hcju/isabelleps huggingface dataset:
nn_integral_T_lfp: fixes g :: "'a cfg \<Rightarrow> ennreal \<Rightarrow> ennreal"   and s :: "'a cfg" assumes "(\<lambda>(x, y). g x y) \<in> borel_measurable (count_space UNIV \<Otimes>\<^sub>M borel)"   and "\<And>s. sup_continuous (g s)"   and "\<And>f s. f \<in> borel_measurable (stream_space (count_space UNIV)) \<Longrightarrow> \<integral>\<^sup>+ \<omega>. g s (f \<omega>) \<partial>MC.T s = g s (integral\<^sup>N (MC.T s) f)" shows "integral\<^sup>N (MC.T s) (lfp (\<lambda>f \<omega>. g (shd \<omega>) (f (stl \<omega>)))) = lfp (\<lambda>f s. \<integral>\<^sup>+ t. g t (f t) \<partial>measure_pmf (K_cfg s)) s"
eq: fixes x :: "'a" shows "ennreal \<bar>f x\<bar> = fp x + fm x"
simps_10: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'c \<Rightarrow> 'b"   and x3 :: "'c" shows "(case dRETURN x3 of dSUCCEEDi \<Rightarrow> f1 | dFAILi \<Rightarrow> f2 | dRETURN x \<Rightarrow> f3 x) = f3 x3"
pnop_vrange: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "pnop A n f" shows "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A"
minus_mult_right: fixes a :: "'a"   and b :: "'a" shows "- (a * b) = a * - b"
totient_formula1: fixes n :: "nat" assumes "0 < n" shows "totient n = (\<Prod>p\<in>prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))"
D5: shows "\<lfloor>\<lambda>w. \<forall>x. x actualizedAt w \<longrightarrow> (\<forall>xa. ((actualizedAt) xa \<^bold>\<rightarrow> x dependsOn xa \<^bold>\<rightarrow> \<^bold>\<box>(E! x \<^bold>\<rightarrow> E! xa)) w)\<rfloor>"
I_asubst: fixes t :: "real \<times> real list"   and a :: "atom"   and xs :: "real list" shows "I\<^sub>R (asubst t a) xs = I\<^sub>R a (eval t xs # xs)"
Suc_Suc_Tr: fixes n :: "nat" assumes "Suc (Suc 0) \<le> n" shows "Suc (n - Suc (Suc 0)) = n - Suc 0"
right_neutral: fixes a :: "'a ppath set" shows "pp_prod a pp_one = a"
bidirected_digraph_axioms: shows "bidirected_digraph G (edge_rev M)"
assms_1: fixes J :: "'j \<Rightarrow> 'j \<Rightarrow> 'j" assumes "finite (Collect (partial_magma.arr J))" shows "has_limits_of_shape J"
P_lnull_ltl_LCons: assumes "lnull (ltl P)" shows "P = LCons v0 LNil"
group_byE2: fixes b :: "'a"   and cs :: "'b set"   and f :: "'c \<Rightarrow> 'a \<times> 'b"   and as :: "'c set" assumes "(b, cs) \<in> group_by f as" obtains a :: "'c" where "a \<in> as" and "fst (f a) = b"
integrable_continuous: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on (cbox a b) f" shows "f integrable_on cbox a b"
has_derivative_within': fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set" shows "(f has_derivative f') (at x within s) = (bounded_linear f' \<and> (\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. 0 < norm (x' - x) \<and> norm (x' - x) < d \<longrightarrow> norm (f x' - f x - f' (x' - x)) / norm (x' - x) < e))"
trail_bound: fixes v :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "finite E"   and "is_trail v ps v'" shows "length ps \<le> card E"
sorted_list_of_map_empty: shows "sorted_list_of_map Map.empty = []"
invfun_def: fixes A :: "'f set"   and B :: "'e set"   and f :: "'f \<Rightarrow> 'e" shows "f\<inverse>\<^bsub>B,A\<^esub> = (\<lambda>y\<in>B. SOME x. x \<in> A \<and> f x = y)"
