = (\<Sum>n. MC_syntax.p K s t n *\<^sub>R z ^ n)"
divide_poly_field: fixes f :: "'a poly"   and g :: "'a poly" shows "f div g = fst (pseudo_divmod (smult (((1::'a) / lead_coeff g) ^ (Suc (degree f) - degree g)) f) g)"
less_sets_imp_strict_mono_sets: fixes A :: "nat \<Rightarrow> 'a set" assumes "\<And>i. A i \<lless> A (Suc i)"   and "\<And>i. 0 < i \<Longrightarrow> A i \<noteq> {}" shows "strict_mono_sets UNIV A"
finite_card_eq_imp_bij_betw: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "card (f ` A) = card A" shows "bij_betw f A (f ` A)"
XY_le_Max_X: fixes y :: "'a" assumes "y \<in> X \<union> Y" shows "y \<le> Max X"
assms_1: shows "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"
sum_outgoing_alt: fixes g :: "nat \<times> nat \<Rightarrow> 'capacity" assumes "\<forall>e. (0::'capacity) \<le> g e \<and> g e \<le> c e" shows "\<forall>v\<in>V. sum g (outgoing v) = (\<Sum>u\<in>V. g (v, u))"
elims_2: fixes x :: "iface"   and xa :: "char list" assumes "match_iface x xa" obtains i :: "char list" and p_iface :: "char list" where "x = Iface i" and "xa = p_iface" and "internal_iface_name_match i p_iface"
cname_of_def: fixes h :: "'heap"   and a :: "'addr" shows "h.cname_of h a = the_Class (ty_of_htype (the (typeof_addr a)))"
block_design_combine: shows "block_design \<V>\<^sup>+ \<B>\<^sup>+ \<k>"
typ_of_subst_bv_no_change: fixes t :: "Core.term"   and u :: "Core.term" assumes "typ_of t \<noteq> None" shows "subst_bv u t = t"
cong_exp_iff_simps_11: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'c \<times> 'c"   and divmod_step :: "num \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c \<times> 'c"   and m :: "num"   and q :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus (num.Bit1 m)) (numeral.numeral one plus (num.Bit0 q)) = modulo (numeral.numeral one plus num.One) (numeral.numeral one plus (num.Bit0 q))) = (modulo (numeral.numeral one plus m) (numeral.numeral one plus q) = zero)"
P1: shows "C B A' CongA D E F \<and> \<not> B C OS A A' \<and> Coplanar A B C A' \<and> A B A' CongA G H I"
lmeasurable_cbox: fixes a :: "'c"   and b :: "'c" shows "cbox a b \<in> lmeasurable"
ps\<^sub>R: shows "sparse (dist (case C\<^sub>0' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case C\<^sub>1' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) ps\<^sub>R"
chain_boundary_sum: fixes p :: "nat"   and g :: "'c \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and I :: "'c set" shows "chain_boundary p (sum g I) = sum (chain_boundary p \<circ> g) I"
arg_cnj_not_pi: fixes z :: "complex" assumes "arg z \<noteq> pi" shows "arg (cnj z) = - arg z"
has_subprob_densityI: fixes f :: "'a \<Rightarrow> ennreal"   and N :: "'a measure"   and M :: "'a measure" assumes "f \<in> borel_measurable N"   and "M = density N f"   and "subprob_space M" shows "has_subprob_density M N f"
assms_2: fixes x :: "'a" assumes "consider' (Inr x)" shows "consider x"
pre: shows "pre @ [Broadcast (r, Expunge e2 mo r)] prefix of k"
simps_1: fixes s :: "('c, 'd) fmap" shows "statelist' s [] = [s]"
steps_star_cycle: fixes us :: "'b list list"   and A :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" assumes "\<forall>u\<in>set us. NAe.accepts A u" shows "(True # start A, True # start A) \<in> NAe.steps (RegExp2NAe.star A) (concat us)"
