hcju/isabelleps huggingface dataset:
the_wall_betw_nempty_1: fixes C :: "'a set"   and D :: "'a set" assumes "THE_default {} (\<lambda>H. H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}}) \<and> separated_by H C D) \<noteq> {}" shows "THE_default {} (\<lambda>H. H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}}) \<and> separated_by H C D) \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}})"
cong: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d" shows "comm_monoid_fset.F f z \<equiv> comm_monoid_fset.F f z"
discI: fixes tree :: "'b tree"   and x21 :: "'b tree"   and x22 :: "'b"   and x23 :: "'b tree" assumes "tree = \<langle>x21, x22, x23\<rangle>" shows "tree \<noteq> \<langle>\<rangle>"
approximating_semantics_iff_fun_good_ruleset: fixes rs :: "'a rule list"   and \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and p :: "'b"   and s :: "state"   and t :: "state" assumes "good_ruleset (rs::'a rule list)" shows "(\<gamma>::('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)),(p::'b)\<turnstile> \<langle>(rs::'a rule list), (s::state)\<rangle> \<Rightarrow>\<^sub>\<alpha> (t::state) = (approximating_bigstep_fun \<gamma> p rs s = t)"
is_tm_tdghmE: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>" obtains "\<Z> \<alpha>" and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>" and "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>" and "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
CH_irc_idem: fixes A :: "'a set" shows "CH (CH A) = CH A"
typs_of_fv_subset_Types: fixes t :: "Core.term" shows "snd ` fv t \<subseteq> Types t"
valid_node_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.valid_node sourcenode targetnode valid_edge n \<equiv> \<exists>a. valid_edge a \<and> (n = sourcenode a \<or> n = targetnode a)"
vdomain_vrange_is_vempty: shows "(\<D>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) = []\<^sub>\<circ>) = (\<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) = []\<^sub>\<circ>)"
wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t_eq_wfrestrictedvars\<^sub>s\<^sub>t: fixes A :: "('fun, 'var) extstrand_step list" shows "wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t A = wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t A"
b_non_zero_imp_v_non_zero: assumes "0 < \<b>" shows "0 < \<v>"
is_constant_constructors_simps2_6: fixes y :: "'a ltln"   and x :: "'a ltln" shows "is_constant (mk_release y x) = is_constant x"
permutes_compose: fixes p :: "'d \<Rightarrow> 'd"   and S :: "'d set"   and q :: "'d \<Rightarrow> 'd" assumes "p permutes S"   and "q permutes S" shows "q \<circ> p permutes S"
countable_UN': fixes X :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "countable X"   and "A ` X \<subseteq> sigma_sets S ((\<lambda>s. {s}) ` S)" shows "\<Union> (A ` X) \<in> sigma_sets S ((\<lambda>s. {s}) ` S)"
size_8: fixes x31 :: "'b set"   and x32 :: "'b com"   and x33 :: "'b com" shows "size (Cond x31 x32 x33) = size x32 + size x33 + Suc 0"
Lit: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and v :: "val" shows "G,A\<turnstile>{Normal (P\<leftarrow>In1 v)} Lit v-\<succ> {P}"
gen_subalgebra_sigma_sets: fixes G :: "'b set set"   and M :: "'b measure" assumes "G \<subseteq> sets M"   and "sigma_algebra (space M) G" shows "sets (gen_subalgebra M G) = G"
block_nominals_branch: fixes block :: "('b, 'c) fm list \<times> 'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list" assumes "block \<in>. branch" shows "block_nominals block \<subseteq> branch_nominals branch"
