hcju/isabelleps huggingface dataset:
simps_6: fixes V :: "char list"   and e :: "char list exp" shows "fv (V:=e) = {V} \<union> fv e"
id: shows "spectrum A = {k. poly CP k = (0::'a)}"
assms_1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> set xs"   and "y \<in> set ys"   and "fst (P x y)" shows "fst (P' x y)"
MapsToCompDef: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and X :: "'a"   and Y :: "'a"   and g :: "'b"   and Z :: "'a" assumes "f maps\<^bsub>C\<^esub> X to Y"   and "g maps\<^bsub>C\<^esub> Y to Z" shows "f \<approx>>\<^bsub>C\<^esub> g"
isCont_pochhammer: fixes z :: "'a"   and n :: "nat" shows "isCont (\<lambda>z. pochhammer z n) z"
distinct_3: shows "T_i32 \<noteq> T_f32"
prems: shows "a # us \<in> lists \<langle>G\<rangle>"
space_P: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "x \<in> space (Pi\<^sub>M {0..<n} M)" shows "space (P n x) = space (M n)"
distinct_conv_nth: fixes xs :: "'b list" shows "distinct xs = (\<forall>i<length xs. \<forall>j<length xs. i \<noteq> j \<longrightarrow> xs ! i \<noteq> xs ! j)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<Phi>'.Y a) F \<Phi>"
typedef_right_total: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and T :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "type_definition Rep Abs A"   and "T \<equiv> \<lambda>x y. x = Rep y" shows "right_total T"
ae_2: shows "AE x in M. nf x = ennreal (- g x)"
set_map: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b fml" shows "set_fml (map_fml f v) = f ` set_fml v"
checker2_def: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real" shows "checker2 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<equiv> let distance = s\<^sub>o - s\<^sub>e; precond = check_precond s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o; safe_dist1 = first_safe_dist v\<^sub>e a\<^sub>e; safe_dist2 = second_safe_dist v\<^sub>e a\<^sub>e v\<^sub>o a\<^sub>o; safe_dist3 = - rel_dist_to_stop (v\<^sub>o - v\<^sub>e) (a\<^sub>o - a\<^sub>e) in if \<not> precond then False else if safe_dist1 < distance then True else if a\<^sub>e < a\<^sub>o \<and> v\<^sub>o < v\<^sub>e \<and> v\<^sub>e - a\<^sub>e / a\<^sub>o * v\<^sub>o < 0 then safe_dist3 < distance else safe_dist2 < distance"
fcomp_comp: fixes f :: "'e \<Rightarrow> 'g"   and g :: "'g \<Rightarrow> 'f" shows "fcomp f g = g \<circ> f"
col_perp2_ncol_col: fixes X1 :: "'p"   and X2 :: "'p"   and A :: "'p"   and B :: "'p"   and Y1 :: "'p"   and Y2 :: "'p" assumes "X1 X2 Perp A B"   and "Y1 Y2 Perp A B"   and "Col X1 Y1 Y2"   and "\<not> Col X1 A B" shows "Col X2 Y1 Y2"
butterfly_adjoint: fixes \<psi> :: "'a"   and \<phi> :: "'b" shows "butterfly \<psi> \<phi>* = butterfly \<phi> \<psi>"
simps_2: fixes p :: "'b \<times> 'b"   and ps :: "('b \<times> 'b) list" shows "equal_pair_count (p # ps) = (if fst p = snd p then Suc (equal_pair_count ps) else 0)"