Inf_closed_cl_op_var: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and cl_op :: "'b \<Rightarrow> 'b"   and Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and X :: "'b set" assumes "class.clattice_with_clop less_eq less cl_op Inf Sup inf sup bot top"   and "\<forall>x\<in>X. x \<in> range cl_op" shows "Inf X \<in> range cl_op"
gbinomial_asymptotic: fixes z :: "'a" shows "(\<lambda>n. (z gchoose n) / ((- (1::'a)) ^ n / exp ((z + (1::'a)) * of_real (ln (real n))))) \<longlonglongrightarrow> inverse (Gamma (- z))"
r: shows "\<forall>D\<in>set (C # Cs). \<forall>(f, g)\<in>fundfoldpairs. r = Abs_induced_automorph f g \<longrightarrow> D \<in> f \<turnstile> \<C>"
simps_4: fixes x2 :: "nat"   and x1 :: "bool" shows "Indet x2 \<noteq> Det x1"
soundnessFEx: fixes x :: "vbl"   and A :: "formula"   and Gamma :: "formula list" assumes "validS (instanceF x A # Gamma)" shows "validS (FAll Neg A # Gamma)"
vsubst_compose2: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and z1 :: "'var"   and y1 :: "'var" assumes "good X"   and "ys \<noteq> zs \<or> y \<notin> {z, z1}" shows "X #[y1 // y]_ys #[z1 // z]_zs = X #[z1 // z]_zs #[y1 @ys[z1 / z]_zs // y]_ys"
listset: fixes xs :: "'a set list" shows "listset xs = {ys. length ys = length xs \<and> (\<forall>i<length xs. ys ! i \<in> xs ! i)}"
bij_betw_imp_surj_on: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw f A B" shows "f ` A = B"
ass_function_Complete_set_non_associates: fixes f :: "'a \<Rightarrow> 'a" assumes "ass_function f" shows "Complete_set_non_associates (range f)"
hd_zip: fixes xs :: "'a list"   and ys :: "'b list" assumes "xs \<noteq> []"   and "ys \<noteq> []" shows "hd (zip xs ys) = (hd xs, hd ys)"
lookup_pp_def: shows "lookup_pp \<equiv> map_fun pp.mapping_of id lookup"
mat_adj_inv: fixes M :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det M \<noteq> 0" shows "mat_adj (mat_inv M) = mat_inv (mat_adj M)"
apartment_min_gallery_pgallery: fixes X :: "'a set set"   and xs :: "'a set list" assumes "X \<in> \<A>"   and "SimplicialComplex.min_maxsimpchain X xs" shows "SimplicialComplex.pmaxsimpchain X xs"
invfun_inj: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "surj_to f A B" shows "inj_on (f\<inverse>\<^bsub>B,A\<^esub>) B"
prefix_to_carriers: fixes xs :: "'evt list"   and i :: "nat" assumes "xs prefix of i" shows "set xs \<subseteq> set (history i)"
periodic_arithmetic_sum_periodic_arithmetic_shift: fixes f :: "nat \<Rightarrow> complex"   and k :: "nat"   and d :: "nat" assumes "periodic_arithmetic f k"   and "0 < k"   and "0 < d" shows "sum f {0..k - 1} = sum f {d..d + k - 1}"
mult_left_sub_dist_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * y \<squnion> x * z \<le> x * (y \<squnion> z)"
coprime_bezout_coefficients: fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and b :: "'a" assumes "comm_monoid_mult_class.coprime f g"   and "bezout_coefficients f g = (a, b)" shows "a * f + b * g = (1::'a)"
real_divr_def: fixes prec :: "nat"   and a :: "real"   and b :: "real" shows "real_divr prec a b = truncate_up prec (a / b)"
SBWriteNonVolatile: fixes a :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'c memref list"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "'d"   and \<O> :: "'e"   and \<R> :: "'f"   and \<S> :: "'g" shows "(Write False a (D, f) A L R W # is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b (is, \<theta>, sb @ [Write\<^sub>s\<^sub>b False a (D, f) (f \<theta>) A L R W], m, \<D>, \<O>, \<R>, \<S>)"
