hcju/isabelleps huggingface dataset:
fbox_def: fixes x :: "'a"   and y :: "'a" shows " |x] y = ad (x \<cdot> ad y)"
dp_consistency_iterator_axioms: shows "dp_consistency_iterator P lookup update cnt nxt sizef"
Start_nsys_xcpts1_4: shows "Start \<noteq> NoClassDefFoundError"
3: fixes f :: "nat \<Rightarrow> real" shows "j < output_size' (insert_weights s m f)"
finite_incidence_system_axioms: shows "finite_incidence_system \<V> \<B>"
cot_gt_zero: fixes x :: "real" assumes "0 < x"   and "x < pi / 2" shows "0 < cot x"
vifunion_vunion_is: fixes I :: "V"   and f :: "V \<Rightarrow> V"   and J :: "V" shows "\<Union>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I) \<union>\<^sub>\<circ> \<Union>\<^sub>\<circ> (VLambda J f `\<^sub>\<circ> J) = \<Union>\<^sub>\<circ> (VLambda (I \<union>\<^sub>\<circ> J) f `\<^sub>\<circ> (I \<union>\<^sub>\<circ> J))"
reduc_calc: shows "reduced_calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F"
that: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"   and "r\<lparr>a\<rparr> = b" shows "thesis"
take_nths_empty: fixes i :: "nat"   and k :: "nat"   and xs :: "'b list" shows "(take_nths i k xs = []) = (length xs \<le> i)"
dim_row_transfer_rule: fixes A :: "'d mat"   and A' :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" assumes "Mod_Type_Connect.HMA_M A A'" shows "dim_row A = CARD('f)"
tiny_semicategoryI: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "\<Z> \<alpha>"   and "vfsequence \<CC>"   and "vsv (\<CC>\<lparr>Comp\<rparr>)"   and "vcard \<CC> = 5\<^sub>\<nat>"   and "tiny_digraph \<alpha> (smc_dg \<CC>)"   and "\<And>gf. (gf \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (\<CC>\<lparr>Comp\<rparr>)) = (\<exists>g f b c a. gf = [g, f]\<^sub>\<circ> \<and> g : b \<mapsto>\<^bsub>\<CC>\<^esub> c \<and> f : a \<mapsto>\<^bsub>\<CC>\<^esub> b)"   and "\<And>b c g a f. \<lbrakk>g : b \<mapsto>\<^bsub>\<CC>\<^esub> c; f : a \<mapsto>\<^bsub>\<CC>\<^esub> b\<rbrakk> \<Longrightarrow> g \<circ>\<^sub>A\<^bsub>\<CC>\<^esub> f : a \<mapsto>\<^bsub>\<CC>\<^esub> c"   and "\<And>c d h b g a f. \<lbrakk>h : c \<mapsto>\<^bsub>\<CC>\<^esub> d; g : b \<mapsto>\<^bsub>\<CC>\<^esub> c; f : a \<mapsto>\<^bsub>\<CC>\<^esub> b\<rbrakk> \<Longrightarrow> h \<circ>\<^sub>A\<^bsub>\<CC>\<^esub> g \<circ>\<^sub>A\<^bsub>\<CC>\<^esub> f = h \<circ>\<^sub>A\<^bsub>\<CC>\<^esub> (g \<circ>\<^sub>A\<^bsub>\<CC>\<^esub> f)" shows "tiny_semicategory \<alpha> \<CC>"
asIMOD_wlsFSw: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "iwlsFSw (asIMOD SEM)"
assms: shows "hom (lead_coeff p) \<noteq> (0::'b)"
mult_closed: fixes A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "ring_closed A"   and "x \<in> A"   and "y \<in> A" shows "x * y \<in> A"
star_def: fixes x :: "'a" shows "quantale_class.star x = Sup (range ((^) x))"
continuous_map_prod_top: fixes X :: "'e topology"   and Y :: "'f topology"   and X' :: "'g topology"   and Y' :: "'h topology"   and f :: "'e \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h" shows "continuous_map (prod_topology X Y) (prod_topology X' Y') (\<lambda>(x, y). (f x, g y)) = (topspace (prod_topology X Y) = {} \<or> continuous_map X X' f \<and> continuous_map Y Y' g)"
cos_arccos_abs: fixes y :: "real" assumes "\<bar>y\<bar> \<le> 1" shows "cos (arccos y) = y"