select_convs_3: fixes rec1 :: "'b"   and rec2 :: "'c"   and more :: "'a" shows "more \<lparr>rec1 = rec1, rec2 = rec2, \<dots> = more\<rparr> = more"
i_aggregate_def: fixes s :: "nat \<Rightarrow> 'b"   and k :: "nat"   and ag :: "'b list \<Rightarrow> 'b" shows "i_aggregate s k ag \<equiv> \<lambda>n. ag (s \<Up> (n * k) \<Down> k)"
hd_coeff_is1_hd_coeff1: fixes m :: "int"   and a :: "atom" shows "hd_coeff (hd_coeff1 m a) \<noteq> 0 \<longrightarrow> hd_coeff_is1 (hd_coeff1 m a)"
mat_of_rows_carrier_2: fixes n :: "nat"   and vs :: "'b vec list" shows "dim_row (mat_of_rows n vs) = length vs"
less_option_None_Some: fixes x :: "'b" shows "None < Some x"
uprod_exhaust: fixes x :: "'b uprod" obtains a :: "'b" and b :: "'b" where "x = Upair a b"
relation_of_iPrefix: fixes c :: "'d \<Rightarrow> 'c"   and i :: "'d \<times> 'd \<Rightarrow> bool"   and j :: "('c, 'd) action \<Rightarrow> ('c, 'd) action"   and S :: "'d \<Rightarrow> 'c set"   and P :: "('c, 'd) action" shows "action.relation_of (iPrefix c i j S P) = (R (true \<turnstile> \<lambda>(A, A'). do_I c (S (alpha_rp.more A)) (A, A') \<and> alpha_rp.more A' = alpha_rp.more A) ;; action.relation_of P)"
emeasure_measure_of: fixes M :: "'c measure"   and \<Omega> :: "'c set"   and A :: "'c set set"   and \<mu> :: "'c set \<Rightarrow> ennreal"   and X :: "'c set" assumes "M = measure_of \<Omega> A \<mu>"   and "A \<subseteq> Pow \<Omega>"   and "positive (sets M) \<mu>"   and "countably_additive (sets M) \<mu>"   and "X \<in> sets M" shows "emeasure M X = \<mu> X"
composite_simps_1: shows "src (f \<star> g \<star> f) = src f"
imp_igWlsInp: fixes MOD :: "('h, 'i, 'j, 'sort, 'k, 'l, 'm, 'n, 'o) model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'm option"   and h :: "'m \<Rightarrow> 'p"   and MOD' :: "('q, 'r, 's, 'sort, 't, 'u, 'p, 'v, 'w) model_scheme" assumes "igWlsInp MOD delta inp"   and "ipresIGWls h MOD MOD'" shows "igWlsInp MOD' delta (lift h inp)"
vmc_path_lset_induction_closed_subset: fixes S :: "'a set"   and T :: "'a set" assumes "\<And>v. \<lbrakk>v \<in> S; \<not> deadend v; v \<in> VV p\<rbrakk> \<Longrightarrow> \<sigma> v \<in> S \<union> T"   and "\<And>v w. \<lbrakk>v \<in> S; \<not> deadend v; v \<in> VV p**; v \<rightarrow> w\<rbrakk> \<Longrightarrow> w \<in> S \<union> T"   and "v0 \<in> S"   and "lset P \<inter> T = {}" shows "lset P \<subseteq> S"
height_hinv: fixes t :: "'a tree" assumes "hinv t" shows "ht t = height (erase t)"
tendsto_powr_complex: fixes a :: "complex"   and f :: "'a \<Rightarrow> complex"   and F :: "'a filter"   and g :: "'a \<Rightarrow> complex"   and b :: "complex" assumes "a \<notin> \<real>\<^sub>\<le>\<^sub>0"   and "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>z. f z powr g z) \<longlongrightarrow> a powr b) F"
oid: shows "\<forall>p n. Deliver p \<notin> set (history n) \<or> Broadcast p \<in> set (history (oid p))"
ideal_gen_by_prod_def: fixes \<aa> :: "'a set"   and \<bb> :: "'a set" shows "ideal_gen_by_prod \<aa> \<bb> \<equiv> additive.subgroup_generated {a \<cdot> b |a b. a \<in> \<aa> \<and> b \<in> \<bb>}"
C3_b: fixes x :: "'a" shows "x r\<rightarrow> (0::'a) = -r x"