Un_Int_assoc_eq: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "(A \<inter> B \<union> C = A \<inter> (B \<union> C)) = order.greater_eq A C"
lindep_same: fixes S :: "'a vec set" assumes "S \<subseteq> mat_kernel A" shows "KA.lin_dep S = MN.lin_dep S"
InitStatus_def: fixes A :: "('a, 'b, 'c) hierauto" shows "InitStatus A \<equiv> Abs_status (A, InitConf A, {}, HAInitValue A)"
split_min_bst: fixes t :: "('a \<times> 'b) tree"   and m :: "'a"   and t' :: "('a \<times> 'b) tree" assumes "split_min t = (m, t')"   and "Tree2.bst t"   and "t \<noteq> \<langle>\<rangle>" shows "Tree2.bst t' \<and> (\<forall>x\<in>Tree2.set_tree t'. m < x)"
lim_shift: fixes t :: "real"   and t' :: "real"   and s :: "'a" shows "K.lim_stream (t + t', s) = distr (K.lim_stream (t, s)) (stream_space S) (smap (\<lambda>(t, s). (t + t', s)))"
IH2: shows "continuous_on UNIV (blin_frechet (good_interp I) (simple_term \<theta>\<^sub>2))"
spmf_rel_mono_strong: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d spmf"   and g :: "'e spmf"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "rel_spmf A f g"   and "\<And>x y. \<lbrakk>x \<in> set_spmf f; y \<in> set_spmf g; A x y\<rbrakk> \<Longrightarrow> B x y" shows "rel_spmf B f g"
P_inv: shows "pState_inv prog \<lparr>pid = pidN, vars = lm.to_map ((STR ''_pid'', Var (VTBounded 0 (integer_of_nat pidN)) (integer_of_nat pidN)) # map modProcArg (zip argDecls (map (exprArith g p) args))), pc = s, channels = [], idx = sidx\<rparr>"
adds_pp_alt: fixes t :: "'a"   and v :: "'t" shows "(t adds\<^sub>p v) = (\<exists>u. v = t \<oplus> u)"
proj_setI: fixes y :: "'c"   and A :: "'c set"   and x :: "'c" assumes "y \<in> A"   and "dist x y \<le> infdist x A" shows "y \<in> proj_set x A"
simps_21: fixes vb :: "val option"   and va :: "(val \<times> val) fset" shows "Abs vb \<sqsubseteq> val.Fun va = False"
atMost_def: fixes u :: "'a" shows "ord.atMost (\<lambda>x y. y \<le> x) u \<equiv> \<up>u"
E_bernoulli: fixes p :: "real"   and f :: "bool \<Rightarrow> real" assumes "0 \<le> p"   and "p \<le> 1" shows "E (map_pmf f (bernoulli_pmf p)) = p * f True + (1 - p) * f False"
strand_sem_subst_2: fixes \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and M :: "('fun, 'var) Term.term set"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> bvars\<^sub>s\<^sub>t S = {}"   and "\<lbrakk>M; S \<cdot>\<^sub>s\<^sub>t \<delta>\<rbrakk>\<^sub>c \<theta>" shows "\<lbrakk>M; S\<rbrakk>\<^sub>c (\<delta> \<circ>\<^sub>s \<theta>)"
While: fixes \<theta> :: "nat \<Rightarrow> nat option"   and e :: "expr"   and s :: "stmt"   and t :: "nat" shows "\<theta>\<turnstile> (While e s, t) \<rightarrow>\<^sub>p ((Cond e (Seq s (While e s)) Skip, t), [])"
top_sorted_cons: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'d"   and l :: "'d list" assumes "top_sorted_abs R (h # l)" shows "top_sorted_abs R l"
refine_IdI: fixes m :: "'b nres"   and m' :: "'b nres" assumes "m \<le> m'" shows "m \<le> \<Down> Id m'"
fgr_1: shows "fgp holomorphic_on cball z fgr"
alternating_list_Suc_Cons: fixes k :: "nat"   and s :: "'a"   and t :: "'a" shows "alternating_list (Suc k) s t = s # alternating_list k t s"