encode_problem_with_operator_interference_exclusion_has_model_then_also_partial_encodings_3: fixes \<A> :: "sat_plan_variable \<Rightarrow> bool"   and \<Pi> :: "'b strips_problem"   and t :: "nat" assumes "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<Pi> t" shows "\<A> \<Turnstile> encode_all_frame_axioms \<Pi> t"
conversep_iff: fixes r :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and a :: "'o"   and b :: "'n" shows "r\<inverse>\<inverse> a b = r b a"
cases: fixes a1 :: "'f LDCFG_node"   and a2 :: "'g"   and Def :: "'f \<Rightarrow> 'g set"   and E :: "'f"   and X :: "'f"   and H :: "'g set"   and L :: "'g set" assumes "(a1, a2) \<in> lift_Def_set Def E X H L" obtains V :: "'g" and n :: "'f" where "a1 = Node n" and "a2 = V__" and "V__ \<in> Def n"   | V :: "'g" where "a1 = Node E" and "a2 = V" and "V \<in> H"
sg5: shows "x t = [] \<or> x t = [sc_ack]"
wtC: fixes c :: "('a, 'b) lit list" assumes "M.wtC c" shows "local.wtC c"
sigma_finite_iff_density_finite: fixes f :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (density M h)" shows "sigma_finite_measure (density (density M h) f) = (AE x in density M h. f x \<noteq> \<infinity>)"
update_tableau_valuated: fixes update :: "nat \<Rightarrow> 'a \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and s :: "('b, 'a) state"   and x :: "nat"   and v :: "'a" assumes "Update update"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x \<notin> lvars (\<T> s)" shows "\<nabla> (update x v s)"
prems: shows "bal_i n (height \<langle>l, y, r\<rangle> + d)"
frechet_derivative_at: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "(f has_derivative f') (at x)" shows "f' = frechet_derivative f (at x)"
inverts_components: fixes a :: "'d \<times> 'c" assumes "DopxC.ide a" shows "S.inverse_arrows (\<Phi>.map a) (\<Psi>.map a)"
map_id: fixes t :: "'g pmf" shows "map_pmf id t = t"
A: fixes n :: "nat"   and run :: "nat"   and b :: "rat" shows "Pri_AgrK (priAK n) \<notin> items (A \<union> spies evsFR4) \<and> (IntMapK (S (Card n, n, run)) = Some b \<longrightarrow> Pri_AgrK b \<notin> items (A \<union> spies evsFR4))"
lessThan_iff: fixes i :: "'a"   and k :: "'a" shows "(i \<in> r.lessThan k) = ((i, k) \<in> r)"
two_lists_cases_Cons_Cons: fixes as :: "'b list"   and bs :: "'a list" obtains ys :: "'a list" where "as = []" and "bs = ys__"   | xs :: "'b list" where "as = xs__" and "bs = []"   | x :: "'b" and xs :: "'b list" and y :: "'a" and ys :: "'a list" where "as = x # xs" and "bs = y # ys"
sWlsDisj_imp_sWlsAbsDisj: fixes SEM :: "('b, 'c, 'd, 'sort, 'e, 'f, 'g) semDom_scheme" assumes "sWlsDisj SEM"   and "sWlsNE SEM" shows "sWlsAbsDisj SEM"
brc_initial_def: fixes \<delta> :: "('a, 'b) ta_rule dlist" shows "brc_initial \<delta> \<equiv> let iq = brc_iq \<delta> in (iq, hs.to_list iq, brc_rcm_init \<delta>)"
FCI2: shows "FCI \<Gamma>2 \<V>2 Tr\<^bsub>ES2\<^esub>"
assms_3: shows "component_of_term ` Keys B1 \<subseteq> component_of_term ` Keys B2"
fun_im: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "{f a |a. a \<in> A} = {b. b \<in> f ` A}"
1: shows "[y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> (x\<^sup>P)) in dw]"