decompose_Some_var_terms: fixes f :: "'a"   and ss :: "('a, 'b) Term.term list"   and g :: "'a"   and ts :: "('a, 'c) Term.term list"   and eqs :: "(('a, 'b) Term.term \<times> ('a, 'c) Term.term) list"   and P :: "(('a, 'b) Term.term \<times> ('a, 'c) Term.term) list" assumes "decompose (Fun f ss) (Fun g ts) = Some eqs" shows "f = g \<and> length ss = length ts \<and> eqs = zip ss ts \<and> (\<Union>(t, u)\<in>set ((Fun f ss, Fun g ts) # P). vars_term t) = (\<Union>(t, u)\<in>set (eqs @ P). vars_term t)"
map_snd_def: fixes f :: "'c \<Rightarrow> 'd" shows "map_snd f = map (\<lambda>(x, y). (x, f y))"
finalStateNonApplicable: fixes state :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set" shows "isFinalState state F0 decisionVars = (\<not> applicableDecide state decisionVars \<and> \<not> applicableUnitPropagate state F0 decisionVars \<and> \<not> applicableBackjump state \<and> \<not> applicableLearn state \<and> \<not> applicableConflict state \<and> \<not> applicableExplain state)"
inject_1: fixes x1 :: "fdecl"   and y1 :: "fdecl" shows "(fdecl x1 = fdecl y1) = (x1 = y1)"
minus_flip: fixes a :: "'c"   and b :: "'c" shows "- (a \<leftrightarrow> b) = (a \<leftrightarrow> b)"
emeasure_bind_const': fixes M :: "'c measure"   and N :: "'d measure"   and X :: "'d set" assumes "subprob_space M"   and "subprob_space N" shows "emeasure (M \<bind> (\<lambda>x. N)) X = emeasure N X * emeasure M (space M)"
insert_DiffM2: fixes x :: "'k"   and M :: "'k multiset" assumes "x \<in># M" shows "M - {#x#} + {#x#} = M"
cong: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "sum g A = sum h B"
weak_sup_pre_fixed_point: fixes f :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "f \<in> carrier L \<rightarrow> carrier L"   and "Mono\<^bsub>L\<^esub> f"   and "A \<subseteq> fps L f" shows "\<Squnion>\<^bsub>L\<^esub>A \<sqsubseteq>\<^bsub>L\<^esub> f (\<Squnion>\<^bsub>L\<^esub>A)"
dense_complement_convex_closed: fixes T :: "'b set"   and S :: "'b set" assumes "aff_dim T < aff_dim S"   and "convex S"   and "closed S" shows "closure (S - T) = S"
cobounded2: fixes b :: "'c \<times> 'd"   and a :: "'c \<times> 'd" shows "less_eq_prod b (sup_prod a b)"
abs_eq: fixes x :: "'a" shows "cblinfun_cinner_right x = CBlinfun ((\<bullet>\<^sub>C) x)"
sel_4: fixes x21 :: "(Term_Class.term \<times> sterm) list"   and x22 :: "(name, value) fmap" shows "un_Vabs2 (Vabs x21 x22) = x22"
closed_segment: fixes a :: "'b"   and b :: "'b" shows "closed (closed_segment a b)"
in_replicate_mset: fixes x :: "'c"   and n :: "nat"   and y :: "'c" shows "(x \<in># replicate_mset n y) = (0 < n \<and> x = y)"
elims: fixes x :: "'b \<Rightarrow> 'a"   and xa :: "'b list"   and xb :: "'a list"   and y :: "'a list" assumes "map_tailrec_rev x xa xb = y" obtains f :: "'b \<Rightarrow> 'a" and bs :: "'a list" where "x = f__" and "xa = []" and "xb = bs__" and "y = bs__"   | f :: "'b \<Rightarrow> 'a" and a :: "'b" and as :: "'b list" and bs :: "'a list" where "x = f" and "xa = a # as" and "xb = bs" and "y = map_tailrec_rev f as (f a # bs)"
vbrelation_Limit_in_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<D>\<^sub>\<circ> f \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<R>\<^sub>\<circ> f \<in>\<^sub>\<circ> Vset \<alpha>" shows "f \<in>\<^sub>\<circ> Vset \<alpha>"
sum_inner_Basis_scaleR: fixes b :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "b \<in> Basis" shows "(\<Sum>i\<in>Basis. inner i b *\<^sub>R f i) = f b"