similar_mat_wit_def: fixes A :: "'a mat"   and B :: "'a mat"   and P :: "'a mat"   and Q :: "'a mat" shows "similar_mat_wit A B P Q = (let n = dim_row A in {A, B, P, Q} \<subseteq> carrier_mat n n \<and> P * Q = 1\<^sub>m n \<and> Q * P = 1\<^sub>m n \<and> A = P * B * Q)"
float_eq: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" assumes "is_finite a"   and "is_finite b" shows "a \<doteq> b = (valof a = valof b)"
basis_in_def: fixes \<E> :: "'a set"   and X :: "'a set" shows "\<E>.basis_in \<E> X = indep_system.basis \<E> (\<E>.indep_in \<E>) X"
stabilizer_contains_P: fixes P :: "'a set"   and p :: "nat" assumes "finite (carrier G)"   and "P \<in> subgroups_of_size p" shows "P \<subseteq> group_action.stabilizer G (conjugation_action p) P"
frechet_derivative_def: fixes f :: "'c \<Rightarrow> 'd"   and net :: "'c filter" shows "frechet_derivative f net = (SOME f'. (f has_derivative f') net)"
exists_element_bw_mix_is_approx: fixes l :: "'a pmf"   and b :: "'a pmf"   and w :: "'a pmf" assumes "l \<in> local.\<P>"   and "b \<in> local.best"   and "w \<in> local.worst" shows "\<exists>\<alpha>\<in>{0..1}. l \<approx>[\<R>] mix_pmf \<alpha> b w"
cnf_disj: fixes R :: "'a literal list" shows "cnf (disj_of_clause R) = {set R}"
matrix_to_iarray_option_def: fixes A :: "(('b, 'c) vec, 'd) vec option" shows "matrix_to_iarray_option A = (if A \<noteq> None then Some (matrix_to_iarray (the A)) else None)"
assms_2: shows "A1 B1 C1 Cong3 A2 B2 C2"
foldl_append_empty: fixes xs :: "'b list"   and f :: "'c \<Rightarrow> 'b list"   and p :: "'c list" assumes "NO_MATCH [] xs" shows "foldl (\<lambda>b x. f x @ b) xs p = foldl (\<lambda>b x. f x @ b) [] p @ xs"
actevent_def: shows "actevent = fst \<circ> action"
rsp: shows "moebius_mat_eq id_moebius_mmat id_moebius_mmat"
inorder_insert: fixes t :: "'a tree"   and x :: "'a" assumes "Sorted_Less.sorted (inorder t)" shows "inorder (Tree_Set.insert x t) = ins_list x (inorder t)"
r_neg: fixes x :: "'a list" assumes "x \<in> carrier (poly_ring R)" shows "x \<oplus>\<^bsub>poly_ring R\<^esub> \<ominus>\<^bsub>poly_ring R\<^esub> x = \<zero>\<^bsub>poly_ring R\<^esub>"
s1: fixes \<psi> :: "'a"   and b :: "'a" shows "cmod (crepresentation basis \<psi> b) \<le> \<parallel>\<psi>\<parallel> * C"
length_MGF1: fixes l :: "nat"   and n :: "nat"   and Z :: "bit list" assumes "l \<le> Suc n * 160" shows "length (MGF1 Z n l) = l"
OTNConc: fixes y :: "('b, 'c) Combinators"   and z :: "('b, 'c) Combinators list" assumes "OnlyTwoNets (y # z)" shows "OnlyTwoNets z"
prems_2: shows "P,E,hp s \<turnstile> es [:] Ts"
add_le_cancel_left: fixes c :: "complex"   and a :: "complex"   and b :: "complex" shows "(c + a \<le>\<^sub>\<complex> c + b) = (a \<le>\<^sub>\<complex> b)"
cone_closed_times: fixes a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "a \<in> cone (h, U)"   and "q \<in> P[U]" shows "q * a \<in> cone (h, U)"
p3: shows "Esigma (h * N) = Esigma h * Esigma N"
simps_8: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and \<nu> :: "'a set" shows "af_letter (\<phi> U\<^sub>n \<psi>) \<nu> = af_letter \<psi> \<nu> or\<^sub>n (af_letter \<phi> \<nu> and\<^sub>n \<phi> U\<^sub>n \<psi>)"
comp_null_1: fixes f :: "'a" shows "A.null \<cdot>\<^sub>A f = A.null"
subset_iff: fixes A :: "'k set"   and B :: "'k set" shows "(A \<subseteq> B) = (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
is_equivalent_to_strict_subcategory: shows "equivalence_of_categories \<F>\<^sub>SC.comp (\<cdot>) \<F>C.D \<F>\<^sub>SC.map \<F>C.\<nu> \<F>C.\<mu>"