get_child_nodes_new_document: fixes ptr' :: "(_) object_ptr"   and new_document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr' \<noteq> cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^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 new_document_ptr"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr"   and "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "r \<in> get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ptr'" shows "r h h'"
interp_ins_subset: fixes op_list :: "('b \<times> 'b option) list" shows "set (interp_ins op_list) \<subseteq> set (map fst op_list)"
AbstFormP_graph_aux_def: shows "AbstFormP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>AbstFormP_aux s v i x x' k. x1 = (v, i, x, x') \<and> x2 = VarP v AND OrdP i AND SyntaxN.Ex s (SyntaxN.Ex k (SeqAbstFormP v i x x' (Var s) (Var k))) \<and> atom s \<sharp> (v, i, x, x', k) \<and> atom k \<sharp> (v, i, x, x'))"
arg_min_if_finite_1: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite S"   and "S \<noteq> {}" shows "arg_min_on f S \<in> S"
assms_7: fixes scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp" assumes "P scrut"   and "P e1"   and "P e2" shows "P (scrut ? e1 : e2)"
map\<^sub>0_in_hom_2: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>\<sigma>\<^sub>0 a : \<sigma>\<^sub>0 a \<Rightarrow>\<^sub>D \<sigma>\<^sub>0 a\<guillemotright>"
get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def: fixes document_ptr :: "(_) document_ptr"   and h :: "(_) heap" shows "get document_ptr h = Option.bind (get (cast document_ptr) h) cast"
eval_fm_graph_aux_def: shows "eval_fm_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>eval_fm_aux e t u. x1 = (e, t IN u) \<and> x2 = (\<lbrakk>t\<rbrakk>e \<^bold>\<in> \<lbrakk>u\<rbrakk>e)) \<or> (\<exists>eval_fm_aux e t u. x1 = (e, t EQ u) \<and> x2 = (\<lbrakk>t\<rbrakk>e = \<lbrakk>u\<rbrakk>e)) \<or> (\<exists>eval_fm_aux e A B. x1 = (e, A OR B) \<and> x2 = (((\<lambda>uu_. uu_) \<circ> eval_fm_aux) (e, A) \<or> ((\<lambda>uu_. uu_) \<circ> eval_fm_aux) (e, B)) \<and> p (e, A) (((\<lambda>uu_. uu_) \<circ> eval_fm_aux) (e, A)) \<and> p (e, B) (((\<lambda>uu_. uu_) \<circ> eval_fm_aux) (e, B))) \<or> (\<exists>eval_fm_aux e A. x1 = (e, Neg A) \<and> x2 = (\<not> ((\<lambda>uu_. uu_) \<circ> eval_fm_aux) (e, A)) \<and> p (e, A) (((\<lambda>uu_. uu_) \<circ> eval_fm_aux) (e, A))) \<or> (\<exists>eval_fm_aux k e A. x1 = (e, SyntaxN.Ex k A) \<and> x2 = (\<exists>x. ((\<lambda>uu_. uu_) \<circ> eval_fm_aux) (finfun_update e k x, A)) \<and> atom k \<sharp> e \<and> HOL.induct_forall (\<lambda>x. p (finfun_update e k x, A) (((\<lambda>uu_. uu_) \<circ> eval_fm_aux) (finfun_update e k x, A)))))"
co1: assumes "dfree t1"   and "Vagree \<nu> \<nu>' (FVT t1)"   and "Iagree I J {Inl x |x. x \<in> SIGT t1}" shows "sterm_sem I t1 (fst \<nu>) = sterm_sem J t1 (fst \<nu>')"
B: shows "\<forall>x<a. poly p x \<noteq> 0"
elims: fixes x :: "('a, 'b) Combinators list"   and y :: "('a, 'b) Combinators list" assumes "removeShadowRules1 x = y" obtains xa :: "('a, 'b) Combinators" and xs :: "('a, 'b) Combinators list" where "x = xa # xs" and "y = (if DenyAll \<in> set xs then removeShadowRules1 xs else xa # xs)"   | "x = []" and "y = []"
Fract_cases_nonzero: fixes q :: "'b fract" obtains a :: "'b" and b :: "'b" where "q = Fraction_Field.Fract a b" and "b \<noteq> (0::'b)" and "a \<noteq> (0::'b)"   | "q = 0"