galois_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<cdot> - y \<le> z) = (x \<le> y + z)"
REFC: fixes it :: "'S set"   and it' :: "'Sa set"   and \<sigma>' :: "'\<sigma>a"   and \<sigma> :: "'\<sigma>" assumes "it \<subseteq> S"   and "it' \<subseteq> S'"   and "\<Phi>' it' \<sigma>'"   and "\<Phi> it \<sigma>"   and "\<forall>x\<in>S - it. \<forall>y\<in>it. RR x y"   and "\<forall>x\<in>S' - it'. \<forall>y\<in>it'. RR' x y"   and "it' = \<alpha> ` it"   and "((it, \<sigma>), it', \<sigma>') \<in> R" shows "c \<sigma> = c' \<sigma>'"
content_dvd_coeff: fixes p :: "'d poly"   and n :: "nat" shows "content p dvd coeff p n"
matrix_to_iarray_nrows: fixes A :: "(('b, 'c) vec, 'd) vec" shows "nrows A = nrows_iarray (matrix_to_iarray A)"
swapcols_mat: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and k :: "nat"   and l :: "nat" assumes "A \<in> carrier_mat nr n"   and "k < n"   and "l < n" shows "swapcols k l A = A * swaprows_mat n k l"
simps_3: fixes F :: "'b formula" shows "depth (\<^bold>\<not> F) = Suc (depth F)"
zless_nat_eq_int_zless: fixes m :: "nat"   and z :: "int" shows "(m < nat z) = (int m < z)"
borel_measurable_ereal_iff_Ioi: fixes f :: "'b \<Rightarrow> ereal"   and M :: "'b measure" shows "(f \<in> borel_measurable M) = (\<forall>a. f -` {a<..} \<inter> space M \<in> sets M)"
0: shows "(0::'a) \<in> affine hull {x + y |x y. x \<in> S \<and> y \<in> T}"
comm_groupE_1: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and y :: "'a" assumes "comm_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"
index_oddpart_decomposition: fixes n :: "nat" shows "n = 2 ^ index n * oddpart n"
distinct_3: fixes x3 :: "int" shows "Unit \<noteq> Intg x3"
param_dfs_variant_step: fixes s :: "('v, 'es) state_scheme" assumes "DFS_invar G param s"   and "cond s"   and "nofail it_dfs" shows "step s \<le> SPEC (\<lambda>s'. (s', s) \<in> param_dfs_variant)"
triangle_left: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" assumes "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "V (H \<epsilon> f) (V (bicategory.\<a>' V H \<a> src trg f g f) (H f \<eta>)) = V (category.inv V (bicategory.lunit V H \<a> \<i> src trg f)) (bicategory.runit V H \<a> \<i> src f)"
elims_2: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "'b list" assumes "sorted_wrt x xa" obtains P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" where "x = Pa__" and "xa = []"   | P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and xb :: "'b" and ys :: "'b list" where "x = P" and "xa = xb # ys" and "Ball (set ys) (P xb) \<and> sorted_wrt P ys"
srev_scons: fixes x :: "'r"   and xs :: "[:'r:]" shows "srev\<cdot>(x :# xs) = srev\<cdot>xs :@ [:x:]"
assms_3: shows "length (half_segments_of_aform X) \<noteq> 1"
card_size_filter_eq: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "finite A" shows "card {a \<in> A. P a} = size (filter_mset P (mset_set A))"
minus_le_self_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "less_eq (uminus a) a = less_eq zero a"
LCons_image_Int_not_lnull: fixes x :: "'c"   and A :: "'c llist set" shows "LCons x ` A \<inter> {xs. \<not> lnull xs} = LCons x ` A"
cInf_insert: 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 X :: "'a set"   and a :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "preorder.bdd_below less_eq X" shows "Inf (insert a X) = inf a (Inf X)"
