hcju/isabelleps huggingface dataset:
SatN: fixes b :: "'d"   and ps :: "('c, 'd) fm list"   and a :: "'d"   and branch :: "(('c, 'd) fm list \<times> 'd) list"   and p :: "('c, 'd) fm"   and A :: "'d set"   and n :: "nat" assumes "(\<^bold>\<not> (\<^bold>@ a p)) at b in (ps, a) # branch"   and "new (\<^bold>\<not> p) a ((ps, a) # branch)"   and "A, Suc n \<turnstile> ((\<^bold>\<not> p) # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
finvimageI: fixes a :: "V"   and b :: "V"   and r :: "V"   and A :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"   and "b \<in>\<^sub>\<circ> A" shows "a \<in>\<^sub>\<circ> r -`\<^sub>\<bullet> A"
assms_4: shows "\<forall>name\<in>dom \<Gamma>. \<exists>t. \<Gamma>,\<gamma>,p\<turnstile> \<langle>the (\<Gamma> name), Undecided\<rangle> \<Rightarrow> t"
igSubstClsSTR_def: fixes MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igSubstClsSTR MOD \<equiv> igSubstIGVar1STR MOD \<and> igSubstIGVar2STR MOD \<and> igSubstIGAbsSTR MOD \<and> igSubstIGOpSTR MOD"
FEXISTS_2: shows "wf_formula n (FEXISTS \<phi>)"
assm: shows "N_Chop(i,i1,i2) \<and> N_Chop(i1,i3,i4)"
t1_range: shows "interpret_floatarith l xs \<in>\<^sub>i range_tm e t1"
f: shows "3 * 2 ^ 7 - 1 = 383"
approx_monad_iff: fixes x :: "'a star"   and y :: "'a star" shows "(x \<approx> y) = (monad x = monad y)"
rec_newrgt2_def: shows "rec_newrgt2 = Cn 2 rec_add [Cn 2 rec_mult [Cn 2 (constn 2) [recf.id 2 0], recf.id 2 1], Cn 2 rec_mod [recf.id 2 0, Cn 2 (constn 2) [recf.id 2 0]]]"
fresh_2: fixes a :: "coname"   and x2 :: "name"   and x1 :: "coname" shows "a \<sharp> Ax x2 x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
powr_divide: fixes x :: "real"   and y :: "real"   and a :: "real" assumes "0 \<le> x"   and "0 \<le> y" shows "(x / y) powr a = x powr a / y powr a"
subprob_space_null_measure: fixes M :: "'c measure" assumes "space M \<noteq> {}" shows "subprob_space (null_measure M)"
fn: fixes x :: "'a"   and y :: "'a"   and M :: "'a trm"   and A :: "'a trm"   and A' :: "'a trm"   and T :: "PreSimplyTyped.type" assumes "x \<noteq> y"   and "y \<notin> fvs M"   and "substitutes A x M A'" shows "substitutes (Fn y T A) x M (Fn y T A')"
Syntax_Arith_axioms: shows "Syntax_Arith var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num zer suc pls tms"
supp: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "VectorSpaceEnd smult V T" shows "supp T \<subseteq> V"
assms_1: fixes i :: "'b" assumes "i \<in> I" shows "F i \<subseteq> events"
pfxes_def: fixes uu :: "'c itself" shows "pfxes uu = map nat [0..int LENGTH('c)]"
suminf_mono_reindex: fixes g :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a" assumes "strict_mono g"   and "\<And>n. n \<notin> range g \<Longrightarrow> f n = (0::'a)" shows "(\<Sum>n. f (g n)) = suminf f"
discr_Seq: fixes c1 :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd" assumes "discr c1"   and "discr c2" shows "discr (c1 ;; c2)"
that: fixes r :: "recf" assumes "recfn 1 r"   and "Partial_Recursive.total r"   and "f = (\<lambda>x. eval r [x])" shows "thesis"
has_density_cong: fixes N :: "'b measure"   and f :: "'b \<Rightarrow> ennreal"   and g :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" shows "has_density M N f = has_density M N g"
HcompNml_Obj_Nml: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Obj t"   and "Nml u"   and "Src t = Trg u" shows "t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u = u"
individually_rational_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "PdXXX.individually_rational_on ds X = (PdXXX.CD_on ds X = X \<and> PdXXX.CH X = X)"
