hcju/isabelleps huggingface dataset:
inject_3: fixes x31 :: "nat set"   and x32 :: "nat set"   and x33 :: "'d"   and y31 :: "nat set"   and y32 :: "nat set"   and y33 :: "'d" shows "(x31\<not>x32:arrive(x33) = y31\<not>y32:arrive(y33)) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33)"
flowD_capacity_OUT: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "flow \<Delta> f" shows "d_OUT f x \<le> d_OUT (capacity \<Delta>) x"
y: shows "(r|labels \<gamma>1''| + r|labels \<gamma>2|, {#fst \<alpha>_step#} + {#fst \<beta>_step#}) \<in> mul r"
lbs_ex_ZleastTr: fixes a :: "int"   and A :: "int set"   and n :: "nat" assumes "a \<in> A"   and "A \<subseteq> Zset"   and "\<nexists>m. m \<in> A \<and> (\<forall>x\<in>A. m \<le> x)" shows "dec_seq A a n \<le> a - int n"
P1: shows "P InAngle D E F \<and> A B C CongA D E P"
\<psi>_eval: fixes i :: "nat" assumes "i < N" shows "\<psi> $ i = complex_of_real (1 / sqrt (real N))"
simps_2: fixes x :: "'a"   and xs :: "'a list"   and e :: "'a" shows "insert_body (x # xs) e = (if x < e then e # x # xs else x # insert_body xs e)"
2: shows "B1 \<noteq> 0 \<and> \<not> 0 < arg B1 \<longrightarrow> 0 < arg (- B1)"
measure_parity: shows "(strict {(x, y). x \<sqsubseteq> y})\<inverse> \<subseteq> measure m_parity"
analz_Un_implSecureSet: fixes G :: "msg set"   and H :: "msg set"   and Ag :: "(agent \<times> agent) set" assumes "Enc_keys_clean (G \<union> H)" shows "analz ({asym.implSecure A B M |A B M. (A, B) \<in> Ag \<and> M \<in> G} \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
SOME_inv_dc: fixes a :: "nat option"   and b :: "bool list" assumes "(a, b) \<in> dc_crypto" shows "(SOME x. inversion x = inversion (a, b) \<and> x \<in> dc_crypto) \<in> dc_crypto"
even_take_bit_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (take_bit n a) = (n = 0 \<or> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
sum_disjoint: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqinter> y = bot" shows "sum_class.sum ((x \<squnion> y) \<sqinter> z) = sum_class.sum (x \<sqinter> z) + sum_class.sum (y \<sqinter> z)"
fw_canonical_subs: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and k :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..k} \<longrightarrow> less_eq zero (linordered_ab_monoid_add.len plus m i i xs)"   and "k \<le> n" shows "linordered_ab_monoid_add.canonical_subs plus less_eq n {0..k} (linordered_ab_monoid_add.fw plus less_eq m n k)"
prems_1: shows "local_iwf t (\<Gamma> \<turnstile> c)"
lookup_combine_with_key: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and m1 :: "('b, 'a) mapping"   and m2 :: "('b, 'a) mapping"   and x :: "'b" shows "Mapping.lookup (Mapping.combine_with_key f m1 m2) x = combine_options (f x) (Mapping.lookup m1 x) (Mapping.lookup m2 x)"
le_less_trans: fixes x :: "('a, 'b) Term.term"   and y :: "('a, 'b) Term.term"   and z :: "('a, 'b) Term.term" assumes "x \<sqsubseteq> y"   and "y \<sqsubset> z" shows "x \<sqsubset> z"
not_knows_conc_knows_elem_not_knows_tail: fixes P :: "specID"   and a :: "Expression"   and e :: "Expression list" assumes "\<not> knows P (a # e)"   and "knows P [a]" shows "\<not> knows P e"
ahm_delete_correct: fixes k :: "'a"   and hm :: "('a, 'b) hashmap" shows "ahm_\<alpha> (ahm_delete k hm) = ahm_\<alpha> hm |` (- {k})"
simps_2: fixes \<sigma> :: "(char list \<times> 'b list) Trace.trace"   and v :: "'b list"   and i :: "nat"   and t1 :: "'b trm"   and t2 :: "'b trm" shows "sat \<sigma> v i (Eq t1 t2) = (eval_trm v t1 = eval_trm v t2)"