frameImpResFreshRight: fixes x :: "name"   and F :: "'b frame" assumes "x \<sharp> F" shows "F \<hookrightarrow>\<^sub>F \<lparr>\<nu>x\<rparr>F"
echelon_form_upt_row_condition1_explicit': fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat"   and i :: "'c"   and j :: "'c" assumes "echelon_form_upt_row A k"   and "mod_type_class.to_nat i < k"   and "is_zero_row i A"   and "i \<le> j"   and "mod_type_class.to_nat j < k" shows "is_zero_row j A"
unlabel_def: fixes ars :: "('a \<times> 'b \<times> 'a) set" shows "unlabel ars = {(a, c). \<exists>b. (a, b, c) \<in> ars}"
r_def: fixes n :: "nat" shows "r n = 1 / 2 * real n / real k"
fresh_PairD_1: fixes a :: "atom"   and x :: "'b"   and y :: "'c" assumes "a \<sharp> (x, y)" shows "a \<sharp> x"
homh: shows "f \<in> hom (reduced_homology_group (int p) (nsphere p)) (reduced_homology_group (1 + int p) (nsphere (Suc p)))"
lm079: fixes x :: "'c"   and f :: "('c \<times> 'd) set" assumes "x \<in> Domain f"   and "runiq f" shows "{x} \<times> f `` {x} = {(x, f ,, x)}"
fds_nth_exp_Suc_0: fixes f :: "'b fds" shows "fds_nth (fds_exp f) (Suc 0) = exp (fds_nth f 1)"
has_left_def: fixes j :: "nat" shows "local.has_left j = (\<exists>i<j. xs ! i < xs ! j)"
Or_def: fixes P :: "('e, 'f, 'g) formula"   and Q :: "('e, 'f, 'g) formula" shows "(P || Q) = ! (! P && ! Q)"
simps_3: fixes Label :: "String.literal"   and Arity :: "nat"   and Guards :: "vname gexp list"   and Outputs :: "vname aexp list"   and Updates :: "(nat \<times> vname aexp) list"   and more :: "'a" shows "Guards \<lparr>Label = Label, Arity = Arity, Guards = Guards, Outputs = Outputs, Updates = Updates, \<dots> = more\<rparr> = Guards"
distinct_77: fixes x5 :: "msg"   and x101 :: "nat"   and x102 :: "msg" shows "Hash x5 \<noteq> Crypt x101 x102"
assms: shows "optim n xs \<Sigma>1 = Ok (ys, \<Sigma>2)"
simps_3: fixes x1 :: "'h"   and x21 :: "'i"   and x22 :: "'j" shows "Pure x1 \<noteq> IO x21 x22"
Class_self: fixes a :: "'a \<times> 'a" assumes "a \<in> R \<times> S" shows "a \<in> rel.Class a"
joinI_left: fixes a :: "'b"   and b :: "'b"   and A :: "('b \<times> 'b) set" assumes "(a, b) \<in> A\<^sup>*" shows "(a, b) \<in> A\<^sup>\<down>"
isSPath_nt_parallel: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat"   and e :: "nat \<times> nat" assumes "isSimplePath s p t"   and "e \<in> set p" shows "prod.swap e \<notin> set p"
cycle_free_diag_equiv: 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 m :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "class.linordered_ab_monoid_add plus zero less_eq less" shows "(\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> less_eq zero (linordered_ab_monoid_add.len plus m i i xs)) = linordered_ab_monoid_add.cycle_free plus zero less_eq m n"
mod_ring_0__integer: fixes pp :: "integer" assumes "p = int_of_integer pp" shows "mod_ring_rel_integer 0 0"
in_idem: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<in> A" shows "min x (Min A) = Min A"
1: shows "v \<in> {v \<in> fmdom' s. v \<in> fmdom' (fst a) \<and> fmlookup (fst a) v = fmlookup s v}"
bind_eq_None_conv: fixes a :: "'d option"   and f :: "'d \<Rightarrow> 'c option" shows "(a \<bind> f = None) = (a = None \<or> f (the a) = None)"