simps_7: fixes M :: "('fun, 'var) Term.term set"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) set"   and uv :: "poscheckvariant"   and t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) stateful_strand_step list" shows "\<lbrakk>M; D; \<langle>uv: t \<in> s\<rangle> # S\<rbrakk>\<^sub>s = (\<lambda>\<I>. (t, s) \<cdot>\<^sub>p \<I> \<in> D \<and> \<lbrakk>M; D; S\<rbrakk>\<^sub>s \<I>)"
E: shows "KBP.k_memb (runJP k_mkAlgAuto t a) (KBP.k_dfs a)"
eq: shows "(1::'a) / c + (1::'a) / d = (c + d) / (c * d)"
Markov_Decision_Process_def: fixes K :: "'a \<Rightarrow> 'a pmf set" shows "Markov_Decision_Process K \<equiv> \<forall>s. K s \<noteq> {}"
IOA_def: fixes ioa :: "('d set \<times> 'd set \<times> 'd set) \<times> 'e set \<times> ('e \<times> 'd \<times> 'e) set" shows "IOA ioa \<equiv> is_asig (asig_of ioa) \<and> starts_of ioa \<noteq> {} \<and> state_trans (asig_of ioa) (trans_of ioa)"
module_mat_def: fixes ty :: "'a itself"   and nr :: "nat"   and nc :: "nat" shows "module_mat ty nr nc \<equiv> \<lparr>carrier = carrier_mat nr nc, mult = (*), one = 1\<^sub>m nr, zero = 0\<^sub>m nr nc, add = (+), smult = (\<cdot>\<^sub>m)\<rparr>"
cross3_simps_11: fixes a :: "'h"   and b :: "'h" shows "a * b = b * a"
mult_inv_non_zero: fixes p :: "int poly fract"   and q :: "int poly fract" assumes "p \<noteq> 0"   and "rat_poly_times p q = 1" shows "q \<noteq> 0"
unique: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "(a, b) \<in> r"   and "(a, c) \<in> r" shows "b = c"
norm_le_zero_iff: fixes x :: "'a" shows "(norm x \<le> 0) = (x = (0::'a))"
eq: shows "a \<otimes> b \<otimes> (inv b \<otimes> inv a) = \<one>"
plus_tree_def: fixes x :: "'a tree"   and y :: "'a tree" shows "x + y = pure (+) \<diamondop> x \<diamondop> y"
has_vector_derivative_within_open: fixes a :: "real"   and S :: "real set"   and f :: "real \<Rightarrow> 'a"   and f' :: "'a" assumes "a \<in> S"   and "open S" shows "(f has_vector_derivative f') (at a within S) = (f has_vector_derivative f') (at a)"
Compositionality_While: fixes b :: "'exp"   and c :: "('exp, 'id) MWLsCom"   and \<iota> :: "nat" assumes "\<forall>d. b \<equiv>\<^bsub>d\<^esub> b"   and "WHATWHERE_Secure [c]"   and "unique_PPc (while\<^bsub>\<iota>\<^esub> b do c od)" shows "WHATWHERE_Secure [while\<^bsub>\<iota>\<^esub> b do c od]"
fmlookup_dom_iff: fixes x :: "'d"   and m :: "('d, 'e) fmap" shows "(x |\<in>| fmdom m) = (\<exists>a. fmlookup m x = Some a)"
awhd_of_awalk: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "S.awalk u p v" shows "hd (pawalk_verts u p) = u"
update_by_fun_def: fixes k :: "'a"   and f :: "'c \<Rightarrow> 'c"   and xs :: "'y" shows "oay.update_by_fun k f xs = oalist_of_listy (update_by_fun_raw k f (list_of_oalisty xs))"
update_another: fixes ip :: "nat"   and dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" assumes "ip \<noteq> dip" shows "update rt dip (dsn, dsk, flag, hops, nhip, pre) ip = rt ip"
eval_fds_zeroD_aux: fixes h :: "'a fds"   and s0 :: "'a" assumes "fds_abs_converges h s0"   and "\<exists>\<^sub>F s in (\<lambda>s. s \<bullet> (1::'a)) going_to at_top. eval_fds h s = (0::'a)" shows "h = 0"