sub_distrib_pconjI: fixes prog :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>P Q. \<lbrakk>unitary P; unitary Q\<rbrakk> \<Longrightarrow> wlp prog P && wp prog Q \<tturnstile> wp prog (P && Q)" shows "sub_distrib_pconj prog"
lfp_iterates: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "healthy (wp body)"   and "bd_cts (wp body)" shows "equiv_trans (wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)) (Sup_trans (range (iterates body G)))"
assms: shows "max_Var_floatarith fa \<le> D"
vcard_vdiff: fixes A :: "V"   and B :: "V" shows "vcard (A -\<^sub>\<circ> B) \<oplus> vcard (A \<inter>\<^sub>\<circ> B) = vcard A"
typing_alt_simps_5: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and c :: "int"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>E Literal (IntegerLiteral c) : \<tau>) = (\<tau> = Integer[1])"
length_last_Cons_code: fixes x :: "'a"   and xs :: "'a list" shows "length_last (x # xs) = fold (\<lambda>x (n, uu_). (n + 1, x)) xs (1, x)"
rel_fun_def: fixes A :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
sorted_RBT_Set_keys: fixes c :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and rbt :: "'c set_rbt" assumes "ID ccompare = Some c" shows "linorder.sorted (le_of_comp c) (RBT_Set2.keys rbt)"
assms_1: shows "\<exists>c>0. \<exists>n. \<forall>m>n. f m \<le> c * g m"
OF_eq: fixes \<gamma> :: "'b set \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "('b, 'd) flow_entry_match list"   and p :: "'c" assumes "no_overlaps \<gamma> f"   and "sorted_descending (map ofe_prio f)" shows "OF_match_linear \<gamma> f p = OF_priority_match \<gamma> f p"
hyps: shows "\<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2)"
lincomb_append: fixes rs :: "'r list"   and ms :: "'m list"   and ss :: "'r list"   and ns :: "'m list" assumes "length rs = length ms" shows "(rs @ ss) \<bullet>\<cdot> (ms @ ns) = rs \<bullet>\<cdot> ms + ss \<bullet>\<cdot> ns"
Ord_less_TC_mem: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "\<beta> \<sqsubset> \<alpha>" shows "\<beta> \<in>\<^sub>\<circ> \<alpha>"
pp_star_commute: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "\<ominus> \<ominus> (star \<circ> f) = star \<circ> \<ominus> \<ominus> f"
unit_group_def: shows "unit_group \<equiv> \<lparr>carrier = UNIV, mult = \<lambda>x y. (), one = ()\<rparr>"
elims: fixes x :: "('a \<times> 'b \<times> 'a) list"   and xa :: "'a \<times> 'b \<times> 'a"   and xb :: "('a \<times> 'b \<times> 'a) list"   and y :: "('a \<times> 'b \<times> 'a) list" assumes "replace_by_UnPath x xa xb = y" obtains uu_ :: "'a \<times> 'b \<times> 'a" and uv_ :: "('a \<times> 'b \<times> 'a) list" where "x = []" and "xa = uu_" and "xb = uv_" and "y = []"   | xc :: "'a \<times> 'b \<times> 'a" and xs :: "('a \<times> 'b \<times> 'a) list" and v :: "'a" and e :: "'b" and v' :: "'a" and ps :: "('a \<times> 'b \<times> 'a) list" where "x = xc # xs" and "xa = (v, e, v')" and "xb = ps" and "y = (if xc = (v, e, v') then ps @ replace_by_UnPath xs (v, e, v') ps else if xc = (v', e, v) then rev_path ps @ replace_by_UnPath xs (v, e, v') ps else xc # replace_by_UnPath xs (v, e, v') ps)"
a1: shows "{(t, u). s setinterleaves ((t, u), A) \<and> (\<exists>a list. t = a # list \<and> a \<notin> A) \<and> u = []} \<subseteq> {(hd s # list, []) |list. tl s setinterleaves ((list, []), A) \<and> hd s \<notin> A}"