unitary_gen_regular: fixes M :: "complex \<times> complex \<times> complex \<times> complex" assumes "unitary_gen M" shows "mat_det M \<noteq> 0"
vwb_lens_mwb: fixes x :: "'e \<Longrightarrow> 'f" assumes "vwb_lens x" shows "mwb_lens x"
remove_child_heap_is_wellformed_preserved_2: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> remove_child ptr child \<rightarrow>\<^sub>h h'"   and "known_ptrs h"   and "type_wf h" shows "known_ptrs h'"
intros_17: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list"   and s :: "'d" assumes "\<Gamma> p = Some bdy" shows "\<Gamma>\<turnstile> (Call p # cs, css, Normal s) \<rightarrow> ([bdy], (cs, Throw # cs) # css, Normal s)"
field_differentiable_ident: fixes F :: "'a filter" shows "(\<lambda>z. z) field_differentiable F"
congruenceSubst_def: fixes P :: "pi"   and Q :: "pi" shows "P \<simeq>\<^sup>s Q \<equiv> (P, Q) \<in> substClosed congruence"
elims_2: fixes x :: "'a local.term" assumes "Nml x" obtains \<mu> :: "'a" where "x = \<^bold>\<langle>\<mu>__\<^bold>\<rangle>\<^sub>0" and "obj \<mu>__"   | \<mu> :: "'a" where "x = \<^bold>\<langle>\<mu>\<^bold>\<rangle>" and "arr \<mu>"   | \<nu> :: "'a" and u :: "'a local.term" where "x = \<^bold>\<langle>\<nu>\<^bold>\<rangle> \<^bold>\<star> u" and "arr \<nu> \<and> Nml u \<and> \<not> is_Prim\<^sub>0 u \<and> \<^bold>\<langle>src \<nu>\<^bold>\<rangle>\<^sub>0 = Trg u"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "S.in_hom f a b = (S.arr a \<and> S.arr b \<and> S.arr f \<and> \<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>)"
fls_subdegree_fls_to_fps: fixes f :: "'b fps" shows "fls_subdegree (fps_to_fls f) = int (subdegree f)"
disj_com: fixes x :: "'b"   and y :: "'b" shows "total_partial.disjoint x y = total_partial.disjoint y x"
gen_subalgebra_is_subalgebra: fixes G :: "'c set set"   and M :: "'c measure" assumes "G \<subseteq> sets M"   and "sigma_algebra (space M) G" shows "subalgebra M (gen_subalgebra M G)"
smult_lincomb: fixes rs :: "'f list"   and s :: "'f"   and ms :: "'v list" assumes "set rs \<subseteq> UNIV"   and "s \<in> UNIV"   and "set ms \<subseteq> V" shows "s \<cdot> rs \<bullet>\<cdot> ms = map ((*) s) rs \<bullet>\<cdot> ms"
expand: fixes list :: "'b list"   and list' :: "'b list" assumes "(list = []) = (list' = [])"   and "\<lbrakk>list \<noteq> []; list' \<noteq> []\<rbrakk> \<Longrightarrow> hd list = hd list' \<and> tl list = tl list'" shows "list = list'"
hyps: assumes "s \<le> n" shows "(\<Prod>j = s..n. a powr 2 ^ j) = a powr sum ((^) 2) {s..n}"
a'_def: shows "a' = (\<lambda>n. 2 / sqrt pi * (if even n then (- 1) ^ (n div 2) / fact (n div 2) else 0))"
sublist_Nil_left: fixes ys :: "'a list" shows "sublist [] ys"
enat_unfold: fixes stop :: "'a \<Rightarrow> bool"   and "next" :: "'a \<Rightarrow> 'a"   and a :: "'a" shows "enat_unfold stop next a = (if stop a then 0 else eSuc (enat_unfold stop next (next a)))"
assms_2: shows "a0 \<in> face_cycle_set a"
A: shows "steps' init (take n qs) (take n Strat) n = steps' init qs Strat n"