abs_eq: fixes xb :: "'e"   and xa :: "'d"   and x :: "('b, 'c, 'd, 'e) Tree" shows "Act\<^sub>\<alpha> xb xa (abs_Tree\<^sub>\<alpha> x) = abs_Tree\<^sub>\<alpha> (tAct xb xa x)"
permutes_surj: fixes p :: "'a \<Rightarrow> 'a"   and s :: "'a set" assumes "p permutes s" shows "surj p"
arg_max_def: fixes f :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool" shows "arg_max f P = (SOME x. is_arg_max f P x)"
emeasure_bind: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b measure"   and N :: "'b measure"   and X :: "'b set" assumes "space M \<noteq> {}"   and "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "X \<in> sets N" shows "emeasure (M \<bind> f) X = \<integral>\<^sup>+ x. emeasure (f x) X \<partial>M"
refine: shows "(icount_impl, icount) \<in> nat_assn\<^sup>k \<rightarrow>\<^sub>a nat_assn"
sum_outgoing_pointwise: fixes g :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat" shows "sum g (cf.outgoing u) = (\<Sum>v\<in>cf.E `` {u}. g (u, v))"
gcd_greatest_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a dvd gcd b c) = (a dvd b \<and> a dvd c)"
process_surj_pair: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set" shows "(FAILURES P, DIVERGENCES P) = P"
\<beta>': fixes x :: "complex" assumes "x \<in> X" shows "rat_of_int (\<beta>' x) = rat_of_int l * \<beta> x"
match_fvi_trm_None: fixes f :: "nat \<Rightarrow> 'a option"   and ts :: "'a MFOTL.trm list"   and xs :: "'a list"   and x :: "nat" assumes "Some f = match ts xs"   and "\<forall>t\<in>set ts. x \<notin> fv_trm t" shows "f x = None"
neg_divideC_less_eq: fixes c :: "complex"   and b :: "'b"   and a :: "'b" assumes "c < 0" shows "(b /\<^sub>C c < a) = (c *\<^sub>C a < b)"
small_vsubset_set: fixes B :: "V set"   and A :: "V set" assumes "small B"   and "A \<subseteq> B" shows "set A \<subseteq>\<^sub>\<circ> set B"
set_add_pairs_iff: fixes ps1 :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> 'd"   and gs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and q :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" assumes "\<And>xs ys. set (comb xs ys) = set xs \<union> set ys"   and "ps1 = apply_ncrit ncrit data gs bs hs (apply_icrit icrit data gs bs hs (np gs bs hs data))" shows "((p, q) \<in> set (add_pairs gs bs ps hs data)) = ((False, p, q) \<in> set ps1 \<or> (p, q) \<in> set ps \<and> \<not> ocrit data hs ps1 p q)"
rep_eq: fixes x :: "unit \<Rightarrow> 'a" shows "Rep_lazy (delay x) = x ()"
ab_2: shows "ball (0::'n) B2 \<subseteq> cbox a b"
equivp_imp_transp: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "equivp R" shows "transp R"
a_eq: shows "\<forall>i<n. \<forall>A\<in>#AAs ! i. A = As ! i"
suffix: shows "suffix i w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>)"
len_not_eq_0: fixes len_of :: "'a itself \<Rightarrow> nat" assumes "class.len len_of" shows "len_of TYPE('a) \<noteq> 0"
F2_def: fixes ds :: "'b set"   and X' :: "'a set" shows "F2 ds X' = - Rf (CD_on ds) X'"
functor_axioms: shows "functor CC.comp (\<cdot>) T"
prefixToLevelElementsElementLevel: fixes e :: "'a"   and level :: "nat"   and M :: "('a \<times> bool) list" assumes "e \<in> set (elements (prefixToLevel level M))" shows "elementLevel e M \<le> level"