Par2B: fixes Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi"   and P :: "pi" assumes "Q \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> Q'"   and "x \<sharp> P" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P \<parallel> Q'"
linepath_translate_left: fixes c :: "'b"   and a :: "'b"   and b :: "'b" shows "linepath (c + a) (c + a) = (\<lambda>x. c + a) \<circ> linepath a b"
comp_inv_arr': fixes f :: "'a arrow_of_spans_data" assumes "V.iso f" shows "V.inv f \<bullet> f = V.dom f"
fold_fun_left_comm: fixes A :: "'a\<^sub>m set"   and z :: "'a\<^sub>m option"   and x :: "'a\<^sub>m" assumes "finite A" shows "Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) z A \<bind> m x = Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) (z \<bind> m x) A"
ATOM_axioms: shows "ATOM aneg anormal I\<^sub>a depends\<^sub>0 decr"
A5_NotSource_level1: shows "isNotDSource level1 sA5"
invariant_1_pos_5: fixes p :: "int poly"   and l :: "rat"   and r :: "rat" assumes "invariant_1 (p, l, r)"   and "0 < r" shows "0 < l"
finite_UnI: fixes F :: "'c set"   and G :: "'c set" assumes "finite F"   and "finite G" shows "finite (F \<union> G)"
SUP_insert: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (f ` insert a A) = sup (f a) (Sup (f ` A))"
dual_inv1: shows "\<partial>\<^sup>- \<circ> \<partial> = id"
substEnv_preserves_good: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "goodEnv rho" shows "goodEnv (rho &[X / x]_xs)"
carr: shows "{mult_col_div_row a k A, A, multrow_mat n k (inverse a), multrow_mat n k a} \<subseteq> carrier_mat n n"
that: fixes p :: "'a \<Rightarrow> 'a" assumes "p permutes A"   and "\<forall>a\<in>A. f a = f' (p a)" shows "thesis"
Gromov_boundary_rep_converging: fixes x :: "'a Gromov_completion" assumes "x \<in> Gromov_boundary" shows "Gromov_converging_at_boundary (rep_Gromov_completion x)"
dilating_fun_image: fixes f :: "nat \<Rightarrow> nat"   and r :: "'b run"   and m :: "nat"   and n :: "nat"   and c :: "clock" assumes "dilating_fun f r" shows "{k. f m \<le> k \<and> k \<le> f n \<and> hamlet (Rep_run r k c)} = f ` {k. m \<le> k \<and> k \<le> n \<and> hamlet (Rep_run r (f k) c)}"
mono2_Step: fixes f :: "char list \<Rightarrow> aexp \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "bexp \<Rightarrow> 'a \<Rightarrow> 'a"   and C1 :: "'a acom"   and C2 :: "'a acom"   and S1 :: "'a"   and S2 :: "'a" assumes "\<And>x e S1 S2. S1 \<le> S2 \<Longrightarrow> f x e S1 \<le> f x e S2"   and "\<And>b S1 S2. S1 \<le> S2 \<Longrightarrow> g b S1 \<le> g b S2"   and "C1 \<le> C2"   and "S1 \<le> S2" shows "Step f g S1 C1 \<le> Step f g S2 C2"
open_UNIV: fixes "open" :: "'d set \<Rightarrow> bool" assumes "class.topological_space open" shows "open UNIV"
trace_def: fixes sig :: "'e signature" shows "trace sig \<equiv> filter (\<lambda>a. a \<in> externals sig) \<circ> schedule"
delete_map_transform_env_delete: fixes x :: "Vars.var"   and t :: "'a \<Rightarrow> exp \<Rightarrow> exp"   and ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "delete x (map_transform t (env_delete x ae) \<Gamma>) = delete x (map_transform t ae \<Gamma>)"
that: fixes x :: "'c"   and ln' :: "'a \<Rightarrow>f nat" assumes "ts t = \<lfloor>(x, ln')\<rfloor>" shows "thesis"