path2_hd: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m" shows "n = hd ns"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "Left_a.in_hom f a b = (Left_a.arr a \<and> Left_a.arr b \<and> Left_a.arr f \<and> \<guillemotleft>f : a \<rightarrow> b\<guillemotright>)"
poincare_line_1: fixes z :: "complex_homo"   and w :: "complex_homo" assumes "z \<noteq> w" shows "on_circline (poincare_line z w) z"
equivp_imp_transp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "equivp R" shows "transp R"
bisimParNil: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<parallel> \<zero> \<sim> P"
ZOretr_sym: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set" assumes "sym theta" shows "sym (ZOretr theta)"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Reinterpret of Convert \<Rightarrow> f1 | Reinterpret \<Rightarrow> f2) = f2"
emeasure_real: fixes M :: "'b pmf"   and A :: "'b set" shows "\<exists>r\<ge>0. emeasure (measure_pmf M) A = ennreal r"
map_of_entries: fixes t :: "('b, 'c) RBT.rbt" shows "map_of (RBT.entries t) = RBT.lookup t"
flincomb_Horbit_induced_veclist_reduce: fixes vs :: "'v list"   and i :: "nat"   and hs :: "'g list"   and css :: "'f list list" assumes "set vs \<subseteq> V"   and "i < length hs"   and "list_all2 (\<lambda>rs ms. length rs = length ms) css (negHorbit_list hs induced_vector vs)"   and "Supgroup.is_rcoset_replist G hs" shows "(concat css \<bullet>\<currency>\<currency> concat (negHorbit_list hs induced_vector vs)) ((1::'f) \<delta>\<delta> hs ! i) = css ! i \<bullet>\<sharp>\<cdot> vs"
delete_conv: fixes k :: "'b"   and al :: "('b \<times> 'a) list"   and k' :: "'b" shows "map_of (AList.delete k al) k' = ((map_of al)(k := None)) k'"
box_real_2: fixes a :: "real"   and b :: "real" shows "cbox a b = {a..b}"
subset_basis: shows "(\<lambda>x. axis x (1::'b)) ` d \<subseteq> cart_basis"
distinct_4: fixes x3 :: "nat"   and x1 :: "nat" shows "Regions.intv.Greater x3 \<noteq> Regions.intv.Const x1"
le_cases: fixes y :: "'d"   and x :: "'d" obtains "y \<le> x"   | "x \<le> y"
intros_1: fixes g :: "'f \<Rightarrow> ('e, 'f) step"   and s :: "'f" assumes "g s = Done" shows "s \<in> terminates_on g"
bi_uniqueI: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "left_unique R"   and "right_unique R" shows "bi_unique R"
lt_eq_tt_monomial: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "is_monomial p" shows "punit.lt p = punit.tt p"
cases: fixes a1 :: "'state"   and a2 :: "'state behaviour" assumes "a1 \<down> a2" obtains s1 :: "'state" and s2 :: "'state" where "a1 = s1a__" and "a2 = Terminates s2a__" and "s1a__ \<rightarrow>\<^sup>* s2a__" and "finished (\<rightarrow>) s2a__" and "final s2a__"   | s1 :: "'state" where "a1 = s1b__" and "a2 = Diverges" and "s1b__ \<rightarrow>\<^sup>\<infinity>"   | s1 :: "'state" and s2 :: "'state" where "a1 = s1" and "a2 = Goes_wrong s2" and "s1 \<rightarrow>\<^sup>* s2" and "finished (\<rightarrow>) s2" and "\<not> final s2"
simps: fixes k :: "name"   and s :: "name"   and y :: "tm" assumes "atom k \<sharp> (s, y)"   and "atom s \<sharp> y" shows "PfP y = SyntaxN.Ex k (SyntaxN.Ex s (PrfP (Var s) (Var k) y))"
H_without_scalar_prod: shows "H = Matrix.mat 2 2 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if i \<noteq> j then 1 / sqrt 2 else if i = 0 then 1 / sqrt 2 else - (1 / sqrt 2)))"
assignment_rhs\<^sub>e\<^sub>s\<^sub>t_append_subst_1: fixes A :: "('fun, 'var) extstrand_step list"   and B :: "('fun, 'var) extstrand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" shows "assignment_rhs\<^sub>e\<^sub>s\<^sub>t (A @ B \<cdot>\<^sub>e\<^sub>s\<^sub>t \<theta>) = assignment_rhs\<^sub>e\<^sub>s\<^sub>t (A \<cdot>\<^sub>e\<^sub>s\<^sub>t \<theta>) \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t (B \<cdot>\<^sub>e\<^sub>s\<^sub>t \<theta>)"