cupcake_list_preserve0: fixes static_cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and env :: "v sem_env"   and es :: "exp0 list"   and rs :: "(v, v) result list"   and vs :: "v list" assumes "cakeml_static_env static_cenv"   and "list_all2_shortcircuit (\<lambda>e r. cupcake_evaluate_single env e r \<and> (cakeml_static_env.is_cupcake_all_env static_cenv env \<longrightarrow> is_cupcake_exp e \<longrightarrow> if_rval (cakeml_static_env.is_cupcake_value static_cenv) r)) es rs"   and "cakeml_static_env.is_cupcake_all_env static_cenv env"   and "list_all is_cupcake_exp es"   and "sequence_result rs = Rval vs" shows "list_all (cakeml_static_env.is_cupcake_value static_cenv) vs"
const: fixes name :: "name" assumes "True" shows "pre_strong_term_class.wellformed (const name)"
scopeExtPar: fixes x :: "name"   and P :: "pi"   and Q :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>(P \<parallel> Q) \<sim>\<^sub>l P \<parallel> <\<nu>x>Q"
add_left_imp_eq: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a + b = a + c" shows "b = c"
nb\<^sub>1: fixes op :: "('a, 'b) sas_plus_operator" assumes "op \<in> set ops" shows "\<exists>op'\<in>set (map (sasp_op_to_strips \<Psi>) ops). op' = \<phi>\<^sub>O \<Psi> op"
IH: fixes gpv :: "('c, 'a, 'b) gpv" assumes "\<I> \<turnstile>g gpv \<surd>" shows "expectation_gpv'' (restrict_gpv \<I> gpv) = expectation_gpv'' gpv"
s_def: shows "s \<equiv> if 0 \<le> p then 1 else - 1"
mrDA_tl: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "applied_rule_rev Cp x p = \<lfloor>DenyAll\<rfloor>"   and "wellformed_policy1_strong p" shows "applied_rule_rev Cp x (tl p) = \<bottom>"
image_Collect_mem: fixes f :: "'e \<Rightarrow> 'd"   and S :: "'e set" shows "{f x |x. x \<in> S} = f ` S"
assms: fixes x :: "'d" assumes "x \<in> A" shows "f x \<in> L F (g x)"
isomorphic_categories_def: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah" shows "isomorphic_categories A B \<equiv> category A \<and> category B \<and> isomorphic_categories_axioms A B"
nn_integral_const: fixes M :: "'c measure"   and c :: "ennreal" shows "\<integral>\<^sup>+ x. c \<partial>M = c * emeasure M (space M)"
filter_tab_SomeI: fixes t :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and x :: "'a"   and C :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "t k = Some x"   and "C k x" shows "filter_tab C t k = Some x"
intros_4: fixes ux :: "Syntax.var"   and uy :: "Syntax.term" shows "value (Rec ux uy)"
compute_float_plus_down_naive: fixes p :: "nat"   and x :: "float"   and y :: "float" shows "float_plus_down p x y = float_round_down p (x + y)"
wadjust_backto_standard_pos_B_no_Oc: fixes m :: "nat"   and rs :: "nat"   and c :: "cell list"   and list :: "cell list" shows "wadjust_backto_standard_pos_B m rs (c, Oc # list) = False"
repeat_mset_replicate_mset: fixes n :: "nat"   and a :: "'b" shows "repeat_mset n {#a#} = replicate_mset n a"
find_shortest_augmenting_spec_def: fixes f :: "nat \<times> nat \<Rightarrow> 'a" shows "find_shortest_augmenting_spec f \<equiv> assert (NFlow c s t f) \<bind> (\<lambda>_. select p. Graph.isShortestPath (cf_of f) s p t)"
Apps_Var_neq_Apps_Abs: fixes T :: "Unary_PCF.type"   and r :: "expr"   and ts :: "expr list"   and xT :: "char list \<times> Unary_PCF.type"   and ss :: "expr list" shows "\<Lambda>\<langle>T\<rangle> r \<bullet> ts \<noteq> \<langle>xT\<rangle> \<bullet> ss"