insert_ops_ref_older: fixes pre :: "('b \<times> 'b option) list"   and oid :: "'b"   and ref :: "'b"   and suf :: "('b \<times> 'b option) list" assumes "insert_ops (pre @ [(oid, Some ref)] @ suf)" shows "ref < oid"
elims: fixes x :: "C"   and y :: "nat \<Rightarrow> C list" assumes "funC x = y" obtains "x = C []" and "y = (\<lambda>i. [C []])"   | ns :: "C list" where "x = C (C [] # ns__)" and "y = (\<lambda>i. replicate i (C ns__))"   | va :: "C" and vb :: "C list" and ns :: "C list" where "x = C (C (va # vb) # ns)" and "y = (\<lambda>i. [C (funC (C (va # vb)) i @ ns)])"
OclIncluding_commute: fixes S :: "'a state \<times> 'a state \<Rightarrow> Set('b)"   and i :: "'a state \<times> 'a state \<Rightarrow> 'b"   and j :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "S->including\<^sub>S\<^sub>e\<^sub>t(i)->including\<^sub>S\<^sub>e\<^sub>t(j) = S->including\<^sub>S\<^sub>e\<^sub>t(j)->including\<^sub>S\<^sub>e\<^sub>t(i)"
unitary_one: fixes n :: "nat" shows "unitary (1\<^sub>m n)"
wf_on_imp_nf_ex: fixes A :: "'a set"   and a :: "'a" assumes "wfp_on (\<rightarrow>)\<inverse>\<inverse> A"   and "dw_closed A"   and "a \<in> A" obtains b :: "'a" where "a \<rightarrow>\<^sup>* b" and "is_final b"
pdo1: shows "partial_density_operator \<rho>\<^sub>1"
dj_supp: fixes a :: "'a" assumes "disjoint TYPE('a) TYPE('b)" shows "supp a = {}"
GrdSymSystem_def: fixes init :: "(nat \<Rightarrow> 'c) set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" shows "GrdSymSystem init r = [: z \<leadsto> (u, x) . u \<in> init \<and> z = x :] \<circ> trs (\<lambda>(u, x). (\<box> lft_rel_st r) u x)"
subspace_cotangent_space: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and p :: "'c" assumes "c_manifold charts k" shows "subspace (c_manifold.cotangent_space charts k p)"
ln_upper_5_neg: fixes x :: "real" assumes "0 < x"   and "x \<le> 1" shows "ln x \<le> ln_upper_5 x"
H: shows "listrelp (\<lambda>x y. P x) xs ys"
Try: fixes Env :: "env"   and B :: "lname set"   and c1 :: "stmt"   and C1 :: "assigned"   and vn :: "vname"   and C :: "qtname"   and c2 :: "stmt"   and C2 :: "assigned"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1"   and "Env\<lparr>lcl := lcl Env(VName vn \<mapsto> Class C)\<rparr>\<turnstile> B \<union> {VName vn} \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2"   and "nrm A = nrm C1 \<inter> nrm C2"   and "brk A = brk C1 \<Rightarrow>\<inter>  brk C2" shows "Env\<turnstile> B \<guillemotright>\<langle>Try c1 Catch(C vn) c2\<rangle>\<guillemotright> A"
IH: fixes E :: "char list \<Rightarrow> ty option" assumes "length Vs = length Ts"   and "length vs = length Ts"   and "\<forall>S\<in>set Ts. is_type P S" shows "(P,E,h \<turnstile> blocks (Vs, Ts, vs, e) : T) = (P,E(Vs [\<mapsto>] Ts),h \<turnstile> e : T \<and> (\<exists>Ts'. map P \<turnstile> typeof\<^bsub>h\<^esub> vs = map Some Ts' \<and> P \<turnstile> Ts' [\<le>] Ts))"
balance_inv1: fixes l :: "('b, 'c) RBT_Impl.rbt"   and r :: "('b, 'c) RBT_Impl.rbt"   and k :: "'b"   and v :: "'c" assumes "inv1l l"   and "inv1l r" shows "inv1 (balance l k v r)"