add_poly_p_add_mset_comb: fixes A :: "(char list multiset \<times> int) multiset"   and Aa :: "(char list multiset \<times> int) multiset"   and r :: "(char list multiset \<times> int) multiset"   and xs :: "char list multiset"   and n :: "int" assumes "add_poly_p\<^sup>*\<^sup>* (A, Aa, {#}) ({#}, {#}, r)" shows "add_poly_p\<^sup>*\<^sup>* (add_mset (xs, n) A, Aa, {#}) ({#}, {#}, add_mset (xs, n) r)"
sorted_list_of_map_def: fixes f :: "'c \<Rightarrow> 'd option" shows "sorted_list_of_map f = map (\<lambda>k. (k, the (f k))) (sorted_list_of_set (dom f))"
bdd_above_mono: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a set"   and A :: "'a set" assumes "class.preorder less_eq less"   and "preorder.bdd_above less_eq B"   and "A \<subseteq> B" shows "preorder.bdd_above less_eq A"
Cidr: fixes C :: "('g, 'h, 'i) Category_scheme"   and f :: "'h" assumes "Category C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "f ;;\<^bsub>C\<^esub> id\<^bsub>C\<^esub> (cod\<^bsub>C\<^esub> f) = f"
w_def: shows "w = (\<chi>i. Re (v $h i))"
update_def: fixes x :: "nat"   and c :: "'a"   and s :: "('c, 'a) state" shows "update x c s \<equiv> \<V>_update (Mapping.update x c (foldl (\<lambda>v' e. Mapping.update (lhs e) (rhs_eq_val (\<V> s) x c e) v') (\<V> s) (\<T> s))) s"
real_RN_deriv: fixes N :: "'a measure" assumes "finite_measure N"   and "absolutely_continuous N N"   and "sets N = N.events" obtains D :: "'a \<Rightarrow> real" where "N.random_variable borel D" and "AE x in N. RN_deriv N N x = ennreal (D x)" and "AE x in N. 0 < D x" and "\<And>x. 0 \<le> D x"
iTILL_iprev_nth: fixes a :: "nat"   and n :: "nat" assumes "a \<le> n" shows "[\<dots>n] \<leftarrow> a = n - a"
monotone_bind_spmf1: fixes g :: "'c \<Rightarrow> 'd spmf" shows "monotone (ord_spmf (=)) (ord_spmf (=)) (\<lambda>y. y \<bind> g)"
spr_jview_length_eq: fixes a :: "'a"   and t :: "'s Trace" shows "tLength (spr_jview a t) = tLength t"
myle: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = ((0::'a) \<le> b - a)"
pos_zdiv_mult_2: fixes a :: "int"   and b :: "int" assumes "0 \<le> a" shows "(1 + 2 * b) div (2 * a) = b div a"
long_ch_by_ord2_def: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set" shows "long_ch_by_ord2 f X \<equiv> \<exists>x\<in>X. \<exists>y\<in>X. \<exists>z\<in>X. x \<noteq> y \<and> y \<noteq> z \<and> x \<noteq> z \<and> ordering2 f betw X"
sums_confine_le: fixes f :: "nat \<Rightarrow> 'a"   and N :: "nat set"   and l :: "'a" shows "confine f N sums l = (\<lambda>n. sum f ({..n} \<inter> N)) \<longlonglongrightarrow> l"
s\<^sub>1_r: shows "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Ref (VE v)])"
fract_collapse_3: fixes k :: "'c" shows "Fraction_Field.Fract k (0::'c) = 0"
conjugate_char_2_pp: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" shows "conjugate f g = (f bot = bot \<and> g bot = bot \<and> (\<forall>x y. f x \<sqinter> y \<le> - - f (x \<sqinter> - - g y) \<and> g y \<sqinter> x \<le> - - g (y \<sqinter> - - f x)))"
simps_2: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and x :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and X :: "'b set" shows "ipurge_fail_aux_t_aux \<lparr>Pol = I, Map = D, Doms = U, List = x # xs, ListOp = Some ys, Set = X\<rparr> = (if \<exists>u\<in>U. (u, D x) \<in> I then ipurge_fail_aux_t_aux \<lparr>Pol = I, Map = D, Doms = U, List = ipurge_tr I D (D x) xs, ListOp = Some ys, Set = ipurge_ref I D (D x) xs X\<rparr> else ipurge_fail_aux_t_aux \<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = Some (ys @ [x]), Set = X\<rparr>)"