lambda_system_Un: fixes x :: "'a set"   and f :: "'a set \<Rightarrow> ennreal"   and y :: "'a set" assumes "x \<in> lambda_system \<Omega> M f"   and "y \<in> lambda_system \<Omega> M f" shows "x \<union> y \<in> lambda_system \<Omega> M f"
return_returns_error: fixes h :: "'a"   and x :: "'c"   and e :: "'b" shows "\<not> h \<turnstile> return x \<rightarrow>\<^sub>e e"
s'_def: fixes t :: "real" shows "other.s' t = (if t \<le> other.t_stop then other.p' t else 0)"
deeper: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat"   and m :: "nat" assumes "\<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> \<Delta> \<down> n + m"
parallel_HSem_ind: fixes P :: "('var \<Rightarrow> 'value) \<Rightarrow> ('var \<Rightarrow> 'value) \<Rightarrow> bool"   and \<rho>\<^sub>1 :: "'var \<Rightarrow> 'value"   and \<Gamma>\<^sub>1 :: "('var \<times> 'exp) list"   and \<rho>\<^sub>2 :: "'var \<Rightarrow> 'value"   and \<Gamma>\<^sub>2 :: "('var \<times> 'exp) list" assumes "adm (\<lambda>\<rho>'. P (fst \<rho>') (snd \<rho>'))"   and "P \<bottom> \<bottom>"   and "\<And>y z. P y z \<Longrightarrow> P (\<rho>\<^sub>1 ++\<^bsub>domA \<Gamma>\<^sub>1\<^esub> \<^bold>\<lbrakk> \<Gamma>\<^sub>1 \<^bold>\<rbrakk>\<^bsub>y\<^esub>) (\<rho>\<^sub>2 ++\<^bsub>domA \<Gamma>\<^sub>2\<^esub> \<^bold>\<lbrakk> \<Gamma>\<^sub>2 \<^bold>\<rbrakk>\<^bsub>z\<^esub>)" shows "P (\<lbrace> \<Gamma>\<^sub>1 \<rbrace>\<rho>\<^sub>1) (\<lbrace> \<Gamma>\<^sub>2 \<rbrace>\<rho>\<^sub>2)"
iIN_div: fixes k :: "nat"   and n :: "nat"   and d :: "nat" assumes "0 < k" shows "[n\<dots>,d] \<oslash> k = [n div k\<dots>,d div k + (n mod k + d mod k) div k]"
multiset_partition: fixes M :: "'b multiset"   and P :: "'b \<Rightarrow> bool" shows "M = filter_mset P M + {#x \<in># M. \<not> P x#}"
unit_char_4: fixes a :: "'b" assumes "B.obj a" shows "\<exists>!\<psi>. \<guillemotleft>\<psi> : HoG.map\<^sub>0 a \<Rightarrow>\<^sub>D HoG.map a\<guillemotright> \<and> D.iso \<psi> \<and> \<psi> \<cdot>\<^sub>D \<i>\<^sub>D[HoG.map\<^sub>0 a] = (HoG.map \<i>\<^sub>B[a] \<cdot>\<^sub>D HoG.cmp (a, a)) \<cdot>\<^sub>D (\<psi> \<star>\<^sub>D \<psi>)"
get_s_get_curr_win: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "low_equal s1 s2" shows "get_S (cpu_reg_val PSR (snd (fst (get_curr_win () s1)))) = get_S (cpu_reg_val PSR (snd (fst (get_curr_win () s2))))"
iIN_finite: fixes n :: "nat"   and d :: "nat" shows "finite [n\<dots>,d]"
real_divl_def: fixes prec :: "nat"   and a :: "real"   and b :: "real" shows "real_divl prec a b = truncate_down prec (a / b)"
step_z_dbm_DBM: fixes A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and l :: "'e"   and D :: "nat \<Rightarrow> nat \<Rightarrow> 'g DBMEntry"   and v :: "'f \<Rightarrow> nat"   and n :: "nat"   and l' :: "'e"   and Z :: "('f \<Rightarrow> 'g) set" assumes "A \<turnstile> \<langle>l, [D]\<^bsub>v,n\<^esub>\<rangle> \<leadsto> \<langle>l', Z\<rangle>"   and "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))" obtains D' :: "nat \<Rightarrow> nat \<Rightarrow> 'g DBMEntry" where "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>v,n\<^esub> \<langle>l', D'\<rangle>" and "Z = [D']\<^bsub>v,n\<^esub>"
rev_path_def: fixes ps :: "('a \<times> 'b \<times> 'a) list" shows "rev_path ps \<equiv> map (\<lambda>(a, b, c). (c, b, a)) (rev ps)"