nullspace_inter_rowspace: fixes A :: "((real, 'b) vec, 'a) vec"   and x :: "(real, 'b) vec" shows "(A *v x = 0 \<and> x \<in> span (rows A)) = (x = 0)"
extendCommute: fixes S :: "('a, 'b) sequent"   and R :: "('a, 'b) sequent"   and c :: "('a, 'b) sequent" shows "extend S (extend R c) = extend R (extend S c)"
rel_sum_sel: fixes R1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'b + 'd"   and b :: "'c + 'e" shows "rel_sum R1 R2 a b = (isl a = isl b \<and> (isl a \<longrightarrow> isl b \<longrightarrow> R1 (projl a) (projl b)) \<and> (\<not> isl a \<longrightarrow> \<not> isl b \<longrightarrow> R2 (projr a) (projr b)))"
simps_3: fixes uu :: "'b"   and pr :: "('a, 'b) Digit"   and m :: "('a, 'b) FingerTreeStruc"   and sf :: "('a, 'b) Digit" shows "FingerTreeStruc.toList (Deep uu pr m sf) = FingerTreeStruc.digitToList pr @ FingerTreeStruc.toList m @ FingerTreeStruc.digitToList sf"
xvecFreshM: shows "(p \<bullet> xvec) \<sharp>* M"
cfun_fun_cong: fixes f :: "'a \<rightarrow> 'b"   and g :: "'a \<rightarrow> 'b"   and x :: "'a" assumes "f = g" shows "f\<cdot>x = g\<cdot>x"
simps_1: fixes p :: "'b" shows "hd_prog p [] = p"
wf_levellist_subset: fixes t :: "dag"   and ll :: "ref list list"   and ll' :: "ref list list"   and var :: "ref \<Rightarrow> nat" assumes "wf_levellist t ll ll' var" shows "set (concat ll') \<subseteq> set (concat ll) \<union> set_of t"
poly_roots_finite: fixes p :: "'a Polynomial.poly" assumes "p \<noteq> 0" shows "finite {x. poly p x = (0::'a)}"
delete_block_wf: fixes b :: "'a set" shows "incidence_system \<V> (del_block b)"
smcf_const_is_semifunctor: fixes \<alpha> :: "V"   and \<CC> :: "V"   and \<DD> :: "V"   and a :: "V"   and f :: "V" assumes "semicategory \<alpha> \<CC>"   and "semicategory \<alpha> \<DD>"   and "f : a \<mapsto>\<^bsub>\<DD>\<^esub> a"   and "f \<circ>\<^sub>A\<^bsub>\<DD>\<^esub> f = f" shows "dghm_const \<CC> \<DD> a f : \<CC> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<DD>"
member_rec_1: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c" shows "List.member (x # xs) y = (x = y \<or> List.member xs y)"
eqvtI: fixes P :: "pi"   and Q :: "pi"   and perm :: "(name \<times> name) list" assumes "P \<approx> Q" shows "perm \<bullet> P \<approx> perm \<bullet> Q"
valid_Act_strong: fixes X :: "'b"   and P :: "'state"   and \<alpha> :: "'act"   and x :: "('c, 'pred, 'act) formula" assumes "finite (supp X)" shows "P \<Turnstile> Act \<alpha> x = (\<exists>\<alpha>' x' P'. Act \<alpha> x = Act \<alpha>' x' \<and> P \<rightarrow> \<langle>\<alpha>',P'\<rangle> \<and> P' \<Turnstile> x' \<and> bn \<alpha>' \<sharp>* X)"
nfoldli_while: fixes l :: "'d list"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and \<sigma> :: "'c"   and I :: "'d list \<times> 'c \<Rightarrow> bool" shows "nfoldli l c f \<sigma> \<le> WHILE\<^sub>T\<^bsup>I\<^esup> (FOREACH_cond c) (FOREACH_body f) (l, \<sigma>) \<bind> (\<lambda>(uu_, \<sigma>). RETURN \<sigma>)"
pol_SOME_1: fixes p :: "'a" assumes "p \<in> carrier R" shows "cf_sol R S X p (SOME f. cf_sol R S X p f)"