igOpIPresIGWlsSTR_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a, 'b, 'c, 'd) model_scheme" shows "igOpIPresIGWlsSTR MOD \<equiv> \<forall>delta inp binp. igWls MOD (stOf delta) (igOp MOD delta inp binp) = (igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp)"
hcomp_in_vhom: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and f :: "'b"   and g :: "'b"   and \<nu> :: "'b"   and h :: "'b"   and k :: "'b" assumes "horizontal_composition V H src trg"   and "partial_magma.in_hom V \<mu> f g"   and "partial_magma.in_hom V \<nu> h k"   and "src h = trg f" shows "partial_magma.in_hom V (H \<nu> \<mu>) (H h f) (H k g)"
cases: fixes x :: "'a \<times> ('t \<Rightarrow>\<^sub>0 'b)" obtains t :: "'a" and p :: "'t \<Rightarrow>\<^sub>0 'b" where "x = (t, p)"
bconfs_def: fixes P :: "'c prog"   and sh :: "sheap"   and es :: "'d exp list"   and b :: "bool" shows "P,sh \<turnstile>\<^sub>b (es,b) \<surd> \<equiv> b \<longrightarrow> (\<exists>C. icheck P C (the (ss_exps es)) \<and> curr_inits P es = \<lfloor>C\<rfloor> \<and> initPD sh C)"
prems_2: shows "reads_consistent False \<O> m (x # xs)"
simps_3: fixes x :: "quat"   and y :: "quat" shows "Im2 (x + y) = Im2 x + Im2 y"
eSwapAbs_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and us :: "'varSort"   and s :: "'sort"   and A :: "'d"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "igWlsAbs MOD (us, s) A" shows "eSwapAbs MOD zs z1 z2 (OK A) = OK (igSwapAbs MOD zs z1 z2 A)"
rep_eq: fixes x :: "uint16"   and xa :: "nat" shows "Rep_uint16 (x >>> xa) = signed_drop_bit xa (Rep_uint16 x)"
fisher_yates_def: shows "fisher_yates = fisher_yates_aux 0"
t_notin_DagsNodesnb: shows "t \<notin> Dags (repc ` Nodes nb levellista) (repc \<propto> low) (repc \<propto> high)"
product_integral_singleton: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'i" assumes "f \<in> borel_measurable (M i)" shows "LINT x|Pi\<^sub>M {i} M. f (x i) = integral\<^sup>L (M i) f"
lm018: fixes R :: "('c \<times> 'd) set"   and X1 :: "'c set"   and X2 :: "'c set" shows "Outside' X2 (Outside' X1 R) = Outside' X1 (Outside' X2 R)"
update_simps_19: fixes T :: "(nat \<times> linear_poly) list"   and s :: "('a, 'b) state" shows "\<T> (\<T>_update T s) = T"
set_list_of_vec: fixes v :: "'b vec" shows "set (list_of_vec v) = set\<^sub>v v"
empty_nextsuffix: fixes s :: "nat \<Rightarrow> 'b" assumes "emptyseq s" shows "nextsuffix s = s"
ord_isom_sym: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e" assumes "Order E"   and "ord_isom D E f" shows "ord_isom E D (f\<inverse>\<^bsub>carrier E,carrier D\<^esub>)"
curry_case_prod: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" shows "curry (\<lambda>(x, y). f x y) = f"
intros_9: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and t :: "('d, 'f) xstate"   and c\<^sub>2 :: "('d, 'e, 'f) com" assumes "s \<in> b"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Cond b c\<^sub>1 c\<^sub>2,Normal s\<rangle> \<Rightarrow> t"
kbo_trans: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and u :: "('f, 'b) Term.term" shows "(kbo'.S s t \<longrightarrow> kbo'.NS t u \<longrightarrow> kbo'.S s u) \<and> (kbo'.NS s t \<longrightarrow> kbo'.S t u \<longrightarrow> kbo'.S s u) \<and> (kbo'.NS s t \<longrightarrow> kbo'.NS t u \<longrightarrow> kbo'.NS s u)"
assms_2: shows "ip \<in> kD (rt \<xi>)"