i_Exec_Comp_Stream_Acc_Output__eq_Msg_State_Idle_conv2': fixes k :: "nat"   and localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'd message_af"   and c :: "'a"   and t :: "nat"   and m :: "'c message_af"   and t0 :: "nat"   and s :: "nat \<Rightarrow> 'a" assumes "Suc 0 < k"   and "State_Idle localState output_fun trans_fun (i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c t)"   and "m \<noteq> NoMsg"   and "t0 = t * k"   and "s = i_Exec_Comp_Stream trans_fun (input \<odot> k) c" shows "(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = m) = ((\<diamond> t1 [0\<dots>,k - Suc 0] \<oplus> t0. output_fun (s t1) = m \<and> State_Idle localState output_fun trans_fun (localState (s t1))) \<or> (\<diamond> t1 [0\<dots>,k - Suc 0] \<oplus> t0. output_fun (s t1) = m \<and> (\<circle> t2 t1 [0\<dots>]. output_fun (s t3) = NoMsg. t3 \<U> t4 [0\<dots>] \<oplus> t2. output_fun (s t4) = NoMsg \<and> State_Idle localState output_fun trans_fun (localState (s t4)))))"
upper_minus: fixes A :: "'b interval"   and B :: "'b interval" shows "upper (A - B) = upper A - lower B"
assms: shows "a \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>)"
eq: shows "p * a * x + p * b * y + q * c * y = u"
var1: fixes x :: "'a"   and y :: "'a"   and M :: "'a trm" assumes "x = y" shows "substitutes (Var x) y M M"
homologous_rel_sym: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "homologous_rel p X S a b = homologous_rel p X S b a"
Max_eq_if: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and B :: "'c set" assumes "class.linorder less_eq less"   and "finite A"   and "finite B"   and "\<forall>a\<in>A. \<exists>b\<in>B. less_eq a b"   and "\<forall>b\<in>B. \<exists>a\<in>A. less_eq b a" shows "linorder.Max less_eq A = linorder.Max less_eq B"
integral_periodic_offset: fixes f :: "real \<Rightarrow> real"   and b :: "real"   and a :: "real"   and c :: "real" assumes "\<And>x. f (x + (b - a)) = f x" shows "LINT x|lebesgue_on {a..b}. f (x + c) = integral\<^sup>L (lebesgue_on {a..b}) f"
range_map_indets: fixes f :: "'e \<Rightarrow> 'c" shows "range (map_indets f) = P[range f]"
carrier_singleton_group: fixes a :: "'a" shows "carrier (singleton_group a) = {a}"
upset_inj: fixes dual :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.order_with_dual dual less_eq less" shows "inj (ord.upset less_eq)"
generate_one_irrel': fixes A :: "'a set" shows "generate G A = generate G (A - {\<one>})"
simps_2: fixes G :: "('c, 'd) graph"   and v :: "'c"   and v1 :: "'c"   and w :: "'d"   and v2 :: "'c"   and p :: "('c \<times> 'd \<times> 'c) list"   and v' :: "'c" shows "is_path_undir G v ((v1, w, v2) # p) v' = (v = v1 \<and> ((v1, w, v2) \<in> edges G \<or> (v2, w, v1) \<in> edges G) \<and> is_path_undir G v2 p v')"
type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap" assumes "l_type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t type_wf"   and "type_wf h" shows "ObjectClass.type_wf h"
mm_equiv_elim: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and c\<^sub>1 :: "'a"   and mds :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>1 :: "'b \<Rightarrow> 'c"   and c\<^sub>2 :: "'a"   and mem\<^sub>2 :: "'b \<Rightarrow> 'c"   and dma :: "'b \<Rightarrow> Sec" assumes "sifum_security stop eval some_val some_val'"   and "(((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>2, mds), mem\<^sub>2) \<in> sifum_security.mm_equiv dma eval" obtains \<R> :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set" where "sifum_security.strong_low_bisim_mm dma eval \<R>" and "(((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>2, mds), mem\<^sub>2) \<in> \<R>"