aezdelta0fun_commutes: fixes c :: "('a, 'b) aezfun"   and b :: "'a" shows "c * (b \<delta>\<delta> (0::'b)) = b \<delta>\<delta> (0::'b) * c"
FDERIV_n_floatariths_def: fixes fas :: "floatarith list"   and xs :: "nat list"   and ds :: "floatarith list"   and n :: "nat" shows "FDERIV_n_floatariths fas xs ds n = ((\<lambda>x. FDERIV_floatariths x xs ds) ^^ n) fas"
list_relE4: fixes l :: "'a list"   and x' :: "'b"   and xs' :: "'b list"   and R :: "('a \<times> 'b) set" assumes "(l, x' # xs') \<in> \<langle>R\<rangle>list_rel" obtains x :: "'a" and xs :: "'a list" where "l = x # xs" and "(x, x') \<in> R" and "(xs, xs') \<in> \<langle>R\<rangle>list_rel"
am3: fixes x :: "'a" shows "ad (ad x) \<cdot> x = x"
assms_4: shows "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"
f_shrink_length: fixes xs :: "'b message_af list"   and k :: "nat" shows "length (xs \<div> k) = length xs div k"
oldDefs_correct: fixes g :: "('b, 'a, 'c) gen_cfg_wf"   and n :: "'b" shows "gen_wf_defs g n = gen_wf_var g ` gen_wf_defs' g n"
FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m_unfold: fixes n :: "nat"   and fs :: "nat list" assumes "0 < n"   and "length fs = Suc n" shows "FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m (Suc n) fs = (FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m n (butlast fs)|||FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m n (fs ! n))"
bit_b''1: fixes x :: "'a"   and y :: "'a"   and x0 :: "'a"   and y0 :: "'a"   and s :: "('a list \<times> bool list \<times> 'a list) pmf"   and qs :: "'a list" assumes "x \<noteq> y"   and "{x, y} = {x0, y0}"   and "BIT_inv s x [x0, y0]"   and "set qs \<subseteq> {x, y}"   and "qs \<in> lang (seq [Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y])" shows "BIT_inv (Partial_Cost_Model.config'_rand BIT s qs) (last qs) [x0, y0] \<and> T\<^sub>p_on_rand' BIT s qs = 75 / 10\<^sup>2 * real (length qs) - 5 / 10"
lcm_factorial_gcd_factorial: 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 a :: "'a"   and b :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.lcm_factorial zero times one normalize a b = normalize (divide (times a b) (factorial_semiring.gcd_factorial zero times one normalize a b))"
nth_default_drop: fixes x :: "'b"   and n :: "nat"   and xs :: "'b list"   and m :: "nat" shows "nth_default x (drop n xs) m = nth_default x xs (m + n)"
sort_key_simps_1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a" assumes "class.linorder less_eq less" shows "linorder.sort_key less_eq f [] = []"
inv_closed: fixes x :: "'a list" assumes "x \<in> carrier (K [X])" shows "\<ominus>\<^bsub>K [X]\<^esub> x \<in> carrier (K [X])"
Id: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" shows "(\<Psi>, P, P) \<in> Rel"
Abs_OrdSum_inverse: fixes y :: "'b \<times> 'c" assumes "y \<in> OrdSum" shows "Rep_OrdSum (Abs_OrdSum y) = y"
PStatImpQ: fixes A\<^sub>R :: "name list"   and \<Psi>\<^sub>R :: "'b" assumes "extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle>"   and "A\<^sub>R \<sharp>* \<Psi>"   and "A\<^sub>R \<sharp>* P"   and "A\<^sub>R \<sharp>* Q" shows "\<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<lessapprox><Rel> Q"
comm: fixes a :: "'a"   and b :: "'a" shows "a * b = b * a"
suffix_commute: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat" shows "\<sigma> |\<^sub>s n |\<^sub>s m = \<sigma> |\<^sub>s m |\<^sub>s n"
scount_eq_0_iff: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "(scount P \<omega> = 0) = alw (\<lambda>xs. \<not> P xs) \<omega>"