tbl: shows "tiTable_SampleT req x stop lose st_in b ack i1 vc st_out"
trivial_pre_depr: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and c :: "Core.term" assumes "term_ok \<Theta> c"   and "typ_of c = Some propT" shows "\<Theta>,{c} \<turnstile> c"
greedy_path_lhd: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c"   and v0 :: "'c"   and v :: "'c"   and P :: "'c llist" assumes "ParityGame G"   and "ParityGame.greedy_conforming_path G p \<sigma> \<sigma>' v0 = LCons v P" shows "v = v0"
sort_remdups: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sort_key less_eq (\<lambda>x. x) (remdups xs) = remdups (linorder.sort_key less_eq (\<lambda>x. x) xs)"
base: fixes P :: "('e, 'f, 'g) psi" shows "\<lparr>\<nu>*[]\<rparr>P = P"
nonzero_mult_divide_mult_cancel_right2: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times a c) (times c b) = divide a b"
poly_gcd_monic: fixes p :: "'b poly"   and q :: "'b poly" assumes "p \<noteq> 0 \<or> q \<noteq> 0" shows "monic (gcd p q)"
interval_lebesgue_integrable_def: fixes M :: "real measure"   and a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'b" shows "interval_lebesgue_integrable M a b f = (if a \<le> b then set_integrable M (einterval a b) f else set_integrable M (einterval b a) f)"
0: shows "\<Union> ((set \<circ> fst \<circ> Ana) ` pair ` setops\<^sub>s\<^sub>s\<^sub>t S) = {}"
farey_fractions_def: fixes N :: "nat" shows "farey_fractions N = {q \<in> {0<..1}. snd (quotient_of q) \<le> int N}"
power_sq_mtx_diag: fixes f :: "'a \<Rightarrow> real"   and n :: "nat" shows "sq_mtx_diag f ^ n = (\<d>\<i>\<a>\<g> i. f i ^ n)"
parents_subset_lgrid: fixes d :: "nat"   and b :: "(nat \<times> int) list"   and p :: "(nat \<times> int) list" shows "parents d b p \<subseteq> lgrid b {d} (level p + 1)"
multiplicity_eq_nat: fixes x :: "nat"   and y :: "nat" assumes "0 < x"   and "0 < y"   and "\<And>p. prime p \<Longrightarrow> multiplicity p x = multiplicity p y" shows "x = y"
events_of_def: fixes P :: "'a process" shows "events_of P \<equiv> \<Union>t\<in>T P. {a. ev a \<in> set t}"
exists_unique_actual_equiv: fixes v :: "i"   and x :: "\<nu>"   and \<psi> :: "\<kappa> \<Rightarrow> \<o>" shows "[(\<^bold>\<exists>!y. \<^bold>\<A>(y \<^bold>= x \<^bold>& \<psi> (x\<^sup>P))) \<^bold>\<equiv> \<^bold>\<A>\<psi> (x\<^sup>P) in v]"
a: shows "X \<in> obj\<^bsub>CatDom F\<^esub>"
D_gt: fixes e :: "real" assumes "0 < e" shows "\<exists>d>0. \<forall>x>t. 0 < dist x t \<and> dist x t < d \<longrightarrow> dist ((f x - f t) / (x - t)) D_gt < e"
node\<^sub>i_height: fixes k :: "nat"   and ts :: "('a btree \<times> 'a) list"   and t :: "'a btree" shows "height_up\<^sub>i (node\<^sub>i k ts t) = height (Node ts t)"
qSwap_preserves_qFresh_distinct: fixes ys :: "'a"   and zs :: "'a"   and y :: "'b"   and z1 :: "'b"   and z2 :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm" assumes "ys \<noteq> zs \<or> y \<notin> {z1, z2}" shows "qFresh ys y (X #[[z1 \<and> z2]]_zs) = qFresh ys y X"