less_def: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "dpo le" shows "dpo.less le x y = (le x y \<and> x \<noteq> y)"
empty_subsetI: fixes A :: "'g set" shows "{} \<subseteq> A"
less_le_not_le: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.preorder less_eq less" shows "less x y = (less_eq x y \<and> \<not> less_eq y x)"
filter_def: shows "DList.filter \<equiv> (id ---> rep_dlist ---> dlist) List.filter"
polymap_times: fixes p :: "'f poly"   and q :: "'f poly" shows "polymap (p * q) = polymap p \<circ> polymap q"
lmeasurable_iff_integrable_on: fixes S :: "'b set" shows "(S \<in> lmeasurable) = ((\<lambda>x. 1) integrable_on S)"
sel_8: fixes x1 :: "real"   and x2 :: "real"   and x3 :: "real"   and x4 :: "real"   and x5 :: "real"   and x6 :: "real"   and x7 :: "real"   and x8 :: "real" shows "Im7 (Octo x1 x2 x3 x4 x5 x6 x7 x8) = x8"
transitive: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "(a, b) \<in> E"   and "(b, c) \<in> E" shows "(a, c) \<in> E"
bidirected_digraphI: fixes arev :: "'b \<Rightarrow> 'b" assumes "\<And>a. a \<notin> arcs G \<Longrightarrow> arev a = a"   and "\<And>a. a \<in> arcs G \<Longrightarrow> arev a \<noteq> a"   and "\<And>a. a \<in> arcs G \<Longrightarrow> arev (arev a) = a"   and "\<And>a. a \<in> arcs G \<Longrightarrow> tail G (arev a) = head G a" shows "bidirected_digraph G arev"
simps_1: fixes s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" shows "transfers [] s = s"
AE_T_iff: fixes P :: "'s stream \<Rightarrow> bool"   and x :: "'s" assumes "Measurable.pred (stream_space (count_space UNIV)) P" shows "almost_everywhere (T x) P = (\<forall>y\<in>set_pmf (K x). AE \<omega> in T y. P (y ## \<omega>))"
birkhoff_sum_small_asymp_neg_nonergodic: fixes f :: "'a \<Rightarrow> real"   and e :: "real" assumes "integrable M f"   and "0 < e" shows "AE x in M. infinite {n. birkhoff_sum f n x \<in> {real n * real_cond_exp M Invariants f x - e..real n * real_cond_exp M Invariants f x}}"
S_def: fixes h :: "'a \<times> 'a"   and e :: "nat" shows "chaum_ped_sigma.S h e = map_spmf (\<lambda>(a, z). (a, e, z)) (S2 h e)"
Catch: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and P :: "'f set"   and c\<^sub>1 :: "('f, 'e, 'g) com"   and Q :: "'f set"   and R :: "'f set"   and c\<^sub>2 :: "('f, 'e, 'g) com"   and A :: "'f set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c\<^sub>1 Q,R"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>R c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Catch c\<^sub>1 c\<^sub>2 Q,A"
strip_map2_acom: fixes C1 :: "'a acom"   and C2 :: "'a acom"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "strip C1 = strip C2" shows "strip (map2_acom f C1 C2) = strip C1"
add_mono: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "a \<le> b"   and "c \<le> d" shows "a + c \<le> b + d"
abs_of_nat: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (semiring_1.of_nat one plus zero n) = semiring_1.of_nat one plus zero n"
ideal_resource_wiring: fixes callee :: "'d \<Rightarrow> 'a \<Rightarrow> ('c \<times> 'd, auth_query, 'b option) gpv"   and s :: "'d" shows "CNV callee s |\<^sub>= 1\<^sub>C \<rhd> channel.res auth_channel.auth_oracle = RES (callee_auth_channel callee \<oplus>\<^sub>O \<dagger>\<dagger>channel.send_oracle \<oplus>\<^sub>O \<dagger>\<dagger>channel.recv_oracle) (s, (), Void)"