list_of_oalist: fixes x :: "('a, 'b) oalist" shows "list_of_oalist x \<in> Collect ko.oalist_inv"
isSucc_pred_diff: fixes r :: "('c \<times> 'c) set"   and i :: "'c" assumes "wo_rel r"   and "isSucc r i" shows "pred r i \<noteq> i"
mult_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<sqinter> b \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
bigo_iff: shows "((\<lambda>_. 1) \<in> O(\<lambda>x. \<Prod>g\<leftarrow>group_primfuns fs. (eval_primfun' \<circ> fst) g x powr snd g)) = nonneg_list (map snd (group_primfuns fs))"
apx_min: fixes S :: "('a \<Rightarrow> real) set"   and U :: "('a \<Rightarrow> real) set set"   and M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and Z :: "('a \<Rightarrow> real) set" assumes "S = \<Union> U"   and "U \<subseteq> \<R>"   and "S = [M]\<^bsub>v,n\<^esub>"   and "dbm_int M n"   and "normalized M"   and "Z \<subseteq> S" shows "Approx\<^sub>\<beta> Z \<subseteq> S"
SCallNone\<^sub>1: fixes P :: "nat exp prog"   and ps :: "nat exp list"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and vs :: "val list"   and s\<^sub>2 :: "heap \<times> val list \<times> sheap"   and C :: "char list"   and M :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>ps,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>map Val vs,s\<^sub>2\<rangle>"   and "\<nexists>b Ts T body D. P \<turnstile> C sees M, b :  Ts\<rightarrow>T = body in D" shows "P \<turnstile>\<^sub>1 \<langle>C\<bullet>\<^sub>sM(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NoSuchMethodError,s\<^sub>2\<rangle>"
int0d: shows "integrable (lebesgue_on {0..d}) (\<lambda>x. \<bar>f (t + x) + f (t - x) - 2 * l\<bar> / x)"
1: fixes y :: "'a" assumes "y \<in> A"   and "r y z" shows "y \<notin> Q"
rep_eq: fixes x :: "'b ttree" shows "carrier x = \<Union> (set ` Rep_ttree x)"
mod_add_self1: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and b :: "'c"   and a :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus b a) b = modulo a b"
sorted_wrt_map: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "sorted_wrt R (map f xs) = sorted_wrt (\<lambda>x y. R (f x) (f y)) xs"
items_le_is_filter: fixes k :: "nat"   and I :: "item set" shows "items_le k I \<subseteq> I"
backward_simulation_inf: fixes i :: "'index"   and s1 :: "'state1"   and s2 :: "'state2" assumes "match i s1 s2"   and "L2.inf_step s2" shows "L1.inf_step s1"
wf_mstate_mstep: fixes \<phi> :: "'a MFOTL.formula"   and \<pi> :: "(char list \<times> 'a list) prefix"   and R :: "'a list set"   and st :: "'a mstate"   and tdb :: "(char list \<times> 'a list) set \<times> nat" assumes "wf_mstate \<phi> \<pi> R st"   and "last_ts \<pi> \<le> snd tdb" shows "wf_mstate \<phi> (psnoc \<pi> tdb) R (snd (mstep tdb st))"
ge2_zmult_pos: fixes m :: "nat"   and z :: "int" assumes "2 \<le> m"   and "0 < z" shows "1 < int m * z"
join_meet_distrib2: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "y \<sqinter>\<^bsub>L\<^esub> z \<squnion>\<^bsub>L\<^esub> x = (y \<squnion>\<^bsub>L\<^esub> x) \<sqinter>\<^bsub>L\<^esub> (z \<squnion>\<^bsub>L\<^esub> x)"
dom_mult_closed: fixes y :: "'a"   and x :: "'a" shows "ar ar (ar ar y \<cdot> ar ar x) = ar ar y \<cdot> ar ar x"
ltln_weak_strong_stable_words_2: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi> = w \<Turnstile>\<^sub>n (\<phi> or\<^sub>n G\<^sub>n \<psi>[\<G>\<F> (\<phi> R\<^sub>n \<psi>) w]\<^sub>\<Pi>\<^sub>1) M\<^sub>n \<psi>"
nat_abs_triangle_ineq: fixes k :: "int"   and l :: "int" shows "nat \<bar>k + l\<bar> \<le> nat \<bar>k\<bar> + nat \<bar>l\<bar>"
hyps_3: shows "pmdl (insert y (G - {p})) = pmdl G"