filter_mset_strict_subset: fixes x :: "'b"   and M :: "'b multiset"   and p :: "'b \<Rightarrow> bool" assumes "x \<in># M"   and "\<not> p x" shows "filter_mset p M \<subset># M"
normal_simps_7: fixes aneg :: "'c \<Rightarrow> 'c fm"   and anormal :: "'c \<Rightarrow> bool"   and I\<^sub>a :: "'c \<Rightarrow> 'd list \<Rightarrow> bool"   and depends\<^sub>0 :: "'c \<Rightarrow> bool"   and decr :: "'c \<Rightarrow> 'c"   and \<phi> :: "'c fm" assumes "ATOM aneg anormal I\<^sub>a depends\<^sub>0 decr" shows "ATOM.normal anormal (ExQ \<phi>) = ATOM.normal anormal \<phi>"
exists_irreducible_divisor: fixes a :: "'a list" assumes "a \<in> carrier (poly_ring R) - {\<zero>\<^bsub>poly_ring R\<^esub>}"   and "a \<notin> Units (poly_ring R)" obtains b :: "'a list" where "b \<in> carrier (poly_ring R)" and "pirreducible (carrier R) b" and "b divides\<^bsub>poly_ring R\<^esub> a"
hd_inf_prefix_length_0: fixes xs :: "'b list"   and ys :: "'b list" assumes "hd xs \<noteq> hd ys" shows "inf_prefix_length xs ys = 0"
Iff_Dis: fixes p :: "'a pfm"   and p' :: "'a pfm"   and q :: "'a pfm"   and q' :: "'a pfm" assumes "\<turnstile>\<^sub>! (p \<^bold>\<longleftrightarrow>\<^sub>! p')"   and "\<turnstile>\<^sub>! (q \<^bold>\<longleftrightarrow>\<^sub>! q')" shows "\<turnstile>\<^sub>! (p \<^bold>\<or>\<^sub>! q \<^bold>\<longleftrightarrow>\<^sub>! p' \<^bold>\<or>\<^sub>! q')"
enat_in_Inf: fixes S :: "enat set" assumes "Inf S \<noteq> top" shows "Inf S \<in> S"
eq_divide_eq_1: fixes b :: "'a"   and a :: "'a" shows "((1::'a) = b / a) = (a \<noteq> (0::'a) \<and> a = b)"
in_diffD: fixes a :: "'d"   and M :: "'d multiset"   and N :: "'d multiset" assumes "a \<in># M - N" shows "a \<in># M"
INT_Int_distrib: fixes A :: "'c \<Rightarrow> 'b set"   and B :: "'c \<Rightarrow> 'b set"   and I :: "'c set" shows "(\<Inter>i\<in>I. A i \<inter> B i) = \<Inter> (A ` I) \<inter> \<Inter> (B ` I)"
col_def: fixes A :: "'a Matrix.mat"   and j :: "nat" shows "Matrix.col A j = Matrix.vec (dim_row A) (\<lambda>i. A $$ (i, j))"
trans_le_add2: fixes i :: "nat"   and j :: "nat"   and m :: "nat" assumes "i \<le> j" shows "i \<le> m + j"
d3: shows "v4 = Matrix.mat 8 1 (\<lambda>(i, j). [[0, 0, 0, 0, 0, 0, - \<phi> $$ (1, 0), \<phi> $$ (0, 0)]] ! j ! i)"
transform_irule_set_iter_def: fixes rs :: "(name \<times> (Term_Class.term list \<times> pterm) fset) fset" shows "Composition.transform_irule_set_iter rs = (transform_irule_set ^^ max_arity rs) rs"
zero_left: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and b :: "'b" assumes "bounded_cbilinear prod" shows "prod (0::'a) b = (0::'c)"
SUP_upper: fixes i :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "i \<in> A" shows "f i \<le> Sup (f ` A)"
that: fixes A :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and q :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite A"   and "A \<subseteq> B"   and "\<And>b. q b \<in> P[X]"   and "p = (\<Sum>b\<in>A. q b * b)" shows "thesis"
bij_betw_def: fixes f :: "'f \<Rightarrow> 'g"   and A :: "'f set"   and B :: "'g set" shows "bij_betw f A B = (inj_on f A \<and> f ` A = B)"
tuple_BC_in_hom: shows "Maps.in_hom tuple_BC (Maps.MkIde (src fg\<^sub>0h\<^sub>1.p\<^sub>0)) (Maps.MkIde (src g\<^sub>0h\<^sub>1.p\<^sub>0))"