assms_2: shows "is_zero_row_upt_k i k A"
le_numeral_extra_2: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less_eq one zero"
divisors_int_1: fixes x :: "int" assumes "x \<noteq> 0" shows "set (divisors_int x) = {i. i dvd x}"
prems_1: shows "cycle_free_up_to m (Suc k) n"
rpo_optim_axioms_def: fixes extf :: "'a \<Rightarrow> (('a, 'b) tm \<Rightarrow> ('a, 'b) tm \<Rightarrow> bool) \<Rightarrow> ('a, 'b) tm list \<Rightarrow> ('a, 'b) tm list \<Rightarrow> bool" shows "rpo_optim_axioms extf \<equiv> \<forall>f. ext_snoc (extf f)"
fair_chainD_1: fixes Rs :: "(('e, 'f) labeled_graph \<times> ('e, 'f) labeled_graph) set"   and S :: "nat \<Rightarrow> ('e, 'g) labeled_graph" assumes "fair_chain Rs S" shows "chain S"
arr_Set_arr_ParD_1: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Set \<alpha> T" shows "arr_Par \<alpha> T"
nhds_order: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b" assumes "class.order_topology less_eq less open" shows "topological_space.nhds open x = inf (INF a\<in>ord.greaterThan less x. principal (ord.lessThan less a)) (INF a\<in>ord.lessThan less x. principal (ord.greaterThan less a))"
divides_primepow_weak: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "dvd.dvd times a (power.power one times p n)" obtains m :: "nat" where "m \<le> n" and "normalize a = normalize (power.power one times p m)"
dim_submatrix_1: fixes A :: "'a mat"   and I :: "nat set"   and J :: "nat set" shows "dim_row (submatrix A I J) = card {i. i < dim_row A \<and> i \<in> I}"
hyps: assumes "r < n" shows "witness X n ! r = (r \<in> X)"
intro: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "semigroup_add_ow S pls"   and "ab_semigroup_add_ow_axioms S pls" shows "ab_semigroup_add_ow S pls"
set_single_hd: fixes vs :: "'b list"   and v :: "'b" assumes "set vs = {v}" shows "hd vs = v"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'a"   and aquery :: "'b aquery" shows "P (case aquery of Look \<Rightarrow> f1 | ForwardOrEdit x \<Rightarrow> f2 x | Drop \<Rightarrow> f3) = ((aquery = Look \<longrightarrow> P f1) \<and> (\<forall>x2. aquery = ForwardOrEdit x2 \<longrightarrow> P (f2 x2)) \<and> (aquery = Drop \<longrightarrow> P f3))"
power_not_zero: fixes a :: "'d"   and n :: "nat" assumes "a \<noteq> (0::'d)" shows "a ^ n \<noteq> (0::'d)"
neg_until_always: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and at :: "'a \<Rightarrow> nat \<Rightarrow> 'a"   and p :: "'a" assumes "class.temporal Inf Sup inf less_eq less sup bot top minus uminus at" shows "uminus (temporal.until Inf Sup inf at p (uminus p)) = temporal.always Inf at p"
ordinal_plus_assoc: fixes x :: "ordinal"   and y :: "ordinal"   and z :: "ordinal" shows "x + y + z = x + (y + z)"
simps_4: fixes p\<^sub>0 :: "int \<times> int"   and v :: "int \<times> int"   and vb :: "int \<times> int"   and vc :: "(int \<times> int) list" shows "closest_pair_bf_code (p\<^sub>0 # v # vb # vc) = (let (\<delta>\<^sub>c, c\<^sub>0, c\<^sub>1) = closest_pair_bf_code (v # vb # vc); (\<delta>\<^sub>p, p\<^sub>1) = find_closest_bf_code p\<^sub>0 (v # vb # vc) in if \<delta>\<^sub>c \<le> \<delta>\<^sub>p then (\<delta>\<^sub>c, c\<^sub>0, c\<^sub>1) else (\<delta>\<^sub>p, p\<^sub>0, p\<^sub>1))"