sturm_adjacent_root_propagate_left: fixes i :: "nat"   and p :: "real poly"   and x :: "real" assumes "i < length (sturm p) - 1"   and "poly (sturm p ! i) x = 0"   and "poly (sturm p ! (i + 1)) x = 0" shows "\<forall>j\<le>i + 1. poly (sturm p ! j) x = 0"
RangeI: fixes a :: "'e"   and b :: "'f"   and r :: "('e \<times> 'f) set" assumes "(a, b) \<in> r" shows "b \<in> Range r"
eq_pred_parent2: fixes i :: "nat"   and k :: "nat" assumes "eq_pred i (s2 k)"   and "i \<noteq> s2 k" shows "eq_pred i k"
sup_absorb1: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "class.semilattice_sup sup less_eq less"   and "less_eq y x" shows "sup x y = x"
fields_def: fixes P :: "'b prog"   and C :: "char list" shows "fields P C \<equiv> THE FDTs. P \<turnstile> C has_fields FDTs"
sub_inf_left_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "- x \<le> - y" shows "- x \<sqinter> - z \<le> - y \<sqinter> - z"
split_list_first_propE: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" obtains ys :: "'a list" and x :: "'a" and zs :: "'a list" where "xs = ys @ x # zs" and "P x" and "\<forall>y\<in>set ys. \<not> P y"
Crypt_notin_initState: fixes K :: "nat"   and X :: "msg"   and B :: "agent" shows "Crypt K X \<notin> parts (initState B)"
norm_mangoldt: fixes n :: "nat" shows "norm (mangoldt n) = mangoldt n"
WhileT: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and c :: "char list exp"   and v\<^sub>1 :: "val"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>3 :: "char list exp"   and s\<^sub>3 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>c,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>2\<rangle>"   and "P \<turnstile> \<langle>while (e) c,s\<^sub>2\<rangle> \<Rightarrow> \<langle>e\<^sub>3,s\<^sub>3\<rangle>" shows "P \<turnstile> \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e\<^sub>3,s\<^sub>3\<rangle>"
minus_add_uminus_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "A - B = A + - B"
prime_cn_reverse: fixes n :: "nat"   and f :: "recf"   and gs :: "recf list"   and m :: "nat" assumes "primerec (Cn n f gs) m" obtains "m = n" and "primerec f (length gs)" and "\<forall>i<length gs. primerec (gs ! i) n"
distinct_1: fixes x2 :: "real" shows "MInfty \<noteq> real_max_bot.R x2"
\<Z>: fixes t :: "symbol \<times> character list" assumes "t \<in> \<Z> k (Suc u)" shows "t \<in> \<Z> k (Suc v)"
functor_axioms: shows "functor C.CC.comp C.CC.comp C.CC.map"
join_lc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
primitiveI: fixes f :: "'a poly" assumes "\<And>x. (\<And>y. y \<in> set (coeffs f) \<Longrightarrow> x dvd y) \<Longrightarrow> x dvd (1::'a)" shows "primitive f"
B_def: shows "B = (\<lambda>i. W \<inter> A i)"
eq_\<I>_gpv_reflI: fixes \<I> :: "('i, 'j) \<I>"   and gpv :: "('h, 'i, 'j) gpv"   and A :: "'h \<Rightarrow> 'h \<Rightarrow> bool" assumes "\<And>x. x \<in> results_gpv \<I> gpv \<Longrightarrow> A x x"   and "\<I> \<turnstile>g gpv \<surd>" shows "eq_\<I>_gpv A \<I> gpv gpv"
tr: shows "trace (S t (i + 1)) (take (length t - (i + 1)) (drop (i + 1) t)) (S t (length t))"
I_def: shows "I \<equiv> \<lambda>n. of_nat n powr (1 - s) / (1 - s)"
simps_1: fixes xs :: "'a list list" shows "[] \<otimes> xs = []"
ptrm_alpha_equiv_reflexive: fixes M :: "'b ptrm" shows "M \<approx> M"
takeWhile_nth: fixes j :: "nat"   and P :: "'g \<Rightarrow> bool"   and xs :: "'g list" assumes "j < length (takeWhile P xs)" shows "takeWhile P xs ! j = xs ! j"
single_zero: fixes k :: "'b" shows "monomial (0::'c) k = 0"
Abs_bset_eqvt: fixes A :: "'c set"   and p :: "perm" assumes "|A| <o natLeq +c |UNIV|" shows "p \<bullet> Abs_bset A = Abs_bset (p \<bullet> A)"