Int_def: fixes A :: "'e set"   and B :: "'e set" shows "A \<inter> B = {x \<in> A. x \<in> B}"
bin'_split_def: shows "bin'_split \<equiv> \<lambda>(a, n) p. heap_WHILET (\<lambda>(low, high). return (low < high)) (\<lambda>(low, high). let mid = (low + high) div 2 in Array.nth a mid \<bind> (\<lambda>s. if p < s then return (low, mid) else if s < p then return (mid + 1, high) else return (mid, mid))) (0, n) \<bind> (\<lambda>(low', high'). return low')"
exists_lub: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set" assumes "class.Join_cpo below" shows "\<exists>u. po.is_lub below S u"
fmap_def: shows "fmap = COERCE((udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'f \<rightarrow> udom\<cdot>'f, ('d \<rightarrow> 'e) \<rightarrow> 'd\<cdot>'f \<rightarrow> 'e\<cdot>'f)\<cdot>fmapU"
y: shows "find (\<lambda>i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \<and> mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) [0..<nrows_iarray (matrix_to_iarray A)] = Some y"
defAss_def: fixes g :: "'a"   and m :: "'node"   and v :: "'val" shows "i.defAss g m v = (\<forall>ns. i.path2 g (Entry2 g) ns m \<longrightarrow> (\<exists>n\<in>set ns. v \<in> i.allDefs g n))"
capacity_finite: fixes e :: "'v vertex \<times> 'v vertex" shows "capacity \<Delta>'' e \<noteq> \<top>"
id: fixes i :: "nat"   and j :: "nat" assumes "i < nr"   and "j < nc" shows "row (map_mat (\<lambda>a. [:a:]) A) i \<bullet> col (map_mat (\<lambda>a. [:a:]) B) j = [:row A i \<bullet> col B j:]"
bounded_by_update_var: fixes xs :: "real list"   and vs :: "float interval option list"   and i :: "nat"   and ivl :: "float interval"   and x :: "real" assumes "bounded_by xs vs"   and "vs ! i = Some ivl"   and "x \<in>\<^sub>r ivl" shows "bounded_by (xs[i := x]) vs"
MASTER_BOUND_postproc_12: fixes a :: "real" shows "CLAMP (MASTER_BOUND'' a) = CLAMP (\<lambda>x. real x powr a)"
connectedin_subtopology: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "connectedin (subtopology X S) T = (connectedin X T \<and> T \<subseteq> S)"
l_inv_ex: fixes x :: "'a" assumes "x \<in> carrier \<G>" shows "\<exists>y\<in>carrier \<G>. y \<otimes> x = \<one>"
assms_4: fixes I :: "'a set"   and a :: "'a"   and b :: "'a"   and J :: "'a set"   and c :: "'a"   and d :: "'a"   and A :: "'a set" assumes "Q I a b"   and "Q J c d"   and "I \<subseteq> A"   and "J \<subseteq> A"   and "A \<in> \<P>"   and "[[a b c]] \<and> a = d" shows "P I J"
simps_8: fixes \<Sigma> :: "nat \<Rightarrow> 'b set"   and project :: "'b \<Rightarrow> 'b"   and lookup :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and r :: "'c rexp" assumes "project \<Sigma> project" shows "project.lang \<Sigma> project lookup n (rexp.Not r) = lists (\<Sigma> n) - project.lang \<Sigma> project lookup n r"
find_first_unique: fixes xs :: "'b list"   and i :: "nat" assumes "distinct xs"   and "i < length xs" shows "find_first (xs ! i) xs = i"
spectral_radius_nonneg: fixes x :: "'b" shows "0 \<le> spectral_radius x"
False_imp_not_eventually: fixes P :: "'b \<Rightarrow> bool"   and net :: "'b filter" assumes "\<forall>x. \<not> P x"   and "net \<noteq> \<bottom>" shows "\<not> eventually P net"
dim: shows "dim_row (mat\<^sub>h A) = dim_row A"
singleton_matrix_zero: fixes j :: "nat"   and i :: "nat" shows "singleton_matrix j i (0::'a) = 0"