card_le: shows "card (f ` T) \<le> nat (\<lfloor>(b - a) / d\<rfloor> + 1)"
opset_def: fixes opset :: "('c \<times> 'd) set"   and deps :: "'d \<Rightarrow> 'c set" shows "opset opset deps \<equiv> (\<forall>oid op1 op2. (oid, op1) \<in> opset \<longrightarrow> (oid, op2) \<in> opset \<longrightarrow> op1 = op2) \<and> (\<forall>oid oper ref. (oid, oper) \<in> opset \<longrightarrow> ref \<in> deps oper \<longrightarrow> ref < oid) \<and> finite opset"
power_swap: fixes G :: "('a, 'b) cyclic_group_scheme"   and \<alpha>0 :: "nat"   and r :: "nat" assumes "Cyclic_Group.cyclic_group G" shows "(\<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> \<alpha>0) [^]\<^bsub>G\<^esub> r = (\<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> r) [^]\<^bsub>G\<^esub> \<alpha>0"
finite: fixes A :: "'g set" assumes "class.finite TYPE('g)" shows "finite A"
get_heap_returns_result: fixes h :: "'l"   and f :: "'l \<Rightarrow> ('l, 'm, 'n) prog"   and x :: "'n" shows "h \<turnstile> Heap_Error_Monad.bind get_heap f \<rightarrow>\<^sub>r x = h \<turnstile> f h \<rightarrow>\<^sub>r x"
yEqAFresh: shows "y = x \<or> qAFresh xs y Y"
asubg_test: fixes H :: "'a set" assumes "H \<subseteq> carrier M"   and "H \<noteq> {}"   and "\<forall>a\<in>H. \<forall>b\<in>H. a \<plusminus> -\<^sub>a b \<in> H" shows "M +> H"
intros: fixes u :: "'d \<Rightarrow> 'e"   and R :: "('d \<Rightarrow> 'e) set"   and \<R> :: "('d \<Rightarrow> 'e) set set"   and R' :: "('d \<Rightarrow> 'e) set"   and t :: "'e" assumes "u \<in> R"   and "R \<in> \<R>"   and "R' \<in> \<R>"   and "(0::'e) \<le> t"   and "R' = [u \<oplus> t]\<^sub>\<R>" shows "R' \<in> Succ \<R> R"
ddh_ext_def: fixes \<G> :: "'b cyclic_group" shows "ddh_ext \<G> \<equiv> cyclic_group \<G>"
lunit_naturality: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "\<mu> \<cdot> PB.lunit (local.dom \<mu>) = PB.lunit (cod \<mu>) \<cdot> (trg \<mu> \<star> \<mu>)"
pne24: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "pid2 \<noteq> pid4"
HPT_Conf_Target_Source_ChiPlus: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d" assumes "TS \<in> HPT ST"   and "S \<in> Conf ST"   and "S \<in> ChiPlus (HA ST) `` Target TS" shows "S \<in> ChiStar (HA ST) `` Source TS"
Conj\<^sub>\<alpha>: fixes tset\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha> set['e]" assumes "finite (supp tset\<^sub>\<alpha>)"   and "\<And>t\<^sub>\<alpha>. t\<^sub>\<alpha> \<in> set_bset tset\<^sub>\<alpha> \<Longrightarrow> hereditarily_fs t\<^sub>\<alpha>" shows "hereditarily_fs (Conj\<^sub>\<alpha> tset\<^sub>\<alpha>)"
B_invariant: fixes a :: "'b"   and b :: "'b" assumes "PB a"   and "B a b" shows "PB b"
\<tau>move1BinOp: fixes P :: "'a prog"   and h :: "'heap"   and v :: "'addr val"   and bop :: "bop"   and v' :: "'addr val" shows "\<tau>move1 P h (Val v \<guillemotleft>bop\<guillemotright> Val v')"