real_powr_at_top: fixes p :: "real" assumes "0 < p" shows "LIM x at_top. x powr p :> at_top"
scalar_matrix_mult_row_code: fixes c :: "'c"   and A :: "(('c, 'b) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and i :: "'d" shows "($h) (scalar_matrix_mult_row c A i) = (\<lambda>j. c * A $h i $h j)"
ipt_case: shows "ipt = SK_EV_SIGNAL EV_SIGNAL_FINISH partner"
axioms_1: fixes G :: "('c, 'd) graph" assumes "connected_graph G" shows "valid_graph G"
p_max_def: shows "delayed_safe_distance.other.p_max = delayed_safe_distance.other.p delayed_safe_distance.other.t_stop"
constant_def: fixes f :: "'a \<Rightarrow> 'b" shows "constant f = (\<forall>x y. f x = f y)"
B: shows "R (P' \<turnstile> Q') \<in> {p. is_CSP_process p}"
hs_size_impl: shows "imp_set_size is_hashset hs_size"
r_neg1: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F"   and "y \<in> carrier F" shows "\<ominus>\<^bsub>F\<^esub> x \<oplus>\<^bsub>F\<^esub> (x \<oplus>\<^bsub>F\<^esub> y) = y"
less_eq_option_def: fixes x :: "'b option"   and y :: "'b option" shows "(x \<le> y) = (case x of None \<Rightarrow> True | Some x \<Rightarrow> case y of None \<Rightarrow> False | Some y \<Rightarrow> x \<le> y)"
assms_5: shows "x \<noteq> \<infinity>\<^sub>h"
bigo_plus_subset: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "O(f + g) \<subseteq> O(f) + O(g)"
pmf_of_set_3way_split: fixes A :: "'a set"   and A1 :: "'a set"   and A2 :: "'a set"   and A3 :: "'a set"   and f :: "'a \<Rightarrow> 'b pmf"   and g :: "'a \<Rightarrow> 'b pmf"   and h :: "'a \<Rightarrow> 'b pmf" assumes "finite A"   and "A \<noteq> {}"   and "A1 \<inter> A2 = {}"   and "A1 \<inter> A3 = {}"   and "A2 \<inter> A3 = {}"   and "A1 \<union> A2 \<union> A3 = A" shows "pmf_of_set A \<bind> (\<lambda>x. if x \<in> A1 then f x else if x \<in> A2 then g x else h x) = pmf_of_set {..<card A} \<bind> (\<lambda>i. if i < card A1 then pmf_of_set A1 \<bind> f else if i < card A1 + card A2 then pmf_of_set A2 \<bind> g else pmf_of_set A3 \<bind> h)"
simps_8: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'c \<Rightarrow> 'b" shows "(case dSUCCEEDi of dSUCCEEDi \<Rightarrow> f1 | dFAILi \<Rightarrow> f2 | dRETURN x \<Rightarrow> f3 x) = f1"
assms: shows "\<forall>a b. p = (a, b) \<longrightarrow> P a b"
nnq: shows "n = 2 ^ s2 * word_of_nat nq"
rbt_comps_2: shows "rbt_comp_lookup ccomp = ord.rbt_lookup cless"
path_ret_ipd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat"   and n :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "\<pi> k \<noteq> IFC_def.return prob"   and "\<pi> n = IFC_def.return prob" obtains l :: "nat" where "k < l" and "l \<le> n" and "\<pi> l = IFC_def.ipd prob (\<pi> k)"
resultant_0_1: fixes p :: "'b poly" assumes "0 < degree p" shows "resultant 0 p = (0::'b)"
arity_1_result_def: fixes op1 :: "b_e" shows "arity_1_result op1 = (case op1 of Unop_i t x \<Rightarrow> t | Unop_f t x \<Rightarrow> t | Testop t x \<Rightarrow> T_i32 | Cvtop t1 Convert x xa \<Rightarrow> t1 | Cvtop t1 _ x xa \<Rightarrow> t1)"
axioms_1: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "invertible_lf s1 s2 f" shows "linear s1 s2 f"
negligible_locally_Lipschitz_image: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "DIM('a) \<le> DIM('b)"   and "negligible S"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>T B. open T \<and> x \<in> T \<and> (\<forall>y\<in>S \<inter> T. norm (f y - f x) \<le> B * norm (y - x))" shows "negligible (f ` S)"
max_dom_def: fixes xs :: "Sec set" shows "max_dom xs \<equiv> if High \<in> xs then High else Low"