val_unit_cond: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "0 < v (1\<^sub>r \<plusminus> -\<^sub>a x)" shows "v x = 0"
hyps: fixes thesis :: "bool" assumes "\<And>lb. \<lbrakk>\<forall>p\<in>set ps. \<forall>x. poly p x = 0 \<longrightarrow> lb < x; \<forall>x\<le>lb. \<forall>p\<in>set ps. sgn (poly p x) = sgn_neg_inf p; lb < b\<rbrakk> \<Longrightarrow> thesis"   and "0 \<notin> set ps" shows "thesis"
basic_just_sp_axioms: shows "basic_just_sp G dist c s num"
slice2_sum_slice2: fixes nx :: "'node SDG_node"   and n :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node" assumes "nx \<in> sum_SDG_slice2 n"   and "n s-p\<rightarrow>\<^bsub>sum\<^esub> n'" shows "nx \<in> sum_SDG_slice2 n'"
normalize_unit_factor: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "normalize (unit_factor a) = (1::'a)"
len_le_Suc: fixes i :: "nat" shows "length (worst_chain (\<lambda>ys xs. length ys \<le> n \<and> length xs \<le> n \<and> ext gt ys xs) (\<lambda>ys xs. length xs < length ys) i) \<le> length (worst_chain (\<lambda>ys xs. length ys \<le> n \<and> length xs \<le> n \<and> ext gt ys xs) (\<lambda>ys xs. length xs < length ys) (Suc i))"
dghm_comp_ArrMap_vdomain: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<D>\<^sub>\<circ> ((\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>ArrMap\<rparr>) = \<AA>\<lparr>Arr\<rparr>"
r: shows "local.gen_shleg_poly n = [:fact n:] * r"
rbt_map_rel_sv: fixes Rk :: "('a \<times> 'b) set"   and Rv :: "('c \<times> 'd) set"   and lt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "single_valued Rk"   and "single_valued Rv" shows "single_valued (\<langle>Rk, Rv\<rangle>rbt_map_rel lt)"
rcond_true_trel: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and b :: "'b upred"   and P :: "'b hrel"   and Q :: "'b hrel" assumes "\<sigma> \<dagger> b = true" shows "(\<sigma>, P \<triangleleft> b \<triangleright>\<^sub>r Q) \<rightarrow>\<^sub>u (\<sigma>, P)"
vector_space_pair_axioms: shows "vector_space_pair (*\<^sub>R) (*\<^sub>R)"
lang_rderiv_lderivs: fixes n :: "nat"   and r :: "'b rexp"   and w :: "'a list"   and a :: "'a" assumes "local.wf n r"   and "wf_word n w"   and "a \<in> \<Sigma> n" shows "lang n (rderiv a (lderivs w r)) = lang n (lderivs w (rderiv a r))"
dnso4: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and domain_op :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.domain_near_semiring_one plus times one domain_op less_eq less" shows "domain_op (plus x y) = plus (domain_op x) (domain_op y)"
simps: fixes A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" shows "diagonal_to_Smith_PQ A bezout = diagonal_to_Smith_aux_PQ [0..<min (nrows A) (ncols A) - 1] bezout (mat (1::'b), A, mat (1::'b))"
StrongRelease_ltlp_def: fixes \<phi> :: "'b pltl"   and \<psi> :: "'b pltl" shows "\<phi> M\<^sub>p \<psi> \<equiv> \<psi> U\<^sub>p (\<phi> and\<^sub>p \<psi>)"
wt_elim_cases_23: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and l :: "jump"   and c :: "stmt"   and x :: "tys" assumes "E,dt\<Turnstile>In1r (l\<bullet> c)\<Colon>x" obtains "x = Inl (PrimT Void)" and "E,dt\<Turnstile>c\<Colon>\<surd>"
match: shows "satisfy_equation p qs [[], [0]] [[1], [- 1]]"
Ints_1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.ring_1 one times plus zero minus uminus" shows "one \<in> ring_1.Ints one plus zero minus"
norm_signed1: shows "norm_signed [\<one>] = [\<one>]"
nth_rev_to_bl: fixes n :: "nat"   and w :: "'b word" assumes "n < LENGTH('b)" shows "rev (to_bl w) ! n = bit w n"