setdist_eq_0_closed: fixes S :: "'c set"   and x :: "'c" assumes "closed S" shows "(setdist {x} S = 0) = (S = {} \<or> x \<in> S)"
real_sqrt_ge_0_iff: fixes y :: "real" shows "(0 \<le> sqrt y) = (0 \<le> y)"
traverse_impl_def: fixes E :: "('c \<times> 'c) list" shows "traverse_impl E = dds_of (\<lambda>visited A. (map snd (filter (\<lambda>(x, y). x \<in> set A \<and> y |\<notin>| visited) E), visited |\<union>| fset_of_list A)) {||}"
scale_eq_0_iff: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'c"   and x :: "'d" assumes "vector_space scale" shows "(scale a x = (0::'d)) = (a = (0::'c) \<or> x = (0::'d))"
arcs_predecessor': fixes a :: "nat"   and b :: "nat"   and i :: "nat"   and j :: "nat"   and x :: "nat"   and xs :: "nat list" assumes "(a, b) \<in> set (arcs i j (x # xs))"   and "(a, b) \<noteq> (i, x)" shows "\<exists>c. (c, a) \<in> set (arcs i j (x # xs))"
LIMSEQ_offset: fixes f :: "nat \<Rightarrow> 'e"   and k :: "nat"   and a :: "'e" assumes "(\<lambda>n. f (n + k)) \<longlonglongrightarrow> a" shows "f \<longlonglongrightarrow> a"
integral_on_1_eq: fixes S :: "'a set"   and f :: "'a \<Rightarrow> (real, 1) vec" shows "integral S f = vec (integral S (\<lambda>x. f x $ 1))"
add: fixes fa :: "'a \<Rightarrow>\<^sub>L 'b"   and b1 :: "'a"   and b2 :: "'a" shows "blinfun_apply fa (b1 + b2) = blinfun_apply fa b1 + blinfun_apply fa b2"
nat_uint_eq: fixes w :: "'k word" shows "nat (uint w) = unat w"
unity_div_num: fixes k :: "nat"   and d :: "nat"   and x :: "nat" assumes "0 < k"   and "0 < d"   and "d dvd k" shows "unity_root k (int (x * (k div d))) = unity_root d (int x)"
call_result: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs :: "'c rule list"   and t :: "state" assumes "matches \<gamma> m p"   and "\<Gamma> chain = Some rs"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Undecided\<rangle> \<Rightarrow> t" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m (Call chain)], Undecided\<rangle> \<Rightarrow> t"
i_cut_singleton_2: fixes a :: "'b"   and t :: "'b" shows "{a} \<down>< t = (if a < t then {a} else {})"
Int_ac_4: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
Restrict_def: fixes A :: "'d set"   and r :: "('d \<times> 'e) set" shows "Restrict A r = r \<inter> A \<times> UNIV"
assms_1: fixes var :: "Vars.var" shows "P (Var var)"
G_def: fixes A :: "(('b, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" shows "fixed_mat.G A = {(i, j). A $h i $h j \<noteq> (0::'b)}"
assoc_props_3: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data"   and h :: "'a arrow_of_spans_data" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "\<guillemotleft>\<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc (\<cdot>) prj0 prj1 f g h, Dom = Dom ((f \<star> g) \<star> h), Cod = Cod (f \<star> g \<star> h)\<rparr> : (f \<star> g) \<star> h \<Rightarrow> f \<star> g \<star> h\<guillemotright>"
total_on_def: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" shows "Restricted_Predicates.total_on P A = (\<forall>x\<in>A. \<forall>y\<in>A. x = y \<or> P x y \<or> P y x)"
w\<^sub>\<mu>: shows "ide (t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) \<and> is_left_adjoint (t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)"
uniform_size_incomp: fixes bl :: "'a set" assumes "\<k> \<le> \<v> - 1"   and "bl \<in># \<B>" shows "incomplete_block bl"
self_factor_iso: shows "(\<lambda>X. the_elem ((\<lambda>x. \<one>) ` X)) \<in> Group.iso (G Mod carrier G) (G\<lparr>carrier := {\<one>}\<rparr>)"