var_infinite_def: fixes uu :: "'d" shows "var_infinite uu \<equiv> infinite UNIV"
l_one: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "Dmonoid prod one" shows "prod one x = x"
delta': fixes a :: "'c"   and g :: "'c \<Rightarrow> 'b" shows "(\<Sum>b. if a = b then g b else (0::'b)) = g a"
in_vars_imp_sub: fixes x :: "'a"   and s :: "('b, 'a) tm" shows "(x \<in> vars s) = sub (Hd (Var x)) s"
betw_a_in_path: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and bc :: "'a set" assumes "[[a b c]]"   and "path bc b c" shows "a \<in> bc"
cardSK_neq_priEK: fixes cardSK :: "nat"   and evs :: "event list"   and C :: "agent" assumes "Key cardSK \<notin> analz (knows Spy evs)"   and "Key cardSK \<in> parts (knows Spy evs)"   and "evs \<in> set_cr" shows "cardSK \<noteq> priEK C"
finite_field_ops64: fixes pp :: "uint64" assumes "p = int_of_uint64 pp"   and "p \<le> 4294967295" shows "field_ops (finite_field_ops64 pp) mod_ring_rel64"
similar_mat_witD_7: fixes n :: "nat"   and A :: "'b mat"   and B :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "Q \<in> carrier_mat n n"
assms_3: shows "\<bar>c\<bar> \<noteq> \<infinity>"
A_3: shows "\<not> P z \<or> \<not> P x"
cobounded1: fixes a :: "'a multiset"   and b :: "'a multiset" shows "a \<inter># b \<subseteq># a"
inject_1: fixes x11 :: "agent"   and x12 :: "agent"   and y11 :: "agent"   and y12 :: "agent" shows "(sharK x11 x12 = sharK y11 y12) = (x11 = y11 \<and> x12 = y12)"
mndet_distrib: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> 'a process" assumes "A \<noteq> {}"   and "B \<noteq> {}" shows "mndet (A \<union> B) P = (mndet A P |-| mndet B P)"
T_sync: fixes P :: "'c process"   and A :: "'c set"   and Q :: "'c process" shows "T (P \<lbrakk>A\<rbrakk> Q) = {s. \<exists>t u. t \<in> T P \<and> u \<in> T Q \<and> s setinterleaves ((t, u), ev ` A \<union> {tick})} \<union> {s. \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)}"
elims: fixes x :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and xa :: "'b option"   and xb :: "'c option"   and y :: "'a option" assumes "lift2_option x xa xb = y" obtains uu_ :: "'b \<Rightarrow> 'c \<Rightarrow> 'a" and uv_ :: "'c option" where "x = uu_" and "xa = None" and "xb = uv_" and "y = None"   | uw_ :: "'b \<Rightarrow> 'c \<Rightarrow> 'a" and v :: "'b" where "x = uw_" and "xa = Some v" and "xb = None" and "y = None"   | f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a" and xc :: "'b" and ya :: "'c" where "x = f" and "xa = Some xc" and "xb = Some ya" and "y = Some (f xc ya)"
sets_Collect_countable_All': fixes I :: "'c set"   and M :: "'d measure"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>i. i \<in> I \<Longrightarrow> {x \<in> space M. P i x} \<in> sets M"   and "countable I" shows "{x \<in> space M. \<forall>i\<in>I. P i x} \<in> sets M"
map_disj_map_le: fixes h\<^sub>0' :: "'c \<Rightarrow> 'd option"   and h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option" assumes "h\<^sub>0' \<subseteq>\<^sub>m h\<^sub>0"   and "h\<^sub>0 \<bottom> h\<^sub>1" shows "h\<^sub>0' \<bottom> h\<^sub>1"
map_leI_left: fixes h :: "'c \<Rightarrow> 'd option"   and h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option" assumes "h = h\<^sub>0 ++ h\<^sub>1"   and "h\<^sub>0 \<bottom> h\<^sub>1" shows "h\<^sub>0 \<subseteq>\<^sub>m h"
fg: shows "of_int_poly f = Polynomial.smult (rat_of_int a / rat_of_int b) (of_int_poly g)"
reachable0: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" shows "system.reachable s\<^sub>0 lstep s\<^sub>0"