bal_list_def2: fixes n :: "nat"   and xs :: "'b list" shows "Root_Balanced_Tree.bal_list n xs = (let (t, ys) = bal n xs in t)"
arrI: fixes f :: "'a \<times> 'a" assumes "arr (fst f) \<and> arr (snd f) \<and> src (fst f) = trg (snd f)" shows "VV.arr f"
simps: fixes blue :: "'a"   and brown :: "'a"   and green :: "'a"   and red :: "'a"   and guru :: "'b"   and n :: "nat"   and p :: "'b"   and w :: "'b \<Rightarrow> 'a" assumes "distinct [blue, brown, green, red]"   and "finite UNIV" shows "Blue_Eyes.leaves blue guru n p w = (\<forall>w'. Blue_Eyes.possible blue guru n p w w' \<longrightarrow> w' p = w p)"
analytic_on_axis_extend: fixes y1 :: "real"   and y2 :: "real"   and f :: "complex \<Rightarrow> complex"   and x :: "real" assumes "y1 \<le> y2"   and "f analytic_on {z. Re z = x \<and> Im z \<in> {y1..y2}}" obtains x1 :: "real" and x2 :: "real" where "x1 < x" and "x < x2" and "f analytic_on cbox (Complex x1 y1) (Complex x2 y2)"
p_def: fixes x :: "nat"   and y :: "nat" shows "p x y = (if c_assoc_have_key (pr_gr y) (c_fst x) = 0 then if c_assoc_value (pr_gr y) (c_fst x) = c_snd x then 0 else 1 else 1)"
valid_cfg_action: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real) set" assumes "cfg \<in> R_G.valid_cfg"   and "s \<in> set_pmf (action cfg)" shows "s \<in> \<S>"
simps_10: fixes f :: "'b \<Rightarrow> 'a"   and x21 :: "'b bdd"   and x22 :: "'b bdd" shows "bdd_map f (Branch x21 x22) = Branch (bdd_map f x21) (bdd_map f x22)"
assms_1: shows "y \<cdot> (0::'a) = (0::'a)"
b_e_check_single_top_not_bot_sound: fixes ts :: "checker_type"   and t_in :: "t list"   and ts' :: "checker_type" assumes "type_update ts (to_ct_list t_in) (TopType []) = ts'"   and "ts \<noteq> Bot"   and "ts' \<noteq> Bot" shows "\<exists>tn. c_types_agree ts tn \<and> suffix t_in tn"
dist_triangle_lt: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and z :: "'a"   and y :: "'a"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "dist x z + dist y z < e" shows "dist x y < e"
take_update_cancel: fixes n :: "nat"   and m :: "nat"   and xs :: "'d list"   and y :: "'d" assumes "n \<le> m" shows "take n (xs[m := y]) = take n xs"
constant_functor_def: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and b :: "'ah" shows "constant_functor A B b \<equiv> category A \<and> category B \<and> constant_functor_axioms B b"
simps_2: fixes onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and p\<^sub>1 :: "net_tree"   and p\<^sub>2 :: "net_tree" shows "opnet onp (p\<^sub>1 \<parallel> p\<^sub>2) = \<lparr>init = {(\<sigma>, SubnetS s\<^sub>1 s\<^sub>2) |\<sigma> s\<^sub>1 s\<^sub>2. (\<sigma>, s\<^sub>1) \<in> init (opnet onp p\<^sub>1) \<and> (\<sigma>, s\<^sub>2) \<in> init (opnet onp p\<^sub>2) \<and> net_ips s\<^sub>1 \<inter> net_ips s\<^sub>2 = {}}, trans = opnet_sos (automaton.trans (opnet onp p\<^sub>1)) (automaton.trans (opnet onp p\<^sub>2))\<rparr>"
y: shows "y = Quad (a2, b2, c2, d2)"
less_eq_char_simp: fixes b0 :: "bool"   and b1 :: "bool"   and b2 :: "bool"   and b3 :: "bool"   and b4 :: "bool"   and b5 :: "bool"   and b6 :: "bool"   and b7 :: "bool"   and c0 :: "bool"   and c1 :: "bool"   and c2 :: "bool"   and c3 :: "bool"   and c4 :: "bool"   and c5 :: "bool"   and c6 :: "bool"   and c7 :: "bool" shows "(Char b0 b1 b2 b3 b4 b5 b6 b7 \<le> Char c0 c1 c2 c3 c4 c5 c6 c7) = (foldr (\<lambda>b k. of_bool b + k * 2) [b0, b1, b2, b3, b4, b5, b6, b7] 0 \<le> foldr (\<lambda>b k. of_bool b + k * 2) [c0, c1, c2, c3, c4, c5, c6, c7] 0)"
interpret_Binop: fixes a :: "floatarith"   and b :: "floatarith"   and xs :: "real list" shows "interpret_floatarith (Binop a b) xs = binop (interpret_floatarith a xs) (interpret_floatarith b xs)"
wf_trm_subst_singleton: fixes t :: "('fun, 'a) Term.term"   and t' :: "('fun, 'a) Term.term"   and v :: "'a" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "wf\<^sub>t\<^sub>r\<^sub>m t'" shows "wf\<^sub>t\<^sub>r\<^sub>m (t \<cdot> Var(v := t'))"
iso_onto_image: fixes H :: "('b, 'c) monoid_scheme"   and f :: "'a \<Rightarrow> 'b" assumes "Group.group H" shows "(f \<in> Group.iso A (subgroup_generated H (f ` carrier A))) = (f \<in> mon A H)"
pentagon': fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and h :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and k :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f"   and "Span.ide g"   and "Span.ide h"   and "Span.ide k"   and "Span.src f = Span.trg g"   and "Span.src g = Span.trg h"   and "Span.src h = Span.trg k" shows "((Span.\<a>' f g h \<circ> k) \<bullet> Span.\<a>' f (g \<circ> h) k) \<bullet> (f \<circ> Span.\<a>' g h k) = Span.\<a>' (f \<circ> g) h k \<bullet> Span.\<a>' f g (h \<circ> k)"
eval_lam: fixes \<rho> :: "(nat \<times> bval) list"   and x :: "nat"   and e :: "exp" shows "\<rho> \<turnstile> ELam x e \<Down> BClos x e \<rho>"
at_swap_simps_3: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "at TYPE('a)"   and "a \<noteq> c"   and "b \<noteq> c" shows "[(a, b)] \<bullet> c = c"
is_processT4_empty: fixes s :: "'c event list"   and X :: "'c event set"   and P :: "'c process" assumes "(s, X) \<in> F P" shows "(s, {}) \<in> F P"
ent_conjE1: fixes A :: "assn"   and C :: "assn"   and B :: "assn" assumes "A \<Longrightarrow>\<^sub>A C" shows "A \<and>\<^sub>A B \<Longrightarrow>\<^sub>A C"
projection_empty_implies_absence_of_events: fixes l :: "'a list"   and X :: "'a set" assumes "l \<upharpoonleft> X = []" shows "X \<inter> set l = {}"
closed_segment_eq_real_ivl: fixes a :: "real"   and b :: "real" shows "{a--b} = (if a \<le> b then {a..b} else {b..a})"
simps_1: fixes x1 :: "tag"   and x2 :: "agent"   and x3 :: "agent"   and x4 :: "msg"   and y1 :: "tag"   and y2 :: "agent"   and y3 :: "agent"   and y4 :: "msg" shows "(Chan x1 x2 x3 x4 = Chan y1 y2 y3 y4) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4)"
is_bound: shows "is_bound rel (SOME n. is_bound rel n)"
lens_put_of_quotient: fixes Y :: "'b \<Longrightarrow> 'c"   and X :: "'d \<Longrightarrow> 'c"   and s :: "'c"   and v\<^sub>2 :: "'b"   and v\<^sub>1 :: "'d" assumes "vwb_lens Y"   and "X \<subseteq>\<^sub>L Y" shows "put\<^bsub>Y\<^esub> s (put\<^bsub>X /\<^sub>L Y\<^esub> v\<^sub>2 v\<^sub>1) = put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> s v\<^sub>2) v\<^sub>1"
axioms_1: fixes f :: "'b \<Rightarrow> 'b"   and T :: "real set"   and S :: "'b set"   and \<phi> :: "real \<Rightarrow> 'b \<Rightarrow> 'b" assumes "local_flow f T S \<phi>" shows "picard_lindeloef (\<lambda>t. f) T S 0"
downsets_upsets_dual: fixes X :: "'a set" shows "(X \<in> downsets) = (\<partial> ` X \<in> upsets)"
strict_iff_order: fixes b :: "'c"   and a :: "'c" shows "(b < a) = (b \<le> a \<and> a \<noteq> b)"
simps: fixes i :: "nat" shows "gso i = fs ! i + sumlist (map (\<lambda>j. - \<mu> i j \<cdot>\<^sub>v gso j) [0..<i])"
cil_subset_il: fixes w :: "'c list"   and \<alpha> :: "'c \<Rightarrow> 'd set \<times> 'd set"   and w' :: "'c list" shows "w \<otimes>\<^bsub>\<alpha>\<^esub> w' \<subseteq> w \<otimes> w'"
F_def: fixes C :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "F C Q = (\<lambda>(ps, n). \<exists>ps1' ps2' m e1 e2. (C, ps) \<Rightarrow>\<^sub>A m \<Down> ps1' + ps2' \<and> ps1' ## ps2' \<and> n = e1 + e2 + m \<and> Q (ps1', e1))"
db\<^sub>s\<^sub>s\<^sub>t_append: fixes A :: "('d, 'e) stateful_strand_step list"   and B :: "('d, 'e) stateful_strand_step list"   and I :: "'e \<Rightarrow> ('d, 'e) Term.term"   and D :: "(('d, 'e) Term.term \<times> ('d, 'e) Term.term) list" shows "db'\<^sub>s\<^sub>s\<^sub>t (A @ B) I D = db'\<^sub>s\<^sub>s\<^sub>t B I (db'\<^sub>s\<^sub>s\<^sub>t A I D)"
sets_measure_pmf: fixes p :: "'b pmf" shows "measure_pmf.events p = UNIV"
weaken: fixes G :: "'b triple set"   and ts' :: "'b triple set"   and ts :: "'b triple set" assumes "G||-ts'"   and "ts \<subseteq> ts'" shows "G||-ts"
dom: shows "\<D>\<^sub>\<circ> vint_uminus = \<int>\<^sub>\<circ>"
cr_spmf_prob_optionT_def: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "('c, 'c spmf) optionT"   and q :: "'d spmf" shows "cr_spmf_prob_optionT A p q = rel_spmf A (run_option p) q"
i_preprocess'_sat: fixes I :: "'a set"   and v :: "nat \<Rightarrow> QDelta"   and s :: "('a \<times> QDelta ns_constraint) list"   and start :: "nat" assumes "(I, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>s set (Atoms (preprocess' s start))"   and "v \<Turnstile>\<^sub>t Tableau (preprocess' s start)"   and "I \<inter> set (UnsatIndices (preprocess' s start)) = {}" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>n\<^sub>s\<^sub>s  set s"
simps_5: fixes x21 :: "'a"   and x22 :: "'a convnet"   and x1 :: "nat" shows "Conv x21 x22 \<noteq> Input x1"
mono: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "(bs ! i * x) powr p * a \<le> (bs ! i * x) powr p * b"
pred_pmf_return: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" shows "pred_pmf P (return_pmf x) = P x"
map_pred_def: fixes f :: "'e \<Rightarrow> 'd" shows "map_pred f = f ---> id"
star_zero_def: shows "0 \<equiv> star_of (0::'a)"
enat_eSuc_iff: fixes y :: "nat"   and x :: "enat" shows "(enat y = eSuc x) = (\<exists>n. y = Suc n \<and> enat n = x)"
heap_wellformed_induct_si: fixes h :: "(_) heap"   and P :: "(_) object_ptr \<Rightarrow> bool"   and ptr :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "\<And>parent. \<lbrakk>\<And>children child. \<lbrakk>h \<turnstile> Shadow_DOM.get_child_nodes parent \<rightarrow>\<^sub>r children; child \<in> set children\<rbrakk> \<Longrightarrow> P (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 child); \<And>shadow_root host. \<lbrakk>parent = cast\<^sub>e\<^sub>l\<^sub>e\<^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 host; h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root\<rbrakk> \<Longrightarrow> P (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^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 shadow_root)\<rbrakk> \<Longrightarrow> P parent" shows "P ptr"
sup_bot_right: fixes sup :: "'l \<Rightarrow> 'l \<Rightarrow> 'l"   and less_eq :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and less :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and bot :: "'l"   and x :: "'l" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup x bot = x"
arrE: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.arr (fst f) \<and> B.VV.arr (snd f) \<and> trg (fst f) = src (fst (snd f))"
setPair: fixes a :: "'b"   and b :: "'b"   and d :: "'b" assumes "{a, b} = {a, d}" shows "b = d"
assms_2: shows "Q \<leadsto><weakBisimulation> P"
sum_iff: fixes u :: "hf"   and A :: "hf"   and B :: "hf" shows "(u \<^bold>\<in> A + B) = ((\<exists>x. x \<^bold>\<in> A \<and> u = HF.Inl x) \<or> (\<exists>y. y \<^bold>\<in> B \<and> u = HF.Inr y))"
seq_char: fixes g :: "'a"   and f :: "'a" shows "Right.seq g f = (Right.arr f \<and> Right.arr g \<and> seq g f)"
assms_3: shows "configuration q (Suc n) \<noteq> {}"